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The differential equation governing the temperature in the bar is an expression of a fundamental physical balance; the rate at which heat flows into any portion of the bar is equal to the rate at which heat is absorbed in that portion of the bar. The terms in the equation are called the flux (flow) term and the absorption term, respectively.
We will first calculate the flux term. Consider an element of the bar lying between the cross sections x = x0 and x = x0 + Ax, where x0 is arbitrary and Ax is small. The
"Actually, k also depends on the temperature, but if the temperature range is not too great, it is satisfactory to assume that k is independent of temperature.
instantaneous rate of heat transfer H(x0, t) from left to right across the cross section x = x0 is given by
H(x0) = — lim kA
u(x0 + d/2, t) — u(x0 — d/2, t)
= -k Aux (x0, t). (2)
The minus sign appears in this equation since there will be a positive flow of heat from left to right only if the temperature is greater to the left of x = x0 than to the right; in this case ux(x0, t) is negative. In a similar manner, the rate at which heat passes from left to right through the cross section x = x0 + Ax is given by
H(x0 + Ax, t) = — kAux(x0 + Ax, t). (3)
The net rate at which heat flows into the segment of the bar between x = x0 and
x = x0 + Ax is thus given by
Q = H(x0, t) — H(x0 + Ax, t) = kA[ux(x0 + Ax, t) — ux(x0, t)], (4)
and the amount of heat entering this bar element in time At is
Q At = kA[ux(x0 + Ax, t) — ux(x0, t)] At. (5)
Let us now calculate the absorption term. The average change in temperature Au,
in the time interval At, is proportional to the amount of heat Q At introduced and inversely proportional to the mass Am of the element. Thus
Au = 1Q-A- = QAt , (6)
s Am sp A Ax
where the constant of proportionality s is known as the specific heat of the material of the bar, and p is its density.12 The average temperature change Au in the bar element under consideration is the actual temperature change at some intermediate point x = x0 + 9 Ax, where 0 < 9 < 1. Thus Eq. (6) can be written as
u(x0 + 9 Ax, t + At) — u(x0 + 9 Ax, t) = — -----------, (7)
sp A Ax
Q At = [u(x0 + 9 Ax, t + At) — u(x0 + 9 Ax, t)]spA Ax. (8)
To balance the flux and absorption terms, we equate the two expressions for Q At:
kA[ux (x0 + Ax, t) — ux (x0, t)] At
= spA[u(x0 + 9 Ax, t + At) — u(x0 + 9 Ax, t)] Ax. (9)
On dividing Eq. (9) by Ax At and then letting Ax ^ 0 and At ^ 0, we obtain the heat conduction or diffusion equation
a2uxx = ut. (10)
The quantity a2 defined by
a2 = k/ps (11)
12The dependence of the density and specific heat on temperature is relatively small and will be neglected. Thus both p and s will be considered as constants.
Chapter 10. Partial Differential Equations and Fourier Series
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 —k Aux (0, t); see Eq. (2). Hence the rate of heat flow out of the bar (from right to left) at x = 0 is kAux (0, t). If this quantity is proportional to the temperature u(0, t), then we obtain the boundary condition
ux(0, t) — h1u(0, t) = 0, t > 0, (14)
where h 1 is a nonnegative constant of proportionality. Note that h 1 = 0 corresponds to an insulated end, while h1 ^ 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.