Use Different Mathematical Methods to Solve Three Dimensional Conduction Heat Equation in Cartesian Coordinate

Abstract


I. INTRODUCTION
Heat transfer problem has big importance in many problems of environmental and industrial. Beforehand , in output energy and transformation applications [6]. The Heat transfer by conduction is mainly concerned with determining the temperature distribution inside the solids [3]. The problems of conduction heat transfer are faced in several engineering applications like the following : Design , Nuclear Reactor Core , Glaciology , Re-entry Shield , Cryosurgery , Rocket Nozzle , Casting , Food Processing [2]. Generally , a heat conduction equation is solved in many cases such as : Quadruple Laplace transform used by Hamood Ur Rehman, Muzammal Iftikhar, Shoaib Saleem, Muhammad Younis and Abdul Mueed to solve heat equation in cartesian coordinate [1].Separation of Variables used to solve conduction heat equation in cartesian coordinate [5] .Xiao-Jun Yang used a new integral transform operator to solve one dimensional heat-diffusion equation in Cartesian coordinate [7].Ranjit R. Dhunde and G.L.Waghmare used double Laplace transform to solve one dimensional heat equation [4]. Xiao-Jun Yang used a new integral transforms to solve a steady heat transfer problem [8].The aim of this paper is use quadruple Laplace transform method and separation of variables method to solve three dimensional heat conduction equation in cartesian coordinate and compare the solution that will be reached in each method and determine the best method .

II. The model and mathematical methods to solve conduction heat equation:
The model is consist of the following equation with initial and boundary conditions [2]: ρ : density of fluid : fluid specific heat k : thermal conductivity of fluid with boundary and initial conditions: We will use the following methods to solve equation (1) : (a) Using separation of variables [5]: The basic idea of the method of separation of variables is

By equal the above equation to :
( ) ( ) After simplify we get : The solution of (3),(5),(6) and (7) are : Substitution above solutions in (2) we get : And we find that : Then the final solution is : Which is solution of equation (1) . (b) Using Quadruple Laplace transform [1]: Quadruple Laplace transform is defined as : By take to both sides of equation (1) : We get : ) Applying initial and boundary condition to above equation we get : ( ) Then :

( )
Which is also solution of equation (1) .

III. Results and Data:
By using Matlab we get the following illustrations which explain the solution of equation (1) : (1-1) Figure shows distribution of temperature by using Separation of Variables method for ( ) , x=1:10 , y=1:10 , z=1 , t=0 .

IIII. Conclusions:
From the illustrations we note the following conclusions : 1-The difference in the solution methodology for the separation of variables method and for the quadruple Laplace transform method leads to a difference in the result of the solution, as shown in the illustrations (1-1),(1-2),(1-3)and(1-4). 2-We note that the temperature distribution in the method of separating the variables is in the form of a dome , also we notice an increase in temperatures as the value of z increases in the same method as shown in the illustrations(1-1)and(1-2) . 3-We note that the temperature distribution in the quadruple Laplace transform method is in an inclined plane , as we notice an increase in temperatures as the value of z increases in the same method as shown in the illustrations(1-3)and(1-4) . 4-We note that quadruple Laplace transform method is better because it reached to the solution by fewer steps compared with separation the variables method .