Page 263 - Modelling in Transport Phenomena A Conceptual Approach

8.1. MOMENTUM TRANSPORT 243

Substitution of Eqs. (8.1-17) and (8.1-19) into Eq. (8.1-22) and noting that
bvZ/~z 0 gives the governing equation for velocity as
=
=o
dr [T(%)] (8.1-23)
The solution of Eq. (8.1-23) is

v, = CI lnr + c2 (8.1-24)
where (71 and Cz are integration constants. The use of the boundary conditions
at T=R v,=O (8.1-25)
at r=r;R vz=V (8.1-26)

gives the velocity distribution as

(8.1-27)

The use of the velocity distribution, Eq. (8.1-27), in Q. (8.1-17) gives the
shear stress distribution as

(8.1-28)

The volumetric flow rate is obtained by integrating the velocity distribution
over the annular cross-sectional area, i.e.,

Q = 1'" lI V, T drd6 (8.1-29)

Substitution of Eq. (8.1-27) into Eq. (8.1-29) and integration gives

(8.1-30)

Dividing the volumetric flow rate by the flow area gives the average velocity as

(8.1-31)

The drag force acting on the rod is

The use of Eq. (8.1-28) in Eq. (8.1-32) gives

(8.1-33)