Physical Chemistry Third Edition

454 10 Transport Processes

Surface of

tube (radius R)

Surface of

imaginary

cylinder

(radiusr)

System of

flowing fluid

Figure 10.5 Flowing Fluid in a Tube.

The magnitude of the hydrostatic force on the liquid in the imaginary cylinder is equal

to the difference of the pressure at its ends times the cross-sectional area of the cylinder:

Fh(P 2 −P 1 )πr^2

whereP 2 is the pressure at one end of the tube andP 1 is the pressure at the other end.

Equating the magnitudes ofFh/AandFf/Agives

(P 2 −P 1 )r

2 Lη



∣

∣

∣

∣

duz

dr

∣

∣

∣

∣

duz

dr

We assume thatuzis negative and conclude thatduz/dris positive since the magnitude

of the flow velocity is larger in the center of the tube. We multiply both sides of this

equation bydrand integrate fromrRtorr′, whereRis the radius of the tube

andr′is some value ofrinside the tube:

∫r′

R

duz

dr

dr

∫uz(r′)

uz(R)

duz

P 2 −P 1

2 Lη

∫r′

R

rdr

If the liquid does not slip on the tube wallsuz(R) will vanish, and the integration yields

uz(r)

P 2 −P 1

4 Lη

(

r^2 −R^2

)

(10.2-23)

where we replacer′byr. This parabolic dependence of the flow velocity on position

is represented in Figure 10.6. The length of each arrow in the figure is proportional to

the flow velocity at its location.

Tube

wall

R
radius

of tube

Fluid flow

velocity

vectors

Tube

wall

Figure 10.6 The Fluid Velocity in

a Tube with Laminar Flow.

The total rate of flow of the liquid through the tube can be computed by considering

a cylindrical shell of thicknessdrand radiusrconcentric with the walls of the tube

and then adding up the contributions of all such shells. The volume of the fluid in this

shell that flows out of the tube in 1 s (the contribution of the shell to the volume rate of

flow) is equal to the cross-sectional area of the shell times a length equal to the distance