Page 436 - Modelling in Transport Phenomena A Conceptual Approach

416 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

modifications of Eqs. (9.1-23) and (9.1-26) for the axial flow in aconcentric annulus
with inner and outer radii of nR and R, respectively, lead to

9.4 For laminar flow of a Newtonian fluid in a circular pipe the velocity profile is
parabolic and Eqs. (9.1-80) and (9.1-84) indicate that

-- - 0.5
(")
Vmax
In the case of a turbulent flow, experimentally determined velocity profiles can be
represented in the form

where n depends on the value of the Reynolds number. Show that the ratio
(vz)/vmax is given as (Whitaker, 1968)

RR n (vz)/vmax
4 x 103 6 0.79
1 x 105 7 0.82
3 x 106 10 0.87

This is the reason why the velocity profile for a turbulent flow is generally consid-
ered "flat" in engineering analysis.

9.5 The steady temperature distribution in a hollow cylinder of inner and outer
radii of 50cm and 80cm, respectively, is given by

T = 5000 (4.073 - 6 r2 + lnr)
where T is in degrees Celsius and r is in meters. If the thermal conductivity is
5 W/ m. K, find the rate of energy generation per unit volume.
(Answer: 6 x lo5 W/ m3)

9.6 Energy generation within a hollow cylinder of inside and outside radii of 60 cm
and 80 cm, respectively, is lo6 W/ m3. If both surfaces are maintained at 55 "C and
the thermal conductivity is 15 W/ m. K, calculate the maximum temperature under
steady conditions.
(Answer: 389.6 "C)