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Analysis of refrigerant-charged solar collectors with phase change
Heat Recovery Systems & CHP Vol. 13, No. 5, pp. 429-439, 1993 Printed in Great Britain
0890-4332/93 $6.00 + .00 © 1993 Pergamon Press Ltd
ANALYSIS OF REFRIGERANT-CHARGED SOLAR COLLECTORS WITH PHASE CHANGE A . M . RADHWAN a n d G . M . ZAKI Faculty of Engineering, King Abdulaziz University, P.O. Box 9027, Jeddah 21413, Saudi Arabia (Received 28 October 1992)
Abstract--This paper presents a mathematical model for natural circulation refrigerant-charged solar collector plates with phase change. The model accounts for the thermal nonequilibrium vapor generation process along fully charged collector tubes. The local vapor generation rate is determined by solving the steady one-dimensional two-phase energy conservation equations in finite difference form along the flow path. The integral momentum balance around the circulation loop determines the circulation flow rate, which shows dependency on the solar flux and the pressure of the heat carrying refrigerant inside the loop. Calculations have been made for a system fully charged by R-11, with temperature-dependent thermophysical properties. The effect of solar insolation, for a controlled pressure system, on the vapor quality, and circulation rate is presented and discussed. Performance of the system is studied for a range of the independent variables, solar flux up to 1000 W m -s, and refrigerant saturation temperatures up to 80°C.
NOMENCLATURE Ab Ac As Cv Cp d D E e f F g gc Gk h I K mt mv rhI n~v nk np Nu P q Qu R Ra Re S T t
cross sectional area of a single pipe (m s) solar energy collection area (m 2) surface area (m2) specific heat at constant volume (W kg- ~K - t) specific heat at constant pressure (W kg -I K -t) density ratio (Pl/Pv) driving force per unit area (N m -2) total energy (W) specific energy (W kg- i ) friction factor friction pressure loss (N m -2) gravitational acceleration (m s -2) conversion factor N (kg m s-2) -I total mass rate at any section (kg s -~) heat transfer coefficient (W m - : K) solar flux (W m - 2) integral momentum per unit volume (kg m s -I ) mass of liquid (kg) mass of vapor (kg) mass rate of liquid (kgs -I) mass rate of vapor (kg s-t) number of sections along a collector tube number of collector pipes Nusselt number pressure (N m -2) heat transfer rate (W m -2) useful energy (W) two-phase friction multiplier Rayleigh number Reynolds number slip ratio temperature (K) time (s)
U0 V v v x z Az
overall heat transfer coefficient (W m -~ K - t ) volume flow rate (m 3 s - t ) specific volume (m 3 kg-i ) velocity (m s- t ) vapor quality length along the riser tubes (m) length of an element k, (m)
Greek letters
~tp ~t ~kk 2 0 p zg
absorptivity of the collector tube surface void friction rate of vapor generation (kg s -t) dynamic viscosity (kg m - i s- i ) error value tilt angle (deg) density (kg m -3) transmissivity of the collector transparent cover
Subscripts i entrance j integer number, 1,2 . . . . k section number, 1, 2 . . . . 1 liquid s saturation t total, for both liquid and vapor v vapor Superscripts
i, j ^
rate integer, 1, 2 . . . . indicating volume
1. I N T R O D U C T I O N Although, thermosyphonic solar water heaters (SWH) have been in use for many years, the concept o f s o l a r w a t e r h e a t e r s c h a r g e d w i t h l o w b o i l i n g p o i n t f l u i d s is v e r y a t t r a c t i v e . T h e r e a r e s e v e r a l advantages for the refrigerant solar water heaters (RSWH) over the classical SWH; these include their fast thermal response, elimination of the fouling, scaling and corrosion in the collector tubes. 429
430
A.M. RADHWANand G. M. ZAK~
Besides this, the RSWH are suitable for cold climate regions where there is no problem of night time freezing. In addition, the high heat transfer coefficients, encountered with phase change processes, can increase the efficiency. These numerous advantages are generally offset by the choking of the flow where the vapor generated in the collector tubes superheats and occupies a large volume, increasing the pressure losses along the tube, and consequently reduces the flow rate. This phenomena is known as vapor locking. In spite of this difficulty the RSWH presents an attractive solar collection system that is worth investigation. Soin et al. [1, 2] investigated the thermal performance of acetone and petroleum-ether charged solar water heaters. The data reported was not encouraging where the efficiency of the two-phase solar collector was 6-11% less than that of a simple similar thermosyphon collector. A1-Tamimi and Clark [3] showed that for a two-phase solar water heater, charged with R-11, the efficiency is higher than that of a flat plate water heater, for the same circulation flow rate of the heat carrying fluid. The efficiency of RSWH, tested by Farrington et al. [4], is of the order of 35% tor R-11 charged collectors. Results of a similar testing, R-il RSWH, by Radhwan et al. [5] showed that the efficiency varies between 20-50% which is in the range of thermosyphonic solar water heaters. The Fanny-Terlizzi [6] correlation developed from indoor simulation testing of RSWH charged with R- 11, is confirmed by the results of Radhwan et al. [5]. Akyurt [7] in an extensive experimental investigation reported that the efficiency of RSWH is slightly lower than conventional open loop thermosyphonic hot water heaters. A model for RSWH was developed by Kamal [8] and verified using data available in the literature. The effect of the liquid level in the tube was investigated [8] showing that a fully charged collector plate gives the highest efficiency. The decrease of efficiency with liquid level was attribued to the poor heat transfer rate to the vapor filled region. It was recommended to adjust the liquid level in the tubes to be as close as possible to the tube outlets [8]. In the present study a mathematical model for the performance of RSWH is presented. A physical model for the vapor generation process with thermal nonequilibrium is considered. The collector tubes are fully charged with low boiling point fluid. Detailed analysis of the two-phase flow, along the collector tubes is a feature of the present study. 2. ANALYSIS 2.1. Governing equations
The closed loop of a solar water heater with boiling fluids consists of a flat plate collector, with multitubes resembling a riser, a manifold with heat exchange surface and a downcomer (Fig. la). The loop is charged with a low boiling point fluid. Phase change takes place as solar energy is absorbed by the collector plate. Vapor leaving the riser condenses on the surface of the manifold and the condensate returns through the downcomer via a lower connecting plenum to the riser plate (Fig. la). The natural circulation rate is determined by the buoyancy forces due to density differences of the hot and cold sides of the loop. In the present investigation phase change takes place along the collector tubes which are fully filled with a heat carrying fluid. Therefore, superheat regions with low heat transfer coefficients do not exist. It is assumed that the flow along the riser is one-dimensional two-phase flow with a nonthermal equilibrium vapor generation process. For an element k (Fig. lb) the one-dimensional mass conservation equations for the liquid and vapor phases can be written as: dml dt
dmldZ=0 dZ
(1)
and dmv dmv dt + - ~ - dZ = O.
(2)
If the vapor generation rate within section k is ~k, then the steady mass conservation equations in finite difference form:
rnl,k= mLk -, - @k
(3)
Analysis of refrigerant-charged solar collectors
IA
A fi'A'
Downcomer
•
(b)
LTer
\qk
431
ptenum
/
d
......
..~
/
,.J
_ _
\
mL,k-I Fig. I. (a) Thermosyphonic refrigerant charged solar collector. (b) Liquid and vapor regions.
and mv,k = r&.,_ l + ~,.
(4)
At any section the total mass flow rate is constant, namely: ml.k + m~,k = rhl.k - I + m~,k - i = rh~,
where rh i is the mass flow rate at the entrance (k = 1). No phase change takes place at this section. The volumetric flow rate at any section k is rhv,k
Pl,k
P~.k
Using equations (3) and (4), the volumetric flow rate at section k is related to that of the preceding section, k - 1 (Fig. lh) by ml'* ~-F/~v,k - - - ml,k- I + r~v,k - l + ~ , ~ , Pt.,
P,.,
Pt.,- l
(6)
P,.*- I
where ~k --'--
-----V[g,k,
where vfg is the difference between the vapor and liquid specific volumes, and the term ?*~k is an average volumetric vapor generation rate.
432
A.M. RADHWANand G. M. ZAKI
The energy balance for section k, is (3el 0ev dE m ~ + m v ~ - + d-~ dZ = dq
(7)
and in steady state difference form (Fig. Ib) (8)
Ek = Ek_ , + qk,
where E is the total energy expressed in general form as E = p~ I?L0e] + Pv l?vev + Pl~i.
(9)
For nonthermal equilibrium conditions the liquid layer adjacent to the hot wall of the tube is at a temperature, Tl, slightly higher than the saturation temperature, therefore, the internal energy, el, is expressed as e, = C v ( T I -
(lO)
Ts).
Combining equations (9) and (10) and substituting for rh~ = Pv" Vv yields an expression for the liquid temperature at section k, as Tl,, = Ts+
Ek -- PVt,
'
- - mv.kev.k
ml,k Cv
;
(11)
making use of equation (4), /'1,, becomes T,,k = Ts +
E , - P Vt,, - the,k_ [ -- @, e~,k
rhLkCv
(12)
The second term of equation (12) represents the degrees of superheat gained by the liquid layer. For thermal equilibrium this term vanishes and TL, = T~. For the present analysis the pressure, P, is taken as constant throughout the circulation loop. TLk can be determined if the value of ¢, is known. For a thermal equilibrium condition, ~,, can easily be determined by applying the energy conservation principle on section k. For the nonequilibrium state it is postulated that the vapor formed on the surface displaces a volume 6 l?L0 from the liquid layer to a fluid bulk region (Fig. 2). The vacant space is then filled with a liquid volume from the bulk region, 617"L~. Therefore, the volume occupied by the vapor is
f 17"v= & l?L0- f lT',j.
(13)
Mass balance on the layer yields that the mass of the vapor and liquid leaving the layer equals the mass of the incoming liquid, that is, Pv 617",~+ Pl 6I~'Lo = Pl 6f/'Li.
LIQUID AT T$
~-~_~--
Fig. 2. Vapor generation with thermal nonequilibrium.
(14)
Analysis of refrigerant-charged solar collectors
433
Energy balance on the liquid layer, A, Fig. 2 of section k, gives:
q, = pv 6 V~hv + p, 6 (/,.oh,.o - p, 6 ~.i h,.i ,
(15)
where q, is the net energy transfer through the tube wall. The enthalpy of the liquid leaving region A, assumed at a mean temperature (Tw + Ts)/2, is
h',° = hs + Cp [ Tw + 2 Ts
Ts] •
(16)
Substitution of h~.0 from equation (16) and setting ht.~= hs into equation (15), gives the surface evaporation rate for section k as qk
hfg+Cp(d-1) { f - ~ -
(17) + T~-T~} k
where d = Pl/Pv. Equation (17) shows that the evaporation rate depends upon qk, the wall temperature, Tw, and the thin liquid layer temperature, TI, which is slightly above the saturation temperature. Further, q, is affected by the wall and ambient temperatures where a portion of the solar energy absorbed by the tube surface is transferred back to the atmosphere (Fig. la). For an overall heat transfer coefficient, U0, between the tube wall and the ambient, qk is:
qklAs,, = ~pr, I - UolTw - Ta],
(18)
where I is the total solar flux received by the tilted surface of the collector plate. As., is the area exposed to solar radiation for section k of a single tube. Equations (12), (17) and (18) are nonlinear functional relations as to the variables T~, Tw and ~b. Solution of these equations for the three unknowns requires additional relations for both the vapor and liquid mass flow rates, thv and rhl. The circulation rate through the closed loop (Fig. 1) is determined by the balance of the driving and resisting forces. The integrated momentum equation around the flow path, can be written as ~d(m i v ) - ~ D - ~ F , dt
(19)
where the left hand side is the integrated rate of change of momentum. D is the driving buoyancy force caused by the fluid density difference in the riser collector plate and the downcomer. It is evaluated as follows: (20) g¢ J where ~k is the void fraction, defined as the ratio between the vapor volume rate and total volumetric flow rate at section k of length AZ. For single phase flow, in the downcomer, • = 0 and equation (20) gives the static head. The maximum limit of ~ is unity, when the collector tubes are fully filled with vapor. While performing the integration, around the circulation loop, (equation (20)) the sign of D changes; being positive for upward flow 0 ~
G, -12 0,., R,
~,j, Lp,~A,j 2gD~,,~,.~,.
(21)
f , is the single phase friction factor and depends upon the Reynolds number, and R, is a two-phase friction multiplier which can be determined from correlations available in two-phase flow literature; for example the following relation is used by Becker et al. [9] TM
R, = 1 + 2400 P is the pressure in bars and x is the vapor quality at section k.
(22)
434
A, M. RADHWANand G. M. ZAKI
The mass rate, Gk, is constant at any section and given by Gk = [PLkVl.~+ Pv,kVg.k].
(23)
For the left hand side of equation (19), the momentum variation with phase change in section k is rn • v = (1 - a)plvl + ~pvVv.
(24)
Since the vapor velocity, v~, is higher than the liquid velocity, v~, the slip ratio is defined as:
Vv S = --. VI
(25)
The total volumetric flow rate at any section is 1), and the superficial flow velocity,
{/~/Ah, is:
Vt/A b - (1 -- 0Qvl + c~vv.
(26)
Substitution of equations (25) and (26) into equation (24) gives the momentum at section k as:
Vt,kPl,k F 1 m •v
-
L
Ab
0~S -I- ( d -
1) 1 + ~ ( S
] -
1) k"
(27)
Assuming negligible variation of p~ and from equation (27), the integrated momentum is:
~
rn- v dZ = ~ I},.k C, = K/o,,i
where
AZ,[
Ck=~
(28)
k=l
S~ ]
l+(d-1)
I+~(S-I)
k'
(29)
The summation in equation (28) starts at the collector entrance (k = 1) where the volumetric flow rate is V~. It is assumed that at the entrance there is no vapor and therefore the volumetric flow rate at any other section with phase change is: k I)t,k = I?i- E Oj(V,,j- Vv,j) (30) )=l Equations (28) and (30) give the circulation velocity at the collector tube entrance as:
Vi Pli - = Vi = -Ab
k= I @ /..a Ck k=l
(31)
To determine v~, complete knowledge of the different variables along the flow path (collector, downcomer and connecting pipes) are needed. Further, to solve the model for the unknowns, T~, Tw and ~, a number of auxiliary relations are needed to determine the parameters, h, S, f and R. These relations are given in the following section. 2.2.
Auxiliary relations
The wall temperature, Tw, at any section can be determined by considering the heat transfer between the tube surface and the fluid bulk temperature. The thin tube wall, made of a good conducting material is assumed to be at a uniform temperature. The net heat transfer rate to the fluid is, then: qk = h [ T w k -
T, k ]
(32)
where qk is given in equation (18). h depends upon the flow regime. For the boiling region along the riser with Freon-11 as the heat carrying fluid, h is given [8 and 10] as: h = [0.875 + 0.0096 T] 4] [Tw - T~]z°3.
(33a)
Analysis of refrigerant-charged solar collectors
435
If the liquid at the riser entrance is subcooled, conventional correlations for free convection inside a tube can be used:
Nu = 0.56Ra 1/4
104 < R a < 10 s
and
(33b)
Nu=O.13Ra I/3 Ra > lOs. The model presented is based on a slip phase change model where the local vapor and liquid velocities are different, S > 1. A simple relation, equation (34), that relates S to ~t, Baksted and Solberg [11] is used.
Sk=--=Vv 1 + 3 . 6 ~ ,
Ctk>0
(34)
Vl
where ~k is the void fraction at section k. The single phase friction factor, fk, in equation (21) can be expressed as: Cl f~ = - Re C2
(35)
where C~=64
and C2=1
for Re<<.2200
C~ = 0.184 and 6'2 = 0.2 for Re > 2200. The two-phase friction multiplier, Rk, is given in equation (22). 2.3. Useful energy and system efficiency Though vapor is generated at each section, at a rate ~kk, it is accumulated at the subsequent sections downstream. The mass of vapor at any section is, therefore, k
rhv.k= ~ ~j. j=l
(36)
The vapor quality at a collector tube exit is nk
xo = ~ ¢j/{pt~ViAb}.
(37)
jffil
The vapor transfers its latent heat to the water in the storage tank and the maximum useful energy that can be gained, Qu, is au = xe {PliviAb}" np,
(38)
where np is the number of collector plate tubes. The efficiency of the collection system is therefore: ~l = Q u / I . Ae.
(39)
3. S O L U T I O N P R O C E D U R E The derived set of equations, (12), (17) and (31), along with the auxiliary relations for, R, h, S and f(equations (22), (33), (34) and (35)) are solved by an iterative method. Calculations start by assuming a circulation velocity vi, and set initial values at the collector tube entrance, ~ = 0, Tj = Ts and ~ = 0. This implies that phase change takes place along the collector tubes (no subcooling or superheat regions). For any subsection, k > 1, a value for ~,~, is assumed, and T[., is calculated (equation (12)). The resulting value is substituted, in equation (17) to find -ri+~ ~tl,k and hence g,~,+~. The values of ~ , and ~ + ~ as well as Ti.k and TfZ~ are compared and the procedure is repeated until an accuracy is obtained. The convergence criteria is in the general form: y i _ yi+t] E=
yi+ i
~< 2,
(40)
436
A.M.
RADHWAN and G, M. ZAKI
where Y is a dummy variable and ). is taken as 10 4. Once the accuracy is obtained calculations proceed to the next section, k + l, using the final values T~.k and ~, as the initial guess for the section k + I. The procedure continues to the last section of a collector tube, k = n k. The void fraction for each section along a collector tube is then calculated, using equation (36) for my and the slip ratio correlation, equation (34), for vv, from which:
(m~/p~)k (mv/p~),
cz, - - -
-
Vv,k " A p
(41 )
Sk vu,. m p "
~tk is calculated through a second iteration where an assumed value for ~ is used in equation (34) to find Sk, Vt is then calculated from
VI.k/Ap
(42)
vl,~ = 1 + ~k(Sk -- 1)' Knowing Sk and v~.k, a new value for the void fraction, e{+ ~, is calculated (equation (41)). The procedure is repeated until equation (40) is satisfied, for every section from k = 2 to k = nk. The resulting distributions of ~, 7"1 and • are then used to calculate the circulation velocity, v~ (equation (31)), the calculated v~ is compared with the assumed value and the calculations are repeated until the specified accuracy (equation (40)) is satisfied for the variable v~. The balance condition (equation (19)) for the steady state is then checked; if it is not satisfied a new value for the loop pressure is selected and the calculations are repeated until the accuracy is satisfied for the pressure inside the collector tubes. If the system pressure is controlled there is no need to perform the last step for adjusting the pressure. 4. R E S U L T S AND D I S C U S S I O N To investigate the performance of RSWH, the geometric data of a system built by Radhwan
et al. [5] is used, and the data is given in Table 1. Thermal nonequilibrium is a main assumption of the present model, therefore, all physical properties of R-11 (p, Cp, Cv, hfg and #) are taken as temperature dependent. Data for R-ll is fitted from different sources [12, 13]. The numerical results cover the condition where phase change starts at the entrance of the collector tubes. The number of sections along a collector tube is taken as 20, and the accuracy, 3. = l0 -4, was found to be sufficient. The collector internal pressure is taken as constant by fixing the saturation temperature. The present calculations cover a range of temperatures between 40 and 80°C. The following parameters have been used: ~p=0.96; Zg=0.9;
U0=8Wm
2K-~.
The air temperature, Ta, has been set constant at 30°C. With the thermal nonequilibrium assumption, the liquid temperature is slightly higher than the saturation temperature. The level of liquid superheating depends upon the pressure inside the collector, and the solar insolation (Fig. 3). This dependency is clear, considering the second term of the right hand side of equation (12) that also shows the variation of superheating with the local mass rates of both the liquid and vapor phases. This explains the increase, though not appreciable, of the liquid superheat with the collector length (Fig. 3). The solar energy reaching the collector determines the natural circulation rate and consequently the evaporation rate along the collector tubes. Figure 4 shows the vapor circulation velocity for R-l I at different solar fluxes and collector pressures. Naturally increasing the flux increases both Table 1. Data for a RSWH Heat carrying fluid Diameter of a collector tube Length of a collector tube Number of collector tubes, nt Plate tilt angle, to the horizontal Downcomer diameter, copper Cover (single glazing), thickness
R-11 0.0125 m 1.0 m 10 3 0 (0 = 60) 0.0125 m 3 mm
437
Analysis of refrigerant-charged solar collectors
0,I
r
r
0 200 W m -2 • 200 Wm -2
~
A 400Win -2 • 400Wm-2
F
~
0.2
r
r't 600 W m-2 I 600 Wm-Z
Ts =
_i~_ .4~._.1:3_.0_ 4:3..~" ~ . 0- / /. . . 0 - - 0 - - - 0 ;.~
•
K
333
.-0- - 0 - -13--El.- -D- -O--C
& ~, ...=,...=....Z/..A_..=,.,.=...,=....a....&.--.&---."'-"a""a""L
"1o
"~ 0.1
~i
Ts=313K /
_
~'/~
_..~.,,.=.=-~,--=--=--,,--=--=-'='~
--~-~-~"~.~.--T
f ~ " - ~ t/,.'r .......
0
_
0
~
--
•
•
•
•
"-...'-..."-"'""~
.,... , - . , , / . . ,
_ _ ./ . . . . . . . . . . . . . . . . . .
_
0.2
0.4
0.6
0.8
1.0
Z/Lr
Fig. 3. Variation of liquid superheat with pressure and solar insolation.
the vapor quality and void fraction, that tends to increase the buoyancy forces (Fig. 4). Lowering the pressure inside the collector plate, for constant heat flux, results in an increase of the circulation velocity caused by the increase of the vapor specific volume (vg). These trends hold only for fully charged collectors. For partially filled collectors the presence of a superheated vapor region lowers the heat transfer rate from the plate to the refrigerant, increasing Tw, which causes an increase in the energy loss (equation (18)). For low values of solar insolation (up to 200 W m -2) the pressure effect upon the circulation velocity is less significant than at high solar fluxes (Fig. 4). This is again due to the low values of both x and ct at low values of the fluxes. Figure 5 shows the variation of ~t with the system pressure along the collector tubes at different saturation temperatures while Fig. 6 shows the quality. The efficiency of the primary loop, for the present conditions, changes with both the solar flux and the system pressure (Fig. 7). Increasing the solar insolation increases the buoyancy forces and consequently the circulation rate, thereby, the useful energy increase results in high efficiency values. The efficiency approaches an asymptotic value, that is determined theoretically by the glass cover transmissivity and plate absorptvity. The present case study, with R-I 1 and the absence of subcooling and superheat regions, gives high loop efficiencies as the losses 3.5
'
I
'
I
13 T$ - 3 3 3 K
~
I
I
OT s -313K
'
I
• Ts= 353 K
3.0
ss S s S
.....
aS
% 2.5 E v
so ~
g,.o
j.
Sf
s S
fss.~ss~ O a ~
~ 1.5
$
//s /
~ I.O
......~.~o-
0.5
200
400
600
800
t000
I100
Solar flux (W m-2)
Fig. 4. Effect of solar insolation and system pressure upon R-11 circulation velocity. HRS 13/5--E
438
A. M. RADHWAN and G. M. ZAK] 05
t
i
T s =313K
....
i
Ts = 5 3 5 K
-- --Ts
i
=353 K
0.4
-1
0.3 "o )~ 0.2
0.1
0 0
0.2
0.4
0.6
08
1.0
Z/Lr
Fig. 5. Void fraction variation along the collector tube.
from the plate to the ambient (at 30°C) are affected by the difference (Tw-Ta). The wall temperature, Tw, is higher than the saturation temperature, that solely is adjusted by presetting the loop pressure. For lower values of the loop pressure the efficiency increases, as a result of the increase in the mass circulation rate, although the latent heat is higher than that for high pressures (Fig. 7). The analysis shows that a primary loop operating with low boiling point heat carrying fluids at low saturation temperatures (40°C) gives a high efficiency (60-80%) which is much higher than values obtained for a partially filled collector at different filling ratios (30-65% for acetone [8], up to 42% for water [1] and up to 45% for R-I1 [5]). It is expected that the efficiency of a complete system, where a secondary loop is to be considered, will result in lower overall efficiencies. 5. C O N C L U S I O N S
A model has been proposed for parametric analysis of the primary loops of solar collectors with phase change. The proposed model is of a general nature where thermal nonequilibrium and dependency of the thermophysical properties upon temperature and pressure are accounted for.
y
0.008
0007
0.006
0005 X
/
QO04
0 0.003 0.002 Ts= 313 K
0.~1 0
-
/
~I 0.2
I 0.4
I 0.6
I 0.8
1.0
Z/Lr
Fig. 6. Variation of the quality along the collector tubes and with the system pressure.
439
Analysis of refrigerant-charged solar collectors I.O
0.9
I
I
|
I
I
.......
0.8
I
I
•.o--. . . . . .
I
I
I
--ta . . . . . . .
I
"~_. . . . . . . , 0
0.7
/
f/
•
I
.~ 0.5 •,,:
[
./'/'
[
""-'
.ff
r, cKI
o 313
/
o 333 A 353
/./,
0.4
0.2 i
0.1 0
Jo0
/
I 200
300
400
500
6oo
700
8o0
900
I
I
Io00 llo0
1200
Sotor f l u x (W rn-2)
Fig. 7. Dependency of primary loop efficiency on solar insolation and system pressure (R-11 fully charged collector: no subcooling).
It is assumed that the collector loop is fully charged by the heat carrying fluid, (liquid level = 100%) to avoid vapor superheating in the collector. Results for a Freon-I 1 charged collector of 1 m 2 area, 10 tubes 12.5 mm diameter each at 30° tilt angle to the horizontal are presented. The results show that the liquid superheating is less than one degree, and insignificantly varies along the collector tubes. However, it is dependent upon the pressure inside the primary loop. The local vapor quality and void fraction vary along the length of the tubes. The circulation rate for the R-11, controlled pressure system, varies linearly with the solar insolation and increases at lower pressures of the primary loop. Increasing the solar insolation raises the mass flow rate through the loop and consequently the useful energy gain. The efficiency of the loop is thereby enhanced and approaches an asymptotic value, theoretically determined by the glass cover transmissivity and collector plate absorptivity. For the present case study, with the absence of liquid subcooling and vapor superheat, the energy loss to the ambient is minimum as the collector plate temperature maintains a constant value above the saturation temperature. REFERENCES I. R. Soin, S. Raghurman and V. Murali, Two-phase water heater model and long term performance. Solar Energy 38, 105-112 (1987). 2. R. Soin, K. S. Rao and D. P. Rao, Performance of flat plate solar collector with fluid undergoing phase change. In Proc. ISES Congr., pp. 952-956 Pergamon Press, New Delhi (1978). 3. A. AI-Tamimi and J. A. Clark, Thermal performance of a solar collector containing a boiling fluid. ASHRAE Trans. 90, 681-696 (1984). 4. R. Farringeton, M. De Angelis, L. Morrison and D. Longberty, Performance evaluation of a refrigerant-charged thermosyphon solar DWH system. In Proc. AS/ISES Annual Meeting, Philadelphia, published by the ISES, American Section, 4, 676-680 (1981). 5. A. Radhwan, G. M. Zaki and J. Arshad, Refrigerant-charged integrated solar water heater. Int. J. Energy Res. 14, 421-432 (1990). 6. A. H. Fanny and C. P. Terlizzi, Testing of refrigerant-charged solar domestic hot water systems. Solar Energy 35, 353-366 (1985). 7. M. Akyurt, AWSWAH, The heat pipe solar water heater. J. Engng appl. Sci. 3, 23-38 (1986). 8. W. Kamal, Performance evaluation of a refrigerant-charged solar collector. In Energex-84, pp. 193-197. Pergamon Press, Regina, Saskatchewan (1984). 9. K. Becker, G. Herenborg and M. Bode, An experimental study of pressure gradients for flow of boiling water in a vertical round duct. AE, pp. 69-70, AB. Aktienbolaget, Atomenergi, Sweden (1962). 10. V. Kishor, M. Ghandi, C. Marquis and K. Rao, Evaluation and testing of flat plate collectors employing phase changing fluids. In Energex 82, pp. 134-139. Pergamon Press, Regina, Saskatchewan (1982). 1 I. K. Baksted and K. O. Solbcrg, A FORTRAN Program for Transient Analysis of Boiling Water Reactors and Boiling Loops. KR-135 Inst. for Atomenergi, Kjeller, Norway (1968). 12. W. C. Reynolds, Thermodynamic Properties in SI, Section 2, pp. 70-72. Dept of Mech. Engng, Stanford Univ., CA (1979). 13. P. Dunn and D. A. Reay, Heat Pipes, p. 274. Pergamon Press, Oxford (1977).
0890-4332/93 $6.00 + .00 © 1993 Pergamon Press Ltd
ANALYSIS OF REFRIGERANT-CHARGED SOLAR COLLECTORS WITH PHASE CHANGE A . M . RADHWAN a n d G . M . ZAKI Faculty of Engineering, King Abdulaziz University, P.O. Box 9027, Jeddah 21413, Saudi Arabia (Received 28 October 1992)
Abstract--This paper presents a mathematical model for natural circulation refrigerant-charged solar collector plates with phase change. The model accounts for the thermal nonequilibrium vapor generation process along fully charged collector tubes. The local vapor generation rate is determined by solving the steady one-dimensional two-phase energy conservation equations in finite difference form along the flow path. The integral momentum balance around the circulation loop determines the circulation flow rate, which shows dependency on the solar flux and the pressure of the heat carrying refrigerant inside the loop. Calculations have been made for a system fully charged by R-11, with temperature-dependent thermophysical properties. The effect of solar insolation, for a controlled pressure system, on the vapor quality, and circulation rate is presented and discussed. Performance of the system is studied for a range of the independent variables, solar flux up to 1000 W m -s, and refrigerant saturation temperatures up to 80°C.
NOMENCLATURE Ab Ac As Cv Cp d D E e f F g gc Gk h I K mt mv rhI n~v nk np Nu P q Qu R Ra Re S T t
cross sectional area of a single pipe (m s) solar energy collection area (m 2) surface area (m2) specific heat at constant volume (W kg- ~K - t) specific heat at constant pressure (W kg -I K -t) density ratio (Pl/Pv) driving force per unit area (N m -2) total energy (W) specific energy (W kg- i ) friction factor friction pressure loss (N m -2) gravitational acceleration (m s -2) conversion factor N (kg m s-2) -I total mass rate at any section (kg s -~) heat transfer coefficient (W m - : K) solar flux (W m - 2) integral momentum per unit volume (kg m s -I ) mass of liquid (kg) mass of vapor (kg) mass rate of liquid (kgs -I) mass rate of vapor (kg s-t) number of sections along a collector tube number of collector pipes Nusselt number pressure (N m -2) heat transfer rate (W m -2) useful energy (W) two-phase friction multiplier Rayleigh number Reynolds number slip ratio temperature (K) time (s)
U0 V v v x z Az
overall heat transfer coefficient (W m -~ K - t ) volume flow rate (m 3 s - t ) specific volume (m 3 kg-i ) velocity (m s- t ) vapor quality length along the riser tubes (m) length of an element k, (m)
Greek letters
~tp ~t ~kk 2 0 p zg
absorptivity of the collector tube surface void friction rate of vapor generation (kg s -t) dynamic viscosity (kg m - i s- i ) error value tilt angle (deg) density (kg m -3) transmissivity of the collector transparent cover
Subscripts i entrance j integer number, 1,2 . . . . k section number, 1, 2 . . . . 1 liquid s saturation t total, for both liquid and vapor v vapor Superscripts
i, j ^
rate integer, 1, 2 . . . . indicating volume
1. I N T R O D U C T I O N Although, thermosyphonic solar water heaters (SWH) have been in use for many years, the concept o f s o l a r w a t e r h e a t e r s c h a r g e d w i t h l o w b o i l i n g p o i n t f l u i d s is v e r y a t t r a c t i v e . T h e r e a r e s e v e r a l advantages for the refrigerant solar water heaters (RSWH) over the classical SWH; these include their fast thermal response, elimination of the fouling, scaling and corrosion in the collector tubes. 429
430
A.M. RADHWANand G. M. ZAK~
Besides this, the RSWH are suitable for cold climate regions where there is no problem of night time freezing. In addition, the high heat transfer coefficients, encountered with phase change processes, can increase the efficiency. These numerous advantages are generally offset by the choking of the flow where the vapor generated in the collector tubes superheats and occupies a large volume, increasing the pressure losses along the tube, and consequently reduces the flow rate. This phenomena is known as vapor locking. In spite of this difficulty the RSWH presents an attractive solar collection system that is worth investigation. Soin et al. [1, 2] investigated the thermal performance of acetone and petroleum-ether charged solar water heaters. The data reported was not encouraging where the efficiency of the two-phase solar collector was 6-11% less than that of a simple similar thermosyphon collector. A1-Tamimi and Clark [3] showed that for a two-phase solar water heater, charged with R-11, the efficiency is higher than that of a flat plate water heater, for the same circulation flow rate of the heat carrying fluid. The efficiency of RSWH, tested by Farrington et al. [4], is of the order of 35% tor R-11 charged collectors. Results of a similar testing, R-il RSWH, by Radhwan et al. [5] showed that the efficiency varies between 20-50% which is in the range of thermosyphonic solar water heaters. The Fanny-Terlizzi [6] correlation developed from indoor simulation testing of RSWH charged with R- 11, is confirmed by the results of Radhwan et al. [5]. Akyurt [7] in an extensive experimental investigation reported that the efficiency of RSWH is slightly lower than conventional open loop thermosyphonic hot water heaters. A model for RSWH was developed by Kamal [8] and verified using data available in the literature. The effect of the liquid level in the tube was investigated [8] showing that a fully charged collector plate gives the highest efficiency. The decrease of efficiency with liquid level was attribued to the poor heat transfer rate to the vapor filled region. It was recommended to adjust the liquid level in the tubes to be as close as possible to the tube outlets [8]. In the present study a mathematical model for the performance of RSWH is presented. A physical model for the vapor generation process with thermal nonequilibrium is considered. The collector tubes are fully charged with low boiling point fluid. Detailed analysis of the two-phase flow, along the collector tubes is a feature of the present study. 2. ANALYSIS 2.1. Governing equations
The closed loop of a solar water heater with boiling fluids consists of a flat plate collector, with multitubes resembling a riser, a manifold with heat exchange surface and a downcomer (Fig. la). The loop is charged with a low boiling point fluid. Phase change takes place as solar energy is absorbed by the collector plate. Vapor leaving the riser condenses on the surface of the manifold and the condensate returns through the downcomer via a lower connecting plenum to the riser plate (Fig. la). The natural circulation rate is determined by the buoyancy forces due to density differences of the hot and cold sides of the loop. In the present investigation phase change takes place along the collector tubes which are fully filled with a heat carrying fluid. Therefore, superheat regions with low heat transfer coefficients do not exist. It is assumed that the flow along the riser is one-dimensional two-phase flow with a nonthermal equilibrium vapor generation process. For an element k (Fig. lb) the one-dimensional mass conservation equations for the liquid and vapor phases can be written as: dml dt
dmldZ=0 dZ
(1)
and dmv dmv dt + - ~ - dZ = O.
(2)
If the vapor generation rate within section k is ~k, then the steady mass conservation equations in finite difference form:
rnl,k= mLk -, - @k
(3)
Analysis of refrigerant-charged solar collectors
IA
A fi'A'
Downcomer
•
(b)
LTer
\qk
431
ptenum
/
d
......
..~
/
,.J
_ _
\
mL,k-I Fig. I. (a) Thermosyphonic refrigerant charged solar collector. (b) Liquid and vapor regions.
and mv,k = r&.,_ l + ~,.
(4)
At any section the total mass flow rate is constant, namely: ml.k + m~,k = rhl.k - I + m~,k - i = rh~,
where rh i is the mass flow rate at the entrance (k = 1). No phase change takes place at this section. The volumetric flow rate at any section k is rhv,k
Pl,k
P~.k
Using equations (3) and (4), the volumetric flow rate at section k is related to that of the preceding section, k - 1 (Fig. lh) by ml'* ~-F/~v,k - - - ml,k- I + r~v,k - l + ~ , ~ , Pt.,
P,.,
Pt.,- l
(6)
P,.*- I
where ~k --'--
-----V[g,k,
where vfg is the difference between the vapor and liquid specific volumes, and the term ?*~k is an average volumetric vapor generation rate.
432
A.M. RADHWANand G. M. ZAKI
The energy balance for section k, is (3el 0ev dE m ~ + m v ~ - + d-~ dZ = dq
(7)
and in steady state difference form (Fig. Ib) (8)
Ek = Ek_ , + qk,
where E is the total energy expressed in general form as E = p~ I?L0e] + Pv l?vev + Pl~i.
(9)
For nonthermal equilibrium conditions the liquid layer adjacent to the hot wall of the tube is at a temperature, Tl, slightly higher than the saturation temperature, therefore, the internal energy, el, is expressed as e, = C v ( T I -
(lO)
Ts).
Combining equations (9) and (10) and substituting for rh~ = Pv" Vv yields an expression for the liquid temperature at section k, as Tl,, = Ts+
Ek -- PVt,
'
- - mv.kev.k
ml,k Cv
;
(11)
making use of equation (4), /'1,, becomes T,,k = Ts +
E , - P Vt,, - the,k_ [ -- @, e~,k
rhLkCv
(12)
The second term of equation (12) represents the degrees of superheat gained by the liquid layer. For thermal equilibrium this term vanishes and TL, = T~. For the present analysis the pressure, P, is taken as constant throughout the circulation loop. TLk can be determined if the value of ¢, is known. For a thermal equilibrium condition, ~,, can easily be determined by applying the energy conservation principle on section k. For the nonequilibrium state it is postulated that the vapor formed on the surface displaces a volume 6 l?L0 from the liquid layer to a fluid bulk region (Fig. 2). The vacant space is then filled with a liquid volume from the bulk region, 617"L~. Therefore, the volume occupied by the vapor is
f 17"v= & l?L0- f lT',j.
(13)
Mass balance on the layer yields that the mass of the vapor and liquid leaving the layer equals the mass of the incoming liquid, that is, Pv 617",~+ Pl 6I~'Lo = Pl 6f/'Li.
LIQUID AT T$
~-~_~--
Fig. 2. Vapor generation with thermal nonequilibrium.
(14)
Analysis of refrigerant-charged solar collectors
433
Energy balance on the liquid layer, A, Fig. 2 of section k, gives:
q, = pv 6 V~hv + p, 6 (/,.oh,.o - p, 6 ~.i h,.i ,
(15)
where q, is the net energy transfer through the tube wall. The enthalpy of the liquid leaving region A, assumed at a mean temperature (Tw + Ts)/2, is
h',° = hs + Cp [ Tw + 2 Ts
Ts] •
(16)
Substitution of h~.0 from equation (16) and setting ht.~= hs into equation (15), gives the surface evaporation rate for section k as qk
hfg+Cp(d-1) { f - ~ -
(17) + T~-T~} k
where d = Pl/Pv. Equation (17) shows that the evaporation rate depends upon qk, the wall temperature, Tw, and the thin liquid layer temperature, TI, which is slightly above the saturation temperature. Further, q, is affected by the wall and ambient temperatures where a portion of the solar energy absorbed by the tube surface is transferred back to the atmosphere (Fig. la). For an overall heat transfer coefficient, U0, between the tube wall and the ambient, qk is:
qklAs,, = ~pr, I - UolTw - Ta],
(18)
where I is the total solar flux received by the tilted surface of the collector plate. As., is the area exposed to solar radiation for section k of a single tube. Equations (12), (17) and (18) are nonlinear functional relations as to the variables T~, Tw and ~b. Solution of these equations for the three unknowns requires additional relations for both the vapor and liquid mass flow rates, thv and rhl. The circulation rate through the closed loop (Fig. 1) is determined by the balance of the driving and resisting forces. The integrated momentum equation around the flow path, can be written as ~d(m i v ) - ~ D - ~ F , dt
(19)
where the left hand side is the integrated rate of change of momentum. D is the driving buoyancy force caused by the fluid density difference in the riser collector plate and the downcomer. It is evaluated as follows: (20) g¢ J where ~k is the void fraction, defined as the ratio between the vapor volume rate and total volumetric flow rate at section k of length AZ. For single phase flow, in the downcomer, • = 0 and equation (20) gives the static head. The maximum limit of ~ is unity, when the collector tubes are fully filled with vapor. While performing the integration, around the circulation loop, (equation (20)) the sign of D changes; being positive for upward flow 0 ~
G, -12 0,., R,
~,j, Lp,~A,j 2gD~,,~,.~,.
(21)
f , is the single phase friction factor and depends upon the Reynolds number, and R, is a two-phase friction multiplier which can be determined from correlations available in two-phase flow literature; for example the following relation is used by Becker et al. [9] TM
R, = 1 + 2400 P is the pressure in bars and x is the vapor quality at section k.
(22)
434
A, M. RADHWANand G. M. ZAKI
The mass rate, Gk, is constant at any section and given by Gk = [PLkVl.~+ Pv,kVg.k].
(23)
For the left hand side of equation (19), the momentum variation with phase change in section k is rn • v = (1 - a)plvl + ~pvVv.
(24)
Since the vapor velocity, v~, is higher than the liquid velocity, v~, the slip ratio is defined as:
Vv S = --. VI
(25)
The total volumetric flow rate at any section is 1), and the superficial flow velocity,
{/~/Ah, is:
Vt/A b - (1 -- 0Qvl + c~vv.
(26)
Substitution of equations (25) and (26) into equation (24) gives the momentum at section k as:
Vt,kPl,k F 1 m •v
-
L
Ab
0~S -I- ( d -
1) 1 + ~ ( S
] -
1) k"
(27)
Assuming negligible variation of p~ and from equation (27), the integrated momentum is:
~
rn- v dZ = ~ I},.k C, = K/o,,i
where
AZ,[
Ck=~
(28)
k=l
S~ ]
l+(d-1)
I+~(S-I)
k'
(29)
The summation in equation (28) starts at the collector entrance (k = 1) where the volumetric flow rate is V~. It is assumed that at the entrance there is no vapor and therefore the volumetric flow rate at any other section with phase change is: k I)t,k = I?i- E Oj(V,,j- Vv,j) (30) )=l Equations (28) and (30) give the circulation velocity at the collector tube entrance as:
Vi Pli - = Vi = -Ab
k= I @ /..a Ck k=l
(31)
To determine v~, complete knowledge of the different variables along the flow path (collector, downcomer and connecting pipes) are needed. Further, to solve the model for the unknowns, T~, Tw and ~, a number of auxiliary relations are needed to determine the parameters, h, S, f and R. These relations are given in the following section. 2.2.
Auxiliary relations
The wall temperature, Tw, at any section can be determined by considering the heat transfer between the tube surface and the fluid bulk temperature. The thin tube wall, made of a good conducting material is assumed to be at a uniform temperature. The net heat transfer rate to the fluid is, then: qk = h [ T w k -
T, k ]
(32)
where qk is given in equation (18). h depends upon the flow regime. For the boiling region along the riser with Freon-11 as the heat carrying fluid, h is given [8 and 10] as: h = [0.875 + 0.0096 T] 4] [Tw - T~]z°3.
(33a)
Analysis of refrigerant-charged solar collectors
435
If the liquid at the riser entrance is subcooled, conventional correlations for free convection inside a tube can be used:
Nu = 0.56Ra 1/4
104 < R a < 10 s
and
(33b)
Nu=O.13Ra I/3 Ra > lOs. The model presented is based on a slip phase change model where the local vapor and liquid velocities are different, S > 1. A simple relation, equation (34), that relates S to ~t, Baksted and Solberg [11] is used.
Sk=--=Vv 1 + 3 . 6 ~ ,
Ctk>0
(34)
Vl
where ~k is the void fraction at section k. The single phase friction factor, fk, in equation (21) can be expressed as: Cl f~ = - Re C2
(35)
where C~=64
and C2=1
for Re<<.2200
C~ = 0.184 and 6'2 = 0.2 for Re > 2200. The two-phase friction multiplier, Rk, is given in equation (22). 2.3. Useful energy and system efficiency Though vapor is generated at each section, at a rate ~kk, it is accumulated at the subsequent sections downstream. The mass of vapor at any section is, therefore, k
rhv.k= ~ ~j. j=l
(36)
The vapor quality at a collector tube exit is nk
xo = ~ ¢j/{pt~ViAb}.
(37)
jffil
The vapor transfers its latent heat to the water in the storage tank and the maximum useful energy that can be gained, Qu, is au = xe {PliviAb}" np,
(38)
where np is the number of collector plate tubes. The efficiency of the collection system is therefore: ~l = Q u / I . Ae.
(39)
3. S O L U T I O N P R O C E D U R E The derived set of equations, (12), (17) and (31), along with the auxiliary relations for, R, h, S and f(equations (22), (33), (34) and (35)) are solved by an iterative method. Calculations start by assuming a circulation velocity vi, and set initial values at the collector tube entrance, ~ = 0, Tj = Ts and ~ = 0. This implies that phase change takes place along the collector tubes (no subcooling or superheat regions). For any subsection, k > 1, a value for ~,~, is assumed, and T[., is calculated (equation (12)). The resulting value is substituted, in equation (17) to find -ri+~ ~tl,k and hence g,~,+~. The values of ~ , and ~ + ~ as well as Ti.k and TfZ~ are compared and the procedure is repeated until an accuracy is obtained. The convergence criteria is in the general form: y i _ yi+t] E=
yi+ i
~< 2,
(40)
436
A.M.
RADHWAN and G, M. ZAKI
where Y is a dummy variable and ). is taken as 10 4. Once the accuracy is obtained calculations proceed to the next section, k + l, using the final values T~.k and ~, as the initial guess for the section k + I. The procedure continues to the last section of a collector tube, k = n k. The void fraction for each section along a collector tube is then calculated, using equation (36) for my and the slip ratio correlation, equation (34), for vv, from which:
(m~/p~)k (mv/p~),
cz, - - -
-
Vv,k " A p
(41 )
Sk vu,. m p "
~tk is calculated through a second iteration where an assumed value for ~ is used in equation (34) to find Sk, Vt is then calculated from
VI.k/Ap
(42)
vl,~ = 1 + ~k(Sk -- 1)' Knowing Sk and v~.k, a new value for the void fraction, e{+ ~, is calculated (equation (41)). The procedure is repeated until equation (40) is satisfied, for every section from k = 2 to k = nk. The resulting distributions of ~, 7"1 and • are then used to calculate the circulation velocity, v~ (equation (31)), the calculated v~ is compared with the assumed value and the calculations are repeated until the specified accuracy (equation (40)) is satisfied for the variable v~. The balance condition (equation (19)) for the steady state is then checked; if it is not satisfied a new value for the loop pressure is selected and the calculations are repeated until the accuracy is satisfied for the pressure inside the collector tubes. If the system pressure is controlled there is no need to perform the last step for adjusting the pressure. 4. R E S U L T S AND D I S C U S S I O N To investigate the performance of RSWH, the geometric data of a system built by Radhwan
et al. [5] is used, and the data is given in Table 1. Thermal nonequilibrium is a main assumption of the present model, therefore, all physical properties of R-11 (p, Cp, Cv, hfg and #) are taken as temperature dependent. Data for R-ll is fitted from different sources [12, 13]. The numerical results cover the condition where phase change starts at the entrance of the collector tubes. The number of sections along a collector tube is taken as 20, and the accuracy, 3. = l0 -4, was found to be sufficient. The collector internal pressure is taken as constant by fixing the saturation temperature. The present calculations cover a range of temperatures between 40 and 80°C. The following parameters have been used: ~p=0.96; Zg=0.9;
U0=8Wm
2K-~.
The air temperature, Ta, has been set constant at 30°C. With the thermal nonequilibrium assumption, the liquid temperature is slightly higher than the saturation temperature. The level of liquid superheating depends upon the pressure inside the collector, and the solar insolation (Fig. 3). This dependency is clear, considering the second term of the right hand side of equation (12) that also shows the variation of superheating with the local mass rates of both the liquid and vapor phases. This explains the increase, though not appreciable, of the liquid superheat with the collector length (Fig. 3). The solar energy reaching the collector determines the natural circulation rate and consequently the evaporation rate along the collector tubes. Figure 4 shows the vapor circulation velocity for R-l I at different solar fluxes and collector pressures. Naturally increasing the flux increases both Table 1. Data for a RSWH Heat carrying fluid Diameter of a collector tube Length of a collector tube Number of collector tubes, nt Plate tilt angle, to the horizontal Downcomer diameter, copper Cover (single glazing), thickness
R-11 0.0125 m 1.0 m 10 3 0 (0 = 60) 0.0125 m 3 mm
437
Analysis of refrigerant-charged solar collectors
0,I
r
r
0 200 W m -2 • 200 Wm -2
~
A 400Win -2 • 400Wm-2
F
~
0.2
r
r't 600 W m-2 I 600 Wm-Z
Ts =
_i~_ .4~._.1:3_.0_ 4:3..~" ~ . 0- / /. . . 0 - - 0 - - - 0 ;.~
•
K
333
.-0- - 0 - -13--El.- -D- -O--C
& ~, ...=,...=....Z/..A_..=,.,.=...,=....a....&.--.&---."'-"a""a""L
"1o
"~ 0.1
~i
Ts=313K /
_
~'/~
_..~.,,.=.=-~,--=--=--,,--=--=-'='~
--~-~-~"~.~.--T
f ~ " - ~ t/,.'r .......
0
_
0
~
--
•
•
•
•
"-...'-..."-"'""~
.,... , - . , , / . . ,
_ _ ./ . . . . . . . . . . . . . . . . . .
_
0.2
0.4
0.6
0.8
1.0
Z/Lr
Fig. 3. Variation of liquid superheat with pressure and solar insolation.
the vapor quality and void fraction, that tends to increase the buoyancy forces (Fig. 4). Lowering the pressure inside the collector plate, for constant heat flux, results in an increase of the circulation velocity caused by the increase of the vapor specific volume (vg). These trends hold only for fully charged collectors. For partially filled collectors the presence of a superheated vapor region lowers the heat transfer rate from the plate to the refrigerant, increasing Tw, which causes an increase in the energy loss (equation (18)). For low values of solar insolation (up to 200 W m -2) the pressure effect upon the circulation velocity is less significant than at high solar fluxes (Fig. 4). This is again due to the low values of both x and ct at low values of the fluxes. Figure 5 shows the variation of ~t with the system pressure along the collector tubes at different saturation temperatures while Fig. 6 shows the quality. The efficiency of the primary loop, for the present conditions, changes with both the solar flux and the system pressure (Fig. 7). Increasing the solar insolation increases the buoyancy forces and consequently the circulation rate, thereby, the useful energy increase results in high efficiency values. The efficiency approaches an asymptotic value, that is determined theoretically by the glass cover transmissivity and plate absorptvity. The present case study, with R-I 1 and the absence of subcooling and superheat regions, gives high loop efficiencies as the losses 3.5
'
I
'
I
13 T$ - 3 3 3 K
~
I
I
OT s -313K
'
I
• Ts= 353 K
3.0
ss S s S
.....
aS
% 2.5 E v
so ~
g,.o
j.
Sf
s S
fss.~ss~ O a ~
~ 1.5
$
//s /
~ I.O
......~.~o-
0.5
200
400
600
800
t000
I100
Solar flux (W m-2)
Fig. 4. Effect of solar insolation and system pressure upon R-11 circulation velocity. HRS 13/5--E
438
A. M. RADHWAN and G. M. ZAK] 05
t
i
T s =313K
....
i
Ts = 5 3 5 K
-- --Ts
i
=353 K
0.4
-1
0.3 "o )~ 0.2
0.1
0 0
0.2
0.4
0.6
08
1.0
Z/Lr
Fig. 5. Void fraction variation along the collector tube.
from the plate to the ambient (at 30°C) are affected by the difference (Tw-Ta). The wall temperature, Tw, is higher than the saturation temperature, that solely is adjusted by presetting the loop pressure. For lower values of the loop pressure the efficiency increases, as a result of the increase in the mass circulation rate, although the latent heat is higher than that for high pressures (Fig. 7). The analysis shows that a primary loop operating with low boiling point heat carrying fluids at low saturation temperatures (40°C) gives a high efficiency (60-80%) which is much higher than values obtained for a partially filled collector at different filling ratios (30-65% for acetone [8], up to 42% for water [1] and up to 45% for R-I1 [5]). It is expected that the efficiency of a complete system, where a secondary loop is to be considered, will result in lower overall efficiencies. 5. C O N C L U S I O N S
A model has been proposed for parametric analysis of the primary loops of solar collectors with phase change. The proposed model is of a general nature where thermal nonequilibrium and dependency of the thermophysical properties upon temperature and pressure are accounted for.
y
0.008
0007
0.006
0005 X
/
QO04
0 0.003 0.002 Ts= 313 K
0.~1 0
-
/
~I 0.2
I 0.4
I 0.6
I 0.8
1.0
Z/Lr
Fig. 6. Variation of the quality along the collector tubes and with the system pressure.
439
Analysis of refrigerant-charged solar collectors I.O
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Fig. 7. Dependency of primary loop efficiency on solar insolation and system pressure (R-11 fully charged collector: no subcooling).
It is assumed that the collector loop is fully charged by the heat carrying fluid, (liquid level = 100%) to avoid vapor superheating in the collector. Results for a Freon-I 1 charged collector of 1 m 2 area, 10 tubes 12.5 mm diameter each at 30° tilt angle to the horizontal are presented. The results show that the liquid superheating is less than one degree, and insignificantly varies along the collector tubes. However, it is dependent upon the pressure inside the primary loop. The local vapor quality and void fraction vary along the length of the tubes. The circulation rate for the R-11, controlled pressure system, varies linearly with the solar insolation and increases at lower pressures of the primary loop. Increasing the solar insolation raises the mass flow rate through the loop and consequently the useful energy gain. The efficiency of the loop is thereby enhanced and approaches an asymptotic value, theoretically determined by the glass cover transmissivity and collector plate absorptivity. For the present case study, with the absence of liquid subcooling and vapor superheat, the energy loss to the ambient is minimum as the collector plate temperature maintains a constant value above the saturation temperature. REFERENCES I. R. Soin, S. Raghurman and V. Murali, Two-phase water heater model and long term performance. Solar Energy 38, 105-112 (1987). 2. R. Soin, K. S. Rao and D. P. Rao, Performance of flat plate solar collector with fluid undergoing phase change. In Proc. ISES Congr., pp. 952-956 Pergamon Press, New Delhi (1978). 3. A. AI-Tamimi and J. A. Clark, Thermal performance of a solar collector containing a boiling fluid. ASHRAE Trans. 90, 681-696 (1984). 4. R. Farringeton, M. De Angelis, L. Morrison and D. Longberty, Performance evaluation of a refrigerant-charged thermosyphon solar DWH system. In Proc. AS/ISES Annual Meeting, Philadelphia, published by the ISES, American Section, 4, 676-680 (1981). 5. A. Radhwan, G. M. Zaki and J. Arshad, Refrigerant-charged integrated solar water heater. Int. J. Energy Res. 14, 421-432 (1990). 6. A. H. Fanny and C. P. Terlizzi, Testing of refrigerant-charged solar domestic hot water systems. Solar Energy 35, 353-366 (1985). 7. M. Akyurt, AWSWAH, The heat pipe solar water heater. J. Engng appl. Sci. 3, 23-38 (1986). 8. W. Kamal, Performance evaluation of a refrigerant-charged solar collector. In Energex-84, pp. 193-197. Pergamon Press, Regina, Saskatchewan (1984). 9. K. Becker, G. Herenborg and M. Bode, An experimental study of pressure gradients for flow of boiling water in a vertical round duct. AE, pp. 69-70, AB. Aktienbolaget, Atomenergi, Sweden (1962). 10. V. Kishor, M. Ghandi, C. Marquis and K. Rao, Evaluation and testing of flat plate collectors employing phase changing fluids. In Energex 82, pp. 134-139. Pergamon Press, Regina, Saskatchewan (1982). 1 I. K. Baksted and K. O. Solbcrg, A FORTRAN Program for Transient Analysis of Boiling Water Reactors and Boiling Loops. KR-135 Inst. for Atomenergi, Kjeller, Norway (1968). 12. W. C. Reynolds, Thermodynamic Properties in SI, Section 2, pp. 70-72. Dept of Mech. Engng, Stanford Univ., CA (1979). 13. P. Dunn and D. A. Reay, Heat Pipes, p. 274. Pergamon Press, Oxford (1977).