J. Cent. South Univ. (2014) 21: 4592−4598 DOI: 10.1007/s11771-014-2465-7
Heat transfer study on solid and porous convective fins with temperature-dependent heat generation using efficient analytical method S. E. Ghasemi1, P. Valipour2, M. Hatami3, D. D. Ganji4 1. Young Reseachers and Elite Club, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran; 2. Department of Textile and Apparel, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran; 3. Department of Mechanical Engineering, Esfarayen University of Technology, North Khorasan, Iran; 4. Department of Mechanical Engineering, Babol University of Technology, Babol, Iran © Central South University Press and Springer-Verlag Berlin Heidelberg 2014 Abstract: A simple and highly accurate semi-analytical method, called the differential transformation method (DTM), was used for solving the nonlinear temperature distribution equation in solid and porous longitudinal fin with temperature dependent internal heat generation. The problem was solved for two main cases. In the first case, heat generation was assumed variable by fin temperature for a solid fin and in second heat generation varied with temperature for a porous fin. Results are presented for the temperature distribution for a range of values of parameters appearing in the mathematical formulation (e.g. N, G, and G). Results reveal that DTM is very effective and convenient. Also, it is found that this method can achieve more suitable results in comparison to numerical methods. Key words: heat transfer; convective fin; solid and porous fin; heat generation; analytical method; thermal analysis
1 Introduction Fins or extended surfaces are widely used in many industrial applications such as air conditioning, refrigeration, automobile, chemical processing equipment and electrical chips. A complete review on this topic is presented by KRAUS et al [1]. Although there are various types of the fins, but the rectangular fin is widely used among all of them, probably, due to the simplicity of its design and its easy manufacturing process. For ordinary fins, the thermal conductivity is assumed to be constant, but when temperature difference between the tip and base of the fin is large, the effect of the temperature on thermal conductivity must be considered. Also, it is very realistic that we consider the heat generation in the fin (due to electric current or etc.) as a function of temperature. AZIZ and BOUAZIZ [2] used the least squares method for a predicting the performance of a longitudinal fin with temperature dependent internal heat generation and thermal conductivity and they compared their results by HPM, VIM and double series regular perturbation method and they found that the least squares method is simple compared with other applied methods. RAZANI and AHMADI [3] considered circular fins with an arbitrary heat source distribution and a nonlinear and temperature-dependent thermal conductivity and
obtained the results for the optimum fin design. UNAL [4] conducted an analytical study of a rectangular and longitudinal fin with temperature dependent internal heat generation and temperature dependent heat transfer coefficient. Another study about this issue (convective fin with both temperature dependent thermal conductivity and internal heat generation) was also performed by SHOUMAN [5]. KUNDU [6] has solved a problem about thermal analysis and optimization of longitudinal and pin fins of uniform thickness subjected to fully wet, partially wet and fully dry surface conditions. DOMAIRRY and FAZELI [7] solved the nonlinear straight fin differential equation by HAM to evaluate the temperature distribution and fin efficiency. Also, temperature distribution for annual fins with temperature-dependent thermal conductivity using HPM was studied by GANJI et al [8]. The effects of temperature-dependent thermal conductivity of a moving fin with considering the radiation losses have been studied by AZIZ and KHANI [9]. BOUAZIZ and AZIZ [10] introduced a double optimal linearization method (DOLM) to get a simple and accurate solution for the temperature distribution in a straight rectangular convective–radiative fin with temperature dependent thermal conductivity. MUSTAFA [11] used HAM to obtain the efficiency of straight fin with temperature dependent thermal conductivity and to determine temperature distribution within it. The concept of
Received date: 2014−01−13; Accepted date: 2014−05−27 Corresponding author: S. E. Ghasemi, PhD Candidate; Tel/Fax: +98−111−3234205; E-mail:
[email protected]
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differential transformation method (DTM) was firstly introduced by ZHOU [12] in 1986 and it was used to solve both linear and nonlinear initial value problems in electric circuit analysis. This method can be applied directly for linear and nonlinear differential equation without requiring linearization, discretization, or perturbation and this is the main benefit of this method. GHAFOORI et al [13] used the DTM for solving the nonlinear oscillation equation. ABDEL-HALIM HASSAN [14] has applied the DTM for different systems of differential equations and has discussed the convergency of this method in several examples of linear and non-linear systems of differential equations. ABAZARI et al [15] have applied the DTM and RDTM (reduced differential transformation method) for solving the generalized Hirota–Satsuma coupled KDV equation. They compared the results with exact solution and they found that RDTM is more accurate than the classical DTM. RASHIDI et al [16] solved the problem of mixed convection about an inclined flat plate embedded in a porous medium by DTM; they applied the Pade approximant to increase the convergence of the solution. ABBASOV et al [17] employed DTM to obtain approximate solutions of the linear and non-linear equations related to engineering problems and they showed that the numerical results are in good agreement with the analytical solutions. BALKAYA et al [18] applied the DTM to analyze the vibration of an elastic beam supported on elastic soil. BORHANIFAR et al [19] employed DTM on some PDEs and their coupled versions. MORADI and AHMADIKIA [20] applied the DTM to solve the energy equation for a temperaturedependent thermal conductivity fin with three different profiles. MORADI [21] applied DTM for thermal characteristics of straight rectangular fin for all types of heat transfer (convection and radiation) and compared it with the results by ADM and numerical method with fourth order Rang-Kutta method using shooting method. He assumes that heat transfer coefficient varies with power-law function of temperature and he finds a good agreement between the methods. KUNDU et al [22] applied DTM for predicting fin performance of triangular and fully wet fins and they noticed that the fin performance of wet fins is almost independent on the relative humidity. Recently, HATAMI et al [23−28] investigated the effect of different geometry of porous fins in dry and wet conditions. Also, recently, some analytical methods have been applied in many engineering problems by GHASEMI et al [29−32]. In the present work, analytical solution of fin temperature distribution with temperature-dependent heat generation and thermal conductivity was studied by differential transformation method. For this purpose, after a brief introduction for DTM and description of the problem,
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DTM was applied to find the approximate solution. Obtaining the analytical solution of the model and comparing with numerical results reveal the capability, effectiveness, simplicity and high accuracy of this method.
2
Differential principles
transformation
method
In this section, the fundamental basic of the differential transformation method is introduced. For understanding method’s concept, suppose that x(t) is an analytic function in domain D, and t=ti represents any point in the domain. The function x(t) is then represented by one power series whose center is located at ti. The Taylor series expansion function of x(t) is in the form as
(t ti )k k! k 0
x(t )
d k x(t ) , t D k dt t ti
(1)
The Maclaurin series of x(t) can be obtained by taking ti=0 in Eq. (1) expressed as t k d k x(t ) k k 0 k ! dt
x(t )
, t D
(2)
t 0
As explained in Ref. [12], the differential transformation of the function x(t) is defined as X (k )
H k d k x(t ) k k 0 k ! dt t 0
(3)
where X(k) represents the transformed function and x(t) is the original function. The differential spectrum of X(k) is confined within the interval t [0, H ] , where H is a constant value. The differential inverse transform of X(k) is defined as x(t )
t
( H )k X (k )
(4)
k 0
It is clear that the concept of differential transformation is based upon the Taylor series expansion. The values of function X(k) at values of argument k are referred to as discrete, i.e., X(0) is known as the zero discrete, X(1) is the first discrete, etc. The more discrete available, the more precise it is possible to restore the unknown function. The function x(t) consists of the T-function X(k), and its value is given by the sum of the T-function with (t/H)k as its coefficient. In real applications, at the right choice of constant H, with the increase of the values of argument k, the discrete of spectrum reduces rapidly. The function x(t) is expressed by a finite series and Eq. (4) can be written as n
x(t ) ( k 0
t k ) X (k ) H
(5)
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Some important mathematical operations performed by differential transform method are listed in Table 1. Table 1 Some fundamental operations of differential transform method Origin function
Transformed function
x(t ) f ( x) g (t )
X (k ) F (k ) G (k )
x(t )
d m f (t ) dt
X (k )
m
(k m)! F (k m) k! k
q* q* (1 (T T ))
1, if k = m 0, if k m
where q* is the internal heat generation at temperature T∞. With the introduction of following dimensionless quantities,
X (k )
x(t ) exp(t )
X (k )
x(t ) sin(t )
X (k )
x(t ) cos(t )
k k!
1 k!
sin(
k
k!
cos(
kπ ) 2 kπ ) 2
dx 2
x 0,
(T T ) x hPL2 , X = , N2 , (Tb T ) L k0 A
G
q* , G (Tb T ) hP (Tb T )
d 2
Consider a longitudinal fin with constant rectangular profile, section area A, length L, perimeter P, thermal conductivity k, and heat generation q*. Fin is attached to a surface with constant temperature Tb and losses heat to the surrounding medium with temperature T∞ through a constant convective heat transfer coefficient h. In this problem, we assume that the temperature variation in the transfer direction is negligible, so heat conduction occurs only in the longitudinal direction (x direction). A schematic of the geometry of described fin (for porous fin) is shown in Fig. 1. For this problem, the governing differential equation and boundary condition can be written as d T
(9)
(10)
Equations (6)−(8) can be rewritten as
3 Description of problem
2
This problem is solved in two main cases using DTM. In the following subsections, the governing equations for these two cases are introduced.
l 0
X ( k ) ( k m) x (t ) t m
(8)
3.1 Solid fin with temperature dependent internal heat generation In the first case, we assume that heat generation in the solid fin varies with temperature as Eq. (9) and the thermal conductivity is constant k0.
X (k ) F (l )G (k l )
x(t ) f (t ) g (t )
x L, T Tb
*
hP q (T T ) 0 kA k
(6)
dT 0 dx
(7)
dX
2
N 2 N 2 G (1 G ) 0
X 0,
d 0 dX
X 1, 1
(11) (12) (13)
Now, we apply differential transformation method (DTM) from Table 1 into Eq. (11) for finding θ(x). (k 1)(k 2) (k 2) N 2 (k ) N 2 G ( (k )
G (k ))=0
(14)
Rearranging Eq. (14), a simple recurrence relation is obtained as
(k 2)
N 2 (k ) N 2G ( (k ) G (k )) (k 1)(k 2)
(15)
where 1, if k =0 0, if k 0
(k )
(16)
Similarly, the transformed form of boundary conditions can be written as
(0) a, (1)=0
(17)
By solving Eq. (15) and using boundary conditions (Eq. (17)), the DTM terms are obtained as 1 2 1 N (0) N 2G (1 G (0)) 2 2 1 2 1 2 (3) N (1) N G (1 G (1)) 6 6
(2)
Fig. 1 Schematic of fin geometry with heat generation source
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1 2 1 N (2) N 2G (1 G (2)) 12 12 1 1 (5) N 2 (1) N 2 G (1 G (1)) 20 20
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For simplifying the above equations, dimensionless parameters are introduced as
(4)
(18)
Now, by applying Eq. (4) to Eq. (18), and by using Eq. (8), the constant parameter “a” will be obtained, so the temperature distribution equation will be estimated. 3.2 Porous fin with temperature dependent internal heat generation For the second case, a porous fin is considered and energy balance can be written as q( x) q ( x x) q * A x m c p [T ( x) T ] h( p x)[T ( x) T ]
(19)
The mass flow rate of the fluid passing through the porous material is [28] m Vw x w (20) The passage velocity from the Darcy’s model is [28] VW
gK (T T ) v
(21)
(T T ) X , X , (Tb T ) L
M2
hPL2 DaxRa L , Sh k0 A kr t
d 2 dX 2
(22)
c p gKw dq q* A [T ( x) T ]2 hp[T ( x ) T ] (23) dx Also, from Fourier’s law of conduction: dT dx
In this work, we study finite-length fin with insulated tip. For this case, the fin tip is insulated so that there will not be any heat transfer at the insulated tip and boundary condition will be d dX
(25)
where φ is the porosity of the porous fin. Substitution Eq. (24) into Eq. (23) leads to
c p gK
hp [T ( x) T ] [T ( x) T ] tk eff v k eff A dx 2 q* 0 keff
2
(26)
It is assumed that heat generation in the fin varies with temperature as [23] q * q* (1 (T T ))
(30)
Now, we apply DTM from Table 1 into Eq. (29) for finding u(t): (k 1)(k 2) (k 2) S h ( (l ) (k l )) l 0
2
( M G GM ) (k ) GM 2 (k ) 0
2
( 31)
k
(k 2) [ S h ( (l ) (k l )) l 0
where A is the cross-sectional area of the fin A=wt and keff is the effective thermal conductivity of the porous fin that can be obtained from following equation [27]
d 2T
0 x 1
Rearranging Eq. (31), we have
(24)
k eff k f (1 )k s
(29)
k
As Δx→0, Eq. (22) becomes
q k eff A
(28)
M 2 M 2G (1 G ) S h 2 0
θ(0)=1,
c p gK w q ( x) q( x x) q* A [T ( x) T ]2 x hp[T ( x) T ]
2
where Sh is a porous parameter that indicates the effect of the permeability of the porous medium as well as buoyancy effect so higher value of Sh indicates higher permeability of the porous medium or higher buoyancy forces. M is a convection parameter that indicates the effect of surface convecting of the fin. Finally, Eq. (26) can be rewritten as
Substitutions of Eqs. (20) and (21) into Eq. (19) yield
some
(27)
where q* is the internal heat generation at temperature T∞.
( M 2 G GM 2 ) (k ) GM 2 (k )] /
[(k 1)(k 2)]
(32)
And boundary condition transformed form is Θ(0)=1, Θ(1)=a
(33)
where a is an unknown coefficient that must be determined. By solving Eq. (32) and using boundary conditions, the DTM terms are obtained as 1 2
1 2
1 2
1 2
(2)= GM 2 Sh M 2 G GM 2 1 3
1 6
1 6 1 1 1 2 (4)= Sh G GM Sh GM 2 Sh2 8 12 12 1 1 1 1 2 4 Sh M G GM Sh a 2 GM 4 8 12 12 24
(3)= Sh a M 2 a M 2 a G G
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1 1 1 M 4 M 4 G G 2 G2 G 2 M 4 24 24 24 1 1 1 (5)= Sh aGM 2 Sh2 a Sh aM 2 20 12 12 1 1 1 Sh a G GM 2 M 4 a a G GM 4 12 120 60 1 a G2 G 2 M 4 120
than N=1 state. Figure 5 shows the errors for N=1 and G=G=0.2, G=G=0.4 and G=G=0.6. As already seen in Fig. 3, maximum error occurs in tip of the fin, and this result also occurs in Fig. 5. The range of the errors reveals that DTM has a good agreement with numerical results. (34)
Now, by substituting Eq. (34) into Eq. (5) and using boundary condition, the “a” coefficient and then θ(X) function can be obtained.
4 Results and discussion 4.1 Case1: Solid fin with temperature dependent internal heat generation Temperature distribution in Case 1 (temperature dependent heat generation and constant thermal conductivity) is shown in Figs. 2−5. It is common that in fin design, the N parameter is considered to be 1. Figure 2 shows the temperature distribution for this state and G=G=0.2, G=G=0.4 and G=G=0.6. This choice of parameters represents a fin with moderate temperature dependent heat generation and the thermal conductivity variation of 20% between the base and the surrounding coolant temperatures that are often used in nuclear rods. As we can see in the figure by increasing in G and G temperature of the fin increased because of increasing in heat generation. By comparing the results and those of numerical method, it was observed that DTM has a good efficiency and accuracy, error of DTM is plotted in Fig. 3 and reveals this fact. As seen in Fig. 3, the maximum error occurs in the tip of the fin. Figure 4 shows a comparison results which pertain to N=0.5 (this choice is used in compact heat exchanger fin design), this figure illustrates that fin temperature in this condition is greater
Fig. 2 Temperature distribution in fin with temperature dependent internal heat generation and constant thermal conductivity for N=1
Fig. 3 Error of DTM in comparison by numerical method for case 1 and N=1
Fig. 4 Temperature distribution in fin with temperature dependent internal heat generation and constant thermal conductivity for N=0.5
Fig. 5 Error of DTM in comparison by numerical method for case 1 and N=0.5
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4.2 Case 2: Porous fin with temperature dependent internal heat generation Figures 6−8 show the temperature distribution in Case 2. The effect of Darcy number on non-dimensional temperature is shown in Fig. 6, while the other parameter is constant. It is illustrated that the magnitude of temperature increases with decreasing the Darcy number.
Fig. 6 Temperature distribution versus Da variation at at G=0.4, M=1, εG=0.6 and Ra=104
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For the effect of Darcy number on temperature profile, the bellow details are mentioned. By attention to the definition of Da=K/t2, the Darcy number reduces when the permeability reduces. Also by attention to the permeability definition, if the bubbles of porous media are very small or if they are poorly connected, the permeability will be low and fluid will not flow through easily. Therefore, when the Darcy number and consequently the permeability reduce, the collision between the fluid flow and pores of porous screen increases. Thus, the passing fluids gave more space to contact with the porous media which has internal heat generation. Consequently, the value of temperature is increased by decreasing the Da number. On the other hand, Sh is connected directly to Da number variation. In addition, increasing the Ra number which represents the buoyancy force has the similar outcome on temperature profile (see Fig. 7). It means that by attention to the Rayleigh number definition (Ra= gβ∆TH3/(αυ)), increasing in the Ra number leads to more effect of bouncy force and consequently heat transfer rate due to convection mechanism. So, higher value of Ra makes enhanced heat transfer between the solid fin and the air flow and subsequently leads to more value of temperature in temperature profile. The effect of temperature dependent internal heat generation value is investigated via Fig. 8. The εG parameter is changed when the other parameters are constant. As mentioned before, the variation of heat generation is directly related to εG. Thus, by increasing this parameter, the heat generation through the fin is increased. Finally, by raising the heat generation parameter, the temperature profile reaches to the higher value. Alternatively, the higher heat generation leads to higher fin temperatures because the fin must dissipate a larger amount of heat to the surroundings under steady state conditions.
Fig. 7 Temperature profile versus Ra variation at G=0.4, M=1, εG=0.6 and Da=10−3
5 Conclusions
Fig. 8 Temperature distribution versus temperature dependent internal heat generation parameter at G=0.4, M=1, Da=10−3, Ra=104
1) Despite DTM simplicity, the solutions match with the numerical results within a maximum error of 0.05%. Obtaining the analytical DTM solution of the problem and comparing with numerical results reveal the facility, effectiveness, and high accuracy of this method. 2) The higher heat generation leads to higher fin temperatures, because the fin must dissipate a larger amount of heat to the surroundings under steady state conditions. 3) The results indicate that the temperature distribution is strongly dependent on the Darcy and Rayleigh numbers for porous fin. 4) It is illustrated that the magnitude of temperature increases with decreasing the Darcy number and consequently Sh parameter.
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