Page 355 - Bird R.B. Transport phenomena

§11.4 Use of the Equations of Change to Solve Steady-State Problems 339
This form of the equation of motion is very useful for heat transfer analyses. It describes
the limiting cases of forced convection and free convection (see Fig. 10.8-1), and the re-
gion between these extremes as well. In forced convection the buoyancy term -pg/3(T - T)
is neglected. In free convection (or natural convection) the term (-Vp + pg) is small, and
omitting it is usually appropriate, particularly for vertical, rectilinear flow and for the
flow near submerged objects in large bodies of fluid. Setting (—Vp + pg) equal to zero is
equivalent to assuming that the pressure distribution is just that for a fluid at rest.
It is also customary to replace p on the left side of Eq. 11.3-2 by p. This substitution has
been successful for free convection at moderate temperature differences. Under these con-
ditions the fluid motion is slow, and the acceleration term Dv/Dt is small compared to g.
However, in systems where the acceleration term is large with respect to g, one must
also use Eq. 11.3-1 for the density on the left side of the equation of motion. This is par-
ticularly true, for example, in gas turbines and near hypersonic missiles, where the term
(p - ~p)Dv/Dt may be at least as important as pg.

USE OF THE EQUATIONS OF CHANGE
TO SOLVE STEADY-STATE PROBLEMS

In §§3.1 to 3.4 and in §§11.1 to 11.3 we have derived various equations of change for a
pure fluid or solid. It seems appropriate here to present a summary of these equations
for future reference. Such a summary is given in Table 11.4-1, with most of the equations
given in both the д I dt form and the D/Dt form. Reference is also made to the first place
where each equation has been presented.
Although Table 11.4-1 is a useful summary, for problem solving we use the equa-
tions written out explicitly in the several commonly used coordinate systems. This has
been done in Appendix B, and readers should thoroughly familiarize themselves with
the tables there.
In general, to describe the nonisothermal flow of a Newtonian fluid one needs

• the equation of continuity
® the equation of motion (containing /JL and к)
© the equation of energy (containing /x, к, and k)
© the thermal equation of state (p = p{p T))
f
e
the caloric equation of state (C p = C (p, T))
p
as well as expressions for the density and temperature dependence of the viscosity, di-
latational viscosity, and thermal conductivity. In addition one needs the boundary and
initial conditions. The entire set of equations can then—in principle—be solved to get the
pressure, density, velocity, and temperature as functions of position and time. If one
wishes to solve such a detailed problem, numerical methods generally have to be used.
Often one may be content with a restricted solution, for making an order-of-magni-
tude analysis of a problem, or for investigating limiting cases prior to doing a complete
numerical solution. This is done by making some standard assumptions:
(!) Assumption of constant physical properties. If it can be assumed that all physical
properties are constant, then the equations become considerably simpler, and
in some cases analytical solutions can be found.
(ii) Assumption of zero fluxes. Setting т and q equal to zero may be useful for (a) adi-
abatic flow processes in systems designed to minimize frictional effects (such as
Venturi meters and turbines), and (b) high-speed flows around streamlined ob-
jects. The solutions obtained would be of no use for describing the situation
near fluid-solid boundaries, but may be adequate for analysis of phenomena
far from the solid boundaries.