Page 386 - Modelling in Transport Phenomena A Conceptual Approach

366 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

Evaluation of Eq. (9.3-32) over the total length gives the bulk temperature at
the exit of the pipe as

(9.3-33)

The rate of heat transferred to the fluid is given by
Q = 7i-h eP(TbOut - (9.3-34)

Substitution of Eq. (9.3-33) into Eq. (9.3-34) yields

Q = (nDL)qw (9.3-35)
9.3.1.1 Thermally developed flow

As stated in Section 8.1, when the fluid velocity is no longer dependent on the axial
direction z, the flow is said to be hydrodynamically fully developed. In the case of
heat transfer. if the ratio
(9.3-36)

does not vary along the axial direction, then the temperature profile is said to be
fully developed.
It is important to note that although the fluid temperature, T, bulk fluid tem-
perature, Tb, and wall temperature, T,, may change along the axial direction, the
ratio given in Eq. (9.3-36) is independent of the axial coordinate2, i.e.,

(9.3-37)

Equation (9.3-37) indicates that

(9.3-38)

Example 9.6 For a themally developed flow of a fluid with constant physical
properties, show that the heat transfer coeficient is constant.

*In the literature, the condition for the thermally developed flow is also given in the form

Note that