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Amount of heat per unit time = ê A | T2 — T1l/d, (1)
where the positive proportionality factor ê is called the thermal conductivity and depends primarily on the material11 of the rod. The relation (1) is often called Fourier’s law of heat conduction. We repeat that Eq. (1) is an empirical, not a theoretical, result and that it can be, and has often been, verified by careful experiment. It is the basis of the mathematical theory of heat conduction.
Now consider a straight rod of uniform cross section and homogeneous material, oriented so that the x-axis lies along the axis of the rod (see Figure 10.A.1). Let x = 0 and x = L designate the ends of the bar.
FIGURE 10.A.1 Conduction of heat in an element of a rod.
We will assume that the sides of the bar are perfectly insulated so that there is no passage of heat through them. We will also assume that the temperature u depends only on the axial position x and the time t, and not on the lateral coordinates y and z. In other words, we assume that the temperature remains constant on any cross section of the bar. This assumption is usually satisfactory when the lateral dimensions of the rod are small compared to its length.
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, ê also depends on the temperature, but if the temperature range is not too great, it is satisfactory to assume that ê 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 ê A
u(x0 + d/2, t) — u(x0 — d/2, t)
= -êAux(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) = —êAux(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) = êA[ux(x0 + Ax, t) — ux(x0, t)], (4)
and the amount of heat entering this bar element in time At is
Q At = êA[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 = 1= 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 + â Ax, where 0 < â < 1. Thus Eq. (6) can be written as
u(x0 + â Ax, t + At) — u(x0 + â Ax, t) = — -----------, (7)
sp A Ax
Q At = [u(x0 + â Ax, t + At) — u(x0 + â Ax, t)]spA Ax. (8)
To balance the flux and absorption terms, we equate the two expressions for Q At:
êA[ux (x0 + Ax, t) — ux (x0, t)] At
= spA[u(x0 + â Ax, t + At) — u(x0 + â 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
“2uxx = ut ¦ (10)
The quantity a2 defined by
a2 = ê/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