(PDF) Aerothermal Performance of a Solar Updraft Tower

Imperial College London,

Department of Aeronautics

MEng Thesis, June 2008

Aerothermal

Performance of a Solar

Updraft Tower

Owen Williams

Supervisor: Dr. N. Waterson

a

ABSTRACT

The objective of this thesis is to simulate the thermo-fluid interaction of a Solar Updraft

Tower (collector, tower and turbine) using a one-dimensional numerical analysis. The aim is to

develop a tool for the accurate prediction of the performance of a Solar Updraft Tower of any

dimension in a timely and accurate manner. An additional objective is the preliminary analysis of the

three dimensional flow under the solar collector.

The one-dimensional simulation is transient and compressible. The simulation was centred on

the flow equations by solving the complete continuity, momentum and energy equations using finite

volumes and the SIMPLE method. Models of frictional, inlet and exit losses as well as losses in the

transition section between the collector and tower were included in the flow simulation. Adjacent to

the flow, the heat storage in the ground was accomplished by solving the heat equation over a set of

one-dimensional domains. A simple model of the collector roof was incorporated into the simulation

to allow for additional heat transfer at the upper surface.

The ground, flow, and cover are all linked via a set of heat fluxes, evaluated using an energy

balance developed from appropriate convective coefficients and radiative exchange between the

ground and collector cover. This simulation was validated using data from the pilot plant at

Manzanares.

The three dimensional analysis uses a commercial CFD code, Cfx5, to model a section of the

solar collector. This investigation focused on identifying local flow regimes and their effect on

convective heat transfer rates from the upper and lower surface. This knowledge was then applied to

the heat transfer correlations used in the one-dimension simulation in order to increase their accuracy.

As a result of this analysis, a set of mixed convective heat transfer correlations were

developed to improve the perceived deficiencies. Using these correlations, the simulation was

validated and found to produce a maximum daily output of 47.6kW with an error of 0.6%. The

addition of the new heat transfer correlations was essential to the accuracy of the simulation as those

for purely forced convection did not even approach the correct output and underestimated the

temperature rise in the collector.

A sensitivity analysis was conducted to evaluate the effect of different types of boundary

conditions on the solution; both constant and time varying. It was concluded that the remaining

discrepancies between the simulation and the pilot plant were a result of differences between the

boundary conditions of the simulation and those of the pilot plant. Further increase in accuracy could

be gained through incorporation of time varying inlet and outlet temperatures and pressures.

The simulation was determined to be a success, demonstrating the benefits solving the

complete flow equations in conjunction with a high fidelity ground model. Further benefits include

the accurate modelling of the adiabatic lapse rate, compressible chimney flow and incorporation of

losses. As a result, this simulation can be applied to the prediction of a full scale solar updraft tower

with confidence in the accuracy of the result.

b

ACKNOWLEDGMENTS

The author would like the thank Dr. Nicholas Waterson for his

invaluable insight, knowledge and support over the course of this project.

Great appreciation is also offered to Dr. Gerhard Weinrebe for providing the test

data from the Manzanares pilot plant used to validate the results of this project

c

TABLE OF CONTENTS

Abstract_________________________________________________________________________ a

Acknowledgments ________________________________________________________________ b

Table of Contents _________________________________________________________________ c

1. Introduction ___________________________________________________________________ 1

1.1 Solar Updraft Towers________________________________________________________ 1

1.2 Project Objectives ___________________________________________________________ 2

2. Background and Literature Review ________________________________________________ 3

2.1 Overview of Solar Updraft Tower Concept ______________________________________ 3

2.2 Solar Updraft Tower System Analysis __________________________________________ 4

2.3 Pilot Plant at Manzanares ____________________________________________________ 7

2.4 Solar Updraft Tower Turbine Characteristics ___________________________________ 8

2.5 Heat Transfer in a Solar Collector _____________________________________________ 9

3. 1D Solar Updraft Tower Analysis _________________________________________________ 10

3.1 Overview _________________________________________________________________ 10

3.2 Flow Simulation ___________________________________________________________ 11

3.2.1 Overview of Assumptions ________________________________________________________ 11

3.2.2 Governing Equations ____________________________________________________________ 11

3.2.3 Staggered Grid _________________________________________________________________ 13

3.2.4 Treatment of Pressure ___________________________________________________________ 14

3.2.5 SIMPLE Method _______________________________________________________________ 14

3.2.6 Treatment of Boundary Conditions _________________________________________________ 15

3.2.7 Treatment of Losses ____________________________________________________________ 15

3.2.8 Validation of Fluid Solution ______________________________________________________ 16

3.3 Ground Simulation _________________________________________________________ 17

3.3.1 Overview of Assumptions ________________________________________________________ 17

3.3.2 Discretisation of Governing Equation _______________________________________________ 17

3.3.3 Treatment of Boundary Conditions _________________________________________________ 18

3.3.4 Solution Procedure _____________________________________________________________ 18

3.3.5 Validation of Ground Solution ____________________________________________________ 19

3.4 Cover Simulation __________________________________________________________ 21

3.4.1 Overview of Assumptions ________________________________________________________ 21

3.4.2 Solution Procedure _____________________________________________________________ 21

3.5 Turbine Model ____________________________________________________________ 21

3.6 Energy Balance ____________________________________________________________ 22

3.7.1 Energy Balance Equations ________________________________________________________ 22

3.7.2 Heat Transfer Correlations _______________________________________________________ 23

3.8 Bringing it All Together _____________________________________________________ 23

4.0 3D Simulation of Collector _____________________________________________________ 24

4.1 3D Simulation _____________________________________________________________ 24

4.2 Mesh Size and Domain ______________________________________________________ 24

5.0 Results _____________________________________________________________________ 26

5.1 Fluid Flow within the Collector _______________________________________________ 26

d

5.2.1 3D Flow within the Collector _____________________________________________________ 26

5.2.2 Assessment of Heat Transfer Correlations ___________________________________________ 27

5.2.3 Development of New Heat Transfer Correlations ______________________________________ 29

5.2 Flow Variation through Plant ________________________________________________ 29

5.1.1 Change in Flow Characteristics with Solar Input ______________________________________ 29

5.1.2 Ground and Cover Temperatures __________________________________________________ 32

5.1.3 Effect of Losses and Turbine Pressure Drop __________________________________________ 32

5.3 1D Simulation of Pilot Plant _________________________________________________ 32

5.3.1 Daily Power Generation _________________________________________________________ 32

5.3.2 Ground Simulation _____________________________________________________________ 34

5.3.3 Sensitivity to Boundary Conditions _________________________________________________ 35

6.0 Discussion __________________________________________________________________ 37

6.1 3D Simulation __________________________________________________________________ 37

6.2 1D Simulation of Pilot Plant________________________________________________________ 37

7.0 Conclusions _________________________________________________________________ 40

References _____________________________________________________________________ 41

Appendix A – Additional Information from Literature Review ______________________________ i

Heat Transfer __________________________________________________________________ i

Conduction __________________________________________________________________________ i

Radiation ___________________________________________________________________________ i

Convection __________________________________________________________________________ i

Computational Methods _______________________________________________________ iii

Method of Residuals and Solving Sets of Equations _________________________________________ iii

Over-relaxation and Under-relaxation ____________________________________________________ iv

TDMA Algorithm ___________________________________________________________________ iv

Solution of Heat Equation _____________________________________________________________ v

Discretising Convective Equations: The Upwind Scheme ____________________________________ vii

Staggered Grids ____________________________________________________________________ viii

Solution of Convective Equations: SIMPLE Method ________________________________________ ix

Appendix B – Data from PIlot Plant At Manzanares ____________________________________ xii

Appendix C – Further Results from 1D Analysis _______________________________________ xv

Flow Simulation ______________________________________________________________ xv

Reasons for Compressible Flow ________________________________________________________ xv

Discretised Flow Equations ___________________________________________________________ xv

Control Volume Analysis for Converting Friction Factor to Pressure Loss ______________________ xvii

Atmospheric Model ________________________________________________________________ xvii

Calculation of Upwind Velocity in Chimney: Test Case 1 __________________________________ xviii

Calculation of Adiabatic Lapse Rate: Test Case 2 _________________________________________ xviii

Ground Simulation __________________________________________________________ xviii

Discretised Heat Equation ___________________________________________________________ xviii

Appendix D – Heat Transfer Correlations ____________________________________________ xx

Forced Convection ____________________________________________________________ xx

Natural Convection ___________________________________________________________ xx

Mixed Convection (Generated from 3D Analysis) ___________________________________ xx

Appendix E – Extended Results ____________________________________________________ xxi

Selection of Domain for 3D Model _______________________________________________ xxi

Fluid Flow within the Collector _________________________________________________ xxii

3D Flow within the Collector _________________________________________________________ xxii

e

Assessment of Heat Transfer _________________________________________________________ xxiii

Heat Transfer at the Upper Surface ____________________________________________________ xxiii

Development of Mixed Convective Heat Transfer Correlation _______________________________ xxvi

Flow Variation through the Plant _______________________________________________ xxvi

Effect of Losses and Turbine Pressure Drop on Flow ______________________________________ xxvi

Comparison with Pilot Plant _________________________________________________ xxviii

Ground Simulation ________________________________________________________________ xxviii

Sensitivity to Boundary Conditions ____________________________________________________ xxix

Appendix F – Assumed Properties of Flow, Ground and Cover __________________________ xxxi

Boundary Condition Functions _________________________________________________ xxxi

Ambient Temperature_______________________________________________________________ xxxi

Incident Solar Radiation _____________________________________________________________ xxxi

Ground Conductivity ________________________________________________________ xxxii

Simulation Input Conditions _________________________________________________ xxxiii

Aerothermal Performance of a Solar Updraft Tower Owen Williams

1

1. INTRODUCTION

1.1 Solar Updraft Towers

With increasing demand and costs of traditional energy producing fuels such as coal and oil,

the world must increasingly turn to alternate forms of energy to meet our needs. A further driver has

been the considerable negative impact that the burning of fossil fuels has had on our environment.

While no form of renewable energy stands poised to take over as the dominant form of energy

generation, adoption of a combination of many systems could help alleviate energy shortages and

make a significant contribution to our environment. These systems also help reduce the impact of

commodity prices on the cost of energy generation, a critical factor in developing nations which rely

on foreign imports of oil and coal.

A traditional drawback of renewable energies such as wind and solar power has been the

lower efficiency and high cost of building a commercial size plant. This drawback will become less of

an issue as the cost of fossil fuels increase, making the lower running cost of renewable energy plants

increasingly economically viable.

Wind energy can make a contribution to power generation but location is essential to its

viability. Additionally, wind energy is only available intermittently, and only when nature chooses to

provide it. Solar panels are still expensive, requiring high-tech industry and solar concentration power

plants require large amounts of direct sunlight for operation.

A solar chimney or solar updraft tower power plant, first proposed by (Gunther 1931), is a

solution to many of these problems, see Fig.1. It utilizes solar energy to heat the ground under a solar

collector which consists of a low circular transparent roof open at the edges. Through convection and

the greenhouse effect, the temperature of the air increases towards the centre of the collector where a

large tower is attached with an airtight seal. The high temperature air at the bottom of the tower is

much less dense than the cold air at the top causing a buoyancy force that creates an updraft which

draws more air in from the periphery of the tower. Thus the thermal energy of the air is converted to

kinetic energy which can be extracted using a turbine and electric generator.

As can be seen, the concept of the Solar Updraft Tower is very simple in principle and it uses

three technologies that have been around for many years, with characteristics that are already well

established; solar collector, tower and turbine. This simplicity makes it easy to maintain and its power

output can be relied upon wherever there is a constant source of solar radiation, either direct or diffuse.

Fig. 1 - Principle of the Solar Updraft Tower; showing glass collector roof, chimney and wind turbines

(Schlaich, Schiel 2000)

Aerothermal Performance of a Solar Updraft Tower Owen Williams

2

There is a close analogy between a solar updraft tower power plant and a hydroelectric dam.

The collector is the equivalent of the water reservoir and the tower gives the change in potential

energy necessary to drive the flow. Their turbines are also very similar in design. Additionally, they

both require a large sum of capital to build but maintenance costs are low. With no fuel costs and the

longevity of these plants, they will continue to operate efficiently years after their cost has been

covered, a great boon to any country in which they are built.

Solar Updraft Towers are currently being developed as a viable and long-lasting renewable

energy source with 200MW plants being planned for areas with plentiful open land and direct sunlight

such as Australia. These plants would have chimneys at least a kilometre high and collectors of a

couple kilometres in diameter. Such large scale plants are a necessity for the generation of electricity

at economical rates of approximately $0.08 per kWh. As no tower has ever been built of such a size, it

is obvious that such an undertaking is not to be taken lightly.

The viability of such power stations is based on projections and models that have been

validated by a 50kW pilot plant built and operated in Manzanares, Spain in the 1980s with promising

results. These models have then been extended to much larger scales to help predict the output of a

full scale plant.

1.2 Project Objectives

This project has two main objectives. The first is to simulate the full thermo-fluid interaction

of a Solar Updraft Tower (collector, tower and turbine) using a one-dimensional finite volume

approach to solve to complete flow equations. The aim is to develop a tool for the accurate prediction

of the performance of a Solar Updraft Tower of any dimension in a timely and accurate manner. An

additional objective is the preliminary analysis of the three dimensional flow under the solar collector

using a commercial CFD code, with the intention of applying this knowledge to increasing the

accuracy of heat transfer in the one-dimensional simulation.

Much research still needs to be conducted to make the construction of the world’s first

economically viable power plant a reality. An accurate and adaptive numerical simulation of the entire

Solar Updraft Tower system capable of providing predictions of flow conditions, power output and

plant efficiencies for different ambient conditions, locations and dimensions would be very useful in

designing such a plant. Such a simulation would, necessitate approximating the Solar Updraft Tower

as one-dimensional, for simplicity and shorter calculation times.

Such simulations have been conducted before, most notably by (Bernardes, Voss et al. 2003).

This model incorporated many simplifying assumptions. The most prominent and possibly least

accurate is the modelling of heat conduction into the ground using an analytical solution for a semi-

infinite flat plate. The ground acts as a storage medium, re-releasing heat at night causing continued

operation of the plant after dark. The applicability of the analytical solution is unclear due to the

cyclic nature of the incident solar radiation. It is also unclear how errors in the modelling of the

ground will affect any solution. Additionally, the conductivity of the ground at Manzanares varied

with depth; a situation more difficult to model with analytical solutions.

Additional numerical simulations for the entire plant have been conducted by (Pasumarthi,

Sherif 1998a), (Pretorius, Kroger 2006b) and (Von Backstrom, Theodor W. 2003). All of these

simulations focus the greatest part of the analysis on the heat balance within the collector, preferring

to use rough analytical formulas for the flow within the chimney, usually involving the Boussinesq

approximation. The author could not find an example of a simulation that solved the full flow

equations over the entire domain.

While these simulations have been generally quite accurate when compared with the pilot

plant at Manzanares, the effect of many of these approximations is unknown when extrapolated to a

large scale plant where additional effects such as compressibility become important.

The one dimensional numerical simulation will be developed using a different approach

centred on the numerical solution of the equations of motion of the fluid using a control volume

analysis. The aim is to develop a program that could be used in future design and performance

analysis of a full scale power station. It will aim to build on the strengths of the previous analyses in

the area of heat transfer, while also addressing some of the perceived deficiencies of the previous

Aerothermal Performance of a Solar Updraft Tower Owen Williams

3

simulations, increasing accuracy and confidence in the solution, especially when applied to the

simulation of a full scale power plant.

As such, a complete energy balance within the collector, including ground heating by solar

radiation, conduction into the ground, convective heating of the flow and radiative exchange with the

collector cover will be incorporated. The model of the conduction into the ground will involve

complete solution of the heat equation, giving a more accurate representation of an entire plant for the

reasons mentioned earlier. The continuity, momentum and energy equations will be solved over the

complete domain, treating the collector and chimney identically. Losses at the entrance, exit and

transition section from collector to chimney as well as skin friction losses will also be included.

Significantly, a model of the turbine will predict power output. This program will be validated based

on the results of the pilot plant in Manzanares, Spain.

Further research must also be conducted into the three dimensional flow in the solar collector.

It can be expected that the flow near the ground will be significantly hotter than near the collector roof.

As a result, it is expected that it will be thermally unstable creating flow patterns similar to Bénard

convection or helical vortex patterns as the flow accelerates toward the centre of the collector.

Heat transfer rates within mixed convective flows (or combined natural and forced convective

flows) have been historically difficult to characterise. (Maughan, Incropera 1987), (Osborne,

Incropera 1985a) and (Osborne, Incropera 1985b) have all investigated mixed convective flow

between two parallel plates heated from below but the additional complication of significant reduction

in cross sectional area could have unforeseen impacts on the flow. The resultant acceleration may alter

heat transfer rates or influence breakdown of the vortical structure, dividing the flow into separate

regions with different characteristics based on Rayleigh and Reynolds numbers.

Additionally, the accuracy of many of the one-dimensional simulations mentioned above will

be highly dependent on the convective heat transfer correlations used almost independently of how

complicated the method of solution of the energy balances. (Beyers, Harms et al. 2001) and (Pretorius,

Kroger 2006a) have conducted evaluations of some of these correlations in the past. (Pretorius,

Kroger 2006a) however, only evaluated the effect of different correlations on power outputs and did

not compare the heat transfer coefficients with collector data.

As a result, the second objective of this project is to investigate the full three-dimensional

flow within the solar collector using a commercial CFD code, CFX5. The nature of the flow will be

explored with the intention of identifying different flow regimes and heat transfer rates. Comparisons

will be made with existing heat transfer correlations, skin friction values and temperature distributions.

An attempt will also be made to develop an accurate mixed heat transfer correlation specific to the

given geometry. The aim is that this analysis will complement the solution of one-dimensional

simulation increasing confidence in its results.

2. BACKGROUND AND LITERATURE REVIEW

2.1 Overview of Solar Updraft Tower Concept

The combination of the major elements of a Solar Updraft Tower into a viable power plant

was first envisioned by (Gunther 1931). At the time, he foresaw that the only way to build a tower of

the required height using contemporary materials was to lean it up against a mountain making the

entire system very impractical.

In the years since, much work has been conducted investigating the efficiency and

effectiveness of such a tower. New materials and construction techniques have brought the Solar

Updraft Tower forward as a financially viable means of power production. Summaries of the state of

Solar Updraft Tower development including an economic analysis are available from (Schlaich,

Bergermann et al. 2005) and (Schlaich, Schiel 2000).

The main benefits of the solar chimney are its simple construction and operation and its

relatively low cost. It is for this reason that the low efficiency, of the order of 2%, is tolerated. It has

other advantages over rival solar power plants. These include the fact that it can operate using both

direct and diffuse radiation, crucial for constant operation and a major drawback of Central Receiver

Solar Power plants. Additionally, solar chimney power plants do not require the use of water, a rare

commodity in most dessert regions that would normally suit solar energy production. This is the

Aerothermal Performance of a Solar Updraft Tower Owen Williams

4

largest drawback of Parabolic Trough power plants. The simplicity of a solar chimney also means that

a locally trained workforce with basic construction techniques can build a solar chimney using local

resources. High technology required for photovoltaic systems is not necessary. The resources needed

to build a solar updraft tower power plant (glass and concrete) are also generally plentifully available

in regions in which a solar chimney would be profitable.

Another benefit of this system is that the ground acts as a thermal storage system, conducting

heat into the deeper levels during the day and then releasing it at night. This enables the plant to

operate even at night making it a good candidate as a base-load power plant. There are also some

disadvantages of this type of power plant. The collector has a large footprint, needing large areas of

cheap flat land. As a result, the land must be low cost with no competing usage. It is also not possible

to build solar chimneys in earthquake prone areas as the cost of the tower would increase drastically.

Other weather effects such as hail or sandstorms can also cause significant problems by damaging the

effectiveness of the collector.

It has been demonstrated that the power output of a solar chimney power plant is proportional

to the product of the collector area and tower height (Schlaich, Bergermann et al. 2005). As such,

there are no optimum dimensions of any future plant. The optimum dimensions will be determined by

material cost and will differ based on location. The economic viability of Solar Updraft Tower power

plants has been discussed by (Schlaich, Schiel 2000), (Schlaich, Bergermann et al. 2005) and (Haaf,

Friedrich et al. 1983). They showed that, while a large capital investment is required to build a solar

chimney power plant, it will become profitable at a large scale. A 200MW power plant was predicted

to generate power at a rate of 0.07 €/kWh which is in-line with the industry rate.

Work on Solar Updraft Towers peaked in the 1980s during the construction and

demonstration of a 50kW pilot plant at Manzanares Spain. The operational principle and construction

were detailed by (Haaf, Friedrich et al. 1983) followed by preliminary test results(Haaf 1984).

(Lautenschlager, Haaf et al. 1985)discussed the extrapolation of these results for large scale plants.

This pilot plant was a success but a full scale plant has yet to be built. Details of this pilot plant will be

discussed in Section 2.1.3

Focus has since shifted to analysing solar updraft tower systems using numerical and

analytical techniques. The intension is to increase the accuracy of predictions involving the power

output of a full scale plant. Given additional confidence in these estimations, it will be much easier to

convince potential investors. Current estimates indicate that these methods are accurate to

10%(Schlaich 2008). These analyses will be detailed more fully in Section 2.2

For a number of years, a 200MW Solar Updraft Tower was being planned for Australia due to

its large expanses of open land and sun throughout the year. This was assisted by their governments

Mandatory Renewable Energy Target (MRET). Unfortunately, funding for this project fell through

and no large Solar Updraft Tower is being planned currently. The next plant is likely to be 50MW and

will once again be constructed in Spain. The donation of a parcel of land and a planned government

subsidy will reduce the cost to such a level that the plant can be built. This plant will not be

commercially viable otherwise and many would like to use it as another test bed to gain further data to

attract investors to a larger more economical version.

2.2 Solar Updraft Tower System Analysis

A preliminary analysis of a solar updraft tower is available from (Schlaich, Bergermann et al.

2005). He gives an estimate of the power output of a plant based on the plant dimensions and the heat

gained in the collector. The driving force within the plant is the pressure difference between the top

and bottom of the tower. This difference is represented by the following equation and will be useful in

determining power output.

( )

dHgP Htower

toweratot ∫−=∆ 0

ρρ

Using the Boussinesq approximation and neglecting losses, he also states that the speed

reached by free convection currents within the chimney, if all the pressure difference is used to

accelerate the air, can be expressed as

Eq.

1

Aerothermal Performance of a Solar Updraft Tower Owen Williams

5

0

max, 2T

T

gHv towertower

∆

=

A tower efficiency and power production are also both defined.

0

TC

gH

AGP

P

Towerplantcolh ==

ηη

The immediate consequence of these equations is that the power output of the solar chimney

is proportional to a cylinder created by the collector area and chimney height. Thus for the same

power output, the dimensions are variable and the optimal proportions of collector to chimney are

solely determined by the relative costs of the construction materials.

(Haaf, Friedrich et al. 1983) present further insights into the performance of a solar chimney.

An equation for the temperature increase in the collector (Eq. 2) was found from the enthalpy

equation.

I

represents the global solar radiation. This equation simultaneously defines the collector

efficiency. This means that the temperature increase and the air mass flow rate are mutually

determinative and that the power generated is proportional to their product. The collector efficiency is

also a function of mass flow rate.

( )

mC

RI

mTT

P

c

c&

&

2

,

π

η

⋅

⋅=∆

A further, more accurate formula for the actual power output including the component

efficiencies and relevant dimensions was also presented by (Haaf, Friedrich et al. 1983). Notice once

again, that the power output is proportional to a cylinder of the collector area and chimney height.

aP

cT

Tfc TC

IRH

gP

2

3

2

π

ηηη

⋅⋅⋅⋅⋅=

(Haaf, Friedrich et al. 1983) also analysed the effect of varying plant dimensions on

efficiencies and total power outputs. He found that increasing the radius of the collector increases

output but also reduces plant efficiency. There comes a point when the temperature rise within the

collector becomes large enough that losses through the cover to the outside environment become

significant. Additionally, expanding the height of the tower increases the chimney efficiency and the

overall efficiency by increasing the mass flow rate. It was also found that increasing the tower radius

reduces friction and can be just as effective at boosting power output as increasing its height. Raising

the height of the collector can have the same effect and enhance the heat transfer to the flow due to the

lower flow velocity.

Many analyses have been conducted of the chimney in isolation. (Padki, Sherif 1999)

developed a set of equations that can be used to get preliminary estimates of power output and

efficiency. Using the Boussinesq approximation, neglecting friction and losses, and assuming steady

flow, they developed a set of equations that could be integrated analytically with some simplifying

assumptions. These equations are stated below and were found to consistently under predict power

output by approximately 4 to 6%. o

his the height of the chimney above the ground.

( )

iae

iaaio

TTA

ATTgh

P∆+

∆

=2

3

2

ρ

2













=

e

i

aP

o

chim A

A

TC

gh

η

Using the same equations, (Padki, Sherif 1988) conducted an analysis demonstrating the

effects of such parameters as envelope shape, height, and entrance to exit area by integrating the

equations numerically. They conclude that there is little benefit gained from having anything other

than a straight sided chimney. They also found that the power output is a strong function of initial

temperature rise within the collector but that the chimney efficiency is almost constant. They also

found that the power and efficiency increase with inlet to exit area ratio. In this analysis, they were

Eq.

2

Eq.

3

Eq.

4

Eq.

5

Aerothermal Performance of a Solar Updraft Tower Owen Williams

6

visualising a chimney of the convergent type. Additionally, the chimney must be limited to 200m

because of compressibility effects.

(von Backstrom, Theodor W., Gannon 2000) aim to address these flaws by conducting a

compressible analysis, examining the viability of a divergent chimney. A full scale solar updraft tower

would have to be approximately 1500m tall. At this height, the temperature will change appreciably

and the flow must be considered as compressible. They develop a set of equations for the governing

variables in terms of Mach number and then integrate them numerically over a number of control

volumes along the length of the chimney. This analysis is more in depth because it takes into account

the losses due to friction, internal obstructions and area changes. They determined that the loss due to

flow acceleration as it travels up the chimney is almost three times that due to friction. For this reason,

they suggest flaring the chimney by approximately 14% to keep the Mach number constant and

reduce these losses.

This is the fundamental debate about the configuration of the solar chimney layout. Both

convergent and divergent types have been shown to have advantages. At this point, neither type is

really being seriously considered for incorporation into a solar updraft tower since current plans wish

to keep construction as simple as possible. Additionally, a tower of the height required is difficult to

build, without the added complication of a divergent section.

A further study by (Von Backstrom, Theodor W. 2003) outlined a numerical method for

calculating the pressure and density variation with a chimney using a compressible set of equations

similar to those used by (von Backstrom, Theodor W., Gannon 2000). In that same year, (Von

Backstrom, Theodor W., Bernhardt et al. 2003) conducted a set of experiments to determine the

pressure drop within a solar chimney due to friction losses or bracing spokes required to reduce the

weight of the structure. It was found that a set of spokes would have a pressure loss coefficient of

approximately 0.0930 but increases to 0.122 in the event of swirling flow. The wall loss coefficient

varied between 0.0101 and 0.0137

Fewer studies have looked at the entire Solar Updraft Tower including the tower, collector

and turbine. These have the advantage of being able to predict the output of the entire system,

allowing for variations in solar radiation and the daily power cycle. One of the most prominent was

the analysis was conducted by (Bernardes, Voss et al. 2003) assuming the flow in the system was one-

dimensional. They simulated the system by solving a set of momentum and energy equations within

the collector in conjunction with analytical formulas for the variation of pressure, density and velocity

within the chimney. The temperature rise within the collector was found by implicitly solving a set of

energy exchanges between the flow, ground, cover and ambient air. The turbine was simulated by

determining the static pressure drop. The entire simulation was validated using experimental data

from the pilot plant at Manzanares and was found to be accurate to within 2%.

Their simulation is the most similar to the one currently being conducted as part of this

project. The main difference is that (Bernardes, Voss et al. 2003) centred their simulation on the

thermal network and energy balance and decided to use approximate equations for the flow in some

situations instead of solving the complete continuity, momentum and energy equations over the entire

system. The aim of the approach in this project is to reliably demonstrate the accuracy of another

method, gaining further insight into the solar updraft tower as a whole and increase confidence in full

scale models.

A further analysis was conducted by (Gannon, von Backstrom, Theodor W. 2000) analysing

the system as using the ideal gas cycle. Analytical formulas for efficiency and specific power output

were developed before adding in loss terms and re-evaluating the formulas. Turbine, friction and exit

losses were all incorporated. The resulting equations were found to consistently overestimate the

power production compared with Manzanares.

(Pasumarthi, Sherif 1998b) constructed another numerical analysis of the solar chimney with

the aim of determining the temperature increase in the collector and the total power output. This was

followed by (Pasumarthi, Sherif 1998c) who compared the results with an experimental chimney

investigation. As with (Bernardes, Voss et al. 2003) a thermal network was the centre of the

investigation except it was solved analytically. Both the experimental and numerical results varied

wildly, due in part to inaccuracies in measurement and the fact that the entire investigation was based

around very small scale chimneys. As such, temperature variations were not very large. When the

same numerical procedure was applied to a plant of the same dimensions as the pilot plant at

Aerothermal Performance of a Solar Updraft Tower Owen Williams

7

Manzanares, it was found to have an error of 20% in exit velocity and 9.5% in power output. Once

again, this points to the need to model the flow equations completely in addition to the heat transfer.

(Pretorius, Kroger 2006a) conducted a critical analysis of previous work by (Pretorius, Kroger

2006c) indicating that their use of a purely forced convection heat transfer correlation resulted in an

overestimation of plant output by 1.3% in yearly output through additional losses to the surroundings.

It is interesting to note that even after this analysis they concluded that the heat transfer from the

ground was dominated by forced convection. This author was unable to obtain a copy of the paper by

(Pretorius, Kroger 2006c) to determine the nature of the simulation.

A useful combination of numerical and experimental results led (Kirstein, Von Backstrom,

Theodor W. et al. 2005) to develop a analytical formula, estimating losses in the flow through section

between the collector and chimney. This formula takes both swirl and inlet guide vane angles into

account and will be used for the optimisation of future plants.

One of the most interesting extensions to the Solar Updraft Tower system was proposed by

(Kreetz 1997) and is summarised by (Schlaich, Bergermann et al. 2005). He suggested the use of

water bags underneath the solar collector. The increased heat capacity of the water would store

additional heat during the day, decreasing peak output but would release that heat at night increasing

night-time output. Through numerical simulation, Kreetz found that this increased heat available at

night coupled with the reduced atmospheric temperature resulted in a almost constant daily output

assuming the ground area was 25% covered with water bags. In this way, it would be possible to

create any output characteristic required. The other benefit of this system is that the water bags do not

have to be refilled as no evaporation can occur. As a result, the plant still does not consume water.

A further concept has also been proposed to eliminate the problems associated with

constructing a concrete tower by a floating structure instead. (Papageorgiou 2007) summarises work

on this proposal. The authors of that paper have also examined the effect of wind loads on such a

tower and the incorporation of turbines.

There are a number of further papers on the subject of Solar Updraft Towers as cited by

papers mentioned above, that this author was unable to obtain a copy. (Yan, Kridii et al. 1991)

constructed a comprehensive analytical model of the solar chimney, developing practical correlations

for airflow rate, power output and thermo-fluid efficiency. (Mullet 1987) conducted an analysis of the

overall efficiency of a solar chimney. A commonly cited numerical analysis was also conducted by

(Kroger, Buys 2002).

2.3 Pilot Plant at Manzanares

There are three main papers that detail the operation and results of the pilot plant at

Manzanares. (Haaf, Friedrich et al. 1983) details the principle and construction of the plant paying

specific attention to costs and plant efficiencies. (Haaf 1984) presents test results for the plant in

continuous operation. Properties of the ground, daily power production and losses were given

particular attention. (Lautenschlager, Haaf et al. 1985) presents further results from the pilot plant and

discuss how they can be extrapolated to large, economically viable plants. For purposes of comparison,

further experimental data was provided by (Weinrebe 1987) giving conditions for a single sample day.

The pilot plant built in Spain was small and as such was still very uneconomical but the main

intention was to investigate the operating principles, thermodynamic performance and storage effect

of the ground. The knowledge gained would help validate further numerical investigations of full

scale plants with further confidence in the power generating potential of the solar updraft tower

system. A secondary objective, but no less important was the verification of the low construction costs

for high slender pipes and light durable translucent covering structures.

The dimensions of the pilot plant were selected to provide reasonable demonstration of

construction methods and to provide accurate results, not for utmost efficiency. As a result, canopy

height and radius were increased so that the change in temperature would be close to 20K. Refer to

(Haaf, Friedrich et al. 1983) for further details surrounding the construction or economic viability of

the pilot plant. A summary of the design criteria and dimensions are given in Table 2 in Appendix B.

(Haaf 1984)found that the conditions at Manzanares provided a very tough test of the pilot

plant. The local wind levels were reasonably high, causing losses, and a quarry went into operation

Aerothermal Performance of a Solar Updraft Tower Owen Williams

8

upwind of the plant not long after it was constructed. As a result irradiation levels above 900 W/m2

were very rare and power output levels were correspondingly lower than expected.

Midday variation in velocity, global horizontal radiation, temperature, upwind velocity and

power output were all presented by (Haaf 1984). The temperature difference within the collector was

rarely as high as predicted prior to building the pilot plant and as such the power production was

lower. A maximum solar radiation of 1040W/m2 produced a power output of 41kW on one sample

day.

Further results from the pilot plant including variations in temperature, power output,

collector efficiency and upwind velocity are all summarised in Appendix B. Measurements of losses

within the chimney, transition section and mechanical losses within the generator were also measured.

The pressure drop at the turbine was found to be approximately 80 Pa at peak operation and the

average daily output was approximately 40kW, much lower than initially expected. Appendix B also

gives measured variations of soil properties such as the conductivity and absorbtivity.

(Haaf 1984) concluded that the demonstration plant was a complete success, demonstrating

the reliable and cost effective operation of the initial concept. Although power generated was less, this

is largely due to the harsh environmental conditions and it was estimated that the output could be

increased by 8% by altering plant dimensions to make it more efficient. Haaf concluded that the initial

principles for the accurate modelling of large scale plants and their construction were validated.

(Lautenschlager, Haaf et al. 1985) details the applicability of lessons learned with the pilot

plant to large scale plants. He found that during night time operation, the flow speed never dropped to

zero and was in fact larger that previously thought. This was partially due to atmospheric radiation

from clouds and partially due to the reduction in ambient pressure giving a greater temperature rise

within the collector.

Further results also indicated that during the day, the ground was heated to such an extent that

it created a small thermal layer near the ground effectively preheating the air on its way into the

collector. Thus, the adiabatic lapse rate was much greater than expected and the driving buoyancy

force correspondingly larger, increasing output. It was also estimated that optimum collector

efficiency for large scale plants of 50% could be obtainable.

Another benefit of a large scale system would be that the tower would be less slender and

wind loads would be lower. With this in mind, a set of tower designs for scales up to 1000m using the

same construction techniques and materials as the pilot plant were detailed.

A summary of pilot plant data, including that which will be used to validate the simulation in

this project is available in Appendix B.

2.4 Solar Updraft Tower Turbine Characteristics

The turbine used for a Solar Updraft Tower power plant is analogous to the turbine in a

hydroelectric dam and as such it has characteristics between both a wind turbine and high speed gas

turbine. It has a different load coefficient and degree of reaction due to the constant updraft and the

fact that it is ducted. As a result, it has more blades than a wind turbine but fewer than a gas turbine.

The blades also adjust in pitch, like a wind turbine instead of a gas turbine for which they are fixed.

Additionally, it is pressure-staged as with gas turbines instead of velocity-staged like a wind turbine.

All of these characteristics bring its design closest to a hydroelectric dam whose operation is well

known with proven reliability records (Schlaich 2008).

A number of different turbine configurations have been proposed for Solar Updraft Tower

power plants. The pilot plant at Manzanares employed a vertical axis turbine at the base of the

chimney. This is the most efficient option and the chimney supports can be shaped to act like inlet

guide vanes to reduce transition losses. A second configuration has been proposed by (Schlaich 2008)

for a full scale solar updraft tower because the shear diameter of the chimney would be such that a

very large turbine would have to be created at additional cost. This second configuration proposes a

set of horizontal axis turbines set between the collector exit and the beginning of the transition section

to the chimney. This configuration has its appeal because not only does power production continue if

a turbine needs repair but they will be sufficiently smaller in size that they can be procured of the

shelf. This option will increase losses but it is thought that the added simplicity and lower cost will

offset this.

Aerothermal Performance of a Solar Updraft Tower Owen Williams

9

The power output from the turbine is proportional to the volumetric flow rate and the pressure

drop at the turbine. Combined with the turbine efficiency (close to 83%) the power produced is given

by Eq. 6 (Haaf, Friedrich et al. 1983).

VPN TT

&

⋅∆⋅=

η

The main difficulty lies in determining the pressure drop at the turbine. Many studies such as

(Bernardes, Voss et al. 2003), (Haaf, Friedrich et al. 1983) and others have assumed a turbine pressure

drop of 2/3 of the total pressure difference for maximum fluid power. This is a total 12.5% higher than

the maximum Betz factor for a free standing turbine. The total pressure difference is defined for a

solar chimney according to Eq. 1, and is the pressure difference that would occur if the turbine was to

extract all power from the flow. This assumption has been shown to work sufficiently well but its use

is largely unjustified in these works.

(von Backstrom, Theodor W., Fluri 2006) examined the validity of this assumption. Using a

power law approach to the variation of pressure potential, they found that the pressure drop as a

fraction of this potential is equal to 2/3 only when it is constant and independent of flow rate. They

showed that this could lead to appreciable underestimation of the performance of a plant. More

importantly, the analysis predicts that maximum fluid power is available at much lower flow rate and

much higher turbine pressure drop than for the 2/3 case. This can lead to over sizing of the flow

passages of the plant. The turbine might also be mistakenly designed with inadequate stall margin and

excessive runaway speed margin.

A very in depth analysis of solar chimney turbine characteristics has also been conducted by

(von Backstrom, Gannon 2004). They presented analytical equations for the variation in turbine

efficiency with load coefficient, degree of reaction and turbine flow. Equations for optimum values of

the above variables are also presented. These equations indicate that for a large scale solar updraft

tower turbine, a total-to-total efficiency of 90% is attainable but not at all operation conditions.

A slightly different type of turbine was evaluated when (Denantes, Bilgen 2006) considered

counter-rotating turbines for on and off-design conditions. Combined with a simplified

thermodynamic model of the collector and chimney, the annual energy output of the system was

estimated. (Denantes, Bilgen 2006) showed that counter-rotating turbines without guide vanes have a

lower design efficiency and higher off-design performance than singular turbines.

(Gannon, Von Backstrom, Theodor W. 2002) and (Gannon, Von Backstrom, Theodor W.

2003) have also conducted considerable work on the design and experimentation of turbines for solar

updraft towers but this author was unable to obtain a copy of these papers for review.

2.5 Heat Transfer in a Solar Collector

Heat from the cover and ground within the collector is dominated by convection as it is

generally assumed that the air does not absorb any radiation. As a result, there is a forced convection

component as the flow moves over the ground and under the cover. What makes this situation

interesting however, is that the ground acts like a heated flat plate and thus will create an unstable

thermal layer near the ground that will cause natural

convection. This will likely form a voritcal pattern within the

flow. Thus convection within the collector is mixed in nature,

a situation in which it is very difficult to estimate heat transfer

because almost all correlations assume purely forced or purely

free convection.

Within the solar collector, the velocity and area are

inversely proportional to the radius. For this reason, the flow

accelerates quite rapidly near the chimney. In this region it is

expected that the flow will be dominated by forced convection.

At the entrance, the flow velocity will be very small, causing

the flow to be dominated by free convective effects.

There are a number of flow regimes that can be

formed with mixed convective flow between parallel plates. As

mentioned previously, at low Rayleigh number, Ra, the thermal layer is stable and no natural

Fig. 2 –

Example of Benard Cells {{53

Incropera,F.P. 2002; }}

Eq.

6

Aerothermal Performance of a Solar Updraft Tower Owen Williams

10

convection can occur. At higher values, 4

1051708 xRaL≤< longitudinal roll cells or Bénard cells

can be formed as shown in Fig. 2. These counter-rotating cells are caused by the buoyant flow. At

even higher Ra, the coherent vortex structure breaks down causing a turbulent fluid motion (Incropera,

DeWitt 2002). It is expected that as the flow speed increases within the collector, a transition between

similar regimes may occur.

Much work has been conducted on mixed convective flows that could be helpful in predicting

flow regimes in the three-dimensional simulation. (Maughan, Incropera 1987) conducted experiments

involving heat transfer into air for the entrance region of a horizontal or inclined channel. For the

horizontal case they found that the entrance region was dominated by forced convection which

showed a rapid decline in Nusselt number, Nu, as a thermal layer formed next to the wall. Thermal

instability would eventually become apparent instituting a secondary flow bringing heated flow into

the free stream creating a region of increased heat transfer. Nusselt number would subsequently peak

and decay to a constant value determined by the Grashof number, Gr. This is due to increased heating

of the free-stream flow causing a more uniform temperature distribution.

Further useful information regarding mixed convective flow in ducts was obtained from

(Osborne, Incropera 1985a) who examined the effect of axis-symmetric heating on laminar flow in

ducts. They found that the top surface was dominated by forced convection while the bottom surface

was dominated by mixed convection that increased heat transfer rates by as much as seven times. It

was also found that the conditions at each surface were mutually exclusive and did not influence each

other.

(Osborne, Incropera 1985b) conducted identical experiments but with transitional and

turbulent flows, for which buoyancy effects are less pronounced. For fully turbulent flows, it was

found that heat transfer at the top surface was once again well correlated with forced convective

relations but the lower surface Nu increased with heat flux. Mixed convection at the lower surface

also greatly increased heat transfer as in the laminar case. Once again, the conditions at each plate did

not influence each other.

None of the above studies examined the effect of flow acceleration on heat transfer rate

within the duct. Only one study came to light that attempted to examine this effect. (Beyers, Harms et

al. 2001) constructed a finite volume analysis of turbulent convective flows so that that they could

investigate heat transfer in radially accelerating ducts. Comparisons were presented between the

computational results and analytical approximations for heat transfer rates and skin friction for both

entry regions and fully developed flow. It was found that such analytical formulas as those proposed

by Gnielinski (Incropera, DeWitt 2002) overestimated the heat transfer in a radially accelerating duct.

3. 1D SOLAR UPDRAFT TOWER ANALYSIS

3.1 Overview

The objective is to develop a program that is responsive and adaptable to any Solar Updraft

Tower dimension or design. Computational time should be sufficiently low so as to be able to use this

program as a development tool to quickly examine the effect of any change to a Solar Updraft Tower

design. It will centre on the numerical solution of the three non-linear flow equations (continuity,

momentum and energy) using a control volume analysis. A single differential equation will govern

heat conduction into the ground while a simple cover model will simulate the top of the collector. The

entire flow will be driven using a full compressible solution and the natural buoyancy of the flow

within the chimney.

An energy balance between radiative, convective and conductive influences will determine

boundary heat fluxes, linking each of these elements together. Heat losses to the outside environment

through the cover will also be evaluated. Convective heat transfer coefficients will be determined

from appropriate correlations.

For accuracy and to evaluate the net power output, a critical objective, the program will

model as many known effects and irreversibility’s as possible. These include friction, inlet and exit

losses. The pressure loss due to the swirl of the flow within the transition section between the

Aerothermal Performance of a Solar Updraft Tower Owen Williams

11

collector and chimney sections will also be estimated. A turbine model will give a close estimate of

power extraction and the pressure drop associated with it.

As the change in solar radiation is slow and thus flow conditions also change slowly, it is

possible that a steady state calculation would be sufficient to generate accurate results at each timestep.

This is the approach used in many of the other numerical analyses of Solar Updraft Towers such as

(Bernardes, Voss et al. 2003) and (Pretorius, Kroger 2006a). This being the case, transient effects may

be significant at turbine start-up and the inclusion of the transient terms will not increase

computational time significantly so in the interest of making the computation as accurate as possible,

all computations will be carried out using a transient analysis.

Additionally, there are assumptions intrinsic in the quasi one-dimensional assumption. It will

be assumed that the flow is identical to that through a variable area duct and that flow properties are

constant across the section. All properties will be mean at any station.

This program will also incorporate a cyclic variation of ambient temperature throughout the

day that will hopefully increase accuracy by increasing output during night time operation.

3.2 Flow Simulation

3.2.1 Overview of Assumptions

(von Backstrom, Theodor W., Gannon 2000) demonstrated that in order for a full scale Solar

Updraft Tower power plant of 200MW to be simulated, the flow must be treated as compressible. A

summary of this calculation is given in Appendix C. For a tall tower, the temperature change between

the entrance and exit is large enough to cause a significant change in density.

Flow Assumptions:

○ Transient, compressible flow with heat transfer in the form of a boundary heat flux.

○ Constant properties across the duct due to quasi-one dimensional approximation

○ Linear variation of collector height and chimney diameter allowing the testing of various

configurations

○ Treat the air as a perfect gas: Cp = 1005 kJ/(kg s), γ = 1.4, kf = 0.0257

○ Assume that the ground and collector cover are approximately parallel even though the height

of the collector can change. This simplifies the mathematics of the friction and pressure terms

because these forces will be always acting in the flow direction as a result. This can be

assumed to be valid because the radius of the collector is large, and change per control

volume will be small.

○ Axisymmetric flow of the air within the collector causing non-uniform heating of the

collector surface due to sun’s angle to be neglected

○ The collector is placed over a plane surface

○ The flow through the entire chimney is turbulent due to the large Reynolds number. This is

not necessarily true near the entrance but is a good approximation over the rest of the plant

3.2.2 Governing Equations

For transient, compressible flow with heat addition, the continuity, momentum and energy

equations must be solved simultaneously. What follows is a brief description of the derivation of each

equation including explanations of each term. Additional terms that were neglected will also be

described, giving reasons for their removal.

Continuity Equation:

The continuity equation is a linear differential equation based upon the principle of

conservation of mass. Considering a control volume analysis, this represents the following in verbal

terms.

(Rate of Change of Mass within the Control Volume) +

(Net Flux of Mass across the Control Volume Boundaries) = 0

Aerothermal Performance of a Solar Updraft Tower Owen Williams

12

The differential equation in one dimension is given below. There are no additional terms and

no terms have been neglected due to the simplicity of this formula (White 2003).

( )

(

)

0=

∂

+

∂

=⋅∇+

∂

x

u

t

u

t

ρ

Momentum Equation:

The momentum equation is an extension of Newton’s Second Law where force equals the rate

of change of momentum. This equation is non-linear and will also be explained using a control

volume analysis. Considering a control volume of differential size, the total balance of momentum

within it can be stated verbally as follows:

(Rate of Change of Momentum in Direction i) =

(Body Forces acting in Direction i) + (Surface Forces Acting in Direction i)

In this simulation there are multiple surface and body forces acting on a control volume of

fluid including the following.

○ Viscous forces – due to the viscosity of the fluid, shear forces form on the edges of the control

volume. These forces are neglected in this simulation

○ Pressure forces – are normal forces, caused by the pressure of the adjacent control volumes

exerting a force on the boundary of the current control volume

○ Buoyancy Forces – are body forces caused by changes in density and are proportional to

gravity. For a one dimensional analysis they are only included when the domain is aligned

with the gravity vector i.e. in the chimney

○ Losses – Additional losses due to irreversibilities such as friction or the turbine are also

included. They contribute to a loss in momentum that can be modelled as an additional force

on a control volume.

The momentum equation in one dimension is shown below (White 2003).

( )

loss

Fg

x

u

x

P

u

x

u

t

+−

∂

+

∂

−=

∂

+

∂

ρµρρ

2

As mentioned previously, the viscous term was neglected in this simulation because friction

losses are modelled through the source terms and diffusion will be small. This being said, it could be

included in future and as such is included in this formula for completeness.

The pressure term requires special treatment and will end up altering the form of the

buoyancy term. This will be discussed in Section 3.2.4

Energy Equation:

The Energy Equation is an extension of the First Law of Thermodynamics and can be found

in many forms. In this simulation, the energy equation will be solved for static enthalpy which is

easily converted to temperature making it much more convenient than solving for total energy.

Applying the first law of thermodynamics to fluid flow through a fixed control volume, then

the energy equation can be stated as follows:

(Rate of Heat Addition to the Fluid from the Surroundings) +

(Rate of Work Done on the Fluid inside the Control Volume) =

(Rate of Change of Energy of the Fluid as it flows through the control Volume)

Inertia

Pressure

Viscous

Buoyancy

Losses

Rate of Change

of Momentum

Eq.

7

Eq.

8

Aerothermal Performance of a Solar Updraft Tower Owen Williams

13

Neglecting the effect of viscous stresses and shaft work, the energy equation in terms of total

energy (internal plus kinetic) is given below in conservation form where f represents the sum of

body forces per unit mass and q

&represents the rate of heat addition (White 2003).

(

)

( ) ( )

ufquP

Dt

ueD ⋅++⋅−∇=

+

ρρρ

&

2

This form of the equation includes both mechanical and thermal energies but only the thermal

form is required as we wish to solve for static enthalpy directly. Subtracting the dot product of the

velocity with the momentum equation from Eq. 9 and rearranging results in the final form of the

energy equation given by Eq. 10 (White 2003).

( ) ( ) ( )

h

SPu

t

P

quhh

t

+∇⋅+

∂

+−∇=⋅∇+

∂

ρρ

Note that the body force term was eliminated in the final equation. In this simulation, it would

represent the potential energy gained by the fluid when rising up the chimney. Potential energy will

not have an effect on thermal energy hence its elimination was expected.

In this derivation, viscous dissipation and shear force terms have also been neglected. This is

acceptable because the effects of friction loses are already being modelled as source terms within the

momentum equation. The viscous dissipation term can also be neglected because this is a low Eckhert

number flow (Bird, Steward et al. 2002).

3.2.3 Staggered Grid

A staggered grid was constructed for the reasons stated in Section2.3.6. The possibility of

using a clustered grid was also included in the formulation. On each end an additional cell was added

to hold the boundary conditions in a simple manner. Staggering the grid means that the program must

keep track of the additional velocity control volume dimensions and the definition stored dimensions

are shown in Fig. 3. An additional cell was added to facilitate the transition between the collector and

chimney. This creates a discontinuity in the solution because its dimensions are much different than

the surrounding cells. The node at Cell 1 corresponds to x = 0

Fig. 3 - Definition of One-Dimensional Staggered Grid

Rate of Heat

Addition

Pressure

Work

Source Terms

Rate of Change

of Enthalpy

Enthalpy

Advection

Cel

l 0

Cell 1

Cell 2

dx_stag

U

0

A0

P

0

ρ0

T0

P

1

ρ1

T1

dx

U

1

A1

U

2

A2

P

2

ρ2

T2

Ground

Cover

Eq.

9

Eq.

10

Aerothermal Performance of a Solar Updraft Tower Owen Williams

14

3.2.4 Treatment of Pressure

The absolute pressure at any point is a large value of the order of 105. As the maximum speed

in a Solar Updraft Tower is relatively small, the changes in pressure within the entire system will be

of the order 102 and the pressure correction terms will be even smaller. It is numerically advantageous

find a method that will solve for small pressure changes instead of the entire absolute pressure.

Otherwise, the small changes can get lost when solving for the large absolute value.

It was decided to split the pressure into three different components; a constant term, a term

proportional to height, and a pressure deviation from the sum of the previous two.

'

PgHPP refO ++=

ρ

As can be seen, the second term is proportional to the buoyant force and is essentially the

hydrostatic portion of the pressure. The definition of the reference density is arbitrary, solely

determining size of the third term. For this reason, we will define the constant and hydrostatic

components so as to simplify the boundary conditions. The constant term will be proportional to the

atmospheric pressure at the inlet, making the inlet pressure boundary condition equal to zero since it

has zero height. The hydrostatic component and reference density will be defined to make the exit

pressure boundary condition also equal to zero.

gH

PP outin

ref

−

=

ρ

The momentum equation must be changed to take account of this change in pressure

definition and it has its greatest impact on the form of the buoyancy force. Differentiating the pressure

with respect to x and substituting this into the momentum equation results in a new momentum

equation, Eq. 13.

( )

lossref Fg

x

u

x

P

u

x

u

t

+−+

∂

+

∂

−=

∂

+

∂

ρρµρρ

2

2'

2

3.2.5 SIMPLE Method

The Semi-Implicit Method for Pressure Linked Equations or SIMPLE method is described in

detail in Appendix A. The momentum equation forms the basis of the velocity-correction equation,

followed by the continuity equation being discretised into the pressure-correction equation. The

energy equation was then solved separately and its effects felt through a change in temperature and

hence density through the equation of state. A summary of the method as stated by (Patankar 1980) is

summarised here while the discretised equations are available in Appendix C. All equations were

solved using the method of residuals and TDMA algorithm also described in Appendix A.

(1) Guess the pressure field *

P

(2) Solve the momentum equation (Eq. 47) for the starred velocities *** ,, wvu

(3) Solve the pressure correction equation for '

P

(4) Calculate the pressure by adding '

P

to *

P

(5) Calculate

w

v

u

,

using the starred velocities and the velocity correction equation.

(6) Solve the discretised equation for temperature. Calculate the effect this has on the flow field

through the equation of state and recalculate the density.

(7) Treat the corrected pressure as the new guessed pressure *

P

and return to step two and

repeat the entire procedure until the constant term, b, in the pressure correction equation is

zero.

Eq.

11

Eq.

12

Eq.

13

Aerothermal Performance of a Solar Updraft Tower Owen Williams

15

3.2.6 Treatment of Boundary Conditions

Boundary values are held in an additional cell on the edge of the domain as shown in Fig. 3.

In this way, the governing equations do not have to be altered to incorporate them.

At the entrance, both the ambient pressure and temperature must be specified whereas at the

exit, only the pressure is required. The other state variables, density and velocity are then specified by

solution of the governing equations. The exit temperature does not need to be specified because the

solution uses an upwind scheme and thus any temperature at the exit would not have an effect within

the chimney. The inlet pressure will be assumed to be constant while all the other boundary variables

will be allowed to vary. The nature of this variation will be left up to the user.

A function was devised for the variation of ambient temperature throughout the day. It has a

sinusoidal pattern for which the average temperature and range can be specified. Definition of this

function is available in Appendix F. The outlet pressure can be tied to this temperature through the

following approximation of atmosphere (Bernardes, Voss et al. 2003).

( )

235.1

1

==











−

−=

−

κ

g

RT

H

z

PzP o

o

3.2.7 Treatment of Losses

There are a number of losses that must be modelled within the fluid simulation to increase

accuracy and provide valid results. All losses will be included in the form of a static pressure drop and

they will account for skin friction, inlet and outlet pressure losses and the loss in pressure due to the

transition section between the collector and the chimney.

Friction Losses

The friction acting on a circular pipe due to turbulent flow is usually stated in terms of a

Darcy friction factor that can be determined from a Moody chart (White 2003). This factor is defined

as:

2

8

V

fw

ρ

τ

=

Using a control volume analysis and a balance of momentum, the change in static pressure

due to the frictional force can be determined and then converted to friction factor as shown in Eq. 48

(White 2003). The calculation can be found in Appendix I.

d

L

fuP 2

2

1

ρ

=∆

For flow in non-circular ducts, the convention is to use the same equations but for an

equivalent hydraulic diameter, h

D. For the collector, the hydraulic diameter is twice the height of a

particular cell or HDh2=. After evaluating the Reynolds number based on hydraulic diameter and

correlating it with the Moody plot the friction factors for the cover, ground and chimney are shown

below. It was assumed that the ground and chimney each had a roughness of 4mm and 2mm

respectively, while the cover was smooth.

01.0011.0008.0

cov === chimneygrounder fff

For increased accuracy, a more advanced method of variable skin friction was also included.

This involved solution of the equation by Haaland as stated in (White 2003). This is an approximation

to the Moody chart, accurate to 2%.

2

11.1

7.3Re

9.6

log8.1

−



































+−= D

f

D

ε

Eq.

14

Eq

.

15

Eq.

16

Eq.

17

Aerothermal Performance of a Solar Updraft Tower Owen Williams

16

Inlet Loss:

Static pressure losses can be estimated for a number of situations including rapid changes in

geometry, valves, inlets or exits. They are generally characterised by a pressure loss coefficient,

K

,

that is a multiple of the dynamic pressure. In such a way, the pressure loss is given by:

(

)

2

1uKP

ρ

=∆

For a sudden contraction, the pressure loss coefficient is generally calculated from Eq. 19,

where d is the diameter of a small section and D is diameter of a large section (White 2003).











−≈ 2

2

142.0 D

d

Ksc

As 0≈Dd for the large inlet 42.0

≈

Kbut is generally rounded up to 0.5. The entrance

must also subtract a full dynamic head from the total pressure. Thus the total loss coefficient will be

1.5 but it must be noted that the actual pressure loss will be small because inlet velocities are low.

Outlet Loss:

The outlet loss can be modelled in a similar way, but in this case we can make a very useful

assumption about the kinetic energy at the exit. It will be assumed that as the flow exits the chimney,

all of the kinetic energy will be lost to the surroundings through dissipation and irreversible processes.

Thus none of the dynamic head is recovered and the static pressure at the exit to the chimney equals

the total pressure at a distance from the exit where the velocity becomes zero.

In this way, the exit loss is already implicitly applied at the exit to the chimney as long as the

total pressure at the altitude of the exit is used. Thus, no loss coefficient is needed.

Transition Loss:

A number of people have made different estimates of the transition loss coefficient between

the collector and chimney. (Kirstein, Von Backstrom, Theodor W. et al. 2005) cite (Kroger, Buys

2002) as using a pessimistic initial estimate of 0.25 for the loss coefficient. This is equivalent to

modelling the transition as a pipe bell mouth where the radius of curvature is 5% of the pipe diameter.

(Von Backstrom, Theodor W., Bernhardt et al. 2003) used experimental data to come up with a value

of 0.161 at model scale.

(Kirstein, Von Backstrom, Theodor W. et al. 2005) used a combination of experimental and

numerical results to derive a loss coefficient that would be applicable for many situations and is a

function of inlet guide vane angle and height of the entrance above the ground. The resulting

equations are shown below, where

β

is the IGV stagger angle,

θ

is the resulting swirl angle, his the

height of the chimney entrance above ground and

D

is its diameter.

D

h

H

K=+













+≈

θ

β

2

tan190.0

sec

00114.00292.0

The program will give the option to estimate the loss coefficient based on either a constant

loss coefficient or from the above set of equations.

3.2.8 Validation of Fluid Solution

Two test situations will be evaluated to determine the solution response relative to known

analytical solutions. For each test, the domain will be simplified to a section of the tower so that the

buoyancy force acts on the fluid. The first test will evaluate the functionality of the buoyancy term.

The second test will evaluate the effect of the pressure work term in the energy equation to determine

if it causes the correct adiabatic lapse rate.

Test Case 1: Buoyancy Functionality

To test the influence of the buoyancy terms, a simulation was set up using only the chimney.

For a constant density flow within the tower, it is expected that the pressure difference will balance

the frictional forces creating a constant velocity flow. Using a momentum balance over a single

Eq.

18

Eq.

19

Eq.

20

Aerothermal Performance of a Solar Updraft Tower Owen Williams

17

control volume, the following relation can be derived for the velocity within the chimney. See

Appendix C for the derivation. hc represents the height of the chimney inlet above ground.

(

)

( )

o

f

refo

RT

P

hcHC

gdH

u=

−

=

ρ

ρρ

2

The height of the chimney was chosen as 194.6m at a height of 1.85m off the ground and a

diameter of 10.16m. The internal density was 1.104 kg/m3 and the reference density was 1.2132 kg/m3.

Most importantly, the skin friction coefficient was set at 0.03. Using these properties, the upwind

velocity through the chimney was calculated at 12.87 m/s. The program returned a constant value of

12.82 m/s giving an error of 0.4%. As a result, it was concluded that buoyant terms were being

applied satisfactorily.

Test Case 2: Adiabatic Lapse Rate

The adiabatic lapse rate within the standard atmosphere is 9.8˚C/km. Within the chimney, the

lapse rate is determined largely through the pressure work term in the energy equation. Assuming a

steady, adiabatic, constant density flow the energy equation can be simplified to give the following

formula for the lapse rate. See Appendix C for this calculation.

H

PP

P

C

P

Toutin

P

−

=∇

∇

=∇

ρ

Once again, a simulation was set up using just the chimney to test this lapse rate. The internal

density of the simulation was 1.104 kg/m3, the inlet pressure was 101325 Pa and the outlet pressure

was 99009 Pa. The specific heat at constant pressure was 1005 J/(kg K). All other dimensions were

identical to Test Case 1. The formula predicts lapse rate of 10.7˚C/km which is exactly identical to the

rate found through simulation thus validating the energy equation and its simulation of the chimney.

3.3 Ground Simulation

3.3.1 Overview of Assumptions

To simulate the ground, the unsteady heat equation was solved over a set of one-dimensional

slabs of ground of a common depth. It was assumed that there was no conduction in the transverse

direction between slabs of ground due to the fact that the temperature gradient in the vertical direction

should dominate.

The theta method will be used to solve the heat equation and iteration will be incorporated to

allow for non-linear boundary conditions that depend on ground temperature. It is this type of

boundary condition that results from the full simulation. Although the full theta method will be

implemented, the final solution will be conducted using the fully implicit condition

(

)

1=f due to its

stability and the fact that it always produces physically relevant solutions. Further information on the

theta method and the heat equation can be found in Appendix A.

Provision was made for the solution to be either steady or unsteady depending on the manner

of the flow solution. Both fixed temperature and fixed heat flux boundary conditions were also

incorporated. The solution was formulated in such a way that it was also valid for a non-equispaced

mesh. Furthermore, it was assumed that density and specific heat capacity can vary with temperature

while the conductivity of the ground varies with depth.

3.3.2 Discretisation of Governing Equation

The unsteady heat equation, neglecting source terms is given by Eq. 23 . It will be discretised

over the domain shown in Fig. 4.













∂

=

∂

x

T

k

xt

T

CP

ρ

Eq.

21

Eq.

22

Eq.

23

Aerothermal Performance of a Solar Updraft Tower Owen Williams

18

Fig. 4 – Detail of a Section of Ground Domain

The process for discretising this equation is given in Appendix C. The resulting equation is shown

below.

( )

(

)

( )

(

)

( )























−

−











−

−

+























−

−











−

=−

∆

w

O

W

O

Pw

e

O

P

O

Ee

w

WPw

e

PEe

O

PPP

x

TTk

x

TTk

f

x

TTk

x

TTk

fTT

t

x

C

δδ

ρ

1

3.3.3 Treatment of Boundary Conditions

There are two general types of boundary conditions that are encountered for the heat

conduction equation; a given boundary temperature or a given boundary heat flux. To incorporate the

boundary, it will be assumed that the final node will be placed directly on the boundary as shown in

Fig. 5. As can be seen, this makes the boundary a half cell which must be treated differently. For the

implicit solution, one equation is generated per cell with the boundary condition only effecting the

equation for the half cell.

Fig. 5 – Detailed of Domain at Boundary

With this domain, if the boundary temperature is given, no problems occur and all the usual

equations can be used without problem. The first equation just gets eliminated from the problem as its

solution has already been given.

With a fixed boundary heat flux, the solution is slightly more complicated. Integration of the

temperature gradient in the heat equation over the half cell results in a balance of the heat flux at each

face.

iB

qq

x

T

k

dx

T

k−=













∂

−











∂

The interface heat flux can be approximated in the usual way, assuming that the temperature

varies linearly across the cell. All that remains is to substitute the given boundary heat flux B

qinto

this equation. Complications exist when including the discretisation with time but the method is

essentially the same. The main difference is the need to keep track of the boundary heat flux at the

previous timestep as well the current one.

3.3.4 Solution Procedure

The method of residuals, described in Appendix A, will be used to calculate a temperature

correction. The formulas for estimating the conductivity at the interfaces between cells using the

harmonic mean are also summarised there. The following method was used to determine a solution for

the ground.

(1) Calculate coefficients of implicit equation for temperature distribution.

P

E

W

∆x

e

w

(δx)

e

(δx)

w

Half Cell

I

B

(δx)

i

Eq.

24

Eq.

25

Aerothermal Performance of a Solar Updraft Tower Owen Williams

19

(2) Calculate Boundary heat fluxes

(3) Add temperature correction to previous solution

(4) Repeat from step one until the difference in the residual of the discretised equation from one

iteration to the next is less than a predefined convergence criteria

The properties of the ground were chosen based on estimation of density and specific heat

capacity. Conductivity was estimated from data available in (Haaf 1984) and summarised in Appendix

F. A logarithmic function for the conductivity as a function of depth was developed, and incorporated

within the program. This function is shown here while its justification can be found in Appendix F.

y

represents the current depth in meters.

(

)

6325.10121.0ln347.0 ++= yk

3.3.5 Validation of Ground Solution

The ground simulation was validated by comparing numerical results with known analytical

solutions. Results were tested in both the steady and unsteady regimes and which both fixed

temperature and fixed heat flux boundary conditions.

Test Case 1: Steady State Fixed Temperature Boundaries

The first test case involved a steady state simulation of a section of ground with an applied

temperature of 350K at the surface and 250K at 100m depth. For any steady state calculation, the

expected solution should be a linear profile between the two boundary temperatures. The numerical

solution was confirmed to lie exactly on this line as shown by Fig. 6

200

225

250

275

300

325

350

375

400

0 10 20 30 40 50 60 70 80 90 100

Depth (m )

Temperature (K)

Numerical Exact

Fig. 6 - Steady Solution of the Heat Equation with Fixed Boundary Temperatures

Test Case 2: Fixed Temperature Boundary on Semi-Infinite Flat Plate

There are a few cases for which an exact analytical solution of the heat equation can be found.

One such solution is for a transient solution over a semi-infinite flat plate. As long as the influence of

the surface boundary does not reach the second, deep, boundary then the numerical domain can be

approximated as semi-infinite and these analytical solutions can be used to test the accuracy of the

numerical solutions.

(Incropera, DeWitt 2002) gives the analytical solution of the heat equation for the above

situation by Eq. 27 where i

T is the initial temperature and s

Tis the surface temperature

(

)

(

)

Psi

s

C

k

t

x

erf

TT

TtxT

ρ

α

=













=

−

2

,

Eq.

26

Eq.

27

Aerothermal Performance of a Solar Updraft Tower Owen Williams

20

The test was conducted with and initial temperature of 300K with a surface temperature of

350K. Taking 4529.4

−

=

e

α

, a comparison of the numerical and exact solutions can be found in

Fig. 7 after three different lengths of time.

It was found that the computational solution was accurate to less than 0.01% error thus

validating the solution. It is also beneficial to note that as internal properties and boundaries were

linear, it took only a single matrix inversion to solve for the solution and the solution algorithm

behaved as expected.

290

300

310

320

330

340

350

360

0 2 4 6 8 10 12 14 16

Depth (m)

Temperature (K)

Computational Solution Exact Solution

t = 4000s

t = 10000s

t = 20000s

Fig. 7 – Comparison of Exact and Numerical Solutions for a Semi-Infinite Flat Plate with Fixed Temperature

Boundary

299.5

300

300.5

301

301.5

302

302.5

0 2 4 6 8 10 12 14 16

Depth (m)

Temperature (K)

Computational Solution Exact Solution

T = 4000s

T = 10000s

T = 26000s

Fig. 8 – Comparison of Exact and Numerical Solutions for a Semi-Infinite Flat Plate with Fixed Heat Flux

Boundary

Test Case 3: Fixed Heat Flux Boundary on Semi-Infinite Flat Plate

An exact analytical solution also exists for the case of a semi-infinite flat plate subjected to a

constant boundary heat flux. Such a solution was also given by (Incropera, DeWitt 2002) and is

shown in Eq. 28 where α is defined as in Eq. 27.

( )

(

)













−











−

=− t

x

erfc

k

xq

t

x

k

tq

TtxT oo

i

α

πα

2

4

exp

2

,

2

1

Eq.

28

Aerothermal Performance of a Solar Updraft Tower Owen Williams

21

The domain and ground properties were set up in an identical manner to the previous test

except that the boundary heat flux, o

q, was 500 W/m2. A comparison of the two solutions is shown in

Fig. 8. The computational solution was once again found to be in close agreement with the exact

solution with a maximum error of less than 0.01%. It is also noteworthy that a constant heat flux is

synonymous with a constant temperature gradient as can be seen clearly in the figure.

3.4 Cover Simulation

3.4.1 Overview of Assumptions

Modelling the cover is similar to the ground as it is a conduction problem but there is a major

difference due the fact that the cover is very thin. It can be assumed that the variation of temperature

through the thickness is very small and that consequently, the temperature gradient is negligible. As a

result, conduction of heat though the cover can also be neglected causing the cover to be modelled as

a lump of mass with boundary heat fluxes. This makes the solution both considerably simpler and can

save a great deal of computational time.

The cover will be discretised in the same way as the ground with one section corresponding to

each cell of the flow simulation. There will be a combination of natural and forced convective heat

fluxes to both cover surfaces but full details of this balance will be discussed in Section 3.6. Losses to

the environment will also be included. The transmittance, emissivity and absorbance of the cover will

be included for accurate prediction of energy balance.

As boundary conditions will be dependent on the cover temperature, they will be non-linear

and some iteration will be required. This will not be included within the cover simulation but will be

included when the entire Solar Updraft Tower solution in assembled.

3.4.2 Solution Procedure

The effect of heat addition on the temperature rise of a lump of mass is governed by Eq. 29. Q

is evaluated by evaluating all heat exchanges in the energy balance. It is a simple case of evaluating

the temperature change based on the properties of material and dimensions of the collector.

TmCq P∆=

3.5 Turbine Model

The modelling of the turbine is a small section of the overall program but very important. It

will be simulated as a force on a single control volume at the base of the chimney proportional to the

turbine static pressure drop.

As mentioned previously, the power produced by the turbine is proportional to the volumetric

flow rate, and pressure drop at the turbine. The constant of proportionality is the turbine efficiency

and will be taken as 83% which was the efficiency of the turbine at the Manzanares pilot plant

VPN TT

&

⋅∆⋅=

η

The turbine pressure drop will be estimated as 2/3 of the total pressure potential of the tower

which is given by Eq. 31.

( )

dHgP Htower

toweratot ∫−=∆ 0

ρρ

This is an approximation to the proportion for maximum turbine power and while it can lead to an

underestimation of power output according to (von Backstrom, Theodor W., Fluri 2006). It will be

used as a conservative first estimate. Thus the source term added to the momentum equation will be

equal to the following:

( )

dHgVS Htower

toweraT ∫−⋅⋅⋅= 0

3

2

ρρη

&

Eq.

29

Eq.

30

Eq.

32

Eq.

31

Aerothermal Performance of a Solar Updraft Tower Owen Williams

22

3.6 Energy Balance

3.7.1 Energy Balance Equations

The energy balance within the collector begins with the incident radiation impacting the cover,

getting transmitted at a slight loss and being absorbed by the ground. The heated ground only

reradiates the energy as long wave radiation and it will then be assumed that the cover is completely

opaque to this radiation, retaining the heat. This is the basis of the greenhouse effect, on which solar

updraft towers operate.

Assuming that the air is “dry” i.e. the humidity is zero, and thus transparent to radiation, the

energy balance involves radiative exchange between the ground and cover and the cover and sky.

There will also be convective exchange between the cover, ground and flow. Forced convective

correlations will determine heat transfer from the upper and lower surfaces into the flow while a

natural convection correlation will determine heat exchange between the ground and collector cover.

The main drawback of this approach is that the natural convection term does not heat the flow directly,

just the cover. As a result, it is likely that heat transfer to the flow will be significantly underestimated.

Attempts will be made to address this issue through examination of heat transfer within the collector

in the three-dimensional analysis. Refer to Fig. 9 for a description of the different heat exchanges.

Fig. 9 – Depiction of Heat Exchanges Taking Place in a Solar Collector

The complete energy balance involves determining the heat flux into the ground, flow and

cover at each control volume. An equation will be described for each boundary heat flux and they

must be simultaneously solved to determine solution

From (Duffie, Beckman 1974) the following set of equations were developed.

(

)

(

)

fccffggfflow TThTThq −+−=

(

)

(

)

(

)

gfgfgcgcgcgcground TThTThTTGq −+−+−+= 44

~

σεατ

(

)

(

)

(

)

(

)

(

)

4444

cov

~

cskyccacacfcfcggccger TTTThTThTThTTq −+−+−+−+−=

σεσε

Note how the cover relation incorporates losses to the outside environment in the form of

radiative and natural convection losses. cf

h and gf

hare forced convection heat transfer coefficients

between the ground cover and flow while gc

his a natural convection heat transfer coefficient between

the cover and the ground. It is assumed that

ε

~

is the combined emissivity for radiative exchange

between two infinite flat plates and is calculated using Eq. 36 from (Duffie, Beckman 1974).

inP TmC ∆outP TmC ∆

Incident Solar Radiation

Forced Convection

Natural Convection

Natural Convection with Ambient Air

Radiative Exchange (Cover and Ground)

Conduction into Ground

Eq.

33

Eq.

34

Eq.

35

Aerothermal Performance of a Solar Updraft Tower Owen Williams

23













−+= 1

11

~

1

cggc

εεε

The clear sky temperature involved in the radiation losses from the collector cover is

calculated using a formula from (Duffie, Beckman 1974)

5.1

0552.0 asky TT =

3.7.2 Heat Transfer Correlations

There are a number of heat transfer correlations that will be incorporated within this program.

The most important of these correlations are those for forced convective flow as they determine the

majority of the heat flux to the flow. A number of different correlations will be used and their

accuracy compared. For each of these equations, all values will be calculated using a characteristic

length of twice the height of the collector which is the hydraulic diameter. All of the relations were

developed for turbulent flow in circular tubes but the hydraulic diameter allows them to be applied to

other geometries.

(

)

h

Perimeter

AreaSurface

Dh2

4==

A list of all the convection correlations used in this project is available in Appendix D. The

first mentioned is a commonly used correlation for smooth tubes called the Ditus-Boelter equation. It

is the simplest presented here and is accurate to 25 %(Incropera, DeWitt 2002). Two further equations

are more complicated but increase accuracy to 10% by taking surface roughness into account. The

first is the Petukov relation (Incropera, DeWitt 2002). For a wider range of Reynolds numbers an

altered version of this equation called the Gnielinski relation is generally preferred. It is the

correlation of choice for many studies into solar updraft towers including (Pretorius, Kroger 2006a)

and (Beyers, Harms et al. 2001). Friction factors used in the above equations were calculated using

the equation of Haaland given in Section3.2.7.

For natural convection, the Globe and Dropkin relation (Incropera, DeWitt 2002)was used to

calculate cf

hfor the convective transfer between the ground and the cover. As mentioned earlier, there

are drawbacks to using this approach because the flow will not be directly heated by this transfer.

For natural convection between the cover and the ambient air, two relations were proposed.

The first was for natural convection from a flat plate from (Incropera, DeWitt 2002) and the second

was a rough approximation of the heat transfer rate from a heated flat plate with wind blowing over

the surface given by (Duffie, Beckman 1974)

3.8 Bringing it All Together

The simulations of cover, ground and flow are linked via the heat fluxes. The ground and

cover incorporate these fluxes as boundary conditions within their solution, while the flow equations

take the heat input as a source term in the energy equation.

As the heat fluxes of the flow, ground and cover solutions are mutually dependent, an

iterative procedure was developed to deal with this non-linearity. All heat transfer coefficients would

be calculated, followed by the flow, ground and cover solutions. This process was then repeated until

the change in the boundary heat flux of each section of the program was less than a pre-set

convergence criterion.

While not very efficient, it was found to be a robust method. In future, its efficiency could

possibly be improved by only calculating the cover and ground solutions every other time-step, as

they are able to withstand this change and remain stable.

It must be noted that the main heat input to the system, the solar radiation only appears within

the heat transfer equation of the ground. As such, none of the other sections of the simulation had to

have any knowledge of the solar input. The function for the solar input was based on data from

(Weinrebe 1987) and modelled as the positive half of a the cosine function. Further information on the

solar input is available in Appendix F.

Eq.

36

E

q.

37

Aerothermal Performance of a Solar Updraft Tower Owen Williams

24





















−

=0,

25.4

25.12

cosmax max

t

II

Simulations could be run using any solar input required, and beginning and any required time

of day. Initial conditions could be from rest or defined from a file holding a previous solution. A full

summary of assumed properties of the flow, ground and cover are available in Appendix F.

4.0 3D SIMULATION OF COLLECTOR

4.1 3D Simulation

Modelling of the collector was conducted using Cfx5 with a mesh generated in Gambit.

Analysis was conducted in a steady state, using a standard k-epsilon turbulence model and standard

buoyancy treatment. A thermal energy method was used for variation in air temperature. Buoyancy

turbulence production and dissipation models were also used, selecting default values of the turbulent

Schmidt parameter and dissipation factor. The fluid within the collector was air as a real gas at 25˚C.

Using an appropriate mesh, a steady state analysis was conducted considering the ground and

collector cover as smooth adiabatic walls. The objective was to obtain an initial condition for the main

calculations involving buoyancy. A pressure difference of 60 Pa was placed across of the entrance and

exit to generate a velocity distribution similar to the one-dimensional model.

The ground was set to a constant temperature, 345K, with a roughness of 3mm while the

cover remained smooth and adiabatic. This was the simplest situation and was conducted to determine

the effect of the mixed convection on the heat transfer coefficient. As the flow velocity varied through

the collector, this gave heat transfer coefficients at a range of Rayleigh and Reynolds numbers. This

was then repeated without the buoyancy model to validate forced convection heat transfer correlations.

A further simulation was conducted with the cover also at a constant temperature. This would

allow an assessment of the applicability of experimental tests by (Osborne, Incropera 1985b) that

suggested that for flow between heated infinite flat plates, the upper surface is dominated by forced

convection only and that the conditions at the two surfaces do not influence each other.

For each of these simulations, the ground was given a roughness of 3mm which is just slightly

more rough than concrete. To generate meaningfully comparative results with the one-dimensional

simulation, all variables were averaged at a given radius.

4.2 Mesh Size and Domain

Modelling the 360˚ of the collector would be a very large domain and would also be

computationally expensive and inefficient because there are likely to be symmetries. As a result, a set

of simulations were conducted with the intention of determining the extent of the domain that should

be modelled as well as its resolution.

Identical simulations were conducted using a domain of five, ten, fifteen and twenty degrees

assuming the collector has the dimensions of the pilot plant. Mesh resolution was 0.25m with

clustering of cells near the cover and ground to give a better accuracy in the boundary layer region.

With a pressure difference of 60 Pa, a maximum velocity of 9m/s was generated, in line with the pilot

plant at maximum output.

It was found that most of the important parameters were identical between simulations

including, velocity, heat transfer coefficient based on the cell near the wall, and wall shear stress.

Plots of their variation with radius are listed in Appendix E.

It was the variation in average flow temperature that was not identical, however. Additionally,

clear patterns in the heat flux appeared at the ground plane due to vortices, for larger domain angles.

More on these vortices will be discussed in subsequent sections, but the effect this had on heat transfer

into the flow can be readily seen through the variation in temperature with radius, Fig. 10.

As can be seen, the temperature plateaus for angles less than twenty degrees. It is possible that

at small angles, the symmetry planes limit the development of longitudinal vortices that enhance the

heat transfer. An attempt was made to increase the size of the domain beyond 20˚ by increasing the

section to thirty and forty-five degrees. Unfortunately, due to computational time and memory

restraints, the resolution of the grid had to be reduced for these larger domains. This led to a

Eq.

38

Aerothermal Performance of a Solar Updraft Tower Owen Williams

25

degradation of the solution as can be seen in Fig. 10. As a result, it was decided that a twenty degree

model would be sufficient for the tasks set out in this project.

296

298

300

302

304

306

308

310

0 20 40 60 80 100 120 140

Radius (m)

Temperature (K)

Theta = 20 Theta = 15 Theta = 10 Theta = 5 Theta = 45

Fig. 10 - Variation of Temperature with Radius for Different Domain Angles.

The mesh was made as fine as possible, while still allowing for limits in memory and

computational time. Brick elements were used in the final mesh, clustered close the walls as stated

previously. Even though this resulted in an elongation of these elements close to the centre of the

collector, tetrahedral elements were judged to be insufficient due to clustering of the mesh near the

centre of the domain causing visibly inaccurate results in these regions. The boundary layer mesh had

an initial cell with size 0.05m and a growth factor of 1.2. The free stream had a resolution of 0.25m.

a) b)

Fig. 11 - Visualisation of the Mesh a) Along Symmetry Plane and b) from above (From Cfx5 by Ansys)

Y-plus values were evaluated briefly to determine the accuracy of the near wall resolution and

if any further refinement was required. Its value was approximately 100. A high Reynolds number

solution using standard wall functions requires the value to be greater than 30 and while it may be

slightly high it was judged to be adequate.

Aerothermal Performance of a Solar Updraft Tower Owen Williams

26

5.0 RESULTS

5.1 Fluid Flow within the Collector

5.2.1 3D Flow within the Collector

Using the three-dimensional model described above, it was possible to visualise the different

flow regimes within the collector. Beginning by examining the heat flux at the ground (Fig. 12a),

clear radial streaks of increased heat transfer are visible, pointing to the development of longitudinal

vortices and mixed convective flow. The natural component of this convection can be visualised by

taking radial contours of velocity. Upwelling of heated flow is clearly visible, creating circular Bénard

Cells. These cells are not present over the full domain, however, as can be seen from the heat flux

contour. At the entrance, they do not exist and once they form, are not completely coherent over the

full domain.

a)

b)

Fig. 12– Visualisation of the Flow with the Collector through a) Contours of Heat Flux at the Ground b)

Contour of Velocity at R=100m

Thus, using this initial assessment is possible to split the domain into four radial sections as

delineated by Fig. 12a. They are; the entrance region, onset of thermal instability, vortex breakdown

and the acceleration region near the chimney. This delineation is supported not only by the heat flux

contour but also by the extent of buoyant flow in each of these regions. Fig. 29 in Appendix E shows

radial contours of velocity within each of these regions. It is clear than near the entrance, buoyant

1

2

3

4

Aerothermal Performance of a Solar Updraft Tower Owen Williams

27

flow does not exist, it peaks within the onset region and then becomes more erratic in the vortex

breakdown section before degrading even further in the acceleration region.

An examination of the wall heat transfer variation (Fig. 13) also supports this delineation of

flow regimes. The Nusselt number decreases from entrance until the onset of thermal instability and

then increases up to a local maximum. It then decreases slightly once again, until the rapid

acceleration near the exit causes a sharp rise in Reynolds number and heat transfer.

The flow in the first three regions is quite similar to what was observed by (Incropera, Knox

et al. 1987) in the entry region of a duct heated from below. In the entrance region, the heat transfer is

dominated by forced convection as the flow enters at a constant temperature. The initial decrease in

heat transfer is due to the flow immediately adjacent to the wall heating up, lessening the temperature

difference and heat transfer.

This thermal boundary layer will eventually pick up enough heat to cause thermal instability

and the onset of secondary flows. This secondary flow is similar to the formation of Bénard cells and

brings cooler air closer to the wall, causing the large increase in heat transfer. It is possible to state

that the heat transfer within this region will be dominated by natural convection due to the small radial

flow.

The vortex breakdown region is complicated as conditions will be transitioning from natural

to forced convection. (Incropera, Knox et al. 1987) found that immediately after the point of

maximum heat transfer, the vortex structure breaks down as the freestream flow becomes increasingly

mixed. This would cause the buoyant upwelling of heated flow to be weaker, reducing the heat

transfer due to secondary flows. This drop is not significant, however, suggesting that the loss in the

heat transfer due to buoyant flow is counterbalanced by flow acceleration, increasing the strength of

forced convection.

Within the acceleration region, it can be seen that the buoyant flow has decreased to a large

extent and that heat transfer is now almost entirely dominated by forced convection. It will be

confirmed in the next section that this is the case.

5.2.2 Assessment of Heat Transfer Correlations

An investigation of heat transfer correlations was carried out to determine the impact of

natural convection on heat transfer from the lower surface of the collector. Multiple forced convective

correlations were compared to asses how closely they approximated this mixed convective flow. Heat

transfer from the cover was also evaluated to determine if it was completely dominated by forced

convection and if flow conditions at the ground have any effect on its solution as predicted by

(Osborne, Incropera 1985b) for turbulent flow between heated flat plates.

The relative strengths of natural and forced convection at the lower surface can be evaluated

through the variation of the parameter 2

ReGr . If this parameter is much greater than one it means

natural convection is dominant, whereas forced convection is dominant if it is much less than one. Fig.

31 in Appendix E shows the variation of this parameter along the duct and hence that natural

convection is important along the entire length of the duct except near the collector exit where forced

convection becomes dominant as the velocity increases.

This can be visibly corroborated when examining the variation of heat transfer coefficient

with radius. Fig. 13a compares the variation of the heat transfer coefficient generated by the 3D

simulation and the three standard forced convection heat transfer correlations stated in Appendix D.

At the entrance, the heat transfer coefficient due to mixed convection is 2.68 times greater

than the closest forced convection approximation, the Gneilinski correlation. It is the most accurate,

becoming almost identical to the 3D estimation close to the collector exit. The Petukov relation is also

quite accurate in the forced convection region but is not as applicable at low Reynolds numbers and so

underestimates the heat transfer compared to the other relations, near the entrance. While the Dittus-

Boelter equation was just as accurate within the mixed convective region as the other correlations, it

was not as accurate in the forced convection region, most likely due to the simplicity of the equation

and the assumption of a smooth surface. The accuracy of the Gnielinksi equation is even more clear

when plotted non-dimensionally, with Reynolds number as in Fig. 13b.

Aerothermal Performance of a Solar Updraft Tower Owen Williams

28

a)

0

5

10

15

20

25

30

0 20 40 60 80 100 120

x (m)

Heat Transfer Coefficient (W/(m2K))

3D Solution Gnielinski Ditus-Boelter Petukov

b)

0

500

1000

1500

2000

2500

3000

3500

4000

0.00E+00 5.00E+05 1.00E+06 1.50E+06 2.00E+06 2.50E+06

Re

Nu

3D Gnielinski Ditus-Boelter Petukov New Mixed Convection

Fig. 13 - Comparison of Forced Convective Correlations with 3D Prediction a) with radius along collector and b)

Non-dimensionally with Reynolds and Nusselt Numbers

All of the above analysis suggests that the Gnielinski correlation, while the best of the forced

heat transfer correlations, also underestimates heat transfer in mixed convective regions. The result

would be a serious underestimation of the temperature rise within the collector, and power output of

the entire plant. The next section will aim to address this issue by developing a mixed convective heat

transfer coefficient for this flow.

Heat transfer at the collector cover was also evaluated to determine to what extent the

unstable thermal layer at the ground influenced its value. (Osborne, Incropera 1985b) found that for

turbulent flow between two heated plates, heat transfer at the upper surface was entirely due to forced

convection and flows at the two surfaces do not influence each other. To investigate the applicability

of this statement, another 3D simulation was conducted, setting the temperature at the upper surface to

310K, an approximation based on data from (Haaf 1984).

Fig. 32 in Appendix E shows that heat transfer at the lower surface was completely unaffected

by this change. A comparison of heat transfer at the upper surface with the Gneilinski equation (Fig.

33, Appendix E) shows that the upper surface is mostly forced convection as expected. This being

said, the Gnielinski equation still underestimates heat transfer, indicating that secondary flows still

enhance the heat transfer slightly along the upper surface. Contours of heat flux along this surface

(Fig. 34) confirm this fact, showing the telltale streaks of enhanced heat transfer by secondary flows.

It is likely that the upper surface is more influenced by the lower surface in this case when compared

Aerothermal Performance of a Solar Updraft Tower Owen Williams

29

with the experiments of (Osborne, Incropera 1985b) because the surfaces are not at the same

temperatures. As a result, the thermal layer next to the cover is not as strong as the one below it and is

less likely to resist temperature and velocity gradients. This being said, it has been shown that

approximating the heat transfer at the upper surface as purely forced convective is still quite accurate

and as such, this is how it will continue to be treated.

5.2.3 Development of New Heat Transfer Correlations

Using the above analysis, it was determined that it would be possible to develop a new

method to incorporate naturally convective heat transfer within the one-dimensional simulation.

The previous method assumed that the heat transfer from the upper and lower surfaces was

purely due to forced convection and was calculated through the Gnielinski correlation. Natural

convection was included by using a natural convection correlation for heat transfer between plates in a

horizontal duct. There are some significant flaws with this method because the flow never gets

directly heated as a result of natural convection. Instead, some heat is transferred from the ground to

the cover. It is very likely that the flow does not pick up any additional heat as a result of this process

and a new, accurate method based on the results of the three-dimensional analysis would be beneficial.

As it was shown that the heat transfer from the upper surface is dominated by forced

convection, the heat transfer will continue to be calculated using the Gnielinski equation. The natural

convection between the upper and lower surfaces will be removed and replaced with a mixed

convective heat transfer coefficient between the ground and the flow.

This mixed heat transfer coefficient will be determined using a combination of forced and free

correlations combined using a power relation of the form shown in Eq. 39. The forced convection

component will once again be calculated from the Gnielinski equation while the natural convection

component will be found using a new correlation with Ra number determined from the three-

dimensional simulation.

n

N

n

F

nNuNuNu +=

By plotting Nusselt number against Reynolds (Fig. 13b) and Rayleigh (Fig. 36, Appendix E)

numbers, it was possible to determine the form of this natural convective component of the equation.

The final natural convection equation is given by Eq.39. It is usually stated that exponent in Eq.39 is 3

or 3.5 depending on the situation. In this case, the best fit with the three-dimensional model resulted

from an exponent of 5.1

=

n.

3

1

06.0 RaNu N=

As can be seen from Fig. 12 and Fig. 37 the new mixed convective correlation for the ground

is far superior to using solely the forced relation. Fig. 37 in Appendix E, shows how natural

convection augments the forced convection as the flow progresses through the collector. As can be

seen, the new correlation is quite accurate and it should lead to a greater temperature difference within

the collector and increased power output. This should bring results more in line with the pilot plant.

Subsequent sections will investigate the effect of this correlation on plant output, temperature

difference and ground temperature.

5.2 Flow Variation through Plant

5.1.1 Change in Flow Characteristics with Solar Input

The variation of velocity, pressure, temperature and density through the entire plant was

generated by time marching to a steady state for different levels of incident solar radiation. All tests

were conducted assuming inlet and exit pressures of 101325Pa and 99009Pa respectively. Ambient

temperature was 298K. The basic heat transfer correlations involved Gnielinski’s equation and was

not augmented using the new set of correlations. Results of this simulation are shown in Fig. 14. It

must be noted that the objective of these simulations was to show trends in the state variables

throughout the domain and not accuracy of total plant output.

Eq.

39

Eq.

40

Aerothermal Performance of a Solar Updraft Tower Owen Williams

30

a)

0

2

4

6

8

10

12

14

0 50 100 150 200 250 300 350

x (m)

Velocity (m/s)

I = 900 W/m2 700 W/m2 500 W/m2 300 W/m2 100 W/m2

b)

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

0 50 100 150 200 250 300 350

x (m)

Relative Pressure (Pa)

I = 900 W/m2 700 W/m2 500 W /m2 300 W/m2 100 W/m2

c)

296

298

300

302

304

306

308

310

312

0 50 100 150 200 250 300 350

x (m)

Temperature (K)

I = 900 W/m2 700 W/m2 500 W/m 2 300 W/m2 100 W/m2

Aerothermal Performance of a Solar Updraft Tower Owen Williams

31

d)

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

0 50 100 150 200 250 300 350

x (m)

Density (kg/m3)

I = 900 W/m2 7 00 W/m2 500 W/m 2 300 W/m2 100 W /m2

Fig. 14 – Variation Fluid Properties through the Plant. a)Velocity b) Pressure Relative to Inlet Pressure and

Hydrostatic Component, '

P

c) Temperature d) Density

Upwind velocity reduces with solar radiation as would be expected, hence reducing output. In

this case, the upwind velocity will not reduce to zero at night when solar radiation also becomes zero.

With the current boundary conditions, an inlet temperature of 298K will be effectively pre-heating the

flow before it enters the collector. Additionally, it is the pressure drop in the chimney that has the

largest influence on upwind velocity, not the temperature difference in the collector. This sensitivity

to boundary conditions will be discussed further in following sections.

Other expected flow attributes are clearly visible. The pressure work term causes the

reduction in temperature at a linear rate throughout the chimney. Also, the velocity in the collector

varies as r1while flow in the chimney is accelerating slightly. This acceleration is associated with

changing density and was predicted by (Von Backstrom, Theodor W. 2003) and causes a loss that can

be almost completely eliminated by flaring the chimney by about 13%. Note also, that the pressure

reduces almost linearly in the chimney.

280

300

320

340

360

380

0 20 40 60 80 100 120

x (m)

Temperature (K)

Gneilinski Formula Ditus-Boelter New Heat Transfer Correlations

Ground

Cover

Flow

Fig. 15 - Comparison of Ground and Cover Temperature Distributions with Different Heat Transfer Coefficients

Aerothermal Performance of a Solar Updraft Tower Owen Williams

32

5.1.2 Ground and Cover Temperatures

The surface temperature of the ground and cover were both determined for midday conditions

and compared with the flow temperature. A comparison was made between the three sets of heat

transfer coefficients; Ditus-Boelter, Gneilinski and the new mixed convection relation. The resulting

temperature distributions are shown below in Fig. 15.

Note that the temperature of the ground and cover reduces rapidly near to the chimney as the

flow velocity is much greater, extracting much more heat from the ground. The inefficiency of the

pilot plant can readily be seen as the flow becomes hotter than the cover near the chimney, increasing

losses to the environment. It is for this reason that many designs would favour the use of double

covers or thin films near the chimney to retain extra heat. Unfortunately, these have been found to be

uneconomical in most cases.

It is also clear that with the use of the new heat transfer correlations, the flow is gaining much

more additional heat, especially near the entrance region. This is encouraging because it means that it

is now feeling the effects of the natural convection terms.

5.1.3 Effect of Losses and Turbine Pressure Drop

It would be beneficial to determine the effect of losses and work extraction by the turbine on

the flow. As a result, a set of simulations were conducted where the effect of each of these losses was

included in succession. Midday conditions with a solar radiation of 900W/m2 were assumed. The

variations of state variables through the domain are plotted in Appendix E. It must be noted that for

the results below, the power stated is the power produced assuming there is no pressure drop at the

turbine until the final simulation in which it is included. Also note that the Ditus-Boelter heat transfer

correlation was used to generate these results as the Gneilinkski relation would alter the heat transfer

to the flow with different friction factors. As such, it would be impossible to separate this effect from

the increased pressure drop due to friction in which we are interested.

Table 1 - Effect of Losses on Flow Properties

∆T Difference Upwind Velocity Difference Power Difference

K % m/s % W %

No Turbine or Losses 8.5387 - 10.9652 - 31701.75 -

Transition Loss 8.7463 2.43 10.0961 -7.93 29845.31 -5.86

Doubling Friction 8.7701 0.27 9.9926 -1.03 29617.86 -0.76

Turbine 9.1443 4.27 8.6152 -13.78 26470.39 -10.63

Total Change 6.97 -22.73 -17.24

The pressure drop at the turbine was found to have the greatest effect on the power output,

upwind velocity and temperature difference in the collector. It was also found that that the effect of

friction was small compared with the losses associated with the transition section between the

collector and the chimney. This was also found by (Von Backstrom, Theodor W., Bernhardt et al.

2003) and (Kirstein, Von Backstrom, Theodor W. et al. 2005) and it also suggests that every effort

should be taken to minimize this loss.

5.3 1D Simulation of Pilot Plant

5.3.1 Daily Power Generation

The entire one-dimensional simulation was validated using the data from a sample day at the

Manzanares pilot plant (Weinrebe 1987). Solar radiation variation was based on this data to make this

simulation as accurate possible. This function can be found in Appendix F. The average inlet pressure

from the data was 92962.5 Pa while the exit pressure was estimated as 90850Pa.

Aerothermal Performance of a Solar Updraft Tower Owen Williams

33

a)

0

10

20

30

40

50

60

0 4 8 12 16 20 24

Time (hrs)

Power Output (kW)

Pilot Plant 15/06/87 Ditus-Boelter Gnielinski Mixed Convective Correlation

b)

0

1

2

3

4

5

6

7

8

9

0 4 8 12 16 20 24

Time (hrs)

Upwind Velocity (m/s)

Pilot Plant 15/06/87 Ditus-Boelter Gnielinski Mixed Convective Correlation

c)

0

5

10

15

20

25

0 4 8 12 16 20 24

Time (hrs)

Temperature Difference In Collector (K)

Pilot Plant 15/06/87 Ditus-Boelter Gnielinski Mixed Convective Correlation

Fig. 16 – Daily Variation of Pilot Plant Parameters Comparing Accuracy of Different Heat Transfer

Correlations. a) Power b) Upwind Velocity c) Temperature Difference in Collector

Aerothermal Performance of a Solar Updraft Tower Owen Williams

34

Ambient temperature was set to the daily average at the pilot plant in Manzanares, 290.15K,

and remained constant for these simulations. Check Appendix F for further assumed properties of the

ground and plant.

A comparison is made between results based on the three different sets of heat transfer

correlations. Fig. 16, above shows the daily variation of power output, temperature rise in collector

and upwind velocity.

As can be seen, the new heat transfer correlations give the most accurate results, with an error

of 0.6% in maximum power output. It was expected that this set of relations would be better as the

two previous sets of relations do not transfer sufficient heat to the flow. This effect is also clearly

visible from the temperature difference within the collector. The Gnielinski equation was also more

accurate than the Ditus-Boelter relation, as expected.

It can be seen that the simulation overestimates power output in the later half of the day and

nightime operation, although the solution during the morning heating period is quite accurate. It is

also beneficial to note that through the incorporation of the mixed heat transfer correlations, the

temperature change within the collector becomes almost equal to the experimental results, at 20.7K.

The upwind velocity was most inaccurate, but it will be shown that this parameter is the most

sensitive to boundary conditions, which are only roughly approximated to the pilot plant with the

above set of solutions.

As a result of the previous simulations, it can be said that the use of the new mixed heat

transfer correlations is validated. This being said, there are regions where the simulation overestimates

power output and the reasons for this, including treatment of boundary conditions will be investigated

further in the following sections.

5.3.2 Ground Simulation

(Haaf 1984) gave the variation of ground temperature at different depths over a sample day

(see Appendix B). Maximum temperatures were clearly visible as was time lag of the heat wave

penetrating the ground. To continue validating this one-dimensional simulation of the Solar Updraft

tower and to demonstrate the benefits of modelling the ground by solving the heat equations, similar

distributions of temperature at different depths were generated. These results were taken from the

simulation using the new heat transfer correlations shown above. Fig. 17 shows the temperature

distributions at a point 50m from the entrance to the collector. Further results for the ground at the

entrance and exit are available in Appendix E.

290

300

310

320

330

340

350

360

370

380

0 4 8 12 16 20 24

Time (hr)

Ground Temperature (K)

0 cm

2 cm

4 cm

8 cm

12 cm

16cm

26 cm

42cm

Fig. 17 - Variation of Ground Temperature through the day at Different Depths

Note that the maximum surface temperature reached is higher in this simulation than for the

data at Manzanares. It is not stated however, at which point in the collector the data was measured,

Aerothermal Performance of a Solar Updraft Tower Owen Williams

35

and its maximum is almost equal to that found in the simulation at the collector exit. Also, the surface

conductivity of the ground within the simulation may be lower than that at the position of the

measurements as (Haaf 1984) found significant spread in measured conductivity. It must also be noted

that both simulation and experiment show an almost constant temperature at 50cm depth and is one of

the major indicators of simulation accuracy.

It can be seen that the ground exhibits the same characteristics as that shown at Manzanares

and thus, its use is justified to increase the accuracy of the solution baring errors in estimated ground

properties.

5.3.3 Sensitivity to Boundary Conditions

Taking the newly devised set of heat transfer correlations as the most accurate, a set of

simulations were conducted to estimate the effect of boundary conditions on the resulting power

output. Three different conditions are compared with the previous simulation that was conducted

assuming constant pressure and temperature boundary conditions. The nature of the boundary

conditions of these subsequent calculations are listed below. They were conducted using combinations

of variable pressure and temperature boundary conditions. The effect of decreasing outlet pressure

was also investigated. Daily variations in power output and flow properties are shown below. All

information related to time varying boundaries can be found in Appendix F.

Boundary types:

(1) Constant ambient temperature and pressure boundary conditions as in previous simulations

(2) Constant inlet and outlet pressures with variable ambient temperature according to function

in Appendix F. KTave 15.290=, KTrange 8.9=

(3) Constant inlet pressure, with variable ambient temperature and outlet pressures. Outlet

pressure given by Eq. 89. KTave 15.290=, KTrange 8.9=

(4) Constant ambient temperature and pressure boundary conditions. Pressure difference

increased by 50 Pa

What is interesting to note is the effect of the first three sets of boundary conditions on the

resulting power output. All three sets of boundary conditions are approximations of the actual

conditions at the pilot plant during the sample day, but they produce vastly different estimates of

power output, upwind velocity and temperature difference. It is impossible to say which of these

solutions best approximates the actual conditions as the variation in exit pressure is unknown and is

very important in determining the final solution.

Each of these conditions are accompanied by their own set of assumptions. For instance, the

formula for variable exit pressure implies that any change in the inlet conditions is immediately felt at

the exit. This is definitely not true, as discussed by (Lautenschlager, Haaf et al. 1985). This being the

case, each of these simulations can provide interesting insight into the effect different boundary

conditions have on the output of a plant.

The reduced night temperature as a result of the variation of ambient conditions creates the

greater temperature difference between the incoming flow and the ground, hence the greater change in

temperature within the collector during night-time operation. The increased ambient temperature

during the day also results in a lower temperature difference within the collector but for opposite

reasons.

Larger temperature differences within the collector result in a much lower density in the

chimney. As a result, there is a much greater total pressure difference and the current turbine model

can extract much more power from the flow. This explains why the second simulation produces more

power at night but also underestimates peak output. The reduced turbine pressure drop during daytime

operation explains the larger upwind velocity through the chimney. The opposite is also true of night-

time operation.

It can be seen that the third simulation, with time varying pressure and temperature boundary

conditions, produces results very similar to the original with constant boundaries. The change in

Aerothermal Performance of a Solar Updraft Tower Owen Williams

36

a)

0

10

20

30

40

50

60

0 4 8 12 16 20 24

Time (hrs)

Power Output (kW)

Pilot Plant 15/06/87 Boundary 1 Boundary 2 Boundary 3 Boundary 4

b)

0

1

2

3

4

5

6

7

8

9

10

0 4 8 12 16 20 24

Time (hrs)

Upwind Velocity (m/s)

Pilot Plant 15/06/87 Boundary 1 Boundary 2 Boundary 3 Boundary 4

c)

0

1

2

3

4

5

6

7

8

9

10

0 4 8 12 16 20 24

Time (hrs)

Upwind Velocity (m/s)

Pilot Plant 15/06/87 Boundary 1 Boundary 2 Boundary 3 Boundary 4

Fig. 18 - Sensitivity Analysis of Boundary Conditions for Overall Simulation

Aerothermal Performance of a Solar Updraft Tower Owen Williams

37

temperature within the collector is once again greater at night and lesser during the day than the first

simulation, but not to the extent as was visible when only the temperature varied. It appears that the

increase in exit pressure cancels out much of the effect of variable temperature at peak operation, as

can be seen from Fig. 40. The upwind velocity at peak output is thus less than that for the constant

boundary condition case. As a result, it can be said that upwind velocity has a greater dependency on

pressure difference than ambient temperature.

The final simulation reduced the outlet pressure, which is equivalent to increasing the height

of the chimney. It is included as an example of the sensitivity of the solution to pressure difference in

isolation

6.0 DISCUSSION

6.1 3D Simulation

The three-dimensional analysis met all of its objectives. Four flow regimes were identified

and reasons given for their formation based on previous experimental results. Most importantly, the

investigation into heat transfer at the upper and lower surfaces led to the formation of a new method

for estimating the heat transfer coefficients.

The assumption of forced heat transfer at the upper surface was validated, proving that it was

insufficient to estimate the entire heat flux, leading to the lower temperature change in the collector,

shown in Fig. 16. The enhancement of this heat transfer with a natural convective correlation with

Rayleigh number seemed the obvious solution. This was shown to improve the output of the

simulation drastically, as can be seen in Fig. 16, not only in terms of its accuracy but also in stability.

This being said, there are issues and errors in the three-dimensional model that must be

identified. From Fig. 37, it is clear that the new set of correlations are most inaccurate within the

entrance as would be expected since they were adapted for fully developed flows only. This source of

error will be less significant when incorporating these equations into the one-dimensional simulation

for the following reason: as the three-dimensional domain only includes the collector, entrance effects

will be much more significant than for an actual plant. The flow adjacent to the ground of a real

collector will be heated before entering, causing the onset of instability to be much closer to the

entrance, possibly removing this region altogether. As a result, it is unlikely that incorporating a

different set of heat transfer relations for the entrance will make the solution significantly more

accurate.

Errors are also possible due to modelling the upper and lower surfaces of the collector at a

constant temperature. This is not realistic and, as shown in Fig. 15, the ground and cover temperature

reduce near the collector as the flow velocity increases. This will affect the Rayleigh number the most

and will have greatest effect on the natural convection. Fortunately, the temperature is almost constant

within the natural convective region, and so it is estimated that the error due to this assumption will be

small.

Further investigations into the heat transfer in the collector should look at extremely low

velocities, like those seen at night, to see if the newly developed correlations are still valid. It would

also be interesting to look at the flow through the collector of a full size plant. The collector would

have a smaller height to radius ratio than the pilot plant and slightly higher velocities. It would be

interesting to identify the effect this large scale would have on the heat transfer correlations. It must

be noted that their validity has been confirmed only for a relatively small range of Reynolds and

Rayleigh numbers, and while this corresponds to the maximum output conditions of the pilot plant,

further investigations are necessary to confirm their validity at different conditions.

6.2 1D Simulation of Pilot Plant

It was found that the flow simulation was much less stable than that for the ground. When run

individually, both the ground and the flow were much more stable that run in conjunction, with the

flow being able to withstand timesteps of greater than 10s. The ground simulation was even more

stable because its solution changes so much more slowly.

Aerothermal Performance of a Solar Updraft Tower Owen Williams

38

When run in conjunction, the change in source terms or boundary conditions led to the

introduction of perturbations in the flow solution that limited its stability. The parameters that had a

large influence on the flow solution were the heat flux from the ground/cover, friction turbine

pressure drop, and boundary conditions. Variation of any of these parameters by too significant a

margin during a single timestep resulted in divergence. As a result, to consistently maintain stability a

timestep of 4s was chosen, including heavy relaxation of approximately 0.3 for temperature, velocity

and pressure.

It was also found that stability was greatly enhanced in accordance with the physical

relevancy of boundary conditions and input parameters, as would be expected. Also, if a relatively

low inlet pressure was chosen with a low inlet temperature, divergence would more than likely result

as this situation rarely happens in practice. Great care had to be taken in the selection of initial

conditions to make sure that they were relevant to the time of day at simulation start-up so as to limit

transients. These transients sometimes resulted in oscillations (stable, damped or divergent) as a result

of poorly chosen initial conditions. An example of a damped oscillation is visible in Fig. 18 for the

solution involving time varying ambient temperature.

The greatest difficulty in maintaining convergence occurred during night-time operation at

low flow velocities when there was no solar input. As a result, the correlations based upon forced

convection heat transfer would not transfer sufficient heat to the flow to maintain the density

difference between the inlet and exit to the chimney. This situation was particularly acute when using

time varying boundary conditions as the density outside the chimney would change significantly,

altering the buoyancy of the flow. This particular stability problem was largely solved by the

introduction of the mixed convection heat transfer correlations as they would allow the flow to gain

additional heat even at low velocity. The natural convection component was only dependant on

Rayleigh number which is determined by the temperature difference between the flow and the ground,

causing heat transfer to continue through the night.

As a result, the final one-dimensional simulation was found to be quite stable and robust with

carefully chosen initial conditions. Even with heavy relaxation and a smaller timestep, it was found

that approximately two hours of simulated time could be modelled in one hour of computational time

on a standard desktop computer. Memory usage was such that two simulations could also be run

simultaneously. This, combined with the ability of the program to easily be adapted to simulate any

solar chimney design, means that this simulation meets its initial objective as a responsive and

adaptable program that can be used to quickly evaluate any design change.

Assessment of the second objective of the one-dimensional simulation, demonstrating the

accuracy of the solution and validating it against pilot plant data, must involve a discussion of the

validity of assumed boundary conditions. Knowledge of their deficiencies will allow an assessment of

the overall accuracy of the solution.

Initial validation of the pilot plant using solely forced convective heat transfer from the upper

and lower surfaces was shown to be quite inaccurate, under-predicting maximum power output by as

much as 53% for the Ditus-Boelter relation and 48% for the Gnielinkski relation. Output was

improved by incorporating the new mixed convective heat transfer relations into the simulation,

achieving a maximum power production of 47.6kW, less than 0.6% from maximum pilot plant output.

It was found that the simulation was most accurate during the intense heating period of the morning

and least accurate during the afternoon and overnight periods where additional power was generated.

It must also be noted that peak output also occurred later in the simulation.

The increased overnight output is due to the higher heat transfer from the ground into the flow

when compared to the pilot plant. As a result, the simulated upwind velocity remains above 2.5m/s,

below which the real turbine was unable to generate power (Haaf 1984).

Both the increased flow velocity and temperature difference can be traced to errors in

approximating the inlet and outlet pressure difference at different times of day. By examining Fig.

41Fig. 1 which shows variation of inlet outlet temperature throughout the day, it is possible to

estimate the variation of inlet to outlet pressure difference as it is proportional to the difference

between the two temperatures. During night-time operation, the difference is very small, indicating

that the pressure difference has also decreased. The difference increases drastically during the day as

the ground heats the air immediately adjacent to it. This makes the lapse rate over-adiabatic, causing

Aerothermal Performance of a Solar Updraft Tower Owen Williams

39

increased pressure difference between inlet and outlet, an effect noted by (Lautenschlager, Haaf et al.

1985).

It was shown in Section 5.3.3 that pressure difference has the greatest impact on upwind

velocity. As the pressure difference of the pilot plant reduced at night, so did the upwind velocity and

power output compared to the simulation.

This effect also contributes to the overestimation of power output during the afternoon.

During this period, the thermal layer next to the ground also becomes larger, increasing outlet

pressure and decreasing the pressure difference. As a result, the output of the pilot plant falls off

drastically compared to the simulation. This can be seen from the variation of upwind velocity, Fig.

16b. Further errors could result from air humidity, an effect not modelled or clouds that appeared over

the pilot plant in the afternoon. During this period the solar radiation was particularly erratic,

supporting this theory and reducing output further.

None of the boundary conditions in the sensitivity analysis approximated the actual

atmospheric conditions of the pilot plant exactly. They were approximations solely based on different

assumptions. Each change made the solution more or less accurate to varying degrees as can be seen

in Fig. 18. Variation of ambient temperature, which should, in theory, increase the accuracy of the

solution, resulted in a sizable decrease in accuracy, especially for upwind velocity and temperature

change distributions. This is mainly due to the inlet and outlet pressures being a function of this

temperature change and without including this dependency, the change in ambient temperature just

artificially cools or heats the flow within the collector, exacerbating differences between night and

day.

Including the dependency of outlet pressure on ambient temperature was a poor

approximation because its effect on outlet pressure is not immediately felt, but involves a time lag as

the thermal layer near the ground becomes thicker throughout the day (see Fig. 41).

As a result, it has been shown that there is no way to fully model the pilot plant using this

simulation procedure without accurate, independent models for the variation of inlet and exit

temperature and pressure. While the simulation behaves as expected for the given inputs, boundary

condition sensitivity is such that large variations in output can be expected if they are improperly

modelled.

A further implication of this realisation is that it is possible to set the inlet and outlet pressures

to obtain a required maximum output. The first set of simulations are still valid as the outlet pressure

was set to a value equal to the standard atmospheric value at the height of the chimney, but it would

have been possible to decrease the outlet pressure slightly to increase the maximum output exactly in

line with experimental results.

Boundary condition sensitivity is likely to be reduced when modelling the full scale plant as

the difference between the inlet and outlet pressures will be greater. Therefore any effect of a thermal

layer near the ground during the day will also be reduced. Additionally, it can be said that the constant

outlet pressure assumption will be increasingly valid as the outlet pressure will be more determined by

large scale atmospheric effects than local heating cycles. However, incorporation of time varying inlet

and outlet pressures would still be beneficial.

Due to this sensitivity to boundary conditions, it is not possible to completely assess the

merits of the full ground simulation and whether it increases the accuracy of the solution. This being

said, it has been shown that the variation of temperature with depth compares favourably with results

from (Haaf 1984) as shown in Appendix B. The time lag of heat as it penetrates the ground is quite

visible and surface temperatures vary as expected. Also, the increased output at night makes it clear

that it acts as an effective storage medium, releasing additional heat when required.

Summing up the above analysis, the simulation was shown to be accurate when compared to

the pilot plant. The solution behaved as expected, considering differences in boundary conditions.

Midday upwind velocity, power output and temperature difference were also found to be nearly

identical to the pilot plant. Additionally, as just mentioned previously, the ground simulation was

shown to work effectively as a heat storage medium. It was found that only the solution with modified

heat transfer coefficients was valid, since the other methods did not pick up sufficient heat from the

ground to approach the correct output. The simulation was also shown to be responsive to different

input conditions, with calculation times fast enough for it to be used as an effective tool in the

modelling of design changes of a future full scale plant.

Aerothermal Performance of a Solar Updraft Tower Owen Williams

40

A number of advantages for using this approach to modelling a solar updraft tower became

apparent. Incorporation of different loss coefficients or friction is quite simple through the source

terms within the momentum equation. Additionally, the solution is fully compressible, calculating the

correct adiabatic lapse rate based on boundary conditions and not assuming a value of 9.8˚/km. With

the incorporation of full compressibility, the simulation of a full scale plant should be more accurate,

using fewer assumptions and providing greater confidence in the result.

The analysis of boundary condition sensitivity indicates an ability for the simulation to

analyse the effect of the local climate of any potential solar updraft tower site on power output, an

outcome that was not expected at the commencement of this project. Given atmospheric models of

sufficient accuracy, it is likely that the effect of local weather conditions could also be estimated using

this simulation.

Separate from the inclusion of these time varying pressure boundary conditions, the program

should be further improved to increase stability and decrease computational time. The stability can be

increased by including the diffusion terms in the momentum equation, likely smoothing perturbations

and allowing for decreased relaxation. Additionally, incorporation of a newer, more efficient

algorithm for the flow equations, such as SIMPLER, would allow for faster reduction of the pressure

residual by including further terms in the pressure-correction equation. This would result in fewer

iterations and shorter computational times.

With further work expanding the validity of the heat transfer correlations, the simulation is

now ready to be applied to a full scale plant, where sensitivity to boundary conditions will likely be

reduced, giving an accurate estimate of overall plant output with a high degree of reliability.

7.0 CONCLUSIONS

Four distinct regions of flow were identified through a three-dimensional CFD investigation

of heat transfer within the solar collector. They were the entrance region, instability onset, vortex

breakdown, and acceleration regions. The extent that each of these regions was influenced by forced,

free or mixed convection was then determined.

The analysis concluded that natural convection plays an important role over the majority of

the collector, with forced convection only dominating at the exit. As a result, forced convection

correlations underestimate heat transfer in the mixed convection region by approximately 2.5 times.

To correct this deficiency, a new set of mixed heat transfer correlations were developed based on the

results of the three dimensional simulation and validated by use within the one-dimensional

simulation.

A basic one-dimensional simulation using these newly developed correlations was found to be

accurate to 0.6% of maximum daily output when compared to sample data from the pilot plant at

Manzanares. Purely forced convective heat transfer coefficients were found to be entirely insufficient.

Discrepancies between simulated and experimental results were analysed and found to be due

to differences in ambient temperature and pressure throughout the day. The simulation overestimated

afternoon power output because the inlet to outlet pressure ratio reduced sharply for the pilot plant as

the effect of ground heating reached the height of the chimney, while for the simulation it remained

constant. These errors were enhanced by the incorporation of a time varying ambient temperature,

thus contributing to the conclusion that the solution is very sensitive to boundary conditions.

It was determined that this sensitivity caused the majority of the error, thus validating the

remainder of the solution that gave correct distributions of velocity, temperature, pressure and density

within the plant. Time variation of ground temperature was also found to agree with data from

Manzanares.

All of the objectives laid out for this project were met. The one-dimensional simulation was

found to be accurate and responsive, with relatively short computation times, making it ideal for

evaluating the effect of design changes on power output. The investigation of heat transfer within the

collector was not only insightful but also resulted in the development of a new set of mixed heat

transfer correlations that were essential to the overall success of the complete simulation. Most

importantly, the strong merits of this approach to the modelling of a solar updraft tower were

confirmed with the accuracy of the flow and ground simulations, showing that it could be reliably

applied to the simulation of a full scale plant with increased confidence in the results.

Aerothermal Performance of a Solar Updraft Tower Owen Williams

41

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Aerothermal Performance of a Solar Updraft Tower Owen Williams

i

APPENDIX A – ADDITIONAL INFORMATION FROM

LITERATURE REVIEW

Heat Transfer

Heat transfer can occur in any of three ways; conduction, convection or radiation. This

section will detail background information on these phenomena including important non-dimensional

parameters and correlations. The individual difficulties about heat transfer in a solar collector will be

examined including results from related studies.

Conduction

Conduction is the energy exchange that takes place between a region of high temperature to a

region of low temperature by the kinetic motion of molecules or the free flow of electrons. In the case

of a fluid or gas, this occurs only when there is no bulk motion of the medium. Conduction is

governed by Fourier’s Law, which states that the heat flux in a given direction is proportional to the

temperature gradient in that direction (Ozisik 1985). The constant of proportionality, k, is the

conductivity in W/(m ˚C).

dx

dT

kq =

Conduction into a solid plate is governed by the heat equation which is closely related to

Fourier’s Law. It will be discussed further in a subsequent section.

Radiation

All bodies emit energy in the form of electromagnetic radiation due to their temperature. It is

generally assumed that the radiation is emitted and absorbed from the surface of a solid object. This is

different for a semi-transparent material where a portion will be absorbed, a portion reflected and a

portion transmitted. In the case of solar radiation, it is emitted over a broad range of the

electromagnetic spectrum but there are peaks at specific wavelengths. This effect on the radiative

energy transfer between bodies is generally ignored (Duffie, Beckman 1974).

The maximum radiation flux emitted by a body at temperature, T is given by the Stefan-

Boltzmann Law where

σ

is the Stefan-Boltzmann constant (Ozisik 1985).

4

TEb

σ

=

Only an ideal radiator or blackbody can emit this emissive power. The same is true for

absorption, where only a blackbody can absorb 100% of incident radiation. As a result, the real

emissive power of a body is given by the blackbody emissive power multiplied by the emissivity,

ε

.

For absorption, the incident heat flux is multiplied by the absorptivity,

α

,to get the proportion of heat

absorbed. It is generally assumed that the absorbtivity is equal to the emissivity for practical reasons

(Ozisik 1985).

4

TEq b

εσε

==

incabs qq

α

=

Convective and radiative processes are generally characterised by a heat transfer coefficient,

giving a heat flux in the form shown in Eq. 44. Even though the emissive power is proportional to the

temperature to the fourth power it is still possible to determine a heat transfer coefficient.

(

)

12 TThq r−=

Convection

Convection takes place when a fluid flows over a solid surface at a different temperature.

Heat transfer occurs at a faster rate than pure conduction because the

bulk motion of the fluid over the surface carries away additional heat. The motion of the fluid can be

artificially induced in the case of a fan or tunnel, which is called forced convection or driven by the

buoyancy of the flow, which is called free or natural convection.

Eq.

42

Eq.

43

Eq.

41

Eq.

44

Aerothermal Performance of a Solar Updraft Tower Owen Williams

ii

Convective heat transfer is generally found using Newton’s Law of Cooling given in Eq. 45.

The order of the temperature terms is solely dependant on the desired direction of heat flow.

(

)

fwc TThq −=

The heat transfer coefficient, c

h, is a complicated function of the type of flow, geometry of

the body, physical properties of the fluid and the average temperatures. Most importantly, it is highly

dependant on whether the flow is free or forced convection. Heat transfer coefficients are usually

given in the form of a correlation in terms of non-dimensional parameters.

Just as a velocity boundary layer develops when a fluid flows over a surface, a thermal

boundary layer will also develop if the flow and wall temperatures are different. This makes the two

governing properties of a boundary layer the skin friction and heat transfer coefficients.

It is appropriate to apply Fourier’s Law at the wall to determine the heat transfer because the

non-slip condition ensures heat transfer only occurs by conduction. Combining Fourier’s Law with

Newton’s Law of Cooling results in the following relationship for the heat transfer coefficient (Ozisik

1985).

fw

y

f

TT

yTk

h−

∂∂−

==0

The difficulty lies in the fact that this equation is dimensional and thus there are many

possible solutions for the temperature gradient with no simple method of determining its value. Thus,

through dimensional analysis, a number of useful dimensionless groups can be identified to

characterise the flow. The first is the Reynolds number or ratio of inertial and viscous forces, which

also plays a role in determining the turbulence of the flow.

Another is the Prandtl number which is the ratio of momentum and thermal diffusivities. This

number is a constant property of a fluid. For air, it has a value of 0.713(Incropera, DeWitt 2002).

α

ν

νρ

=≡

k

CP

Pr

The most important non-dimensional number in heat transfer is the Nusselt number. It is a

non-dimensional measure of the heat transfer coefficient and is useful because, as a non-dimensional

number it can be used much more readily in heat transfer correlations.

f

k

hL

Nu =

These non-dimensional numbers are sufficient to determine heat transfer correlations for

forced convective flow. This is generally done by plotting Nu against Re on log-log plots for variable

Prandtl number giving correlations of the form shown in Appendix D. There are many correlations of

this form for forced convective flow, some of which will be used in this project. Correlations will be

different based on laminar or turbulent flow and some can even take surface roughness into

account(Incropera, DeWitt 2002).

nm

LL CNu PrRe=

Naturally convective flows are more complicated because fluid movement is due to buoyancy

forces caused by density gradients. Heat transfer rates are smaller due to the lower velocities involved

and the stability of thermal layers becomes a factor. Considering a heated flat plate, the fluid

immediately adjacent to it will reduce in density as its temperature increases. As a result, it would

wish to rise into higher density fluid due to a buoyancy force. For the fluid above the plate, the fluid

will rise in plumes causing natural convection. Below the plate, it is obstructed by the plate itself. As

a result, it forms a stable layer and naturally convective flow does not occur. The opposite occurs if

the plate is cooling(Ozisik 1985).

Slightly different similarity parameters are found to be important for naturally convective

flows. One such parameter is the Grashof number, which is the ratio of buoyant to viscous forces

given by Eq. 51.

β

is the volumetric thermal expansion of the fluid and for a perfect gas is equal to

1/T. The Rayleigh number is also a measure of buoyant to viscous forces and is usually used to

correlate the transition to turbulence in a free convection boundary layer(Incropera, DeWitt 2002).

Eq.

45

Eq.

46

Eq.

47

Eq.

48

Eq.

49

Aerothermal Performance of a Solar Updraft Tower Owen Williams

iii

(

)

2

3

ν

β

LTTg

Gr s∞

−

=

Pr

GrRa

=

It has been found that a heated layer can remain stable with no cells of rising fluid at low Ra

numbers less than 1708. At these values, the viscous forces are sufficient to dampen out buoyant

perturbations and the thermal layer remains stable. Thus, with these conditions, natural convection

does not occur and heat transfer is by pure conduction through the fluid medium.

Using these non-dimensional numbers, naturally convective heat transfer correlations are

generally correlated using functions of the following form, using the Rayleigh number instead of the

Reynolds number.

n

L

CRaNu =

It is useful to determine the relative strengths of natural and forced convection. The ratio

2

Re/ LL

Gr can be used for this purpose. If the ratio is large, ie 1Re/ 2>>

LL

Gr then buoyant effects

are dominant and forced convection can be neglected. For 1Re/ 2<<

LL

Gr , the opposite is true and

free convection is negligible but it if the ratio is close to unity then neither can be neglected and the

results is a mixed convection flow. This is the type of flow that dominates within a solar collector

(Incropera, DeWitt 2002).

When characterising mixed convective flows, it is difficult to determine a combined heat

transfer coefficient unless the correlation was developed specifically for the geometry and situation to

be analysed. A common approximation, used when this situation arises is given by Eq. 53. Whether

the Nusselt numbers are added or subtracted depends on whether the natural convection is assisting or

opposing the main convective flow. The best correlation is commonly obtained with an exponent of 3

but for flow between flat plates, 3.5 may be better suited (Incropera, DeWitt 2002).

n

N

n

F

nNuNuNu ±=

Computational Methods

Method of Residuals and Solving Sets of Equations

This project involves the solution of non-linear equations using a semi-implicit method. As

mentioned previously, this involves solving for successively more accurate approximations to the

correct solution.

This being said, it is clear that it would be numerically beneficial to solve for a correction to

the previous guess instead of a completely new solution. It is both more efficient and allows for a

more accurate solution. If the numerical of the solution is large, and the difference from the correct

solution is small, solving for the small number that is the difference would allow faster convergence

and is numerically superior (Peiro 2007).

The method of residuals is one method used to solve for this correction. Consider a

differential equations (linear or non-linear) of the form given below, where L is a differential function

of variable u, the correct solution to the equation.

(

)

0=uL

On the other hand, if an approximate solution to the equation, u’ was substituted into the

same equation, the right hand side would not equal zero but would leave a residual where n is the

number of the current step. So,

(

)

n

ruL ='

Taking a Taylor expansion of the residual term with respect to the solution u’ gives:

(

)

21 uOu

u

r

rr nn ∆++∆

∂

+=

+L

We will assume that we wish to drive the residual to zero over with each successive step. Thus taking

rn+1 = 0 and neglecting higher order terms, (Peiro 2007)

Eq.

51

Eq.

52

Eq.

53

Eq.

50

Eq.

54

Aerothermal Performance of a Solar Updraft Tower Owen Williams

iv

n

ru

u

r−=∆













∂

Thus by evaluating the residual at the previous timestep, a set of linear equations can be

constructed. This set of equations can then be solved implicitly for the correction to the solution by

inverting the matrix formed by the derivative term on the left hand side.

Solution of a set of linear algebraic equations can be accomplished in a multitude of ways

such as Gaussian-elimination, depending on the complexity of the equations involved. Fortunately,

the method of residuals stated above leads to a set of simple equations that form a tridiagonal matrix,

or a matrix in which the non-zero coefficients lie along the three diagonals. This can be solved by a

useful process called the TriDiagonal Matrix Algorithm (TDMA). Its benefit is that for a domain of N

nodes it only uses computer storage and computational time proportional to N instead of N2 or N3 like

other matrix solving algorithm (Patankar 1980)

Over-relaxation and Under-relaxation

The solution of non-linear equations necessitates the use of iteration to apply a correction to a

previous guess at the correct answer. The method for finding this guess can vary but in this project it

is arrived at by solving an implicit set of equations by the residual method detailed in Section 2.3.2.

After a number of iterations, the solution converges to the correct answer.

It can become necessary to either speed up or slow down the change in the governing variable

by increasing or decreasing the value of the applied correction. These processes are called

overrelaxation and underrelaxation respectively.

Overrelaxation is useful when wishing to increase the convergence speed of a very stable

algorithm. It is often a useful addition to the Gauss-Sidel method, resulting the in what is known as a

Successive Over-Relaxation (SOR) scheme (Patankar 1980).

Underrelaxation is necessary for the solution of non-linear equations to avoid divergence and

hence will the only relaxation utilised in this project. The applied corrections are generally too large

and it can be analogous to a poor driver trying to drive a car down the centre of the road. As the car

starts to stray from the centre line, a small correction can bring it back into line while an

inexperienced driver would steer too far in the other direction and could lose control.

There are many methods for incorporating relaxation into a solution as can be found in

(Patankar 1980) but only the simplest form called proportional relaxation will be discussed here. A

correction, '

P

u, is multiplied to a constant of proportionality,

α

, before it is applied to an old guess,

O

P

u. The value of

α

can be varied to obtain the convergence characteristics required.

tionOverrelaxa

ationUnderrelax

uuu P

O

PP

α

≤

+= 1

10

'

TDMA Algorithm

Consider a domain of grid points numbered 1, 2, 3, … , N. The discretisation of a generic

equation over the domain using the above method could result in an equation of the form:

iiiiiii ducubua ++= −+ 11

where ui is related to the neighbouring values ui+1 and ui-1 creating the tridiagonal matrix when the N

equations are assembled.

At the boundaries the equations take on a trivial form c1 and bN both equal zero making sure

that uo and uN+1 do not play a role in the calculations. Thus the first equation is gives u1 as a function

of u2. This can then be substituted into the second equation giving u2 as a function of u3 and so on

until the final equation which gives a the value of uN as uN+1 does not play a role. In this way, the

values can be back-substituted into all the previous equations to give the full solution (Patankar 1980).

A recurrence relation can be developed from this process. Assuming we are looking for an

equation of the following form in the forward-substitution phase,

iiii QuPu += +1

having just found an equation of the form,

Eq.

55

Eq.

56

Eq.

57

Aerothermal Performance of a Solar Updraft Tower Owen Williams

v

111 −−− += iiii QuPu

Subbing into the original equation gives

(

)

iiiiiiiii dQuPcubua +++= −−+ 111

which can be rearranged to give recurrence relations for the coefficients Pi and Qi.

1

a

d

Q

Pca

Qcd

Q

a

b

P

Pca

b

P

iii

i

iii

i

=

−

+

=

−

=

−

Also noting that bN = 0 then PN = 0 and the final value uN can now be found. Thus the backward

substitution phase can begin (Patankar 1980).

Solution of Heat Equation

Conduction of heat into any solid medium is governed by the heat equation which is an

example of the general equation with the convective terms eliminated. In general, source terms can

also be included, though they will play no part in the ground simulation of this project. The heat

equation is a linear ODE which is in one-dimension is:

S

x

T

k

xt

T

CP+













∂

=

∂

ρ

The one-dimensional finite volume domain for this solution of this equation is shown in

Fig.23. Note that nodes need not be equidistant and that interfaces between cells are midway between

adjacent nodes.

Fig. 19 - Domain of One-Dimensional Solution of Heat Equation (Patankar 1980)

Using this domain, the discretised form of the above equation forms a set of equations of the

form shown in Eq. 60. As can be seen, this is the standard type of equations that can be solved using

the TDMA as was discussed in the previous section. In order to reach this form of the equation, the

heat equation must be integrated with respect to both distance and time. The complete computation

will be given in Section 3.3 but sections of the procedure will be given here to illustrate important

factors in the solution of this equation.

bTaTaTa wwEEPP ++=

There are three main complications that arise in the discretisation of this equation. The first is

the due to the fact that the profile for the variation of temperature over a cell must be assumed when

discretising in space. This is due to the integration of the heat equation with respect to space which

leaves temperature gradients that must be evaluated at the interfaces between cells. The simplest

profile assumption for which the temperature gradient is defined is for a piecewise linear profile as a

step function is not defined at the boundaries. More complicated assumptions can also be employed

but the piecewise linear solution will be employed in this project because it leads to a set of equations

that can be solved by the TMDA (Patankar 1980). Using this assumption, the right hand side of the

heat equation, when discretised with respect to space, becomes the difference in heat flux at the two

interfaces of a cell.

(

)

( )

(

)

( )

we

w

WPw

e

PEe qq

x

TTk

x

TTk −==











−

−











−0

δδ

P

E

W

∆x

e

w

(δx)

e

(δx)

w

Eq.

58

Eq.

59

Eq.

60

Eq.

61

Aerothermal Performance of a Solar Updraft Tower Owen Williams

vi

It is possible to assume that the density, specific heat capacity and conductivity are all

variables dependent on temperature or depth due to non-homogeneity of the material. If this is the

case, it adds much greater complication to the discretisation procedure. For the simulation of the

ground under a Solar Updraft Tower it will be assumed that the conductivity varies strongly with

depth from the surface, as was shown to be the case at the pilot plant in Manzanares. It is assumed

that density and specific heat capacity are both constant for simplicity.

Thus the second major difficulty in discretising the heat equation lies in determining the

interface conductivity at the boundary between cells.

Fig. 20 - Detail of Cell Interface (Patankar 1980)

The easiest way to interpolate the interface conductivity is to assume a linear variation

between nodes. This leads to the relation given in Eq. 62 but this is a simple minded approach that

does not work well for rapid changes in conductivity.

( )

(

)

( )

e

eEePee x

x

fkfkfk

δ

+

≡−+= 1

Thus a different approach is often used which aims to generate the correct heat flux at the

boundary not just the conductivity. Assuming that two adjacent cells have conductivities kP and kE

respectively, then a steady one-dimensional analysis without sources leads to

( ) ( )

E

e

P

e

EP

ekxkx

TT

q// +− +

−

=

δδ

which gives

1

1−











+

−

=

E

e

P

e

ek

f

k

f

k

A major difference between these two approaches is that when the interface is equidistant

from each node

(

)

5.0=

e

f, the first equation gives the conductivity as the arithmetic mean of the

nodal values whereas the second approach gives the harmonic mean. The second approach will be

more preferred because it is more robust (Patankar 1980).

Treatment of the transient term in the heat equation can take many forms. Just as discretising

in space requires integrating with respect to the coordinate directions, the equation must also be

discretised in time by integrating each term. Thus an assumption must also be made about how

temperature varies with time. There are many possibilities, but one common choice is

( )

[

]

tTfTfdtT O

PP

tt

tP∆−+=

∫∆+ 1

1

where f is a weighting factor between 0 and 1. In essence, the assumption is that the temperature

varies as a weighted average of the old and new time steps. The interesting fact is that this is actually

just an extension of the well known implicit

(

)

1=f and explicit schemes

(

)

0=f.

The implicit scheme assumes that the new value of T1 prevails over the entire timestep

whereas the explicit scheme assumes the opposite (T0 prevails over the whole timestep). Another

method called the Crank-Nicolson Method assumes that the temperature varies linearly from one

timestep to the next which corresponds to

(

)

5.0=f. This would seem like the best option at first

glance but (Patankar 1980) makes the case that the implicit solution is in fact better in this situation.

The Crank-Nicolson Method is usually unconditionally stable, just like an implicit solution

but there is a key difference. The Crank-Nicolson Method will not always converge to a physically

relevant solution at large timesteps as an implicit method would. Additionally, iteration would be

P E e

(δx)

e

(δx)

e+

+

(δx)

e-

+

Eq.

62

Eq.

63

Eq.

64

Eq.

65

Eq.

66

Aerothermal Performance of a Solar Updraft Tower Owen Williams

vii

required for a solution of this type as the previous timestep is also included. The Crank-Nicolson

Method is more accurate than an implicit solution for small timesteps but for the advantages of both

schemes and none of the disadvantages, an exponential solution should be used instead.

Discretising Convective Equations: The Upwind Scheme

(Patankar 1980) presents a discussion on the solution of convective equations governed by the general

convective-diffusive equation for variable

φ

, Eq. 67. The diffusion term (right hand side) can be

treated in the same manner as the heat equation in Section2.3.3 because they have the same form. The

solution is made more complicated by the convective term. The reasons for this will be discussed as

well as possible solutions.

( )

S

xx

u

xt jj

j

+













∂

Γ

∂

=

∂

+

∂

φ

φρρφ

Considering control volume shown in Fig. 20 and integrating over its full length, neglecting

unsteady and source terms, the discretised equations are as shown in Eq. 62. The diffusion term is in

the exact form of Eq. 61 hence it can be treated in the same manner as the heat equation and will not

be discussed here any further.

( ) ( )

we

we dx

d

dx

d

uu 









Γ−











Γ=−

φφ

φρφρ

The problem with the convective term occurs when estimating the value of

φ

at each

boundary. Using a piecewise linear profile, as with the diffusion terms, causes the value at the

interface to be the average of the two adjacent cells. This is also sometimes called a central difference

scheme and is the natural outcome of a Taylor-series formulation. Using this scheme, the discretised

equation will be of the same general form as Eq. 69.

WEP

w

wW

e

eE

WWEEPP

aaa

F

Da

F

Da

x

D

uF

aaa

+=

+=−=

Γ

=

+=

22

δ

ρ

φφφ

(Patankar 1980) explains the limitations of this discretisation. He states that there are a set

number of fundamental rules that must be obeyed for a stable solution. One of these is that all

coefficients of the above equation must remain positive at all times for a physically realistic solution.

As the equations now involve fluid flow, the flux uF

ρ

=

can become positive or negative based on

the direction of flow. With negative coefficients, it implies that P

awhich equals

∑

nb

a does not

equal

∑

nb

a. This is a violation of the Scarborough Criterion . Due to the limitation of this scheme

many early attempts to solve convective equations were limited to low Reynolds numbers or small

ratios of F/D.

To address the limitations of this scheme, Courant, Isaacson and Rees (1952) proposed the

Upwind Scheme which is now widely used for its stability and ease of implementation. They

proposed that the value of

φ

at an interface should be equal to its value at the grid point on the

upwind side of the face. In other words, for the left face,

0

<=

>

=

eEe

ePe

Fif

φφ

φ

A similar formulation can be constructed for the other face. A more compact for of this equation is

given in Eq. 71 where BA, shall be defined as the greater of A or B.

0,0, eEePee FFF −−=

φφφ

Eq.

67

Eq.

68

Eq.

69

Eq.

71

Eq.

70

Aerothermal Performance of a Solar Updraft Tower Owen Williams

viii

Using this method, the coefficients will always be positive giving physically relevant

solutions and satisfying the Scarborough Criterion. This scheme is still often used due to its simplicity

and the fact that it is easily converged. It also closely resembles the physical situation where the

properties of the upwind cell are carried across the interface. All of these advantages are balanced by

the fact that it is only first order accurate. For this reason, many other higher order schemes are often

used by commercial solvers. Additionally, an unavoidable drawback of the Upwind Scheme is that it

is equivalent to replacing the diffusion

Γ

in the central difference scheme with 2/xu

δ

ρ

+

Γ

. This is

called false diffusion and unfortunately cannot be helped in most cases using just this scheme.

(Patankar 1980) also details many other possible schemes, each with their own advantages

and disadvantages. The Exact Solution estimates interface values using an exact solution analytical

solution based on Peclet number, or the ratio of convection to diffustion. The Exponential Scheme

approximates the total flux (diffusive ad convective) between the two cells using the exact solution

and results in an equation that involves exponentials of Peclet number. While accurate the benefit is

usually outweighed by the computation cost of computing the exponentials. Two further schemes are

also presented that exhibit the benefits of the exponential scheme but that are easy to compute. They

are called the Hybrid Scheme and the Power-Law Scheme and they approximate the central-

difference scheme at low Peclet numbers when it is not unstable and the upwind scheme otherwise.

Staggered Grids

Several difficulties arise as a result of the discretisation of pressure within the momentum

equation and the solution of the continuity equation. (Patankar 1980) highlights the problems and

why they necessitate the use of a special type of grid.

The difficulty partial lies within the discretisation of the x-component of the momentum

equation. Integrating the pressure gradient term, dxdP−, over the normal control volume shown in

Fig. 21 results in the following discretised form in terms of nodal pressures, assuming a piece-wise

linear profile.

2

EW

Ep

PW

ew

PP

PP −

=

+

−

+

=−

Notice that the pressure equation is only between alternating nodal pressures and not adjacent

ones. Not only does this make the effective pressure grid twice as coarse as a normal grid but the a

zig-zag pressure field or checkerboard pattern. With an alternating pressure field, the corresponding

pressure difference EW PP −is zero and no pressure force will be felt in the x-direction. Thus, for any

true solution, an infinite number of checkerboarding fields can be superimposed upon it and the

momentum equation will remain unaffected.

The same phenomenon occurs with the discretisation of the continuity equation. Considering

the one-dimensional incompressible equation, its discretised form results in the following equation

0

2

0=−=

+

−

+

=−⇒=WE

PW

EP

we uu

uu

dx

du

Once again, the solution of these equations will permit checkerboarding solutions that must

be eliminated in order to have any confidence in the solution. (Patankar 1980) suggests that there are

a number of methods available to remove these unwanted solutions by special treatment of boundary

conditions or underelaxation with respect to a smooth initial condition but these do not treat the

original root of the problem.

He suggests that the best method is to use a staggered grid. This later becomes the basis of his

SIMPLE method. The benefits of a staggered mesh result from not solving for all of the variables

solely at the nodes. By solving for and storing, the velocities at cell faces, the checkerboarding

solutions can be entirely eliminated. u-velocities are stored at the faces normal to the x-direction as

shown in Fig. 21

Eq.

72

Eq.

73

Aerothermal Performance of a Solar Updraft Tower Owen Williams

ix

Fig. 21 - Staggered Positions for Velocity Field

One benefit of this grid is that the mass fluxes at the boundaries can be calculated without any

interpolation. The main benefit, however, stems from the fact that the continuity equation will now be

calculated using adjacent velocity nodes and a checkerboarding velocity field cannot result.

Chekerboarding pressure fields will also no longer be felt as a uniform field but the reasons are less

simple. The pressure difference between two adjacent nodes becomes the natural driving force for the

velocity component at those nodes.

Staggered meshes are more complicated to implement in many ways and the program

implementing them must keep track the indexing and geometric information of the locations of the

velocity components. There are also complications implementing them in multiple dimensions. Still,

the benefits can far outweigh the complications in many situations as the alternatives have drawbacks

of their own

Solution of Convective Equations: SIMPLE Method

The solution of convective equations is quite complicated and as such, Appendix A should be

consulted for information on discretising convection equations and the reasons for using a staggered

grid.

Using this information, it is possible to descritise the momentum and continuity equation

which are just special cases of the general equation, Eq. 67. The non-linearity of the momentum

equation is also not a problem because an iterative solution can be found such that given an initial

guess of the velocity field, progressively more accurate solutions can be generated until arriving at the

correct solution.

The problem lies in the fact that the momentum equation is heavily influenced by pressure

gradients and thus a method for estimating the pressure field must be developed to fully solve the

equations. Once the pressure field is known there is no particular difficulty in solving the momentum

equations for the velocity field.

The pressure field is implicitly specified by the continuity equation since when the correct

pressure field is substituted into the momentum equation, the resulting velocity field satisfies the

continutity equation. In some cases it is possible to treat the density as the dependant variable in the

continuity equation but this approach does not work for constant flows it which the effect of pressure

of the velocity field is of primary importance

A widely used method for estimating the pressure field from the continuity equation is

detailed by (Patankar 1980) and is called the Semi-Implicit Method of Pressure-Linked Equations

(SIMPLE) and this is the one that will be used in this simulation.

Beginning by considering the discretised form of the momentum equations, it becomes clear

that it is just an example of the general equation. All of the difficulties in discretising the equation

have already been solved. As the SIMPLE method uses a staggered grid for the reasons set out in

Section 2.3.6, and the dependent variable of the momentum equation is the velocity, a slightly

different control volume must be used. A visualisation of this control volume is shown in Fig.647.

The resulting discretised form of the equation will be of the form shown in Eq.305 and the

number of adjacent values that will be used will depend on the dimensionality of the problem; two

adjacent cells for one-dimensional, four for two-dimensional and so on. Notice that the pressure

gradient term is now dependant on adjacent cells. Further equations can be found for the other

coordinate directions.

(

)

∑

−++= EEPPnbnbee PAPAbuaua

Using a guessed pressure field, *

P

, the velocity field approximate velocity field, *

u, can be

found using Eq.1234. The problem lies in the fact that unless the exact pressure field is employed, the

solution will not satisfy the continuity equation. Thus, it is necessary to determine a set of pressure

and velocity corrections so that the answer converges to the correct solution.

P E W

e

w

Eq.

76

Eq.

75

Eq.

74

Aerothermal Performance of a Solar Updraft Tower Owen Williams

x

(

)

∑

−++= ****

PEPPnbnbee PAPAbuaua

(Patankar 1980) suggests solving for a pressure correction '

P

which will be added to the

guessed pressure field to obtain the correct solution. The velocity corrections can be introduced in the

same manner.

'*'* uuuPPP +=+=

The equation for the velocity correction will be derived by subtracting Eq.23 from Eq.235.

This results in the following equation.

(

)

''''

EEPPnbnbee PAPAuaua −+=

∑

To complete the equation, the terms related to the adjacent cells are neglected resulting in the

final velocity-correction equation which is solely in terms of the pressure corrections.

(

)

e

EEPP

a

PAPA

uu

''

*−

+=

Neglecting the terms involving the adjacent terms can be justified for a number or reasons, all

of which can be found in (Patankar 1980). The main reason neglecting them is that if they were

included, their solution would involve including their neighbours, who involve their neighbours until

the velocity correction equation would be a function of every cell in the domain. This would be

unmanageable and neglecting these terms makes it possible to cast this equation in the same form as

the general equation

Apart from the convenience, it would not be possible to neglect these terms if it altered the

solution. It turns out that because these equations are just solving for a correction to the previous

solution, as long as the solution converges, it will be the correct solution. Omission of the extra terms

just slows down the convergence process and it will take a few extra iterations to reach the final

solution.

Neglecting the

∑

'

nbnb ua terms give the SIMPLE method its semi-implicit nature. They

represent the implicit influence of the pressure corrections at adjacent cells on the velocity correction

and their omission means that the solution to follow will only be partial implicit and not fully.

The pressure correction equation is derived from the continuity equation because the correct

pressure field is implied in the satisfaction of this equation.

(

)

0=

∂

+

∂

x

u

t

ρ

Discretising this equation over a normal control volume centred around node P where the

pressure is stored gives the following formula when using an implicit time discretisation

(

)

( ) ( )

0=−+

∆

−

w

e

O

PP uAuA

t

V

ρρ

The final pressure-correction equation is determined by subbing in the various velocity-correction

equations and rearranging into the following form.

bPaPaPa WWEEPP ++= '''

The constant term, b, contains all references to the previous timestep or iteration and is essential the

negative of the left hand side of Eq.24356 evaluated with the starred velocities of the previous

iteration. In this way, if it becomes zero, then continuity is satisfied and the correct solution has been

arrived at.

The derivation of all the nessisary equations has been completed. What follows is the step by step

procedure of the SIMPLE method as stated in (Patankar 1980)

(5) Guess the pressure field *

P

(6) Solve the momentum equations (Eq.345) for the starred velocities *** ,, wvu

(7) Solve the pressure correction equation for '

P

Eq.

77

Eq.

78

Eq.

79

Eq.

80

Eq.

81

Eq.

82

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xi

(8) Calculate the pressure by adding '

P

to *

P

(9) Calculate

w

v

u

,

using the starred velocities and the velocity correction equation.

(10) Solve the discretised equation for other

φ

such as temperature, concentration another scalar

variable. If these properties have an effect on the flow field throught the fluid properties,

incorporate its effect at this point.

(11) Treat the corrected pressure as the new guessed pressure *

P

and return to step 2 and repeat

the entire procedure until the constant term, b, in the pressure correction equation is zero.

It must be noted that the pressure-correction equation is just an intermediate equation the

form of which, if the solution converges will not affect the final answer. This being said if too many

terms are omitted the solution may diverge. It may also diverge if there is insufficient relaxation of

the pressure and velocity corrections. An underrelaxation of 0.5 for the velocity and 0.8 for the

pressure have been shown to be sufficient for most applications. Also, within this solution procedure,

the density is treated as a known quantity. Its value will be calculated through the equation of state for

each iteration once the temperature has been found in Step 6.

Note that while this method will always give the correct answer, there are further

enhancements that can be made to this algorithm to make the solution converge faster. The omission

of the

∑

'

nbnb ua terms result in exaggerated pressure corrections that result in the pressure field

converging much slower than the velocity field. The revised methods aim to increase the rate of

convergence of the pressure field. Some such algorithms are called SIMPLE or SIMPLEC.

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xii

APPENDIX B – DATA FROM PILOT PLANT AT MANZANARES

Table 2 - Typical Dimensions of Solar Chimneys (Haaf, Friedrich et al. 1983)

Plant size (MW) 5 30 100

Collector Diameter (m) 1110 2200 3600

Chimney Height (m) 445 750 950

Chimney Diameter (m) 54 84 115

Temperature rise in Collector (˚C) 25.6 31.0 35.7

Up-draught velocity (m s

-

1

) 9.1 12.6 15.8

Total Pressure head (Pa) 383.3 767.1 1100.5

Average efficiency

Collector (%) 56.24 54.72 52.62

Chimney (%) 1.45 2.33 3.10

Turbine (%) 77.00 78.30 80.10

Whole System 0.63 1.00 1.31

Table 3 - Technical Data and Design Criteria of the Pilot Plant (Haaf, Friedrich et

al. 1983)

Tower height, H

T

(m) 194.6

Tower radius, R

T

(m) 5.08

Mean collector radius, R

c

(m) 122

Average canopy height (m) 1.85

No. of turbine blades 4

Blade radius (m) 5.0

Operating Modes a) stand-alone operation

b) grid connection mode

Turbine speed in grid connection mode (RPM) 100

Gear Ratio 1:10

Design irradiation, I (W/m

2

) 1000

Design fresh-air temperature, T

a

(K) 302

Temperature increase, mean for model assumptions

at design point, ∆T (K) 20

Collector efficiency, mean for model assumptions at

design point, η

c

0.32

Turbine efficiency, η

T

0.83

Friction loss factor, η

f

0.9

Upwind Velocity under Load Conditions (m/s) 9

Upwind Velocity on release (m/s) 15

Power output, mean for model assumptions at

design point (kW) 50

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xiii

Updraft velocity and electric output vs. time of day

0

10

20

30

40

50

60

0:00 3:00 6:00 9:00 12:00 15:00 18:00 21:00 0:00

time of the day

electric power [kW], updraft velocity

[m/s]

0

200

400

600

800

1000

1200

global solar insolation [W/m²]

Pel, measured

Updraft velocity, measured

Gh, measured

Fig. 22 - Results from Pilot Plant 15/06/87 (Weinrebe 1987)

updraft velocity and electric output vs. solar input

0

10

20

30

40

50

60

0 100 200 300 400 500 600 700 800 900 1000 1100

Global Horizontal insolation [W/m²]

power output [kW]

0

1

2

3

4

5

6

7

8

9

Pel, measured

Updraft velocity, measured

Updraft velocity, measured (linear fit)

Pel ,measured (linear fit)

Fig. 23 - Trends in Power Output and Upwind Velocity at Pilot Plant 15/06/87 (Weinrebe 1987)

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xiv

Fig. 24 – Time Variation of Ground Temperature at Various Depths (Haaf 1984)

Fig. 25 – Data from Pilot Plant including turbine pressure drop, power output and upwind velocity for 02/09/82

(Haaf 1984)

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xv

APPENDIX C – FURTHER RESULTS FROM 1D ANALYSIS

Flow Simulation

Reasons for Compressible Flow

The Mach number within the chimney is always below 0.3. As such, the Mach number changes the

density less than 5%, the generally accepted boundary between compressible and incompressible

flows. The flow must be treated as compressible, howerever, because the temperature decreases at the

adiabatic lapse rate of 9.8 K/km. For a tower 1500m high, as required for a 200MW plant, would

yield a 15˚C or 5% drop with an inlet temperature of 300K. Assuming adiabatic conditions and air as

a perfect gas, the isentropic relation gives a change in density of 12% as shown below. Thus the flow

cannon be treated as incompressible (Von Backstrom, Theodor W. 2003).

( )

88.095.0

1

2

1

24.0

1

=













==











−

ρ

γ

T

Discretised Flow Equations

Momentum Equation

The velocity-correction equation is derived from the momentum equation integrated over a

staggered control volume such as that found in the figure below. The current discretisation will

involve the upwind scheme.

( )

lossref F

x

u

g

x

P

u

x

u

t

+

∂

−−+

∂

+

∂

+

∂

2

µρρρρ

Term 1 (rate of change of momentum):

(

)

(

)

( )

[ ]

o

i

o

j

o

jijj

I

o

i

o

iii

Iuu

t

V

t

uu

V

t

u

11

2

1

−− −−+

∆

=

∆

−

⇒

∂

ρρρρ

ρρ

ρ

Term 2 (momentum flux): uF

ρ

=

(

)

( ) ( )

1

2

−

−⇒

∂

jj FuAFuAu

x

ρ

Scalar mass fluxes at centre of each cell from upwind scheme

Cell I

dx_stag

Pj-1

Ρj-1

Tj-1

dx

Ui

Ai

Pj

Ρj

Tj

Te

rm2

Term3

Term4

Term5

Losses

Term1

Eq.

83

Eq.

84

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xvi

0,0,

1111

1

++++

−

−−=

iiiii

uuF

ρρ

Convective fluxes at edge of each cell

(

)

(

)

11

2

1

++

+= iiiij FAFAFA

Thus,

(

)

(

)

(

)

( ) ( ) ( )

0,0,

11

1

−−

−

+

−−=

j

i

j

i

j

jijij

FAuFAuFAu

Term 3 (Pressure Gradient):

( ) ( )

11

2

1

2

1

−−+ +−+⇒

∂

jiijii PAAPAA

x

P

Term 4 (Buoyancy Term):

(

)

(

)

(

)

gVg jjrefIref

ρρρρρ

+−⇒−−1

2

1

Term 5 (Viscous Term):

istag

idx

d

µ

=

( )( ) ( )( )

( )

11

2

1

111

2

1

2

−−+++ −+−−+⇒

∂

iiiiiiiiii uuAAduuAAd

x

u

µµ

These equations can be rearranged into the standard form which can be solved for a velocity

correction using the residual method.

buauaua iiiiii ++= −−++ 1111

Velocity-Correction Equation

Assuming that the velocity-correction is of the same form as the momentum equation,

''

11

'

11

'

11

'

jjjjiiiiii PAPAuauaua −++= −−−−++

Neglecting terms involving adjacent cells, the velocity correction equation is shown below in terms of

correcting pressures from the pressure-correction equation.

(

)

ijjjjii aPAPAuu ''

11

*−+= −−

Pressure-Correction Equation

The pressure correction equation is calculated from the continuity equation by integrating it

over a regular control volume centred on point j and using the upwind scheme.

(

)

0=

∂

+

∂

x

u

t

ρ

(

)

( ) ( )

0

1=−+

∆

−

+ii

o

jj

IuAuA

t

V

ρρ

Subbing in the velocity correction equation and re-arranging results in the pressure-correction

equation

( ) ( ) ( )













−+

∆

−

−=













−













−











+

+

+++

−

+

++

ii

o

jj

I

j

i

jii

j

i

jii

j

i

jii

i

jii

AuAu

t

V

P

a

AA

P

a

AA

P

a

AA

a

AA

*

1

*

'

1

111

'

1

'

1

11

ρρ

ρρρρ

Eq.

85

Eq.

86

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xvii

The fluxes are by first order upwind method

(

)

[

]

( )

[ ]

0,0,

1111

1

*

1

*

++++

+

−

−−=

iiiii

i

iiiii

i

uuAAu

ρρρ

Energy Equation

The energy equation is discretised over a normal control volume allowing a solution for static

enthalpy.

( ) ( ) ( )

h

SPu

t

P

quhh

t

+∇⋅+

∂

+−∇=⋅∇+

∂

ρρ

Integrating over control volume and treating pressure work term as a source term,

(

)

( ) ( )

(

)

0

2

11

1=











−

−

∆

−

−+−+

∆

−+−

+∑

istag

ii

o

ii

I

ii

o

j

o

jjj

Idx

PP

u

t

PP

VAqFAhFAh

t

hh

V

ρρ

Enthalpy Fluxes:

(

)

(

)

(

)

( ) ( ) ( )

0,0,

1

1111

iiiii

FAhFAhFAh

−−=

−

++++

Heat Fluxes:

(

)

TThq w−=

Control Volume Analysis for Converting Friction Factor to Pressure Loss

Atmospheric Model

( )

235.1

1

=











−

−=











−

−=











−

−=

−

κ

ρρ

κ

g

RT

H

z

H

z

PzP

H

z

TzT

o

Eq.

87

Eq.

88

Eq.

89

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xviii

Calculation of Upwind Velocity in Chimney: Test Case 1

(

)

oocinoutref RTPgHPP =−=

ρρ

From Momentum Control Volume Analysis and balance of forces:

( )

dhHuC

d

gHPA cfrefoc

πρ

π

ρρ

−=−=∆ 2

2

1

2

4

Where c

his the height of the collector above the ground.

Rearranging the above equation,

(

)

( )

hcHC

gdH

u

f

refo

−

=

ρ

ρρ

2

Calculation of Adiabatic Lapse Rate: Test Case 2

The energy equation in terms of static enthalpy is given as

( ) ( ) ( )

h

SPu

t

P

quhh

t

+∇⋅+

∂

+−∇=⋅∇+

∂

ρρ

Assuming steady, adiabatic flow

(

)

PuTuCP∇⋅=∇

ρ

Applying continuity

(

)

0=∇−∇∇⋅ PTCu P

ρ

If u is finite then the following must be true

0=∇−∇ PTCP

ρ

Thus the adiabatic lapse rate must be

P

C

P

T

ρ

∇

=∇

For the classic atmospheric adiabatic lapse rate take

kmC

C

g

TgP

P

/76.9 °==∇=∇

ρ

Ground Simulation

Discretised Heat Equation

In order to discretise the equation over a finite volume, the equation must be integrated with respect

to time and space. The domain that will be used is shown in Fig. 19. Starting with the unsteady term

on the left hand side of the equation

(

)

O

PPP

e

w

tt

t

PTT

t

x

Cxt

t

T

C−

∆

=∂∂

∂

∫ ∫ ∆+

ρρ

Integration of the temperature gradient term is slightly harder. Assuming that the temperature varies

linearly across a cell and that then incorporating a proportional variation with time (see Eq. 93) as

discussed in Section 2.3.3 gives

ρ

c

oin PP =

gHPP refoout

ρ

−=

H

h

c

Eq.

91

Eq.

92

Eq.

93

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xix

( )

( ) ( )

( )























−

−











−

−+























−

−











−

=























−

−











−

=























−













=























∂

∫

∫∫ ∫

∆+

∆+∆+

w

O

W

O

Pw

e

O

P

O

Ee

w

WPw

e

PEe

tt

tw

WPw

e

PEe

tt

twe

tt

t

e

w

x

TTk

x

TTk

f

x

TTk

x

TTk

f

dt

x

TTk

x

TTk

dt

dx

dT

k

dx

dT

kdxdt

x

T

k

x

δδδδ

δδ

1

All together the discretised equations are of the following form

( )

(

)

( )

(

)

( ) ( )

(

)

( )

(

)

( )























−

−











−

−+























−

−











−

=−

∆

w

O

W

O

Pw

e

O

P

O

Ee

w

WPw

e

PEe

O

PPP x

TTk

x

TTk

f

x

TTk

x

TTk

fTT

t

x

C

δδδδ

ρ

1

Eq.

94

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xx

APPENDIX D – HEAT TRANSFER CORRELATIONS

Forced Convection

Ditus-Boelter Equation:

(

)

( )

000,10Re

700,16Pr7.0

3.0

4.0

PrRe023.0 54

>

<<







=<

=>

=

Dms

ms

n

DD nTT

nTT

ifNu

Petukov Relation:

(

)

( )

64

32

21 105Re10

2000Pr5.0

1Pr87.1207.1

PrRe8

x

f

Nu

D

D<<

<<

−+

=

Gneilinski Formula:

(

)

(

)

( )

6

32

21 105Re3000

2000Pr5.0

1Pr87.121

Pr1Re8

x

f

Nu

D

D<<

<<

−+

−

=

All of these equations were obtained from (Incropera, DeWitt 2002)

Natural Convection

Globe and Dropkin relation for natural convection between two plates (Incropera, DeWitt 2002):

9574.031 107103Pr069.0 xRaxRaNu DDD <<=

For natural convection from a heated plate facing upwards or a cold plate facing down, the following

relation is given by (Incropera, DeWitt 2002)

11731

7441

101015.0

101054.0

<<=

LLL

RaRaNu

For natural convection from a heated flat plate exposed to the wind, (Duffie, Beckman 1974) give the

following approximation.

ww vh 8.37.5 +=

Mixed Convection (Generated from 3D Analysis)

Mixed Convective Heat transfer coefficient of the following form with 5.1

=

n

N

n

F

nNuNuNu +=

Forced convection by Gneilinski Formula:

(

)

(

)

( )

6

32

21

,105Re3000

2000Pr5.0

1Pr87.121

Pr1Re8

x

f

Nu

D

DF <<

<<

−+

−

=

Natural Convection by new correlation with Rayleigh number

3

1

06.0 RaNu N=

Eq.

95

Eq.

96

Eq.

97

Eq.

98

Eq.

99

Eq.

100

Eq.

101

Eq.

102

Eq.

103

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xxi

APPENDIX E – EXTENDED RESULTS

Selection of Domain for 3D Model

The following three plots were generated by Cfx5 and demonstrate the impact of different domain

sizes on velocity, heat transfer coefficient and wall shear stress. As can be seen, there is minimal

impact on these variables, but there was a significant impact on temperature distribution (Fig. 10)

0

2

4

6

8

10

12

0 20 40 60 80 100 120 140

Radius (m)

Velocity (m/s)

Theta = 20 T heta = 15 Theta = 10 T heta = 5

Fig. 26 - Variation of Collector Velocity with Radius

0

5

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140

Radius (m)

Heat Transfer Coefficient (Wm^-2K^-1)

Theta = 20 Theta = 15 Theta = 10 Theta = 5

Fig. 27 - Variation of Wall Heat Transfer Coefficient with Radius

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xxii

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 20 40 60 80 100 120 140

Radius (m)

Wall Shear (Pa)

Theta = 20 Theta = 15 Theta = 10 Theta = 5

Fig. 28 - Variation in Wall Shear Stress with Radius in Collector

Fluid Flow within the Collector

3D Flow within the Collector

a) Entrance Region

b) Instability Onset

c) Vortex Breakdown

d) Acceleration Region

Fig. 29 - Local Contours of Velocity within Each Flow Region at a) R=120 b) R=100 c) R=60 d) R=10

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xxiii

The following figure shows streamlines within the collector, clearly showing onset of thermal

instability and then the change to forced convection.

Fig. 30 - Streamlines of Flow within the Collector

Assessment of Heat Transfer

Heat Transfer at the Upper Surface

The following figure demonstrates the fact that heat transfer at the lower surface is unaffected

by a heat transfer at the lower surface.

0

5

10

15

20

25

30

35

0.00E+00 2.00E+01 4.00E+01 6.00E+01 8.00E+01 1.00E+02 1.20E+02

x (m )

Gr/(Re*Re)

Fig.

31

-

Relative Strength of Natural and Forced Convection

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xxiv

0

500

1000

1500

2000

2500

3000

3500

4000

0 20 40 60 80 100 120 140

x (m)

Nusselt Number

With Heated Upper Surface W ithout

Fig. 32 - Heat Transfer at the Ground, with and without a heated cover

The following graph demonstrates the fact that heat transfer at the upper surface is less affected by

secondary flows and can be approximated by fully forced convection

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 20 40 60 80 100 120 140

x (m)

Nu

3D Gnielinski

Fig. 33 - Comparison of Heat Transfer at the Upper Surface with Forced Convective Heat Transfer

This contour of heat flux at the upper surface demonstrates that secondary flows only have a

small influence on the heat flux at the upper surface. The variation at the lower surface is also

included for comparison.

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xxv

Fig. 34 - Heat Flux at the Cover when Heated

Fig. 35 - Variation of Heat Flux at the at the Ground

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xxvi

Development of Mixed Convective Heat Transfer Correlation

The following plot of Nusselt number against Rayleigh number demonstrates the accuracy of

the naturally convective component of the new mixed convective heat transfer correlation.

0

500

1000

1500

2000

2500

3000

3500

4000

1.7E+11 1.8E+11 1.9E+11 2E+11 2.1E+11 2.2E+11 2.3E+11

Ra

Nu

3D New Mixed Convection

Fig. 36 - Mixed Convective Heat Transfer Coefficient with Rayleigh Number

The following plot shows the enhancement of the new mixed heat transfer correlation over

the Gnielinski equation

0

5

10

15

20

25

30

0 20 40 60 80 1 00 120

x (m)

Heat Transfer Coefficient (W/m2K)

3D Solution Gnielinski Combined

Fig. 37 - Natural Convection Augmentation of Forced Convective Heat transfer Coefficient within Collector.

Flow Variation through the Plant

Effect of Losses and Turbine Pressure Drop on Flow

The following plots show the variation of velocity, temperature, pressure and density through

the domain as generated with the one-dimensional program. Loss terms were successively added,

giving an estimate of their affect on the flow.

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xxvii

a)

0

2

4

6

8

10

12

14

16

0 50 100 150 200 250 300 350

x (m)

Velocity (m/s)

Nothing Minor Losses Double Friction With Turbine

b)

-250

-200

-150

-100

-50

0

0 50 100 150 200 250 300 350

x (m)

Pressure (Pa)

Nothing Minor Losses Double Friction W ith Turbine

c)

297

298

299

300

301

302

303

304

305

306

307

308

0 50 100 150 200 250 300 350

x (m)

Temerature (K)

Nothing Minor Losses Double Friction With Turbine

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xxviii

d)

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

0 50 100 150 200 250 300 350

x (m)

Density (kg/m3)

Nothing Minor Losses Double Friction W ith Turbine

Fig. 38 – Variation Fluid Properties through the Plant with inclusion of Losses. a)Velocity b) Pressure

Relative to Inlet Pressure c) Temperature d) Density

Comparison with Pilot Plant

Ground Simulation

a)

290

300

310

320

330

340

350

360

370

380

0 4 8 12 16 20 24

Time (hr)

Ground Temperature (K)

0 cm

2 cm

4 cm

8 cm

12 cm

16cm

26 cm

42cm

b)

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xxix

290

300

310

320

330

340

350

360

370

380

0 4 8 12 16 20 24

Time (hr)

Ground Temperature (K)

0 cm

2 cm

4 cm

8 cm

12 cm

16cm

26 cm

42cm

c)

290

300

310

320

330

340

350

360

370

380

0 4 8 12 16 20 24

Time (hr)

Ground Temperature (K)

0 cm

2 cm

4 cm

8 cm 12 cm

16cm

26 cm

42cm

Fig. 39 – Temperature Variation Depth and Time at a) Collector Entrance b) x=50m c) Collector Exit

Sensitivity to Boundary Conditions

The following is the resulting variation in inlet of exit pressure difference for the simulation involving

variable ambient temperature and exit pressure. The exit pressure was estimated using Eq.234. This

graph is used to justify results showing a significant reduction in plant output for this set of boundary

conditions.

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xxx

2070

2080

2090

2100

2110

2120

2130

2140

2150

0 4 8 12 16 20 24

Time (hrs)

Pressure Difference (Pa)

Fig. 40 – Variation of inlet to exit pressure difference for variable ambient temperature and exit pressure case.

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xxxi

APPENDIX F – ASSUMED PROPERTIES OF FLOW,

GROUND AND COVER

Boundary Condition Functions

Ambient Temperature

The ambient temperature function was adapted from ambient temperature data provided by

(Weinrebe 1987) and based on a sinusoidal function. The best fit coincides with the following

function with an average temperature of 290.15 K and a range 9.8K. A closer fit can be generated

with the data but it will not have a period of 24hrs, a requirement of the program.

( )

mean

range

aTt

T

T+











+= 5.15

12

sin

2

π

282

284

286

288

290

292

294

296

0 5 10 15 20

time (h)

Ambient Temperature (K)

1D Simulation Pilot Plant (1m) 15/06/87 Pilot Plant (195m)

Fig. 41 - Variation of Ambient Temperature through a sample day

Incident Solar Radiation

The incident solar radiation variation was also based upon data from (Weinrebe 1987) for a

sample day at the pilot plant at 15/06/87. It was designed to correspond to the positive sections of a

sinusoidal curve with a maximum of 1026.17W/m2. This maximum value could be specified in an

input file. The best fit with experimental data occurred with the following function





















−

=0,

25.4

25.12

cosmax max

t

II

Eq.

104

Eq.

105

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xxxii

0

200

400

600

800

1000

1200

0 2 4 6 8 10 12 14 16 18 20 22 24

hr

I (W/m2)

1D Simulation Pilot Plant (15/06/87)

Fig. 42 - Daily Variation of Incident Solar Radiation with a Maximum at 900 W/m2

Ground Conductivity

Ground conductivity was made a function of depth based on experimental data from (Haaf,

Friedrich et al. 1983). The resulting function is given below where the depth x, is in meters and the

function is designed to give a conductivity of 0.1 at the surface.

(

)

6325.10121.0ln347.0 ++= xk

y = 0.347Ln(x) + 1.6325

R

2

= 0.9362

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Depth (m)

K

Pilot Plant Data Simulated Distribution Log. (Pilot Plant Data)

Fig. 43 – Variation of Ground Conductivity with Depth

Eq.

106

Aerothermal Performance of a Solar Updraft Tower Owen Williams

xxxiii

Simulation Input Conditions

Table 4 - Technical Data and Design Criteria of the Pilot Plant (Haaf, Friedrich et

al. 1983)

Tower height, H

T

(m) 194.6

Tower radius, R

T

(m) 5.08

Mean collector radius, R

c

(m) 122

Average canopy height (m) 1.85

Ground Density (kg/m

3

) 1320

Ground Cp (J/kg K) 920

Ground Absorbtivity 0.8

Ground Emissivity 0.8

Cover Emissivity 0.9

Cover Thickness (m) 0.001

Cover Density (kg/m

3

) 920

Cover Transmissivity 0.9

Ground Roughness (mm) 4

Cover Roughness (mm) Smooth

Max irradiation, I (W/m

2

) 1000

Ambient Temperature, T

a

(K) 302

Inlet Pressure (Pa) 92962.5

Outlet Pressure (Pa) 90840

Turbine efficiency, η

T

0.83

Velocity Relaxation 0.2

Pressure Relaxation 0.34

Temperature Relaxation 0.2

Density Relaxation 0.35

Aerothermal Performance of a Solar Updraft Tower - MEng Thesis 2008

Figures

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