Page 258 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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50 Fluid Dynamics (Chemical Engineering)

can show that the rate of increase of specific entropy is τ =−2µ a D, (36)
given by
where µ a is the apparent viscosity or viscosity function
Ds q 1 1
ρ =−∇ · + −τ : ∇v + q · ∇T + ˙ r CR , and D is the symmetric part of ∇v given by
Dt T T T
T
1
D = (∇v +∇v ). (37)
(31) 2
In general, µ a is a complex and often nonlinear function
where s is the specific entropy. Equation (31) is in the
of II D , the second principal invariant of D;II D is given by
Lagrangian form of Eq. (3) with ψ = ρs and where the
eqation of continuity has been invoked. Thus, we recog- 1 2
−II D = [(∇· v) − D : D]. (38)
2
nize the term −∇ · (q/T ) as the diffusive influx of entropy
and the production or generation of entropy as the remain- In the special case of a Newtonian fluid, µ a = µ is a con-
ing three terms on the right side of Eq. (31). In the absence stant called the viscosity of the fluid and Eq. (36) becomes
of chemical reactions, the principle of entropy production Newton’s “law” of viscosity. In a great many practical
(or “postulate of irreversibility,” as Truesdell has called it) cases of interest to chemical engineers, however, the non-
states that Newtonian form of Eq. (36) is encountered.
1 1 The formulation of proper constitutive relations is a
(−τ : ∇v) + q · ∇T ≥ 0. (32) complex problem and is the basis of the science of rheol-
T T 2
ogy, which cannot be covered here. This section presents
From Eq. (32) it follows that only the part −τ : ∇v of
only four relatively simple constitutive relations that have
the stress work contributes to the production of entropy;
provedtobepracticallyusefultochemicalengineers.Elas-
hence, it is the “irreversible” or nonrecoverable work. The
tic fluid behavior is expressly excluded from considera-
remainder of the stress work, expressed by v · (∇ · T), is
tion. The following equations are a listing of these consti-
“reversible” or recoverable, as already described.
tutive relations; many others are possible:
Bingham Plastics
F. Constitutive Relations
τ 0 1 2
The generic balance relations and the derived relations τ =−2 µ ∞ ± √ −2II D D, 2 τ :τ >τ 0 (39)
2

presented in the preceding section contain various diffu-
sion flux tensors. Although the equation of continuity as 0 = D, 1 2 τ : τ ≤ τ 0 2 (40)
presented does not contain a diffusion flux vector, were it
Ostwald–DeWael or Power Law
to have been written for a multicomponent mixture, there
would have been such a diffusion flux vector. Before any n−1
t =−2k 2 −2II D D (41)

of these equations can be solved for the various field quan-
tities, the diffusion fluxes must be related to gradients in Herschel–Bulkley or Yield Power Law
the field potentials φ.
n−1 τ 0
In general, the fluxes are related to gradients of the τ =−2 k 2 −2II D ± √ D

specific concentrations by relations of the form 2 −2II D
1 2
j Dψ =−β∇φ (33) τ :τ >τ (42)
2 0
or 0 = D, 1 τ :τ ≤ τ 2 (43)
2 0
j dψ =−B ·∇φ. (34)
Casson
In the form of Eq. (33) β is a scalar parameter called a τ ±τ 0 µ ∞
transport coefficient. In the form of Eq. (34) B is a ten- 1/2 =−2 + 1/2 D
|2 − 2II τ | |2 − 2II D | |2 − 2II D |
sor, the elements of which are the transport coefficients.
1 2
In either form the transport coefficients may be complex 2 τ :τ >τ 0 (44)
nonlinear functions of the scalar invariants of ∇φ. 1 2
0 = D, τ :τ ≤ τ 0 (45)
For isotropic fluids the heat flux vector q takes the form 2
When these constitutive relations are coupled with the
q =−k T ∇T, (35)
stress distributions derived from the equations of motion,
where k T is the thermal conductivity. Equation (35) is details of the velocity fields can be calculated, as can the
known as Fourier’s law of conduction. The momentum overall relation between pressure drop and volume flow
flux tensor τ is expressed in the form rate.