1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

214 Chapter 2 The Heat Equation

Definingu=S+P, find the boundary and initial conditions foru,and

solve completely. Then findP(x,t)asu(x,t)−S(x,t).

36.Refer to Exercises 34 and 35. In order to determine the response time

of the enzyme electrode, one wants to know the functionP( 0 ,t).Ap-

proximate this, using in your solution only steady-state terms and the

first term of each infinite series. Sketch. Find the “time constants,” the

multipliers oftin the exponential functions.

37.Consider a steel plate that is much larger in length and width (x-andz-

directions) than in thickness (y-direction), and suppose the plate is free

to expand or contract under the effects of heating. Assume that the tem-

peratureTin the plate is a function ofyandtonly. Timoshenko and

Goodier (Theory of Elasticity, pp. 399–403) derive the following expres-

sion for the stresses due to thermal effects:

σx=σz=−

αTE

1 −ν+

1

2 c( 1 −ν)

∫+c

−c

αTE dy

+^3 y

2 c^3 ( 1 −ν)

∫+c

−c

αTEy dy.

The parameters, and their values for steel are as follows:αis the co-

efficient of expansion, 6. 5 × 10 −^6 per degree F;Eis Young’s modulus,

28 × 106 lb/in.^2 ;νis Poisson’s ratio, 0.7; and 2cis the thickness of the

plate. Note that the origin is located so that the plate lies betweeny=c

andy=−c.

a. Show that ifT(y)=T 0 +Sy,whereT 0 andSare constants, then the

thermal stress is 0. (This is a typical steady-state temperature distri-

bution.)

b.Suppose that the plate is initially at temperature 500◦F throughout

and that the temperature on the facey=cis suddenly changed to

200 ◦whilethetemperatureaty=−cremains at 500◦.FindT(y,t).

c.Assume the initial and boundary conditions given inb. Use your un-

derstanding of the functionT(y,t)to explain why the thermal stress

near the facey=cis large just after time 0.