Question

Solve Matrix by hand. No matrix solvers

A two-dimensional corner of a brass plate has the following finite difference representation as shown. Compute the node temperatures (1, 2, 3, 1', and 2'). adli a batic 200 100 200 100 100 2' 100

Answer 1

This is the problem related to heat transfer through conduction process between different nodes.

As we know fourier's law of heat conduction Q= kA( T1-T2)/L

k is thermal conductivity of brass ( 109 w/mk) is constant for the whole surface.

A is the area through which heat has to be transferred. ( constant for all nodes)

L is the characteristics length (m)

Let us consider Node 1 first - Here temperatures 200 degree celcius and 100 degree celcius.

Q1= kA ( T1-T)....1

Q2= kA(T-T2).....2

Q1=Q2

T1-T = T-T2

200-T = T-100

2T= 300

T= 150 degree celcius

Temperature at node 1 = 150 degree celcius

Similarly at Node 2 and Node 1'= 150 degree celcius.

Heat transfer between node 2 and 3 is taken into consideration:

Q2=kA(T2-T3)/L .........3

Q3= kA (T3-T)/L..........4

Q2=Q3

T2-T3 = T3-T

150- T3= T3-100

2 T3= 250

T3= 125 degree celcius

Similarly T1'-T2' = T2'-T3

150-T2' = T2'-125

2T2' = 275

T2'= 275/2

T2'= 137.5 degree celcius