Adiabatic Wall Conditions (Adiabatic Wall Temperature)

Let us imagine the following situation. Assume that the flow illustrated in Figure 16.5 is established. We have the parabolic temperature profile established as shown, and we have heat transfer into the walls as just discussed. However, both wall temperatures are considered fixed, and both are equal to the same constant value. Question: How can the wall temperature remain fixed at the same time that heat is transferred into the wall? Answer: There must be some independent mechanism that conducts heat away from the wall at the same rate that the aerodynamic heating is pumping heat into the wall. This is the only way for the wall temperature to remain fixed at some cooler temperature than the adjacent fluid. For example, the wall can be some vast heat sink that can absorb heat without any appreciable change in temperature, or possibly there are cooling coils within the plate that can carry away the heat, much like the water coils that keep the engine of your automobile cool. In any event, to have the picture shown in Figure 16.5 with a constant wall temperature independent of time, some exterior mechanism must carry away the heat that is transferred from the fluid to the walls. Now imagine that, at the lower wall, this exterior mechanism is suddenly shut off. The lower wall will now begin to grow hotter in response to qw, and Tw will begin to increase with time. At any given instant during this transient process, the heat transfer to the lower wall is given by Equation (16.24), repeated below.

At time t = 0, when the exterior cooling mechanism is just shut off, hw = hc, and qw is given by Equation (16.35), namely,

However, as time now progresses, Tu, (and therefore hw) increases. From Equa­tion (16.24), as hy, increases, the numerator decreases in magnitude, and hence qw decreases. That is,

Hence, as time progresses from when the exterior cooling mechanism was first cut off at the lower wall, the wall temperature increases, and the aerodynamic heating to the wall decreases. This in turn slows the rate of increase of Tw as time progresses. The transient variations of both qw and Tw are sketched in Figure 16.6. In Figure 16.6a, we see that, as time increases to large values, the heat transfer to the wall approaches zero—this is defined as the equilibrium, or the adiabatic wall condition. For an adiabatic wall, the heat transfer is, by definition, equal to zero. Simultaneously, the wall temperature Tw approaches asymptotically a limiting value defined as the adiabatic wall temperature Taw, and the corresponding enthalpy is defined as the adiabatic wall enthalpy haw.

The purpose of this discussion is to define an adiabatic wall condition; the ex­ample involving a timewise approach to this condition was just for convenience and edification. Let us now assume that the lower wall in our Couette flow is an adiabatic wall. For this case, we already know the value of heat transfer to the wall—by defi­nition, it is zero. The question now becomes, What is the value of the adiabatic wall enthalpy haw, and in turn the adiabatic wall temperature Taw‘l The answer is given by Equation (16.23), where qw = 0 for an adiabatic wall.

In turn, the adiabatic wall temperature is given by

[16.40]

Clearly, the higher the value of ue, the higher is the adiabatic wall temperature.

The enthalpy profile across the flow for this case is given by a combination of Equations (16.16) and (16.40), as follows. Setting hw = haw in Equation (16.16), we

obtain

[16.43]

Equation (16.43) gives the enthalpy profile across the flow. The temperature profile follows from Equation (16.43) as

[16.44]

This variation of T is sketched in Figure 16.7. Note that Tuw is the maximum tem­perature in the flow. Moreover, the temperature curve is perpendicular at the plate for v = 0; that is, the temperature gradient at the lower plate is zero, as expected for an adiabatic wall. This result is also obtained by differentiating Equation (16.44):

which gives З T/9у = 0 at у = 0.