2013 台灣化學工程學會 60 週年年會論文
Heat Transfer Enhancement of Power-Law Fluids in a Parallel-Plate Channel for
Improved Device Performance under Asymmetric Wall Temperatures
(何啟東) Chii-Dong Ho*, (林國賡) Gwo-Geng Lin, (林承毅) Cheng-Yi Lin
and (林立邦) Li-Pang Lin
Energy and Opto-Electronic Materials Research Center, Department of Chemical and Materials
Engineering, Tamkang University, Tamsui, New Taipei, Taiwan 251
(淡江大學化學工程與材料工程學系、能源與光電材料研究中心)
Abstract —The heat transfer of a Graetz problem to a double-pass parallel-plate heat exchanger under
uniform wall temperatures to enhance the device performance improvement is investigated
theoretically. The theoretical mathematical model is solved analytically using the separation of
variables, superposition principle and an orthogonal expansion technique in extended power series.
The theoretical results show that the power-law fluids results in the significant heat-transfer efficiency
improvement as compared with those in an open conduit (without an impermeable resistless sheet
inserted), especially when the double-pass device was operated in large Graetz number. The results
show that the good agreement between the experimental results and theoretical prediction. The
effects of impermeable-sheet position and power consumption increment for power-law fluids have
also been presented.
Keywords: Power-law fluids; Uniform wall temperatures; Double-pass operations; Nusselt number;
Performance improvement.
*
Corresponding author. Tel: 886-26215656 ext. 2724 Fax: 886-2-26209887;
E-mail: [email protected]
1. Introduction
The study of the thermal response of the
conduit wall and fluid temperature distributions
to the Newtonian fluid in laminar heat transfer
problems with negligible axial conduction is
known as the Graetz problem [1,2]. Other
engineering
applications
were
devoted
to
with thickness W and (1-W ,with temperature
T1 and T2, as shown in Figure 1.
(Constant wall temperature)
δ
the
non-Newtonian materials with the power-law
constants in the appropriate shear rate range due
to their high viscosity levels. The purposes of the
present study are to extend the heat transfer problem
in a double-pass with recycle design parallel-plate
heat exchanger to asymmetrical wall boundary
conditions [3] and to non-Newtonian fluids [4] in
aiming to solve analytically by means of
orthogonal expansion techniques [5].
2. Theoretical treatments
The design of parallel-plate heat exchanger
with height W, length L, and width B (B>>W), the
system of open duct be inserted an impermeable
sheet, which thickness can be neglected, to divide
two channels, subchannel a and subchannel b,
(Constant wall temperature)
Feed flow
Heat transfer by Heat transfer by Heat flux through
convection
conduction
plate
Fig. 1 Double-pass heat exchangers
The following assumptions were made to
simplify the system analysis: fully-developed
laminar flow with power law index in each
subchannel, constant physical properties of fluid,
neglecting end effects, impermeable sheet
thermal resistance and axial heat conduction, and
well-mixing at the inlet and outlet of fluid. The
2013 台灣化學工程學會 60 週年年會論文
dimensionless velocity distributions may be
written as
1
1 
V 1  2 
 

(1)
va 
1

2


1
a




Wa B  1   

1
1 
 V 1  2 
 

(2)
vb 
1

2


1
b




Wb B  1   

In this study, using the power series to expansion
the third term of velocity profile are
1
1 

1  2  1  


 X 1  X 2 2  X 3 3  X 4 4  X 5 5
(3)
The equations of energy in dimensionless form
for double-pass heat exchangers with uniform
wall fluxes may be obtained as:
 2 a ( a ,  ) Wa2 va ( a )   a ( a ,  )
(4)


 a2

 GzL 
 2 b (b ,  ) Wb2 vb (b )   b (b ,  )
(5)


b2

 GzL 
The definition of dimensionless groups in the Eqs.
(1) to (5) are
z
VW
, Gz 
, Wa  W , Wb  (1  )W ,

LGz
BL
x
Wa

T  Ti

, a  a
,  b  Tb  Ti ,  a  a ,
Wb 1  
Wa
T1  Ti
T1  Ti
x
T T
b  b ,   2 i
(6)
T1  Ti
Wb
The corresponding boundary conditions for
solving Eq. (4) and Eq. (5) are
 a 0,    1
(7)
 b (0,  )  
 a (1,  )   b (1,  )
 a (1,  )
  b (1,  )

 a
1    b
(8)
(9)
solutions of a ( a ,  ) and b ( b ,  ) as follows
a  a ,   

Fa ,m  a Gm  
(15)
 b  b ,     S b,m Fb,m  b Gm  
(16)
S
m0

a ,m
m 0
Substituting Eqs. (15) and (16) into the governing
equations of aa,and bb, gives
Gm ( )  e
m (
1
 )
Gz
1
1 
 1  2 
Fa,m ( a )  m Δ
1  2 a  1   Fa ,m ( a )  0
 1   

(17)
1 
 1  2 
Fb,m ( b )  m (1  Δ)
1  2 b  1   Fb,m ( b )  0
 1   

1
(18)
In which the eigenfunctions Fa,m(a) and Fb,m(b)
were assumed to be polynomials to avoid the loss
of generality.

Fa.m     d m,n a , d m,0  1 , d m,1  0
(19)
Fb,m     em,n b , em,0  1 , em,1  0
(20)
n 0

n
n
n 0
Combination of Eq. (9) and Eq. (10) results in

(21)
S a ,m Fa,m (1)  
S b,m Fb,m (1)
1 
All the coefficients d m,n and em,n for
Fa,m ( a ) and Fb,m (b ) , can be obtained in
terms of eigenvalue  m by substituting Eq. (19)
and Eq. (20) into Eq. (17) and Eq. (18) Also, the
constants of S a, m and S b,m in Eq. (15) and Eq.
(16) can be calculated from orthogonality
(10) conditions when n  m :
1
W 2 v ( ) 
The complete solutions of Eq. (4) and Eq. (5) Wb   a a a  S a,m S a,n Fa,m Fa,n d a
LGz 
0
can be separated into the linear superposition of
1
an asymptotic solution,  ( ,  ) , and a
W 2 v ( ) 
 Wa   b b b  S b,m S b,n Fb,m Fb,n d b
(22)
homogeneous solution,  ( ,  ) , as follows:
LGz 
0
 a  a ,     a  a ,    a  a ,  
(11) The overall energy balance of whole system
1
 b  b ,     b  b ,    b  b ,  
(12)
BL  a (0,  )
V (0   F )   Gz Gz
d
0
The asymptotic solutions are
Wa
 a
 a  a   A1 a  A2
(13)
1
BL  b (0,  )
Gz Gz

d
(23)
θb ηb   B1ηb  B2
(14)
0
Wb
 b
Using the separation of variables to express the
The average dimensionless outlet temperature
2013 台灣化學工程學會 60 週年年會論文
 F may be obtained at the end of subchannel b
by

m
Ih 
Nu  Nu0
x 100%
Nu0
(28)
I P due to the friction losses ( w f ,a and w f ,b
 F  
F 1  F 0
for double pass while w f ,0 for single pass) in
m  0 1    m
the conduits
1 1  2
 (
)( X 1 b1  X 2 b2  X 3 b3  X 4 b4  X 5 b5 ) 
(w f ,a  w f ,b )  (w f ,0 )
0 1 
Ip 
w f ,0
((  1)(  1) b   )d b
(24)
 1

1
 1
(29)
The average dimensionless temperatures at the   2 1 
2 1 
Δ
(1  Δ)


end of the subchannels a and b are
3. Experimental system
 S
 aL    a ,m Fa',m 1  Fa',m 0
Polymer solutions with two different
m  0  m
concentrations were prepared using as the
1 1  2
1
2
3
4
5
 (
)( X 1 a  X 2 a  X 3 a  X 4 a  X 5 a )  working fluid in the heat transfer module for
0 1 
experimentation to verify the theoretical analysis
of this work. The first one is the 1000ppm
(25)
((  1) a  1)d a
aqueous poly-acrylic acid (Carbopol 934 from

S b,m
Lubrizol) solution and the other was of 2000ppm.
 bL   
Fb',m 1  Fb',m 0
m 0 1    m
All the experiments conducted in this work were
1 1  2
performed using the heat transfer equipment
 (
)( X 1 b1  X 2 b2  X 3 b3  X 4 b4  X 5 b5 ) 
0 1 
depicted in Fig. 1. Both the upper and lower
aluminum plates were heated indirectly by water
(26)
((  1)(  1) b   )d b
The average Nusselt number for double-pass at different temperature under thermostat to keep
parallel-plate heat exchangers under uniform wall constant wall temperature, and the fabrication of
the double-pass heat exchanger module was
fluxes may be calculated as:
demonstrated in Fig. 2. The whole experimental
hW
VW  F GZ  F
consisted
of
the
pumps,
Nu 


(27) apparatus
k
2BL  w
2 w
temperature-controlled fluid storage tanks, heat
The improvement of the heat-transfer exchanger module and measuring units for the
efficiency, I h , based on the heat transfer of a liquid flow rate and fluid temperature.
single-pass device

e
Gz
 S b,m

'
b,m
'
b,m





C
(A) Thermostat
B
E
A
(B) Pump
D
D
(C) Flow meter
A
B
(D) Temperature indicator
B
(E) Heat exchanger module
A
Fig. 2 Experimental setup
4. Results and conclusions
The mathematical modeling equations for
counterflow double-pass heat-exchanger equation
of power-law fluids with uniform wall
temperatures have been formulated and solved
analytically
using
orthogonal
expansion
technique for the homogeneous part and an
asymptotic solution for the non-homogeneous
2013 台灣化學工程學會 60 週年年會論文
part. As show in Fig. 3, the average Nusselt
number Nu increases with increasing Gz .
Besides, the smaller the power-law index  is,
the higher is the average Nusselt number,
especially for the higher volumetric flow rate.
Figure 3 shows the relation between the theoretical
average Nusselt numbers Nu and Graetz number
Gz with the wall temperature ratio  and
power-law index  as parameters. The average
Nusselt number of the double-pass device Nu
increases with increasing wall temperature ratio 
and power-law index  .
Acknowledgement
The authors wish to thank the National
Science Council of the Republic of China for its
financial support.
Nomenclature
Gz Graetz number, VW / BL
heat transfer enhancement
Ih
power consumption increment
Ip
m
consistency in power-law model
Nu average Nusselt number
temperature of fluid, K
T
inlet temperature of fluid in conduit, K
Ti
wall temperature in lower plate, K
T1
wall temperature in upper plate, K
T2
in backward flow channel
at the outlet of a double-pass device
At the end of the channel
at the wall surface
References
[1] Shah, R.K., A.L. London, Laminar Flow
Forced Convection in Ducts, Academic
Press, New York, USA(1978).
[2] Dang, V.D., M. Steinberg, “Convective
diffusion
with
homogeneous
and
heterogeneous reaction in a tube,” J. Phys.
Chem. 84, 214-219 (1980).
[3] Ho, C.D., Y.C. Chen, H. Chang and S.W.
Kang, “Heat Transfer Modeling of
Conjugated Graetz Problems in Double-Pass
Parallel-Plate Heat Exchangers under
Asymmetric Wall Temperatures,” J. of the
Chinese Institute of Engineers, 36, 647-657
(2013).
[4] Lin, G.G., C.D. Ho, J.J. Huang and Y.C.
Chen, “An analytical study of power-law
fluids in double-pass heat exchangers with
external recycle,” Int. J. Heat and Mass
Transfer, 55, 2261-2267(2012).
[5] Bharti, R.P., R.P. Chhabra, V. Eswaran,
“Steady forced convection heat transfer
from a heated circular cylinder to
power-law fluids,” Int. J. Heat and Mass
Transfer 50, 977-990 (2007).
16
input volume flow rate of conduit, m3 / s
velocity distribution of fluid, m / s
conduit height, m
longitudinal coordinate
transversal coordinate
Greek symbols

thermal diffusivity of fluid, m 2 / s
shear rate
γ

longitudinal coordinate, x / W
m eigenvalue

wall temperature ratio
Impermeable-sheet thickness, m


transversal coordinate, z / LGz
Dimensionless temperature

power-law index

Subscripts
0
at the inlet or for the single pass
a
in forward flow channel
= 1.0
14
= 0.8
Double-pass device
= 3
= 5
12
10
Nu
V
v
W
x
z
b
F
L
w
= 0.4
8
6
4
2
0
1
10
100
1000
Gz
Fig. 3 Nu vs. Gz with and  as parameters.