Page 354 - Bird R.B. Transport phenomena

338 Chapter 11 The Equations of Change for Nonisothermal Systems

(ii) For a fluid flowing in a constant pressure system, Dp/Dt = 0, and

2
C j^ = kV T (11.2-8)
P p
]
(iii) For a fluid with constant density, (d In p/д In T) = 0, and
p
2
C j^ = kV T (11.2-9)
P p
(iv) For a stationary solid, v is zero and

2
C ^ = kV T (11.2-10)
P p
These last five equations are the ones most frequently encountered in textbooks and re-
search publications. Of course, one can always go back to Eq. 11.2-5 and develop less re-
strictive equations when needed. Also, one can add chemical, electrical, and nuclear
source terms on an ad hoc basis, as was done in Chapter 10.
Equation 11.2-10 is the heat conduction equation for solids, and much has been writ-
2
ten about this famous equation developed first by Fourier. The famous reference work
by Carslaw and Jaeger deserves special mention. It contains hundreds of solutions of this
equation for a wide variety of boundary and initial conditions. 3

§11.3 THE BOUSSINESQ EQUATION OF MOTION
FOR FORCED AND FREE CONVECTION
The equation of motion given in Eq. 3.2-9 (or Eq. В of Table 3.5-1) is valid for both
isothermal and nonisothermal flow. In nonisothermal flow, the fluid density and viscos-
ity depend in general on temperature as well as on pressure. The variation in the density
is particularly important because it gives rise to buoyant forces, and thus to free convec-
tion, as we have already seen in §10.9.
The buoyant force appears automatically when an equation of state is inserted into
the equation of motion. For example, we can use the simplified equation of state intro-
duced in Eq. 10.9-6 (this is called the Boussinesq approximation) 1
p(T) = p-pp(T-f) (11.3-1)
in which /3 is —{\/р){др/дТ) р evaluated at T = T. This equation is obtained by writing
the Taylor series for p as a function of T, considering the pressure p to be constant, and
keeping only the first two terms of the series. When Eq. 11.3-1 is substituted into the pg
term (but not into the p(Dv/Dt) term) of Eq. В of Table 3.5-1, we get the Boussinesq equation:

(11.3-2)
~ T)

1
The assumption of constant density is made here, instead of the less stringent assumption that
{d In piд In T) = 0, since Eq. 11.2-9 is customarily used along with Eq. 3.1-5 (equation of continuity for
p
constant density) and Eq. 3.5-6 (equation of motion for constant density and viscosity). Note that the
hypothetical equation of state p = constant has to be supplemented by the statement that {др/дТ) =
р
finite, in order to permit the evaluation of certain thermodynamic derivatives. For example, the relation
1 ( д In p\ (dp\
г -Q V = --[ - — (11.2-9a)
' P \d In T) P \dT) p
for the "incompressible fluid" thus defined.
leads to the result that C f , = C v
2
J. B. Fourier, Theorie analytique de la chaleur, CEuvres de Fourier, Gauthier-Villars et Fils, Paris (1822).
H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, 2nd edition (1959).
3
J. Boussinesq, Theorie Analytique de Chaleur, Vol. 2, Gauthier-Villars, Paris (1903).
1