Chemical Engineering 6453 Heat Transfer Prof. Geoff Silcox Spring 2005 Solution to Homework Assignment 3 Due Wednesday, 2 February, by 17:00.

Problem 1 In class on Monday, 24 January, we estimated the time scale of an infinite plate of thickness 2L as

( )2 22

L BiBi

τα

+=

where Bi = hL/k and α is the thermal diffusivity, k/ρc. Test the validity of this approximation using a fully implicit, node-centered, numerical solution to the following equations.

( )( )

( ) ( )

2

2

,0

0,0

,,

i

T Tt x

T x T

T tx

T L tk h T L t T

x

α

∞

∂ ∂=

∂ ∂=

∂=

∂∂

⎡ ⎤− = −⎣ ⎦∂

Perform your calculations for Bi = 0.01, 0.1, 0.2, 0.5, 1, 10, 100, 1000. In each case, use the amount of energy conducted to the plate (W/m2) as the function whose maximum slope you are estimating in order to obtain the time scale via

max min

max

Q QdQdt

τ−

=

Include a summary of all calculations and a listing of your MATLAB code with your solution, with sample output. Solution: Numerical calculations performed with the m-file, tplate.m, at Bi = 0.01, 1, and 1000, verify the validity of the first approach for all Bi. The second approach begins to fail when Bi is much greater than 1. In fact, for Bi = 1000, the second approach gives a time scale of about 3 s when the actual value is 50 s. The reasons for this are clearly visible in the plot below that shows the fractional

heat loss as a function of Fo for the numerical solution (green) and the one-term analytical solution. There are clearly multiple time scales for the green function - the initial slope is extremely steep and this agrees with the time scale of 3 s.

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Heat loss profile (green-num, red-one-term)

Fourier number

Frac

tiona

l hea

t los

s

Problem 2 Read the article by Yovanovich. It is on electronic reserve in the library. Use the code developed in Problem 1 to critically evaluate the one term approximation for the flat plat that he develops. The heart of that approximation is Equation 41 and the critical Fourier numbers, Foc, given in the introduction. The value of Foc that he reports for the infinite plate is 0.24. Is that value reasonable? What is the error involved in using the one-term approximation when Foc = 0.24? Solution: The value of Foc = 0.24 is consistent with the figure above with Bi >> 1, but I am not sure the two significant figures are justified. At Bi = 1, Foc = 0.2 is still appropriate. At Bi = 0.01, Foc is actually about 2, but at these conditions you would want to use the lumped analysis anyway, so it doesn’t really matter.