Latent heat thermal storage for solar dynamic power generation

Solar Energy Vol. 51, No. 3, pp. 169-173, 1993

0038-092X/93 $6.00 + .00 Copyright © 1993 Pergamon Press Ltd.

Printed in the U.S.A.

LATENT HEAT THERMAL STORAGE FOR SOLAR DYNAMIC POWER GENERATION C. BELLECCI* and M. CONTI** *Dipartimento di Fisica, Universita' delia Calabria, 87030 Rende, Italy, **Dipartimento di Matematica e Fisica, Universita' di Camerino, 62032 Camerino, Italy Abstract--Latent heat thermal storage is very attractive for space-based solar dynamic power systems. In this article, we describe a solar receiver unit (SRU) integrated with a phase change thermal storage facility. A two-dimensional model has been set up to simulate the transient behavior of the system. The resulting equations have been solved numerically by the finite difference method, and a parametric study on the performances of the SRU has been conducted; the results are presented and discussed.

1. I N T R O D U C t i O N

Space-based activities in low earth orbits are marked by a strong increase of the electric power demand. The "Freedom" space station is projected at 75 kW; the European "Columbus Free Flyer" will start with 25 kW. Due to the low conversion efficiency, photovoltaic generation is not attractive for plants of that size: the large absorber area results in too high drag losses. Moreover, the storage of the electric energy requires too heavy and short-life batteries. A better solution is the solar dynamic power generation via high temperature thermodynamical cycles (Brayton, Stirling)[l,2]. In this case a proper thermal storage facility is a crucial point of the conversion system, in order to bridge the eclipse phases. The operating storage temperatures lie in the range 900-1200 K; the transient times for the heat charge and release are in the order of 1 h. Thermal storage as latent heat in a solid-liquid phase change has been proved an interesting way in two respects. 1. It allows high storage density--an essential point in space applications; and 2. heat is charged and released at a fixed temperature and the thermodynamical cycle can take advantage of low temperature fluctuations. A basic solar receiver unit (SRU) integrated with latent heat thermal storage is sketched in Fig. 1. A tube is surrounded by an annular envelope, filled with a phase change material (PCM). The receiver collects the solar radiation focused by a concentrator. A working fluid flows through the tube and then is supplied to the heat engine in a closed hydraulic loop. During sunlight the PCM melts; heat propagates inward and is partly stored in the phase transition, and partly delivered to the working fluid. During the eclipse phase, the PCM solidifies and the stored heat is extracted by the fluid. The SRU is the basic module of a photothermal conversion system conceived for the "Freedom" sta-

tion[l]. In this article, a numerical model to describe the transient behavior of the SRU is presented. The solid-liquid interface in the PCM moves continuously with time, and the problem cannot be reduced to a simple solution of the Fourier equation. The difficulty has been removed by the application of the standard enthalpy method[3-7].. The fluid velocity has been regarded as an independent variable as is usual in turbulent flow forced convection problems. Hence, the continuity and momentum equations dropped out, and we had to deal with the energy equation alone. The SRU is supposed to be lighted with uniform and constant intensity during the active phase, and heat diffusion in the PCM is assumed axisymmetric. The model has been solved by the finite difference method, and the influence of some design parameters on the performances of the SRU has been analyzed. 2. MATHEMATICALMODEL The working fluid (an ideal gas) is assumed to be incompressible, and viscous heating is neglected. The heat transfer inside the SRU is governed by the energy equations, written for the fluid and the PCM: 2.1 Fluid ___

OTF

OTF t- v . . Ot Ox

. 2 . .r . h .. r l PF" CF" A

[ir(r = r , ) - TF].

(1)

+ ~x

(2)

2.2 P C M

Ot

r Or k . r.-~r

-~x

"

H in eqn (2) is the specific enthalpy of the PCM, i.e., the enthalpy per unit volume, and is related to the temperature field via H(~=ps'Cs'?

(~ < TM)

H('P) = os" cs" TM + OL" X

* ISES member.

+ PL" CL" ( ? -- TM)(? > TM).

169

(3)

170

C . BELLECCI a n d M . C O N T I

]

FCM

]

F L U I D ~_..~--~.............................................. q L _ ~ . . . T~N t

~M

I

Tout

I

I

0

L

" x

Fig. 1. Solar receiver unit (SRU) module.

The advantages of this approach (the enthalpy method) are: 1. no boundary conditions have to be satisfied at the solid-liquid interface; 2. the solution does not require an accurate tracking of the interphase boundary. The fluid flow in eqn (1) is assumed radially uniform; moreover the axial velocity v is regarded as an independent variable. Dimensional analysis of the Fourier equation shows that heat diffusion is not substantially influenced by the container walls when c~,o. r >> 62

(4)

where a~ and 6~ are the thermal diffusivity of the wall and the wall thickness, respectively, and 7 is the transient time. In our case, we can assume a~ = 3.9. 10 -6 m2/s (stainless steel walls), 6~ ~ l0 -3 m and r ~ l03 s; the condition [eqn (4)] is quite well satisfied. The initial and boundary conditions for eqns (1) and (2) are defined as follows: 2.3 I n i t i a l c o n d i t i o n s TF(X, O) = To

(5)

~(r, x, O) = To

(6)

The initial temperature To of the SRU is assumed to be uniform. The working fluid enters into the SRU at the constant temperature TIN. Equations (8), (9), and (10) represent the continuity of the heat flux at the fluid-PCM interface and at the outer walls, respectively. The solar radiation intensity I on the SRU is assumed to be uniform and constant with time during the sunlight phase; due to the space application only radiative losses are taken into account. No heat exchange is assumed through the walls normal to the SRU axis as it is expressed by eqn (I 1). In normal operative conditions the flow is turbulent inside the tube (Re > 104); due to the high length to diameter ratio ( L / D = 120) and with a Prandtl number 0.7, we can assume a fully developed temperature profile in the most part of the fluid flow[8]; in these conditions the heat transfer coefficient h can be evaluated from the Colburn equation[8]: Nu = 0.023. Pr °'3. Re °'8.

(13)

Equations (1) and (2) have been approximated by an implicit finite difference scheme. The resulting algebraic equations have been solved by the successive overrelaxations technique. The coefficient matrix is updated at each time step, and adjusted for the local transport conditions. As the enthalpy-temperature dependence is nonlinear, some iterations are required to solve for H at each time step. No experimental data are available to validate the numerical results, and the consistency of the computational scheme has been proved by performing an overall energy balance at each time step. The absorbed solar radiation is compared with the enthalpy stored in the PCM plus the thermal losses and the enthaipy gain of the working fluid. The percentage error never exceeds 0.1%.

3. N U M E R I C A L

RESULTS

The numerical calculations have been carried out with reference to the design characteristics as shown in the Appendix. The working fluid (air) is operated at a pressure of 6. l05 P a - - m i n i m u m value in the hy-

TOuT(K)

2.4 B o u n d a r y c o n d i t i o n s

1200 ro= 0 )24 m

TF(O, l) = TIN

(7) 1150

0T h. [T(r, x , t) - TF(X, t)] = k . ~ -r (r = rl)

Or = 13. I - tr. ~ . f ~ (r = ro, sunlight phase)

k Of "ar =-~'"~ OT -0 0x

aTF Ox

-0

(8) (9)

1100

1050

a

1000

(r = ro, eclipse phase)

(10) 950

(x=0, x=L)

(11)

(x=L).

(12)

I 5OOO

,

I 1130OO

15000

20000

25000

time (s) Fig. 2. Fluid temperature at the outlet of the SRU, ro = 0.024 m. Curve a: rn = 0.012 kg/s; b: rn = 0.016 kg/s; c: rn = 0.020 kg/s; d: rn = 0.024 kg/s.

Thermal storage for power generation

2. economy in the PCM mass and size is a central point for space-based applications. Now we are going to see how the oscillations of Tour are affected by the PCM mass. The influence of the fluid mass flow rate will also be shown. The inner radius rl is an external condition of the S R U design, hence the mass of the PCM is determined by the choice of the outer radius ro and this has been selected as the representative variable to show the results. In Fig. 2, the oscillations of Tour are represented versus the time at different values of the gas flow rate; ro is set at 0.024 m, that means a PCM mass of 9.34 kg. The S R U shows a small thermal inertia: after just two simulated orbits steady cyclic conditions are attained. Low slope ranges can be observed: here heat is absorbed and extracted as latent heat in the phase changes; high slope ranges indicate, however, that overheating and/or subcooling of the PCM occurs during the cycle. As rh increases Tour lowers and all the S R U works on the average at lower temperatures. A quite different situation is depicted in Fig. 3 where ro = 0.052 m, corresponding to a PCM mass of 35.4 kg. Here we see a high thermal inertia of the SRU: after four orbits steady cyclic reproduction is not yet attained. The oscillations have been strongly dumped; the drawback is the high value of the PCM mass. In Fig. 4, curves a and b represent the m a x i m u m and m i n i m u m values of Tour, respectively, versus the fluid flow rate in a steady cycle. The window between the two curves decreases as ro increases. At each to, the flow rate determines the range allowed for Toc,r. A proper selection of rh and ro results in a narrow range for the outlet temperature with a not too high PCM mass. A better understanding of the problem can be achieved if we note that for an efficient operation of

Tour(K) 1200 a ro = 0.052 m

b

1150

c,

d 110o

e

1050

10o0

950

I

i

5000

I

,

10000

I

i

15000

I

,

20000

171

25000

time (s) Fig. 3. Fluid temperature at the outlet of the SRU. ro = 0.052 m. Curve a: rh = 0.020 kg/s; b: rh = 0.028 kg/s; c: rh = 0.036 kg/s; d: rh = 0.044 kg/s; e: rn = 0.052 kg/s.

draulic loop. Hence even a moderate fluid velocity allows high mass flow rate, i.e., fast response of the storage system. Lithium fluoride has been selected for the PCM, due to the high heat of fusion and a proper melting temperature. The geometric features of the SRU shown in the Appendix as well as the solar radiation intensity correspond to design characteristics fixed within the " F r e e d o m " project. The working fluid is injected at a fixed temperature TtN into the SRU, where it is heated (Fig. 1). The outlet temperature Tour oscillates with time, due to the alternating sunlight and eclipse phases. Two leading criteria should be followed for a proper thermal storage design. 1. The oscillations of Tour should be kept in a narrow range in order to optimize the heat engine performances, but:

TOuT(K) 1200 1150 1100 1050 1000

\ro=

0.024 m

b~

~

950

b

b

900 1150

ro= 0.032 m

.028 m

.

ro= 0.036 m

I

i

I

i

I

i

I

1

I

i

I

i

ro= 0.052 m

ro= 0.040 rn

1100 1050

1000

b

b

b

95O

9000 ' 1; ' 2b ' 3b '4; 'sb'

't; '2; ' 3o' '4; 'sb '

' lO' '2; '3; ' 40' '5; '

M A S S F L O W R A T E (kg/s .10 .3 ) Fig. 4. Fluid temperature at the outlet of the SRU vs. the mass flow rate. Curves a and b represent the maximum and the minimum values, respectively, in a steady cycle.

172

C. BELLECCIand M. CONT|

F ro= 0.024 m

1.5

1

ro= 0.028 m

ro= 0.032 m

~a

-..<

"d

"-4

0.5 0 -0.5 -1

.

ro= 0.040 m

ro= 0.036 m

1.5

.

i

.

i

.

i

,

i

.

ro= 0.052 m

-..<

-..<

1

i

0.5

b

0 -0.5 -1

0 '1; '2; '3; '4; 50

"1; 2 ;

3;

"40 " 5 ;

1 ; "2; 3 ;

413 "5; 6 0

M A S S F L O W R A T E (kg/s .10 -3 ) Fig. 5. F values versus the mass flow rate. F is defined by eqn (14). Curves a and b represent the maximum and minimum values, respectively, in a steady cycle.

the thermal storage heat must be stored as latent heat in the phase change: overheating as well as subcooling of the PCM has to be avoided. A nondimensional storage density can be defined as F -

M.X

(14)

w h e r e / 1 represents the enthalpy stored in the PCM:

I ~ = fvec~, ( H -

ps. Cs. TM).dxdydz.

(15)

If, in a cycle, FMIN < 0 and/or FMAX > 1, extraction and/or storage of sensible heat in the PCM is being performed. Subcooling and/or overheating occurs and Tour runs away. If FMIN > 0 and/or FMAx < 1 results; it means that only a fraction of the PCM is involved in the phase change. In this case an excessive a m o u n t of the PCM has been utilized. In a proper design, the storage density F oscillates between 0 and 1. In Fig. 5 curves a and b represent the m a x i m u m and m i n i m u m values of F, in a steady cycle, versus the fluid flow rate. As expected, the excursions of F decrease as ro increases. Sensible heat operation cannot be prevented at ro = 0.024 m and ro = 0.028 m; furthermore the graphs show that for ro > 0.036 m, part of the P C M is excluded from the phase change. However, it can be noted that for ro = 0.032 m and rh = 0.023 kg/s, F ranges exactly between 0 and l; it should be the best operative condition.

4. CONCLUSIONS

Solar dynamic power generation is attractive for space-based applications. Stability of the thermal power

supplied to the heat engine, as well as of the operating temperatures, require a proper design of the thermal storage system. Thermal storage as latent heat in a solidliquid phase transition might be a promising solution. The numerical model that has been presented simulates the behavior of a S R U with a latent heat thermal storage facility. The model has been solved, and the influence of some design parameters on the performances of the system has been analyzed. The storage mass is a critical point for space applications. The results we have presented indicate some criteria to determine the optimal size of the storage system. NOMENCLATURE A c D F h H /] I k L M th Nu Pr Re r,, ro t TTr Tm TM To Tour x, r a~ #

area of the fluid flow cross section, 7r. R~ specific heat internal diameter of the solar receiver unit (SRU) nondimensional storage density, as defined in eqn (14) convective heat transfer coelficient enthalpy per unit volume enthalpy stored in the PCM, as defined in eqn (15) solar radiation intensity on the SRU outer walls thermal conductivity of the PCM length of the SRU PCM mass mass flow rate of the fluid flow Nusselt number, defined as h. D/kF Prandtl number, defined as cr" #/kr Reynolds number, defined as #r" v. D/# inner and outer radius of the SRU, respectively time temperature in the PCM and in the fluid, respectively fluid temperature at the inlet of the SRU melting temperature of the PCM initial temperature of the SRU fluid temperature at the outlet of the SRU axial and radial coordinates, respectively thermal diffusivity of the SRU walls absorptivity of the SRU outer walls

Thermal storage for power generation 6~ thickness of the SRU walls emissivity of the SRU outer walls emissivity of the SRU outer walls latent heat of the PCM fluid viscosity 0 density Stefan-Boltzman constant

173

mesh for enthalpy formulations of phase change problems, IMA J. Numerical Analysis 11, 55 (1991). 8. W. M. Kays and H. C. Perkins, Forced convection, internal flow in ducts. In: W. M. Rohsenow and J. P. Hartnett (eds.), Handbook of heat transfer, McGraw-Hill, New York (1973). APPENDIX

Subscripts F fluid L liquid phase of the PCM MAX maximum value MIN minimum value S solid phase of the PCM

REFERENCES

1. H. J. Strumpf and M. G. Coombs, Solar receiver experiment for the space station Freedom Brayton engine, J. Sol. Energy Eng. 112, 12 (1990). 2. C. Bellecci and M. Conti, Thermal energy storage in a porous medium: An entropy generation approach in a power production perspective (to be published in II Nuovo Cimento C). 3. R. M. Furzeland, A comparative study of numerical methods for moving boundary problems, J. Inst. Math. Appl. 26, 411 (1980). 4. V. Voller and M. Cross, Accurate solutions of moving boundary problems using the enthalpy method, Int. Z Heat Mass Trans. 24, 545 (1981). 5. F. Civan and C. M. Sliepcevich, Efficient numerical solution for enthalpy formulation of conduction heat transfer with phase change, Int. ,L Heat Mass Trans. 27, 1428 (1984). 6. Y. Cao and A. Faghry, Performance characteristics of a thermal energy storage module: A transient PCM/forced convection conjugate analysis, Int. J. Heat Mass Transfer 34, 93 (1991). 7. D. B. Duncan, A simple and effective self-adaptive moving

Design parameters Working fluid (air): Molecular mass Pressure (minimum value in the loop) Temperature at the inlet of the SRU

0.029 kg/kmol 6.105 Pa 950 K.

PCM (lithium fluoride): Melting temperature Latent heat Density Density Specific heat Specific heat Thermal conductivity Thermal conductivity Initial temperature

(solid phase) (liquid phase) (solid phase) (liquid phase) (solid phase) (liquid phase)

l 122 K 1.04- 106 J/kg 2330 kg/m 3 1800 kg/m 3 2350 J/(kg. K) 2450 J/(kg. K) 4 W/(m.K) 1.73 W/(m. K) 1100 K.

SRU." Length Radius of the inner tube Emissivity Radiation intensity on the outer surface

Operating cycle. Duration of the sunlight phase Duration of the eclipse phase

2800 s 2800 s.

2.4 m 0.01 m 0.04 2. l04 W/m 2.