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

Dr. R. Nagarajan

Professor

Dept of Chemical Engineering

IIT Madras

Advanced Transport PhenomenaModule 2 Lecture 4

Conservation Principles: Mass Conservation

CONSERVATION EQUATIONS

FORM OF EQUATION IN FIXED, MACROSCOPIC CV

where ( ) applies to:Mass,Momentum,Energy, orEntropy

( )

( ) ( )

( )

Rate of Net outflow rate Net inflow rate

accumulation of by convection of by diffusion

in CV across CS across CS

Net source

of within CV

USES OF MACROSCOPIC CV EQUATIONS CONTD…

To test predictions or measurements for overall conservation

To solve “black box” problemsTo derive finite-difference (element) equations

using arbitrary, coarse meshesAs a starting point for deriving multiphase flow

conservation equationsAt discontinuities, provide “jump conditions” and

appropriate boundary conditionse.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 0e.g., in cartesian coordinates, divide by x y z

Local “divergence” of ( ) is defined by:

0

1im ( ) " " ( ) ( )L

Net outflow

L associated with Local divergence of divVacross CS

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

SimplestCannot be created or consumed by

chemical reactionsCannot diffuse

Conservation equation is, therefore, simplified to two terms:

0

Rate of mass Net outflow rate

accumulation of mass by convection

in CV across CS

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

0v v

dV dA

v.n

FIXED (EULERIAN) CONTROL VOLUME

TOTAL MASS CONSERVATION CONTD…Formulation in differential CV (local PDE):

“continuity” equationAlso applies across “surface of discontinuity”,

which may itself be moving:e.g., premixed flame front

Expressed per unit area of surfaceUsually, accumulation term negligible

0divt

v

Eq. for surface of discontinuity simplifies to:

0

Net outflow rate

of mass by convection

relative to the CS

TOTAL MASS CONSERVATION CONTD…

CHEMICAL SPECIES MASS CONSERVATIONMass transport can occur by diffusion as well as

convectionNet production (generation – consumption) is a

result of all homogeneous reactionsConservation equation in Fixed CV:

inflow

Rate of accumulation Net outflow rate

of species i mass of species i mass by convection

within in CV across CS

Net rate Net chemical source

of species i mass by diffusion strength o

across CS

f species i

mass within CV

Definitions:

Convective flux of species mass = i v = i v

Total local flux of species i =

Diffusion flux of species i, ji” = - i v

Net rate of production of species i mass per unit

volume (via homogeneous chemical reactions)

=

CHEMICAL SPECIES MASS CONSERVATION CONTD…

im

im

ri

In PDE form:

CHEMICAL SPECIES MASS CONSERVATIONCONTD…

''. .i i is VdV dA dA r dV

t

''is

v.n j n

( ( 1,2....., )ii idiv div r i N

t

''iv) j

“Jump condition” for surface of discontinuity:

inflow

Net outflow rate Net rate

of species i mass by convection of species i mass by diffusion

relative to the CS across CS

Net chemical source

of species i per unit

area of discontinuity


“Pillbox” Control Volume


All but one of N species mass balance equations

are independent of total mass balance.

1 1 1

, 0, 0N N N

i ii i i

r

''ij

'' ''

1 1

'' 0N N

i ii i

then

m m v, j


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.


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”

0 ( 1,2,...., )elemkr k N

CHEMICAL ELEMENT MASS CONSERVATION CONTD…

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

( ) ( ) .k kV s sdV dA ndA

t

''

kv.n j

CHEMICAL ELEMENT MASS CONSERVATION

''kj

kth element conservation law in local PDE form:

“Jump condition” for kth element mass transfer across surface of discontinuity:

inflow

the element by diffusion th th

Net outflow rate of the Net rate of

k element by convection k

relative to CS across CS

( ) ''( )(kk kdiv div

t

v) j


All but one of Nelem element mass balance

equations are independent of total mass balance.

''( ) ( )

1 1

, 0elem elemN N

k kk k

j


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