Page 328 - Bird R.B. Transport phenomena

312 Chapter 10 Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

Fig. 10.8-2. Heating of a fluid in laminar flow through a cir-
cular tube, showing the annular ring over which the energy
balance is made.
Fluid inlet
temperature T^

The energy balance is obtained by summing these contributions and equating the
sum to zero. Then we divide by 2тг Ar Az to get
е.'
( r e ) \ - ( г £ > ) | г + Д г е \ - ^zlz-ь
r
7 zlz
г
?
r
Ar + ' r Az + PVgT = 0 (10.8-7)
z z
In the limit as Ar and Az go to zero, we find
(10.8.8)
The subscript z in g z has been omitted, since the gravity vector is acting in the +z direc-
tion.
Next we use Eqs. 9.8-6 and 9.8-8 to write out the expressions for the r- and z-compo-
nents of the combined energy flux vector, using the fact that the only nonzero compo-
nent of v is the axial component v :
z
— v<LL (10.8-9)
e = T v + q = - ^ K
r n z r v d r dr
e = ikpvl)v + pHv + r v + q
z z z zz z z
dv.
(p - f)v + C (T - T)v -\2n-±\v - (10.8-10)
z P p z dz z dz
Substituting these flux expressions into Eq. 10.8-8 and using the fact that v depends only
z
on r gives, after some rearrangement,

The second bracket is exactly zero, as can be seen from Eq. 3.6-4, which is the z-component
of the equation of motion for the Poiseuille flow in a circular tube (here & = p — pgz). The