Dr. R. Nagarajan

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Advanced Transport Phenomena Module 2 Lecture 7
Conservation Equations Alternative Formulations

• Dr. R. Nagarajan
• Professor
• Dept of Chemical Engineering
• IIT Madras

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• Conservation Equations Alternative Formulations

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EQUIVALENCE BETWEEN LAGRANGIAN EULERIAN
FORMULATIONS

• To express this relation, we introduce here the
  notion that each field quantity f(x, t) (
  including vectors) possesses a local spatial
  gradient defined such that the projection of the
  vector grad f in any direction gives the spatial
  derivative of that scalar f in that direction
  thus,

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EQUIVALENCE BETWEEN LAGRANGIAN EULERIAN
FORMULATIONS CONTD

• Where is the unit vector in the direction of
  increasing coordinate is the length
  increment associated with an increment
  in the coordinate

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EQUIVALENCE BETWEEN LAGRANGIAN EULERIAN
FORMULATIONS CONTD

• If we define the local material derivative of f
  in the following reasonably way
• and expand in terms of f(x, t) using a
  Taylor series about x, t, that is

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EQUIVALENCE BETWEEN LAGRANGIAN EULERIAN
FORMULATIONS CONTD

• This valuable kinematic interrelation17 now
  allows each of the above-mentioned primitive
  conservation equations to be re-expressed in an
  equivalent Eulerian form.

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EQUIVALENCE BETWEEN LAGRANGIAN EULERIAN
FORMULATIONS CONTD

• Then it follows from equation
• that an observer moving in that local fluid
  velocity v(x, t) will record

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EQUIVALENCE BETWEEN LAGRANGIAN EULERIAN
FORMULATIONS CONTD

• In particular, each of the local species mass and
  element. Mass balance equations
• can now be expressed

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EQUIVALENCE BETWEEN LAGRANGIAN EULERIAN
FORMULATIONS CONTD

• Note that there is a nonzero species i mass
  convective term only when the vectors and
  are both perpendicular, there is no
  convective contribution to the species i mass
  balances despite the presence of total mass
  convection

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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE

• The local material derivative also provides a
  convenient shorthand for making changes in
  dependent variables, as shown below. The
  distributive property of differentiation makes it
  clear that if we derive an equation for
  and subtract it from the equation for
  we can construct an equation for
  .

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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE
CONTD

• The latter could then be used to generate an
  equation for by the addition of
  ,
• etc.
• If the linear- momentum conservation (balance)
  Eq. is multiplied, term by term, by v (scalar
  product ), we obtain the following equation for
  

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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE
CONTD

• Subtracting this from the equation of energy
  conservation , Eqn.
• permits us to write
• where we have introduced the short hand notation

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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE
CONTD

• Equation therefore governs the rate of change of
  the specific internal energy of a fluid parcel
• By adding
  
  to both the RHS and
  lhs of this equation , we obtain an equation
  governing the rate of change of specific enthalpy
  of a fluid
  parcel

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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE
CONTD

• This equation can be simplified by rewriting it
  in terms of that part of the contact stress(T)
  left after subtracting the local thermodynamics
  pressure

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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE
CONTD

• In terms of this socalled extra stress, Eq.
• becomes

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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE
CONTD
At this point we reiterate that the specific
enthalpy, , of the mixture includes
chemical (bond-energy) contributions, and must be
calculated from a constitutive reaction of the
general form where the values are the
partial specific enthalpies in the prevailing
mixture.
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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE
CONTD

• For a mixture of ideal gases this relation
  simplifies considerably to
• Alternatively, in terms of mole fractions

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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE
CONTD

• Here the are the
  absolute molar enthalpies of the pure
  constituents, that is,
• Where is the molar heat of
  formation of species i that is,

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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE
CONTD
the enthalpy change across the stoichiometric
in which one mole of species i as formed from its
constituent chemical elements in some
(arbitrarily chosen) reference states (e.g.
H2(g), O2(g) and C (graphite) at Tref
298 K.
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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE
CONTD

• Because of the (implicit) inclusion in of the
  heat of formation in h, the local energy
  addition term appearing on the RHS of the
  PDE, (Eq.)
• is not associated with chemical reactions (this
  would give rise to a double-counting error).

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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE
CONTD

• An explicit chemical-energy generation term
  enters energy equations only when expressed in
  terms of a sensible- (or thermal-) energy
  density dependent variable, such as
• (or T itself).

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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE
CONTD
Using Eqs and a PDE for is readily
derived (Eq. ),
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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE
CONTD
and its RHS indeed contains (in addition to )
the explicit chemical-energy source
term Where
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ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE
CONTD

• Finally, addition of the equation for
  allows us to construct the following PDE for
  the total enthalpy

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MACROSCOPIC MECHANICAL-ENERGY EQUATION
(GENERALIZED BERNOULLI EQUATION)

• A widely used macroscopic mechanical energy
  balance can be derived from our equation for
  
• (Eq. ) by combining
• term-by-term volume integration, Gauss theorem,
  and the rule for differentiating products.

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MACROSCOPIC MECHANICAL-ENERGY EQUATION
(GENERALIZED BERNOULLI EQUATION) CONTD

• We state the result for the important special
  case of incompressible flow subject to gravity
  as the only body force, being expressible in
  terms of the spatial gradient of a time
  independent potential function , that is

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MACROSCOPIC MECHANICAL-ENERGY EQUATION
(GENERALIZED BERNOULLI EQUATION) CONTD

• Then for any fixed macroscopic CV
• The corresponding results for a variable-density
  fluid (flow) (Bird, et al. (1960)) are rather
  more complicated than Eq. above, and not,
  exclusively, mechanical in nature.

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MACROSCOPIC MECHANICAL-ENERGY EQUATION
(GENERALIZED BERNOULLI EQUATION) CONTD

• In contrast to Eq.
• note that Eq. (from 59)
• makes no reference to changes in thermodynamic
  internal energy, nor surface or volume heat
  addition hence the name mechanical
  energy equation.

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MACROSCOPIC MECHANICAL-ENERGY EQUATION
(GENERALIZED BERNOULLI EQUATION) CONTD

• Common applications of Eq.
• are to the cases of
• Passive steady-flow component (pipe length,
  elbow, valve, etc.) on the control surfaces of
  which the work done by the extra stress can be
  neglected.

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MACROSCOPIC MECHANICAL-ENERGY EQUATION
(GENERALIZED BERNOULLI EQUATION) CONTD
Then there must be a net inflow of
to compensate for the
volume integral of T grad v, a positive
quantity shown in Section to be the local
irreversible dissipation rate of mechanical
energy ( into heat).
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MACROSCOPIC MECHANICAL-ENERGY EQUATION
(GENERALIZED BERNOULLI EQUATION) CONTD

1. Steady-flow liquid pumps, fans, and turbines,
   relating the work required for unit mass flow to
   the net outflow of .
   

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MACROSCOPIC MECHANICAL-ENERGY EQUATION
(GENERALIZED BERNOULLI EQUATION) CONTD
For (b), in cases with a single inlet and single
outlet Eq. may be rewritten in the
engineering form.
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MACROSCOPIC MECHANICAL-ENERGY EQUATION
(GENERALIZED BERNOULLI EQUATION) CONTD

• Here the indicating sum (RHS) accounts for all
  viscous dissipation losses in fluid-containing
  portions of the system ( other than those
  contained in the excluded pumping-device
  shown in Figure), and , given by
• is the rate at which the mechanical work is done
  on the fluid by the indicated pumping Device.

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MACROSCOPIC MECHANICAL-ENERGY EQUATION
(GENERALIZED BERNOULLI EQUATION) CONTD
Note that the work requirement per-unit-mass-
flow is the sum of that required to
change and that required to
overcome the prevailing viscous dissipation
losses throughout the system. With a suitable
change in signs, this equation can clearly also
be used to predict the output of a turbine
system for power extraction from the fluid.