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Analysis of three-fluid separate type heat pipe exchanger
Heat Recovery Systems & CHP Vol. 12, No. 4, pp. 317-322, 1992 Printed in Great Britain
ANALYSIS
0890-4332/92 $5.00 + .00 Pergamon Press Ltd
OF THREE-FLUID SEPARATE PIPE EXCHANGER
TYPE
HEAT
SHI CHENGMING, S I N MINGDAO a n d CrmN YUANGUO Institute of Engineering Thermophysics Chongqing University, Chongqing, Sichuan 630044, China (Received 20 November 1991)
A ~ e t - - T h i s paper puts forward an analytical model of heat transfer of a three-fluid separate type heat pipe exchanger. A universal basic temperature transfer matrix equation is obtained from the analysis of heat transfer. The equation is suitable for predicting performance of the parallel- or counter-flow type exchanger in that heat is transferred between one hot fluid and two cold fluids or between two hot fluids and one cold fluid. Finally, a calculated engineering example is taken, and the proposed model and result is proved to be reasonable.
NOMENCLATURE A B
C C
m
Q R
T x
Subscr~ 1,2,3 i e
b O n
p v
temperature transfer matrix in parallel flow temperature transfer matrix in counter flow integrating constant specific heat (J/kg, K) flow rate (kg/s) heat rate (W) thermal resistance (K/W) temperature (°C) relative position, dimensionless three fluids, respectively at x = O atx=l inlet outlet nth fluid, nth iteration mean steam
INTRODUCTION
The three-fluid separate type heat pipe exchanger, a new member of the heat exchanger family, is a new type of heat exchanger developed in the 1980s. Besides possessing advantages of the waste heat recovery package of a conventional heat pipe exchanger and thermal medium type exchanger, it also has the following advantages [1-3]: 1. It can be applicable to the large-scale package. 2. It can keep apart cold and hot fluids to prevent them from leaking into each other. 3. It can realize heat exchange within a long distance. 4. Its heat transfer surface areas are adjustable. 5. It is convenient to realize the mixing arrangement of the parallel- and counter-flow modes of fluids. Due to its advantages mentioned above this kind of heat exchanger has found wide application prospect in the waste heat recovery of low or middle temperature fields. Since 1985, five three-fluid separate type heat pipe exchangers have been applied in engineering in China. One of them is used for hot-blast stove in a blast furnace at Shanghai BouShan steel works in order to preheat the air and the coal gas. The quantity of heat to be transferred is about 20,000 kW. All of these five heat exchangers have realized heat transfer from one hot fluid to two cold fluids. Some analyses have been performed on three-fluid plate-fin and shell-tube type heat exchangers [4-7]. Since these heat exchangers are different from the three-fluid separate heat pipe exchanger 317
318
SHI CHENGMINGet al.
J
[ I I I IIII
I
conde~
"
I I .Ill
'2 1 I lllllll
TI
'
denser
j tl.*eom
l
'
'l
I I
I I
evaporter~
e~roter condensate
la) one hot. fluid via t w o cold fluids
(b) two cold fluids via one hot. fluid
Fig. 1. Heat exchanger unit.
in construction and principle of heat transfer, the analytical models and results of the former are not applicable to the problem discussed in this paper. Unfortunately, so far, a reliable and practical analysis applicable to modelling and designing the three-fluid separate type heat pipe exchanger has been lacking. The purpose of this paper is to provide an analysis and method of calculation which could be used in practice for prediction of performance and for the design of such a heat exchanger. C O N S T R U C T I O N AND HEAT T R A N S F E R F E A T U R E A schematic of the three-fluid separate type heat pipe exchanger unit is shown in Fig. 1. Figure l(a) expresses that heat is transferred between one hot fluid and two cold fluids. Figure l(b) expresses that heat is transferred between two hot fluids and one cold fluid. A unit of this heat exchanger is composed of separated evaporation and condensation sections. Each section has two headers, the upper and the lower. Some finned tubes are welded between them. The upper headers are connected through steam tubes, the lower ones through condensate tubes. The heat exchanger package consists of a certain number of units. Figure 2 shows the principle of heat transfer of this kind of heat exchanger. As shown in Fig. l(a), a quantity of steam (or quantity of heat) flowing separately from the evaporation section of a unit into the condensers is determined according to the coupling design of three sections and not by adjustment of the regulating valves. Thus, there are two difficulties to be solved to analyse heat transfer of this kind of heat exchanger. (a) The basic heat transfer surface areas for each fluid are different. (b) It is unknown how much steam from the evaporation section flows into one of two condensers. That is, one cannot determine the proportion of area of the evaporation section attributed to each condenser in advance.
_r]__ T2i.j" -I _ -LJ
fluid 2 _ ~ . ° . . o
fluid 3
.....
-.[
OQQOQ
1st unit jth unit. terminal unit Fig. 2. Schematicdiagram of the heat exchanger.
Analysis of three-fluid separate type heat pipe exchanger
319
HEAT TRANSFER ANALYSIS In order to analyse heat transfer of the three-fluid separate type heat pipe exchanger, one must solve the two difficulties mentioned above. To do this, the thermal circuit as shown in Fig. 3 is assumed and the heat pipe exchanger is modelled as a shell-tube heat exchanger. The analytical model of heat transfer is shown in Fig. 4, in which x expresses the relative position of full flow pass of the heat exchanger and is between 0 and 1. No heat transfer is assumed between fluid 1 and fluid 2, that is, Q~2 = 0. Heat transfer from each fluid to the working medium of the heat pipe is considered to be one heat transfer process, whose heat resistances are R~, R2 and R3, respectively. R31 in R3 and Rs constitute a heat transfer branch, and R32 in R3 and R2 forms the other branch, as shown in Fig. 3. Then, we can obtain differential equations as follows:
dTl
d'-'x-+ k]3(Tl -- T3) = 0
dT2
d'--x"q" k23(T2 - T3)
dT3
0
=
+ k3l ( T 3 - T I ) -q- k32(T3 - T2) = 0
(0~
T,(0)=T,i,
T 2 ( 0 ) = T2i,
T3(0)=T3i
atx=l,
T,(1)=T~e,
T2(1)--- T2e, T3(1)=T3~.
Heat transfer between the exchanger and the surroundings is neglected. From the heat balance, yield m, cl dTl + m2¢2 dT2 + m 3 c 3 dT3 -- 0.
(2)
Assuming m c to be constant, the following equation can be obtained by integrating equation (2) and applying the boundary condition of equation (1), ml Cl (Tli - - T1 ) + m 2 c 2 ( T 2 i - T2) + m 3 c 3 ( T 3 i - T3) = 0.
(3)
Differentiating equation (1) and using equation (3), we get dT~ d2T~dx 2 )- (kl3 + k23 + k31 + k32) ~ + YT~ -- Z = 0
dT2
d2T2dx 2 I- (k~3+ k23 + k3~ + k32) ~ d2T3dx 2
b
r. . . . .
RI
T3p I!
dT3
(kl3 + k23 + k31 + k32) ~
05.R ~
'-H
0
:;
L. . . .
.J
IT. "
+ YT3 - Z = 0
T+i
------
T21
',
I R2
Rs
Fig. 3. Thermal circuit. HRS 12/4--C
Tip
+ YT2 - Z = 0
T2
(4)
~
[ Tz)
0
I
lit
x Fig. 4. Heat transfer analysis model.
G
320
SHI CHENGMING et al.
where Y = k13k23 + k13k32 q- k23k31 Z = k31 k23 Tli -I- kl3k32 T2i + k13k23 T3i.
Equation (4) is a second order linear, n o n - h o m o g e n e o u s differential equation with constant coefficients. A general solution is given by Tn = Cn, exp(al x) + Cn2 exp(a2x) + To
(5)
where subscript n = 1, 2 and 3 which explain three fluids, respectively, ro - z / Y
al = [ - k + (k 2 - 4Y)t/2]/2 a2 = [ - k - (k 2 - 4Y)J/2]/2 k = k13 -I- k23 -t- k3! -1- k32.
Applying b o u n d a r y conditions, at x -- 0, T. = T.i; at x -- 1, T. -- T.~; integrated constants C., and Cn2 can be obtained as follows: C.l = [Tn~ - To - (Tni - To) exp(a2)]/[exp(al) -- exp(a2)]
(6)
Cn2 = [T~ - To - (Tni -- To) exp(al )]/[exp(a2) - exp(al )].
(7)
Equations (5), (6) and (7) are the general analytical solution for the temperature distribution in the three-fluid separate type heat pipe exchanger. Differentiating equation (5), then from equation (1) and b o u n d a r y conditions, yield C!! =
kt3[(T ~ -- Tie ) - (T3i - Tli) exp(a2)] a~ [exp(al)
C!2 =
-
exp(a2)]
k!3 [(T3, - Tl,) -- (Tai - T!i) exp(al )] - a2 [exp(a! ) - exp(a2)]
(8)
(9)
C2! = k23 [(T3e - T2~) - (T3i - T2i) exp(a2)] a! [exp(a! ) - exp(a2)]
(10)
C22 = k23 [(T3~ - T2~) - (T3i - T2i) exp(at )] - a2 [exp(al ) - exp(a2)]
(1 1)
C3! = k3! [Tie -- T3~ - ( T . - T3i) exp(a2)] + k32 [T2c - T3~ - (T~ - T3i) exp(a2)] a! [exp(a, ) - exp(a2)]
(12)
C32 = k3! [T!~ - T3~ - (T!i - T3i) exp(at )] + k32 [T2e - T3e - (T2i - T3i) exp(al )] - a2 [exp(al ) - exp(a2)]
(1 3)
Equations
(6),
(7) and
(8)-(13)
can be solved to give the matrix equation as follows:
Te= ATi
(14)
where: T, = [T!~, T2e, T~]', T, is outlet temperatures o f fluids; T~ = [Tti, T~a, T3i]', T~ is inlet temperatures o f fluids, A m a y be called a 'temperature transfer matrix'. Elements o f the matrix A are a!! = (k23k31e! + e2Y)/eo al2 = kt3k32el/eo a13 -- {k!3k23e! -I- k13Y[exp(al) - exp(a2)]}/eo a21 = k23k31 el/Co
Analysis of three-fluid separate type heat pipe exchanger
321
a22 = (kt3k32et + e3Y)/eo a23 = {kt3k23et + k23Y[exp(al ) -- exp(a2)]}/eo a3t = {k23k3t el + Y[exp(al ) - exp(a2)]k31 }/Co a32 = {kt3k32et + Yk32[exp(al) -- exp(a2)]}/eo a33 = (kl3k23e j + Ye4)/eo
eo = Y(al - a2) e, = al [1 - exp(a2)] - a2[1 - exp(al)] e2 = (at + k13) exp(a2) - (a2 + kt3 ) exp(al)
e3 = (al + k23) exp(a2) - (a2 + k23) exp(al ) e4 = (at + k32 + k3t ) exp(a2) - (as + k32 + k3l ) exp(al ).
Now, the temperature transfer matrix equation has been derived for a three-fluid separate type heat pipe exchanger in the condition o f parallel flow. In the above analysis, we do not appoint which fluid must be the hot one, or the cold one. Thus equation (14) is suitable for heat transfer between one hot fluid and two cold fluids or between a cold fluid and two hot fluids. F o r counter flow, such as between fluid 3 and fluids 1 and 2, only a proper transformation o f equation (14) is needed and the heat capacity rate o f fluid flowing along the smaller x direction is made negative. Then: To = BTb
(15)
where To = [Tie, T~e, T3i]', To is the outlet temperatures of three fluids; Tb = [T,i, T2i, T3o]', Tb is the inlet temperatures o f three fluids. The elements of matrix B are: bit = all -- a13a31/a33
bl2 = at2 -- a~3a32/a33 bt3 = a13/a33 b2, = a2t - a23a3~/a33 b22 = a22 - a23 a32/a33 b23 = a23/a33 b3j = - a3,/a33 b32 = - a32/a33 b33 = 1/a33.
It is worth pointing out that R3t and R32 a r e considered known in the above derivation. In fact, they are u n k n o w n in advance and must be determined by iteration at Q~2 = 0 and using the heat transfer feature. The iterating equations are
Q32,n/Q31,n)
(16)
R32, n + I --- R3/(1 - R 3/R31,n + 1)
(1 7)
R31,n+l = R3(1 +
where Q3t,o and Q32, n a r e the values obtained from the former calculation. Then each element o f A or B can be computed, and the outlet temperature in the condition o f parallel or counter flow can be determined from equation (14) or (15). CALCULATING
EXAMPLE
Here is an example showing the application o f the proposed analysis result. The calculated problem is from waste heat recovery engineering o f a steel plant. The involved fluids are air(fluid 1), coal gas(fluid 2), flue gas(fluid 3). The given parameters are listed in Table 1.
SHI CHENGMING et al.
322
Parameters Flows Inlet temp. O.D. of tubes I.D. of tubes
Fin thickness Fin height Fin pitch
Table 1. Given parameters Unit Air Coal gas Nm3/h °C mm mm mm mm mm
188812 20 38 31 1.2 17 5
Flue gas 2 4 9 8 0 8 403702 30 250 38 48 31 41 1.2 1.2 17 21 5 5.5
Table 2. The calculated results Parameters Unit Air Coal gas Thermalresist. °C/kW 0.006625 0.004750 Outlet temp.
Heat rate
°C 135.0 kW 8600.0
138.0 9780.0
Flue gas 0.0028625 134.0 18400.0
First, assume the outlet temperatures of fluids 1, 2 and 3, respectively, calculate the values of heat capacity rate of fluids and the Q31,I and Q32.~ of two heat transfer branches at the mean temperature, and choose the basic structure parameters of the heat exchanger as shown in Table 1. Then, calculate the thermal resistance R~, R2 and R3. Next, find out R3~.1 and R32, t from equations (1 6) and (1 7). And then, calculate the temperature transfer matrix B from equation (1 5) to obtain outlet temperature To. With the calculated To being used as the initial value of the next calculation, redetermine R31.n+~ and R32.n+l from equations (16) and (17) by repeating the above steps. Continue iterating until the given accuracy is satisfied. Table 2 lists the calculated results. CONCLUSIONS The exact explicit formulae for the temperature fields and outlet temperatures for three-fluid separate type heat pipe exchanger are proposed. These can be used to predict the performance of the exchanger. A calculated engineering example proves the proposed method satisfactory. REFERENCES 1. M. C. Jin and Y. G. Chen, Heat Pipe and Heat Pipe Exchanger. Chongqing University Press (1986). 2. I. K o h t a k a et al., Development, design and operation of large scale separate type heat pipes, Proc. of Sixth Int. Heat Pipe Conf., France, Vol. 3, p. 624 (1987). 3. C. M. Shi and M. D. Xin, Design and correction method of double-preheating separate type heat pipe exchanger, Proc. of 2nd National Heat Pipe Conf., p. 181 (1988). 4. N. C. Willis Jr, Analysis of Three-Fluid, Crossflow Heat Exchanger. NASA, TR. R-284. 5. B. S. Baclic et al., Performances of three-fluid single pass crossflow heat exchanger, heat transfer-1982. Hemisphere 6, 167, (1982). 6. J. C. Chato et al., Analyses of parallel flow, multi-stream heat exchanger, Int. J. Heat Mass Transfer 14, 1691 (1971). 7. D. D. Aulds and R. F. Barron, Three-fluid heat exchanger effectiveness, Int. J. Heat Mass Transfer 10, 1457 (1967).
ANALYSIS
0890-4332/92 $5.00 + .00 Pergamon Press Ltd
OF THREE-FLUID SEPARATE PIPE EXCHANGER
TYPE
HEAT
SHI CHENGMING, S I N MINGDAO a n d CrmN YUANGUO Institute of Engineering Thermophysics Chongqing University, Chongqing, Sichuan 630044, China (Received 20 November 1991)
A ~ e t - - T h i s paper puts forward an analytical model of heat transfer of a three-fluid separate type heat pipe exchanger. A universal basic temperature transfer matrix equation is obtained from the analysis of heat transfer. The equation is suitable for predicting performance of the parallel- or counter-flow type exchanger in that heat is transferred between one hot fluid and two cold fluids or between two hot fluids and one cold fluid. Finally, a calculated engineering example is taken, and the proposed model and result is proved to be reasonable.
NOMENCLATURE A B
C C
m
Q R
T x
Subscr~ 1,2,3 i e
b O n
p v
temperature transfer matrix in parallel flow temperature transfer matrix in counter flow integrating constant specific heat (J/kg, K) flow rate (kg/s) heat rate (W) thermal resistance (K/W) temperature (°C) relative position, dimensionless three fluids, respectively at x = O atx=l inlet outlet nth fluid, nth iteration mean steam
INTRODUCTION
The three-fluid separate type heat pipe exchanger, a new member of the heat exchanger family, is a new type of heat exchanger developed in the 1980s. Besides possessing advantages of the waste heat recovery package of a conventional heat pipe exchanger and thermal medium type exchanger, it also has the following advantages [1-3]: 1. It can be applicable to the large-scale package. 2. It can keep apart cold and hot fluids to prevent them from leaking into each other. 3. It can realize heat exchange within a long distance. 4. Its heat transfer surface areas are adjustable. 5. It is convenient to realize the mixing arrangement of the parallel- and counter-flow modes of fluids. Due to its advantages mentioned above this kind of heat exchanger has found wide application prospect in the waste heat recovery of low or middle temperature fields. Since 1985, five three-fluid separate type heat pipe exchangers have been applied in engineering in China. One of them is used for hot-blast stove in a blast furnace at Shanghai BouShan steel works in order to preheat the air and the coal gas. The quantity of heat to be transferred is about 20,000 kW. All of these five heat exchangers have realized heat transfer from one hot fluid to two cold fluids. Some analyses have been performed on three-fluid plate-fin and shell-tube type heat exchangers [4-7]. Since these heat exchangers are different from the three-fluid separate heat pipe exchanger 317
318
SHI CHENGMINGet al.
J
[ I I I IIII
I
conde~
"
I I .Ill
'2 1 I lllllll
TI
'
denser
j tl.*eom
l
'
'l
I I
I I
evaporter~
e~roter condensate
la) one hot. fluid via t w o cold fluids
(b) two cold fluids via one hot. fluid
Fig. 1. Heat exchanger unit.
in construction and principle of heat transfer, the analytical models and results of the former are not applicable to the problem discussed in this paper. Unfortunately, so far, a reliable and practical analysis applicable to modelling and designing the three-fluid separate type heat pipe exchanger has been lacking. The purpose of this paper is to provide an analysis and method of calculation which could be used in practice for prediction of performance and for the design of such a heat exchanger. C O N S T R U C T I O N AND HEAT T R A N S F E R F E A T U R E A schematic of the three-fluid separate type heat pipe exchanger unit is shown in Fig. 1. Figure l(a) expresses that heat is transferred between one hot fluid and two cold fluids. Figure l(b) expresses that heat is transferred between two hot fluids and one cold fluid. A unit of this heat exchanger is composed of separated evaporation and condensation sections. Each section has two headers, the upper and the lower. Some finned tubes are welded between them. The upper headers are connected through steam tubes, the lower ones through condensate tubes. The heat exchanger package consists of a certain number of units. Figure 2 shows the principle of heat transfer of this kind of heat exchanger. As shown in Fig. l(a), a quantity of steam (or quantity of heat) flowing separately from the evaporation section of a unit into the condensers is determined according to the coupling design of three sections and not by adjustment of the regulating valves. Thus, there are two difficulties to be solved to analyse heat transfer of this kind of heat exchanger. (a) The basic heat transfer surface areas for each fluid are different. (b) It is unknown how much steam from the evaporation section flows into one of two condensers. That is, one cannot determine the proportion of area of the evaporation section attributed to each condenser in advance.
_r]__ T2i.j" -I _ -LJ
fluid 2 _ ~ . ° . . o
fluid 3
.....
-.[
OQQOQ
1st unit jth unit. terminal unit Fig. 2. Schematicdiagram of the heat exchanger.
Analysis of three-fluid separate type heat pipe exchanger
319
HEAT TRANSFER ANALYSIS In order to analyse heat transfer of the three-fluid separate type heat pipe exchanger, one must solve the two difficulties mentioned above. To do this, the thermal circuit as shown in Fig. 3 is assumed and the heat pipe exchanger is modelled as a shell-tube heat exchanger. The analytical model of heat transfer is shown in Fig. 4, in which x expresses the relative position of full flow pass of the heat exchanger and is between 0 and 1. No heat transfer is assumed between fluid 1 and fluid 2, that is, Q~2 = 0. Heat transfer from each fluid to the working medium of the heat pipe is considered to be one heat transfer process, whose heat resistances are R~, R2 and R3, respectively. R31 in R3 and Rs constitute a heat transfer branch, and R32 in R3 and R2 forms the other branch, as shown in Fig. 3. Then, we can obtain differential equations as follows:
dTl
d'-'x-+ k]3(Tl -- T3) = 0
dT2
d'--x"q" k23(T2 - T3)
dT3
0
=
+ k3l ( T 3 - T I ) -q- k32(T3 - T2) = 0
(0~
T,(0)=T,i,
T 2 ( 0 ) = T2i,
T3(0)=T3i
atx=l,
T,(1)=T~e,
T2(1)--- T2e, T3(1)=T3~.
Heat transfer between the exchanger and the surroundings is neglected. From the heat balance, yield m, cl dTl + m2¢2 dT2 + m 3 c 3 dT3 -- 0.
(2)
Assuming m c to be constant, the following equation can be obtained by integrating equation (2) and applying the boundary condition of equation (1), ml Cl (Tli - - T1 ) + m 2 c 2 ( T 2 i - T2) + m 3 c 3 ( T 3 i - T3) = 0.
(3)
Differentiating equation (1) and using equation (3), we get dT~ d2T~dx 2 )- (kl3 + k23 + k31 + k32) ~ + YT~ -- Z = 0
dT2
d2T2dx 2 I- (k~3+ k23 + k3~ + k32) ~ d2T3dx 2
b
r. . . . .
RI
T3p I!
dT3
(kl3 + k23 + k31 + k32) ~
05.R ~
'-H
0
:;
L. . . .
.J
IT. "
+ YT3 - Z = 0
T+i
------
T21
',
I R2
Rs
Fig. 3. Thermal circuit. HRS 12/4--C
Tip
+ YT2 - Z = 0
T2
(4)
~
[ Tz)
0
I
lit
x Fig. 4. Heat transfer analysis model.
G
320
SHI CHENGMING et al.
where Y = k13k23 + k13k32 q- k23k31 Z = k31 k23 Tli -I- kl3k32 T2i + k13k23 T3i.
Equation (4) is a second order linear, n o n - h o m o g e n e o u s differential equation with constant coefficients. A general solution is given by Tn = Cn, exp(al x) + Cn2 exp(a2x) + To
(5)
where subscript n = 1, 2 and 3 which explain three fluids, respectively, ro - z / Y
al = [ - k + (k 2 - 4Y)t/2]/2 a2 = [ - k - (k 2 - 4Y)J/2]/2 k = k13 -I- k23 -t- k3! -1- k32.
Applying b o u n d a r y conditions, at x -- 0, T. = T.i; at x -- 1, T. -- T.~; integrated constants C., and Cn2 can be obtained as follows: C.l = [Tn~ - To - (Tni - To) exp(a2)]/[exp(al) -- exp(a2)]
(6)
Cn2 = [T~ - To - (Tni -- To) exp(al )]/[exp(a2) - exp(al )].
(7)
Equations (5), (6) and (7) are the general analytical solution for the temperature distribution in the three-fluid separate type heat pipe exchanger. Differentiating equation (5), then from equation (1) and b o u n d a r y conditions, yield C!! =
kt3[(T ~ -- Tie ) - (T3i - Tli) exp(a2)] a~ [exp(al)
C!2 =
-
exp(a2)]
k!3 [(T3, - Tl,) -- (Tai - T!i) exp(al )] - a2 [exp(a! ) - exp(a2)]
(8)
(9)
C2! = k23 [(T3e - T2~) - (T3i - T2i) exp(a2)] a! [exp(a! ) - exp(a2)]
(10)
C22 = k23 [(T3~ - T2~) - (T3i - T2i) exp(at )] - a2 [exp(al ) - exp(a2)]
(1 1)
C3! = k3! [Tie -- T3~ - ( T . - T3i) exp(a2)] + k32 [T2c - T3~ - (T~ - T3i) exp(a2)] a! [exp(a, ) - exp(a2)]
(12)
C32 = k3! [T!~ - T3~ - (T!i - T3i) exp(at )] + k32 [T2e - T3e - (T2i - T3i) exp(al )] - a2 [exp(al ) - exp(a2)]
(1 3)
Equations
(6),
(7) and
(8)-(13)
can be solved to give the matrix equation as follows:
Te= ATi
(14)
where: T, = [T!~, T2e, T~]', T, is outlet temperatures o f fluids; T~ = [Tti, T~a, T3i]', T~ is inlet temperatures o f fluids, A m a y be called a 'temperature transfer matrix'. Elements o f the matrix A are a!! = (k23k31e! + e2Y)/eo al2 = kt3k32el/eo a13 -- {k!3k23e! -I- k13Y[exp(al) - exp(a2)]}/eo a21 = k23k31 el/Co
Analysis of three-fluid separate type heat pipe exchanger
321
a22 = (kt3k32et + e3Y)/eo a23 = {kt3k23et + k23Y[exp(al ) -- exp(a2)]}/eo a3t = {k23k3t el + Y[exp(al ) - exp(a2)]k31 }/Co a32 = {kt3k32et + Yk32[exp(al) -- exp(a2)]}/eo a33 = (kl3k23e j + Ye4)/eo
eo = Y(al - a2) e, = al [1 - exp(a2)] - a2[1 - exp(al)] e2 = (at + k13) exp(a2) - (a2 + kt3 ) exp(al)
e3 = (al + k23) exp(a2) - (a2 + k23) exp(al ) e4 = (at + k32 + k3t ) exp(a2) - (as + k32 + k3l ) exp(al ).
Now, the temperature transfer matrix equation has been derived for a three-fluid separate type heat pipe exchanger in the condition o f parallel flow. In the above analysis, we do not appoint which fluid must be the hot one, or the cold one. Thus equation (14) is suitable for heat transfer between one hot fluid and two cold fluids or between a cold fluid and two hot fluids. F o r counter flow, such as between fluid 3 and fluids 1 and 2, only a proper transformation o f equation (14) is needed and the heat capacity rate o f fluid flowing along the smaller x direction is made negative. Then: To = BTb
(15)
where To = [Tie, T~e, T3i]', To is the outlet temperatures of three fluids; Tb = [T,i, T2i, T3o]', Tb is the inlet temperatures o f three fluids. The elements of matrix B are: bit = all -- a13a31/a33
bl2 = at2 -- a~3a32/a33 bt3 = a13/a33 b2, = a2t - a23a3~/a33 b22 = a22 - a23 a32/a33 b23 = a23/a33 b3j = - a3,/a33 b32 = - a32/a33 b33 = 1/a33.
It is worth pointing out that R3t and R32 a r e considered known in the above derivation. In fact, they are u n k n o w n in advance and must be determined by iteration at Q~2 = 0 and using the heat transfer feature. The iterating equations are
Q32,n/Q31,n)
(16)
R32, n + I --- R3/(1 - R 3/R31,n + 1)
(1 7)
R31,n+l = R3(1 +
where Q3t,o and Q32, n a r e the values obtained from the former calculation. Then each element o f A or B can be computed, and the outlet temperature in the condition o f parallel or counter flow can be determined from equation (14) or (15). CALCULATING
EXAMPLE
Here is an example showing the application o f the proposed analysis result. The calculated problem is from waste heat recovery engineering o f a steel plant. The involved fluids are air(fluid 1), coal gas(fluid 2), flue gas(fluid 3). The given parameters are listed in Table 1.
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Parameters Flows Inlet temp. O.D. of tubes I.D. of tubes
Fin thickness Fin height Fin pitch
Table 1. Given parameters Unit Air Coal gas Nm3/h °C mm mm mm mm mm
188812 20 38 31 1.2 17 5
Flue gas 2 4 9 8 0 8 403702 30 250 38 48 31 41 1.2 1.2 17 21 5 5.5
Table 2. The calculated results Parameters Unit Air Coal gas Thermalresist. °C/kW 0.006625 0.004750 Outlet temp.
Heat rate
°C 135.0 kW 8600.0
138.0 9780.0
Flue gas 0.0028625 134.0 18400.0
First, assume the outlet temperatures of fluids 1, 2 and 3, respectively, calculate the values of heat capacity rate of fluids and the Q31,I and Q32.~ of two heat transfer branches at the mean temperature, and choose the basic structure parameters of the heat exchanger as shown in Table 1. Then, calculate the thermal resistance R~, R2 and R3. Next, find out R3~.1 and R32, t from equations (1 6) and (1 7). And then, calculate the temperature transfer matrix B from equation (1 5) to obtain outlet temperature To. With the calculated To being used as the initial value of the next calculation, redetermine R31.n+~ and R32.n+l from equations (16) and (17) by repeating the above steps. Continue iterating until the given accuracy is satisfied. Table 2 lists the calculated results. CONCLUSIONS The exact explicit formulae for the temperature fields and outlet temperatures for three-fluid separate type heat pipe exchanger are proposed. These can be used to predict the performance of the exchanger. A calculated engineering example proves the proposed method satisfactory. REFERENCES 1. M. C. Jin and Y. G. Chen, Heat Pipe and Heat Pipe Exchanger. Chongqing University Press (1986). 2. I. K o h t a k a et al., Development, design and operation of large scale separate type heat pipes, Proc. of Sixth Int. Heat Pipe Conf., France, Vol. 3, p. 624 (1987). 3. C. M. Shi and M. D. Xin, Design and correction method of double-preheating separate type heat pipe exchanger, Proc. of 2nd National Heat Pipe Conf., p. 181 (1988). 4. N. C. Willis Jr, Analysis of Three-Fluid, Crossflow Heat Exchanger. NASA, TR. R-284. 5. B. S. Baclic et al., Performances of three-fluid single pass crossflow heat exchanger, heat transfer-1982. Hemisphere 6, 167, (1982). 6. J. C. Chato et al., Analyses of parallel flow, multi-stream heat exchanger, Int. J. Heat Mass Transfer 14, 1691 (1971). 7. D. D. Aulds and R. F. Barron, Three-fluid heat exchanger effectiveness, Int. J. Heat Mass Transfer 10, 1457 (1967).