Page 356 - Modelling in Transport Phenomena A Conceptual Approach

336 CHAPTER 9. STEADY MICROSCOPIC BALANCES WTH GEN.

where C1 and C2 axe integration constants.
The center of the tube, i.e., r = 0, is included in the flow domain. However,
the presence of the term Inr makes v, -+ - 00 as r --f 0. Therefore, a physically
possible solution exists only if Cl = 0. This condition is usually expressed as “v, is
finite at r = 0.” Alternatively, the use of the symmetry condition, i.e., dv,/dr = 0
at r = 0, also leads to C1 = 0. The constant Cz can be evaluated by using the
no-slip boundary condition on the surface of the tube, Le.,
at r=R v,=O (9.1-78)

so that the velocity distribution becomes

(9.1-79)

The maximum velocity takes place at the center of the tube, i.e.,

(9.1-80)

The use of Eq. (9.1-79) in Eq. (9.1-65) gives the shear stress distribution as

(9.1-81)

The volumetric flow rate can be determined by integrating the velocity distribution
over the cross-sectional area, i.e.,

Q=rlRv,rdr&3 (9.1-82)

Substitution of Eq. (9.1-79) into Eq. (9.1-82) and integration gives

(9.1-83)

which is known as the Hagen-Poiseuille law. Dividing the volumetric flow rate by
the flow area gives the average velocity as

(9.1-84)