dr. r. nagarajan professor dept of chemical engineering iit madras
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Advanced Transport Phenomena Module 2 Lecture 4. Conservation Principles: Mass Conservation. Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras. CONSERVATION EQUATIONS. FORM OF EQUATION IN FIXED, MACROSCOPIC CV. where ( ) applies to: Mass, Momentum, Energy, or - PowerPoint PPT PresentationTRANSCRIPT
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Dr. R. Nagarajan
Professor
Dept of Chemical Engineering
IIT Madras
Advanced Transport PhenomenaModule 2 Lecture 4
Conservation Principles: Mass Conservation
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CONSERVATION EQUATIONS
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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
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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.
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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
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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…
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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
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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
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MASS CONSERVATION
Total Mass Conservation
Chemical Species Mass Conservation
Chemical Element Mass Conservation
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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
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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
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FIXED (EULERIAN) CONTROL VOLUME
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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
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Eq. for surface of discontinuity simplifies to:
0
Net outflow rate
of mass by convection
relative to the CS
TOTAL MASS CONSERVATION CONTD…
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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
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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
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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
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“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
CHEMICAL SPECIES MASS CONSERVATIONCONTD…
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“Pillbox” Control Volume
CHEMICAL SPECIES MASS CONSERVATIONCONTD…
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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
CHEMICAL SPECIES MASS CONSERVATIONCONTD…
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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 CONSERVATIONCONTD…
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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
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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…
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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
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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
CHEMICAL ELEMENT MASS CONSERVATION CONTD…
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All but one of Nelem element mass balance
equations are independent of total mass balance.
''( ) ( )
1 1
, 0elem elemN N
k kk k
j
CHEMICAL ELEMENT MASS CONSERVATION CONTD…