If the energy balance equation is developed for a case with no generation, constant properties, and conduction (molecular transport) as the only flux term, it becomes
Solution of this PDE is complex, depending on the geometry, boundary conditions, etc., and is the subject of entirre books. In this course, we will look at a few simplified applications.
By convention, the Fourier Number is given by:
The characteristic length depends on the shape of the system. Typically, we use the radius of a cylinder or a sphere and the "semithickness" of a slab (so MSH use s in their equations).
The Fourier number is the dimensionless time for a temperature change to occur.
The Biot number represents the ratio of heat transfer resistance in the interior of the system (L/k) to the resistance between the surroundings and the system surface (1/h). Therefore, small Bi represents the case were the surface film impedes heat transport and large Bi the case where conduction through and out of the solid is the limiting factor.
We'll begin by looking at the limiting case where the Biot number approaches zero. This applies to systems where the rate of heat transfer is determined by the resistance between the system surroundings and the outer surface of the object. Inside the object, there is very little resistance (1/k = 0), so the entire object may be treated as a single "lump" of uniform temperature.
These "lumped" models treat solid objects in the same way a perfectly- mixed fluid system would be, and the entire object heats and cools at once.
The energy balance is:
The lumped model assumes that conduction within the object is much faster than transfer of heat to the object. The object is treated as "uniform" and the problem reduces to an ODE.
If this isn't true, it is necessary to solve the PDE. Solutions are often tabulated, producing results such as
Simpler forms can be found if we look only at the second major limiting case: there is no film resistance at the surface, the Biot number becomes very large, and conduction within the object is dominant.
MSH present (in Eqs. 10.20, 10.21, and 10.22, Fig 10.6) solutions to this case. In all of these, the assumption is that the surface temperature has suddenly been changed to some new, constant value. If the elapsed time is large enough, these can be truncated to a single term and rearranged to solve for the time (Eqs. 10.23, 10.24, and 10.25).
Be careful! These solutions only apply if the film resistance between the surroundings is non-existent, or if we know the object surface temperature explicitly.
The Fourier equation has been solved for many geometries and sets of conditions. A set of general solutions has been plotted for use in obtaining reasonably good sollutions with less work. These are the Gurney-Lurie or Heisler Charts. MSH Figure 10.8 is a simple version of such a chart for spheres.
Use of the charts is restricted to cases where:
To use the charts, some variables need to be defined. Various versions of the charts are slightly different in how the variables are defined:
In a typical problem, one would
Since the unsteady diffusion equation is mathematically the same as the unsteady conduction equation, these same charts can be used to solve problems in mass transfer. One simply needs to use slightly different definitions of the chart variables:
References:
R.M. Price
Original: 3/1/2000
Modified: 4/18/2000, 3/5/2003; 1/5/2004
Copyright 2004 by R.M. Price -- All Rights Reserved
