International Journal of Applied Sciences and Smart Technologies


International Journal of Applied Sciences and Smart Technologies 

Volume 3, Issue 1, pages 111–124 

p-ISSN 2655-8564, e-ISSN 2685-9432 

  
111 

 

  

 
Obtaining the Efficiency and Effectiveness of Fin 

in Unsteady State Conditions Using Explicit 

Finite Difference Method 
 

Petrus Kanisius Purwadi1,*, Budi Setyahandana1, R.B.P. Harsilo1 

 
1Department of Mechanical Engineering, Faculty of Science and 

Technology, Sanata Dharma University, Yogyakarta, Indonesia 
*Corresponding Author: pkpurwadi1966@gmail.com 

 

(Received 10-04-2021; Revised 10-05-2021; Accepted 17-05-2021) 

 

Abstract 

This paper discusses the search for fin efficiency and effectiveness in 

unsteady state conditions using numerical computation methods. The 

straight fin under review has a cross-sectional area that changes with the 

position x. The cross section of the fin is rectangular. The fins are composed 

of two different metal materials. The computation method used is the 

explicit finite difference methods. The properties of the fin material are 

assumed to be fixed, or do not change with changes in temperature. When 

the stability requirements are met, the use of the explicit finite difference 

methods yields satisfactory results. The use of the explicit finite difference 

methods can be developed for various other fin shapes, which are composed 

of two or more different materials, time-varying convection heat transfer 

coefficient, and the properties of the fin material that change with 

temperature. 

Keywords: fin, efficiency, effectiveness, finite-difference, unsteady state 

 



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1 Introduction 

In the design of fins, the important thing to know is the efficiency and effectiveness 

of the fins. There are many ways to know the value of fin efficiency and fin 

effectiveness. For certain fin shapes, the value of the efficiency and effectiveness of the 

fins can be found by using existing charts. For fins that do not yet exist, other ways are 

needed to get them. For cases in unstable state, the solution becomes more complicated. 

One way that can be done is by using numerical computation methods. 

Several articles [1-7] related to the efficiency and fin effectiveness of fins in the 

unsteady state have helped in in solving this problem.  In this case, the case discussed is 

a fin which is composed of two different materials and is in an unsteady state. The 

shape and cross-section of the fins are different from those that have been used as the 

object of the previous discussion. In this case, the cross-sectional area of the fin changes 

with the change in position x. The shape of the fin chosen is a truncated rectangular 

pyramid. Figure 1 presents an image of the fin shape to be discussed. With the total 

length of the fin 𝐿, where 𝐿 = 𝐿1 + 𝐿2 and 𝐿1 = 𝐿2  

Figure 1. Straight fin with a truncated rectangular pyramid shape  

The mathematical model for this problem is expressed by equation (1): 

 

𝜕2𝑇(𝑥, 𝑡)

𝜕𝑥2
+

ℎ𝑃

𝑘𝐴𝑝
(𝑇(𝑥, 𝑡) − 𝑇∞) =  

1

∝

𝜕𝑇(𝑥, 𝑡)

𝜕𝑡
 

0 < 𝑥 < 𝐿, 𝑥 ≠ 𝐿1, 𝑡 > 0      (1) 

𝛼 =
𝑘

𝜌𝑐
  

(2) 

with initial condition: 



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113 

 
  

𝑇(𝑥, 0) = 𝑇𝑖 = 𝑇𝑏  0 ≤ 𝑥 ≤ 𝐿, 𝑡 = 0              (3) 

and boundary condition: 

𝑇(0, 𝑡) = 𝑇𝑏 𝑥 = 0, 𝑡 > 0  (4) 

𝑘2
𝜕𝑇(𝑥, 𝑡)

𝜕𝑥
= ℎ

𝐴𝑠

𝐴𝑝1
(𝑇(𝑥, 𝑡) − 𝑇∞) 𝑥 = 𝐿, 𝑡 > 0  

(5) 

𝑘1
𝜕𝑇(𝑥, 𝑡)

𝜕𝑥
= 𝑘2

𝐴𝑝2
𝐴𝑝1

𝜕𝑇(𝑥, 𝑡)

𝜕𝑥
+ ℎ𝐴𝑠

𝐴𝑠

𝐴𝑝1
(𝑇∞ − 𝑇(𝑥, 𝑡)) 𝑥 = 𝐿1, 𝑡 > 0  

(6) 

 

In equations (1)-(6) we have used the notations: 

x : Stated the position on fin, m  

t : Stated the time, seconds  

T(x,t) : Temperature at the x position, at the t time, oC  

T∞ : Fluid temperature, 
oC.  

Ti : Initial temperature of the fin, 
oC 

Tb : Temperature at the base of the fin, 
oC 

L1 : The length of the fin with material 1or the length of the fin with material 1, m  

L2 : The length of the fin with material 1or the length of the fin with material 2, m  

L : The length of the fin, L = L1+L2, m  

Ap : Cross-sectional of the fin, m
2 

P : Around the cross section, m  

k : Conduction heat transfer coefficient of fin material 1 or material 2, W/(m.oC) 

k1 : Conduction heat transfer coefficient of fin material 1, W/(m.
oC)  

k2 : Conduction heat transfer coefficient of fin material 2, W/(m.
oC)  

h : Convection heat transfer coefficient, W/(m2 oC)  

ρ : Density of the material, kg/m3 

c : Specific heat of the material, kJ/(kg.oC) 

c1 : Specific heat of the material 1, J/(kg.
oC) 

c2 : Specific heat of the material 2, J/(kg.
oC) 

α : Thermal diffusivity of the material, m2/s 

 

1.1.  Calculation Steps 

The steps used to calculate the efficiency and effectiveness of the fin in an unsteady 

state condition using the numerical method are as follows:   

1. Determine fixed variable variables, such as: ℎ, 𝜌1, 𝜌2, 𝑐1, 𝑐2, 𝑘1, 𝑘2, 𝑎1, 𝑏1 

𝑎2, 𝑏2, 𝑇∞, 𝑇𝑖, 𝑇𝑏, 𝐿1, 𝐿2, 𝐿, 𝑃, 𝐴𝑠𝑖, 𝐴𝑝𝑖 , 𝑚, ∆𝑥, ∆𝑡. 



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2. Perform temperature calculations for each control volume from time to time from 

volume control 1 to control up to volume control to m. Volume control 1 is on the 

bottom of the fin and volume control m is on the end of the fin. The total volume of 

the control is m. 

3. Calculate the actual heat flow rate (𝑞𝑎𝑐𝑡) released by the fins, the ideal heat flow rate 

released by the fins (𝑞𝑖𝑑𝑒𝑎𝑙 ), and the heat flow rate released if there are no fins 

(𝑞𝑛𝑜,𝑓𝑖𝑛). Temperature calculations are carried out from time to time. 

4. Perform calculations of fin efficiency (η) from time to time 

5. Perform calculations of fin effectiveness (ϵ) from time to time 

 

Figure 2 presents a flow chart to get the efficiency and effectiveness of the fins. 

 

Figure 2. Flow chart to get the efficiency and effectiveness of the fins 

 

Some of the assumptions used in this calculation are: 

1. The fluid temperature and the convection heat transfer coefficient are uniform and 

constant. 

2. Material properties (density, specific heat, thermal conductivity of the material) are 

uniform and constant.  

3. The shape of the fins is fixed and does not change in volume during the process. 



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Volume 3, Issue 1, pages 111–124 

p-ISSN 2655-8564, e-ISSN 2685-9432 

  
115 

 
  

4. The transfer of heat by radiation is negligible. 

5. The connection on the two fin materials is assumed to be perfectly connected. 

6. No energy generated in the fins. 

7. The heat transfer by conduction in the fins is assumed to take place in only one 

direction, the x direction. 

8. The entire surface of the fins is in contact with the fluid around the fins. 

 

 

1.2 Control volume and energy balance on the control volume 

Figure 3 shows a fin image divided into many controls. The total control volume is 

m. Volume control 1 is at the base of the fin, and volume control 𝑚 is at the end of the 

fin. The volume control 𝑝 is located at the border of the two fin materials. In the control 

volume 𝑝, the half control volume is made of material 1, and the half control volume is 

made of material 2. The numbering of the control volume starts from the bottom of the 

fin to the end of the fin in sequence, with the names of the volume controls 1, 2, 3, … , 𝑚. 

The distance between control volumes is ∆𝑥. At the control volume 𝑖 = 1, 2, 3, … , 𝑚 −

1, the control volume performs a convection heat transfer process through its blanket 

surface. At the control volume 𝑚, the process of convection heat transfer outside the 

blanket surface, also carries out the process of convection heat transfer through the 

cross-sectional area of the fins. The thickness of the volume control at the base of the 

fin and at the tip of the fin is 0,5 ∆𝑥. The control volume thickness of 𝑖 =

1, 2, 3, … , 𝑚 − 1 is ∆𝑥. 

 

In Figure 4, Figure 5 and Figure 6, the convection heat flow is represented by 𝑞𝑐. The 

conduction heat flow from the control volume in position 𝑖 − 1 to the control volume at 

𝑖, is represented by 𝑞𝑖−1. The conduction heat flow from the control volume in position 

𝑖 + 1 to the control volume in position 𝑖, expressed as 𝑞𝑖+1. 

 

 

 



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Figure 3. Distribution of volume control on the fins 

 

 

 

The energy balance of the control volume on the fins, without energy generation, in the 

unsteady state condition can be stated by the following statement: 

[
all energy entering  

the control volume during
the time interval ∆t

] =  [

the energy change 
in the control volume 

during in the time interval ∆t
] 

It can also be expressed by equation (7): 

∑ 𝑞𝑖

𝑖=𝑘

𝑖=1

= 𝜌𝑐𝑉
∆𝑇

∆𝑡
 (7) 

 

 

Figure 4 presents the energy balance at the control volume inside the fin body, or at 

the control volume at position 𝑖 = 2, 3, 4, … , 𝑝 − 3, 𝑝 − 2, 𝑝 − 1 (between the base of 

the fin with the border of the two fin materials) and in position 𝑖 = 𝑝 + 1, 𝑝 + 2, 𝑝 +

3, … , 𝑚 − 2, 𝑚 − 1 (between the border of the two fin materials with the tip of the fin). 

The energy balance in this control volume can be expressed by equation (8): 

𝑞𝑖−1 + 𝑞𝑖+1 +  𝑞𝑐 = 𝜌𝑐𝑉
𝑇𝑖

𝑛+1−𝑇𝑖
𝑛

∆𝑡
 .  

(8) 

 

 

 



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117 

 
  

 

 

 

 

 

 

 

 

 

 

 

Figure 4. The energy balance at the control volume at 𝑖 = 2, 3, 4, … , 𝑝 − 2, 𝑝 − 1 and at the 
control volume at 𝑖 = 𝑝 + 1, 𝑝 + 2, 𝑝 + 3, … , 𝑚 − 2, 𝑚 − 1 

 

 

In equations (8)-(10), 𝑇𝑖
𝑛 is the temperature in control volume 𝑖 at time 𝑡 = 𝑛 or at time 

𝑡, and 𝑇𝑖
𝑛+1 is at control volume 𝑖 at time 𝑡 = 𝑛 + 1 or when 𝑡 = 𝑡 + ∆𝑡. Assume that 

the control volume has a uniform temperature. The variable 𝑉 represents the volume, ∆𝑡 

and represents the time interval. 

 

 

 

 

 

 

 

 

 
Figure 5. Energy balance on volume control at 𝑖 =  𝑝 

 

 



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Figure 5 presents the energy balance at the control volume at the boundary of the two 

materials or at 𝑖 = 𝑝. At the control volume at 𝑖 = 𝑝, the control volume is composed of 

two different materials. The thickness of the control volume for material 1 is 0.5 ∆𝑥, the 

thickness of the control volume for material 2 is 0.5 ∆𝑥. The energy balance at the 

control volume in position 𝑖 = 𝑝 can be expressed by equation (9): 

𝑞𝑖−1 +  𝑞𝑖+1 +  𝑞𝑐 = 𝜌1𝑐1𝑉𝑖.
𝑇𝑖

𝑛+1−𝑇𝑖
𝑛

∆𝑡
+ 𝜌2𝑐2𝑉2 

𝑇𝑖
𝑛+1−𝑇𝑖

𝑛

∆𝑡
. 

 (9) 

 

 

 

 

 

 

 

 

 

Figure 6. Energy balance on the control volume di 𝑖 = 𝑚 

 

Figure 6 presents the energy balance at the control volume at the tip of the fin or at 

𝑖 = 𝑚. In the control volume at the fin tip, the process of convection heat transfer 

through the blanket surface and the cross-sectional surface of the fin tip. The thickness 

of the volume control at the tip of the fin is 0.5 ∆𝑥 . The energy balance at the control 

volume at position 𝑖 = 𝑚 can be expressed by equation (10): 

𝑞𝑖−1 +  𝑞𝑐1 + 𝑞𝑐2 = 𝜌2𝑐2𝑉2
𝑇𝑖

𝑛+1 − 𝑇𝑖
𝑛

∆𝑡
 

(10) 

1.3 Temperature Distribution at the Fin 

By using equation (7), the equation used to calculate the temperature at the control 

volume 𝑖 = 2, 3, 4, … , 𝑚 − 2, 𝑚 − 1, 𝑚, when 𝑡 > 0 can be found. The equation for 

calculating the temperature at the control volume at 𝑖 = 2, 3, 4, … , 𝑝 − 2, 𝑝 − 1, when 

𝑡 > 0 is expressed by equation (11): 



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𝑇𝑖
𝑛+1 =  

∆𝑡

∝1 ∆𝑥
2

[(𝑇𝑖−1
𝑛 − 2𝑇𝑖

𝑛 + 𝑇𝑖+1
𝑛 ) + 𝐵𝑖1(𝑇∞ − 𝑇𝑖

𝑛)] + 𝑇𝑖
𝑛 . 

(11) 

The stability requirements for equation (11) are expressed by equation (12): 

∆𝑡 ≤
𝜌1𝑐1∆𝑥

2

𝑘1 (2 + 𝐵𝑖1
∆𝑥𝐴𝑠𝑖

𝐴𝑝
)

 . 
(12) 

The equation for calculating the temperature at the control volume at 𝑖 = 𝑝 when 𝑡 > 0, 

can be expressed by equation (13): 

𝑇𝑖
𝑛+1 =  

∆𝑡

(𝜌1𝑐1𝑉1 + 𝜌2𝑐2𝑉2)
[(𝑘1𝐴𝑝

𝑇𝑖−1
𝑛 − 𝑇𝑖

𝑛

∆𝑥
+ 𝑘2𝐴𝑝

𝑇𝑖+1
𝑛 − 𝑇𝑖

𝑛

∆𝑥
)

+ ℎ𝐴𝑠(𝑇∞ − 𝑇𝑖
𝑛)] + 𝑇𝑖

𝑛 . 

    

(13) 

The stability requirements for equation (13) are expressed by equation (14): 

∆𝑡 ≤
(𝜌1𝑐1𝑉1 + 𝜌2𝑐2𝑉2)

(
𝑘1𝐴𝑝

∆𝑥
+

𝑘2𝐴𝑝

∆𝑥
+ ℎ𝐴𝑠)

 . 
(14) 

The equation for calculating the temperature at the control volume at 𝑖 = 𝑝 when 𝑡 > 0, 

can be expressed by equation (15): 

𝑇𝑖
𝑛+1 =  

∆𝑡

∝2 ∆𝑥
2

[(𝑇𝑖−1
𝑛 − 2𝑇𝑖

𝑛 + 𝑇𝑖+1
𝑛 ) + 𝐵𝑖2(𝑇∞ − 𝑇𝑖

𝑛)] + 𝑇𝑖
𝑛 . 

(15) 

The stability requirements for Equation (15) are expressed by equation (16): 

∆𝑡 ≤
𝜌2𝑐2∆𝑥

2

𝑘2 (2 + 𝐵𝑖2
∆𝑥𝐴𝑠𝑖

𝐴𝑝
)

 . 
(16) 

The equation for calculating the temperature at the control volume at 𝑖 = 𝑚 when 𝑡 >

0, can be expressed by equation (15): 

𝑇𝑖
𝑛+1 =  

∆𝑡

0.5∝2∆𝑥
2

[(𝑇𝑖−1
𝑛 − 𝑇𝑖

𝑛) + 𝐵𝑖2(𝑇∞ − 𝑇𝑖
𝑛) + (𝐵𝑖2

𝐴𝑠

𝐴𝑝
(𝑇∞ − 𝑇𝑖

𝑛))] + 𝑇𝑖
𝑛 .  

(17) 

The stability requirements for Equation (17) are expressed by equation (18): 

∆𝑡 ≤
0.5 𝜌2𝑐2∆𝑥

2

𝑘2 (1 + 𝐵𝑖2 + 𝐵𝑖2
𝐴𝑠

𝐴𝑝
)

 . 
(18) 

  



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1.4.  Heat flow rate, efficiency and effectiveness 

The amount of actual heat released by the fins in the unsteady state condition can be 

calculated by equation (19): 

𝑞act
𝑛 = ∑ ℎ𝐴𝑠𝑖(𝑇𝑖

𝑛 − 𝑇∞)
𝑚

𝑖=1
 (19) 

The amount of ideal heat released by the fins can be calculated by equation (20): 

𝑞ideal = ∑ ℎ𝐴𝑠𝑖 (𝑇𝑏 − 𝑇∞)
𝑚

𝑖=1
. (20) 

The amount of heat released by the bottom of the fin, if the length of the fin is 

considered zero, can be calculated by equation (21): 

𝑞no fin = ∑ ℎ𝐴𝑏(𝑇𝑏 − 𝑇∞)
𝑚

𝑖=1
. (21) 

The efficiency of the fin at an unsteady state condition can be calculated by equation 

(22): 

𝜂𝑛 =
𝑞act

𝑛

𝑞ideal
=

∑ ℎ𝐴𝑠𝑖(𝑇𝑖
𝑛 − 𝑇∞)

𝑚
𝑖=1

∑ ℎ𝐴𝑠𝑖 (𝑇𝑏 − 𝑇∞)
𝑚
𝑖=1

 . (22) 

The effectiveness of the fin at an unsteady state condition can be calculated by equation 

(23): 

∈𝑛=
𝑞act

𝑛

𝑞no fin
=

∑ ℎ𝐴𝑠𝑖(𝑇𝑖
𝑛 − 𝑇∞)

𝑚
𝑖=1

∑ ℎ𝐴𝑏(𝑇𝑏 − 𝑇∞)
𝑚
𝑖=1

 . (23) 

  

2 Research Methodology 

The search for efficiency and effectiveness is carried out using numerical methods. 

The numerical method used is the explicit finite difference methods. The selected fin 

shape is a truncated rectangular pyramid (Figure 1). Fins are composed of two different 

materials. The properties of the fin material are presented in Table 1. The total fin 

length is 𝐿, where 𝐿 = 10 𝑐𝑚. The length of the fin with material 1 is the same as 

length p with material 2, where 𝐿1 = 𝐿2 = 5 𝑐𝑚. The section of the fin at the base of 

the fin, has 𝑎1 width and 𝑏1 height, where 𝑎1 = 1 𝑐𝑚 and 𝑏1 = 0.5 𝑐𝑚. The section of 

the fin at the end of the fin, has 𝑎2 width and 𝑏2 height, where 𝑎2 = 0.5 𝑐𝑚 and 𝑏2 =

0.25 𝑐𝑚. The sum of the control volume on the fin is 𝑚, where 𝑚 = 25. The distance 



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between the control volume is ∆𝑥, and ∆𝑥 = 0.004167 𝑐𝑚. The time interval is ∆𝑡, 

where ∆𝑡 = 0.05 second. The temperature of the fluid around the fin is 𝑇∞, where 𝑇∞ =

30℃ The base temperature of the fin remains equal to 𝑇𝑏, with 𝑇𝑏 = 100℃. The initial 

temperature of the fin is Ti, where 𝑇𝑖 = 𝑇𝑏 = 100℃. The convection heat transfer 

coefficient is ℎ, where ℎ = 100𝑊/𝑚2. ℃. 

Table 1. Properties of the material 

(Y.A. Cengel, Heat Transfer A Practical Approach, pp 868-870, see [8]) 

Material 
Density (ρ) 

(kg/m3) 

Thermal Conductivity (k) 

(W/m.oC) 

Specific Heat (c) 

(J/kg.oC) 

Copper (Cu) 8933 401 385 

Aluminum (Al) 2702 237 903 

Zinc (Zn) 7140 116 389 

Nickel (Ni) 8900 90.7 444 

Iron (Fe) 7870 80.2 447 

 

 

3 Results and Discussion 

The results of the calculation of temperature distribution, actual heat flow rate, 

efficiency and effectiveness of fins are presented in Figure 7, Figure 8, Figure 9 and 

Figure 10. The solution using explicit finite difference methods gives satisfactory 

results. As long as the stability requirements are met, the results are realistic. The 

truncated rectangular pyramid shape is an example of the shape of the fin. Thus, in the 

same way, the use of the explicit finite difference methods can be developed for the 

calculation of efficiency and effectiveness with other fin shapes. 

Figure 7 presents the temperature distribution that occurs in the fins with various 

variations in the composition of the material. At the boundary of the two materials, the 

temperature distribution looks broken. The fins with an iron-copper composition were 

more likely to be broken than those with an iron-nickel composition. The results of the 

temperature distribution in the unsteady state are influenced by the properties of the 

material such as mass density, thermal conductivity and specific heat. Figure 8, Figure 9 

and Figure 10 present the actual heat flow rate of fin removal, fin efficiency and fin 

effectiveness. The results of this calculation, it all depends on the temperature 

distribution that occurs in the fin. 



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Figure 7.  The temperature of the fin at time 

t= 100 seconds 

 

 

 

 

 

 

 

 

 

 

 

 
Figure 8.  The heat flow rate of the fin 

 

 

 

 

 

 

 

 

 

 

 

 
Figure 9. Fin efficiency  

 

 

 

 

 

 

 

 

 

 

 

 

 
Figure 10.  Fin Effectiveness 

 

The use of the explicit finite different methods can also be developed for the case of 

conduction heat flow which is not only for the one-dimensional case, but also the two- 

dimensional unsteady state case. The two-dimensional case is when the conduction heat 

20

40

60

80

100

1 7 13 19 25

T
e
m

p
e
ra

tu
re

, 
o
C

Control volume

Fe-Cu

Fe-Al

Fe-Zn

Fe-Ni

4

7

10

13

16

0 25 50 75 100
H

e
a
t 

fl
o

w
 r

a
te

, 
w

a
tt

Time t, second

Fe-Cu

Fe-Al

Fe-Zn

Fe-Ni

0

0,25

0,5

0,75

1

0 25 50 75 100

F
in

 e
ff

ic
ie

n
c
y

Time t, second

Fe-Cu

Fe-Al

Fe-Zn

Fe-Ni

0

15

30

45

60

0 25 50 75 100

F
in

 e
ff

e
c
ti

v
e
n
e
ss

Time t, second

Fe-Cu

Fe-AL

Fe-Zn

Fe-Ni



International Journal of Applied Sciences and Smart Technologies 

Volume 3, Issue 1, pages 111–124 

p-ISSN 2655-8564, e-ISSN 2685-9432 

  
123 

 
  

flow in the object takes place in two directions: the x direction and the y direction. The 

two-dimensional case occurs for fins with a greater length and width than the thickness 

of the fin. It can also be developed for the three-dimensional case, with three-directions 

of conduction heat flow: x, y, and z directions.  

The fin object discussed in this issue is for fins which are composed of two different 

materials. If the fins are composed of three or more different fin materials, the use of the 

explicit finite difference methods can also be used. In this case, the fin material used 

does not change with changes in temperature. The use the explicit finite difference 

methods can also be used for fins with temperature changing material properties. 

Material properties that change due to temperature, such as density, specific heat and 

thermal conductivity of the fin material. 

4 Conclusion 

As long as the stability requirements are met explicit finite difference methods can be 

used well to calculate the efficiency and effectiveness of fins in the unsteady state 

condition. The use of the explicit finite difference methods can be developed for various 

other fin shapes, which are composed of two or more different materials, the time-

changing convection heat transfer coefficient value, and the temperature-changing 

properties of the fin material. 

Acknowledgements  

This research was conducted at Sanata Dharma University. The authors thank Sanata 

Dharma University for supporting this research. 

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International Journal of Applied Sciences and Smart Technologies 

Volume 3, Issue 1, pages 111–124 

p-ISSN 2655-8564, e-ISSN 2685-9432 

  
124 

 
  

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