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

1. Advanced Transport Phenomena Module 2 Lecture 4 Conservation Principles: Mass Conservation Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

2. CONSERVATION EQUATIONS

3. FORM OF EQUATION IN FIXED, MACROSCOPIC CV where ( ) applies to: Mass, Momentum, Energy, or Entropy

4. USES OF MACROSCOPIC CV EQUATIONS CONTD… To test predictions or measurements for overall conservation To solve “black box” problems To derive finite-difference (element) equations using arbitrary, coarse meshes As a starting point for deriving multiphase flow conservation equations At discontinuities, provide “jump conditions” and appropriate boundary conditions e.g., shock waves, flames, etc.

5. FORM OF EQUATION IN FIXED, DIFFERENTIAL CV Divide each term in macroscopic CV equation by V, and pass to the limit V  0 e.g., in cartesian coordinates, divide by Dx Dy Dz Local “divergence” of ( ) is defined by:

6. div ( ) = local outflow associated with the flux of ( ), calculated on a per-unit-volume basis - div ( ) = net inflow per unit volume PDE’s result FORM OF EQUATION IN FIXED, DIFFERENTIAL CV CONTD…

7. USES OF DIFFERENTIAL CV EQUATIONS Predict detailed distribution of flow properties within region of interest Extract flux laws/ coefficients from measurements in simple flow systems Provide basis for estimating important dimensionless parameters governing a chemically reacting flow

8. USES OF DIFFERENTIAL CV EQUATIONS CONTD… Derive finite-difference (algebraic) equations for numerically approximating field densities Derive entropy production expression and provide guidance for proper choice of constitutive laws

9. MASS CONSERVATION Total Mass Conservation Chemical Species Mass Conservation Chemical Element Mass Conservation

10. Simplest Cannot be created or consumed by chemical reactions Cannot diffuse Conservation equation is, therefore, simplified to two terms: TOTAL MASS CONSERVATION

11. TOTAL MASS CONSERVATION CONTD… Or, mathematically, as the following integral constraint: where rv . ndA mass flow through area ndA per unit time, and Integral  summation over all such control surface elements in overall CS

12. FIXED (EULERIAN) CONTROL VOLUME

13. TOTAL MASS CONSERVATION CONTD… Formulation in differential CV (local PDE): “continuity” equation Also applies across “surface of discontinuity”, which may itself be moving: e.g., premixed flame front Expressed per unit area of surface Usually, accumulation term negligible

14. TOTAL MASS CONSERVATION CONTD… • Eq. for surface of discontinuity simplifies to:

15. CHEMICAL SPECIES MASS CONSERVATION Mass transport can occur by diffusion as well as convection Net production (generation – consumption) is a result of all homogeneous reactions Conservation equation in Fixed CV:

16. Definitions: Convective flux of species mass = riv = wirv Total local flux of species i = Diffusion flux of species i, ji” = - riv Net rate of production of species i mass per unit volume (via homogeneous chemical reactions) = CHEMICAL SPECIES MASS CONSERVATION CONTD…

17. In PDE form: CHEMICAL SPECIES MASS CONSERVATION CONTD…

18. “Jump condition” for surface of discontinuity: CHEMICAL SPECIES MASS CONSERVATION CONTD…

19. CHEMICAL SPECIES MASS CONSERVATION CONTD… “Pillbox” Control Volume

20. All but one of N species mass balance equations are independent of total mass balance. CHEMICAL SPECIES MASS CONSERVATION CONTD…

21. When some chemical species are ionic in nature (e.g., solution electrochemistry, electrical discharges in gases, etc.), principle of “electric charge conservation” comes into effect. CHEMICAL SPECIES MASS CONSERVATION CONTD…

22. Used widely in analysis of chemically reacting flows: Fewer in number Conservation equations identical in form to those governing inert (e.g., tracer) species CHEMICAL ELEMENT MASS CONSERVATION

23. Similar in structure to species conservation equation, except that…. For conventional (extra-nuclear) chemical reactions, no element can be locally produced, however complex the reaction. Elements can “change partners” CHEMICAL ELEMENT MASS CONSERVATION CONTD…

24. kth element conservation equation for a fixed macroscopic CV is thus “source-free”: = diffusion flux of kth element = weighted sum of fluxes of chemical species containing element k CHEMICAL ELEMENT MASS CONSERVATION

25. kth element conservation law in local PDE form: “Jump condition” for kth element mass transfer across surface of discontinuity: CHEMICAL ELEMENT MASS CONSERVATION CONTD…

26. All but one of Nelem element mass balance equations are independent of total mass balance. CHEMICAL ELEMENT MASS CONSERVATION CONTD…