07-02-2012, 12:08 PM
Numerical Solution of radial heat conduction in an infinitely long cylinder
[attachment=17230]
INTRODUCTION
Conduction problems can be solved in various methods but in this paper we have solved a simple radial heat conduction problem and compared the results obtained in each method (FDM, FVM, FEM) for different meshes. We have also supported our solution with sensible reasons.
This problem explains the temperature distribution on the radial direction of the pipe. The practical application is in the electrical wires where heat is generated in the core of the wire and the same heat it needed to be dissipated to the surrounding so that the melting of the wire is avoided.
Here we are assuming the cylinder to be infinitely long, hence by having such a assumption we are neglecting any heat conduction in the axial direction of the pipe. Hence what we will be studying will only be the radial heat conduction in a cross-section of the pipe. We have assumed the cylinder to be open to the atmosphere with Temperature as T∞ and Convective heat transfer as h∞, hence we will apply symmetric boundary condition at the center (i.e. at r = 0) of the cylinder and Robbin’s boundary condition on the surface (i.e. at r = R).
THEORY:
Governing Differential Equation:
The general heat conduction problem in a 3 dimensional plane with generation and with time variation is given by the equation given below.
Governing Differential Equation:
K{dT/dx+dT/dy+dT/dz}+g=ρC dT/dt
The similar governing differential Equation for cylinder is:
K{1/r d/dr r dT/dr}+g=ρC dT/dt
Here since we have neglected the heat conduction in the axial direction the equation is only in one dimension.
The similar governing differential equation for sphere is given as:
K{1/r^2 d/dr r^2 dT/dr}+g=ρC dT/dt
For our study we will mainly focus on the GDE for cylinder.
Boundary Conditions:
Symmetric Boundary Condition
This treatment of the boundary condition corresponds to the physical assumption that, on the two sides of the boundary, the same physical processes exist. The variable values at the same distance from the boundary at the two sides are the same. The function of such a boundary is that of a mirror that can reflect all the fluctuations generated by the simulation region.
In our problem we have this boundary condition at the center i.e. at r=0.
Hence,
Boundary Condition 1:
At r = 0,
dT/dr=0
Robbin’s Boundary Condition:
This treatment of the boundary is done when the boundary is exposed or open to the atmosphere. Hence in this kind of boundary condition the conductive heat flux and convective heat flux are made equal. This kind of boundary condition is also called as mixed type boundary condition.
In our problem we will be applying this boundary condition at the lateral surface i.e. r = R of the cylinder since the boundary is open to atmosphere.
Hence,
Boundary Condition 2:
At r = R,
K dT/dr=h(T_s-T_∞ )
FDM:
Finite difference method is numerical method for approximating the solution of differential equation using finite difference equations to approximate derivatives.
According to our problem the FDM Equations are:
Assuming the radius is divided into n equals parts, hence we will have n+1 node.
m=n+1
[attachment=17230]
INTRODUCTION
Conduction problems can be solved in various methods but in this paper we have solved a simple radial heat conduction problem and compared the results obtained in each method (FDM, FVM, FEM) for different meshes. We have also supported our solution with sensible reasons.
This problem explains the temperature distribution on the radial direction of the pipe. The practical application is in the electrical wires where heat is generated in the core of the wire and the same heat it needed to be dissipated to the surrounding so that the melting of the wire is avoided.
Here we are assuming the cylinder to be infinitely long, hence by having such a assumption we are neglecting any heat conduction in the axial direction of the pipe. Hence what we will be studying will only be the radial heat conduction in a cross-section of the pipe. We have assumed the cylinder to be open to the atmosphere with Temperature as T∞ and Convective heat transfer as h∞, hence we will apply symmetric boundary condition at the center (i.e. at r = 0) of the cylinder and Robbin’s boundary condition on the surface (i.e. at r = R).
THEORY:
Governing Differential Equation:
The general heat conduction problem in a 3 dimensional plane with generation and with time variation is given by the equation given below.
Governing Differential Equation:
K{dT/dx+dT/dy+dT/dz}+g=ρC dT/dt
The similar governing differential Equation for cylinder is:
K{1/r d/dr r dT/dr}+g=ρC dT/dt
Here since we have neglected the heat conduction in the axial direction the equation is only in one dimension.
The similar governing differential equation for sphere is given as:
K{1/r^2 d/dr r^2 dT/dr}+g=ρC dT/dt
For our study we will mainly focus on the GDE for cylinder.
Boundary Conditions:
Symmetric Boundary Condition
This treatment of the boundary condition corresponds to the physical assumption that, on the two sides of the boundary, the same physical processes exist. The variable values at the same distance from the boundary at the two sides are the same. The function of such a boundary is that of a mirror that can reflect all the fluctuations generated by the simulation region.
In our problem we have this boundary condition at the center i.e. at r=0.
Hence,
Boundary Condition 1:
At r = 0,
dT/dr=0
Robbin’s Boundary Condition:
This treatment of the boundary is done when the boundary is exposed or open to the atmosphere. Hence in this kind of boundary condition the conductive heat flux and convective heat flux are made equal. This kind of boundary condition is also called as mixed type boundary condition.
In our problem we will be applying this boundary condition at the lateral surface i.e. r = R of the cylinder since the boundary is open to atmosphere.
Hence,
Boundary Condition 2:
At r = R,
K dT/dr=h(T_s-T_∞ )
FDM:
Finite difference method is numerical method for approximating the solution of differential equation using finite difference equations to approximate derivatives.
According to our problem the FDM Equations are:
Assuming the radius is divided into n equals parts, hence we will have n+1 node.
m=n+1