Page 257 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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Encyclopedia of Physical Science and Technology En006G-249 June 27, 2001 14:7

Fluid Dynamics (Chemical Engineering) 49

D. Total Energy Balance Spherical Polar
Two types of energy terms must be considered: (1) ther- ∂v r 1 ∂v θ v r
mal and (2) mechanical. The specific internal energy is (τ : ∇v) = τ rr ∂r + τ θθ r ∂θ + r
u = C v T , where C v is the heat capacity and T is the tem-
2
perature of the fluid. The specific kinetic energy is v /2. 1 ∂v φ v r v θ cot θ
+ τ φφ + +
2
Thus, the total energy density is ρ(u + v /2) . Thermal en- r sin θ ∂φ r r
ergy diffuses into the fluid by means of a heat flux vector
∂v θ 1 ∂v r v θ
q. Mechanical energy diffuses in by means of work done + τ rθ + −
∂r r ∂θ r
against the stresses v · (−T). Energy may be produced in-

ternally in the fluid by chemical reactions at a rate ˙ r CR and ∂v φ 1 ∂v r v φ
+ τ rφ + −
by the action of external body forces v · ρg. Thus, Eq. (2) ∂r r sin θ ∂φ r
can be written as Eulerian-form total energy balance as
1 ∂v φ 1 ∂v θ cot θ
2
2
+ τ θφ + − v φ
∂ v v r ∂θ r sin θ ∂φ r
ρ u + =−∇ · ρ u + v
∂t 2 2 (28)
− ∇ · q − ∇ · [v · (−T)]
Equation (23) represents the total energy balance or first
+ v · ρg + ˙ r CR . (23) law of thermodynamics. It includes all forms of energy
transport. An independent energy equation, which does
By appropriate manipulation as before, this can be written
not represent a generic balance relation, is obtained by
in Lagrangian form as
performing the operation v · (equations of motion) and is
D v 2
ρ u + =−∇ · q − ∇ · [v · (−T)] D v 2
Dt 2 ρ =−v · ∇p − v · (∇ · τ) + v · ρg. (29)
Dt 2
+ v · ρg + ˙ r CR . (24)
This relation, called the mechanical energy equation, de-
By using Eq. (12) the term −∇ · [v · ( − T)] can be written
scribes the rate of increase of kinetic energy in a fluid
as
element as a result of the action of external body forces,
−∇ · [v · (−T)] =−∇ · (pv) − v · (∇· τ) − τ : ∇v. (25) pressure, and reversible stress work.
When Eq. (29) is subtracted from Eq. (24), one obtains
In Eq. (25) the term v · (∇· τ) represents reversible stress
work, while τ : ∇v represents irreversible or entropy- Du
producing stress work. The following are expressions for ρ =−∇ · q − p∇ · v = τ :∇v + ˙ r CR , (30)
Dt
the latter quantity in rectangular Cartesian, cylindrical po-
lar, and spherical polar coordinates: which is called the thermal energy equation. It describes
Cartesian the rate of increase of thermal internal energy of a fluid
element by the action of heat fluxes, chemical reactions,

∂v x ∂v y ∂v z
(τ : ∇v) = τ xx + τ yy + τ zz volumetric expansion of the fluid, and irreversible stress
∂x ∂y ∂z
work.

∂v x ∂v y ∂v y ∂v z Clearly, only two of the three energy equations are in-
+ τ xy + + τ yz + dependent, the third being obtained by sum or difference
∂y ∂x ∂z ∂y
from the first two. The coupling between Eqs. (29) and

∂v z ∂v x (30) occurs by means of Eq. (25), which represents the
+ τ zx + (26)
∂x ∂z total work done on the fluid element by the stress field.
Cylindrical Polar Neither Eq. (29) nor Eq. (30) is a balance relation by it-
self, but the sum of the two, Eq. (24), is.

∂v r 1 ∂v θ v r ∂v z
(τ : ∇v) = τ rr + τ θθ + + τ zz
∂r r ∂θ r ∂z
E. Entropy Production Principle
∂ v θ 1 ∂v r

+ τ rθ r +
∂r r r ∂θ We cannot write down a priori a generic balance relation
for the entropy of a fluid. We can, however, derive a result

1 ∂v z ∂v θ ∂v z ∂v r
+ τ θz + + τ rz + that can be placed in the same form as Eq. (3) and therefore
r ∂θ ∂z ∂r ∂z
recognized as a balance relation. By working with the
(27) combined first and second laws of thermodynamics, one