5.6 Internal flows 157

We define such that = .

Tm Et mcvTm

For incompressible flow in a circular tube with constant cv , it follows that

The bulk mean temperature, Tm is also called the mixing cup temperature.

Newton’s law of cooling

Once Tm is defined, we can use Newton’s law of cooling for heat transfer in this geometry (tube/duct flow)

where h is the local heat transfer coefficient, and Ts is the surface or wall temperature. There is an essential difference between external and internal flow. While in external flow T∞ is a constant, here Tm must vary in the flow direction. Stated explicitly, for heat transfer to occur, dTm /dx ≠ 0. dTm /dx is +ve, if the fluid is getting heated and

dTm /dx is -ve, if the fluid is getting cooled.

Fully developed conditions

In the hydrodynamic case, ∂u/ ∂x = 0 is valid for fully developed flow. But if heat transfer occurs, neither dTm /dx nor ∂T / ∂x equals zero at any radius r. T (r) changes continuously with x. However, this seeming paradox may be reconciled by working with a dimensionless temperature (TS (x)− T (r, x)) / (TS (x)− Tm (x)) and examining if this quantity becomes independent of x beyond a certain x. In fact, though T (r)

changes with x, (TS (x)− T (r, x)) / (TS (x)− Tm (x)) is not a function of x in the fully developed region. Hence, the mathematical condition for fully developed flow from

the viewpoint of convective heat transfer is

We can work a little with the dimensionless temperature considered above at r = r0, as follows.

158 CHAPTER 5  Forced convection

We know that

or q / k (5.224)

= − ∂T

∂r r =r0

Similarly, from Newton’s law of cooling, TS (x) − Tm (x) = q/h. Substituting these results in, Eq. (5.222)

This is indicated in Fig. 5.10 for a constant k.

Hence, in the thermally fully developed flow of a fluid with constant properties, h ≠ f (x) and is a constant. One way of coming to peace with these “uneasy” arguments is to first state that from measurements, we know that Eq. (5.221) is valid, and using all our definitions and ideas of convection, we get to the fact that h ≠ f (x) in the fully developed region. If this too causes discomfiture, one can start with the argument that measurements show that h ≠ f (x) beyond a certain x and we would like to know what mathematical condition (Ts (x)− Ts (r, x)) must satisfy in order for this to be true; this will logically lead us back to Eq. (5.221).

FIGURE 5.10

Variation of heat transfer coefficient, h, along the length of the tube.