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

CONSERVATION EQUATIONS

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

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.

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  x  y  z  Local “divergence” of ( ) is defined by:

 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…

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

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

MASS CONSERVATION  Total Mass Conservation  Chemical Species Mass Conservation  Chemical Element Mass Conservation

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

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

FIXED (EULERIAN) CONTROL VOLUME

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

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

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:

 Definitions:  Convective flux of species mass =  i v =  i  v  Total local flux of species i =  Diffusion flux of species i, j i ” = -  i v  Net rate of production of species i mass per unit volume (via homogeneous chemical reactions) = CHEMICAL SPECIES MASS CONSERVATION CONTD…

 In PDE form: CHEMICAL SPECIES MASS CONSERVATION CONTD…

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

“ Pillbox” Control Volume CHEMICAL SPECIES MASS CONSERVATION CONTD…

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

 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…

 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

 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…

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

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

 All but one of N elem element mass balance equations are independent of total mass balance. CHEMICAL ELEMENT MASS CONSERVATION CONTD…