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Temperature and pressure variations within compressed air energy storage caverns

R. Kushnir, A. Dayan, A. Ullmann

⇑

School of Mechanical Engineering, Tel Aviv University, Tel Aviv 69978, Israel

article info

Article history:

Received 8 December 2011

Received in revised form 20 May 2012

Accepted 21 May 2012

Available online 10 July 2012

Keywords:

Compressed air energy storage (CAES)

Underground storage

Cavern reservoirs

Periodic heat transfer

abstract

In the present work, the thermodynamic response of underground cavern reservoirs to charge/discharge

cycles of compressed air energy storage (CAES) plants was studied. During a CAES plant operation, the

cyclical air injection and withdrawal produce temperature and pressure ﬂuctuations within the storage

cavern. Predictions of these ﬂuctuations are required for proper cavern design and for the selection of

appropriate turbo-machinery. Based on the mass and energy conservation equations, numerical and

approximate analytical solutions were derived for the air cavern temperature and pressure variations.

Sensitivity analyses were conducted to identify the dominant parameters that affect the storage temper-

ature and pressure ﬂuctuations and the required storage volume. The heat transfer at the cavern walls

was found to highly affect the air temperature and pressure variations as compared to adiabatic condi-

tions. In essence, heat transfer reduces the temperature and pressure ﬂuctuations during cavern charge

and discharge and effectively leads to a higher storage capacity. Additionally, for realistic conditions, in

each cycle, few percents of the injected energy are lost by conduction into the rocks. The principal ther-

mal property that governs the heat transfer process is the rock effusivity. To reduce the required storage

volume preference must be given to sites of rocks that have the largest thermal effusivity. Lower injected

air temperatures also reduce the required storage volume, but increase the cooling costs. The injected

temperature can also be used to control the cycle temperature extreme limits. It is evident from the

results that the storage pressure ratio has a dominant effect on the required storage volume and should

preferably range between 1.2 and 1.8.

Ó2012 Elsevier Ltd. All rights reserved.

1. Introduction

Compressed air energy storage (CAES) is a promising venue to

supply peaking power to electric utilities. A CAES plant provides

the advantage of compressing air during off peak hours to a rela-

tively inexpensive underground reservoir, at the low cost of excess

base-load electrical power. Later, during peak hours, the com-

pressed air is released, heated (ﬁred) and then driven to the gas

turbine expansion, which in turn run the electrical power genera-

tors. The technology has the potential of improving the power pro-

duction economics while reducing both, pollution emissions and

fossil fuel depletion. Three geological types of underground reser-

voirs are feasible for the compressed air storage: porous rock res-

ervoirs (such as depleted gas reservoirs or aquifers [1,2]), salt

caverns and hard rock caverns.

The present study addresses the air storage in cavern reservoirs.

During operation, the cavern air temperature and pressure ﬂuctu-

ate between maxima and minima values owing to the cyclical air

injections and withdrawals. Accurate predictions of the reservoir

air pressure and temperature ﬂuctuations are essential to deter-

mine the required storage volume, and to assure that the reservoir

will operate within safe pressure and temperature limits. Addition-

ally, the selection of the compression equipment is one that must

meet the maximal storage pressure, whereas the minimum storage

pressure essentially determines the turbine inlet pressure.

1.1. Operational data

To date, there are two operational CAES plants in the world: the

290 MW plant (later up-rated to 321 MW) at Huntorf, Germany,

built in 1978 [3], and the 110 MW plant in McIntosh, Alabama,

USA, commissioned in 1991 [4]. Both plants are using salt caverns

as their underground reservoir. The two plants provide valuable

data on the temperature and pressure variations of their caverns,

during injection and withdrawal actions. In the trial runs of the

Huntorf plant, extensive measurements of temperature and pres-

sure were carried out. Results of such daily measurements of the

temperature and pressure in the cavern and at the wellhead are

presented by Quast and Crotogino [5]. During the diurnal cycle

the cavern was charged and discharged several times a day. The

most important measurement ﬁnding implies that temperature

variations during injection and withdrawal were much smaller

0017-9310/$ - see front matter Ó2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05.055

⇑

Corresponding author. Tel.: +972 3 640 7829; fax: +972 3 640 7334.

E-mail address: ullmann@eng.tau.ac.il (A. Ullmann).

International Journal of Heat and Mass Transfer 55 (2012) 5616–5630

Contents lists available at SciVerse ScienceDirect

International Journal of Heat and Mass Transfer

journal homepage: www.elsevier.com/locate/ijhmt

than those predicted in the design phase. In fact, the temperature

ﬂuctuations were substantially smaller than those calculated by

adiabatic cavern assumption. Consequently, the plant storage

capacity is larger than anticipated.

Performance test results from the McIntosh plant are provided

by Nakhamkin et al. [6]. To verify the net energy storage available,

a complete cavern discharge test was performed. During the test

the plant delivered a power of 100 MW for 27 h and 46 min until

it reached the minimum design cavern pressure (exceeded the

guaranteed 26 h). After approximately three hours, the cavern

was recharged for 40 h and 50 min (to restore the air consumed

during the total discharge test). The temperatures and pressures

measured at the wellhead during the tests are shown in the paper.

It follows that the temperature transient assumptions of the design

phase were conservative. Thus, similar to the Huntorf plant, the

plant storage capacity is larger than anticipated.

1.2. Theoretical studies

Although, a considerable amount of studies on CAES exist, only

few consider the temperature and pressure aspects of CAES reser-

voirs. In this context, the temperature and pressure variations

within adiabatic caverns of CAES plants were studied by Kushnir

et al. [7]. Solutions for the air cavern temperature and pressure

variations were derived and applied to three different gas state

equations, namely ideal, real and a self-developed simpliﬁed gas

models. It is demonstrated that the air thermodynamic properties

can adequately be represented by a simpliﬁed real gas model. This

is in contrast to an ideal gas model that yields smaller pressure

ﬂuctuations and storage volume requirements. Nonetheless, for

practical conditions, the deviations from an ideal gas behavior

are conﬁned within few percent spans.

The study of adiabatic reservoirs reveals the basic thermody-

namic variations associated with a CAES plant operation. However,

to accurately predict the temperature and pressure ﬂuctuations in

the cavern, heat transfer through the cavern walls must be consid-

ered. Heat transfer from the air to the rocks, during the charge cy-

cle, cools the air and therefore requires both, a smaller storage

volume and lower compression work. Likewise, during the dis-

charge cycle, heat transfer from the rocks to the air yields both, a

higher discharge pressure and require a smaller storage volume

to expel a given air volume (as compared to adiabatic conditions

with a similar pressure ﬂuctuation).

Langham [8] was the ﬁrst to model the pressure and tempera-

ture transients of CAES caverns assuming ideal gas behavior. He

calculated the temperature and pressure within a hard rock hori-

zontal tunnel subjected to a daily cycle. The calculations accounted

for both, heat conduction and air leakage assuming one dimen-

sional radial processes in homogenous rocks. The results showed

that the rock properties inﬂuence the maxima tunnel temperatures

Nomenclature

a

i

i=1...4, coefﬁcients deﬁned in Eq. (B.5)

A

c

Cavern walls surface area

b

i

i=1...4, coefﬁcients deﬁned in Eq. (B.6)

Bi Biot number based on cavern radius, h

c

R

w

/k

R

Bi

⁄

Biot number based on penetration depth, hct1=2

p=eR

e

Bi Dimensionless number, Biq

avt

1=mr

1=2

c

1

Coefﬁcient deﬁned in Eq. (32)b

c

p

Constant-pressure speciﬁc heat

c

v

Constant-volume speciﬁc heat

CD Charging/Discharging time ratio

eThermal effusivity

F

i

,F

e

Dimensionless mass ﬂow-rates, see Eq. (4)

Fo Fourier number, aRtp=R2

w

e

Fo Dimensionless number, Fo q

avt

1=mr

hSpeciﬁc enthalpy

h

c

Heat transfer coefﬁcient

kThermal conductivity

_

mcAir mass ﬂow rate through the compressor

m

r

Injected to initial cavern air mass ratio, _

mct1=ðq0VÞ

pPressure

P

0

Initial air pressure in the cavern

p

⁄

Dimensionless pressure, p/P

0

q

r

Dimensionless heat transfer parameter, hcAc=ð_

mccv0Þ

QEnergy crossing the cavern walls by conduction

_

QHeat transfer rate across the cavern walls

rRadial coordinate

r

⁄

Dimensionless radial coordinate, r/R

w

RSpeciﬁc air constant

R

⁄

Dimensionless group, RZ

0

/c

v

0

R

pDimensionless penetration radius, R

p

/R

w

R

w

Cavern radius

sLaplace transform parameter

tTime

t

p

Time period of the cycle

t

i

i=1...3, process duration times, see Fig. 2

t

⁄

Dimensionless time, t/t

p

TTemperature

T

0

Initial air temperature in the cavern

T

i

Injected air temperature at the cavern port

T

⁄

Dimensionless temperature, T/T

0

uSpeciﬁc internal energy

U

qDimensionless derivative of uwith respect to

q

at initial

state, RT

0

Z

T0

/c

v

0

xParameter deﬁned in Eq. (B.7)

VCavern volume

ZAir compressibility factor

Z

T

Derivative of Zwith respect to T

Greek Symbols

a

Thermal diffusivity

bStretching parameter, see Eq. (18)

c

Speciﬁc heat ratio at initial state, c

p0

/c

v

0

g

Dimensionless coordinate for numerical calculation

n

n

Parameter deﬁned in Eq. (A.4)

q

Density

q

0

Initial air density in the cavern

q

a

v

Average air density in the cavern

q

⁄

Dimensionless density,

q

/

q

0

Subscripts

0 Initial state

C Compression stage

eExit

G Generation (discharge) stage

iInlet

RRock

Rw Cavern walls

sSteady periodic cycle

S Storage stage

Superscript

0 Ideal gas

R. Kushnir et al. / International Journal of Heat and Mass Transfer 55 (2012) 5616–5630 5617

and that the tunnel radius affects the air leakage from the tunnel

and thereby the air temperature.

The cavern temperature and pressure ﬂuctuations were obvi-

ously modeled by the constructors of the two existing CAES plants.

Their publications, however, contain only results related to the spe-

ciﬁc operation of the Huntorf and McIntosh plants, and do not pro-

vide mathematical formulations. In this context, the KBB (Kavernen

Bau und Betriebs-GmbH) model of the Huntorf plant is brieﬂy dis-

cussed by Quast and Crotogino [5]. The cavern temperature mea-

surements taken at the plant trial run were used to calibrate the

model. The ESPC (Energy Storage and Power Consultants) model

of the McIntosh plant is discussed by Nakhamkin et al. [9,10].

Recently, Raju and Khaitan [11] modeled the temperature and

pressure variations within CAES caverns. Their heat transfer calcu-

lations were based on the assumptions of a constant wall temper-

ature and a certain variable heat transfer coefﬁcient. The constant

wall temperature assumption is justiﬁable only for perfectly con-

ducting rocks, and therefore limits its applicability. Separately,

Kim et al. [12] investigated the candidacy of shallow depth lined

rock caverns for CAES applications. A two dimensional numerical

model was developed for an underground cavern of circular cross

section. The analysis showed that the principal property responsi-

ble for long term air tightness is the permeability, both of the con-

crete lining and the surrounding rocks. A more comprehensive

description of the pertinent literature on CAES reservoirs thermo-

dynamics is presented in [13].

1.3. Summary

Only a few studies relate to the thermodynamic response of

CAES caverns, whereas most investigations address the calcula-

tions of the cavern temperature and pressure to speciﬁc conditions.

As such, those investigations do not reveal the general sensitivity

of the cavern temperature and pressure to the operating conditions

and the reservoir characteristics. Additionally, all heat transfer

considerations addressed only the ﬁrst few cycles. The current

study is aimed to provide adequate computational tools to calcu-

late the cavern thermodynamic conditions during CAES cycles.

The model is used to examine the inﬂuence of each parameter on

the cavern temperature and pressure variations and reveals the re-

quired cavern volume. The calculations are not limited to the ﬁrst

cycles, but address subsequent cycles in reference to steady peri-

odic conditions. The solutions for the temperature and pressure

variations within the storage cavern were developed for typical

conditions of constant air mass ﬂow rates during both, the charge

and discharge stages. It is also assumed that the air is cooled to a

certain temperature prior to storage. The developed model can

be applied to any set of such operating condition required by the

above-ground facilities.

2. Formulation of the problem

Consider an underground storage cavern of constant volume V,

located at a certain depth below the surface, which is initially ﬁlled

with compressed air at a pressure P

0

and temperature T

0

(equaling

surrounding rock temperature). The cavern is either vertical (salt

cavern) or horizontal (hard rock cavern), as illustrated in Fig. 1.

During a CAES plant operation, air ﬂows into and out of the cavern

cyclically.

2.1. Cavern thermodynamics

Deﬁning the cavern port and walls as the boundaries of a con-

trol volume, the mass and energy conservation equations become

identical to those obtained for adiabatic caverns [7], except for

an added term that represents the heat transfer across the cavern

boundaries. Air leakage is assumed to have a negligible effect on

the cavern temperature and pressure and therefore is uncounted

for. This assumption is valid for both, salt and low permeability

rock caverns. Likewise, the kinetic and potential energy changes

are insigniﬁcant even in tall caverns [7], and therefore are ignored.

Consequently, the mass and energy conservation equations, sub-

ject to the generalized gas state equation, are:

Vd

q

dt ¼ðF

i

þF

e

Þ_

m

c

ð1Þ

V

q

c

v

dT

dt ¼F

i

_

m

c

h

i

hþZRT

q

@u

@

q

T

þF

e

_

m

c

ZRT

q

@u

@

q

T

þ_

Qð2Þ

p¼Z

q

RTð3Þ

uand hare the speciﬁc internal energy and enthalpy, and c

v

and Z

are the constant volume speciﬁc heat and compressibility factor of

the air. _

Q stands for the heat transfer rate across the cavern walls.

The subscript idenotes the control volume inlet air conditions

and the subscript edesignates the outlet air conditions. p,

q

, and

Twhich represent the instantaneous pressure density and temper-

ature of the air within the cavern, are assumed to be uniform

throughout the storage space. This is a reasonable assumption ow-

ing to both air circulation and slow rates of temperature variations.

The product ðF

i

þF

e

Þ_

m

c

represents the momentary air mass ﬂow

rate at the cavern port, where _

m

c

is the compressor ﬂow rate, and

the sum F

i

+F

e

is a dimensionless periodic function with a cycle

time t

p

.Fig. 2 shows the variations of F

i

+F

e

of a CAES plant operat-

ing with compressor and turbine constant mass ﬂow rates. The indi-

cated time intervals are: t

1

for the charging time, t

2

t

1

for the

storage time, t

3

t

2

for the power generation time, and CD repre-

sents the discharging to charging mass ﬂow ratio (equal also to

Fig. 1. Schematic of horizontal and vertical underground air storage caverns in rock

formation.

Fig. 2. The dimensionless air mass ﬂow-rate at the cavern port during a CAES plant

cycle.

5618 R. Kushnir et al. / International Journal of Heat and Mass Transfer 55 (2012) 5616–5630

the charging to discharging time ratio). Accordingly, the functions F

i

and F

e

are deﬁned as

F

i

¼1 during charge

0 otherwise

;F

e

¼CD during discharge

0 otherwise

ð4Þ

Solutions for the cavern air temperature and pressure are

dependent on the air thermodynamic properties. In general, the

properties state equations are functions of both temperature and

density (or temperature and pressure). However, certain terms of

the state equations vary little within the expected cycle tempera-

ture and pressure ranges. Consequently, it was found that the air

thermodynamic properties can adequately be represented by the

following substitutions [7]:

ZZ

0

;c

v

c

v

0

;h

i

hc

p0

ðT

i

TÞ;@u

@

q

T

RT

2

0

Z

T0

q

0

ð5Þ

where T

i

is the air temperature entering the cavern at the charging

stage (which not necessarily equal to the after-cooler temperature).

Z

0

,c

v

0

,c

p0

, and Z

T0

are all evaluated at the initial state condition

(

q

0

,T

0

), where Z

T0

denotes the derivative of Zwith respect to T.

All properties are calculated by Sychev et al. model [14]. Note that

by setting Z

0

=1,Z

T0

=0,c

v

0

¼c

0

v

0

, and c

p0

¼c

0

p0

(where the super-

script 0 denotes ideal gas), an ideal gas representation is obtained.

2.2. Heat convection at the cavern walls

At the cavern boundaries heat is exchanged between the com-

pressed air and the cavern walls by processes of convection and

heat conduction. Assuming that the cavern wall surface is nearly

isothermal, the total heating rate of the air by means of convective

heat transfer from the walls is

_

Q¼h

c

A

c

ðT

Rw

TÞð6Þ

where h

c

is the average heat transfer coefﬁcient, A

c

the cavern walls

surface area, and T

Rw

the cavern walls surface temperature. In gen-

eral, the heat transfer coefﬁcient is a function of the air properties,

the cavern shape and size, and the air to wall temperature differ-

ence. Estimates of the heat transfer coefﬁcient should be based on

both, reasonable assumptions regarding the air ﬂow within the cav-

ity, and on the operational data collected from the two working

CAES plants.

In the current models, the heat transfer coefﬁcient dependence

on temperature differences was neglected. It is a reasonable

assumption since the air to wall temperature difference is expected

to be small. Furthermore, the natural convection heat transfer coef-

ﬁcient is weakly dependent on that difference. Additionally, air

movement owing to cavern charge or discharge could theoretically

induce somewhat higher heat transfer coefﬁcients than those of

the storage periods. However, due to the wall conductive heat

resistance and the large storage volume (that substantially limits

the needed internal air velocity requirements) the heat transfer

coefﬁcient was simply represented by a mean constant value. This

assumption proved to be adequate owing to the successful repro-

duction of the Huntorf plant operational data (Section (5.1)). Nev-

ertheless, the numerical model presented herein can also

accommodate a time dependent heat transfer coefﬁcient.

2.3. Thermal conduction in the rocks

To calculate the air temperature and pressure, the cavern con-

servation equations coupled with the heat conduction equation

of the surrounding rocks are to be solved. Assuming a long cylindri-

cal cavern shape (vertical or horizontal), the heat conduction in the

rocks can be considered as one-dimensional and radial for condi-

tions of short temperature penetration depths. Consequently, the

equation describing the rocks temperature is

q

R

c

pR

dT

R

dt ¼1

r

@

@rk

R

r@T

R

@r

ð7Þ

where ris the radial distance from the cavern center. The subscript

Rdenotes the rocks properties (assuming homogeneous formation),

where

q

,c

p

, and kare the density, speciﬁc heat, and thermal con-

ductivity, respectively. In the calculations, a constant rock thermal

conductivity is used owing to the expected small temperature ﬂuc-

tuations. The appropriate boundary conditions are

r¼R

W

;k

R

@T

R

@r¼h

c

ðTT

Rw

Þð8Þ

r!1;T

R

!T

0

ð9Þ

where R

w

is the cavern radius. The second boundary condition im-

plies that the rocks formation extends beyond the temperature pen-

etration depth.

2.4. Dimensionless form of the equations

Upon substituting Eqs. (5) and (6) into Eq. (2), and when T

0

,

q

0

,

t

p

, and R

w

are the temperature, density, time, and length scales

respectively, the dimensionless form of Eqs. (2) and (7) are ob-

tained as follows:

t

1

m

r

q

dT

dt

¼F

i

c

T

i

þR

c

ðÞT

þU

q

q

þF

e

R

T

þU

q

q

þq

r

T

Rw

T

ð10Þ

dT

R

dt

¼Fo

r

@

@r

r

@T

R

@r

ð11Þ

The dimensionless initial and boundary conditions are

t

¼0;T

¼T

R

¼1ð12Þ

r

¼1;@T

R

@r

¼Bi T

Rw

T

ð13Þ

r

!1;T

R

!1ð14Þ

and the periodic density variations, obtained form Eq. (1), are

q

¼1þm

r

t

t

1

;06t

6t

1

ð15aÞ

q

¼1þm

r

;t

1

<t

6t

2

ð15bÞ

q

¼1þm

r

t

3

t

t

3

t

2

;t

2

<t

6t

3

ð15cÞ

q

¼1;t

3

<t

61ð15dÞ

The pressure is calculated by the generalized gas state equation

p

¼Z

Z

0

q

T

ð16Þ

where

T

¼T

T

0

;T

R

¼T

R

T

0

;T

i

¼T

i

T

0

;p

¼p

P

0

;

q

¼

q

q

0

;t

¼t

t

p

;r

¼r

R

W

m

r

¼

_

m

c

t

1

q

0

V;

c

¼c

p0

c

v

0

;R

¼RZ

0

c

v

0

;U

q

¼RT

0

Z

T0

c

v

0

q

r

¼h

c

A

c

c

v

0

_

m

c

;Fo ¼

a

R

t

p

R

2

W

;Bi ¼h

c

R

W

k

R

;Bi

¼h

c

t

1=2

p

e

R

ð17Þ

a

R

=k

R

/(

q

R

c

pR

) and e

R

=(k

R

q

R

c

pR

)

1/2

are the rock thermal diffusivity

and effusivity, respectively. To improve the air pressure calculation

accuracy, the instantaneous compressibility factor is used in Eq.

(16). Additionally, in order to use Eqs. (15) a–d at any cycle, t

⁄

must

represent the dimensionless elapsed time from the beginning of

that cycle.

R. Kushnir et al. / International Journal of Heat and Mass Transfer 55 (2012) 5616–5630 5619

Heat transfer effects are expressed by the dimensionless groups

q

r

,Fo and Bi. The latter two have the form of the Fourier and Biot

numbers but not their usual deﬁnition, since their characteristic

length is the cavern radius. The product B

i

ðFoÞ

1=2

¼h

c

t

1=2

p

=e

R

Bi

, which represents a thermal resistance ratio (conduction in

the rocks versus convection within the cavern) is the preferred rep-

resentation of the Biot number. It is subsequently demonstrated

that at relatively short temperature penetration depths the cavern

air and wall temperatures depend only on Bi

⁄

(instead of both, Fo

and Bi).

3. Numerical model

To develop a numerical scheme, Eqs. (10)–(14) were converted

to a system of initial value ordinary differential equations, using

central differences representation of the spatial derivatives. In or-

der to conduct an effective numerical computation, the grid points

were arranged in increasing intervals as the distance from the cav-

ern walls increases (owing to the steep temperature gradients near

the wall). To develop a central difference numerical scheme, the r

⁄

coordinate is transformed to a uniform grid size variable by the

transformation [15]

g

¼1

ln bþ1

r

1

R

p

1

.

b1þ

r

1

R

p

1

hi

ln

bþ1

b1

;1<b<1ð18Þ

The physical domain 1;R

p

hi

in the r

⁄

plane, with its clustered grid

points near the cavern walls, is transformed into a uniform grid

computational domain [0,1] in the

g

plane. The stretching parame-

ter, b, clusters more points near the cavern walls (within the r

⁄

do-

main) as b?1. Applying the transformation to Eqs. (11)–(14),

yields the following set of equations

@T

R

@t

¼Fo

g

02

@

2

T

R

@

g

2

þ

g

00

þ

g

0

r

@T

R

@

g

() ð19Þ

t

¼0;T

¼T

R

¼1ð20Þ

g

¼0;

g

0

@T

R

@

g

¼Bi T

Rw

T

ð21Þ

g

¼1;@T

R

@

g

¼0ð22Þ

where

g

0

and

g

00

are the ﬁrst and second derivatives of

g

with re-

spect to r

⁄

. For the numerical computation, the boundary condition

(14) is applied at R

p

, which represents the dimensionless tempera-

ture penetration radius. For a central difference derivatives repre-

sentation, the semi discrete form of Eq. (19) is

@T

Rj

@t

¼Fo

g

02

T

Rjþ1

2T

Rj

þT

Rj1

D

g

2

þ

g

00

þ

g

0

r

T

Rjþ1

T

Rj1

2

D

g

r

¼r

j

;

j¼1;2;...;N1ð23Þ

where

D

g

¼

1

N

;

g

j

¼j

D

g

r

j

¼1þR

p

1

bþ1ðb1Þ

bþ1

b1

ðÞ

1

g

j

1þ

bþ1

b1

ðÞ

1

g

j

;j¼0;1;...;Nð24Þ

Upon combining the discrete form of the boundary conditions and

Eq. (23), the boundary grid point equations are obtained as follows:

@T

R0

@t

¼Fo 2

g

02

T

R1

T

R0

D

g

2

þBi

g

00

g

0

þ1

r

2

g

0

D

g

T

R0

T

r

¼r

0

jð25Þ

@T

RN

@t

¼Fo 2

g

02

T

RN1

T

RN

D

g

2

r

¼r

N

jð26Þ

The cavern equation (Eq. (10)) becomes

t

1

m

r

q

dT

dt

¼F

i

c

T

i

þðR

c

ÞT

þU

q

q

þF

e

R

T

þU

q

q

þq

r

T

R0

T

ð27Þ

Eqs. (23)–(27) constitute a complete set from which the air temper-

ature T

⁄

, and the rock temperatures T

Rj

ðj¼0...NÞcan be calculated.

They are solved for a uniform initial condition of unity. The compu-

tation was performed with the problem solving environment Maple

[16], based on the default stiff method, which is an implicit Rosen-

brock third–fourth order Runge–Kutta method. The set of equations

were solved for N= 20. The stretching parameter bwas chosen by

matching the numerical solutions to known analytical solutions.

The dimensionless temperature penetration radius, R

p

, was calcu-

lated by an appropriate formula (see Appendix A for more details).

4. Analytical analyses

Heat transport across the cavern boundary signiﬁcantly affects

the analytical analyses complexity as compared to adiabatic condi-

tions. Nevertheless, with some reasonable approximations the

equations can still be treated analytically. The importance of the

approximate analytical approach is in its ability to reveal explicitly

the inﬂuence of each parameter, and also to provide a reference

solution for the numerical model. Being easier to analyze, the lim-

iting cases of perfectly conducting rocks (k

R

?1) and adiabatic

cavern walls (k

R

?0orh

c

?0) are considered ﬁrst.

4.1. Perfectly conducting rocks or adiabatic caverns

In the limiting cases of perfectly conducting rocks (T

R

= const) or

adiabatic caverns, the air temperature variations are directly de-

rived from Eq. (10), as described in Appendix B. The results reveal

the dependence of the required storage volume on the cavern min-

imum and maximum pressures. In particular, for an ideal gas rep-

resentation, simple volume expressions are obtained for

isothermal (q

r

?1) and adiabatic (q

r

= 0) caverns:

Vj

isothermal

¼

_

m

c

t

1

RT

R

p

s;max

p

s;min

ð28Þ

Vj

adiabatic

¼

_

m

c

t

1

R

c

0

T

i

p

s;max

p

s;min

ð29Þ

Fig. 3. The dimensionless storage volume dependence on the storage pressure ratio

for a cavern surrounded by perfectly conducting rocks (ideal gas model, T

i

/T

R

= 1.05,

c

0

= 1.4, t

1

¼12=24, t

2

¼18=24, t

3

¼21=24).

5620 R. Kushnir et al. / International Journal of Heat and Mass Transfer 55 (2012) 5616–5630

The interrelationship of the storage pressure ratio to the re-

quired storage volume, as expressed by Eqs. (28) and (29), is dem-

onstrated in Fig. 3. The case of q

r

= 30 is also plotted in the ﬁgure.

Obviously, smaller pressure ratios (smaller p

s,max

’s) would require

larger storage volumes. It is seen that the optimal pressure ratio

should preferably lie between 1.2 and 1.8. At pressure ratios smal-

ler than 1.2, a slight increase of the ratio, entails small increases of

both, the required compression work and compressor cost but sub-

stantially decreases the storage volume and its cost. On the other

hand, at pressure ratios larger than 1.8, a decrease of the ratio do

not affect much the storage volume, but reduces the required com-

pression work and compressor cost. Consequently, the selection of

the storage pressure ratio and its corresponding storage volume

should be based on both, design considerations and economical

criteria.

It is also seen in Fig. 3 that larger heat transfer rates (larger q

r

)

reduce the required cavern volume for a given pressure ratio, or

alternatively, reduce the pressure ratio for a given storage volume.

Hence, an increase of the cavern wall surface area reduces the re-

quired storage volume (or pressure ratio). Theoretically, by

decreasing the cavern diameter and increasing its height (or

length), it is possible to increase the cavern wall surface area for

any given cavern volume, and thereby diminish the pressure ratio.

Note that, solutions of real cases are likely to lie in between the

two bounds of perfectly conducting rocks and adiabatic caverns.

4.2. Finite rock conductivity

In reality, the rocks have a considerable heat conduction resis-

tance. Therefore, the cavern conservation equations and the rock

heat conduction equation should be solved simultaneously. Since

the combined problem is relatively quite complex, an approxima-

tion is introduced to obtain a useful analytical solution. During

each cycle, the dimensionless air density

q

⁄

varies from a mini-

mum value of 1 to a maximum value of 1 + m

r

. For small values

of m

r

, it is reasonable to represent

q

⁄

by a constant average value.

With this simpliﬁcation, the equations are solved through the La-

place transform.

Compression stage temperatures-the exact solution derivations of

Eqs. (10)–(14) (with

q

¼

q

a

v

¼1þm

r

=2) for the ﬁrst cycle com-

pression stage, are outlined in Appendix C. Those solutions can

be adequately represented by their asymptotic expression, namely

T

C

¼1þc

1

q

r

Z

1

0

1e

n2mrt

q

a

v

t

1

w

2

1

ðnÞþw

2

2

ðnÞdn

ð30Þ

T

R;C

¼1þc

1

ﬃﬃﬃﬃ

r

p

Z

1

0

w

1

ðnÞsin

nðr

1Þ

ﬃﬃﬃﬃ

e

Fo

p

!

þw

2

ðnÞcos

nðr

1Þ

ﬃﬃﬃﬃ

e

Fo

p

! !

1e

n2mrt

q

a

v

t

1

!

w

2

1

ðnÞþw

2

2

ðnÞ

dn

n

ð31Þ

where

w

1

ðnÞ¼e

Bi

c

R

n

2

;w

2

ðnÞ¼n

c

R

þq

r

n

2

ð32aÞ

c

1

¼2

pc

T

i

c

þR

þU

q

q

a

v

e

Bi ð32bÞ

e

Fo ¼Fo

q

a

v

t

1

m

r

;e

Bi ¼Bi

q

a

v

t

1

m

r

1=2

ð32cÞ

The subscript C in T

⁄

and T

R

indicates compression stage tempera-

tures. Eqs. (30) and (31) are valid when t

⁄

1/Fo. In geometric

terms, it implies that the temperature penetration depth into the

rocks, being on the order of (

a

R

t)

1/2

, should be substantially smaller

than the cavern radius, R

w

. Evidently, for such cases the heat ﬂow

can be considered as planar one dimensional (rather than radial).

Hence, the air temperature asymptotic solution (Eq. (30)) can also

be obtained from a planar one dimensional heat conduction

equation. At the asymptotic conditions the air and wall tempera-

tures depend only on Bi

⁄

(instead on both Fo and Bi).

To estimate the temperature penetration depth, thermal proper-

ties data of CAES-suitable rocks, adopted from [17], are presented in

Table 1. Each rock is represented by ranges of plausible thermal con-

ductivities and diffusivities. These rangesare primarily a consequence

of variations in mineral content, porosity and cracks presence. The

rock thermal properties dependence on temperature is negligible

for the applicable ranges of interest. The properties are therefore ta-

ken at the local rock temperature which is typically slightly above

room temperature [18]. At such temperatures, the thermal diffusivity

of the rocks isof the order of 10

6

m

2

/s. Thus, fora time span of 24 h,

the penetration depth that is on the order of (

a

R

t)

1/2

0.3 m is indeed

substantially smaller than typical cavern radii.

To examine the heat transfer effects and the validity of the ana-

lytical approximations, calculated air temperatures during the ﬁrst

compression stage are presented in Fig. 4. The temperature

changes are for Fo =410

4

and Bi = 200 which represent typical

salt cavern properties. The limiting cases of adiabatic and perfectly

conducting rocks are also shown in the ﬁgure. As seen, the approx-

imate analytical solution is in close agreement with the numerical

model results and the exact solutions of the limiting cases. Obvi-

ously, owing to the ﬁnite rock conductivity, the temperature rise

during compression is higher than in caverns surrounded by per-

fectly conducting rocks. Nonetheless, it is substantially smaller

than one of adiabatic compression, as seen in the ﬁgure. Hence,

the cycle maximum temperature is signiﬁcantly reduced owing

to the heat transfer process.

Full cycle temperatures-the full ﬁrst cycle temperatures can be

constructed from the compression stage solution, as outlined in

Appendix C. A comparison of the numerically calculated ﬁrst cycle

air temperature solution against that of the approximate analytical

solution is presented in Fig. 5. The calculation covers different val-

ues of m

r

and q

r

which reﬂects different masses of stored air. As ob-

served, the approximate temperature variations follow closely the

numerical results, and fully coincide at the cycle maximum and

minimum temperatures. The differences between both solutions

are apparent only during the storage periods and at large air den-

sity changes (larger m

r

). This is expected since the approximation

is based on an average density, while the storage period is repre-

sented by extreme density values.

Inspection of the temperature proﬁles seen in Fig. 5 reveals that

the cycle average air temperature is larger than the undisturbed

rock temperature, T

0

. Therefore, two processes of heat ﬂow are dis-

tinguished, one governed by temperature ﬂuctuations (of short

penetration distance), and the other reﬂecting the continuous heat

ﬂow caused by the higher average air temperature versus that of

the distant rock temperature. Notice that, in theory, the latter pro-

cess in a one dimensional radial heat ﬂow representation would

eventually raise any distant rock temperature. However, in reality

the heat conduction process would preferably drive that heat to-

ward the surface (in a three dimensional manner) rather than to

extremely distant rock locations. Consequently, the one dimen-

sional model is applicable for the period in which the temperature

penetration depth is still relatively small.

5. Results and discussion

Representative ranges of the studied cavern characteristics,

operating conditions, and their corresponding dimensionless

parameters are presented in Table 2. The reservoirs principal heat

transfer parameters for conditions of short temperature penetra-

tion depths are q

r

, and Bi

⁄

. As seen in the table, the study covers

a wide range of those parameters. These ranges, in practice are

narrower when addressing a particular family of rock formation

R. Kushnir et al. / International Journal of Heat and Mass Transfer 55 (2012) 5616–5630 5621

combined with their corresponding cavern geometry characteris-

tics (large cavern radii for salt and small for hard rocks).

5.1. Comparison to operational data

During the ﬁrst ﬁll of the Huntorf plant caverns, and in the fol-

lowing trial runs, extensive measurements of temperature and

pressure were carried out. Results of such measurements for a dai-

ly cycle are presented in Fig. 6. At the cycle trial initiation, the air

and rocks were at 40 °C. Subsequently, the cavern was partially

ﬁlled or partially emptied several times within a day span. For

comparison, calculated air temperatures and pressures, based on

Eqs. (23)–(27), are also plotted. The calculations were performed

for an input data that fully matches the tested charging and dis-

charging ﬂow rates (functions F

i

and F

e

). The cavern was assumed

to be cylindrical with an average radius of 20 m, a volume of

141,000 m

3

and surface area of 25,000 m

2

. Since the temperature

ﬂuctuations penetration depth is relatively small the cavern radius

(or precise shape) does not affect the results. In contrast, the cavern

volume and surface area are of considerable signiﬁcance.

As observed, calculated temperatures and pressures are in good

agreement with the measured ﬁeld data. Yet, the span of the mea-

sured temperature variations during charge and discharge are

somewhat smaller than the calculated one, which implies that

the heat ﬂow between the air and the surrounding rocks is larger

than the model prediction. Notice that both, the KBB model (Hun-

torf plant) and the ESPC model (McIntosh plant) also predicted lar-

ger temperature ﬂuctuations than the actual plant data [10]. The

discrepancy is likely to emanate from the uncertainty in the value

of both, the cavity surface area and the heat transfer coefﬁcient.

Naturally, solution mining techniques are known to produce wavy

cavern walls. The surface bulges act like ﬁns and thereby enhance

the wall heat transfer rates. This effect could be accounted for by

an enlargement of the cavity surface area as compared to that of

a smooth cylindrical cavern. In this respect, in the KBB model

and here, a surface area of 25,000 m

2

is adopted. Furthermore,

the action of air injection and withdrawal on non-smooth walls

could in theory produce some turbulence and thereby entail some-

what larger apparent mean heat transfer coefﬁcients as compared

to smooth walls coefﬁcients.

5.2. Temperature and pressure characteristics

To closely inspect the compressed air thermodynamic response

to charge/discharge cycles, calculated temperature and pressure

Table 1

Data on the thermal properties of characteristic CAES rocks (measured at room temperature; n- number of samples; Adopted from [17]).

Rock type Thermal conductivity, kW/(mK) Speciﬁc heat, c

p

kJ/(kgK) Thermal diffusivity,

a

10

7

m

2

/s

nrange mean nrange mean range mean

Granite 356 1.25...4.45 3.05 102 0.67...1.55 0.958

174 1.34...3.69 2.4 84 0.74...1.55 0.946 3.33...15.0 9.27

Granodiorite 89 1.35...3.40 2.65 11 0.84...1.26 1.093

23 1.64...2.48 2.11 10 0.74...1.26 1.057 3.05...7.5 5.15

Diorite 185 1.72...4.14 2.91 3 1.13...1.17 1.14

43 1.38...2.89 2.20 3 1.12...1.17 1.14 3.32...8.64 6.38

Gabbro 71 1.62...4.05 2.63 9 0.88...1.13 1.005

7 1.80...2.83 2.28 0.88...1.13 1.01 9.32...12.2 9.72

Quartzite 186 3.10...7.60 5.26 8 0.71...1.34 1.013

9 2.68...7.60 5.26 8 0.72...1.33 0.991 13.6...20.9 17.9

Gneiss 388 1.16...4.75 2.44 55 0.46...0.92 0.75

40 0.94...4.86 2.02 7 0.75...1.18 0.979 6.30...8.26 7.32

Dolomite 29 1.60...5.50 3.62 21 0.84...1.55 1.00

72 1.63...6.50 3.24 35 0.65...1.47 1.088

Limestone 487 0.62...4.40 2.29 38 0.82...1.72 0.933

216 0.92...4.40 2.4 92 0.75...1.71 0.887 3.91...16.9 11.3

Salt 70 1.40...

7.15 4.00 0.84

1.67...5.50 11.2...17.7

Fig. 4. Calculated cavern air temperature variations during the ﬁrst cycle com-

pression stage (m

r

= 0.3, q

r

= 15, T

i

¼1:04, t

1

¼8=24, T

0

= 310 K, P

0

= 45 bar).

Fig. 5. First cycle air temperature variations: comparison between the average

density approximation results and the numerical solution for Fo =410

4

,

Bi = 200, T

i

¼1:04, t

1

¼8=24, t

2

¼14=24, t

3

¼18=24, T

0

= 310 K, P

0

= 45 bar.

5622 R. Kushnir et al. / International Journal of Heat and Mass Transfer 55 (2012) 5616–5630

variations during the ﬁrst and the ﬁfteenth cycles, are presented in

Fig. 7 (for the indicated set of parameters). The effects of the heat

transfer between the air and the cavern wall are clearly apparent.

The temperature variations during air injection and withdrawal are

substantially smaller than those of adiabatic compression/expan-

sion. Additionally, in contrast to the adiabatic case, the air is cooled

by the cavern walls after a charge period and heated after a dis-

charge period. The cycle pressure ratio (maximum to minimum)

is about 1.44. The corresponding pressure ratio for an adiabatic cy-

cle is 1.48 and for the perfectly conducting rocks 1.38.

As observed, the cycle average air temperature is larger than the

undisturbed rock temperature. Therefore, in each cycle, a certain

amount of heat is lost by conduction into the rocks. As previously

discussed, that heat would preferably be driven toward the surface

in a three dimensional manner. However, as long as the tempera-

ture penetration depth is relative small, the process can be consid-

ered as one dimensional. In this respect, the ﬁfteen cycle air

temperature and pressure were calculated (Fig. 7b) based on the

one dimensional model and by incorporation of an appropriate

penetration depth adjustment (see Appendix A). As seen, the ﬁf-

teenth cycle results are close to those of steady conditions.

To further examine the nature of the temperature penetration

into the rocks, calculated temperature variations at different rock

locations during the ﬁrst and the ﬁfteenth cycles are illustrated

in Fig. 8. One can clearly distinguished between the temperature

ﬂuctuations that fade away within a short distance, and a mean

temperature rise that penetrates into deeper rock locations. At

the ﬁfteenth cycle the temperature penetration depth is well be-

yond the ﬂuctuation penetration distance.

As seen in Figs. 7 and 8, at the ﬁfteenth cycle the air and rock

temperatures approach steady conditions. Evidently, since the heat

build up in the rocks would cease only when the penetration

reaches the ground surface and owing to the substantial cavern

depth, fully steady conditions would be reached only after a long

time. In practice, the plant would not operate continuously for suf-

ﬁcient number of cycles to fully reach steady condition. Hence, the

ﬁfteenth cycle conditions will be considered as representative

operational conditions. At that cycle, the temperature penetration

depth is still relatively small and the one dimensional numerical

model is applicable.

Table 2

Representative ranges of cavern characteristics, operating conditions, and their corresponding dimensionless parameters.

Variable Deﬁnitions Minimum value Maximum value Units Comment

T

0

Local rock temperature 20 60 °C Data from [18]

P

0

First ﬁll cavern pressure 20 70 bar According to the desired turbine inlet pressure

Z

0

Air compressibility factor 0.99 1.01 –

_

m

c

Flow rate through the compressor 50 150 kg/s

R

W

Cavern radius 5 30 m

A

c

Cavern surface area 5 10

3

10

5

m

2

h

c

Heat transfer coefﬁcient 10 150 W/(m

2

K)

k

R

Rock thermal conductivity 1 7 W/(mK) See Table 1

a

R

Rock thermal diffusivity 0.3 10

6

310

6

m

2

/s See Table 1

e

R

Rock thermal effusivity, k

R

=a

1=2

R

550 13,000 Ws

1/2

/(m

2

K)

p

min

Minimum cavern operational pressure 20 70 bar According to the desired turbine inlet pressure

p

max

/p

min

Operational cavern pressure ratio 1.2 1.8 – Subject to geological constraints and economics

T

i

/T

0

Relative injected air temperature 1 1.2 – Subject to geological constraints and economics

m

r

_

m

c

t

1

=ðq

0

VÞ0.1 0.55 – Based on Fig. 3

q

r

h

c

A

c

=ð_

m

c

c

v

0

Þ1 150 –

Fo a

R

t

p

=R

2

w

310

5

0.01 –

Bi h

c

R

w

/k

R

10 4500 –

Bi

⁄

h

c

t

1=2

p

=e

R

0.2 75 –

t

1

t

1

/t

p

6/24 12/24 – Compression

t

2

t

1

(t

2

t

1

)/t

p

2/24 8/24 – Storage

t

3

t

2

(t

3

t

2

)/t

p

2/24 10/24 – Power generation

Fig. 6. Temperature, pressure, and port ﬂow-rate of cavern NK1 versus time:

comparison between measured data [5] and numerical results for q

r

= 54.7,

Fo = 6.4 10

4

,Bi = 285.7, and T

i

¼1:035, 1.019, 1.029, respectively (for each of

the charging stages, based on the measured wellhead temperatures).

R. Kushnir et al. / International Journal of Heat and Mass Transfer 55 (2012) 5616–5630 5623

5.3. Sensitivity analysis

Since the storage temperature and pressure variations affect the

overall plant performance and economics, it is of interest to exam-

ine their dependence on the cavern characteristics and operating

conditions. That examination addresses both, the ﬁrst and the ﬁf-

teenth cycle conditions.

Rock properties-the analytical analysis revealed that for short

temperature penetration depths the air temperature depends only

on Bi

¼h

c

t

1=2

p

=e

R

(instead of Fo and Bi). Calculations indicate that

within the ranges given in Table 2, the dependence holds quite well

even at the ﬁfteenth cycle. Therefore, the principal rock property

that governs the heat transfer process is the thermal effusivity

(appearing in Bi

⁄

). To illustrate the effect of the effusivity, calcu-

lated temperature variations for different values of Bi

⁄

are pre-

sented in Fig. 9. As observed, larger effusivities (smaller Bi

⁄

’s)

enhance the heat transfer between the air and the rocks, and there-

by reduce the temperature changes during air injection and

withdrawal.

Fig. 10 illustrates the dimensionless heat transfer rates crossing

the cavern walls. It is clearly seen that larger e

R

’s increase both, the

rate of heat losses to the rocks during air injection, and the rate of

heat gain from the rocks during air withdrawal (positive sign

represents heat transfer entering the cavern). In total, a certain

amount of heat per cycle is lost by conduction into the rocks, since

the heat losses are greater than the gains. Numerical values of the

cycle energy losses, calculated by the integration of _

Q over a cycle

period, are given in Table 3. The values are divided by _

m

c

c

p0

T

i

t

1

which represents the total energy injected into the cavern during

the charging phase. As seen, even for perfectly conducting rocks

only a few percent of the injected energy is lost into the rocks in

each cycle. The energy losses are also calculated for both the four-

teenth and ﬁfteenth cycles. The differences in losses between these

cycles are quite small, implying that the cavern is near steady

conditions.

The effect of the rock effusivity on the required storage volume

is shown in Fig. 11. The ﬁrst cycle calculations were obtained both

numerically and analytically. An excellent agreement exists be-

tween the solutions for all pressure ratios. It may be observed that

larger effusivities reduce the required storage volume for any given

storage pressure ratio. A diminution of the storage volume is par-

ticularly advantageous for hard rock caverns, since they are rela-

tively expensive to excavate. Therefore, preference must be given

to sites of rocks that have the largest thermal effusivity.

Heat transfer coefﬁcient-air temperature variations for different

values of heat transfer coefﬁcient are presented in Fig. 12. Since

Fig. 7. Calculated cavern temperature and pressure variations (m

r

= 0.3, q

r

= 25, Fo =610

4

,Bi = 250, T

i

¼1:04, t

1

¼8=24, t

2

¼14=24, t

3

¼18=24, T

0

= 310 K, P

0

= 45 bar).

(a) during the ﬁrst cycle; (b) during the ﬁfteenth cycle.

Fig. 8. Calculated temperature variations at different rock locations (m

r

= 0.3, q

r

= 25, Fo =610

4

,Bi = 250, T

i

¼1:04, t

1

¼8=24, t

2

¼14=24, t

3

¼18=24, T

0

= 310 K,

P

0

= 45 bar). (a) during the ﬁrst two cycles; (b) during the ﬁfteenth and sixteenth cycles.

5624 R. Kushnir et al. / International Journal of Heat and Mass Transfer 55 (2012) 5616–5630

both Bi

⁄

and q

r

are directly proportional to h

c

, changes of h

c

affect

Bi

⁄

and not q

r

/Bi

⁄

. As observed, larger heat transfer coefﬁcients (lar-

ger Bi

⁄

’s) reduce the temperature changes during air injection and

withdrawal up to a certain limit. At the limit of Bi

⁄

?1and

q

r

?1, the air temperature is equal to the cavern surface temper-

ature and the heat transfer rate is determined solely by the rocks

thermal conduction. The temperature variations at the limit de-

pend on q

r

/Bi

⁄

. In principle, stronger heat transfer coefﬁcients

and larger rock effusivities have similar effects. At large Bi

⁄

num-

bers, heat transfer rates are controlled by conduction and therefore

the effusivitiy shows stronger effects. At small Bi

⁄

’s, the convective

heat transfer resistance is dominating and therefore its coefﬁcient

is the dominant parameter.

Temperature of the injected air-the injected air temperature is a

design parameter that can be controlled by the after-cooler device.

Calculated cavern air temperature variations for three different in-

jected air temperatures are shown in Fig. 13. Clearly, the cycle

average temperature is substantially affected by the injected air

temperature. Higher injected air temperatures produce higher

cavern temperatures. Changing the injected air temperature is

therefore a method to raise or lower the cycle maximum and min-

imum temperatures. In salt caverns it is recommended that the

cavern air temperature would not exceed 80 °C[19]. It implies

that for a local rock temperature of 40 °C, T

⁄

is to be kept below

1.13. For the indicated set of parameters of Fig. 13,T

i

should be

less than 1.2 to meet the limitation. It is also noticeable from

Fig. 9. Calculated cavern temperature variations for different rock effusivities (m

r

= 0.3, q

r

= 30, Fo =10

4

,T

i

¼1:05, t

1

¼8=24, t

2

¼14=24, t

3

¼18=24, T

0

= 310 K, P

0

= 45 bar).

(a) during the ﬁrst cycle; (b) during the ﬁfteenth cycle.

Fig. 10. Dimensionless heat transfer rates crossing the cavern walls for different rock effusivities (m

r

= 0.3, q

r

= 30, Fo =10

4

,T

i

¼1:05, t

1

¼8=24, t

2

¼14=24, t

3

¼18=24,

T

0

= 310 K, P

0

= 45 bar). (a) during the ﬁrst cycle; (b) during the ﬁfteenth cycle.

Table 3

Conductive energy losses (per injected energy) during various cycles for different rock effusivities (m

r

= 0.3, q

r

= 30, Fo =10

4

,T

i¼1:05, t

1¼8=24, t

2¼14=24, t

3¼18=24,

T

0

= 310 K, P

0

= 45 bar).

Bi

⁄

First cycle Q=ð_

m

c

t

1

c

p0

T

i

ÞFourteenth cycle Q=ð_

m

c

t

1

c

p0

T

i

ÞFifteenth cycle Q=ð_

m

c

t

1

c

p0

T

i

Þ

0 0.06579 0.06572 0.06572

2 0.06779 0.05754 0.05733

6 0.06701 0.04402 0.04361

30 0.04479 0.01541 0.01506

100 0

R. Kushnir et al. / International Journal of Heat and Mass Transfer 55 (2012) 5616–5630 5625

the ﬁgure that the differences between the ﬁrst cycle and the ﬁf-

teenth cycle temperatures become apparent at high injected

temperatures.

Calculated cycle energy losses for different injected air temper-

atures, are given in Table 4. The injected air temperature highly af-

fects the cycle energy losses. As seen, the energy losses can be

Fig. 11. The dimensionless storage volume dependence on the storage pressure ratio for different rock effusivities (q

r

= 30, Fo =10

4

,T

i

¼1:05, t

1

¼8=24, t

2

¼14=24,

t

3

¼18=24, T

0

= 310 K, P

0

= 45 bar). (a) based on the ﬁrst cycle; (b) based on the ﬁfteenth cycle.

Fig. 12. Calculated cavern temperature variations for different heat transfer coefﬁcients (m

r

= 0.3, q

r

/Bi

⁄

=5,Fo =10

4

,T

i

¼1:05, t

1

¼8=24, t

2

¼14=24, t

3

¼18=24, T

0

= 310 K,

P

0

= 45 bar). (a) during the ﬁrst cycle; (b) during the ﬁfteenth cycle.

Fig. 13. Calculated cavern temperature variations for different injected air temperatures (m

r

= 0.3, q

r

= 30, Fo =10

4

,Bi

⁄

=6, t

1

¼8=24, t

2

¼14=24, t

3

¼18=24, T

0

= 310 K,

P

0

= 45 bar). (a) during the ﬁrst cycle; (b) during the ﬁfteenth cycle.

5626 R. Kushnir et al. / International Journal of Heat and Mass Transfer 55 (2012) 5616–5630

diminished through the reduction of the injected air temperature.

Evidently, lower injected air temperatures also reduce the re-

quired storage volume. However, it is important to note that low-

ering the injected air temperatures entails an associated cooling

cost.

Duration of compression and power generation periods-the

compression and power generation time periods are deter-

mined from local power demands and production capacity.

Realistic bounds for these time intervals are shown in Table 2.

At adiabatic conditions and for a given amount of injected air

ð_

m

c

t

1

¼constÞ, the duration of the compression and power gen-

eration periods do not affect the air temperature and pressure

variations. For non adiabatic conditions, prolongation of the

compression and power generation periods reduces the air

temperature ﬂuctuations as seen in Fig. 14(a) and (b), respec-

tively. In all cases the same amount of air is injected and

withdrawn.

The results indicate that for longer compression periods more

heat is transferred from the air to the rocks, thereby dampening

the temperature rise. Conversely, for longer discharge periods

more heat is transferred from the rocks to the air, thus inhibiting

the extent of temperature fall. Consequently, in similar to porous

reservoirs [1,2], for a given amount of injected air it is advanta-

geous to expand the duration of compression and power genera-

tion periods as much as feasible.

6. Conclusions

A combined analytical and numerical study of the thermody-

namic response of underground caverns to CAES plants charge/dis-

charge cycles was conducted. Calculated temperatures and

pressures were found to be in close agreement with the measured

ﬁeld data of the Huntorf plant. The following conclusions were

drawn from the investigation:

Two processes of heat ﬂow within the rocks are distinguished,

one governed by temperature ﬂuctuations (of short penetration

distance), and the other reﬂecting the continuous heat ﬂow

caused by the higher average cavern air temperature versus that

of the distant rock temperature. It turns out that the cavern air

temperature is near steady conditions when the temperature

penetration depth is relatively small and therefore can be calcu-

lated by a one dimensional model.

The two principal dimensionless parameters that characterize

the heat transfer process are q

r

¼h

c

A

c

=ð_

m

c

c

v

0

Þand Bi

¼

h

c

t

1=2

p

=e

R

. The study reveals that, for practical conditions, heat

transfer at the cavern walls plays an important role in reducing

the air temperature and pressure variations and therefore

enhancing the storage capacity. Those assertions are supported

by the operational data of the Huntorf and McIntosh plants.

For realistic operating conditions and reservoir characteristics,

in each cycle, few percents of the injected energy are lost by

conduction into the rocks. These losses can be diminished by

reducing the injected air temperature.

The thermal property that rules the heat transfer process is the

rock effusivity. As it turns out, to reduce the required storage

volume, preference must be given to rock sites where the ther-

mal effusivity is the largest.

The injected air temperature substantially affects the storage

average temperature and provides a method to control the cycle

temperature extreme limits. Smaller injected air temperatures

also reduce the required storage volume, however require higher

cooling costs.

In similar to porous rock reservoirs, for a given amount of

injected air ð_

m

c

t

1

¼constÞ, it is advantageous to expand the

duration of compression and power generation periods as much

as feasible.

The optimal storage pressure ratio should preferably lie

between 1.2 and 1.8, and its selection should be based on both,

design considerations and economical criteria.

Table 4

Conductive energy losses (per injected energy) during various cycles for different injected air temperatures (m

r

= 0.3, q

r

= 30, Fo =10

4

,Bi

⁄

=6,t

1¼8=24, t

2¼14=24, t

3¼18=24,

T

0

= 310 K, P

0

= 45 bar).

T

i

First cycle Q=ð_

m

c

t

1

c

p0

T

i

ÞFourteenth cycle Q=ð_

m

c

t

1

c

p0

T

i

ÞFifteenth cycle Q=ð_

m

c

t

1

c

p0

T

i

Þ

1.0 0.03061 0.01171 0.01163

1.05 0.06701 0.04402 0.04361

1.1 0.10010 0.07339 0.07268

1.2 0.15801 0.12479 0.12357

Fig. 14. Calculated cavern temperature variations during the ﬁfteenth cycle (m

r

= 0.3, q

r

=t

1

¼90, Fo =10

4

,Bi

⁄

=6,T

i

=1.05, T

0

= 310 K, P

0

= 45 bar). (a) for different duration

of compression periods; (b) for different duration of power generation periods.

R. Kushnir et al. / International Journal of Heat and Mass Transfer 55 (2012) 5616–5630 5627

Acknowledgements

The authors thank Mr. Fritz Crotogino from Kavernen Bau und

Betriebs GmbH, who provided valuable data on the Huntorf plant

caverns, which made the comparison of the current study model

to the Huntorf ﬁeld data feasible.

Appendix A. Useful heat conduction solutions

To clearly understand the nature of the heat transfer processes

within the rocks, some relevant heat conduction solutions are pre-

sented. These solutions are incorporated to calculate the tempera-

ture penetration depth and to calibrate the numerical model

(described in Section (3)).

A.1. Constant air temperature

Consider a large rock formation with a uniform initial tempera-

ture (say T

0

) that is bounded internally by a cylindrical cavern of

radius R

w

. The cavern is abruptly ﬁlled with air of temperature

T=T

1

. Under the assumption of constant air temperature, the un-

steady temperature in the rock formation for one dimensional heat

conduction is [20]

T

R

T

1

T

0

T

1

¼2Bi

p

Z

1

0

Y

0

ðnr

Þ

W

J

ðnÞJ

0

ðnr

Þ

W

Y

ðnÞ

W

2

J

ðnÞþ

W

2

Y

ðnÞ

e

n

2

Fot

ndnðA:1Þ

where

W

J

ðnÞ¼nJ

1

ðnÞþBiJ

0

ðnÞ

W

Y

ðnÞ¼nY

1

ðnÞþBiY

0

ðnÞðA:2Þ

J

i

and Y

i

are the Bessel functions of the ﬁrst and second kind of order

i, respectively. Evidently, if T

1

is higher than T

0

, heat ﬂows contin-

uously into the rocks and eventually raises even distant rock tem-

peratures. The temperature penetration depth moves deeper into

the rocks as time progresses.

A.2. Periodic air temperature

Consider the case of section A.1, but with a periodic air temper-

ature that ﬂuctuates around the initial rock temperature, that is

T=T

0

+

D

T

1

sin(2

p

nt

⁄

+h

n

). In this case, the rock temperature can

be constructed from two separate parts, a transient part and a stea-

dy periodic part. The latter can be obtained by the complex combi-

nation method, and is

T

R;s

T

0

D

T

1

¼N

0

ðn

n

r

Þ

jZ

1

jsin 2

p

nt

þh

n

þ/

0

ðn

n

r

ÞargðZ

1

ÞðÞðA:3Þ

where

n

n

¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2

p

n

Fo

rðA:4Þ

N

0

ðxÞ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Ker

2

0

ðxÞþKei

2

0

ðxÞ

q;/

0

ðxÞ¼argðKer

0

ðxÞþKei

0

ðxÞiÞ;

Z

1

¼Ker

0

ðn

n

Þ n

n

ﬃﬃﬃ

2

pBi ðKer

1

ðn

n

ÞþKei

1

ðn

n

ÞÞ

þKei

0

ðn

n

Þþ n

n

ﬃﬃﬃ

2

pBi Ker

1

ðn

n

ÞKei

1

ðn

n

ÞðÞ

iðA:5Þ

Ker

i

and Kei

i

are the Kelvin functions of order i. Eq. (A.3) represents

a temperature wave which oscillates everywhere at the same fre-

quency as the cavern air. The oscillations amplitude, appearing as

an expression of Kelvin functions, diminishes rapidly as the radial

distance from the cavern increases. The dimensionless radius R

p

at

which the temperature amplitude reduces to a fraction

e

from that

of the cavern walls, satisﬁes the equation

N

0

n

n

R

p

¼

e

N

0

ðn

n

ÞðA:6Þ

Repeating the derivation procedure of Eq. (A.6) for planar one

dimensional heat conduction (rather than radial), yields

R

p;planar

¼1þﬃﬃﬃ

2

p=n

n

lnð1=

e

ÞðA:7Þ

It follows that higher disturbances frequency (larger n) produce

shorter penetration radius R

p

. As it turns out, Eqs. (A.6) and (A.7)

give identical penetration radii for any realistic CAES Fo number

(for n= 1). Thus, the temperature ﬂuctuations in CAES caverns are

expected to disappear at short depth within the surrounding rocks.

A.3. Numerical model calibration

In order to use the numerical model one must ﬁrst determine

the external boundary location, R

p

. For the ﬁrst cycle solution R

p

can be calculated by Eq. (A.7). At the ﬁfteenth cycle the tempera-

ture penetration will be larger owing the continuous heat ﬂow

caused by the higher average air temperature versus that of the

distant rock temperature. Nevertheless, it was found that R

p

can

be calculated through Eq. (A.7) by replacing t

p

(in the Fo number)

with 15t

p

. In all calculation

e

= 0.5%.

The stretching parameter bwas chosen such that the numerical

model will ﬁt all the analytical solutions of the simpliﬁed cases

(Eqs. (A.1),(A.3) and (C.3),(C.4)). In each comparison, Eq. (27) of

the numerical model was adjusted to fully match the equivalent

analytical problem (e.g. Eq. (27) was replaced by T=T

1

for the

comparison with Eq. (A.1)). It follows that bis insensitive to the

form of Eq. (27).

Appendix B. Solutions for limiting cases

In the limiting case of perfectly conducting rocks, the rock tem-

perature remains constant owing to the formation immense heat

capacity. Hence, the air temperature variations are directly derived

from the cavern energy balance equation. The nth cycle tempera-

ture variations obtained from Eq. (10), are therefore as follows:

T

T

R

¼a

1

þa

2

q

þT

0n

T

R

a

1

a

2

q

ðb

1

Þ

;06t

6t

1

ðB:1Þ

T

T

R

¼1þTðt

1

Þ

T

R

1

e

b

2

tt

1

ðÞ

t

2t

1

ðÞ

;t

1

<t

6t

2

ðB:2Þ

T

T

R

¼a

3

þa

4

q

þTðt

2

Þ

T

R

a

3

a

4

x

q

x

b

3

;t

2

<t

6t

3

ðB:3Þ

T

T

R

¼1þTðt

3

Þ

T

R

1

e

b

4

tt

3

ðÞ

1t

3

ðÞ

;t

3

<t

61ðB:4Þ

where

a

1

¼

c

T

i

=T

R

þq

r

c

R

þq

r

;a

2

¼U

q

c

R

þq

r

þ1;a

3

¼q

r

CDR

þq

r

;

a

4

¼U

q

1R

q

r

=CD ðB:5Þ

b

1

¼

c

R

þq

r

;b

2

¼q

r

m

r

t

2

t

1

ð1þm

r

Þt

1

;b

3

¼R

þq

r

CD ;

b

4

¼q

r

m

r

1t

3

t

1

ðB:6Þ

x¼1þm

r

ðB:7Þ

T

0n

is the initial air temperature of the nth cycle, and t

⁄

is the dimen-

sionless elapsed time from the beginning of that cycle. Note that, for

q

r

= 0 the solutions reduce to those of adiabatic caverns, whereas for

q

r

?1, the wall heat transfer resistance is absent and the air tem-

perature remains constant (at T

R

).

5628 R. Kushnir et al. / International Journal of Heat and Mass Transfer 55 (2012) 5616–5630

Consider a series of cycles beginning with T

01

T

0

=T

R

at the

ﬁrst cycle. The temperature variations at subsequent cycles can

be determined by substituting the prior cycle end temperature as

initial condition for the current cycle. Evidently, steady periodic

variations are reached when the cycle initial and end temperatures

are identical. Thus, the initial temperature of a steady periodic cy-

cle, T

0s

, is obtained by setting t

⁄

=1,T=T

0s

and T

0n

=T

0s

in Eq. (B.4)

and solving for T

0s

, which yields

T

0s

T

R

¼1

ð1a

1

Þx

b

1

1

a

2

x

b

1

þ1

1

þð1a

3

Þx

b

1

e

b

2

x

b

3

1

a

4

x

b

1

þ1

e

b

2

x

b

3

1

1

x

b

1

þb

3

e

b

2

þb

4

1

ðB:8Þ

The most interesting values of the temperature are the maxi-

mum and minimum values, occurring at the end of charging stage

and discharge stage, respectively. Hence, the maximum and mini-

mum steady state temperatures are

T

s;max

T

R

¼a

1

þa

2

xþT

0s

T

R

a

1

a

2

x

b

1

ðB:9Þ

T

s;min

T

R

¼a

3

þa

4

þa

1

1þa

2

xþT

0s

T

R

a

1

a

2

x

b

1

e

b

2

þ1a

3

a

4

x

x

b

3

ðB:10Þ

The corresponding pressures, following Eq. (16), are

p

s;min

P

0

¼Z

min

Z

0

T

s;min

T

R

ðB:11Þ

p

s;max

P

0

¼ð1þm

r

ÞZ

max

Z

0

T

s;max

T

R

ðB:12Þ

where Z

min

=Z(

q

0

,T

s,min

) and Z

max

=Z(

q

0

(1 + m

r

), T

s,max

). For a given

set of operating conditions and reservoir characteristics the re-

quired cavern volume (appearing in m

r

) and initial ﬁll pressure P

0

can be derived from Eqs. (B.11) and (B.12).

Appendix C. Approximate ﬁrst cycle solution

An approximate analytical solution for the air and rock temper-

ature variations during the ﬁrst cycle is presented herein. The solu-

tion is for a constant average air density representation.

C.1. Compression stage solution

The Laplace transform of Eqs. (10)–(14) (with

q

¼

q

a

v

), for the

ﬁrst cycle compression stage, yields the following solutions

T

ðsÞ¼

cT

i

cþR

þU

q

q

a

v

BiK

0

ﬃﬃﬃﬃ

s

Fo

p

þﬃﬃﬃﬃ

s

Fo

pK

1

ﬃﬃﬃﬃ

s

Fo

p

sBi

q

a

v

t

1

s

m

r

þcR

K

0

ﬃﬃﬃﬃ

s

Fo

p

þ

q

a

v

t

1

s

m

r

þcR

þq

r

ﬃﬃﬃﬃ

s

Fo

pK

1

ﬃﬃﬃﬃ

s

Fo

p

ðC:1Þ

T

R

ðr

;sÞ¼

cT

i

cþR

þU

q

q

a

v

BiK

0

ﬃﬃﬃﬃ

s

Fo

pr

sBi

q

a

v

t

1

s

m

r

þcR

K

0

ﬃﬃﬃﬃ

s

Fo

p

þ

q

a

v

t

1

s

m

r

þcR

þq

r

ﬃﬃﬃﬃ

s

Fo

pK

1

ﬃﬃﬃﬃ

s

Fo

p

ðC:2Þ

where T

ðsÞand T

R

(r

⁄

,s) denote the Laplace transforms of T

⁄

(t

⁄

)1

and T

R

(r

⁄

,t

⁄

)1, and K

0

and K

1

are the modiﬁed Bessel functions of

the second kind of order zero and one, respectively. The solutions

T

⁄

(t

⁄

) and T

R

ðr

;t

Þare obtained by applying the residue theorem

to the inversion integral. Consequently, the air and rock tempera-

tures during the ﬁrst cycle compression stage 0 6t

6t

1

are

T

C

¼1þc

1

q

r

ﬃﬃﬃﬃﬃﬃ

e

Fo

q2

p

Z

1

0

1e

n2mrt

q

a

v

t

1

w

2

Y

ðnÞþw

2

J

ðnÞ

dn

nðC:3Þ

T

R;C

¼1þc

1

Z

1

0

Y

0nr

ﬃﬃﬃﬃ

e

Fo

p

!

w

J

ðnÞJ

0nr

ﬃﬃﬃﬃ

e

Fo

p

!

w

Y

ðnÞ

!

ð1e

n2mrt

q

a

v

t

1

Þ

w

2

Y

ðnÞþw

2

J

ðnÞ

dn

n

ðC:4Þ

where

w

Y

ðnÞ¼w

1

ðnÞY

0

n

ﬃﬃﬃﬃﬃﬃ

e

Fo

p

!

þw

2

ðnÞY

1

n

ﬃﬃﬃﬃﬃﬃ

e

Fo

p

!

w

J

ðnÞ¼w

1

ðnÞJ

0

n

ﬃﬃﬃﬃﬃﬃ

e

Fo

p

!

þw

2

ðnÞJ

1

n

ﬃﬃﬃﬃﬃﬃ

e

Fo

p

! ðC:5Þ

and

w

1

(n),

w

2

(n), c

1

,e

Fo and e

Bi are all deﬁned in Eqs. (32)a–c. By

substituting r

⁄

=1in (C.4) and utilizing the Bessel functions prop-

erty relation Y

0

(z)J

1

(z)J

0

(z)Y

1

(z) = 2/(

p

z), the cavern wall temper-

ature is obtained as follows:

T

Rw;C

¼1þc

1

ﬃﬃﬃﬃﬃﬃ

e

Fo

q2

p

Z

1

0

w

2

ðnÞ1e

n2mrt

q

avt

1

!

w

2

Y

ðnÞþw

2

J

ðnÞ

dn

n

2

ðC:6Þ

It should be noted that the inversion procedure was not applied di-

rectly to T

(s) and T

R

(r

⁄

,s), since the latter would produce integrals

that require complex numerical calculations. Instead, Eqs. (C.3) and

(C.4) were obtained by applying the inversion procedure to sT

(s)

and sT

R

(r

⁄

,s), and subsequently using the Laplace transforms prop-

erty (where L

1

denotes the inverse Laplace transform operator)

T

¼Z

t

0

L

1

fsT

ðsÞgdt

ðC:7Þ

As it turns out, the solutions for typical CAES conditions can be

signiﬁcantly simpliﬁed trough the introduction of the Bessel func-

tion asymptotic approximations. The asymptotic expansions of

K

0

(z) and K

1

(z) for large values of zare

K

0

ðzÞ ﬃﬃﬃﬃ

p

2z

pe

z

1

1

8z

þ

9

128z

2

K

1

ðzÞ ﬃﬃﬃﬃ

p

2z

pe

z

1þ

3

8z

15

128z

2

þ

ðC:8Þ

Typically, the values of Fo are small enough (see Table 2)soasto

represent the Bessel functions of Eqs. (C.1) and (C.2) by their ﬁrst

asymptotic term. The substitution of the asymptotic approxima-

tions of K

0

and K

1

, yields

T

ðsÞ¼

c

T

i

c

þR

þU

q

q

a

v

Bi þﬃﬃﬃﬃ

s

Fo

p

sBi

q

av

t

1

s

m

r

þ

c

R

þ

q

av

t

1

s

m

r

þ

c

R

þq

r

ﬃﬃﬃﬃ

s

Fo

p

ðC:9Þ

T

R

ðr

;sÞ¼

Bi

c

T

i

c

þR

þU

q

q

a

v

e

ﬃﬃﬃ

s

Fo

p

ðr

1Þ

ﬃﬃﬃﬃ

r

psBi

q

av

t

1

s

m

r

þ

c

R

þ

q

av

t

1

s

m

r

þ

c

R

þq

r

ﬃﬃﬃﬃ

s

Fo

p

ðC:10Þ

By repeating the derivation procedure of Eqs. (C.3) and (C.4), the

asymptotic formulas of Section 4.2 are obtained.

C.2. Full cycle solution

The solution for the compression stage temperatures can serve

as a building block for the construction of the full ﬁrst cycle tem-

peratures. This is accomplished by representing the temperature

solutions as superposition of similar functions, namely

T

C

¼1þT

1

ðt

Þ;06t

6t

1

T

R;C

¼1þT

R1

ðr

;t

Þ;06t

6t

1

ðC:11Þ

T

S1

¼T

C

ðt

ÞT

2

t

t

1

;t

1

<t

6t

2

T

R;S1

¼T

R;C

ðr

;t

ÞT

R2

r

;t

t

1

;t

1

<t

6t

2

ðC:12Þ

T

G

¼T

S1

ðt

ÞT

3

t

t

2

;t

2

<t

6t

3

T

R;G

¼T

R;S1

ðr

;t

ÞT

R3

ðr

;t

t

2

Þ;t

2

<t

6t

3

ðC:13Þ

T

S2

¼T

G

ðt

ÞþT

4

t

t

3

;t

3

<t

61

T

R;S2

¼T

R;G

ðr

;t

ÞþT

R4

ðr

;t

t

3

Þ;t

3

<t

61ðC:14Þ

R. Kushnir et al. / International Journal of Heat and Mass Transfer 55 (2012) 5616–5630 5629

When these expressions are substituted into Eqs. (10)–(14) (with

q

¼q

a

v

), it follows that all the functions, T

1

through T

4

, satisfy

the same differential equation form, however with different coefﬁ-

cients. Hence, the temperatures T

2

through T

4

and T

R2

through T

R4

can be determined from the ﬁrst compression stage solutions sub-

ject to a simple adjustment of the appropriate parameters. By that

technique the following asymptotic expressions are derived

T

2

ðt

Þ¼

c

T

i

þU

q

q

a

v

ð

c

R

ÞT

C

t

1

þt

2

2

p

Z

1

0

q

r

e

Bi 1e

n2mrt

q

avt

1

!

dn

e

Bi

2

n

4

þn

2

q

r

n

2

2

ðC:15Þ

T

R2

ðr

;t

Þ¼ cT

i

þU

q

q

a

v

cR

ðÞT

C

t

1

þt

2

2~

Bi

p

ﬃﬃﬃﬃ

r

p

Z

1

0

e

Bin

2

sin

nðr

1Þ

ﬃﬃﬃﬃ

e

Fo

p

!

þnðq

r

n

2

Þcos

nðr

1Þ

ﬃﬃﬃﬃ

e

Fo

p

! !

1e

n2mrt

q

a

v

t

1

!

e

Bi

2

n

4

þn

2

q

r

n

2

2

dn

n

ðC:16Þ

T

3

t

ðÞ¼ U

q

q

a

v

þR

T

S1

t

2

þt

2

2CD

p

Z

1

0

q

r

e

Bi 1e

n2mrt

q

avt

1

!

dn

e

Bi

2

CDR

n

2

2

þn

2

CDR

þq

r

n

2

2

ðC:17Þ

T

R3

ðr

;t

Þ¼ U

q

q

a

v

þR

T

S1

t

2

þt

2

2CDe

Bi

p

ﬃﬃﬃﬃﬃ

r

p

Z

1

0

e

Bi CDR

n

2

sin

nðr

1Þ

ﬃﬃﬃﬃ

e

Fo

p

!

þnCDR

þq

r

n

2

cos

nðr

1Þ

ﬃﬃﬃﬃ

e

Fo

p

! !

1e

n2mrt

q

a

v

t

1

!

e

Bi

2

CDR

n

2

2

þn

2

CDR

þq

r

n

2

2

dn

n

ðC:18Þ

T

4

ðt

Þ¼ U

q

q

a

v

þR

T

G

t

3

þt

2

2CD

p

Z

1

0

q

r

e

Bi 1e

n2mrt

q

avt

1

!

dn

e

Bi

2

n

4

þn

2

q

r

n

2

2

ðC:19Þ

T

R4

ðr

;t

Þ¼ U

q

q

a

v

þR

T

G

t

3

þt

2

2CDe

Bi

p

ﬃﬃﬃﬃ

r

p

Z

1

0

e

Bin

2

sin

nðr

1Þ

ﬃﬃﬃﬃ

e

Fo

p

!

þnðq

r

n

2

Þcos

nðr

1Þ

ﬃﬃﬃﬃ

e

Fo

p

! !

1e

n2mrt

q

a

v

t

1

!

e

Bi

2

n

4

þn

2

q

r

n

2

2

dn

n

ðC:20Þ

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