Applied Energy 30 (1988) 131-146

Steady Heat Transfers through a Phase-Changing Fluid in a Solar Collector Circuit

J. Prakash* & M. Bertel~_

Dipartimento di Energetica, UniversitY. di Firenze, Via di S. Marta-3, 50139 Florence, Italy

A BS TRA C T

In this paper we have presented the design and steady-state analysis o f a solar thermal system employing a phase-changing f luid as a heat carrier for transporting thermal energy f rom the collection side to the user side, without the use o f an external mechanical device. The connection between the behaviour o f the system and the various geometrical parameters which are representative o f the system performance is also established. It was found that Freon 11 has all the necessary requisite properties to be a candidate as the working fluid for such a system.

NOTATION

a

b c

C1 d D e

f g G h i

A constant (45"584 98, dimensionless). A constant (4173.274 K). A constant (3.512 109, dimensionless). Specific heat of Freon 11 liquid at 295 K (858 J/kg K). A constant (291 930 J/kg). Diameter of tubes (m). A constant (370 J/kg K). Friction factor (dimensionless). Acceleration due to gravity (9-81 m/s2).

Mass flow rate (kg/s). Height with respect to z = 0 (m). Specific enthalpy (J/kg).

* On study leave from the Department of Physics, Ramjas College, University of Delhi, Delhi-110007, India.

131 Applied Energy 0306-2619/88/$03.50 © 1988 Elsevier Applied Science Publishers Ltd, England. Printed in Great Britain

132 J. Prakash, M. BertelcJ

L~ = L v = L Length of tubes (m). p Static pressure (Pa). q~ Inlet power (W). qo Outlet power (W). R v Gas constant (58.6 J/kg K). R e 1 = Dwl /v~ Reynolds number for liquid phase (dimensionless). R e v = D w M / v v Reynolds number for vapour phase (dimensionless). S Cross-sectional area of tubes (m2). T Temperature (K). w Velocity (m/s). y A function (J/kg), z A coordinate (mt.

G r e e k l e t t e r s

0q Coefficient for liquid friction term (mZ/kg/s). % A design parameter (dimensionless). 2v Coefficient for vapour friction term (m 4 s - 3 6 4 5 K 1.645).

F Friction factor term (J/kg). A Height difference between the two interfaces (m).

A design parameter (dimensionless). 2 Latent heat of evaporation of Freon 11 (J/kg). v~ Kinematic viscosity of Freon 11 liquid at 295 K (0"286 x 10 -6 mZ/s). v v Kinematic viscosity of Freon 11 vapour at 295 K (1"88 x 10 - 6 m2/s). tq Density of Freon 11 liquid at 295 K (1525 kg/m3).

S u b s c r i p t s

A, B, C, E, F, M Indicate the positions in the circuit where the quantities are considered.

1 Liquid. v Vapour.

I N T R O D U C T I O N

In most solar thermal systems heat is transported by means of a liquid which is heated at the collection side and is cooled at the load side through a heat exchanger. The circulation of the liquid is assured either by an external pump or by the thermal gradients which are established in it, as in the case of thermosyphon flow. Whereas the pump needs an external source of power, a check valve must be used in the second case against the reverse flow from the store during the night, which would otherwise cause a loss of energy through the collector. In both the processes mentioned, only the sensible heat of the liquid is transferred. For the transportat ion of large quantities of heat, either the temperature gradient in the working fluid or its mass flow rate, or both,

Steady heat transfers 133

are to be considerably increased, but the choice of the increase in mass flow rate is possible only if the system is pump-assisted; in the case of the thermosyphon, the only way to transfer more energy is to increase the temperature of the fluid at the energy-collection side. This in turn results in larger heat losses from the collector surface and the connecting tubes.

To overcome this unfavourable situation, many workers 1'2 have suggested the use of a phase-changing fluid for heat transport. Some of its advantages are immediately seen: (i) lower operating temperatures because of the high latent heat of vaporisation; (ii) heat is transferred at an almost constant temperature; and (iii) no need to employ a check valve to avoid the reverse flow in the night.

In the present paper, the steady-state operation of a system with a phase- changing fluid as the heat carrier is studied in order to obtain information about its performance. The layout of the apparatus is shown schematically in Fig. 1. Computations permit the ranges of variability for the quantities which govern the operation of the system to be determined. Then the convenience, the advantages and/or the possible drawbacks of such a heat transport system can be assessed.

Results for the steady-state operation of the system can be obtained by the macroscopic mass and energy balance equations.

DESCRIPTION OF THE SYSTEM

A schematic diagram of the system using a phase-changing fluid as the heat carrier, which is discussed and analysed in the present study, is shown in Fig. 1. It consists of a solar collector and a condenser connected by tubes to form a closed circuit. Both the collector and the condenser are half filled with a phase-changing fluid. It is supposed that most of the energy gathered by the

{

VAPOUR

LI<}U!D

C qo

E

Fig. h Schematic diagram of the present system.

134 J. Prakash, M. Bertelh

solar collector is used for the phase change of the working fluid. It moves as a vapour from A to C; there it condenses and moves as a liquid from C to A, where it again evaporates. Pumps are not needed for the accomplishment of these movements as they are caused by the pressure difference which is established between the solar collector and the condenser. The latter is maintained at a constant pressure, Pc, while T c is the corresponding saturation temperature. The same connection holds between PA and T A. Freon 11 was chosen as the working fluid because of its very low boiling point (BP). (At a pressure p = 1.011 × 105 Pa its BP equals 23"7°C. 3) This feature is particularly suitable for solar devices. Further, small variations of temperature (say 5 K) around the BP yield neither too high pressures, with structural implications, nor too low depressions, with consequent sealing problems. Specific and latent heats of Freon 11 are rather large and these are other attributes of the design. Therefore this fluid is potentially a highly suitable heat-carrier for the present system.

Both the thermally-insulated connecting tubes for the liquid (from E to F) and vapour (from B to M) are horizontal. However, real geometrical situations different from this might be requested. Although most of the calculations were performed for the configuration sketched in Fig. 1, the behaviours of the system in two other configurations were also investigated.

A knowledge of the mass flow rates which may occur in such a system and their connections with the pressures and temperatures which are established in it is necessary. The mass flow rate gives information about the magni- tude of the heat transport by this method, while the pressures and temperatures at the collector and the condenser sides are the design parameters for the system.

Before the start of the analysis, it was clear by inspection that the behaviour of the system depends on several geometrical parameters, such as the diameters and the lengths of the vapour and liquid connecting tubes. After a few preliminary calculations, it was seen that the two diameters could have the same value, i.e. Dj = D,. = D. Also the two lengths were assumed to be equal (i.e. L~ = L v = L) for simplicity, even if they are not shown equal in Fig. 1. It was likewise assumed that hv = hE and ha = hM. Since the tubes are thermally insulated, T~ = Tv = To, which is the lowest temperature in the cycle.

ANALYSIS

A one-dimensional analysis of the system was undertaken. The power qj enters the system through the collector, while the power extracted from the condenser is qo. It is assumed that the geometry of the apparatus is known:

Steady heat transfers 135

the physical properties of the working fluid are supposed to be independent of temperature except when it is explicitly declared. They are evaluated at T c. The input data are ql, qo, Tc and Pc, while the unknown quantities to be determined are G, TA, PA, To and A. They are sufficient to describe the operation of the system. A indicates the difference between the interface levels in the collector and in the condenser; with reference to orientation, it is positive when hc > hA.

The macroscopic energy balance equation per unit mass for the vapour phase between B and M (Fig. 1) may be written as follows:

;/ 0"5(W2M -- W 2) + g(hM -- hB) + dp/pv + F v = 0 (1)

It was found that, near T c = 295 K andpc = 0"953 × 105 Pa, the perfect-gas equation may be used for Freon 11 vapour, provided that the gas constant, Rv, is computed by a suitable fitting. This allows us to transform the kinetic- energy terms in eqn (1) into functions of temperature and mass flow rate. Using G = pvSwv and introducing the realistic assumptions TH ~- TA, TM ~ Tc, PB -~ PA and PM ~ PC, we have

w 2 2 2 2 2 _ - = RvG (T~/pc T2/p2)/S 2 (2)

To evaluate the integral in eqn (1), isothermal flow at temperature T c is assumed; then

fn ~ dpv/Pv = RvTcln(pM/pB) R~Tc (3) ln(pC/PA)

The friction loss, F v, between B and M is given by

F v = 0.5JvLW~/D

It is convenient to also express it in terms of G. A very accurate fitting for./~, for a smooth tube in terms of the Reynolds number, Re~, has yielded

j~ = Rev °'355 5 × 104 < Re v < 5 × l0 s (4)

Using eqn (4) and expressing PM ~- Pvc by the perfect-gas equation, then

F v = 0.744vv°355 Rvl.OaS(Tc/Pc) 1.64SG1.645L/D4.64s

which for Freon 11 becomes

F~=5"586(Tc/Pc)l"645GI"645L/D4"645 or Vv=Otv(Tc/Pc)"6'*SG 1"645 (5)

where ~v = 5"586L/D4645. Substitutions from eqns (2), (3) and (5) into eqn (1) yield

2 2 2 05Rv(T~ /p c T2A/pZ)G2/S 2 + RvTc(lnpc lnp A) + Otv(GTc/Pc) ~645 = 0 (6)

136 J. P r a k a s h , M . B e r t e h i

The macroscopic mechanical-energy balance equation for the liquid phase between E and F (Fig. 1), assuming the flow to be isothermal at temperature To, is

0 " 5 ( W 2 - - w2) + (PA - - Pc)/P, -- gA + F, = 0 (7)

For reasons of continuity, the velocity of the liquid phase is uniform all along the tube; thus

,~'~ = w E = w , ( 8 )

The friction loss in terms of G is

F t = 40"774(v , / p , )GL/D 4 (9)

In writing eqn (9), we have made use of / I = 64~Re 1 since the liquid flow is laminar. For Freon 11, F1 may be written as

F 1 = ~IG (10)

where ~1 = 7"65 x 1 0 - g L / D 4. Using eqns (8) and (10), eqn (7) becomes

PA - - Pc -- gPl A + ~ lp tG = 0 (11)

The thermal-energy balance equations at the condenser and the collector are given by

G C , ( T c - To) + G2c = qo (12)

and

respectively, while GC,(T/ , - To) + G)o A = qi (13)

lnpA= a - b / T A - cln TA (14)

is a classical relation between the saturation pressure and the temperature; the constants present in it were computed for Freon 11 through an accurate fitting.

Five equations, namely (6) and (11)-(14), are at one's disposal for the determination of the five unknown quantities cited earlier. As qo expresses the amount of energy requested by the user, it must be considered. It should also satisfy G2 c <_ qo as qo is not obtained completely by the condensation of the vapour. We may write qo = ~,G2 c with :% > 1. It should be chosen in such a way that the difference (Tc -- To) is not very large (say 5 K) because too low a temperature To is not convenient for heat withdrawal.

Substituting for qo in eqn (12) and rearranging, we get

T O = T c - ( ~ q -- 1)2c/C , (15)

The mass flow rate can be computed immediately by

G : qo/(~,2c) (16)

Steady heat transfers 137

Let us put

qi = eqo = e~q2cG (17)

with e. > 1. The value of e is an index of the performance of the system and it permits the evaluation of qi, which in turn gives the size of the collector surface. It must be determined. For this task, it is convenient to express 2A in terms of T A. This was performed through the relationship

2 = d - e T (18)

where d and e were evaluated by accurate curve-fitting for 2 in the neighbourhood of T c. Substituting for To, G, qi and 2 a from eqns (15)-(18), respectively, in eqn (13) and rearranging, one gets for e

~: = T A ( C ! - e)/~¢,2 c -- (C, T c - d)/~q)o c + (~q - - 1)/¢Xq (19)

Equat ion (19) shows that e is linear in TA and this should be such that the corresponding PA is j ust capable of maintaining the desired mass flow rate G. The link between G and TA is supplied by eqn (6); its left-hand side was put equal to y and the function y =f(TA) was plotted against T A with known values of G and the geometrical parameters. The roots of eqn (6) are the values of T A for which y = 0. Only one root of eqn (6) was found for 200 K < TA < 500 K in all the conditions considered. Once the root o feqn (6) is known, e and subsequently qi can be computed easily. This allows one to design the solar collector for the desired energy output qo. Then the values of PA and A corresponding to a specific TA are computed by the use ofeqns (14) and (11), respectively. Since the coefficients in eqn (6) depend on the geometrical parameters D and L, different solutions for T A are obtained for the same G in the various geometrical conditions.

RESULTS

Many solutions to the set of eqns (6) and (11)-(14) were computed. Of course the starting point was always the qo value. As G characterises the performance of the system, its dependence on qo was shown in Fig. 2 for two values of ~q (= 1-03 and 1.05) or T 0, according to eqn (15). Since there are restrictions to the T o change, the two straight lines in the figure are quite close. Anyway G does not appear to be highly sensitive to T o variations. This could also be deduced from eqn (12).

As stated earlier, the link between G and TA, expressed by eqn (6), is also influenced by the geometrical parameters of the system. Figure 3 shows the dependence of T A on G for four different D values and one value of L. TA is not highly sensitive to D changes and it must be noted that a small rise of T A above T c is sufficient to maintain the desired mass transfer in the cycle. For a

138 J. Prakash, M. Bertel5

50

40

30

"~ 20 v

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% =288.6 K

..... 270=284.4 K

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qo (w)

Fig. 2. Var ia t ion of the mass flow rate G with the energy ou tpu t qo for two values of T 0.

given G, T A increases as D is reduced, since a larger pressure difference is required to transfer the same mass through a smaller D tube. This is also shown by Fig. 4, from which the smallness of the pressure difference necessary for the vapour movement can be appreciated.

The influence of L on the function T A = TA(G ) is shown in Fig. 5. T A for a given G increases with L because of the linear dependence between F v and the length of the connecting pipe.

The parameter e defined in eqn (17) is plotted against T n in Fig. 6. The function e depends on % or T o and T c. e presents a low sensitivity to To variations.

The last unknown quantity, A, is plotted against G in Fig. 7 for four different D values and one L value. As A is the difference in height between the two vapour- l iquid interfaces, a knowledge of it is useful for the design of the apparatus. The function A depends on the same geometrical parameters as TA. The dependence of A on D is shown in the figure; the smaller D is, the larger A must be for the same mass flow rate G in the liquid phase. It must be remembered that always PA >Pc.

TA also depends on Tc (eqn (6)), as e and A do. Figure 8 sets out that dependence only for TA, for one value of D and L. The two curves

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approximately overlap after about a 2 K rigid translation parallel to the TA axis; more precisely, the dashed line would put itself just a little under the solid one after the translation. But we have to be aware that this situation could not be repeated if Tc were largely different from 295 K.

The configuration of the apparatus with inclined connecting tubes was also investigated. No difficulty is caused to the system operation if the slope of the connecting tubes is positive because the condenser tends to be completely filled with vapour. If the difference in height is such that the l iquid-vapour interface is under the point E (Fig. 1), the cooling of the liquid from T c to To cannot occur. For a negatively increasing slope, however, the condenser would tend to be completely filled with liquid and it cannot carry out its task in that condition. This is when the interface level should fall above M. Thus an acceptable lowering of the condenser with respect to the collector must be estimated, taking account of the value of A.

C O N C L U S I O N S

The first aim of the present study was to design a simple system for transferring moderate amounts of energy from a solar collector to a user without mechanical help to move the heat carrier fluid. The second aim was to describe the operation of such a system, pointing out the connections and the dependences among the various quantities which govern its behaviour in order to assess what performances may be expected.

The apparatus considered is certainly very simple and it appears to be capable of transferring even large quantities of energy with small diameter connecting tubes. Small differences of pressure could maintain relatively large mass flow rates. But some restrictions must be imposed on the relative positions of the collector and the condenser. It was found that if the condenser is put at a sufficiently lower level than the collector, the system cannot work.

A very suitable working fluid could be found. The thermodynamic behaviour of Freon 11 permits the pressure inside the system to be within a narrow range in the neighbourhood of the atmospheric value; this simplifies structural problems and reduces plant costs.

A C K N O W L E D G E M E N T S

One of the authors (J. Prakash) is thankful to Professor G. Furlan of ICTP for providing the grants, and to Professors S. Stecco and E. Carnevale for providing necessary facilities for the work. The work was supported by a

146 J. Prakash, M. Bertel~

grant from the ICTP Programme for Training and Research in Italian Laboratories, Trieste, Italy, for the years 1984-86.

R E F E R E N C E S

1. R. S. Soin, K. Sangameshwar Rao, D. E Rao and K. S. Rao, Solar Energy, 23 (1979), p. 69.

2. V. V. N. Kishore, M. R. Gandhi and K. S. Rao, Applied Energy, 17 (1984), pp. 13349.

3. R. C. Weast (Ed.), CRC Handbook of Chemistry and Physics, CRC Press, Ohio, 1969, p. E28.