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Solution of the diffusion equation

Dimensional analysis can also be used to solve certain types of partial differential equations. If this seems too good to be true, it isn't. Here we will concentrate on the solution of the diffusion equation; we will encounter this equation many times in the remainder of the course, so it will be useful to work out some of its properties now.

We'll start by deriving the one-dimensional diffusion, or heat, equation.gif1 Let tex2html_wrap_inline1258 represent the temperature of a metal bar at a point x at time t (I'll use tex2html_wrap_inline1264 to avoid confusion with the symbol for the dimension of time, T). The first step is the derivation of a continuity equation for the heat flow in the bar. Let the bar have a cross sectional area A, so that the infinitesimal volume of the bar between x and tex2html_wrap_inline1272 is tex2html_wrap_inline1274 . The quantity of heat contained in this volume is tex2html_wrap_inline1276 , with tex2html_wrap_inline1278 the specific heat at constant pressure per unit volume; it has dimensions tex2html_wrap_inline1280 . In a time interval dt this heat changes by an amount tex2html_wrap_inline1284 due to the change in temperature. This change in the heat must come from somewhere, and is the result of a flux of heat q(x,t) through the area A (q is the heat flowing through a unit area per unit time). Into the left side of the volume an amount of heat q A dt flows in a time dt; on the right hand side of the volume a quantity tex2html_wrap_inline1296 flows out in a time dt, so that the net accumulation of heat in the volume is tex2html_wrap_inline1300 . Equating the two expressions for the rate of change of the heat in the volume tex2html_wrap_inline1274 , we find


which is the equation of continuity. It is a mathematical expression of the conservation of heat in the infinitesimal volume tex2html_wrap_inline1274 . We supplement this with a phenomenological law of heat conduction, known as Fourier's law: the heat flux is proportional to the negative of the local temperature gradient (heat flows from a hot reservoir to a cold reservoir):


with tex2html_wrap_inline1306 the thermal conductivity of the metal bar. The thermal conductivity is usually measured in units of tex2html_wrap_inline1308 , and has dimensions tex2html_wrap_inline1310 . See the Table on p. 118 of the PQRG for the thermal conductivities of some materials. Combining Eqs. (1.22) and (1.23), we obtain the diffusion equation (often called the heat equation)


where tex2html_wrap_inline1312 is the thermal diffusivity of the metal bar; it has dimensions tex2html_wrap_inline1314 , as it should. Eq. (1.24) is the diffusion equation for heat. The diffusion equation will appear in many other contexts during this course. It usually results from combining a continuity equation with an empirical law which expresses a current or flux in terms of some local gradient.

Suppose that the bar is very long, so that we can consider the idealized case of an infinite bar. At an initial time t=0, we add an amount of heat H (with dimensions tex2html_wrap_inline1320 ) at some point of the bar, which we will arbitrarily call x=0. We could do this, for instance, by briefly holding a match to the bar. The heat is conserved at all times, so that


How does this heat diffuse away from x=0 as a function of time t; i.e., what is tex2html_wrap_inline1258 ? We first identify the important parameters. The temperature tex2html_wrap_inline1264 certainly depends upon x, t, and the diffusivity D; we see from Eq. (1.25) that it also depends upon the initial conditions through the combination tex2html_wrap_inline1338 . What are the dimensions? We have [x]=L, [t]=T, tex2html_wrap_inline1344 , and tex2html_wrap_inline1346 , so that n=4. These dimensions are not independent, for the quantity tex2html_wrap_inline1350 is dimensionless, so that k=3. We will choose as our independent quantities (t,D,Q). Now express tex2html_wrap_inline1264 in terms of these variables:


We find


which has the solution a=-1/2, b=-1/2, c=1. Therefore, dimensional analysis tells us that the solution of the diffusion equation is of the form


with tex2html_wrap_inline1070 a function which we still need to determine. The important point is that tex2html_wrap_inline1070 is only a function of the combination tex2html_wrap_inline1350 , and not x and t separately. To determine tex2html_wrap_inline1070 , let's introduce the dimensionless variable tex2html_wrap_inline1376 . Now use the chain rule to calculate various derivatives of tex2html_wrap_inline1264 :




Substituting Eqs. (1.30) and (1.31) into the diffusion equation (1.24), and canceling various factors, we obtain a differential equation for tex2html_wrap_inline1070 ,


Dimensional analysis has reduced the problem from the solution of a partial differential equation in two variables to the solution of an ordinary differential equation in one variable! The normalization condition, Eq. (1.25), becomes in these variables


You might think that Eq. (1.32) is hard to solve; however, it turns out that it is an exact differential,


which we can integrate once to obtain


However, since any physically reasonable solution would have both tex2html_wrap_inline1382 and tex2html_wrap_inline1384 as tex2html_wrap_inline1386 , the integration constant must be zero. We now need to solve a first order differential equation, which we do by dividing Eq. (1.35) by tex2html_wrap_inline1070 , multiplying by dz, and integrating, with the result that tex2html_wrap_inline1392 , or


with C a constant. To determine C, we use the normalization condition, Eq, (1.33):


where the integral (known as a Gaussian integral) can be found in integral tables. Therefore tex2html_wrap_inline1398 . Returning to our original variables, we have


This is the complete solution for the temperature distribution in a one-dimensional bar due to a point source of heat gif2. Here are pictures:

Click here to see an animation (controlled by mouse buttons) of the one-dimensional diffusion process described by eq. (1.38). The animation starts at some arbitrary time t0 (zero time corresponds to an infinitely sharp peak) and ends at 25t0, and x is in units of the square root of Dt0.


gif1 I prefer the term diffusion equation, since we are just describing the diffusion of heat.

gif2 For the mathematically sophisticated, I'll mention that the same solution can be obtained using the method of Fourier transforms applied to the diffusion equation.

next up previous
Next: Similarity, modeling, ... Up: Examples Previous: Energy in a nuclear explosion

V. Celli
Thu Jul 10 16:27:59 EDT 1997