Isothermal and Isoflux Boundary Conditions

In analytical conduction problems, two commonly used boundary conditions are isothermal, meaning that the temperature at the boundary is fixed, and isoflux, which means that the heat flux at the boundary is fixed. The two are mutually exclusive (you can’t specify both the heat flux and the temperature on a single boundary) and lead to very different thermal behaviors inside the body. In this post, we’ll look at the effects of these two boundary conditions.
First, let’s discuss briefly the physical situations that these two mathematical boundary conditions might represent. It is important to note that it would be impossible to strictly maintain either of these theoretical boundary conditions exactly and indefinitely, but as a practical matter, they approximate some physical situations reasonably well.

The isothermal case is easy to visualize—a surface is held at a fixed temperature regardless of how much heat flow occurs because of that temperature. A solid surface exposed to a boiling liquid, or the surface under an impinging liquid jet at constant temperature might both be situations that could be approximated very well by an isothermal boundary condition. The isoflux case is the converse—a constant heat flux (in or out) is maintained at the surface regardless of how high (or low) the temperature gets because of that heat flux. A very absorbing surface exposed to a steady source of infrared light might be approximated well by an isoflux boundary condition. 

As an illustration, this figure shows temperature profiles within a semi-infinite body for both isothermal (solid lines) and isoflux (dashed lines) boundary conditions. It is assumed that

initially the body was at a uniform temperature, then suddenly either a fixed temperature or fixed heat flux was imposed at the surface. The temperature distribution inside the body is shown for three different times for each of the two boundary conditions. Notice that in the isothermal case, the surface temperature (boundary) never changes. Of course, you would expect the heat flux at the surface in that case to be very high at first, then to gradually decline as the body starts to warm up toward the surface temperature. In the isoflux case, the same heat flux is forced into the body at all times, and the surface temperature has to rise in order to accommodate that heat flux.

Fourier’s law describing one-dimensional conduction shows that the heat flux is proportional to the temperature gradient: 

This is visible in the isothermal profiles where the slope at the surface is very steep at first, then gets less steep with time, indicating that the heat flux is decreasing. In the isoflux profiles the slope at the surface is the same for all profiles. 

This figure explicitly shows the surface temperature as a function of time for both boundary conditions. 

This figure shows the heat flux at the surface as a function of time for both boundary conditions.