Develop a Matlab program to implement Liebmann\'s method for a rectangular plate

Question

Develop a Matlab program to implement Liebmann's method for a rectangular plate with Dirichlet boundary conditions to solve temperature distribution of the heated plate. The side walls are maintained at the constant given temperatures. Employ overrelaxation with a value of 1.5 for the weighting factor and iterate to Es = 1%. Based on calculated temperature distribution calculate the heat flux distribution for the heated plate. The aluminum plate has the size 40x40cm and the conductivity K = 0.49 cal/(s*cm*deg.C). Display your results in a nice and tidy way.

Explanation / Answer

The lower edge of the plate is insulated. This means the flux of heat is zero at that edge.
This is a condition of the rate of c hange of the dependen t v ariable T at those p oin ts. This is called Nuemann b oundary conditions.This means we now have an additional 3 grid points that we do not know the tempurature at
those are the points T1; T2; T3; T4.
The tempreature at each one of those edge points is found using the method of images.
To allow ending T at the insulated edge, and to b e able to use the 4-p oin t di erence equation, one of the 4 points must then be outside the physical plate boundaries. Such a point is interoduced momenterraly , then solved for in terms of the inside point, resulting in a new 4-points dierecne equation that can be used only to evaluate the T for those points on this special edge.

Looking at the diagram, this is done as follows. Assume we want to end the 4-point dierence equation for the point T2 , then as normally , it is T2 = T4 + T3 + T1 + Tx/4 , where T x is the imaginary outside point. Now the ux at the point T2 in the y-direction is defined as T1.

so,
dt/dy = T1 - Tx/2h
Hence solve for Tx results in
Tx - T1-2h*dt/dy

For insulated edge, the flux at the point is zero. This means no change of T emreature at the
dt/dy = 0
Tx = T1
So,
now looking back at the original 4-point dierence equation we get
T2 = T4+T3+T1+T1/4
T2 = T4+T3+2T1/4
This means that for the insulated edge, use the above modified dierence equation to solve for T at each point on that edge.

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.