Chapter 2 The Heat Equation 141
When we rearrange Eq. (17) and take the limit as xapproaches 0, it becomes
−∂q
∂x+g=∂u
∂t. (18)
In these equations,gis a “generation rate”([g]=m/tL^3 ),afunctionthatac-
counts for any gain or loss of the substance from the layer by means other than
movement in thex-direction. For example, the substance may participate in a
chemical reaction with the medium at a rate proportional to its concentration
(a first-order reaction) so that in this case the generation rate is
g=−ku(x,t). (19)The concentration and the mass flux are linked by a phenomenological re-
lation calledFick’s first law, written in one dimension as
q=−D∂u
∂x. (20)In words, the diffusing substance moves toward regions of lower concentra-
tion at a rate proportional to the gradient of the concentration. The coefficient
of proportionalityD, usually constant, is called thediffusivity. By combining
Fick’s law with Eq. (18) arising from the conservation of mass, we obtain the
diffusion equation
∂^2 u
∂x^2+g
D=^1
D
∂u
∂t. (21)
At a boundary of the medium, the concentration of the diffusing substance
may be controlled, leading to a condition like Eq. (6), or the flux of the sub-
stance may be controlled, leading via Fick’s law to a condition like Eq. (7); an
impermeable surface corresponds to zero flux. If a boundary is covered with a
permeable film, then the flux through the film is usually taken to be propor-
tional to the difference in concentrations on the two sides of the film. Suppose
that the surface in question is atx=a. Then these statements may be expressed
symbolically as
q(a,t)=h(
u(a,t)−C(t))
, (22)
where the proportionality constanthis called the film coefficient andCis the
concentration outside the medium. Using Fick’s law here leads to the equation
−D
∂u
∂x(a,t)=hu(a,t)−hC(t). (23)This equation is analogous to Eq. (10) and can be put into the form of Eq. (8).