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Heat Transfer project using Matlab
A metallic object is attached to a hot surface and immersed in cold fluid as shown in Figure 1(a)
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below. The dimension of the metallic object is L *H* w mm The width, W, is much larger than
L or H (w>> L, H) so that temperature (7) distribution in Z direction can be ignored. The hot
surface (surface A) has a uniform temperature of 400 K (Ta -400K), and the cold fluid has a
temperature of 300 K (T. - 300K). Assuming sufficiently large fluid heat capacity and
convection heat transfer coefficient, all the surfaces other than the hot surface have the same
temperature of the cold fluid. In this project, using the 2D finite difference method with / (# of
rows) m (# of columns) mesh network as in Figure 1(b), we will analyze thermal properties of
this metallic object, and compare the numerical results with the analytical solution. Here, you
can choose any thermal conductivity (k) value between 80 and 500 W/m-K. Use MATLAB for
analyses and plotting in the following parts.
T 300 K, T 400 K
Ixm nodal network
T,
Surface A
T,
W
L
(a)
(b)
Figure 1
(a) We define / x m nodal network as in Fig. 1(b), and nodal point index of i-th row and j-th
column can be expressed as n - m* (i-1)+j(i-1,2, 1;j 1,2, m). Since all the
nodes are inside the metal block, the finite-difference equation for node n will be Tut Tr
+ Tz+ Tr 4Th - 0, where subscripts U, B, L, and R represent upper, below, left and right
nodes, respectively. For numerical code, i) decide subscripts U and B in terms of n, m
and/or prescribed T 'depending on i;1) - 1,2) I, and 3) 1 and R depending onj; 1)j 1, 2)j m, and 3) 1 m.
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(b) (i) Suppose that the dimension of the metal block is 100 *100* w mm and we use
mesh size of x -y 20 mm, use a Gauss-Seidel (GS) approach to determine a
numerical solution for the nodal temperatures. ii) Find an appropriate convergence
criterion (8) in GS algorithm. (Check and compare nodal temperatures by reducing E).
iii) Plot the 2D temperature distribution (color heat map).
(c) In the textbook, we have analytical solution for 2D heat conduction. Using the solution
(Eq. 4.19 on p.234), calculate the temperatures on the nodal points defined in part
(b).Here, you will include only the first p terms of the infinite series. For an
accuracy of 0.1 K, how many terms do we need to consider? By increasing p (- 1,
3,5, find p which makes the maximum difference fromp + 2 smaller than 0.1 K.
Plot the 2D T-distribution Try a smaller accuracy criterion, repeat the above
questions.
(d) i) Find the error of numerical approach by comparing the results from part (c) with
part (b). (Calculate the maximum T difference between two approaches.) ii) By
reducing mesh size, we want to calculate a more accurate distribution. To have
temperature difference from analytical approach less than 1 K at all nodal points, how
small mesh size do we need to use (<1 K)? Plot the 2D T-distribution Try a smaller
maximum difference.
(e) Using the numerical results from part (d), calculate the heat flux vectors at nodal
points, and show the heat flux vector distribution (by adding arrows). What is the
total heat loss from surface A? Change the thermal conductivity (k) and discuss
its effect on Tand q distribution.

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