(Original post by Maths boy)
Can someone help me with this maths problem or know anyone that can. thank you :-)
In steady state heat transfer the temperature T satisfies the Laplace equation. In two dimensions with Cartesian coordinates x and y this equation is d2T/dx2 +d2T/dY2=0
This is a partial differential equation, and general methods of solving it are beyond the scope of this module - wait for the second and final year! However there are ways of getting an approximate solution to the equation by realising that it is a mathematical statement of the law of conservation of energy - in this case heat energy. One such method is the finite difference method.
One way to see how such methods work, is to consider a square area ABCD of a heat-conducting plate:
The values T0, T1, T2, T3, and T4 represent respectively the temperatures at the central point (or node) and the four surrounding equally distant nodes. T0 is at the centre of the sqaure, T1 goes through the middle of the points AD and conects to T0 on the sqaure. T3 goes through the middle ofthe points CB and connects to T0. T2 goes through the middle of the points DC and connects to T0 and T4 goes through the middle of the points AB and connects to T0. For the sake of simplicity suppose the length of side of the square is h, and that each of the temperature nodes are a distance h apart.
The law of conservation of energy in this case says that the net heat flowing into ABCD must equal the net heat flowing out. The heat flowing across a boundary of ABCD is proportional to the temperature gradient and the length of that boundary. Thus conservation of energy says
Heat in + Heat in = Heat out + Heat out
across AC across CD across AB across BD
Approximately this gives
(T0-T1/h)h +(T0-T4/h)h = (T2-T0/h)h+(T3-T0/h)h
or rearranging T0=1/4(T1+T2+T3+T4) .
ie temperature at one node = average of temperatures at four surrounding nodes.
Why is this approximate, and how can it be made more accurate? Write some words on this, and to see if you’ve really understood, do the following:
Work out the equation when ABCD is a rectangle but not necessarily a square, with the lengths
AB = CD = distance from T1 to T0 = distance from T0 to T3 = h
and the lengths
AC = BD = distance from T4 to T0 = distance from T0 to T2 = k
A simple case
A flat plate as shown in Figure 2 has the temperature on three edges held at T = 0; the left hand side and the top and bottom the fourth edge is held at a temperature of T = 100, which is on the right hand side. T1 and T2 are on the central nodes of the grid.
An approximate value of the temperature at certain points can be obtained as follows:
• Place a regular square grid over the plate as shown.
• Name the temperatures at the two central nodes of the grid T1 and T2.
• Then according to the finite difference model above, the values of T1 and T2, which are approximate values of the temperature at those points, can be found by noting that the temperature at each node is the average of the temperatures at the four surrounding nodes.
This results in two simultaneous equations which can be solved for T1 and T2. The equations are:
Solve these equations in two ways. Verify your answers.