Page 315 - Bird R.B. Transport phenomena

§10.4 Heat Conduction with a Viscous Heat Source 299

Top surface moves with velocity v = ЯП Щ- 10.4-2. Modification of a portion of the flow
b
system in Fig. 10.4-1, in which the curvature of the
is neglected.
bounding surfaces
J
"t 2 / \ Л(х)

Stationary surface

As the outer cylinder rotates, each cylindrical shell of fluid "rubs" against an adja-
cent shell of fluid. This friction between adjacent layers of the fluid produces heat; that
is, the mechanical energy is degraded into thermal energy. The volume heat source re-
sulting from this "viscous dissipation," which can be designated by S , appears automat-
v
ically in the shell balance when we use the combined energy flux vector e defined at the
end of Chapter 9, as we shall see presently.
If the slit width b is small with respect to the radius R of the outer cylinder, then the
problem can be solved approximately by using the somewhat simplified system de-
picted in Fig. 10.4-2. That is, we ignore curvature effects and solve the problem in Carte-
sian coordinates. The velocity distribution is then v z = v {x/b), where v b = UR.
b
We now make an energy balance over a shell of thickness Ax, width W, and length L.
Since the fluid is in motion, we use the combined energy flux vector e as written in Eq.
9.8-6. The balance then reads

WLe \ - WLe \ = 0 (10.4-1)
x x x x+Ax
Dividing by WL Ax and letting the shell thickness Ax go to zero then gives

(10.4-2)
T
This equation may be integrated to give dx
e = Q (10.4-3)
x
Since we do not know any boundary conditions for e x/ we cannot evaluate the integra-
tion constant at this point.
We now insert the expression for e x from Eq. 9.8-6. Since the velocity component in
the x direction is zero, the term (\pv 2 + pLOv can be discarded. The x-component of q is
-k(dT/dx) according to Fourier's law. The x-component of [т • v] is, as shown in Eq.
9.8-1, T V + T v y + T V . Z Since the only nonzero component of the velocity is v z and
xy
XX
XZ
X
since r xz = -\xidvjdx) according to Newton's law of viscosity, the x-component of [т • v]
is -ixv (dv /dx). We conclude, then, that Eq. 10.4-3 becomes
z
z
(10.4-4)
When the linear velocity profile v z = v (x/b) is inserted, we get
b
(10.4-5)
2
in which /ji(v /b) can be identified as the rate of viscous heat production per unit volume S . v
b
When Eq. 10.4-5 is integrated we get
т - - г Ч л - [ (10.4-6)