Negligible Viscous Dissipation

To some extent, we have already discussed this case in regard to the local heat flux at any point within the flow. If ue is very small, hence г is very small, then the amount of viscous dissipation is negligibly small, and Equation (16.21) becomes

/he – hw Pr V D

Clearly, for this case, the heat flux is constant across the flow. Moreover, the enthalpy profile given by Equation (16.16) becomes

h = hw + {he — hw)~ [16.27]

Since h = cpT, the temperature profile is identical to the enthalpy profile:

T = Tw + (Te — Tw)~ [16.28]

Note that the temperature varies linearly across the flow, as sketched in Figure 16.4. The case shown here is for the upper wall at a higher temperature than the lower wall. The heat transfer at the lower wall is obtained from Equation (16.24) with a negligible

The heat transfer at the upper wall is similarly obtained as

Equations (16.29) and (16.30) are identical; this is no surprise, since we have already shown that the heat flux is constant across the flow, as shown by Equation (16.26), and therefore the heat transfer at both walls should be the same. Equations (16.29) and (16.30) can also be written in terms of temperature as

[16.31]

Examining Equations (16.29) to (16.31), we can make some conclusions which can be generalized to most viscous flow problems, as follows:

1. Everything else being equal, the larger the temperature difference across the viscous layer, the greater the heat transfer at the wall. The temperature difference (Te — Tw) or the enthalpy difference (he — hw) takes on the role of a “driving potential” for heat transfer. For the special case treated here, the heat transfer at the wall is directly proportional to this driving potential.

2. Everything else being equal, the thicker the viscous layer (the larger D is) the smaller the heat transfer is at the wall. For the special case treated here, qw is inversely proportional to D.

3. Heat flows from a region of high temperature to low temperature. For negligible viscous dissipation, if the temperature at the top of the viscous layer is higher than that at the bottom, heat flows from the top to the bottom. In the case sketched in Figure 16.4, heat is transferred from the upper plate into the fluid, and then is transferred from the fluid to the lower plate.