Application of Homotopy Perturbation Method (HPM) for Nonlinear Heat Conduction Equation in Cylindrical Coordinates Milad Boostani * - Sadra Azizi - Hajir Karimi Department of Chemical Engineering, Yasouj University, Yasuj, Iran ABSTRACT In this paper, heat equation in cylindrical coordinates is solved by using the homotopy perturbation method (HPM) in two cases, uniform and non-uniform heat generation. In addition, thermal conductivity is temperaturedependent which makes nonlinear terms in the heat equation. The comparison of obtained results from HPM solution with exact solution indicated that the maximum and the minimum accuracy of HPM, observe on the surface and in axis of cylinder, respectively. Furthermore, the effect of the various parameters on temperature profile is investigated. It is observed that the HPM results have good agreement with the exact solution. Keywords: Homotopy perturbation method, Cylindrical Coordinates, Nonlinear heat equation.. Introduction In general, heat transfer phenomenon in cylinders with heat generation occurs in many processes such as cooling cylindrical catalyst with chemical reaction, cylindrical nuclear reactors, tubular chemical reactors, combustion chambers and all type of the wire elements passing electric current. In many materials, thermal conductivity varies with temperature. This dependency changes heat conduction equation into nonlinear equation. Nonlinear differential equations are usually solved with numerical methods while some of them can be solved by analytical methods such as the homotopy perturbation method (HPM). Generally, this procedure is applied for both linear and nonlinear differential equations []. By the HPM, nonlinear equations are changed into an unlimited number of simple equations without applying the perturbation technique []. This approach is proper for cylindrical and spherical coordinates. Compared to amount of works on the HPM in the Cartesian coordinates, a limited number of studies on other coordinates exist. Several investigators used the HPM as a convenient technique to solve the nonlinear heat equations. Ganji [3] applied the HPM to nonlinear heat equations and showed that the HPM is reliable procedure for all of the nonlinear differential equations with high order of nonlinearity. Rajabi et al. [4] employed this method for the temperature distribution in lumped system of combined convection radiation and also a nonlinear equation of the steady conduction in a slab. The HPM results were the same as the perturbation method (PM) results. Ganji et al. [6] obtained temperature distribution of a straight fin with a temperature-dependent thermal conductivity by using the HPM. Arslanturk [7] studied the radiating fins with respect to thermal analysis by the HPM and compared the obtained results with numerical solutions and indicated that the maximum difference between the HPM and the numerical results is.5%. Sheikholeslami et al. [] applied this method for three-dimensional problem of condensation film on inclined rotating disk and the obtained results indicated that this method is a powerful procedure for solving the governing nonlinear differential equations. The HPM was also studied to solve the nonlinear differential equations in cylindrical coordinates by some researcher. Fereidoon et al. [9] employed the HPM and the Adomian decomposition method (ADM) to investigate the distributions of stresses and radial displacement at the thick-wall cylinder. They found that the rate of convergence in the HPM is higher than the ADM and the HPM is simple for solving the governing equations. Moghimi et al. [] investigated the HPM for MHD Jeffery Hamel problem in cylindrical polar coordinates and indicated that this method is in excellent agreement with numerical solution. In this study, we investigate the heat equation in cylindrical coordinates using the homotopy perturbation method in cases of uniform and non-uniform heat generation.. Homotopy Perturbation Method [4] Consider the following nonlinear differential equation,, A u f r r () www.iccce.ir
With following boundary conditions B u, u / n, r () where A is a general differential operator, B is a boundary operator, f(r) is a known analytic function, and Γ is the boundary of the domain Ω. The operator A may be separated into two parts L and N, where L is linear and N is nonlinear, therefore, Eq. can be rewritten as follows, L u N u f r (3) The homotopy can be constructed as H v, p p L v L u p A v f r Or (4) H v, p L v L u p L u p N v f r (5) where p is called homotopy parameter. The solution of Eq. 5 may be expressed as v v pv p v (6) By setting p = eventuates in the approximate solution of Eq. u limv v v v (7) p 3. The application of HPM in heat transfer equation There, we consider the heat conduction equation in cylindrical coordinates for the cases of uniform and nonuniform heat generation. 3. Uniform heat generation (Case ) In this case, we consider equation of heat conduction for a long cylinder with uniform heat generation. By assuming that the temperature distribution is negligible in the axial direction T k T r q r r r (8) In whitch k is thermal conductivity and q is heat source. Boundary conditions of the governing equation can be expressed as T at r r T T at r R () Where T is ambient temperature and R is cylinder radius. The thermal conductivity is supposed to be a linear function of temperature as k T k T () Where β is a constant. Since k(t) is dependent on temperature, so the Eq. 8 is nonlinear. In order to simplify this equation, the following dimensionless variables are defined T r q R,, T, T R k T In which θ and ξ are dimensionless temperature and dimensionless radius, respectively. Also δ and Φ are two dimensionless constants. Rewriting Eq. 8 in terms of the dimensionless variables and according to the boundary conditions (9) www.iccce.ir
d d d d d d d d d d () d d at (3) at (4) d d d d d d d d d d. Where L, N and f r 3... Homotopy perturbation method for case : Letting d / d from the Eq. 3, with C, then we get d at and C at d (5) Substituting Eq. 6 into Eq. 5, we get d v pv d v pv H v, p d d d v pv d v pv v pv d v pv p v pv d d d (6) Equating the same powers of p, we get d v dv d d p : dv d at and v C at (7b) and, d v dv d v dv dv p : v v (8a) d d d d d dv at and v at d (8b) and so on. We can now solve these equations for v, v, v etc. For five terms of series when p,we get v v v v v (9) 3 4 Hence 3 3 3C 4 C C C C 4 3 3 Constant C represents temperature at the axis of cylinder and can be evaluated from the second boundary condition given in Eq. 4. (7a) () www.iccce.ir 3
3. Non-uniform heat generation (Case ) This time we consider the previous problem under a condition the heat generation is to be function of cylinder radius T ' r k T r q r r r R Boundary conditions are similar to the first case. The following dimensionless variables are defined ' T r qr,, T, T R k T Hence d d d d d d d d d d () d d at (3) at (4) 3... Homotopy perturbation method for case : Applying a procedure similar to case for Eq., we get d v dv d d p : dv d at and v C at (5b) and, d v dv d v dv dv p : v v (6a) d d d d d dv at and v at d (6b) and so on. Finally, with continuing this procedure, temperature distribution for this case with five terms of series is obtained as follows () (5a) C C 3 C C 4 3C C C C 3C 3C 6 64 5 3 4 6 8 (7) 4. Results and discussion The results of the HPM are compared with the exact solution for cases and in Tables and, respectively. These tables also present relative errors for both cases. As can be seen from these Tables, accuracy of case is higher than case and maximum errors occur in the axis (ξ = ) of cylinder. www.iccce.ir 4
Table. Comparison of exact and HPM solution for case (δ =., Φ =.3) ξ HPM Exact Relative Error.898744.9888.9758 e-4..879776.89968.95349 e-4..884.8438.968 e-4.3.779.74648.844 e-4.4.59549.6385.769 e-4.5.444.43995.54879 e-4.6.5375.96.3485 e-4.7.968554.97966. e-4.8.68377.69755 7.9784 e-5.9.3688.3653 4.3347 e-5.. Table. Comparison of exact and HPM solution for case (σ =., Φ =.3) ξ HPM Exact Relative Error.34564.439954 9.976 e-4..37448.47958 9.4 e-4..7355.3647 8.9398 e-4.3.8567.763 8.4568 e-4.4.66497.45493 7.886 e-4.5.976.994546 6.95893 e-4.6.7578.85478 5.9846 e-4.7.567584.649963 4.75 e-4.8.374994.4869 3.3399 e-4.9.8566.9.74 e-4.. Also, the effect of various parameters on the temperature distribution is represented in Figs. and. Fig. shows effect of parameters δ and Φ on the temperature distribution for case, and Fig. illustrates effect of σ and Φ on the temperature distribution for case. As can be seen maximum temperature occurs in the axis of cylinder due to existence of energy generation. Since δ and σ are proportional to energy generation term, with increasing these parameters, temperature increases. It can be also seen that with increasing Φ, temperature decreases. The results for different steps of HPM are shown in Fig. 3. As is obvious from this figure, with increasing the number of steps, error of the HPM tends towards zero. www.iccce.ir 5
(a) (b) Fig.. Effect of varying (a) δ when Φ =.3 and (b) Φ when δ =.5 on the temperature distribution for case. (a) (b) Fig.. Effect of varying (a) σ when Φ =.3 and (b) Φ when δ =.5 on the temperature distribution for case. Fig. 3. Effect of increasing the number of HPM steps on the accuracy of results (for case ). www.iccce.ir 6
5. Conclusions The heat conduction equation in cylindrical coordinates with the heat generation term was solved using the homotopy perturbation method (HPM). The comparison of the obtained results with the exact solution indicated that the HPM is an appropriate tool for finding the approximate analytical solutions of nonlinear problems. It was found that the HPM for the uniform case has higher accuracy than the non-uniform case. Also, the obtained results indicated that with increasing σ and δ, temperature in the axis of cylinder increases and that decreases with increasing Φ. The maximum and the minimum accuracy occurred on the surface and in the axis of cylinder, respectively. REFERENCES [] Sharma, P.R., Methi, G., (). Solution of coupled nonlinear partial differential equations using homotopy perturbation method, Int. J. Appl. Math. Mech. vol. 6, no., p. 33-49. [] Hosseinnia, S.H., Ranjbar, A., Ganji, D.D., Soltani, H., Ghasemi, J., (9). Homotopy perturbation based linearization of nonlinear heat transfer dynamic, Appl. Math. Comput. vol. 9, p. 63-76. [3] Ganji, D.D., (6). The application of He s homotopy perturbation method to nonlinear equations arising in heat transfer, Phys. Lett. vol. 355, p. 337-34. [4] Rajabi, Ganji, D.D., Taherian, H., (7). Application of homotopy perturbation method in nonlinear heat conduction and convection equations, Phys. Lett., vol. 36, p. 57-573. [5] Chowdhury, M.S.H., Hashim, I., Abdulaziz, O., (9). Comparison of homotopy analysis method and homotopyperturbation method for purely nonlinear fin-type problems, Commun. Nonlinear. Sci. vol. 4, p. 37-378. [6] Ganji, D.D., Afrouzi, G.A., Talarposhti, R.A., (7). Application of variational iteration method and homotopy perturbation method for nonlinear heat diffusion and heat transfer equations, Phys. Lett. vol. 368, p. 45-457. [7] Arslanturk, C., (). Performance analysis and optimization of radiating fins with a step change in thickness and variable thermal conductivity by homotopy perturbation method, Heat. Mass. Transfer. vol. 47, p. 3-38. [8] Siddiqui, A.M., Zeb b, A., Ghori, Q.K., Benharbit, A.M., (8). Homotopy perturbation method for heat transfer flow of a third grade fluid between parallel plates, Chaos. Soliton. Fract. vol. 36, p. 8-9. [9] Esmaeilpour, M., Ganji, D.D., (7). Application of He s homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate, Phys. Lett. vol. 37, p. 33-38. [] Sheikholeslami, M., Ganji, D.D., (3). Heat transfer of Cu-water nanofluid flow between parallel plates, Powder. Technol. vol. 35, p. 873-879. [] Domairry, G. Nadim, N., (8). Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation, Int. Commun. Heat. Mass. Transfer. vol. 35, p. 93-. [] Sheikholeslami, M., Ashorynejad, H.R., Ganji, D.D., Yıldırım, A., (). Homotopy perturbation method for three-dimensional problem of condensation film on inclined rotating disk, Scientia. Iranica. vol. 9, no. 3, p. 437-44. [3] Ganji, D.D., Hosseini, M.J., Shayegh, J., (7). Some nonlinear heat transfer equations solved by three approximate methods, Int. Commun. Heat. Mass. Transfer. vol. 34, p. 3-6. [4] He, J.H., (999). Homotopy perturbation technique, Comput. Meth. Appl. Mech. Eng. vol. 78, p. 57-6. [5] He, J.H, (3). Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput. vol. 35, p. 73-79. [6] Barari, T., Barari, A., Momeni, M., Vahdatirad, M.J., () Numerical analysis of coupled system of nonlinear diffusion-reaction problem, Int. Journal. Appl. Math. Comput. vol., no. 4, p. 55-6. [7] Abdous, F., Ghaderi, R., Ranjbar, A., Hosseinnia, S.H., Soltani, H., Ghasemi, J., (9). Asymptotical time response of time varying state space dynamic using homotopy perturbation method, J. Appl. Math. Comput. vol. 3, p. 5-63. [8] Zhou, S., Wu, H., (). Analytical solutions of nonlinear Poisson Boltzmann equation for colloidal particles immersed in a general electrolyte solution by homotopy perturbation technique, Colloid. Polym. Sci. 9 65 8. [9] Fereidoon, A., Rostamiyan, Y., Davoudabadi, M.R., Farahani, S.D., Ganji, D.D., (). Analytic approach to investigation of distributions of stresses and radial displacement at the thick-wall cylinder under the internal and external pressures, Middle-East. J. Sci. Res. vol. 5, no. 5, p. 3-38. [] Moghimi, S.M., Ganji, D.D., Bararnia, H., Hosseini, M., Jalaal, M., (). Homotopy perturbation method for nonlinear MHD Jeffery Hamel problem, Comput. Math. Appl. vol. 6, p. 3-6. www.iccce.ir 7