Equations of Change ChE 131

8/13/2019 Equations of Change ChE 131

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Equations of Change

ChE 131


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Shell Balance Approach. . .

Is a tool to train the student invisualizing transport problems.Introduces the Equations of Change sothat the student understands thephysical significance of each term.

Can be a tedious exercise if done forevery problem we encounter.


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Equations of Change

General Balance Equations (PDEs) that canbe applied to any problem

Steady- or unsteady-stateSimple or complexCartesian, cylindrical, spherical coordinatesystems

It is important to understand the physicalsignificance of each term (from shellbalance exercises) in order to write theappropriate equation for a system


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Equations of Change forIsothermal Systems

Equation of ContinuityEquation of Motion

Equation of Mechanical Energy(Kinetic Energy)


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Equation of Continuity

Total mass balance:

t

z y xt z y x

Note that if density were constant,

0

0

t

Divergence of ordiv

For incompressible fluids

Or,


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Equation of Motion

x-, y-, z-component momentum balances:

z zz yz xz

y zy yy xy

x zx yx xx

g z y x

g z y x

g z y xt


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Equation of Motion in vectornotation

Per component of velocity:

z y xi g t iii

,,,

Add all 3 components:

g t

recall:

p


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Equation of Motion

g pt

Rate ofmomentum

accumulation/volume

Rate of momentumaddition by

convection/ volume

Rate of momentumaddition bymolecular

transport/ volume

Externalforces/volume


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Equation of MechanicalEnergy

Dot product of equation of motion

g pt

Equation of change for kinetic energy

g

p pt

:

2212

21


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Different Time Derivatives

Consider that there are different ways of measuring theconcentration change with time in a tubular reactor

t c

1. Fix the sensor at one point in the reactor will give us the partialtime derivative

2. Install the sensor on a mount that can be moved up and downthe reactor independent of the fluid motion will give us the totaltime derivative dc/dt.

3. Let the sensor drift with the fluid as it flows down the reactor,

i.e., the sensor and the fluid move at the same velocity. Thisgives us the substantial time derivative

ct c

z c

yc

xc

t c

Dt Dc

z y x


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Special Cases

Navier-Stokes equation (constant , )

g p Dt

D 2

Stokes or Creeping flow equation (substantialderivative ~ 0 )

g p

20Euler or inviscid flow equation ( ~0)

g p Dt D

Flow throughporous media,particle motion

in fluids

Flow around bluffobjects (airplane

wings, piers, tanks)


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Equations of Change for Single-Component Non-IsothermalSystems

Equation of EnergyEquation of Change for Internal EnergyEquation of Change for EnthalpyEquation of Change for Temperature


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Forms of Energy Accountedfor in the Equation of Energy

Kinetic energy, K.E. = 2

Internal energy, U = U( , T)

Potential energy will be accounted forby the work done by gravityHeat by conduction

Work done by molecular mechanismsand by external forces (e.g. gravity,centrifugal, electromagnetic, etc.)


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Equation of Energy

g eU t

221

Recall that

q H qU e )

()

( 2

212

21


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Equation of Energy

g p

qU U t

2212

21

g

q H U t

2212

21

or

Note: nuclear, radiative, electromagnetic and chemicalsources of energy are not included. These are added assource terms or as boundary conditions as the case maybe.


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Equation of Internal EnergySubtract the Equation of Mechanical Energy from the

Equation of Energy

g

p pt

:

2

212

21

g

q H U t

2

212

21

: pqU U t


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Equation of Internal Energy interms of substantial derivative

: pqU Dt D

Recall that :

pU H

pU H

or


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Equation of Enthalpy

Dt Dp

q H Dt D

:

T k q

T k q2

Recall that:

And if k wereconstant,

Also,

:


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Using derived thermodynamicrelations,

dP T

dT C H d

dP

T T

dT C H d

dP T V

T V dT C H d

dP P H

dT T

H H d

T P H H

P p

P

p

P

p

T P

lnln1

1

11

,


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Equation of Change forTemperature

Dt Dp

T T k

Dt DT

C P

p

lnln


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Special Forms of the Equation ofEnergy

for systems with constant k at low to moderate fluidvelocities

For an ideal gas, ( ln/lnT) P=1

Dt Dp

T k Dt DT

C p 2

For fluid flowing at constant pressure, Dp/Dt=0

T k Dt DT

C p2


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Special Forms of the Equation ofEnergy

for systems with constant k at low to moderate fluidvelocities

For a fluid with constant density, ( ln/lnT) P= 0

T k Dt DT

C p2

For stationary solid, = 0

T k Dt DT

C p2


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Boussinesq Equation of Motion forForced and Free Convection

In a fluid where there are temperature gradients, the fluid density andviscosity will not be uniform resulting in natural or free convection

The equation of state may be written as a Taylor series expansiontruncated to two terms (a.k.a Boussinesq approximation)

T T T

T T g g p Dt D

Boussinesq Equation

Neglected inforced

convection

Neglected infree

convection


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Equations of Change forMulticomponent Mixtures

Mass and Molar Equations ofChange


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Equations of Continuity forMulticomponent MixturesOne equation of continuity may be written for each

specie in a mixture composed of N components

r nt

r jt

convection diffusionreaction

Note that if we add all the equations of continuity for all the energy definitions,

we will get the equation of continuity for a mixture.

Mass balance


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Equations of Continuity forMulticomponent MixturesOne equation of continuity may be written for each

specie in a mixture composed of N components

R N ct

R J cct **

convection diffusionreaction

Note that if we add all the equations of continuity of all the species in the

mixture, we will get the equation of continuity for a mixture.

Mole balance


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Special Cases of BinarySystems

A A AB A A r w Dwt

w

2

B A A B A AB A A

R x R x xcD xt

x

c

2

Mass balance for a binary system

Mole balance for a binary system


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Special Cases of BinarySystems

A A AB A A r w Dwt

w

2 Systems with zero flow, and where density and viscosity are constant

A AB

A c Dt

c 2

Or,

Ficks second law of diffusion

Used for diffusion in solids,stationary liquids and equimolar

counterdiffusion.


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How to use Equations of Changeto solve Transport Problems

Isothermal Newtonian Flow ProblemsNon-isothermal steady-state problemsSimultaneous heat and mass transfer

in a flow reactor


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Equations that Describe IsothermalNewtonian Flow Problems

Equation of ContinuityEquation of Motion

Components of the Shear Stress Tensor An Equation of State [p = p( )]Equation(s) for viscosity

[ = ( ), = ( )]Boundary and Initial conditionsNavier-Stokes Equation


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What the solution will give

Velocity distributionPressure distributionDensity distribution


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Equations that describe nonisothermalflow of a Newtonian Fluid

Equation of ContinuityEquation of Motion ( explicit in and )

Equation of Energy ( explicit in , and k )Thermal Equation of State (p = p( , T))Caloric Equation of State ( CP=C P ( , T) )

Viscosity Equations [ = ( ), = ( )]Thermal Conductivity Equations (k = k(T))Boundary and Initial Conditions


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Solution will yield

Temperature distribution, T(x,y,z,t)Pressure distribution, P(x,y,z,t)Density distribution, (x,y,z,t)Velocity distribution, (x,y,z,t)


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Equations to describe multicomponent,nonisothermal, Newtonian flow systems andgravity as the only external force

Equations of ContinuityEquation of MotionEquation of EnergyEquation of StateEquations for Cp, , , kBoundary and Initial conditions

Equations of Change ChE 131

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