Computational Fluid Dynamics

Running head: COMPUTATIONAL FLUID DYNAMICS
COMPUTATIONAL FLUID DYNAMICS
Name of the Student
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Author Note

COMPUTATIONAL FLUID DYNAMICS2
Introduction:
The finite volume method is popularly used for solving heat conduction equation in 1
dimensional plane. Here, the general transport equation is discretized into several system of
linear equations. The linear equations are then solved using numerical methods or specifically
using Gauss-Siedel method and TDMA numerical method in MATLAB. The heat conduction
equation can be derived using the Energy balance equation in specific node. In the first
problem heat convection through a rod of 2 m is evaluated at different nodes where the end
temperatures are 200 °C and 600 °C respectively. In the second problem the heat convection
is evaluated for a plate of 2.5 cm, where, the two ends are maintained at temperatures 200 °C
and 600 °C. In the third problem the convection-diffusion problem is solved using Analytical
method and assuming suitable initial conditions for the property of interest. Generally, the
convection-diffusion has two variable the length x and time t and differential equation is
second order partial differential equation. However, in this case the problem is identified for
one variable, length x and the time variable t is assumed constant. Thus the convection-
diffusion equation becomes an ordinary differential equation of second order.
Question 1:
The 1D heat conduction equation is given by,
( d
dx )( k∗dT
dx )=0
Here, k is thermal conductivity.
T = local temperature
x = spatial co-ordinate
The dynamic temperature of the rod is found in 10 nodes.

COMPUTATIONAL FLUID DYNAMICS3
Rod length = 2 m, rod cross section A = 10^(-2) m^2.
Thermal conductivity k = 1500 W/Km.
Temperature at end A = 200 .°C
Temperature at end B = 600 °C.
The 1D heat equation is solved in MATLAB and the Output is shown below.
Plot:
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
L [m]
200
250
300
350
400
450
500
550
600
T [oC]
1D Conduction through the rod
Question 2:
Now, the heat equation has a source term q and the equation is given by,
( d
dx )( k∗dT
dx )+q=0

COMPUTATIONAL FLUID DYNAMICS4
Heat is conducted within a flat plate of thickness 2.5 cm in which the ends of the plate are at
200 and 300 respectively.°C °C
Heat source q= 1000 kW/m^3.
Thermal conductivity k = 1.5 W/Km.
The plot of dynamic heat flow through 10 nodes is shown below.
Plot:
0 0.005 0.01 0.015 0.02 0.025
L [m]
200
220
240
260
280
300
320
T [oC]
1D Conduction through plate
Question 3:
The 1D convection diffusion problem is given by,