292 Chapter 4 The Potential Equation
Figure 4 Exercise 38.of the lower strip (see Fig. 4). Of course,Vis bounded asy→∞.Solve
this boundary value problem forVin terms off(x).
39.The authors of the article cited in Exercise 38 say that any measurement
ofV(x,y)made at a distanceygreater than 5Lis independent ofx.Ex-
plain this statement, and determine what value (in terms off) would be
measured.
40.In the article “A production-planning and design model for assessing the
thermal behavior of thick steel strip during continuous heat treatment”
[W.D. Morris,Journal of Process Engineering(2001): 53–63] the author
models the temperatureT(x,y)of a long steel strip that comes out of
an oven atx=0, moving to the right, where it is exposed to coolant air.
These equations enter into the modeling (see Table 1 and Fig. 5):
a. Conservation of energy/steady-state heat equation, derived by con-
sidering conservation of energy for a rectangle of dimensions xby
ythat is fixed in space (see Sections 5.1 and 5.2):v
k∂T
∂x=∂^2 T
∂x^2 +∂^2 T
∂y^2 ,^0 <x, −b<y<b;b. Symmetry condition:∂T
∂y(x, 0 )= 0 , 0 <x;c.Cooling by convection at the surface (see Section 2.1, Eq. (10)):−κ∂T
∂y(x,b)=h(
T(x,b)−Ta)
, 0 <x;