Draw the element with internal sections as follows,
Step 2 of 6
Express the finite difference equation at Node-1 (
) from equation (1).
Express the finite difference equation at Node-2 (
) from equation (1).
Express the finite difference equation at Node-3 (
) from equation (1).
Express the finite difference equation at Node-4 (
) from equation (1).
The bottom surface temperature is?
.
Step 3 of 6
Express the finite difference equation at Node-5 (
) from equation (1).
Interior node and the bottom surface temperature is?
.
Express the finite difference equation at Node-6 (
) from equation (1).
Express the finite difference equation at Node-7 (
) from equation (1).
Express the finite difference equation at Node-8 (
) from equation (1).
The bottom surface temperature is?
.
Step 4 of 6
Calculate value of time step as follows,
Calculate the upper limit of time step from the stability criteria as follows,
Substitute?
?for
,?
?for
, 0.015 m for?
, and?
?for?
.
Take the time step as 15 seconds.
Consider
, and?
.
Calculate?
?as follows,
Substitute?
?for
,?
?for?
, and 0.015 m for?
.
Step 5 of 6
Calculate the number of time steps (n) as follows
After 2 min
Substitute 2 min for total time, and 15 s for?
.
After 5 min
Substitute 5 min for total time, and 15 s for?
.
After 60 min
Substitute 60 min for total time, and 15 s for?
.
Step 6 of 6
Substitute 0.015 m for?
, and?
?0.2133 for?
,?
?for?
?, 15 s for?
,?
?for
, 0.015 m for?
,
?for?
,
for?
?and?
?for?
. In the 8 equations and solve them by EES software.
Therefore, The temperature at node-3 after 2 min, 5 min, 30 min are?
?,?
?and?
?.
