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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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