Page 372 - Bird R.B. Transport phenomena

354 Chapter 11 The Equations of Change for Nonisothermal Systems

Just as the dimensional analysis discussion in §3.7 provided an introduction for the
discussion of friction factors in Chapter 6, the material in this section provides the back-
ground needed for the discussion of heat transfer coefficient correlations in Chapter 14.
As in Chapter 3, we write the equations of change and boundary conditions in dimen-
sionless form. In this way we find some dimensionless parameters that can be used to
characterize nonisothermal flow systems.
We shall see, however, that the analysis of nonisothermal systems leads us to a
larger number of dimensionless groups than we had in Chapter 3. As a result, greater re-
liance has to be placed on judicious simplifications of the equations of change and on
carefully chosen physical models. Examples of the latter are the Boussinesq equation of
motion for free convection (§11.3) and the laminar boundary layer equations (§12.4).
As in §3.7, for the sake of simplicity we restrict ourselves to a fluid with constant /x,
k, and С . The density is taken to be p = p — pj8(T — T) in the pg term in the equation of
p
motion, and p = ]э everywhere else (the "Boussinesq approximation"). The equations of
change then become with p + ~pgh expressed as 2P,
Continuity: (V • v) = 0 (11.5-1)
Motion: ^Щ = ~ V<3> + ^ 2у + ^ ( Т ~ ^ (И.5-2)

2
Energy: pC ^ = kV T + /хФ, (11.5-3)
p
We now introduce quantities made dimensionless with the characteristic quantities (sub-
script 0 or 1) as follows:

i = £ y = l 2 = 1 } = V A (Ц.5-4)
'0 *0 '0 *0
£ # ^ £ t IZ?L (Ц.5-5)
Т,-Т о
R: (11.5-6)

Here / , v , and SP are the reference quantities introduced in §3.7, and T and^ Tj are
0
0
o
0
temperatures appearing in the boundary conditions. In Eq. 11.5-2 the value T is the
temperature around which the density p was expanded.
In terms of these dimensionless variables, the equations of change in Eqs. 11.5-1 to 3
take the forms
Continuity: (V • v) = 0 (11.5-7)
Motion: Щ=-*Ф + Ш * * - pWT,- Wflf - f) (11.5-8)
Dt II ^ II 11 l J I W
2
1
Energy: Щ- = \~К-Ь Т + - ^ ^ к (11.5-9)
Dt 4 VopC i KpCpiT, - T )Л
Q
o
p
The characteristic velocity can be chosen in several ways, and the consequences of the
choices are summarized in Table 11.5-1. The dimensionless groups appearing in Eqs.
11.5-8 and 9, along with some combinations of these groups, are summarized in Table
11.5-2. Further dimensionless groups may arise in the boundary conditions or in the
equation of state. The Froude and Weber numbers have already been introduced in §3.7,
and the Mach number in Ex. 11.4-7.
We already saw in Chapter 10 how several dimensionless groups appeared in the
solution of nonisothermal problems. Here we have seen that the same groupings appear
naturally when the equations of change are made dimensionless. These dimensionless
groups are used widely in correlations of heat transfer coefficients.