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Dynamic behaviour of baffled solar air heaters
Renewable Energy, Vol. 13, No. 2, pp. 153 163, 1998
~
Pergamon
© 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain P I I : S 0 9 6 0 - 1 4 8 1 (97)00093-1 0960-1481/98 $19.00 + 0.00
DYNAMIC BEHAVIOUR OF BAFFLED SOLAR AIR HEATERS M. A. AL-NIMR* and R. A. DAMSEH Mechanical Engineering Department, Jordan University of Science and Technology, Irbid, Jordan
(Received 26 September 1997 ; accepted 14 November 1997)
Abstract--A
mathematical model describes the dynamic thermal behaviour of a baffled solar air heater is presented. The transient behaviour of the heater results from sudden changes in the intensity of the incident solar radiation and the inlet fluid temperature. In terms of the presented model, analytical solutions for the fluid and solid domains are derived. Also, expression for the thermal efficiency of the heater is presented. The effects of different design parameters on the thermal performance of the heater are investigated. The validity of the theoretical model is verified experimentally where it is found that both theoretical and experimental results are in a good agreement. © 1998 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION One of the most potential application of solar energy is the supply of hot air for drying of agricultural, textile and marine product, and heating of buildings to maintain a comfortable environment especially in the winter season. There are various possible configurations of solar air heaters which could offer a wide range of performance efficiencies. Examples of such heaters are air collectors, conventional duct-type [9, 3], first and second kinds [5], matrix air heaters [1,2] and baffled air heaters [4, 7, 8, 5]. All of the previous designs have been proposed to increase the heat-transfer coefficient between the absorber and the flowing air. This goal is attained by using porous material as in matrix air heaters, corrugated absorber, offset-fin-plate-heat exchanger and finned absorber air heater as in baffled solar air heater. The aim of the present study is to investigate the dynamic performance of baffled solar air heaters where the steady state performance of such heaters is considered in [7~ 8, 6]. The investigation is done by introducing a mathematical model which describes the dynamic thermal behaviour of baffled heaters. In terms of this model, the temperature distribution within the fluid and solid domains is found analytically and the predictions of this model are verified experimentally where it is found that both results are in a good
* To whom all correspondence should be addressed. 153
M. A. AL-NIMR and R. A. DAMSEH
154
agreement. The analytical expression for the temperature distribution within the heater are used to find explicit form for the thermal efficiency of the heater. As a result, the effect of different dimensionless design parameters on the thermal performance of the heaters may be investigated.
2. A MATHEMATICAL FORMULATION OF THE MODEL The mathematical model describing the dynamic thermal behaviour of the solar air heater is based on the following assumptions : 1. There are two separate phases, fluid and solid, which are no in thermal equilibrium. This is the case for low thermal conductivity fluid (air) and low volumetric heat transfer coefficient between the fluid and solid domains. Also, this assumption is verified during the early stages of time of the heater dynamic response. 2. The fluid is one dimensional plug flow. 3. The fluid is laminar and Newtonian. 4. The thermal properties of the fluid and solid domains are constant and uniform. 5. Thermal and mass diffusion are assumed to be negligible. 6. Transient response is resulted from sudden change in the intensity of the incident solar radiation and/or in the inlet fluid temperature. Now, referring to Fig. 1 and using the dimensionless parameters defined in the nomenclature, the dimensionless governing energy equation for the solid and fluid domains are given as :
GI
•
Absorbingplate
,
1~ . . . . . . .
. . . . . . . .
~ 1:12 ....................... Insular:ion l ~ ........ Fig. 1. Solar air heater with fins.
X
Dynamic behaviour of baffled solar air heaters 00~ ~r
-
155 (1)
I~-E, Os-E2(O~-O0
00f 00f t-~ -~- 0-X = E3 (0s - Of) - E4 OK.
(2)
where Ic=
E2 =
W" I" L" pf" Af T~ • p~ • A~ " Cs " m f ' Ul " b" L" pf" Af - p~ " As" C~ " mf '
E~=
E~ "
Uu " W" L" pf" Af Ps " As "C~" rhf
U1 " b" L rhf. Cf
,
E4 -
Us " d" L rhf" Cr
Equations (1) and (2) are subjected to the following initial and boundary conditions: o f ( o , x ) = o~(o,x) = o Of(r,O) = 0 o ( r )
fort>0
(3)
(4)
It is worth mentioning that the axial thermal diffusion in both fluid and solid domains is neglected in eqns (1) and (2). This is a good assumption, especially for flows of height Peclet numbers which are the operating conditions for most practical cases. In order to get solutions for 0r and 0~, the Laplace transform technique will be used. Now with the notation that L { O ( r , x ) } = W ( S , X ) , the Laplace transform of eqns (1) and (2) yields : SW. = ~-E, (~rv,. swf-I- ~--=
2W~+E2Wf
(5)
E3W~-E34W f
(6)
where E12 = Et + Ez,
E34 = E3 --1-E4
The boundary condition given in eqn (4) is reduced to : W,.(S, O) = Wo (S)
(7)
Now, solving for W~ from eqn (5), substituting the result into eqn (6) and rearranging yields" ~Wf
(
E5 ) E31~ --]--E34 Wf S + E~2 S ( S + E12)
0X -~" S - - -
where E 5 = E2E 3
(8)
156
M.A.
AL-NIMR and R. A. DAMSEH
Now, solving eqn (8) for Wf, applying eqn (7) and rearranging yields :
S+E,2E5 +E34)X1
+~¢(~exp[--(S-The term
S(S+E~z)(S--[Es/(S+EIz)]+E34)
can be written as
S(S+20(S+~2),
(9)
where
~'1 +42 = E12-]-E34, "~1"42= E12E34-Es. So eqn (9) can be rewritten as:
Ic
-- ~xS(S-~l~l)(S-~-~2)J
~x(S-]-~l)(S-]-/~2)
-}- S(S-[-~l)(S--~l~2)
/ E5 X~
rexp (-- SX) exp (-- E34X) exp ~ S ~ 2 ~ 2 )]
Wo(S)(S+e,2))[,c In terms of eqns (5) and (10), Ws(S,x)ic
s+E,2
(10)
W~(S,X) is given as:
(S+~l)(S+,~2) + s ( s + , ~ , ) ( s + ~ 2 ) j
Wo(S).
s(s+,~-ff(s+,~O
-exp ( - S X ) exp (--E34) exp
I~ J
S+E12
The fluid and solid domain temperatures can be found by considering the following inversion formulas [I0]
L-'{exp(--ns)h(s)} =H(z-n)R(z--n)=
{O(~-a)
if ~>~a if z < a
L - ' {_exp(a/(s~'+ b)) ~ =
~b
~-, J exp(-bz)(z/a)~-I~ , (2x/~),
L-' {f(s)g(s)} = fl F(z-- u)G(u) du {
1
}
L ' (s + ~) (s + IS) (s + ~,) =
(7--fl)exp(-az)+(a--,)exp(-flr)+(fl-a)exp(--,v) (~ - IS) (IS- ~) (~ - ~)
{ 1 } exp ( - Isz)-exp (--c~) L --1 -(S "Jr-~i(S -1--IS) = (0~ -- IS) where
R(v--n)
is the unit step function. Now define z* = (v-X), a = Es, b = E12, v = 1,
Dynamic behaviour of baffled solar air heaters
157
n = X. Before inverting the solution to the time domain it is clear that the exact form of temperature distribution depends on the inlet fluid temperature. As a special case let us consider that the inlet fluid temperature is constant 0o(~)=0o
00 Wo(S)=-S
and a s a r e s u l t
In terms of the above consideration, eqns (10) and (11) are inverted to yield" 0f(v, X) = I~[W, - R, e-~E3,x+~2~*~0, +
R2e-(E34x+)~,~*)O2 -- R3e E~4xo3]
+O0[e-(E~,x+E,~*)BS+E,2e-E~4x03] for ~* /> 0 = IcUd, for r* < 0 O~('c, X) = I~['ad2 M1 e -(E~'x+~' ~*)02 + M2e-(e~"x+~')O, -
-
q-Oo[e-E34X03]
-
M3e E~'xo3]
-
for r* ~> 0 =/cW2
for r* < 0
where
E3(22--E,2) R,-
E3()~l--E12)
22(21_~2) ,
R 2 --
E3EI2 R3 = 2,22
21(21_22),
E3 22(21--,~2)
E3 R4 =/].l (~.l--'~.2) '
R5
E5 M, - '~1('~,--'~2)'
E5 M2 - ,~2(,~, _--~),
Es M3 = "~1'~2
/~2 --E34 M4 - 22(21 - - 2 2 ) '
21 --E34 M5 - 21(2~--22)'
E34 M6 - )~1~2
R6
E3 2t22
also
tlJ 1 = R 4 e - ) ~ - - R s e 8s
~2~--1-R6'
IJ~2 =
M4e-'~,~--Mse ;~2*+M 6
= I0
O1 --
fi
e-~uI0(2~)
du
O2 =
e-~, "Io ( 2 ~ )
du
O3 =
e-e12UIo ( 2 ~ )
du
where cq = E12- 21 and 72 = E12- 22. To simplify the computational procedure, the term • = m a y be rewritten as"
fi
02)
e - a2PIo(2 a x / ~Xp) dp
(13)
158
M. A. AL-NIMR and R. A. DAMSEH
a2 J
.
.
.
.
o ( k - n ) ~ (azz*)k-"
3. H E A T E R E F F I C I E N C Y
The heater efficiency is defined as : E2 q= ~ [0f(z, 1) - Of(z, 0)]
(14)
where the fluid temperature at the outlet Of(z, 1) can be obtained from eqn (12).
4. EXPERIMENTAL INVESTIGATION To verify the validity of the theoretical model proposed in this study, an experimental set-up is built and tested. The transient response of the heater is resulted only from sudden change in the inlet fluid temperature where the incident solar radiation is assumed to be absent. The heater is tested under the following operating conditions :
Tf(O,z) = Ts(0, z) = T~ = 17"CTf(z, 0) = To = 61°C, U, = 52.4(W/m 2 K), Ps = 7865(kg/m3), pf = 100(kg/m3), L = l(m),
Uu = 4.22(W/m 2 K),
Us = 2(W/m 2 K)
Cs = 460(J/kg K), Cf = 4175(J/kg K),
W = 0.2(m),
mf = 0.004(kg/s)
As = 6.6(cm) 2 Af -- 54.84(cm) 2
d = 0.26(m),
b = 0.32(m)
5. RESULT AND DISCUSSION To verify the validity of the proposed theoretical model, the analytical solution with absent incident solar radiation (i.e., with Ic = 0) is compared with the experimental results. In this case, it is assumed that the heater transient response is resulted from sudden change in the inlet fluid temperature. The comparison between theoretical and experimental results is shown in Fig. 2. The comparison reveals that both results are in a good agreement especially in locations near the inlet of the heater. However, and as clear from the same figure, experimental results predict that inlet conditions have deeper axial thermal penetration than the predictions of the theoretical results. This is due to the fact that the theoretical model neglects the axial thermal diffusion. The main source of error causes deviation between theoretical and experimental results in other locations is the uncertainty in estimating different heat transfer coefficients. In estimating these coefficients we rely on empirical formulae which assumed that these coefficients are of constant values. In fact, this is not the case where these coefficients vary with the variation of different operating conditions. Another source causes the deviation is the time constant of the measurement devices used to since the temperature response. It is clear that the working fluid used in the solar heater, which is water, has low heat transfer coefficient with the hot junction of the
Dynamic behaviour of baffled solar air heaters
159
3.00
- ***** ooooo
Theoretical
El=.5 E2=13.36 E4= 1.7gE-2
Experimental
=====
Ic=0
2.50
2.00 ~) r~
21.50 I-
1.00
-I II
r-.
11
E k5 0.50
(./I
o
A o
0.00
Illllll[IJllll
0.00
IIII
0.20
III
0.40
II
IliJ
IIIII
0.60
i,,,,,~,~i,=~,=, 0.80
1.00
,~, 1.20
Dimensionless Axial Distance ( X ) Fig. 2. Theoretical and experimental temperature distribution of fluid within the duct.
thermocouple. As a result, there is a certain delay in the ability of the thermocouple for fast detection of the dynamic response. Figure 3 shows the fluid temperature distribution with the axial location of the heater. These results are obtained from the theoretical solution with the assumption that the dynamic response is resulted only from sudden change in the incident solar radiation. As clear from this figure, the temperature profile attains its linear shape when the system reaches the steady state performance. This is predicted since both incident radiation and thermal losses to the ambient are uniform with the axial location. Figure 4 shows the variation in thermal efficiency with time. In the early stages of time, the system has low thermal efficiency. The reason for this is that during warming up stages, a large part of the incident solar radiation is stored in the solid structure of the heater. As a result, the working fluid will not benefit from this incident energy. In the literature, researchers do not judge the performance of such systems based on its efficiency during its dynamic response. In fact such a judgement will give misleading conclusions. As a result, our concern is only focused on the heater thermal efficiency at steady state conditions. Figure 5 investigates the effect of different design parameters on the steady state thermal efficiency. Again, the results of this figure are estimated theoretically with the assumption that the heater dynamic response is resulted from sudden change in the incident solar radiation. It is clear from this figure that increasing E2 improve the heater thermal efficiency. This is a predicted result since E2 is a dimensionless parameter which measures mainly the heat transfer coefficient between the working fluid and the solid domain. Increasing such a parameter eases the flow of energy, absorbed by the solid wall, to the working fluid. It is clear from this curve that there is a critical value for E2 beyond which there is no detectable
160
M . A . A L - N I M R and R. A. D A M S E H
0.60
"fheoreti E4= Ic=6. E E2= 1=.c501ol6 1.79Eresults -23,36
~ ~ e. ~
¢) 00=0
v
P 0.40 --i
G) 0_
E I--
0.20 C
E
k5
I.=. 2
0.00 [ [ l l l l l l l l l l r l | l l l l l l l [ l l I I I I I I I I I I I I I I I I I I 0.20 0.40 0.60 0.80 0.00 Dimensionless Axial Distance ( X )
I I l l l
1.00
Fig. 3. Theoretical temperature distribution of fluid within the heater
1.00
0.80
~,
Theoretical results E1=.5 E2=13.36 E4=l.79E-2 IC=6.06 0o=0
0.60
o E ¢1
:_~ 0.40 LLI
0.20
0.00
=
0.00
=
i
=
v
¢
I
I
i
I
=
i
=
=
i
v
=
=
i
I
a
i
=
1.00 2.00 Dimensionless Time ( r ) Fig. 4. Efficiency variation with time.
v
v
=
i
=
=
3.00
Dynamic behaviour of baffled solar air heaters 1.00 --r
161 .--,
E2
Theoretical results 0.80
Steady state E1=.5 E2=13.36 E4=1.79E-2 IC=6.06 0 o =0
~" 0.60
gc
._~
~
0.40
bJ
0.20
0,00 0.00
10.00 20.00 D[mensionless parameters
30.00
Fig. 5. Efficiency variation with dimensionless parameters.
improvement in the heater efficiency. The reason for that is due to the fact that as the heat transfer coefficient between the solid and fluid increases, both solid and fluid at their interface region will attain the same temperature. This implies that the internal heat transfer convective resistance between the solid and fluid is absent. Any further increase in the heat transfer coefficient, or in E2, will not make any further reduction in the convective resistance. Also, it is clear from Fig. 5 that increasing both E1 and E4 leads to a reduction in the system thermal efficiency. It is clear from the definition of Ej or E4 that these parameters represent mainly the heat loss coefficient from the solid or fluid domains to the ambient. It is understood now that increasing these parameters enhances the thermal losses from the system and as a result causes a reduction in the system thermal efficiency. 6. CONCLUSIONS A mathematical model describing the dynamic thermal behaviour of a baffled solar air heater is presented. The dynamic response of the heater resulted from sudden changes in the inlet fluid temperature and/or in the incident solar radiation. The presented mathematical model is described by two coupled partial differential equations which are solved exactly using Laplace transformation technique. In terms of the obtained solutions, the transient temperature distribution in both solid (walls and fins of the heater) and fluid domains is found. The effect of different design parameters on the thermal efficiency of the heater are investigated where it is found that there are mainly three design parameters affecting the thermal performance of the heater. One of these parameters measures the coupling between the solid and the working fluid and as a result it measures the ease with
162
M. A. AL-NIMR and R. A. DAMSEH
which the absorbed solar radiation flows from the absorber plate to the working fluid. The other two parameters measures the heat losses from the fluid and solid domains to the ambient, respectively. To verify the adequacy of the presented theoretical model, an experimental set-up is built and different experiments are conducted under different operating conditions. It is found that both theoretical and experimental results are in good agreement. NOMENCLATURE Af
A, b C I-
C~ d hl h2 I L S t
T, L T~ Ul
u~ Uu W X
X
Greek letters q Or O, Oo P~ Pr T
cross sectional area of fluid passage cross sectional area of solid domain inner perimeter of the upper plate with its fins specific heat of fluid at constant pressure specific heat of solid at constant pressure length of sides and bottom (2h 1+ h2) height of the cross sectional area width of the cross sectional area intensity of radiation at any location length of"the heater mass flow rate of fluid Laplacian domain time fluid temperature solid temperature ambient temperature heat transfer coefficient from fluid to solid lower heat loss coefficient upper heat loss coefficient width of the heater axial coordinate dimensionless axial distance, x/L heater efficiency dimensionless fluid temperature (Tf--T~)/T~ dimensionless solid temperature (T~- T~)/Tj dimensionless inlet fluid temperature ( Tr(t, O) - T,~) / T j density of solid density of fluid dimensionless time (mft/pfAf/L).
REFERENCES
1, A1-Nimr, M. A., Transient behaviour of a matrix solar air heater. Energy Convers. M g m t . , 1993, 34(8), 649-656. 2. A1-Nimr, M. A., Dynamic behaviour of a cylindrical solar air heater. Renewable Energy, 1994, 4(5), 579 583. 3. Biondi, P., Performance analysis of solar air heaters of conventional design, solar air heaters of conventional design. Solar Energy, 1988, 41(1), 101-107. 4. Duffle, J. and Beckman, W. Solar Engineerin9 o f Thermal Processes, 2nd edn, ch. 6. John Wiley and Sons, New York. 5. Garg, H. P., Sharma, V. K., Bandy Opadhyay, B. and Bhargava, A. K., Transient analysis of a solar air heater of the second kind. Energy Convers. Mgrnt., 1982, 22, 4753.
Dynamic behaviour of baffled solar air heaters
163
6. Garg, H. P., Jha, R., Choudhury, C. and Datta, Gouri, Theoretical analysis on a new finned type solar air heater. Energy, 1991, 16(10), 1231-1238. 7. Ho-Ming, Yeh, Theory of baffled solar air heaters. Energy, 1992, 17(7), 697~702. 8. Ho-Ming, Yeh, Efficiency of upward-type baffled solar air heaters. Energy, 1994, 19(4), 443~448. 9. Jallut, C., Jemni, A. and Lallemand, M., Steady-state and dynamic characterization of an array of air flat-plate collectors. Solar and Wind Technology, 1988, 5(5), 573 579. 10. Roberts, G. E. and Kaufman, H., Table of Laplace transform. W. B. Saunders, Philadelphia, 1966.
~
Pergamon
© 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain P I I : S 0 9 6 0 - 1 4 8 1 (97)00093-1 0960-1481/98 $19.00 + 0.00
DYNAMIC BEHAVIOUR OF BAFFLED SOLAR AIR HEATERS M. A. AL-NIMR* and R. A. DAMSEH Mechanical Engineering Department, Jordan University of Science and Technology, Irbid, Jordan
(Received 26 September 1997 ; accepted 14 November 1997)
Abstract--A
mathematical model describes the dynamic thermal behaviour of a baffled solar air heater is presented. The transient behaviour of the heater results from sudden changes in the intensity of the incident solar radiation and the inlet fluid temperature. In terms of the presented model, analytical solutions for the fluid and solid domains are derived. Also, expression for the thermal efficiency of the heater is presented. The effects of different design parameters on the thermal performance of the heater are investigated. The validity of the theoretical model is verified experimentally where it is found that both theoretical and experimental results are in a good agreement. © 1998 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION One of the most potential application of solar energy is the supply of hot air for drying of agricultural, textile and marine product, and heating of buildings to maintain a comfortable environment especially in the winter season. There are various possible configurations of solar air heaters which could offer a wide range of performance efficiencies. Examples of such heaters are air collectors, conventional duct-type [9, 3], first and second kinds [5], matrix air heaters [1,2] and baffled air heaters [4, 7, 8, 5]. All of the previous designs have been proposed to increase the heat-transfer coefficient between the absorber and the flowing air. This goal is attained by using porous material as in matrix air heaters, corrugated absorber, offset-fin-plate-heat exchanger and finned absorber air heater as in baffled solar air heater. The aim of the present study is to investigate the dynamic performance of baffled solar air heaters where the steady state performance of such heaters is considered in [7~ 8, 6]. The investigation is done by introducing a mathematical model which describes the dynamic thermal behaviour of baffled heaters. In terms of this model, the temperature distribution within the fluid and solid domains is found analytically and the predictions of this model are verified experimentally where it is found that both results are in a good
* To whom all correspondence should be addressed. 153
M. A. AL-NIMR and R. A. DAMSEH
154
agreement. The analytical expression for the temperature distribution within the heater are used to find explicit form for the thermal efficiency of the heater. As a result, the effect of different dimensionless design parameters on the thermal performance of the heaters may be investigated.
2. A MATHEMATICAL FORMULATION OF THE MODEL The mathematical model describing the dynamic thermal behaviour of the solar air heater is based on the following assumptions : 1. There are two separate phases, fluid and solid, which are no in thermal equilibrium. This is the case for low thermal conductivity fluid (air) and low volumetric heat transfer coefficient between the fluid and solid domains. Also, this assumption is verified during the early stages of time of the heater dynamic response. 2. The fluid is one dimensional plug flow. 3. The fluid is laminar and Newtonian. 4. The thermal properties of the fluid and solid domains are constant and uniform. 5. Thermal and mass diffusion are assumed to be negligible. 6. Transient response is resulted from sudden change in the intensity of the incident solar radiation and/or in the inlet fluid temperature. Now, referring to Fig. 1 and using the dimensionless parameters defined in the nomenclature, the dimensionless governing energy equation for the solid and fluid domains are given as :
GI
•
Absorbingplate
,
1~ . . . . . . .
. . . . . . . .
~ 1:12 ....................... Insular:ion l ~ ........ Fig. 1. Solar air heater with fins.
X
Dynamic behaviour of baffled solar air heaters 00~ ~r
-
155 (1)
I~-E, Os-E2(O~-O0
00f 00f t-~ -~- 0-X = E3 (0s - Of) - E4 OK.
(2)
where Ic=
E2 =
W" I" L" pf" Af T~ • p~ • A~ " Cs " m f ' Ul " b" L" pf" Af - p~ " As" C~ " mf '
E~=
E~ "
Uu " W" L" pf" Af Ps " As "C~" rhf
U1 " b" L rhf. Cf
,
E4 -
Us " d" L rhf" Cr
Equations (1) and (2) are subjected to the following initial and boundary conditions: o f ( o , x ) = o~(o,x) = o Of(r,O) = 0 o ( r )
fort>0
(3)
(4)
It is worth mentioning that the axial thermal diffusion in both fluid and solid domains is neglected in eqns (1) and (2). This is a good assumption, especially for flows of height Peclet numbers which are the operating conditions for most practical cases. In order to get solutions for 0r and 0~, the Laplace transform technique will be used. Now with the notation that L { O ( r , x ) } = W ( S , X ) , the Laplace transform of eqns (1) and (2) yields : SW. = ~-E, (~rv,. swf-I- ~--=
2W~+E2Wf
(5)
E3W~-E34W f
(6)
where E12 = Et + Ez,
E34 = E3 --1-E4
The boundary condition given in eqn (4) is reduced to : W,.(S, O) = Wo (S)
(7)
Now, solving for W~ from eqn (5), substituting the result into eqn (6) and rearranging yields" ~Wf
(
E5 ) E31~ --]--E34 Wf S + E~2 S ( S + E12)
0X -~" S - - -
where E 5 = E2E 3
(8)
156
M.A.
AL-NIMR and R. A. DAMSEH
Now, solving eqn (8) for Wf, applying eqn (7) and rearranging yields :
S+E,2E5 +E34)X1
+~¢(~exp[--(S-The term
S(S+E~z)(S--[Es/(S+EIz)]+E34)
can be written as
S(S+20(S+~2),
(9)
where
~'1 +42 = E12-]-E34, "~1"42= E12E34-Es. So eqn (9) can be rewritten as:
Ic
-- ~xS(S-~l~l)(S-~-~2)J
~x(S-]-~l)(S-]-/~2)
-}- S(S-[-~l)(S--~l~2)
/ E5 X~
rexp (-- SX) exp (-- E34X) exp ~ S ~ 2 ~ 2 )]
Wo(S)(S+e,2))[,c In terms of eqns (5) and (10), Ws(S,x)ic
s+E,2
(10)
W~(S,X) is given as:
(S+~l)(S+,~2) + s ( s + , ~ , ) ( s + ~ 2 ) j
Wo(S).
s(s+,~-ff(s+,~O
-exp ( - S X ) exp (--E34) exp
I~ J
S+E12
The fluid and solid domain temperatures can be found by considering the following inversion formulas [I0]
L-'{exp(--ns)h(s)} =H(z-n)R(z--n)=
{O(~-a)
if ~>~a if z < a
L - ' {_exp(a/(s~'+ b)) ~ =
~b
~-, J exp(-bz)(z/a)~-I~ , (2x/~),
L-' {f(s)g(s)} = fl F(z-- u)G(u) du {
1
}
L ' (s + ~) (s + IS) (s + ~,) =
(7--fl)exp(-az)+(a--,)exp(-flr)+(fl-a)exp(--,v) (~ - IS) (IS- ~) (~ - ~)
{ 1 } exp ( - Isz)-exp (--c~) L --1 -(S "Jr-~i(S -1--IS) = (0~ -- IS) where
R(v--n)
is the unit step function. Now define z* = (v-X), a = Es, b = E12, v = 1,
Dynamic behaviour of baffled solar air heaters
157
n = X. Before inverting the solution to the time domain it is clear that the exact form of temperature distribution depends on the inlet fluid temperature. As a special case let us consider that the inlet fluid temperature is constant 0o(~)=0o
00 Wo(S)=-S
and a s a r e s u l t
In terms of the above consideration, eqns (10) and (11) are inverted to yield" 0f(v, X) = I~[W, - R, e-~E3,x+~2~*~0, +
R2e-(E34x+)~,~*)O2 -- R3e E~4xo3]
+O0[e-(E~,x+E,~*)BS+E,2e-E~4x03] for ~* /> 0 = IcUd, for r* < 0 O~('c, X) = I~['ad2 M1 e -(E~'x+~' ~*)02 + M2e-(e~"x+~')O, -
-
q-Oo[e-E34X03]
-
M3e E~'xo3]
-
for r* ~> 0 =/cW2
for r* < 0
where
E3(22--E,2) R,-
E3()~l--E12)
22(21_~2) ,
R 2 --
E3EI2 R3 = 2,22
21(21_22),
E3 22(21--,~2)
E3 R4 =/].l (~.l--'~.2) '
R5
E5 M, - '~1('~,--'~2)'
E5 M2 - ,~2(,~, _--~),
Es M3 = "~1'~2
/~2 --E34 M4 - 22(21 - - 2 2 ) '
21 --E34 M5 - 21(2~--22)'
E34 M6 - )~1~2
R6
E3 2t22
also
tlJ 1 = R 4 e - ) ~ - - R s e 8s
~2~--1-R6'
IJ~2 =
M4e-'~,~--Mse ;~2*+M 6
= I0
O1 --
fi
e-~uI0(2~)
du
O2 =
e-~, "Io ( 2 ~ )
du
O3 =
e-e12UIo ( 2 ~ )
du
where cq = E12- 21 and 72 = E12- 22. To simplify the computational procedure, the term • = m a y be rewritten as"
fi
02)
e - a2PIo(2 a x / ~Xp) dp
(13)
158
M. A. AL-NIMR and R. A. DAMSEH
a2 J
.
.
.
.
o ( k - n ) ~ (azz*)k-"
3. H E A T E R E F F I C I E N C Y
The heater efficiency is defined as : E2 q= ~ [0f(z, 1) - Of(z, 0)]
(14)
where the fluid temperature at the outlet Of(z, 1) can be obtained from eqn (12).
4. EXPERIMENTAL INVESTIGATION To verify the validity of the theoretical model proposed in this study, an experimental set-up is built and tested. The transient response of the heater is resulted only from sudden change in the inlet fluid temperature where the incident solar radiation is assumed to be absent. The heater is tested under the following operating conditions :
Tf(O,z) = Ts(0, z) = T~ = 17"CTf(z, 0) = To = 61°C, U, = 52.4(W/m 2 K), Ps = 7865(kg/m3), pf = 100(kg/m3), L = l(m),
Uu = 4.22(W/m 2 K),
Us = 2(W/m 2 K)
Cs = 460(J/kg K), Cf = 4175(J/kg K),
W = 0.2(m),
mf = 0.004(kg/s)
As = 6.6(cm) 2 Af -- 54.84(cm) 2
d = 0.26(m),
b = 0.32(m)
5. RESULT AND DISCUSSION To verify the validity of the proposed theoretical model, the analytical solution with absent incident solar radiation (i.e., with Ic = 0) is compared with the experimental results. In this case, it is assumed that the heater transient response is resulted from sudden change in the inlet fluid temperature. The comparison between theoretical and experimental results is shown in Fig. 2. The comparison reveals that both results are in a good agreement especially in locations near the inlet of the heater. However, and as clear from the same figure, experimental results predict that inlet conditions have deeper axial thermal penetration than the predictions of the theoretical results. This is due to the fact that the theoretical model neglects the axial thermal diffusion. The main source of error causes deviation between theoretical and experimental results in other locations is the uncertainty in estimating different heat transfer coefficients. In estimating these coefficients we rely on empirical formulae which assumed that these coefficients are of constant values. In fact, this is not the case where these coefficients vary with the variation of different operating conditions. Another source causes the deviation is the time constant of the measurement devices used to since the temperature response. It is clear that the working fluid used in the solar heater, which is water, has low heat transfer coefficient with the hot junction of the
Dynamic behaviour of baffled solar air heaters
159
3.00
- ***** ooooo
Theoretical
El=.5 E2=13.36 E4= 1.7gE-2
Experimental
=====
Ic=0
2.50
2.00 ~) r~
21.50 I-
1.00
-I II
r-.
11
E k5 0.50
(./I
o
A o
0.00
Illllll[IJllll
0.00
IIII
0.20
III
0.40
II
IliJ
IIIII
0.60
i,,,,,~,~i,=~,=, 0.80
1.00
,~, 1.20
Dimensionless Axial Distance ( X ) Fig. 2. Theoretical and experimental temperature distribution of fluid within the duct.
thermocouple. As a result, there is a certain delay in the ability of the thermocouple for fast detection of the dynamic response. Figure 3 shows the fluid temperature distribution with the axial location of the heater. These results are obtained from the theoretical solution with the assumption that the dynamic response is resulted only from sudden change in the incident solar radiation. As clear from this figure, the temperature profile attains its linear shape when the system reaches the steady state performance. This is predicted since both incident radiation and thermal losses to the ambient are uniform with the axial location. Figure 4 shows the variation in thermal efficiency with time. In the early stages of time, the system has low thermal efficiency. The reason for this is that during warming up stages, a large part of the incident solar radiation is stored in the solid structure of the heater. As a result, the working fluid will not benefit from this incident energy. In the literature, researchers do not judge the performance of such systems based on its efficiency during its dynamic response. In fact such a judgement will give misleading conclusions. As a result, our concern is only focused on the heater thermal efficiency at steady state conditions. Figure 5 investigates the effect of different design parameters on the steady state thermal efficiency. Again, the results of this figure are estimated theoretically with the assumption that the heater dynamic response is resulted from sudden change in the incident solar radiation. It is clear from this figure that increasing E2 improve the heater thermal efficiency. This is a predicted result since E2 is a dimensionless parameter which measures mainly the heat transfer coefficient between the working fluid and the solid domain. Increasing such a parameter eases the flow of energy, absorbed by the solid wall, to the working fluid. It is clear from this curve that there is a critical value for E2 beyond which there is no detectable
160
M . A . A L - N I M R and R. A. D A M S E H
0.60
"fheoreti E4= Ic=6. E E2= 1=.c501ol6 1.79Eresults -23,36
~ ~ e. ~
¢) 00=0
v
P 0.40 --i
G) 0_
E I--
0.20 C
E
k5
I.=. 2
0.00 [ [ l l l l l l l l l l r l | l l l l l l l [ l l I I I I I I I I I I I I I I I I I I 0.20 0.40 0.60 0.80 0.00 Dimensionless Axial Distance ( X )
I I l l l
1.00
Fig. 3. Theoretical temperature distribution of fluid within the heater
1.00
0.80
~,
Theoretical results E1=.5 E2=13.36 E4=l.79E-2 IC=6.06 0o=0
0.60
o E ¢1
:_~ 0.40 LLI
0.20
0.00
=
0.00
=
i
=
v
¢
I
I
i
I
=
i
=
=
i
v
=
=
i
I
a
i
=
1.00 2.00 Dimensionless Time ( r ) Fig. 4. Efficiency variation with time.
v
v
=
i
=
=
3.00
Dynamic behaviour of baffled solar air heaters 1.00 --r
161 .--,
E2
Theoretical results 0.80
Steady state E1=.5 E2=13.36 E4=1.79E-2 IC=6.06 0 o =0
~" 0.60
gc
._~
~
0.40
bJ
0.20
0,00 0.00
10.00 20.00 D[mensionless parameters
30.00
Fig. 5. Efficiency variation with dimensionless parameters.
improvement in the heater efficiency. The reason for that is due to the fact that as the heat transfer coefficient between the solid and fluid increases, both solid and fluid at their interface region will attain the same temperature. This implies that the internal heat transfer convective resistance between the solid and fluid is absent. Any further increase in the heat transfer coefficient, or in E2, will not make any further reduction in the convective resistance. Also, it is clear from Fig. 5 that increasing both E1 and E4 leads to a reduction in the system thermal efficiency. It is clear from the definition of Ej or E4 that these parameters represent mainly the heat loss coefficient from the solid or fluid domains to the ambient. It is understood now that increasing these parameters enhances the thermal losses from the system and as a result causes a reduction in the system thermal efficiency. 6. CONCLUSIONS A mathematical model describing the dynamic thermal behaviour of a baffled solar air heater is presented. The dynamic response of the heater resulted from sudden changes in the inlet fluid temperature and/or in the incident solar radiation. The presented mathematical model is described by two coupled partial differential equations which are solved exactly using Laplace transformation technique. In terms of the obtained solutions, the transient temperature distribution in both solid (walls and fins of the heater) and fluid domains is found. The effect of different design parameters on the thermal efficiency of the heater are investigated where it is found that there are mainly three design parameters affecting the thermal performance of the heater. One of these parameters measures the coupling between the solid and the working fluid and as a result it measures the ease with
162
M. A. AL-NIMR and R. A. DAMSEH
which the absorbed solar radiation flows from the absorber plate to the working fluid. The other two parameters measures the heat losses from the fluid and solid domains to the ambient, respectively. To verify the adequacy of the presented theoretical model, an experimental set-up is built and different experiments are conducted under different operating conditions. It is found that both theoretical and experimental results are in good agreement. NOMENCLATURE Af
A, b C I-
C~ d hl h2 I L S t
T, L T~ Ul
u~ Uu W X
X
Greek letters q Or O, Oo P~ Pr T
cross sectional area of fluid passage cross sectional area of solid domain inner perimeter of the upper plate with its fins specific heat of fluid at constant pressure specific heat of solid at constant pressure length of sides and bottom (2h 1+ h2) height of the cross sectional area width of the cross sectional area intensity of radiation at any location length of"the heater mass flow rate of fluid Laplacian domain time fluid temperature solid temperature ambient temperature heat transfer coefficient from fluid to solid lower heat loss coefficient upper heat loss coefficient width of the heater axial coordinate dimensionless axial distance, x/L heater efficiency dimensionless fluid temperature (Tf--T~)/T~ dimensionless solid temperature (T~- T~)/Tj dimensionless inlet fluid temperature ( Tr(t, O) - T,~) / T j density of solid density of fluid dimensionless time (mft/pfAf/L).
REFERENCES
1, A1-Nimr, M. A., Transient behaviour of a matrix solar air heater. Energy Convers. M g m t . , 1993, 34(8), 649-656. 2. A1-Nimr, M. A., Dynamic behaviour of a cylindrical solar air heater. Renewable Energy, 1994, 4(5), 579 583. 3. Biondi, P., Performance analysis of solar air heaters of conventional design, solar air heaters of conventional design. Solar Energy, 1988, 41(1), 101-107. 4. Duffle, J. and Beckman, W. Solar Engineerin9 o f Thermal Processes, 2nd edn, ch. 6. John Wiley and Sons, New York. 5. Garg, H. P., Sharma, V. K., Bandy Opadhyay, B. and Bhargava, A. K., Transient analysis of a solar air heater of the second kind. Energy Convers. Mgrnt., 1982, 22, 4753.
Dynamic behaviour of baffled solar air heaters
163
6. Garg, H. P., Jha, R., Choudhury, C. and Datta, Gouri, Theoretical analysis on a new finned type solar air heater. Energy, 1991, 16(10), 1231-1238. 7. Ho-Ming, Yeh, Theory of baffled solar air heaters. Energy, 1992, 17(7), 697~702. 8. Ho-Ming, Yeh, Efficiency of upward-type baffled solar air heaters. Energy, 1994, 19(4), 443~448. 9. Jallut, C., Jemni, A. and Lallemand, M., Steady-state and dynamic characterization of an array of air flat-plate collectors. Solar and Wind Technology, 1988, 5(5), 573 579. 10. Roberts, G. E. and Kaufman, H., Table of Laplace transform. W. B. Saunders, Philadelphia, 1966.