User:Egm6936.s09/A thermal problem

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The transient heat transfer[edit]

The dimensional thermal problem[edit]

The governing equation has the following form:

(1)

where are material thermal conductivity, specific heat and density proportionally.

The initial condition:

(2)

The boundary conditions:

(3)

Using the Galerkin's Finite Element Method (FEM) as appendix 1 for a transient heat transfer problem, we obtain the global equations:

(4)

where vector is temperature at n nodes, that obtains by solving the original differential equation (4) with the initial condition (2).

We will demonstrate the above procedure through an example.

Example 1: Solving the 1D following heat equation in a cylindrical rod having diameter d and length L:

with material parameters

and geometric parameters

Heat generation:

Initial condition:

The boundary condition:

Solution

Dividing the rod into four elements, each has length . Thus, we have five nodes with five freedom degrees:

FEM equations for each element are given as appendix 2

Subsituting numerical values into the global equation (4) we have:

To obtain solutions for first-order ordinary differential equations, we use the finite difference method or specifically the forward difference method. This method approximate the time derivative of the nodal temperature matrix as:

Substituting the above equation, the global equation becomes

In this example, this equation yields two cases:

a) From : heat generation is non-zero

b) From : heat generation is zero

Therefore, nodal temperatures are:

With , we plot values versus .

DimenT.jpg

The nondimensional thermal problem[edit]

Now we nondimensionalize the heat transfer equation (1). For this equation, there are four fundamental units and ten physical quantities which are listed below together with their corresponding dimensional formula for the unit in terms of the fundamental units:

Table 1: Heat transfer equation: Physical quantities
Physical quantity Symbol Dimensional formula
Initial temperature
Temperature at time t
The spatial coordinate
The length of rod
Time
the last time
The heat conductivity
The specific heat and mass density
The heat flux
The heat source

Thus, the dimension matrix is:

It can find the nullspace of matrix :

From the above result, we have six independent dimensionless groups of the original variables.

Alternatively we have 6 nondimensional variables which are provided as below:

In terms of these nondimensional variables the simplified thermal problem can be written in the following nondimensional form (the details of nondimensionalizing the heat problem are found in appendix 3):

Heat conduction equation:

Initial condition:

Boundary condition:

where

The complete set of nondimensional variables needed for the problem is given in equations (5)-(10). The nondimensional temperature can be expressed function of the nondimensional distance and the nondimensional time as well as function of four other nondimensional parameters.

The physical interpretation of the remaining four nondimensional parameters in equations (8)-(10) is the following. , the Fourier number or a nondimensional thermal diffusivity, is the ratio of the rate of heat conduction and the rate of heat storage(thermal energy storage). is the ratio of the incident heat flux and the rate of heat conduction, or can be seen as effective heat flux. Finally, is the ratio of the heat source and the rate of heat storage, can interpreted as effective heat source.

Example 2: Solving the problem in example 1 with the nondimensional form. Using Galerkin's FEM, similarly to dimensional form, we have the desired FEM equation for one element:

where

Using assembly procedure to obtain the global equations:

Dividing the rod into four element, each element has length , we have 5 nodes with 5 freedom degree:

where

Substituting the values of parameters, it can easily calculate conductance and capacitance matrix, forcing function vectors as

The graph of nondimensional temperature versus nondimensional time is below:

NondimenT.jpg

Example 3: Now we want to generate a graph that will give us the maximum value of the temperature in a rod as the example 1(max temperature over both time and space) for any value of . We do this this by solving the non-dimensional equation (11)-(13) for a range of these values. It can easily find that the maximum value of the nondimensional temperature depends on three independent nondimensional variables:

.

From equation (15), we can note that the maximum temperature will be linear in so that we do not need this as a parameter. To prove this conclusion, we scale the solution by changing variable:

At that time, the global equation (15) becomes:

or

Thus, dividing both sides of above equation to , we get

where is determined as equations (16),(17) and with the initial condition

Summing up, the maximum nondimensional temperature only depends on two independent nondimensional variables The graph of for a range of these two parameters is below: Ximax.jpg

Appendix[edit]

Appendix 1: Finite element procedures for a thermal problem[edit]

Here the Galerkin's Finite Element Method (FEM) is applied for the heat transfer equation to obtain the element equations. A two-node element with linear interpolation functions is used and the temperature distribution in an element expressed as

(20)

where and are the temperature at nodes 1 and 2, which define the element, and the interpolation functions and are given by ,

Substitution of the discretized solution into the governing differential Equation results in the residual intergrals:

(21)

where we note that the integration is over the volume of the element, that is, the domain of the problem, with

or

(22)

Intergratig the first term by parts and rearranging,

(23)

Substituting for from (20) yields

(24)

The two equations represented by equation (24) are conveniently combined into a matrix form by rewriting as

(25)

and substituting to obtain

(26)

Equation (26) is in the desired finite element form:

(27)

where is the conductance matrix defined as

(28)

and is the element capacitance matrix defined by

(29)

The forcing function vectors on the right-hand side of equation (27) include the internal heat generation and boundary flux terms. These are given by

(30)
(31)

Using assembly procedure for a transient heat transfer problem, we obtain the global equations:

(32)

where vector is temperature at n nodes, that obtains by solving the differential equation (32) with the initial condition (2).

Appendix 2: Finite element implementation of a dimensional problem[edit]

FEM equations for each element are given as (27)

Element 1:

or

Element 2:

Element 3:

Element 4:

The global equation (4) on the system assembly is:

where

Appendix 3: Nondimensionalize the thermal problem[edit]

To clarify the nondimensionalizing process, we change variables:

Therefore:

Substituting the above quantities into the heat transfer problem (1):

or

and the final form is:

where

The initial condition:

From

we have boundary conditions:

Appendix 4: Analytical solution of a heat transfer problem[edit]