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is called the thermal diffusivity, and is a parameter depending only on the material of the bar. The units of a2 are (length)2/time. Typical values of a2 are given in Table 10.5.1.
Several relatively simple conditions may be imposed at the ends of the bar. For
example, the temperature at an end may be maintained at some constant value T. This
might be accomplished by placing the end of the bar in thermal contact with some reservoir of sufficient size so that any heat that may flow between the bar and reservoir does not appreciably alter the temperature of the reservoir. At an end where this is done the boundary condition is
u = T. (12)
Another simple boundary condition occurs if the end is insulated so that no heat passes through it. Recalling the expression (2) for the amount of heat crossing any cross section of the bar, we conclude that the condition of insulation is that this quantity vanish. Thus
ux = 0 (13)
is the boundary condition at an insulated end.
A more general type of boundary condition occurs if the rate of flow of heat through an end of the bar is proportional to the temperature there. Let us consider the end x = 0, where the rate of flow of heat from left to right is given by —ê Aux (0, t); see Eq. (2). Hence the rate of heat flow out of the bar (from right to left) at x = 0 is ê Aux (0, t). If this quantity is proportional to the temperature u(0, t), then we obtain the boundary condition
ux(0, t) — hxu(0, t) = 0, t > 0, (14)
where h j is a nonnegative constant of proportionality. Note that h j = 0 corresponds to an insulated end, while hx ^ to corresponds to an end held at zero temperature.
If heat flow is taking place at the right end of the bar (x = L), then in a similar way we obtain the boundary condition
ux(L, t) + h2u(L, t) = 0, t > 0, (15)
where again h2 is a nonnegative constant of proportionality.
Finally, to determine completely the flow of heat in the bar it is necessary to state the temperature distribution at one fixed instant, usually taken as the initial time t = 0. This initial condition is of the form
u(x, 0) = f (x), 0 < x < L. (16)
The problem then is to determine the solution of the differential equation (10) subject to one of the boundary conditions (12) to (15) at each end, and to the initial condition (16) at t = 0.
Several generalizations of the heat equation (10) also occur in practice. First, the bar material may be nonuniform and the cross section may not be constant along the length of the bar. In this case, the parameters ê, p, 5, and A may depend on the axial variable x. Going back to Eq. (2) we see that the rate of heat transfer from left to right across the cross section at x = x0 is now given by
H(x0, t) = -k(x0)A(x0)ux(x0, t)
with a similar expression for H (x0 + Ax, t). Ifwe introduce these quantities into Eq. (4) and eventually into Eq. (9), and proceed as before, we obtain the partial differential equation
(ê Aux )x = sp Aut. (18)
We will usually write Eq. (18) in the form
r (x )ut = [p(x )ux ]x, (19)
where p(x) = ê (x) A(x) and r (x) = s (x)p(x) A(x). Note that both these quantities are intrinsically positive.
A second generalization occurs if there are other ways in which heat enters or leaves the bar. Suppose that there is a source that adds heat to the bar at a rate G (x, t, u) per unit time per unit length, where G (x, t, u) > 0. In this case we must add the term G(x, t, u) Ax At to the left side ofEq. (9), and this leads to the differential equation
r (x)ut = [p(x)ux]x + G(x, t, u). (20)
If G (x , t, u) < 0, then we speak of a sink that removes heat from the bar at the rate G(x, t, u) per unit time per unit length. To make the problem tractable we must restrict the form of the function G. In particular, we assume that G is linear in u and that the coefficient of u does not depend on t. Thus we write
G(x, t, u) = F(x, t) — q(x)u. (21)
The minus sign in Eq. (21) has been introduced so that certain equations that appear
later will have their customary forms. Substituting from Eq. (21) into Eq. (20), we
r (x)ut = [p(x)ux]x — q(x)u + F(x, t). (22)
This equation is sometimes called the generalized heat conduction equation. Boundary value problems for Eq. (22) will be discussed to some extent in Chapter 11.