الرحيم الرحمن الله بسمSUDAN UNIVERISTY OF SCIENCE AND TECHNOLOGY

COLLEGE OF POST GRADUATE STUDIESM.SC. PROGRAMME IN POWER ENGINEERING

SIMINER:(2 DIMENSION CONDUCTION )

BY:

JASSIM SALAH MOHAMMED HAMMAD

Outlines:Introduction.Discretization.Domain DefinitionSolution Algorithm Results and Discussion :

The effect of the Mesh The effect of the boundary conditionThe effect Error T he effect of the Coefficient of conductivity Sensitivity

Introduction.

Computational fluid dynamics (CFD) : The science of predicting fluid flow, mass transfer, heat transfer, chemical reactions, and related phenomena by solving the mathematical equations which govern these processes using a numerical process .

The Heat Diffusion Equation:

Energy storage :Energy generation:

Conduction heat rate in the surface:Conduction heat rate out the volume

Equation of energy balance:

Discretization.

Discretization :

It is a method to translate Partial differential equations (PDE) to Algebraic

equations by integration.

Methods:

1- Finite Volume Method (FVM).

2-Finite Element Method (FEM).

3-Finite Difference Method (FDM).

4-Spectral methods.

5-Boundary Elements Methods…

Typical control volume

Integrating over CV (Double integration):

For (X) Term:

Equidistence Mesh:

Same for (Y) direction:

yields:

Algebraic form :

Comparing :

Domain Definition.

.conductivity (k) = 17.5 Length (L) = 4.5

Height (H) = 1.5

Source term (S)

Boundary (4) Boundary (3) Boundary (2) Boundary (1) CASE .NO

10 20c dt/dy=0 40c 50c 11

Solution Algorithm

Solution Algorithm

1. Input Data.2. Mesh dimensions with Source

calculation.3. Initial temperature and error guesses.4. Coefficients calculation.5. Temperature calculation. 6. Check e=( abs(ap*(Told-T)))/F.7. Repeat from 5 to 6.8. Results.

13/10/201217 Task1 2D conduction

Set Solution Parameters

Initialize

Enable Solution Monitors

Calculate a Solution

Modify the Solution Parameters or the MeshConverged?

No

No

Yes

Yes

Accurate?

Stop

Results and Discussion

The effect of the boundary condition

Explicit

The effect of the Mesh

E=0.1 , Mesh (10*10 ) ,K=17.5

E=0.1 , Mesh (20*20 ) ,K=17.5

E=0.1 , Mesh (40*40 ) ,K=17.5

E=0.1 , Mesh (10*10 ) ,K=100K

E=0.1 , Mesh (10*10 ) ,K=K/100

The effect Error

k

17.5

er

0.1

0.001

Re

T

N E T N E*e-4

10*10

0.0156

47

0.0939 0.0156

98

9.6123

20*20

0.0936

178

0.0988

0.1716

380

9.905

40*40

1.1544

716

0.0998

2.34 1520

9.9465

m

T he effect of the Coefficient of conductivity

er 0.1

K 100K K/100

Re T N E T N E

10*10 .0156 47 .0924 .0156 50 .0945

20*20 .078 175 .0988 .0936 194 .0986

40*40 1.076 682 .0988 1.295 768 .0994

m

error

0.001

K

100K K/100

Re T N E*e-4

T N E*e-4

10*10

0.0156

98

9.4662

0.0156

101

9.9642

20*20

0.1716 377

9.888

0.0156

396

9.8693

40*40

2.262 1585

9.952

2.574

1571

9.9907

m

T=time required m=iteration

k=conductivity factorE= actual error

Implicit

The effect of the Mesh

E=0.1 , Mesh (10*10 ) ,K=17.5

E=0.1 , Mesh (20*20) ,K=17.5

E=0.1 , Mesh (40*40) ,K=17.5

E=0.1 , Mesh (10*10) ,K=100K

E=0.1 , Mesh (10*10) ,K=K/100

The effect Error

k

17.5

er

0.1

0.001

Re

T

N E T N E*e-4

10*10

.0156

47

.0939 .0156

98

9.612

20*20

.0936

178

.0988

.1716 380

9.905

40*40

1.1544

716

.0998

2.2776

1520 9.9465

m

T he effect of the Coefficient of conductivity

error 0.1

K 100K K/100

Re T N E T N E

10*10 .0156 47 .0924 .0156 50 .0945

20*20 .078 175 .0988 .0936 194 .0981

40*40 1.0608 682 .0998 1.2324 768 .0994

m

error

0.001

k

100K K/100

Re

T

N E*e-4 T N E*e-4

10*10

.0156 98 9.4662 .0156 101 9.6942

20*20

.156 377 9.888 .1872 396 9.8693

40*40

2.1372 1485 9.9521 2.280 1571 9.9907

m