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‫ دﮐﺘﺮ اﻓﺴﺎﻧﻪ ﻣﺠﺮي‬:‫اﺳﺘﺎد درس‬

(‫ )روش اﺧﺘﻼف ﻣﺤﺪود‬1 ‫ﺗﻤﺮﯾﻦ درس دﯾﻨﺎﻣﯿﮏ ﺳﯿﺎﻻت ﻣﺤﺎﺳﺒﺎﺗﯽ‬

1. Solve the boundary-value problem +3

(0) = 1,

+8 ( )=0 (1) = 2

at 9 interior points. (Using n = 10 and therefore ∆t = 0.1) * MATLAB code

2. Consider a large plane wall of thickness L=0.2 m, thermal conductivity k=1.2 W/m·°C, and surface area A= 15 m2. The two sides of the wall are maintained at constant temperatures of T1=120°C and T2=50°C. Determine the variation of

temperature within the wall and the value of temperature at x = 0.1 m under steady conditions.

* Check the effect of reducing ∆x by developing a code.

3. Consider a large Uranium plate of thickness L=4 cm and thermal conductivity k=28

W/m·°C in which heat is generated uniformly at a constant rate of g·= 5 * 106 W/m3. One side of the plate is maintained at 0°C by iced water while the other side is

subjected to convection to an environment at T∞=30°C with a heat transfer coefficient of h= 45 W/m2 · °C. Considering a total of three equally spaced nodes in

the medium, two at the boundaries and one at the middle, estimate the exposed

surface temperature of the plate under steady conditions using the finite difference approach.

4. Consider the heat equation

1

‫ دﮐﺘﺮ اﻓﺴﺎﻧﻪ ﻣﺠﺮي‬:‫اﺳﺘﺎد درس‬

(‫ )روش اﺧﺘﻼف ﻣﺤﺪود‬1 ‫ﺗﻤﺮﯾﻦ درس دﯾﻨﺎﻣﯿﮏ ﺳﯿﺎﻻت ﻣﺤﺎﺳﺒﺎﺗﯽ‬

=

,

0
0

on an interval of length l, with constant thermal diffusivity dependent Dirichlet boundary conditions: ( , 0) = ( ),

( , ) = ( ),

> 0. We impose time-

>0

fixing the temperature at the ends of the interval, along with the initial conditions: (0, ) = ( ),

0≤

≤

specifying the initial temperature distribution. Fix the diffusivity

= 1 and the interval

length l = 1. Take a spatial step size of ∆x = 0.1 and ∆ = (∆ ) . Initial condition is expressed as follows:

⎧ − , ⎪ 2 (0, ) = ( ) = − , 5 ⎨ ⎪ ⎩1− ,

Extract the algebric set of equations.

2

0≤

≤

1 5

1 7 ≤ ≤ 5 10 7 ≤ ≤1 10