Equivalence of Lagrangian and Eulerian Formulations in Conservation Equations

1. Advanced Transport Phenomena Module 2 Lecture 7 Conservation Equations: Alternative Formulations Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

2. Conservation Equations: Alternative Formulations

3. 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,

4. 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

5. 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

6. 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.

7. 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:

8. EQUIVALENCE BETWEEN LAGRANGIAN & EULERIAN FORMULATIONS CONTD… In particular, each of the local species mass and element. Mass balance equations can now be expressed:

9. 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

10. 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 .

11. 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

12. 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

13. 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:

14. 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

15. ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE CONTD… In terms of this so–called “extra stress,” Eq. becomes:

16. 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.

17. ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE CONTD… For a mixture of ideal gases this relation simplifies considerably to : Alternatively, in terms of mole fractions

18. 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,

19. 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.

20. 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).

21. 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).

22. ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE CONTD… Using Eqs and a PDE for is readily derived (Eq. ),

23. ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE CONTD… and its RHS indeed contains (in addition to ) the explicit “chemical”-energy source term: Where

24. ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE CONTD… Finally, addition of the equation for allows us to construct the following PDE for the “total” enthalpy

25. 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.

26. 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

27. 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.

28. 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.

29. 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.

30. 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).

31. MACROSCOPIC MECHANICAL-ENERGY EQUATION (GENERALIZED BERNOULLI EQUATION) CONTD… • Steady-flow liquid pumps, fans, and turbines, relating the work required for unit mass flow to the net outflow of .

32. 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”.

33. 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.

34. 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.