EML4534, Computer Applications in ME 2, Section 2 Dr. BoberProject 3 Spring 2014

P5.2. The temperature distribution of a thick flat plate, initially at a uniform temperature,

T0, and which is suddenly immersed in a huge bath at a temperature T∞, is given by

(P5.2a)

where

L = 1/2 of the plate thickness

= the thermal diffusivity of the plate material

δn are the roots of the equation:

(P5.2b)

where

h = the convective heat transfer coefficient for the bath

k = the thermal conductivity of the plate material.

There are an infinite number of roots to Equation (P5.2b). This can be seen in Figure

P5.2b.

The roots being δ1, δ2, δ3, … δn. Note that δ1 lies between 0 and π/2, δ2 lies between π and

3π/2, δ3 lies between 2π and 5π/2, etc. Since it can readily determine the interval in which δj

lies, no search method is required to find the interval containing the root.

Subtracting from Equation (P5.2a) and dividing by , we obtain Equation (P5.2c).

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(P5.2c)

A plot of TRATIO vs. time, for several different values of x/L should appear as shown in

Figure P5.2c.

Finally, the heat transfer ratio, Qratio, from the plate to the bath in time t is given by:

(P5.2d)

where

Q(t) = the amount of heat transferred from the plate to the bath in time t.

Q0 = the amount of heat transferred from the plate to the bath in infinite time.

= the change in internal energy in infinite time.

1. Write a computer program that will solve for the roots, δ1, δ2, …., δ50 using

MATLAB’s fzero function. Print out the δ values in 10 rows and 5 columns. Also

print out the functional values at the roots, i.e.; f (δn).

Note: Only 50 δ values were asked to be computed.

2. Solve Equation (P5.2c) for TRATIO for x/L = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 and

t = 0, 10, 20, … 200s . Print out results in table form as shown in Table P5.2.

Also use MATLAB to produce a plot similar to Figure P5.2c.

3. Construct a table for QRATIO vs. t for times 0, 10, 20, 30, ..…, 200s.

4. Use MATLAB to produce a plot of QRATIO vs. t.

Use the following values for the parameters of the problem:

T0 = 300C, T = 30C, h = 45 w/m2-C,

k = 10.0 w/m-C, L = 0.03 m, = 0.279 x 10-5 m2/s

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