![THE FIGURE ABOVE PRESENTS A SPECIFIC USE OF THE KIM AND MUDAWAR [12] CRITERIA FOR ANNULAR TO PLUG/SLUG TRANSITIONS FOR CONDENSATION OF PURE FC-72 (Dh = 8 mm AND pin = 100 kPa) AND PRESENTS THE TRANSITION CRITERIA IN THE − PLANE FOR FIXED VALUES OF ( ⁄ , ⁄ , ⁄ , , ) = (0.0307, 0.008, 0.0234, 6.44e6, 6.36e6).](https://www.researchgate.net/profile/Amitabh-Narain/publication/277880875/figure/fig1/AS:354076807188481@1461429897306/THE-FIGURE-ABOVE-PRESENTS-A-SPECIFIC-USE-OF-THE-KIM-AND-MUDAWAR-12-CRITERIA-FOR-ANNULAR_Q320.jpg)
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Review Article – JTEN – 2014 – 80
307
Journal of Thermal Engineering
Yildiz Technical University Press, Istanbul, Turkey
Vol. 1, No. 4, pp. 307-321, October, 2015.
http://eds.yildiz.edu.tr/journal-of-thermal-engineering/Articles
Manuscript Received December 19, 2014; Accepted December 25, 2014
This paper was recommended for publication in revised form by Regional Editor Mohammed Sajjad Mayeed
FUNDAMENTAL ASSESSMENTS AND NEW ENABLING PROPOSALS FOR HEAT TRANSFER
CORRELATIONS AND FLOW REGIME MAPS FOR SHEAR DRIVEN CONDENSERS IN THE
ANNULAR/STRATIFIED REGIME
*A. Narain
Michigan Technological University
Houghton, Michigan, United States
R. R. Naik
Michigan Technological University
Houghton, Michigan, United States
S. Ravikumar
Michigan Technological University
Houghton, Michigan, United States
S. S. Bhasme
Michigan Technological University
Houghton, Michigan, United States
Keywords: mm-scale condensers, annular condensation, heat-transfer rate correlations, flow regime transition
* Corresponding author: A. Narain, Phone: (906) 487-2555, Fax: (906) 487-2822
E-mail address: narain@mtu.edu
ABSTRACT
Modern-day applications need mm-scale shear-driven flow
condensers. Condenser designs need to ensure large heat transfer
rates for a variety of flow conditions. For this, good estimates for
heat-transfer rate correlations and correlations for the length of
the annular regime (beyond which plug-slug flows typically
occur) are needed. For confident use of existing correlations
(particularly the more recent ones supported by large data sets)
for shear-pressure driven internal condensing flows, there is a
great need to relate the existing correlation development
approaches to direct flow-physics based fundamental results
from theory, computations, and experiments. This paper
addresses this need for millimeter scale shear driven and annular
condensing flows. In doing so, the paper proposes/compares a
few new and reliable non-dimensional heat-transfer coefficient
correlations as well as a key flow regime transition
criteria/correlation.
INTRODUCTION
There is a critical need for thermal management
systems with reduced mass and size which are capable of
handling ever-increasing high heat loads at small to moderate
temperature differences and high heat-flux values ([1, 2]). This
need is particularly evidenced in requirements of high-power
consumption electronics industry. To achieve this, one needs to
develop innovative condensers and boilers (see [3-5]) that
operate in the annular regime and are shear driven with
negligible effects of gravity. By shear driven, it is meant that the
liquid condensate is driven by interfacial shear whereas the
overall flow is driven by inlet mass flow rate and device-level
pressure-differences. This paper focuses on addressing base-
design needs for operating millimeter scale shear driven
condensers in a non-pulsatile steady mode. Advantages of
operating these condensers in a pulsatile mode are discussed in
[3, 4]. Such millimeter scale condensers (and boilers [3]) are also
of great value in the design of next generation space-based
thermal systems, and gravity insensitive aircraft-based systems
(including avionics cooling).
Traditional macro-scale internal flow condensers
operating in vertical or inclined orientations relative to Earth’s
gravitational vector give rise to what are known as gravity-driven
flows. Macro-scale horizontal flow condensers often (unless the
flow rate is sufficiently fast [6]) operate in gravity-assisted mode
because azimuthal direction condensate motion is significantly
gravity driven in the annular/stratified mode. Gravity-driven or
gravity-assisted annular/stratified flows are generally more
stable. Furthermore available empirical/computational results
for gravity driven/assisted flow condensers are well known and
reasonably reliable (see [6, 7]).
The absence of flow direction gravity component in the
shear driven condensers causes problematic non-annular flow
regimes – which are thermally and hydro-dynamically
inefficient – to appear and cover a significant portion of the
condenser length (particularly if complete condensation is a
Review Article – JTEN – 2014 – 80
308
design requirement). These inefficient non-annular flow regimes
are commonly categorized as plug flow, slug flow, and bubbly
flow ([8-13]). Since analogous flow regimes are observed in
adiabatic gas-liquid flows inside a duct, flow-regime map results
([6, 8-11]), heat-transfer coefficient (HTC) correlation results
([12]), and pressure-drop results ([13]) for phase-change flows
largely employ terminologies (such as quality X, the ratio of
vapor or gas-phase mass flow rate to total mass flow rate) that
are prevalent and natural to large diameter in-tube adiabatic air-
water type gas-liquid flow studies ([8, 9]).
A look at internal steady condensing and boiling flows
(see [3-7], and Figs. 1-3) quickly establishes the fact that quality
X strongly depends on physical distance from the inlet and
thermal boundary conditions (i.e. the method of cooling/heating
for condensers/boilers). Though there exist procedures ([14]) to
relate existing flow-regime maps ([6, 8-11]), heat-transfer
coefficient (HTC) correlations ([12]), and pressure-drop results
([13]) for a phase-change flow to a device’s downstream
physical distance and thermal boundary conditions; the
descriptions are indirect and masked by prevalence of adiabatic
two-phase flow terminologies that raise doubt on the validity of
the heat-transfer correlations or introduce opacity in the direct
use of flow regime maps. Furthermore many flow regime maps
are often not properly non-dimensional or suitably differentiated
by flow-physics – making it difficult to use them for tube-
diameters and/or fluids different than the ones used in the
underlying experimental data (even if their repeatability and
reliability is assumed).
For mm-scale shear driven condensers, this paper
addresses the need for proper flow-physics based new HTC
correlations, annular to non-annular flow regime transition
maps, and the need to relate them to existing correlation and
criteria (particularly the more recent ones supported by large data
sets). The approach’s relationship to direct flow-physics based
fundamental results from theory, computations, and experiments
are described.
The flow morphologies or liquid-vapor configurations
play a critical role in the determination of heat and mass transfer
rates associated with internal condensing flows. The goal of a
high performance condenser is to lengthen its annular/stratified
flow regime (e.g., through introduction and control of
recirculating vapor flow rates as shown in [3]) and to operate
entirely in this regime. Hence it is important to predict and avoid
the conditions that are associated with transition from annular to
non-annular regimes. Several flow regime maps are available in
literature ([8-11], etc.) that can provide estimates with regard to
which flow regime is likely to occur for different device
operation conditions – but this is only possible if they are
properly non-dimensionalized, categorized, and extended (also
see [15, 16]) to make them applicable to different fluids,
diameters, cooling conditions, etc. Specifically, with regard to
the length x
A
of the annular zone in Fig.3, fundamental stability-
theory based estimates are obtained from direct numerical
simulations ([16-18]) and are used here to supplement existing
empirical knowledge ([12, 13]) to present more confident
estimates (both for horizontal earth-based and zero-gravity shear
driven annular condensing flows).
Once the conditions favorable for realizing
annular/stratified flows are identified and implemented ([3]),
estimates for the heat-transfer coefficient (HTC) and pressure-
drops is required for designing and sizing condensers for the
operating heat loads. There are several semi-empirical
correlations available in the literature ([12, 19-21]) that present
equations to calculate the heat-transfer coefficient. Most of these
studies focus on providing HTC estimates for the
annular/stratified regime since that is the primary flow regime of
interest. A key objective of this paper is to provide a structure
and approach for examining the validity of existing semi-
empirical heat-transfer correlations and, also, for future
development of such correlations (including other regimes of
condensing flows as well as boiling flows). An assessment of the
structure of what non-dimensional parameters (or equivalent sets
of numbers) should affect these HTC correlations is provided.
The results/correlations from the studies available in literature
([12, 19-21]) are compared with results/correlations that are
obtained and proposed in this study. This is done by using a
combination of experimental data ([3, 4]), an approximate 1-D
condensing flow simulation tool ([7]), and a nearly exact
scientific 2-D steady/unsteady simulation tool ([16-18]). The
mathematical models and the two simulation tools used in aiding
this development are described in separate papers ([22, 23]).
The effect of the presence of wave-structures on the
proposed HTC correlations, the factors causing the wave-
structures, and the methodology to obtain empirical correction
factors to incorporate these effects are also discussed.
2. PHYSICS BASED UNDERSTANDING AND MODEL
STRUCTURES FOR HEAT TRANSFER COEFFICIENTS
AND FLOW REGIME MAPS
This study focuses primarily on shear-driven annular
flow condensation on: the bottom wall of horizontal channels,
any one of the walls of a channel in zero-gravity, or on the inner
wall of a circular-tube with the flow being shear driven and
annular (as in mm-scale, moderate to large mass-flux, and
negligible gravity terrestrial cases or in zero-gravity). The
definition of annular and other condensing flow regimes
(plug/slug, dispersed, bubbly, etc.) are available in [7-9] etc. A
representative annular or stratified internal condensing two-
dimensional flow in a channel (which is approximately the same
as bottom wall condensation in rectangular cross-section ducts
of small height ‘h’ to width ‘w’ ratios) is shown in Fig.1. The
condensation is on the bottom wall which is kept at a temperature
lower than the saturation temperature (i.e., allowing for
flow direction variations in wall temperature,
where ). Film-wise condensation
is realized because the surface is assumed to be wetting and
is assumed to be sufficiently large at all x locations (say
> 2-3
o
C).
In Fig.1, there is no condensation on the upper plate
because its temperature
is kept above (say by 5
o
C) the
saturation temperature
. If all the conditions for the flow
in Fig.1 are retained except that the upper-surface temperature
is changed to be equal to the lower surface temperature, then
Review Article – JTEN – 2014 – 80
309
there will be condensation on the upper plate as well - and the
condensate motion may also be influenced by the presence of
transverse-gravity. For inlet vapor velocities
below a certain
critical speed
, the presence of transverse-gravity does not
allow film-wise condensation on the upper surface of the
condenser and liquid will drip down (compare Fig. 1 and Fig. 2A
with its channel height satisfying: h
II
= h). Therefore, if flow in
Fig. 1 corresponding to the flow in Fig. 2A is realized
for
, the flow is said to be “stratified” as transverse
gravity plays a role in stabilizing the condensate motion (just as
it plays a role in destabilizing the upper condensate motion in
Fig. 2A) . However, for inlet vapor velocity
above a certain
critical speed
with
, separated vapor-liquid
flow (i.e. film condensation on both the upper and lower
surfaces) is known to prevail for a substantial length of the
condenser (see Fig. 2B. Therefore, if flow in Fig.1 corresponding
to the flow in Fig. 2A is realized for
, the flow is said
to be “annular” or “wavy annular” as interfacial shear and not
transverse gravity plays the more significant role in stabilizing
the condensate motion. The inlet speed range
is said to correspond to the “transition” zone and is
sometimes marked as shown in the Baker map ([9]) for larger
diameter tubes and discussed later on in this paper for mm-scale
shear driven condensers. For purposes of this paper, we term the
flow regime in Fig. 1 “annular/stratified” and the term means to
include all the aforementioned regimes termed “stratified,”
“annular,” and “transition.”
FIGURE 1: SCHEMATIC OF AN ANNULAR/STRATIFIED
INTERNAL CONDENSING FLOW INSIDE A HORIZONTAL
RECTANGULAR CROSS-SECTION CHANNEL.
For millimeter-scale condensers of primary interest, if
the length of the channel is sufficiently long (see Fig. 3), then
the annular/stratified condensing flow regime typically
transitions (see [12] and [3]) to plug/slug regimes at downstream
distances x >
. As in Fig. 3, the end of the annular regime will
be modeled by a sharp location x ≈
with the understanding
that, in reality, there is a transition zone between the annular and
the plug/slug regimes.
(A)
(B)
FIGURE 2: SCHEMATIC OF A SYMMETRICALLY COOLED
HORIZONTAL CHANNEL WITH CONDENSATION
ON BOTH
UPPER AND LOWER SURFACES SHOWING:(A) FOR
, A SEPARATED ANNULAR/STRATIFIED FLOW IS NOT
POSSIBLE; (B) FOR
, BUT LESS THAN SOME
OTHER CRITICAL SPEED [9], A SEPARATED FLOW
(TERMED ANNULAR OR WAVY ANNULAR FLOW) IS
POSSIBLE.
q”w(x)
Gap
Height
Interfacial Wave Motion Non-Annular
Zone
Vapor Liquid
M
in
(A)
xA
Wavy Annular
Top
View
Plug/Slug and
Bubbly
(B)
xSteady film
FIGURE 3: (A) SCHEMATIC OF FLOW TRANSITION FROM
ANNULAR/STRATIFIED TO PLUG/SLUG FLOWS IN A MM-
SCALE AND SUFFICIENTLY LONG CHANNEL CONDENSER
WHOSE BOTTOM PLATE IS COOLED. (B) EXPERIMENTAL
PHOTOGRAPHS ([4]) OF A TYPICAL REALIZATION.
The flow variables and fluid properties for the liquid
and vapor phases of the condensing flows illustrated above are
denoted with the subscript (where or L is for the liquid
phase and or V is for the vapor phase). In the description
of low Mach number flow condensation in this paper, the fluid
properties symbols for either of the two phases are: density,
Review Article – JTEN – 2014 – 80
310
dynamic viscosity, specific heat
and thermal
conductivity. Over the entire length of the condenser under
consideration, the above fluid properties are modeled as
approximate constants – though they take separate and different
values for each of the two phases (subscripted by ).
Inlet velocity is parallel to the condenser plates (x-axis in Fig. 1)
and its averaged magnitude at inlet is denoted by
, inlet
pressure by
, and inlet temperature by
even if it is
slightly superheated (this is because, as is well known in [22, 23],
theory and experiments establish that 2-10
o
C superheat in the
inlet vapor temperature lead to behavior very close to saturated
vapor flows). The temperature fields in the interior of each phase
are denoted by
, pressure fields by
, and the velocity fields
by
(where and are unit vectors along the x and
y axes shown in Fig.1). The film thickness is denoted by and
the local interfacial mass flux per unit area (in kg/m
2
/s) by
.
As noted earlier, the local average temperature at the x-location
of the bottom condensing plate is
,
where represents the local characteristic
temperature-difference for the x-location. Different spatially
varying steady wall temperature function
may arise from
different “methods of cooling” for the condensing surface ([3]).
The spatially averaged temperature
can be
used to define a mean temperature difference
. Under the conditions specified above, a particular “method
of cooling” is defined by a particular function
. For all cooling methods that lead to a uniform
condenser surface temperature
, i.e.
constant
over , the “method of cooling” is said to be the same
and is characterized by the constant function. From
here and henceforth, unless otherwise specified (as is typical of
most studies), only those “methods of cooling” classes will be
considered for which an assumption of approximate uniform
wall temperature, i.e.
or, is adequate as
far as development of the primary HTC correlations and flow-
regime maps are concerned.
Later on we discuss an approach by which significant
deviations from cases can still be adequately
addressed with regard to spatial variation effects on flow
variables (heat-flux or quality X) of interest. This is possible
because quality based engineering HTC correlations that are
developed (as discussed in section-3 of this paper) exhibit the
feature that they themselves are not significantly affected by
significant departures from the tentative modeling assumption of
.
When variations in local interfacial values of saturation
temperature
and heat of vaporization
are
allowed in computational simulations ([22, 23]) through
computed variations in local x-dependent interfacial pressure
, it is found that - for applications with large values of steady
inlet pressure
(typically in 50 – 1000 kPa range) - pressure
differences over the channel are sufficiently small relative to the
steady inlet pressure
. As a result, well-known
approximations
and
are found to be adequate approximations for typical fluids and
flows under consideration. For wavy interface annular/stratified
flows of interest, because of presence of liquid-vapor surface
tension σ (at the representative temperature
) in one of
the critical interface conditions ([22, 23]), surface tension
parameter σ is expected to be relevant – particularly in values of
variables at locations where steady interfacial curvatures are
large (e.g. σ is likely to be important ahead of or around the
location
in Fig. 3).
In Fig. 1, the length of the channel is denoted by L, the
channel height by h, and channel width by w (with h/w << 1).
The characteristic length h is used for non-dimensionalization of
flow variables. The hydraulic diameters (
, where
is the duct cross-section and
is the wetted perimeter) for
characteristic length are also used and are to be defined and used
differently for assessing heat-transfer rates through non-
dimensional HTC (or Nusselt number) correlations available in
the literature as well as for assessing vapor-phase turbulence
through critical vapor-phase Reynolds number definitions
available in the literature.
At any location x, the heat flow per unit area
is
determined by the flow geometry, the average inlet speed
(or
inlet mass flow rate), relevant fluid properties of each phase
(including saturation temperature and heat of vaporization), and
the cooling conditions (as defined by the characteristic
temperature difference and the actual values of).
Because heat flow rate from the nearly saturated vapor core to
the interface can be ignored in comparison to: (i) the latent heat
release rate at the interface, and (ii) the heat flow from the
interface to the condensing surface; the vapor’s thermal
conductivity and specific heat are not expected to be important
in the determination of wall heat-flux. Therefore, the defining
relationship between heat-flux
and local HTC
is
chosen to be:
(1)
where, at most, the following list of variables are expected to
affect the local HTC
:
(2)
Similarly, because the length
of the annular regime
is self-determined by the flow, it is expected that:
(3)
Furthermore, it is commonly and correctly assumed in
available empirical - and rather general (attempting to cover
different flow-physics) - correlations ([12, 13]) that the form of
the functions in Eqs. (1-2) depend on laminar or turbulent nature
of the vapor and the liquid flows. What is often missed (as in
[12]) is that, for mm-scale shear driven condensers, the shear
driven liquid condensate motion is thin, slow, and nearly always
laminar (discounting low amplitude interfacial turbulence
restricted to the random nature of interfacial waves – that are
typically found superposed on the steady laminar solutions). The
adjoining near-interface vapor flow (with its streamlines bending
Review Article – JTEN – 2014 – 80
311
and piercing the interface (as shown in [16-18, 22, 23]) with an
effective suction) is also laminar – even if the far-field vapor core
is turbulent.
Either application of the well-known Pi-theorem ([24])
or consideration of the non-dimensional differential forms of the
governing equations (see [16-18, 22, 23]) imply the following
non-dimensional version of the dependencies given in Eqs. (2-
3):
(4)
(5)
where
and
are unknown functions that are to be
experimentally and/or theoretically/computationally
determined. These functions may depend on: ,
,
,
,
,
,
, and
. Note
, and
are
respectively called local Nusselt number, inlet Reynolds
number, Jakob number, liquid Prandtl number, Suratman
number, and a non-dimensional transverse gravity number.
Furthermore, for most applications of interest,
condensate film thickness is found to be sufficiently small
relative to h and one can typically ignore convection in the liquid
film’s differential form of energy equation (as in [7]). As a result,
linear temperature variations in the condensate are known to be
a very good approximation - both for wavy quasi steady and
strictly steady flows ([22, 23]). This fact combines the
aforementioned (in Eqs. (4-5)) separate dependences on Jakob
and liquid Prandtl
numbers to a single number
. Also, as discussed in later sections, the transverse gravity
parameter
and surface-tension parameter , has a weak
impact on
(except at x ≈ x
A
) and also on factors determining
the lower bound of
(with impact being greater at x >
).
Therefore, for flows of interests here, Eqs. (4-5) can be
simplified to:
(6)
(7)
Furthermore, at any location of a quasi-steady flow
realization, it should be noted that mass balance for a liquid-
vapor control volume between = 0 and a location in Fig. 1
can be written as:
(8)
where
and
are respectively cross sectional mass
flow rates (kg/s) of the separated liquid/vapor flow. It is common
practice to define vapor quality as:
(9)
Thus, from mass balance Eq. (9), one obtains
(10)
Energy balance for the liquid-vapor control volume in
Fig. 1 (which is of width along the flow and depth across
the flow) is easily shown to yield (see [15]):
(11)
Using and other introduced non-dimensional
numbers, along with the assumption that the “method of cooling”
is modeled by, it is easily shown that the non-dimensional
version of Eq. (11) is:
(12)
where X(0) = 1 but, because of a singularity (infinite value) in
Nu
x
correlations at X ≈ 1, it is best to solve Eq. (12) with suitably
chosen ε ≈ 0 that require X(ε) = X*, where X* is close to 1 (say
0.99).
For the flow in Fig. 1, whether one uses experimental
or computational approach to obtain the correlation for
and
in Eqs. (6-7), it is clear that the chosen correlations should
be consistent with critical physics constraints - such as the
satisfaction of mass, momentum, and energy balance equations
(and, therefore, Eq. 12).
Substitution of the dependence in Eq. (6) in the energy-
balance Eq. (12) makes it an explicit linear ordinary differential
equation (ODE) which, when solved under the known condition
of and assumed condition, clearly
yields the expected dependence:
(13)
3. FORMS OF DEPENDENCES ASSUMED IN THE
LITERATURE FOR OBTAINING NON_DIMENSIONAL
HTC CORRELATIONS AND FLOW-REGIME
TRANSITION MAPS
The fundamental non-dimensional forms of
correlations for HTC
applicable to annular/stratified shear
driven condensing flows, as given by Eq. (4) or Eq. (6), is
proposed in the literature (e.g. see relevant correlations of Shah
[20], Kim and Mudawar [12], Wang [21], Dobson and Chato
[[19]], etc.) in forms that are equivalent to:
(14)
where and function
is implicitly defined by the
available correlations ([12, 19-21]) with hydraulic diameter
replacing h in the earlier definitions for
and . Note that
the tube-diameter D equals
for shear-driven in-tube
condensation.
Review Article – JTEN – 2014 – 80
312
The
definition, as far as heat transfer rates are
concerned, is
for the single-sided condensing flow in
Fig. 1, and
for the double-sided condensing flow case
in Fig. 2B. With all other flow and fluid parameters kept identical
between the two channel flows in Fig. 1 and Fig. 2, use of
in Fig. 2B doubles the inlet mass flow rate with respect to the
flow in Fig.1 (as
remains the same for the two cases) and
makes the condensing flow in each of its symmetric halves
approximately the same as in Fig. 1. This leads to the expectation
that, under these conditions, at the same physical distance x, the
local heat-flux
, and the local HTC
are going to be the
same and, therefore, the correlations for the two flow situations
should satisfy (and relate to the definition in Eq. (14)) through
the relationship:
(15)
If curvature effects could be ignored (which is
questionable for mm-scale condensers), an in-tube (inner
diameter D) shear driven condensing flow would be equivalent
to the flow in Fig. 2B (with width w = πD/2) and one would
expect (in addition to Eq. (15)):
(16)
Continuing to use while repeating the
arguments for Eq. (12) - with the help of
and
definitions in Eq. (14) instead of the
and
definition
in Eq. (4), one obtains:
(17)
Using vast quantities of existing experimental data for
mm-scale tubes and square/rectangular ducts ([12]), simulation
capabilities for channels ([16-18, 22, 23]), and experimental data
([3, 4]) for channels (i.e. small aspect ratio rectangles with h/w
<< 1), this paper intends to propose HTC correlations and test
the equivalence efficacy of Eqs. (15-17).
Substitution of Eq. (14) in the one-dimensional energy
balance Eq. (17) makes it an implicit non-linear ordinary
differential equation in - which is solvable (despite the
singularity at ) and yields an explicit dependence of quality
on non-dimensional downstream distance . It is again clear
that the solution gives the quality X in a dependence whose form
is consistent with the one in Eq. (13). Clearly any substitution of
the resulting correlation for quality (obtained/developed by
experimental and/or theoretical/computational approach) in the
original Nusselt number dependence of Eq. (14) implies that the
resulting argument list is equivalent to the original fundamental
argument list in Eq. (4). However, this implication is not the
same as the one in which one obtains an explicit fundamental
Nusselt-number and quality correlations in their respective
fundamental forms (Eq. (4) and Eq. (13)) and then shows that
these can be combined to obtain the popular/conventional form
exemplified by Eq. (14). An argument implying this is
approximately true is given in section-5 of this paper – and
although this justifies the literature’s approach embodied in Eq.
(14), better and more accurate correlation approaches may retain
Ja/Pr
1
dependence.
Additionally, semi-empirical flow regime maps are
available in the literature. They are either used indirectly for
condensing flows as they are originally based on flow regimes
observed inside large diameter tubes with air-water type
shear/pressure driven adiabatic flows (Baker [9], Taitel and
Dukler [8], Coleman and Garimella [10], etc.) or they are directly
developed/used on the basis of consideration of shear/pressure
driven condensing flows (e.g., for millimeter scale channels, see
transition criteria of Kim and Mudawar [12], etc.). The one in
[12] and our own theoretical/computational results in [16-18]
hold the promise of identifying the annular/stratified to plug/slug
transition of interest here by developing explicit dependence in
the form given by Eq. (7). To show that this promise can be
realized, it is to be noted that semi empirical flow regime maps
are typically obtained from a hypothesis that is tacitly assumed
for condensing flows because, perhaps, the hypothesis is rather
obvious for air-water flows. The hypothesis states that all the
flow-regime transitions (annular to slug/plug, plug/slug to
bubbly, annular to dispersed-liquid, etc.) may be marked by
various curves in the parameter space
of
. These various curves can be
denoted as
(say for annular/stratified to plug/slug,
for wavy annular to dispersed-liquid, and for plug/slug
to bubbly, etc.) and can be obtained as curves in the above
identified parameter space. Such transition-marking curves can
be expressed as a single curve or a combination of single curves
in the form:
(18)
where the subscript in
takes the values of 1,2,3, etc. The
curve
which marks the transition from annular/stratified to
plug/slug is of primary interest here.
3.1 Examples of Semi-Empirical HTC
Correlations
and their Approximate Equivalence to the Representation
in Eq. (14)
For shear driven condensing flows (with h as in Fig.1
and
), there are well known correlations in [12, 19-21],
etc. These correlations are available in the literature and are well
summarized and assessed in a recent article of Kim and
Mudawar ([12]). For reasons of brevity, only
correlation
from Kim and Mudawar ([12]) is primarily considered in this
paper and is shown to be approximately equivalent to the
standardized structure of Eq. (14). A cursory look at these
correlations may appear to suggest that they use different non
dimensional parameters such as: Lockhardt Martinelli Parameter
that explicitly depends on quality X, a two-phase multiplier
Review Article – JTEN – 2014 – 80
313
representing a certain ratio of well-defined pressure
gradients, etc. (see Tables 2 and 4 in [12]). This paper intends to
show that if one substitutes the acceptable correlations in one-
dimensional energy balance (Eq. (17)) and one solves the
resulting ODE (subject to X(ε) = X*); one can computationally
obtain explicit form of and
– and such dependences
are effectively characterized by the arguments list in Eq. (6), and
Eq. (13).
The HTC correlation recently proposed by Kim and
Mudawar ([12]) is:
(19)
where
and
are defined in Table 4 of [12]. Clearly, this
correlation does not explicitly show any dependence on the
thermal boundary condition as uniform surface temperature class
of is considered adequate though, if needed,
may still be used in Eq. (17).
For a representative condensing flow of FC-72 vapor,
use of FC-72 thermodynamic properties at p
in
= 100 kPa allows
one to specify the raw argument list through
(1 m/s, 0.002 m,
1603 kg/
, 13.06 kg/
, 0.000494 kg/m-s, 1.16e-5 kg/m-s,
1.142 kJ/kg-K, 0.07041 W/m-K, 83.54 kJ/kg, 15 °C, 0.008329
N/m.) to evaluate Eq. (19) and substitute it in the ODE Eq.(17).
One then obtains the ODE’s numerical solution on MATLAB.
This specific flow situation is equivalently characterized by the
non-dimensional parameters list/
= (9006, 0.0307, 0.008,
0.0234, 6.44e6). For this correlation and flow situation, the
- X variation and are respectively shown in Figs.
4A and 4B (for ).
Next one artificially changes fluid properties and FC-
72 condensing flow situation from their representative values of
to a new raw argument list given by
= (0.833 m/s, 0.0024 m,
1923.6 kg/
, 15.672 kg/
, 5.928e-4 kg/m-s, 1.39e-5 kg/m-s,
1.142 kJ/kg-K, 0.08449 W/m-K, 83.54 kJ/kg , 15 °C, 0.008329
N/m) but the values of non-dimensional parameters:
= (9006, 0.0307, 0.008,
0.0234, 6.44e6) are kept the same. The resulting
curve
for this second situation, as shown in Figs. 4A-B, remains
approximately the same as the first situation. This verifies that
Kim and Mudawar correlation (and perhaps some other “good”
correlations) has effectively the same dependence as the
fundamental one described in Eq. (14).
Despite the above feature, a fundamental limitation of
such empirical correlations are that they deal directly with the
raw arguments list/ rather than the one associated with the
non-dimensional parameters list/ . In an attempt to cover
a range of non-dimensional values towards overcoming this
limitation, one typically considers/specifies a set of working
fluids, hydraulic diameters, and operating conditions to first
define the range of values of interest for the argument list/.
For example, this paper chooses/defines its argument list/
through the choices given in Table-1.
(A)
(B)
FIGURE 4: FOR THE REPRESENTATIVE CONDENSING
FLOW OF FC-72 SPECIFIED BY SET “NA,” THE FIGURES
SHOW: (A) NUSSELT NUMBER
VERSUS QUALITY ,
and (B) QUALITY VERSUS NON-DIMENSIONAL
DISTANCE VARIATIONS.
As a result of the above choice for the range of flows,
one can typically obtain heat transfer coefficient
data
(experimentally or computationally) only at a set of discrete
points in the
space of Eq. (4)
or
space of Eq. (14). Such discrete
point-wise raw variable data, when combined with experimental
challenges in accessing the desired range, often lead to discrete
point-wise data of non-dimensional values that are not conducive
to developing reasonably accurate empirical correlations. For
example the choice in Table-1 has limited experimental
accessibility in the associated
space described in Table-
2.
0.10.20.30.40.50.60.70.80.9
10
20
30
40
50
Quality, X
Non-dimensional HTC, Nu
x
Argument list - set A
Argument list - set A*
A
A*
0 20 40 60 80 100
0.86
0.88
0.9
0.92
0.94
0.96
0.98
Non-dimensional distance,
Quality, X
Argument list - set A
Argument list - set A*
A
A*
^
x
Review Article – JTEN – 2014 – 80
314
TABLE-1: RANGE OF FLUIDS AND FLOW CONDITIONS CONSIDERED FOR FLOW IN FIG. 1
Working fluids FC-72 R113 R113 R134a
Inlet Pressure (kPa) 100 25 225 150
Saturation temperature (
) 55.94 11.1 73.86 -17.15
Hydraulic Diameter (h = 0.002 m) 2h 2h 2h 2h
Transverse Gravity 0 ≤ |
| ≤ g 0 ≤ |
| ≤ g 0 ≤ |
| ≤ g 0 ≤ |
| ≤ g
Inlet Vapor Speed (U) (m/s) 0.35 – 9.79 2.07 – 57.16 0.35 – 9.79 0.54 – 14.9
(
) 2.93 – 12.30 8.69 – 36.50 2.25 – 9.45 3.45 – 14.45
(A)
(B)
(C)
FIGURE 5: FOR CORRELATIONS DEVELOPED IN THIS
PAPER, THE
SPACE
THAT IS COVERED IS VISUALIZED THROUGH: (A) A
THREE-DIMENSIONAL
SPACE, (B) A
THREE-DIMENSIONAL
SPACE, AND (C) A
TWO-DIMENSIONAL
SPACE.
In this paper, by combining experimental and
computational approaches, the range in Table-2 is more
judiciously covered for any chosen boxed zone (dashed lines in
Fig. 5A-C) – e.g. including its boundaries and interior (center,
diagonal, etc.) - to develop meaningful correlations for a chosen
range.
TABLE-2: RANGE OF NON-DIMENSIONAL PARAMETERS
ASSOCIATED WITH TABLE-1
800 ≤
≤ 23000
0.005 ≤
≤ 0.021
0.0013 ≤
≤ 0.011
0.012 ≤
≤ 0.0343
6.63 *
≤
≤ 2.55*
≤
≤ 1.33*
Though the boxed zone can be extended or several such
boxed zones (e.g. see water condensation cases in [15]) may be
considered, this paper’s focus is for the well-defined range in
Table-2.
3.2 Examples of Semi-Empirical Flow-Regime Maps and
their Typical Representation
Shear driven annular condensing flows’ experimental
realization at low to moderate inlet Reynolds number
(or
inlet mass-flux, as in [12]) typically show (see Carey [6],
Kivisalu et al. [3], Kim and Mudawar [12]) transitions indicated
in Fig. 3 i.e. annular/stratified to plug/slug to bubbly. Similar
transitions are indicated in large diameter adiabatic air/oil flow
experiments of Baker ([9]) whose flow-regime map, though not
properly non-dimensional, is typically considered along with
other popular adiabatic flow-regime maps (Taitel and Dukler [8],
etc.) - even for condensing flows ([6]). This is based on the
assumption that the instabilities associated with the transition
curves in these maps are mechanical in nature - i.e. only mass
and momentum balances are involved once energy Eq. (12)
yields the quality. Furthermore, maps or transition criteria
specific to condensing flows ([12], etc.) exist. It is found (in
[15]) that Taitel and Dukler ([8]) map, though properly non-
dimensional, does not relate to mm-scale condensing flow
transitions observed in [3] and [12] – perhaps as a result of very
large diameter tubes and different orientations considered in the
creation of the map. Similarly, in [16], we find that, after proper
non-dimensionalization, Baker-map ([9]) does yield
qualitatively similar transitions as those observed in mm-scale
Review Article – JTEN – 2014 – 80
315
shear driven condensing flows (as in Fig. 3). Despite this,
because of the larger diameter tubes (several cms) involved, the
trends of the transition curves are quite different than those
directly obtained for mm-scale condensers – either in
experiments ([3]) or through theory and computations (results
discussed later on in this paper on the basis of our CFD-based
stability results of [16-18]). For the above reasons, in this
section, we primarily discuss annular/stratified to plug/slug
empirical transition criteria of Kim and Mudawar ([12]).
3.2.1 Kim and Mudawar Criteria for Annular to Plug/Slug
Transition
Using FC-72 condensing flow data in horizontal mm-
scale tubes and square/rectangular cross-section ducts, Kim and
Mudawar ([12]) provide an annular to plug/slug transition
criteria and correlation. Their criteria of We
*
=7X
tt
0.2
at transition,
with definitions of We
*
and X
tt
, is more fully defined in [12].
This annular (We
*
7X
tt
0.2
) to non-annular plug/slug
(We
*
7X
tt
0.2
) transition criteria is used for the specific cases of
pure FC-72 condensation (D
H
= 8 mm and p
in
= 100 kPa)
considered in Fig. 6. The results (for a relevant range of Re
in
|
Dh
)
for fixed values of
, are depicted in the
Re
in
|
Dh
- X plane of Fig. 6.
FIGURE 6: THE FIGURE ABOVE PRESENTS A SPECIFIC
USE OF THE KIM AND MUDAWAR [12] CRITERIA FOR
ANNULAR TO PLUG/SLUG TRANSITIONS FOR
CONDENSATION OF PURE FC-72 (D
h
= 8 mm AND p
in
= 100
kPa) AND PRESENTS THE TRANSITION CRITERIA IN THE
PLANE FOR FIXED VALUES OF
(
) = (
0.0307, 0.008, 0.0234,
6.44e6, 6.36e6
).
Again as in Fig. 4, when one artificially changes fluid
properties for a representative FC-72 condensing flow situation
from raw argument set A to set A* in a way that the non-
dimensional parameters set NA remains the same, the resulting
curve for this second situation remains approximately
the same as before. This fact is shown in Fig. 6. This verifies that
Kim and Mudawar transition criteria ([12]) has effectively the
same dependence as the fundamental one described in Eq. (4).
However, this does not mean that the criteria has as much
experimental support as their
correlations. This is
because experimental data for x
A
is limited (i.e., not measured
for all the experimental runs cited in [12]).
4. COMPUTATIONAL APPROACH FOR ANNULAR
SHEAR DRIVEN CONDENSING FLOW REGIME
Steady and unsteady solutions are computationally
obtained and correlated to propose non-dimensional HTC
and length
estimates through their dependences ascertained in
the format of Eqs. (4-5). The computational approach currently
limits itself to estimates of
applicable to steady and wave-
free cases (i.e. quasi-steady and low-amplitude waves) – though
an empirical correction approach for wave effects is described.
In section 5, it is shown that the proposed correlations in the
format of Eqs. (4-5) can also be re-cast in the frequently used
correlation formats given in Eqs. (14-15).
The governing equations for the computational
approach is described in ([16-18, 22, 23]) whereas the more
recent and successful steady/unsteady simulation algorithm are
described in ([16-18]).
The simulation capability uses an approach of
separately solving, on COMSOL, the unsteady liquid and vapor
domain governing equations over their respective domains that
result from an assumed "sharp" interface location. Accurately
locating the interface at time “t,” the approach tentatively treats
the two domains to be “fixed” over a short time interval of time
[t, t+∆t]. The interface location for time “t+∆t” is then iteratively
relocated by tracking the interface with the help of numerical
solution of its evolution equation (which is obtained from one of
the interface conditions [16-18, 22]) on MATLAB. This
procedure employs a moving grid that takes into account the
nature of wave propagation on the interface. Concurrently, the
approach solves the unsteady liquid and vapor domain governing
equations while suitable values of interfacial variables (speeds,
stresses, temperatures, etc.) are imposed at the interface as
boundary conditions. Later on, these assumed interfacial values
of the flow variables are iteratively changed in the computations
until the convergence towards satisfying all the remaining
unsteady and well known interface conditions [16-18].
COMSOL’s fluid flow and heat transfer modules are used for
separately solving the liquid and vapor domain governing
equations in the interior while the two solutions’ are made to talk
to one another, along with judicious but concurrent interface re-
locations, with the help of algorithms (see [16-18]) implemented
on MATLAB (that links with COMSOL). Interface evolution
equation is a wave equation which is solved (with the help of the
well-defined characteristics equation underlying this problem)
with 4th order accuracy in time. This kind of unique approach
ensures accurate prediction of interface location and interface
variables towards accurate satisfaction of all the time-varying
interface conditions.
The approach and solutions in [16-18] are aided by
improvements in implementation schemes relative to the ones
described in our earlier papers ([22, 23]). As far as obtaining and
correlating solutions for h
x
and x
A
are concerned, the signatures
of instability that determines x
A
in Fig. 3 is first identified
through unsteady simulations and next related to its signature in
the energy transfer mechanisms and flow variables associated
00.10.20.30.40.50.60.70.80.91
0
1
2
3
4
5x 10
5
Quality, X
Inlet Reynolds number, Re
in
Argument list - set A
Argument list - set A*
A
A*
Review Article – JTEN – 2014 – 80
316
with the steady solutions. The h
x
, x
A
, etc. values obtained from
the 2-D solution method and aided by a much faster 1-D solution
technique described in [7], are correlated over the parameter
space described in Table-2 and Fig. 5.
Consistency of the steady solutions obtained from the
two independent approaches (2-D solver and 1-D solver) are
shown in [7] and their consistency with gravity driven and shear
driven flows’ experimental results are respectively shown in [7]
and [16, 17].
It suffices here to note that all the physical variables (in
SI units unless mentioned otherwise) mentioned in the earlier
paragraph can be non-dimensionalized. Furthermore non-
dimensional form of the governing equations are considered by
introducing the following non-dimensional variables ([16-18]):
(20)
In Eq. (20) above
and
are the non-
dimensional forms of
. The governing
equations and solutions for condensation of saturated vapor for
the flow in Fig.1 are reconsidered after their non-
dimensionalization through Eq. (20). The steady solutions
indicate that thin liquid condensate flows under negligible
effects of the convection term in the differential form of the
condensate energy equation. Under this approximation the non-
dimensional governing equations ([16-18, 22, 23]) clearly
indicate the significance of the following non-dimensional
parameters (some have been defined earlier):
(21)
For the “black-box” empirical approach recommended
for developing correlations within structures proposed in Eqs.
(4-5), or Eq. (14) and Eq. (18), it is convenient to restrict inertia
effects associated with the inlet mass flow rate to the inlet
Reynolds number
. It was for this reason Suratman number
and transverse gravity number
were introduced in the earlier section. Though
the parameters in Eq. (21) more closely control the actual flow
physics, the observation that
and
allows one to see the equivalence between the parameter set
in Eq. (21) with the one used earlier and given below:
(22)
The results given here and elsewhere [15-16] indicate
insignificant impact of transverse gravity (through
) on
but somewhat more significant impact on
. These
investigations [6, 16-18, 22-23] also establish that, for mm-scale
condensers, the approximate argument list in Eqs. (6-7) is well
supported by rigorous steady and unsteady solutions for shear
driven annular/stratified flows.
5. RESULTS AND DISCUSSION
5.1 Computationally Obtained Values for
(or
),
Quality, and Length
along with their Correlations
and Assessments
Steady solutions of the full 2-D governing equations
were obtained in the parameter space defined by Table-2 and Fig.
5. It was found that the presence or absence of transverse gravity
has negligible effects on steady solutions and associated heat-
transfer rates (i.e.
). As discussed in
[16-18], unsteady solutions based on onset of instability analyses
yield the length of the annular zone
which is
somewhat affected by the presence or absence of transverse
gravity. Furthermore, the signature of the onset of instability
(associated with annular to plug/slug transitions) can be
approximately identified/correlated ([16-18]) with the steady
solutions’ features dealing with spatial variations in mechanical
energy transfer mechanisms and characteristic speeds/wave
speeds.
Using natural log of the computed values of variables
(both by 2-D and 1-D solvers) on either side of Eqs. (6-7) and
employing linear regression with least square fits ([25]), the
following correlations (with 5.7% average error for Nu
x
) are
obtained:
(23)
Based on the steady and unsteady analyses ([16-18]),
the length of annular regime in absence of transverse gravity is
identified by
and in the presence of transverse gravity is
identified by
. Through the analyses, an approximate value
for the difference between the two lengths is correlated with
measured features of the steady solution distances given by
and
. The relationship among
,
and
is
given in Eqs. (24-25). The actual values of
or
, though
somewhat uncertain because of the inherent behavior of a
transition zone, are identified by a careful analysis of the
unsteady CFD simulation tool results and is reported in [16-18].
A good estimate is given by:
(24)
and
(25)
It is typically found that in Eq. (24), i.e.
and
in Eq. (25), i.e.
.
Correlations for
(with 6.433 % average error) and
(with 5.791 % average error) are presented below through
Eqs. (26-27).
(26)
Review Article – JTEN – 2014 – 80
317
(27)
Recall that the thin condensate film flow
approximations (see [22, 23]) yield:
(28)
(29)
Using Eqs. (28-29), it is easy to show that the reported
correlations in Eqs. (23-25) are consistent with the following
correlations for the non-dimensional film thickness
and quality:
(30)
(31)
Substitution of
, with
coming from
the appropriate correlation in Eq. (26), into Eq. (29) above yields
one of the relationships described in Eq. (18). For example, one
gets:
(32)
The dependence on
term in Eq. (32) is weak –
because its value ranges from 0.46 to 0.60 (see
range
inTable-2) - and one can approximately replace it by the form
used in the literature and described in Eq. (18). This more
approximate form is:
(33)
For
correlations similar to the ones in Eqs. (32-
33) can be obtained and reported – and this is done elsewhere
([16, 17]). Using
values for identifying the annular to
plug/slug transition curve for FC-72 condensation at
,
,
and
or
, transition curves are plotted in the
plane of Fig. 7. Thus, in Fig. 7, one obtains another
estimate (this time by a theoretical/computational approach) -
besides those obtained from the empirical criteria of Kim and
Mudawar ([12]). The differences between
theoretical/computational and Kim and Mudawar are discussed
in [15-17]. It suffices to note that: (i) the differences are not
entirely accounted by differences in curvature effects (with
aspect ratio AR ≡ h/w, AR ≈ 0 and AR ≈ 1), and (ii) our
theoretical/computational estimates are supported by our AR ≈ 0
experimental results of [3-4].
FIGURE 7: THE FIGURE PRESENTS CFD-PREDICTED AND
KIM AND MUDAWR [12] CRITERIA PREDICTED ANNULAR
TO PLUG/SLUG TRANSITIONS CURVES FOR
CONDENSATION OF PURE FC-72 (h = 2 mm AND p
in
= 100
kPa ). THE CFD TRANSITION CRITERIA IS IN THE
PLANE AND KIM AND MUDAWR [12] PREDICTIONS ARE
IN
- X PLANE.
5.1.1 Theoretical support for quality-based correlations in
the form of Eqs. (14-15) and their assessments
Note that, using Eq. (29) to obtain an expression for
in terms of in the proposed HTC theoretical/computational
correlation given in Eq. (23), one obtains a quality based
correlation:
(34)
for with
in their ranges given in Table-2.
The dependence on
in Eq. (34) is somewhat
weak – because the
term contribution ranges from 0.24
to 0.4 (see
range in Table-2). The dependence on , and
was already assessed to be weak through direct inspection of
the steady solutions ([16-18]). Therefore one may approximately
replace Eq. (34) by the form used in the literature and described
in Eq. (14). This more approximate form is:
(35)
Also note that the reduced correlation in Eq. (35),
though more approximate than the one in Eq. (34), is of the
widely used ([12, 13]) form suggested through Eq. (14). The
above argument along with the earlier argument of implied
0.40.50.60.70.80.9
2000
4000
6000
8000
10000
Vapor quality, X
Inlet Reynolds number,
1/4 Re
Dh
/Re
in|h
Kim and Mudawar (AR ~ 1)
X
cr|0g
- CFD (AR ~ 0)
X
cr|1g
- CFD (AR ~ 0)
Review Article – JTEN – 2014 – 80
318
consistency with the fundamental structure proposed in Eq. (6),
establishes the approximate equivalence of the two
representations, for HTC correlations.
The results in Eqs. (34-35) are for rectangular channels
with aspect ratio AR ≡ h/w ≈ 0 whereas Kim and Mudawar
correlations are for circular and rectangular cross-sections with
AR = D/D
h
= w/h ≈ 1. Yet if D was large and the curvature effects
could be ignored, the channel correlations
and tube-
correlations - theoretical ([16-18]) or empirical (Kim and
Mudawar [12]) - would satisfy the approximate equalities in Eq.
(16). To clearly assess the effects of curvature or AR, a
theoretical/computational result is obtained for a shear driven in-
tube case with the help of 1-D/2-D CFD approach ([7, 16-18])
whose correlations are reported elsewhere.
To assess the adequacy of the representation in Eq. (34),
its result is compared, in Fig. 8 for an FC-72 condensing flow
situation, against theoretical/computational results and Kim and
Mudawar and others’ correlations ([12, 19-21]) applicable for
shear driven annular condensing flows. Figure 8 is plotted for a
specific case of flow condensation of pure FC-72 in a horizontal
channel (h =2 mm,
=100 kpa, and ∆T = 15 °C) or a tube with
D = 4h. The result in Fig. 8A clearly establishes the consistency
of AR = 0 channel flow computational predictions and the
significance of curvature in comparison to in-tube (AR=1)
computational results. It should be observed that channel results
are approximately 90% higher than Kim and Mudawar results
([12]) whereas in-tube results are approximately 30% higher
than the Kim and Mudawar empirical results. Since Kim and
Mudawar empirical results also cover square and rectangular
channel data, and given the ± 30% nature of their “curve-fit” with
a very large amount of data – the comparisons establish the
fidelity of both types of correlations (the computational ones as
well as the empirical ones). However our proposals (channel and
cylinder) are more reliable on the following counts: (i) our
channel flow CFD predictions are in agreement (reported in [16,
17]) with our own experimental results ([3, 4]) for small aspect
ratio rectangular cross-sections, and (ii) our flow-physics
assumptions of laminar condensate (despite interfacial waves)
and laminar vapor flow near interface is consistent with low
liquid Reynolds numbers (i.e. Re
L
(x) ≡
= Re
in
·(µ
2
/µ
1
)·(1-
X(x) < 800) and vapor-suction at the interface – and these
assumptions are superior to the turbulent liquid flow
assumptions of Dobson and Chato ([19]) used by Kim and
Mudawar to develop their “curve-fit” correlation. Fig. 8B shows
the ability to make reliable and consistent “quality X versus
distance ” predictions with the help of the models in Fig. 8A
and the energy equation in Eq. (12) or Eq. (17). Fig. 8C compares
our and Kim and Mudawar ([12]) predictions with predictions
from correlations of Dobson and Chato ([19]), Wang et al. ([21]),
and Shah ([20]). The results in Fig. 8C reinforce the assessment
that, for AR ≈ 1 mm-scale condensers, the Kim and Mudawar
curve-fit is more consistent and superior to other empirical
curve-fits. Because the in-tube theoretical/computational
correlation is close to the more recent correlation of Kim and
Mudawar and this work has extensive discussions associated
with their correlation in relation to others’, it is not necessary for
this paper to focus on merits/demerits of various other
correlations (including the ones in Fig. 8C). It should also be
noted that the much higher HTC and lower expected pressure
drop for channel over in-tube flows suggest that multiple channel
condensers with AR ~ 0 (as in plate and fin heat exchangers) are
superior to multiple millimeter scale rectangular/square channels
of AR ~ 1.
(A)
(B)
(C)
FIGURE 8: THE FIGURE PRESENTS: (A) Nu
X
VERSUS
QUALITY X CURVES OBTAINED FROM COMPUTATIONS-
BASED CORRELATIONS FOR CHANNELS AND
VERSUS X CURVES FOR TUBES [7] AND KIM AND
MUDAWAR CORRELATION [12]. (B) THE QUALITY X
VERSUS DISTANCE
CURVE IS OBTAINED FOR THE
MODELS CONSIDERED IN (A). (C) SAMPLE COMPARISONS
OF PROPOSED 2-D THEORY BASED CHANNEL
CORRELATIONS FOR
WITH
VALUES
OBTAINED FOR KIM AND MUDAWAR [12], AND OTHERS’
CORRELATIONS.
0.30.40.50.60.70.80.9
0
20
40
60
80
Quality, X
Non-dimensional HTC, Nux
1/4 Nux|Dh Kim and Mudawar
1/4 Nux|Dh 1-D Circular Tube
2D Channel Correlation
2D Channel CFD Data
0 20 40 60 80 100
0.4
0.5
0.6
0.7
0.8
0.9
1
Non-dimensional distance,
Quality, X
Kim and Mudawar
1D Circular Tube
2D Channel Correlation
2D Channel CFD Data
^
x
0.30.40.50.60.70.80.9
0
20
40
60
80
Quality, X
Non-dimensional HTC, Nux
Nux|h 2D Channel Correlation
1/4 Nux|Dh Kim and Mudawar
1/4 Nux|Dh Dobson and Chato
1/4 Nux|Dh Wang
1/4 Nux|Dh Shah
Review Article – JTEN – 2014 – 80
319
The importance of good correlations for non-
dimensional HTC and
also come from the fact that the next
generation of shear driven condensers ([3, 4]) are likely to
experience annular condensing flows over the entire length of
the device. This is because controlled recirculation of vapor in
the innovative devices (see Fig. 9) allow adjustment of vapor
flow rates (and thus within the device) in a way that the
device-length in Fig. 9 always satisfies
. That is, the
controlled rate of vapor recirculation allows one to operate the
shear driven condenser in the annular/stratified portions of the
flow regime map in Fig.7.
q”
w
(x)
Vapor
Recirculating Vapor
Inlet
x
L
FIGURE 9: THE FIGURE DEPICTS AN INNOVATIVE ALL
ANNULAR NON-PULSATILE CONDENSER OPERATION.
BOTH PULSATILE AND NON-PULSATILE OPERATIONS
ARE DESCRIBED IN [3].
5.1.2 Comments on Possible Uses of Other Non-
Dimensional Parameters
Besides the more physics-based non-dimensional
parameter set in Eq. (21) and its equivalent “typically followed”
empirical correlation structure in Eq. (22), other non-
dimensional parameter sets are also possible. Since the non-
dimensional governing equations hold for a large class of flows
and one is often interested in particular variables (such as Nusselt
number
, etc.) or particular limiting flow classes (thin, thick,
or wavy condensate in annular flows), it is conceivable that some
other non-dimensional numbers - obtainable from simple power-
law combinations of the ones in Eq. (21) – may work and better
elucidate the physics underlying a particular situation of interest.
5.2 Relationship between Proposed Popular Quality-
Based Maps and Maps in the Physical Parameter Space
As mentioned in section 2, a flow regime map in the
space depicts the transition between
the different flow regimes in a much more intuitive fashion than
the one in
space marked by transition
curves of the type given by Eq. (15). This is because, for a given
pure fluid, diameter
,
and inlet pressure
; a condenser sees
approximately fixed values of
– whereas the
non-dimensional downstream distance (), cooling condition
(
), and inlet mass flow rate (
) are the variables that
naturally control non-dimensional HTC values (
) and the
length of the annular regime (
) in experimental observations
([3, 4]) as well as in computational approaches ([16-18]).
However, empirical or theoretical flow regime maps (such as the
one in the
space in Fig. 7) can be converted to the
desired map in the three dimensional space of
for
the representative values of
,
,
and
. For example, to re-map the
non-dimensional transition criteria map in Fig. 6 with the help of
the
correlation in Eq. (19), the X() solution of the nonlinear
ODE energy equation given in Eq. (12) or Eq. (17) is obtained
for a range of
and
values. The 3-D map for Fig. 6, based
on Eq. (16), Eq. (17) and Eq. (19) is shown in Fig. 10. Such maps
are helpful in understanding the flow-physics and can always be
obtained by the above described procedure and can be presented
for our own more reliable ([15-18]) transition criteria in Fig. 7.
FIGURE 10: THE ABOVE IS A RE-PLOT OF FIG. 6 IN
SPACE FOR
,
,
AND
. THE ZONES ARE LABELED
AS STRATIFIED ANNULAR, WAVY ANNULAR, PLUG,
SLUG, ETC.
5.4 Role of Empirical Correction Factors and its
Relationship to Superposed Wave-Structures
It is expected that steady wave-free annular flow CFD
solutions and correlations based on them - such as the one in Eq.
(23) or in Eq. (35) – may underestimate the HTC value because
they ignore wave-effects. Typically 0-20% enhancements are
expected ([26]), depending on the amplitude of the superposed
wave-structures. Therefore, for specific condenser operations at
low to medium
, an empirical correction factor can be
used/developed to adjust the HTC
values over its wave-free
values through the relation:
(36)
For a particular shear-driven condenser operation, can
be adjusted to ensure that the experimental values of heat-flux
are closer than the one predicted by Eqs. (34)-(35). Though
is commonly expected, it helps to know the factors that
typically govern wave-structures and thus affect the value of .
Review Article – JTEN – 2014 – 80
320
5.4.1 Vapor Core Turbulence
The correlations presented here clearly cover parameter
space (see Table-2) where
and vapor core
away from the interface is turbulent while the shear driven flow
itself is annular/stratified. In most of the shear driven condenser
applications, the liquid condensate is thin and liquid film is
laminar in the interior – with
[26]. However the
interface locations exhibit laminar turbulence in the sense that
wave structures have a statistical non-deterministic nature in
some of their details. A significant zone of the vapor flow near
the interface of the thin condensate is expected to be laminar –
this is not only because of the continuity of tangential velocities
(which is valid for annular flow boiling and adiabatic two-phase
flows as well) but, also, because the normal component of
interfacial vapor velocity represents suction of the vapor into the
laminar condensate (with significantly reduced normal velocity
on the liquid side). Because, for 2D CFD, cell-by-cell velocity
determination near the interface is not affected by turbulence in
the far field, it correctly simulates the near interface laminar
vapor flow. This is the reason why heat-transfer correlations
based on laminar-laminar CFD theory work with regard to
predicting condensate motion (not far field vapor-core pressure-
drops) and heat-transfer rates even for
(see[16, 17]). The vapor core’s turbulence typically affects the
amplitude of the wave-structure for non-pulsatile cases but does
not significantly affect the length
of the annular regime. This
fact is discussed in [16, 17] where it is shown that destabilizing
energy contents on the laminar film (whether it is within
or
) are rather “sharp” and “peaky” at a particular de-
stabilizing spatial frequency identified as the most dangerous
wave-length whereas turbulence related energy content is
diffused and spread out over a range of length and time scales.
5.4.2 Role of Exit Conditions
In an actual shear driven condenser or experiments (see
[3-5]), a condenser’s exit conduction is more likely to be
something similar to the arrangement in Fig. 8 rather than the
one shown in Fig. 1. The speed with which interfacial waves
move forward, i.e. whether they speed up or slow down when
they reach the exit, depend on exit conditions (see [16, 17]) and
this affects the long time wave-structure and its energy-content
present on the mean-interface location. For typical instantaneous
film thickness values at wave-troughs (if they are greater than
100 μm), this wave structure dependence on exit condition in not
too large and simple modeling of, with in Eq.
(36) will suffice.
5.4.3 Pulsatile Flows’ Impact on Wave Structures
In the experimental results reported in ([3-5]), if the
inlet flow rate in Fig. 8 is made deliberately pulsatile (i.e. there
is significant externally imposed amplitude and frequency in the
inlet pressure/flow rate), there are large amplitude interfacial
waves. This leads to a situation where large amplitude waves are
caused by pulsatile vapor dynamics and, concurrently, wave-
troughs start “sticking” on a wetting heat-exchange surface (this
is observed particularly when the local film-thickness values
become less than). This “sticking” phenomenon leads
to very high heat-fluxes, (see [3]). For such pulsatile cases,
needs to be separately modeled - as is not typical
(instead is more common as shown in [3, 5]).
5.5 Pressure-Drop Correlations
For the mm-scale shear driven annular condensation
considered here, engineering literature also proposes several
pressure-drop correlation (see, e.g. Lockhardt-Martinelli [27]),
Friedel [28], etc.). Also there are flow-condensation specific
adaptations of pressure-drop correlations in Kim & Mudawar
([12]). Though experiments and theory based synthesis of
pressure drop estimates are also possible by an approach similar
to the one described here for HTC, such estimates are outside the
scope of this paper and are being reported elsewhere (see [15]).
6. CONCLUSIONS
i. For shear driven annular condensing flows, this paper
proposed a first-of-its-kind properly non dimensional heat
transfer co-efficient (HTC) correlations. Besides proposing
such a correlation based on a fundamental physics-based
and theoretical/computational approach, the paper also
provides/validates the key foundational ideas for
development and assessment of empirical HTC correlations.
ii. This paper proposed, for the first time, properly non-
dimensionalized correlation/map approach for
developing/validating criteria for transition from annular to
plug/slug flows for shear driven annular condensing flows.
Besides assessing existing correlations/maps, it proposes a
new criteria which is based on a theoretical/computational
instability theory (which is presented elsewhere).
iii. The importance of coupling quality-based correlation with a
one–dimensional energy equation solver - to go back and
forth between a reduced indirect parameter space
(employing quality) and the physical parameter space in
which the flows are observed (involving physical distance)
- has been clearly elucidated.
iv. The proposed correlations are based on steady laminar
liquid and laminar vapor CFD solutions for
annular/stratified flows. Despite far field (away from the
interface) vapor core turbulence, this flow-physics is
demonstrated to be effective in estimating heat-transfer
rates. Parameters that affect wave-structure and that need
development of a well-defined empirical correction factor
have been identified.
v. Because of significant new understanding of flow-physics
and CFD tool development (reported elsewhere), the
approach presented here provides the foundation for its
extension that will also allow development of correlations
for pulsatile flows – which are experimentally shown to be
promising for development of next generation high heat-
flux shear driven condensers.
ACKNOWLEDGMENTS
This work was supported by NSF-CBET-1402702 and NSF-
CBET-1033591 grants.
Review Article – JTEN – 2014 – 80
321
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