PROBLEMA 3 3

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PROBLEM 3.3 
Give an example of a practical problem in which the variation of thermal conductivity 
with temperature is significant and for which a numerical solution is therefore the only 
viable solution method. 
SOLUTION 
From Figure 1.6, the thermal conductivity of stainless steel (either 304 or 316) is a fairly strong 
function of temperature. For example 
 kss 316 (100°C) = 14.2 (W/m K) 
 kss 316 (500°C) = 19.6 (W/m K) 
which is about a 38% difference. 
Suppose a stainless steel sheet is to receive a heat treatment that involves heating the sheet to 500°C 
and then plunging it into a water bath. The water near the sheet would probably boil producing a sheet 
surface temperature near 100°C while the interior of the sheet would be at 500°C, at least for a short 
time. One would expect the large variation in thermal conductivity to be important in this type of 
problem. 
PROBLEM 3.4 
Discuss advantages and disadvantages of using a large control volume. 
SOLUTION 
The advantages of a large control volume are 
(1) the numerical solution can be carried out quickly 
(2) manual calculation for all control volumes are feasible for the purpose of verifying the numerical 
calculation 
(3) energy will be conserved 
Disadvantages are 
(1) large temperature gradients cannot be accurately represented with large control volumes 
(2) it is difficult to accommodate all but rectangular geometries. 
PROBLEM 3.5 
For one-dimensional conduction, why are the boundary control volumes half the size of 
interior control volumes? 
GIVEN 
One-dimensional conduction 
EXPLAIN 
(a) Why the boundary control volume is half the size of internal control volumes 
SOLUTION 
There is a node on the boundary as well as one a distance x to the interior of the boundary. Since the 
interior nodes are centered within a control volume of width x, the control volume associated with 
the first non-boundary node comes within x/2 of the boundary. So, there is a volume of only x/2 left 
over for the boundary node. 
PROBLEM 3.6 
Discuss advantages and disadvantages of two methods for solving one-dimensional steady 
conduction problems.