Analytical and Numerical investigation of natural convection in a heated cylinder using Homotopy Perturbation Method

Homotopy Perturbation Method (HPM) has been applied to solve a nonlinear heat transfer problem. Natural convection around an isothermal horizontal cylinder was studied. Heat transfer coefficient and specific heat coefficient were assumed to be dependent on temperature. Outcomes were compared with solution of heat transfer equation with constant properties. Solutions of HPM were compared with numerical results for different cases, and variation of Nusselt number was obtained and investigated.


Introduction
Many familiar heat transfer application involve natural convection as the primary mechanism of heat transfer.Some examples are cooling of electronic equipments such as power transistors, TVs and VCRs; heat transfer in electronic baseboard heaters or steam radiators; heat transfer phenomena in the refrigeration coils and power transmission lines.The fluid velocities associated with natural convection are low; typically less than 1 m/s therefore, the heat transfer coefficient encountered in natural convection are usually much lower than those encountered in forced convection (CHENG, 2009;GANGI;SADIGHI, 2007;JONES et al., 1988).
So, studying natural convection that rises in applicable engineering problems, help to manipulate better natural convection as one of the heat transfer mechanism.Cooling cylindrical fin with natural convection is a good application of this heat transfer mechanism.The boundary layer over a hot horizontal cylinder start to develop at the bottom, increasing in thickness along the circumference, and forming a rising plum at the top, as shown in Figure (1).Therefore, the local Nusselt number is highest at the bottom and lowest at the top of cylinder when the boundary layer flow remains laminar (BEJAN, 2004).Most of problems arising in heat transfer area are nonlinear and through the majority of them only a limited numbers of them have exact analytical solution so these nonlinear equations should be solved using other methods.Other methods include numerical and semi exact methods, scientists believe that the combination of these two methods can be more cost effective method and also lead to useful results.
One of the semi-exact methods is the homotopy perturbation method (HPM), which is established by by He (HE, 1999).This method has been applied by many authors to solve a wide variety of scientific and engineering problems.Esmaeilpour and Ganji (2007), Ganji et al. (2007) andRajabi et al. (2007) use this method and other semi exact methods to solve nonlinear heat transfer problems.Ghasemi et al. (2007) solve a nonlinear and inhomogeneous twodimensial wave equation problem by HPM.It was shown by many authors such as Hoseinnia et al. (2008) and He (2005) that this method provides improvements over existing numerical techniques.In this paper, the mathematical model of this method is introduced and then its application in natural convection flow over a horizontal hot cylinder is studied.
The aim of this study is to consider the variation of temperature with the time in an isothermal horizontal cylinder that has been cooled with the natural convection of airflow.In recent years, much attention has been devoted to the newly developed methods to construct an analytic solution of some heat transfer equation; such methods include the HPM (ESMAEILPOUR; GANJI, 2007;RAJABI et al., 2007).
Therefore in the present work the influence of heat transfer coefficient, h, and specific heat coefficient, c, when they are variable with temperature or they are constant on isothermal cylinder and how long it takes to be cooled are studied, We find solution for these kinds of problems by HPM and compare it with numerical method (NM).

Analysis of the Homotopy Perturbation Method
The Homotopy perturbation method is a combination of the classical perturbation technique and Homotopy technique.To explain the basic idea of the HPM for solving nonlinear differential equations we consider the following nonlinear differential equation: Subject to boundary condition: where: A is a general differential operator, B a boundary operator, In case that the nonlinear Equation (1) has no 'small parameter', we can construct the following Homotopy: where, In Equation ( 7),   0 , 1 p  is an embedding parameter and 0 u is the first approximation that satisfies the boundary condition.We can assume that the solution of Equation ( 4) can be written as a power series in p, as following: And the best approximation for solution is: When, Equation (4) correspond to Equation ( 1) and ( 7) becomes the approximate solution of Equation (1).Some interesting results have been attained using this method.Convergence and stability of this method is shown in (GHASEMI et al., 2007).

Description of the problem
The aim of this study is to consider the temperature variation of a small hot isothermal horizontal cylinder in Figure 1.With diameter and length of 1 cm that is being cooled with natural convection of air flow.In this article, 3 cases have been investigated which are presented in Table 1.In each case parameters and the following equations have been introduced: In this case heat transfer coefficient, h, and specific heat coefficient, c, are variable with temperature.Equation ( 8) represents the heat equation of a lump system (INCCROPERA; DEWITT, 2002).
Which c is the quality of temperature dependency of specific heat on temperature.
The average Nusselt number over the entire surface can be determined from Churchill and Chu (1975) for an isothermal horizontal cylinder: where Rayleigh number and heat transfer coefficient are as follow: 3 ( ) From Equation ( 13), the variation of h against θ could be found as follow: Substituting Equation ( 13) and ( 9) in (8), we have: where: Solving this equation for a real condition with the air as a cooling flow so we reach to the following coefficients and equation: Case 2 Consider heat transfer in a lumped system, Equation ( 8), with constant h and variable c.The specific heat coefficient varies linearly with temperature as shown in Equation ( 9).
Substituting Equation (9) in Equation ( 8), we have: where: For a real condition with the air as a cooling flow, we reached the following coefficients: Case 3 In this case both h and c are constant.We solve this equation for a real condition with the air as a cooling flow.So, we reach the following coefficients and equation: -4 1.13097 10 hA   (30) Solution using Homotopy Perturbation Method Case 1 In this section, we will apply the HPM to nonlinear ordinary differential Equation ( 21).According to the HPM, we can construct a homotopy of Equation ( 21 Substituting Equation (33) into Equation (32) and collect ( , ) and then put the coefficients of p equal to zero, we have: By applying the HPM to nonlinear ordinary differential Equation( 28) According to the HPM, we can construct a homotopy of Equation ( 28 Substituting Equation (53) into Equation (52) and collect ( , ) H p  and then put the coefficients of p equal zero, we have: (54) Solving Equation (54-56) with initial conditions, we have: So we have:

Results and discussion
In Figure 2 the temperature distribution is compared with numerical solution.The heat transfer coefficient and heat specific is taken variable.It can be seen that there is good agreement between them.In Figure 4 the temperature distribution for case of constant heat transfer coefficient and variable specific heat is drawn.For this case the temperature gradient is depicted also.It is depicted in Figure 5.In Figure 6 the temperature distribution for the case of variable heat transfer coefficient and constant specific heat with both Numerical method and HPM is depicted.With growth of time the temperature of cylinder closes to ambient temperature.For this case the temperature gradient is depicted also in Figure 7.In Figures 8 and 9 the result of constant properties and variable properties is compared .itcan be seen in the case of variable properties cylinder reach the ambient temperature at a shorter time and temperature gradient tend to zero faster.
In   It can be seen in the case of variable properties cylinder reach the ambient temperature at a shorter time and temperature gradient tend to zero faster.
In Figure 12 average Nusselt number in each time is shown.At the beginning time because of high gradient temperature Nusselt number is great.With growth of time Nusselt number tend to zero, because of zero temperature gradient.
For case of constant heat transfer properties the Nusselt number is zero because it related to heat transfer coefficient and thermal conductivity and they are constant Figre 13.

Conclusion
In the present work, natural convection flow over a hot isothermal horizontal cylinder has been analyzed.The influence of variable h and c is studied, and the nonlinear equation that is extracted by He's Homotopy Perturbation Method (HPM) is solved.These considered equations are easily solved by mentioned analytical method.Consequently, these equations are solved by the numerical method (Runge-Kutta fourth-order) using the software Maple 12 ® and the results of the HPM and NM are compared in Figures 2,6,10,11,12 and 13.Then effects of h and c when they are variable or constant are shown in Figures 3,4,5,7,8 and 8. Also the variations of Nusselt number are shown in Figure 14 and 15.So the following results are obtained: (I) The effect of c is stronger than h when temperature is decreasing.
(II) With increasing h and decreasing c, the time of cooling approach will be decreased.
(III) The natural convection is not appropriate in industry that time is an important parameter.
(IV) Obtained results from case 1, 2 and 3 are approximately similar.
a known analytical function, Γ is the boundary of domain Ω and u n   denotes differentiation along the normal drawn outwards from Ω .The operator A can, generally speaking, be divided into two parts: a linear part L and a nonlinear part N. Equation (1) therefore can be rewritten as follows:

Figure
Figure 9. Compare ( ) , , , d t t h c cteand h c cte dt  

Table 1 .
Coefficients in different cases.