# The Two-Dimensional Heat Transfer Analysis in Arrayed Fins with the Thermal Dissipation Substrate

Authors: Hai-Ping Hu
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

The aim of the present study is to investigate the two-dimensional heat transfer analysis in arrayed fins with thermal dissipation substrate. The governing equations for the fins and the substrate are expressed with Laplace equations, and the boundary conditions around the fins and substrate are Robin conditions. The present investigation first aims to provide a solution with regard to the geometry models by a series truncation method. Then the research will compare the results of the series truncation method with the point-matching method. Furthermore, the present study will also discuss the effects of dimension and Biot number of the fins on local dimensionless temperature, mean temperature, and heat transfer rate.


---

## Body

## 1. Introduction

Fins are used mainly to increase the extent of the heat transfer surface area in order to enhance the overall heat transfer or heat flux. The application of the fins and the thermal substrate widely ranges with regard to the larger scale and small scale of the heat cooling, power plant, chemical reactors, air condition equipment, and many other units where thermal heat is generated and must be transferred from high temperature to low temperature. Some important researches in the early stages have been based on one-dimensional or two-dimensional, steady state analysis. For example, Levitsky [1] investigated the criteria for validity of the one-dimensional fin approximation. The analysis of temperature distribution and heat flux in fins customarily makes use of a one-dimensional fin approximation. Irey [2] compared the errors between the one-dimensional and two-dimensional fin solutions, and the ratios of the interior to exterior resistance and the Biot number. The results showed that only for a small Biot number is the one-dimensional solution a satisfactory approximation. Lau and Tan [3] investigated the errors in one-dimensional heat transfer analysis in straight and annular fins. Compare one-dimensional analysis with two-dimensional analysis. The presented error decreases with decreasing values of thermal conductivity. Sparrow and Lee [4] researched the effects of fin base-temperature depression in a multifin array.In the present approach, the temperature in the tube wall was determined via a solution of Laplace’s equation. Suryanarayana [5] analyzed the two-dimensional effects on heat transfer rates from an array of straight fins and examined the errors involved in computing the heat transfer rates from fins on the basis of uniform base temperature. Heggs and Stones [6] investigated the effects of dimensions on the heat flow rate through extended surfaces, comparing one- and two-dimensional heat flows through longitudinal and annular fin assemblies for a wide range of system parameters. Heggs et al. [7] investigated the two-dimensional analysis of fin assembly heat transfer by a series truncation method. They showed that the series truncation method yields accurate solutions even for problems for which the finite-difference and finite-element methods fail to provide acceptable results.Manzoor et al. [8] researched the accuracy of perfect contact fin assembly analysis. Their work can easily be extended to annular geometry and to include fins with tapered or curved fin profile. Recently, Wood et al. [9] researched the performance indicators for steady-state heat transfer through fin assemblies; they presented several models describing steady-state heat flow through an assembly consisting of a primary wall and attached extended fin. For transient analysis of heat conduction, Wang et al. [10] established a new computing method for the cubic spline difference method for the heat conduction problem. Next, Wang et al. [11] developed a highly accurate numerical method named the cubic spline difference method and investigated the transient heat conduction problems. Mabood et al. [12] researched the series solution for steady heat transfer in a heat-generating fin and applied the optimal homotopy asymptotic method for the approximate solution of steady state of heat-generating fin with simultaneous surface convection and radiation.Since the fins are widely applied in thermal and energy engineering, the aim of the present study is to investigate the two-dimensional heat transfer analysis in arrayed fins with thermal dissipation substrate. First, the present investigation focuses on the substrate and fin by a series truncation method. Next, the research compares the results from the series truncation method with the point-matching method and compares the precision by using both methods. Furthermore, the study also discusses in detail the different parameters, for example, dimension of the fins and substrate, Biot number on the local dimensionless temperature, mean temperature, and heat transfer rate.


## 2. Analysis

Consideration of the heat transfer of thermal dissipation substrate and arrayed fins is under the assumption of steady state, constant thermal conductivities, uniform heat transfer coefficients, and perfect thermal dissipation of substrate-to-fin contact. The physical model is shown in Figure1. The geometrical symmetry of an assembly of equally spaced longitudinal rectangular fins attached to a thermal dissipation substrate indicates that it is only necessary to examine that direction of the thermal dissipation substrate and fin, shown schematically in Figure 2. The governing equation for temperature distribution in the thermal dissipation substrate can be expressed as
(1)
∇
2
T
1
x
,
y
=
0
.
And the boundary conditions of (1) are
(2)
∂
T
1
∂
y
=
0
,
y
=
p
,
-
w
≤
x
≤
0
,
∂
T
1
∂
y
=
0
,
y
=
0
,
-
w
≤
x
≤
0
,
-
k
1
∂
T
1
∂
x
=
h
∞
1
T
-
T
∞
1
,
-
x
=
w
,
0
≤
y
≤
p
.
The governing equation for temperature distribution in the fin can be expressed as
(3)
∇
2
T
2
x
,
y
=
0
.
And the boundary conditions of (3) are
(4)
-
k
2
∂
T
2
∂
y
=
h
∞
2
T
2
-
T
∞
2
,
y
=
t
,
0
≤
x
≤
l
,
∂
T
2
∂
y
=
0
,
y
=
0
,
0
≤
x
≤
l
,
-
k
2
∂
T
2
∂
x
=
h
∞
2
T
2
-
T
∞
2
,
x
=
l
,
0
≤
y
≤
t
.
The following dimensionless variables and equations are now defined:
(5)
X
=
x
p
,
Y
=
y
p
,
θ
1
=
T
1
x
,
y
-
T
∞
2
T
∞
1
-
T
∞
2
,
θ
2
=
T
2
(
x
,
y
)
-
T
∞
2
T
∞
1
-
T
∞
2
,
L
=
l
p
,
W
=
w
p
,
T
=
t
p
,
k
=
k
2
k
1
,
B
i
1
=
h
∞
1
p
k
1
,
B
i
2
=
h
∞
2
p
k
1
.
Furthermore, the governing equations and the boundary conditions of the thermal dissipation substrate can be rewritten as dimensionless equations in the following form:
(6)
∇
2
θ
1
(
X
,
Y
)
=
0
,
(7)
∂
θ
1
∂
Y
=
0
,
Y
=
1
,
-
W
≤
X
≤
0
,
∂
θ
1
∂
Y
=
0
,
Y
=
0
,
-
W
≤
X
≤
0
,
∂
θ
1
∂
X
=
-
B
i
1
1
-
θ
1
,
X
=
-
W
,
0
≤
Y
≤
1
.
The governing dimensionless equation (Figure 3) for the fin is
(8)
∇
2
θ
2
(
X
,
Y
)
=
0
,
(9)
∂
θ
2
∂
Y
=
-
B
i
2
k
θ
2
,
Y
=
T
,
0
≤
X
<
L
,
∂
θ
2
∂
Y
=
0
,
Y
=
0
,
0
≤
X
<
L
,
∂
θ
2
∂
X
=
-
B
i
2
k
θ
2
,
X
=
L
,
0
≤
Y
≤
T
. Substitute the boundary conditions of (7) and (9) into the governing equations (6) and (8), the temperature distribution of the fin and the substrate can be solved and obtained as follows:(10)
θ
1
X
,
Y
=
1
+
b
0
1
+
B
i
1
W
+
X
+
∑
n
=
1
∞
b
n
cosh
⁡
μ
n
W
+
X

+

B
i
1
/
μ
n
sinh
⁡
⁡
μ
n
W
+
X
cosh
⁡
μ
n
W
+
B
i
1
/
μ
n
sinh
⁡
μ
n
W
×
cos
⁡
μ
n
Y
,
(11)
θ
2
X
,
Y
=
∑
n
=
1
∞
a
n
cosh
⁡
λ
m
L
-
X
+
B
i
2
/
k
λ
n
sinh
⁡
λ
n
L
-
X
cosh
⁡
λ
n
L
+
B
i
2
/
k
λ
n
sinh
⁡
λ
n
L
×
cos
⁡
⁡
λ
n
Y
,
where the eigenvalues, respectively, are
(12)
μ
n
=
n
π
,
λ
n
tan
λ
n
T
=
B
i
2
k
.
One more boundary condition for (8) is
(13)
∂
θ
1
∂
X
=
-
B
i
2
θ
1
,
X
=
0
,
T
≤
Y
≤
1
.
Substitute the boundary condition into (11), and the following equation can be obtained:
(14)
B
i
2
+
b
0
B
i
1
+
B
i
2
1
+
B
i
1
W
+
∑
n
=
1
∞
b
n
B
i
2
+
μ
n
tanh
μ
n
W
+
B
i
1
/
μ
n
1
+
B
i
1
/
μ
n
tanh
μ
n
W
cos
⁡
μ
n
Y
=
0
,
T
≤
Y
≤
1
.
Next, the temperature and the heat transfer of the two regions of the L-shaped domain can be matched along the common boundary. The conditions are given as
(15)
θ
1
0
,
Y
=
θ
2
0
,
Y
,
0
≤
Y
≤
T
,
∂
θ
1
0
,
Y
∂
X
=
k
∂
θ
1
0
,
Y
∂
X
,
0
≤
Y
≤
T
.
Substituting the boundary conditions into (10) and (11) yields
(16)
1
+
b
0
1
+
B
i
1
W
+
∑
n
=
1
∞
b
n
cos
⁡
μ
n
Y
=
∑
n
=
1
∞
a
n
cos
⁡
λ
n
Y
,
0
≤
Y
≤
T
,
(17)
-
k
∑
n
=
1
∞
a
n
λ
n
tanh
λ
n
L
+
B
i
2
/
k
λ
n
1
+
B
i
2
/
k
λ
n
tanh
λ
n
L
cos
⁡
λ
n
Y
=
b
0
B
i
1
+
∑
n
=
1
∞
b
n
μ
n
tanh
μ
n
W
+
B
i
1
/
μ
n
1
+
B
i
1
/
μ
n
tanh
μ
n
W
cos
⁡
μ
n
Y
,
0
≤
Y
≤
T
.
For calculating the unknown coefficients a
n and b
n, this study will first use the series truncation method and truncation to N terms. The process steps are as follows. Multiplying (17) by cos
⁡
λ
m
Y and then integrating over the range from 0 to T define the relation between the a
n and b
n; namely,
(18)
a
m
=
∫
0
T
cos
⁡
λ
m
-
b
0
B
i
1
sin
λ
m
T
λ
m
-
∑
n
=
1
∞
b
n
μ
n
tanh
μ
n
W
+
B
i
1
/
μ
n
1
+
B
i
1
/
μ
n
tanh
μ
n
W
×
∫
0
T
cos
⁡
λ
m
Y
cos
⁡
μ
n
Y
d
Y
b
0
B
i
1
sin
λ
m
T
λ
m
×
tanh
λ
m
L
+
B
i
2
/
k
λ
m
1
+
B
i
2
/
k
λ
m
tanh
λ
m
L
k
4
λ
m
2
λ
m
T
+
sin
2
λ
m
T
λ
m
×
tanh
λ
m
L
+
B
i
2
/
k
λ
m
1
+
B
i
2
/
k
λ
m
tanh
λ
m
L
-
1
,
where
m
=
1,2
,
3
,
…
.
Thus, it remains to determine the coefficients b
n. However, the combined complexity of the governing relations, namely, (14), (16), and (18), precludes the possibility of obtaining an explicit expression for the coefficients b
n. In fact, these coefficients can only be approximately determined. The temperature distributions θ
1
(
X
,
Y
) and θ
2
(
X
,
Y
) are approximated by the first N terms in each of the series expansions, (10) and (11), respectively, and the relations (14) and (16) are accordingly modified to
(19)
1
+
b
0
*
1
+
B
i
1
W
+
∑
n
=
1
N
b
n
*
cos
⁡
μ
n
Y
+
b
0
*
B
i
1
sin
λ
n
T
λ
n
-
∑
m
=
1
N
b
m
*
μ
m
tanh
μ
m
W
+
B
i
1
/
μ
m
1
+
B
i
1
/
μ
m
tanh
μ
m
W
b
0
*
B
i
1
sin
λ
n
T
λ
n
×
∫
0
T
cos
⁡
λ
n
Y
cos
⁡
μ
m
Y
d
Y
b
0
*
B
i
1
sin
λ
n
T
λ
n
×
b
0
*
B
i
1
sin
λ
n
T
λ
n
k
4
μ
n
2
μ
n
T
+
sin
2
μ
n
T
tanh
λ
n
-
1
cos
⁡
λ
n
Y
=
0
,
0
≤
Y
≤
T
,
(20)
B
i
2
+
b
0
*
B
i
1
+
B
i
2
1
+
B
i
1
W
+
∑
n
=
1
N
b
n
*
B
i
2
+
μ
n
tanh
μ
n
W
+
B
i
1
/
μ
n
1
+
B
i
1
/
μ
n
tanh
μ
n
W
cos
⁡
μ
n
Y
=
0
,
T
≤
Y
≤
1
.
Equations (19) and (20) can be further multiplied by cos
⁡
μ
n; (19) is integrated over the range 0
≤
Y
≤
T and (20) is integrated over the range T
≤
Y
≤
1. Then the coefficients b
n
* of N terms can be determined.Figure 1
Physical model and coordinate system.Figure 2
The L-shaped regions and boundary conditions.Figure 3
The L-shaped regions with the dimensionless governing equations and the boundary conditions.Besides, the coefficientsa
n
* have been replaced using the approximate form of (18); namely,
(21)
a
m
*
=
∫
0
T
cos
⁡
λ
m
-
b
0
B
i
1
sin
λ
m
T
λ
m
-
∑
n
=
1
N
b
n
*
μ
n
tanh
μ
n
W
+
B
i
1
/
μ
m
1
+
B
i
1
/
μ
m
tanh
μ
m
W
×
∫
0
T
cos
⁡
λ
m
Y
cos
⁡
μ
n
Y
d
Y
b
0
B
i
1
sin
λ
m
T
λ
m
×
k
4
μ
m
2
λ
m
T
+
sin
2
λ
m
T
λ
m
×
k
4
μ
m
tanh
λ
m
L
+
B
i
2
/
(
k
λ
m
)
1
+
B
i
2
tanh
λ
m
L
/
(
k
λ
m
)
k
4
μ
m
-
1
,
m
=
1,2
,
3
,
…
,
N
.
Next, to compare the results obtained from the series truncation method, the present research uses the point-matching method [13] for calculating the coefficients of a
n
* and b
m
*. Choose N points along the boundary at X
=
0, and set
(22)
y
i
=
i
-
1
N
,
where
i
=
1,2
,
3
,
…
,
N
.
Truncate b
n
* to N terms and a
m
* to M terms where
(23)
M
=
floor
function
(
N
T
)
+
1
.
Equations (14), (16), and (17) can be rewritten as the following equations:
(24)
B
i
2
+
b
0
*
B
i
1
+
B
i
2
1
+
B
i
1
W
+
∑
n
=
1
N
b
n
*
B
i
2
+
μ
n
tanh
μ
n
W
+
B
i
1
/
μ
n
1
+
B
i
1
/
μ
n
tanh
μ
n
W
cos
⁡
μ
n
Y
i
=
0
,
i
=
M
+
1
to
N
,
1
+
b
0
*
1
+
B
i
1
W
+
∑
n
=
1
N
b
n
*
cos
⁡
μ
n
Y
i
=
∑
m
=
1
M
a
m
*
cos
⁡
λ
m
Y
i
,
i
=
1
to
M
,
-
k
∑
m
=
1
M
a
m
*
λ
m
tanh
λ
m
L
+
B
i
2
/
(
k
λ
m
)
1
+
B
i
2
tanh
λ
m
L
/
(
k
λ
m
)
cos
⁡
λ
m
Y
i
=
b
0
*
B
i
1
+
∑
n
=
1
N
b
n
*
μ
n
tanh
μ
n
W
+
B
i
1
/
μ
n
1
+
B
i
1
/
μ
n
tanh
μ
n
W
cos
⁡
μ
n
Y
i
,
i
=
1
to
M
.
The mean temperature, θ
m, of the entire thermal dissipation substrate and fin can be obtained by the following equation:
(25)
θ
m
=
1
W
∫
0
-
W
∫
0
1
θ
1
X
,
Y
d
Y
d
X
+
1
L
T
∫
0
L
∫
0
T
θ
2
(
X
,
Y
)
d
Y
d
X
.
The mean temperature can be rewritten in a more detailed form as follows:
(26)
θ
m
=
W
+
b
0
*
W
+
W
2
B
i
1
2
+
∑
n
=
1
N
b
n
*
cosh
⁡
μ
n
W
+
B
i
1
μ
n
sinh
⁡
μ
n
W
-
1
1
μ
n
2
sinh
⁡
μ
n
W
+
B
i
1
μ
n
3
cosh
⁡
μ
n
W
-
B
i
1
μ
n
3
×
cosh
⁡
μ
n
W
+
B
i
1
μ
n
sinh
⁡
μ
n
W
-
1
sin
μ
n
P
.
The heat transfer rate for the fin is derived as
(27)
Q
=
-
∫
0
T
∂
θ
2
∂
X
X
=
0
d
Y
.
Integrating (27) gives
(28)
Q
=
∑
m
=
1
N
a
m
*
tanh
λ
m
L
+
B
i
2
/
k
λ
m
1
+
B
i
2
/
k
λ
m
tanh
λ
m
L
sin
λ
m
T
.The method of numerical analysis was the Gauss-Seidel method and the program was aided by VisualC
+
+. The coefficients for temperature field in (10) and (11) were evaluated with the aid of (19)-(20) obtained by series truncation method and (24) obtained by point-matching method. And then local dimensionless temperature distributions θ
1 and θ
2 can be obtained. Then mean temperature θ
m and heat transfer rate Q can be obtained in (26) and (28).


## 3. Results and Discussion

In order to estimate the series convergence, the present research tests the term of the coefficientsN. When the term of the coefficients N is set as N
=
60, the precision of the coefficient converges to five decimal places. Therefore, the present investigation uses N
=
80 to calculate the results from Figures 4, 5, 6, 7, 8, and 9 in order to ensure convergence. Table 1 shows the coefficient obtained by both the series truncation method and point-matching method and compares the precision by using the two different methods. The result shows that the point-matching method obtained the same degree of precision with the series truncation method, with easier calculation than series truncation method.Table 1
Estimating the precision of the coefficient under two different methods.
T
=
0.5
,
L
=
0.5
,
W
=
0.5
,
Bi
1
=
100
,
Bi
2
=
0.01
,
k
=
0.1
,
N
=
80
Series truncation method/point-matching method
b
0
*  −0.000429/−0.000427
b
1
*  −0.0098/−0.00097
b
2
* 0.004091/0.004087
b
3
* 0.004331/0.004326
b
5
* 0.000741/0.000735
b
10
* 0.000827/0.000806
b
20
* 0.000344/0.000331
b
30
* 0.000269/0.000265
b
40
* 0.000087/0.000081
b
50
* 0.000383/0.0000377
b
60
* 0.000093/0.000091
b
80
* 0.000003/0.000002
a
1
*  −0.00702/−0.00688
a
2
* 0.007064/0.007011
a
3
* 0.000121/0.000113
a
9
* 0.000479/0.000471
a
12
* 0.000230/0.000298
a
13
*  −0.000086/−0.000086
a
18
* 0.000041/0.000039
a
19
*  −0.000027/−0.000026
a
39
*  −0.000005/−0.000003Figure 4
The isotherms forT
=
0.5, L
=
0.5, W
=
0.5, B
i
1
=
100, B
i
2
=
1, and k
=
0.01.Figure 5
The isotherms forT
=
0.5, L
=
0.75, W
=
0.5, B
i
1
=
10, B
i
2
=
1.0, and k
=
0.1.Figure 6
Effects of the thermal conductivities ratio on mean temperature under differentB
i
1.Figure 7
Effects of the length of the fin on mean temperature under differentB
i
2.Figure 8
Effects of the length of the fin on heat transfer rate under different thermal conductivity ratios.Figure 9
Effects of theB
i
2 on heat transfer rate under different heights of the fin.Figures4 and 5 show the isotherms of the substrate and fins obtained by both the series truncation method and point-matching method. As the figures show, under the different parameter conditions and dimensions, the trends of the isotherm exhibit good agreement obtained by the two different methods. The temperature distribution from the left-hand side of the high temperature of the substrate gradually decreases to the right-hand side of the lower temperature of the fin. Besides, based on the boundary conditions, the isotherms are both symmetric to Y
=
0 and Y
=
1.Figure6 shows the effects of the thermal conductivities ratio (k
=
k
2
/
k
1) on the mean temperature under different B
i
1. The condition of B
i
2 set in the figure is B
i
2
=
0.01 (where B
i
2
=
h
∞
2
P
/
k
1). Owing to the B
i
2 being a small value, 0.01, it is under the setting value of the case, and the value of k
1 is larger than the value of h
∞
2
P. The ability of conduction heat transfer in the substrate is greater than the ability of convection heat transfer on the fin side. Besides, an increase in the k will bring about a decrease in the k
1 and increase the values of B
i
1 (where B
i
1
=
h
∞
1
P
/
k
1). This will lead to the increased values of h
∞
1 enhancing the ability of convection heat transfer on the thermal dissipation substrate side and increasing the mean temperature. Besides, since the definition of the B
i
1 is h
∞
1
P
/
k
1, the larger value of B
i
1 will lead to an increased heat transfer rate and elevated mean temperature.Figure7 shows the effects of the length of the fin on mean temperature under five different values of B
i
2. The mean temperature is defined as (26), when the increased length of the fin L will lead to an increased area of the fins and achieve better heat dissipation; that is, the mean temperature decreases as the length of the fin L increases. Furthermore, since B
i
2
=
h
∞
2
P
/
k
1, the larger values of B
i
2 mean better convection heat transfer on the fin side, better heat dissipation, and decreased mean temperature.Figure8 shows the effects of the length of the fin L on heat transfer rate Q under five different values of thermal conductivities ratio k. The condition of L
=
0 means the case without fin and only with the substrate. The smaller values of k mean that the conduction heat transfer of the substrate is larger, leading to an increased heat transfer rate. As the same situation of L
=
0, under the range of 1
≤
L
≤
5, the smaller values of k increase the Q. Besides, along with the L increase, the Q will increase to a maximum value. Then the increased L will have no influence on the Q (e.g., under k
=
0.01, the maximum Q is 0.32471); that is, when L reaches the length of finishing heat dissipation, increasing the length of the fin will not influence the heat transfer rate.Figure9 shows the effects of the B
i
2 on the heat transfer rate between the interface of thermal dissipation substrate and fins (X
=
0, 0
≤
Y
≤
T) under five different values of the height of the fin. Both the horizontal axis and the vertical axis are logarithmic coordinates. As the figure shows, increasing the B
i
2 will lead to an increased heat transfer rate. Furthermore, the larger value of the height of the interface between substrate and the fin will increase the heat transfer area, leading to an increased heat transfer rate.Figure10 shows the effects of the length of the fin L on fin performance under five different values of thermal conductivities ratio k, where the Q
s are defined as dimensionless heat transfer rate without fins and only with the substrate. As the figure shows, under the case of L
=
0, the Q
/
Q
s are all 1.0. Under the range of 0
<
L
≤
5, the smaller values of k increase the fin performances. Besides, along with the L increase, the fin performances will increase to a maximum value. Then the increased L will have no influence on the fin performances; that is, when L reaches the length of finishing heat dissipation, increasing the length of the fin will not influence the fin performances.Figure 10
Effects of the length of the fin on fin performances under different thermal conductivity ratios.


## 4. Conclusions

The following conclusions can be drawn from the results of the present theoretical study.(1)
The results show that the point-matching method obtained the same degree of precision as that of the series truncation method. However, the point-matching method is more easily calculated than the series truncation method.
(2)
An increase in thermal conductivities ratio,k, will increase the ability of convection heat transfer of the substrate side and increase the mean temperature. Besides, the larger value of Bi1 will lead to an increased heat transfer rate as well as an increased mean temperature.
(3)
The mean temperature decreases as the length of the finL increases. Furthermore, the larger values of B
i
2 will bring about better heat dissipation and decrease the mean temperature.
(4)
The smaller values ofk will increase heat transfer rate. Besides, the L increase will also increase the heat transfer rate to a maximum value. Then increasing the L will no longer influence the heat transfer rate.
(5)
The increasedB
i
2 will lead to an increased heat transfer rate, Q. Furthermore, the larger value of the height of the interface between substrate and the fin will increase the heat transfer area and lead to an increased heat transfer rate.



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*Source: 716352-2015-02-23.xml*