Page 44 - Modelling in Transport Phenomena A Conceptual Approach

2.2. DIMENSIOIVLESS NUMBERS 25

2.2 DIMENSIONLESS NUMBERS

Newton’s “law” of viscosity, Fourier’s “law” of heat conduction, and Fick’s first
“law” of diffusion, in reality, are not laws but defining equations for viscosity, p,
qy,
thermal conductivity, k, and diffusion coefficient, DAB. The fluxes (T~~, j~,)
and the gradients (dw,/dy, dT/dy, dpA/dy) must be known or measurable for the
experimental determination of p, k, and DAB.
Newton’s law of viscosity, &. (2.1-2), Fourier’s law of heat conduction, Eq.
(2.1-4), and Fick’s first law of diffusion, Eqs. (2.1-7) and (2.1-9), can be generalized
as
Molecular Transport Gradient of
driving force (2.2-1)
Although the constitutive equations are similar, they are not completely analogous
because the transport properties (p, k, VAB) have different units. These equations
can also be expressed in the following forms:

Pd
ryx = - --(PV2) p = constant ,owz = momentum/volume
P dY
(2.2-2)
k d
(pCp~) pep = constant
=
%I=--- pCp~ energy/volume
pcp dY
(2.2-3)
dpA
j~~ = -DAB- p = constant pA = mass of A/volume
dY
(2.2-4)
The term p/p in Eq. (2.2-2) is called momentum diflwiwity or kinematic vis-
cosity, and the term k/p(?p in &. (2.2-3) is called themal diffwivity. Momentum
and thermal diffusivities are designated by u and a, respectively. Note that the
terms u, a, and DAB all have the same units, m2/s, and Eqs. (2.2-2)-(2.2-4) can
be expressed in the general form as
Molecular Gradient of
( flu ) = (Diffusivity) ( Quantity/Volume (2.2-5)

The quantities that appear in Eqs. (2.2-1) and (2.2-5) are summarized in Table
2.1. Since the terms u, a, and VAB all have the same units, the ratio of any two
of these diffusivities results in a dimensionless number. For example, the ratio of
momentum diffusivity to thermal diffusivity gives the Prandtl number, Pr:

CPP
Prandtl number = Pr = - = - (2.2-6)
a k
The Prandtl number is a function of temperature and pressure. However, its ds
pendence on temperature, at least for liquids, is much stronger. The order of
magnitude of the Prandtl number for gases and liquids can be estimated as