Page 383 - Modelling in Transport Phenomena A Conceptual Approach

9.3. HEAT TRANSFER WTH CONVECTION 363

where ria is the mass flow rate given by
ria = p(vZ)rR2 (9.3-14)

On the other hand, since ET/& = 0 as a result of the symmetry condition at
the center of the tube, the integral on the right-side of Eq. (9.3-11) takes the form

(9.3-15)

Substitution of Eqs. (9.3-13) and (9.3-15) into Eq. (9.3-11) gives the governing
equation for the bulk temperature in the form

(9.3-16)

The solution of Eq. (9.3-16) requires the boundary conditions associated with
the problem to be known. The two most commonly used boundary conditions are
the constant wall temperature and constant wall heat flux,

Constant wall temperature
Constant wall temperature occurs in evaporators and condensers in which phase
change takes place on one side of the surface. The heat flux at the wall can be
represented either by Fourier’s law of heat conduction or by Newton’s law of cooling,
1.e..
(9.3-17)

It is implicitly implied in writing Eq. (9.3-17) that the temperature increases in the
radial direction. Substitution of Eq. (9.3-17) into EQ. (9.3-16) and rearrangement
yields

(9.3-18)

Since the wall temperature, T,, is constant, integration of Eq. (9.3-18) yields

mcp In ( Tw - = rD(h),z (9.3-19)
Tw - Tb
in which (h)” is the average heat transfer coefficient from the entrance to the point
z defined by
(h)” = J”hdz (9.3-20)
0
If Eq. (9.3-19) is solved for Tb, the result is

(9.3-21)