NUMERICAL METHODS FOR 2-D HEAT TRANSFER

KARTHIKA M202112010

CHEMICAL ENGINEERING19.04.2013

• Due to the increasing complexities encountered in the development of modern technology, analytical solutions usually are not available.

• For these problems, numerical solutions obtained using high-speed computer are very useful, especially when the geometry of the object of interest is irregular, or the boundary conditions are nonlinear.

Numerical methods are necessary to solve many practical problems in heat conduction that involve:

– complex 2D and 3D geometries– complex boundary conditions– variable properties

An appropriate numerical method can produce a useful approximate solution to the temperature field T(x,y,z,t); the method must be– sufficiently accurate– stable– computationally efficient

General Features

A numerical method involves a discretization process, where the solution domain is divided into subdomains and nodes

The PDE that describes heat conduction is replaced by a system of algebraic equations, one for each subdomain in terms of nodal temperatures

A solution to the system of algebraic equations almost always requires the use of a computer

As the number of nodes (or subdomains) increase, the numerical solution should approach the exact solution

Numerical methods introduce error and the possibility of solution instability

1. The Finite Difference Method (FDM)

– subdomains are rectangular and nodes form a regular grid network

– nodal values of temperature constitute the numerical solution; no interpolation functions are included

– discretization equations can be derived from Taylor series expansions or from a control volume approach

Types of Numerical Methods

2. The Finite Element Method (FEM)– subdomain may be any polygon shape, even with

curved sides; nodes can be placed anywhere in subdomain

– numerical solution is written as a finite series sum of interpolation functions, which may be linear, quadratic, cubic, etc.

– solution provides nodal temperatures and interpolation functions for each subdomain

• In heat transfer problems, the finite difference method is used more often and will be discussed here.

• The finite difference method involves:Establish nodal networksDerive finite difference approximations for the

governing equation at both interior and exterior nodal points

Develop a system of simultaneous algebraic nodal equations

Solve the system of equations using numerical schemes

The Nodal Networks

The basic idea is to subdivide the area of interest into sub-volumes with the distance between adjacent nodes by Dx and Dy as shown. If the distance between points is small enough, the differential equation can be approximated locally by a set of finite difference equations. Each node now represents a small region where the nodal temperature is a measure of the average temperature of the region.Example:

m,n

m,n+1

m,n-1

m+1, nm-1,n

Dy

Dx

m-½,nintermediate points

m+½,nx=mDx, y=nDy

Finite Difference Approximation

2

P

2

1Heat Diffusion Equation: ,

kwhere = is the thermal diffusivity

C

No generation and steady state: q=0 and 0, 0t

First, approximated the first order differentiation

at intermediate

q TT

k t

V

T

1, ,

( 1/ 2, ) ( 1/ 2, )

, 1,

( 1/ 2, ) ( 1/ 2, )

points (m+1/2,n) & (m-1/2,n)

T

x

T

x

m n m n

m n m n

m n m n

m n m n

T TT

x x

T TT

x x

Finite Difference Approximation (cont.)

21/ 2, 1/ 2,

2

,

21, 1, ,

2 2

,

2

Next, approximate the second order differentiation at m,n

/ /

2

( )

Similarly, the approximation can be applied to

the other dimension y

m n m n

m n

m n m n m n

m n

T x T xT

x x

T T TT

x x

T

, 1 , 1 ,2 2

,

2

( )m n m n m n

m n

T T T

y y

Finite Difference Approximation (cont.)

2 21, 1, , , 1 , 1 ,

2 2 2 2

,

2

2 2

( ) ( )

To model the steady state, no generation heat equation: 0

This approximation can be simplified by specify x= y

and the nodal

m n m n m n m n m n m n

m n

T T T T T TT T

x y x y

T

1, 1, , 1 , 1 ,

equation can be obtained as

4 0

This equation approximates the nodal temperature distribution based on

the heat equation. This approximation is improved when the distance

m n m n m n m n m nT T T T T

between the adjacent nodal points is decreased:

Since lim( 0) , lim( 0)T T T T

x yx x y y

A System of Algebraic Equations

• The nodal equations derived previously are valid for all interior

points satisfying the steady state, no generation heat equation.

For each node, there is one such equation.

For example: for nodal point m=3, n=4, the equation is

T2,4 + T4,4 + T3,3 + T3,5 - 4T3,4 =0

T3,4=(1/4)(T2,4 + T4,4 + T3,3 + T3,5)

• Derive one equation for each nodal point (including both

interior and exterior points) in the system of interest. The result

is a system of N algebraic equations for a total of N nodal points.

Matrix Form

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

The system of equations:

N N

N N

N N NN N N

a T a T a T C

a T a T a T C

a T a T a T C

A total of N algebraic equations for the N nodal points and the system can be expressed as a matrix formulation: [A][T]=[C]

11 12 1 1 1

21 22 2 2 2

1 2

= , ,

N

N

N N NN N N

a a a T C

a a a T Cwhere A T C

a a a T C

Numerical Solutions

Matrix form: [A][T]=[C].

From linear algebra: [A]-1[A][T]=[A]-1[C], [T]=[A]-1[C]

where [A]-1 is the inverse of matrix [A]. [T] is the solution vector.

• Matrix inversion requires cumbersome numerical computations and is not efficient if

the order of the matrix is high (>10).

• For high order matrix, iterative methods are usually more efficient. The famous

Jacobi & Gauss-Seidel iteration methods will be introduced in the following.

Iteration

1

1 1

31 1 32 2 33 3 1 1

1( ) ( ) ( 1)

1

General algebraic equation for nodal point:

,

(Example : , 3)

Rewrite the equation of the form:

i N

ij j ii i ij j ij j i

N N

iij ijk k ki

i j jj j iii ii ii

a T a T a T C

a T a T a T a T C i

a aCT T T

a a a

1

N

• (k) - specify the level of the iteration, (k-1) means the present level and (k) represents the new level.• An initial guess (k=0) is needed to start the iteration.• By substituting iterated values at (k-1) into the equation, the new values at iteration (k) can be estimated• The iteration will be stopped when maxTi

(k)-Ti(k-1), where specifies a

predetermined value of acceptable error

Replace (k) by (k-1)for the Jacobi iteration

CASE STUDY Finite Volume Method Analysis of Heat Transfer inMulti-Block Grid During SolidificationEliseu Monteiro1, Regina Almeida2 and Abel Rouboa31CITAB/UTAD - Engineering Department ofUniversity of Tr´as-os-Montes e Alto Douro, Vila Real2CIDMA/UA - Mathematical Department ofUniversity of Tr´as-os-Montes e Alto Douro, Vila Real3CITAB/UTAD - Department of Mechanical Engineering andApplied Mechanics of University of Pennsylvania, Philadelphia, PA1,2Portugal3USA

The governing differential equation for the solidification problem may be written in the following conservative form

∂ (ρCPφ)∂t = · (∇ k∇φ) + q˙ (1)

where ∂(ρCPφ) ∂t represents the transient contribution to the conservative energy equation (φ temperature); · (∇ k∇φ) is the diffusive contribution to the energy equation and q˙ represents the energy released during the phase change.

Boundary conditions

Numerical solution method

• Finite volume method

Finite difference method

Results and discussion

• Iterative performance of the three different numerical methods are also given.

Concluding remarksA multi-block grid generated by bilinear interpolation was successfully applied in

combination with a generalized curvilinear coordinates system to a complex

geometry in a casting solidification scenario. To model the phase change a

simplified two dimensional mathematical model was used based on the energy

differential equation. Two discretization methods: finite differences and finite

volume were applied in order to determine, by comparison with experimental

measurements, which works better in these conditions. For this reason a coarse grid

was used. A good agreement between both discretization methods was obtained with

a slight advantage for the finite volume method. This could be explained due to the

use of more information by the finite volume method to compute each temperature

value than the finite differences method. The multi-block grid in combination with a

generalized curvilinear coordinates system has considerably advantages such as:

• better capacity to describe the contours through a lesser number of elements, which considerably reduces the computational time;

• - any physical feature of the cast part or mold can be straightforwardly defined and obtained in a specific zone of the domain;

• - the difficulty of the several virtual interfaces created by the geometry division are easily overcome by the continuity condition

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