Transport Phenomena.pdf

§10.9 Free Convection 317
fluid in the upward-moving stream is the same as that in the downward-moving
stream. The plates are presumed to be very tall, so that end effects near the top and bottom
can be disregarded. Then for all practical purposes the temperature is a function of
у alone.
An energy balance can now be made over a thin slab of fluid of thickness Ay, using
the y-component of the combined energy flux vector e as given in Eq. 9.8-6. The term
containing the kinetic energy and enthalpy can be disregarded, since the y-component of
the v vector is zero. The y-component of the term [т • v] is r yz
v z
= —fji{dv z
/dy)v z
, which
would lead to the viscous heating contribution discussed in §10.4. However, in the very
slow flows encountered in free convection, this term will be extremely small and can be
neglected. The energy balance then leads to the equation
--^ = 0 or k ^ = 0 (10.9-1)
dy dy 2
for constant k. The temperature equation is to be solved with the boundary conditions:
B.C. 1: at у - -В, Т = T 2
(10.9-2)
B.C. 2: at у = +B, Т = Т, (10.9-3)
The solution to this problem is
T T
2 AT B
(10.9-4)
in which AT = T 2
- T }
is the difference of the wall temperatures, and T = \(T^ + T 2
) is
their arithmetic mean.
By making a momentum balance over the same slab of thickness Ay, one arrives at a
differential equation for the velocity distribution
d 2 v z
dp
*-&'1 + "
(m9- 5)
Here the viscosity has been assumed constant (see Problem 10B.11) for a solution with
temperature-dependent viscosity.
The phenomenon of free convection results from the fact that when the fluid is
heated, the density (usually) decreases and the fluid rises. The mathematical description
of the system must take this essential feature of the phenomenon into account. Because
the temperature difference AT = T 2
- Т г
is taken to be small in this problem, it can be expected
that the density changes in the system will be small. This suggests that we should
expand p in a Taylor series about the temperature T = \{T^ + T 2
) thus:
d
p
_(T-T)
т=т
= p- pp(T - T) + • • • (10.9-6)
Here p and /3 are the density and coefficient of volume expansion evaluated at the temperature
T. The coefficient of volume expansion is defined as
\ __I(*P
We now introduce the "Taylor-made" equation of state of Eq. 10.9-6 (keeping two terms
only) into the equation of motion in Eq. 10.9-5 to get
^
+
~ pg ^ ( T f ) (m9 ~ 8)