reacting systems new

7/28/2019 Reacting Systems New

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Chemical Kinetics

1

Global vs. Elementary Reactions

Law of Mass Action

Arrhenius Law

Relation between forward and reverse reaction rates

Steady State approximation

from S.R Turns


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Hydrocarbon Combustion (Turns p. 157)

2


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H2-O2 Combustion (Turns p. 117)

3


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4

Reactor models

Constant pressure reactor

T = T(t)

[Xi] = [Xi](t)

V = V(t)

Constant volume reactor

T = T(t)

[Xi] = [Xi](t)

P = P(t)


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5

Reactor models (contd.)

Well-stirred reactor

T = constant[Xi] = constant

P = constant

Plug-flow reactor

T = T(x)[Xi] = [Xi](x)

P = P(x)

V = V(x)


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Assumptions:

No temperature gradients

No composition gradientsi.e. T and [Xi] are functions of time

Known:

0

0

)0(

)0(

ii XtX

TtT

Constant Pressure Reactor


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Conservation of energy (First Law of Thermodynamics):

dt

dvP

dt

dhm

dt

dvmP

dt

dumWQ

dt

dh

m

Q

Constant Pressure Reactor

Expressing system chemical enthalpy in terms of chemical composition,

m

hN

m

Hh i

ii


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8

i i

ii

ii

dt

hdN

dt

dNh

mdt

dh 1

Assuming Ideal Gas behavior (h is a function of T only) :

dt

dT

cdt

dT

T

h

dt

hdip

ii

,

Constant Pressure Reactor (contd.)


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N

j

i

M

k u

akkkikii

kjk X

TR

ETA

11

'

,

"

,

,'

exp

The rate of change ofNi : [ ]i

i i i

dNN V X Vdt

Where,

Substituting these expressions in the First Law,

i

pi

i

ii

icX

hVQ

dt

dT


1


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10

Where,

i

ii MWX

mV

T

T

ipifi

ref

dTchh ,0,


dt

dV

VXdt

dV

VNdt

dN

Vdt

V

Nd

dt

Xdiii

i

i

i 1112

The rate of change of[Xi] is given by :


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dt

dT

TX

X

dt

Xd

ii

i

i

iii 1

By using the ideal gas law :

dt

dT

Tdt

dN

Ndt

dV

V i

i

i

i

111


2

i ui

PV N R T Differentiating


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Solution Methodology:

System ofFirst order ODEs

Integration routine capable of handling stiff equations


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Assumptions:

No temperature gradients

No composition gradients

i.e. T and [Xi] are functions of time

Known:

0

0

)0(

)0(

ii XtX

TtT

Constant Volume Reactor


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Conservation of energy (First Law of Thermodynamics):

dt

dumWQ

dt

du

m

Q

Constant Volume Reactor

Expressing system chemical internal energy in terms of chemical

composition,

i

upi

i

ii

i

iu

RcX

hTRVQ

dt

dT

i

1


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Constant Volume Reactor (contd.)

Also, the rate of change of pressure (using Ideal Gas Law):

The rate of change of[Xi] is given by :

ii

dt

Xd

iiu

iiu dt

dT

XRTRdt

dP

2


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Well-stirred reactor

Conservation of species for the integral CV :

Assumptions: Steady state operation

Steady flow operation

System isperfectly mixedand

homogenous in compositionini

ini

h

Y

m

,

,

outi

outi

h

Y

m

,

,

outiinii

cvi mmVmdt

dm,,

"',

Rate of

accumulation

of mass i inCV

Rate of

generation of

mass i in CV

Mass

flow ofi

into CV

Mass flow

ofi out of

CV

Known: iniYVm ,,,


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17

Well-stirred reactor (contd.)

Since diffusional mass flow rate is negligible,

0)( ,,"' outiinii YYmVm

N

j

i

M

k u

akkkikiii

kjk XTR

ETBMWm11

'

,

"

,

,'

exp

Where,

ii Ymm Hence, conservation of mass :

fori = 1,2,,N

Since the reactor is homogeneous, the mass fraction at the outlet is

equivalent to that inside the reactor

TXfTXfVmouticvii

,,"'


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Conservation of Species providesNequations withN+1 unknowns

Additional equation from conservation of energy

inout hhmQ

N

j

jj

ii

i

MWX

MWX

Y

1

Where, [Xi]and Y

iare related as

Conservation of energy for steady state,

steady flow conditions:

In terms of individual species :

N

i

iniini

N

iiouti

ThYThYmQ1

,

1

,


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T

T

ipifi

ref

dTchTh ,0,Where,

Conservation of mass and conservation of energy are simultaneously

solved forTand Yi,out

Solution Methodology:

Coupled non-linear algebraic equations, rather than system ofODEs

Mass generation rate depends only on Yior[X]i

Generalized Newtons methodused to solve the system

"'im

Mean residence time, tR

mVtR /


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Plug flow reactor

Assumptions:

Steady state and steady flow operation

No mixing in the axial direction

Uniform properties in direction perpendicular to flow

Ideal frictionless flow

Ideal gas behavior

X

x


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Plug flow reactor (contd.)

Conservation of mass :

Known quantities :

", ( ), ( ) (nozzle or diffuser), ( )i

m k T A x Q x

0

dx

uAd

Conservation of momentum: 0dx

duu

dx

dP

Conservation of energy:

2

"

2 0m

ud h

Q Pdx m

(Pm = perimeter)

Conservation of species: "'

ii m

dx

uYd

fori = 1,2,,N


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Ideal gas equation of state:

1

1

N

i i

imix

mix

u

MW

YMW

MW

TR

P

Where,

Hence,N+4 unknowns (Yis,, P, T, u)

withN+4 equations

Usable forms of equations

0111

dx

dA

Adx

du

udx

d

From conservation of mass


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0 dx

duudx

dP

0"

m

PQ

dx

duu

dx

dh

dx

dYh

dx

dTc

dx

dh iN

i

ip

1

From ideal gas calorific

equation of state h=h(T, Yi)

01111 dx

dMWMWdx

dTTdx

ddxdP

Pmix

mix

From conservation of momentum

From conservation of energy

From ideal gas law

dx

dY

MW

MW

dx

dMW iN

i i

mixmix

1

2 1 FromMWmix


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Combustion system modeling

Conceptual drawing of a Gas turbine combustor

Fuel

Air

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