CHE+3054S+Multiple+Steady+State+ST+Ver+1+ Print+Friendly

7/27/2019 CHE+3054S+Multiple+Steady+State+ST+Ver+1+ Print+Friendly

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Multiple Steady States in a

CSTR


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Stability and Multiplicity

Stability tendency of a system to remain in a certain state

Multiplicity a system to have more than one state available to it, e.g.

temperature


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Multiple Steady State in a CSTR

For first-order reaction

From mole balance:

From the energy balance (CSTR with heat exchange):

Neglecting shaft work in CSTR and setting 0 ..

Substituting mole balance into energy balance:

. .

() X + ( 0)

=

.( ) ( 0)

=+ ()


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Define the ratio of the amount of heat transferred relative to the amount of

heat used to heat up the reaction mixture as:

The simplified form:

.( ) )

= 0 +()

.=

. (

) )

=1 + 0


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We recognize on the left hand side of the equation, we have heat generated

by the chemical reaction, G(T):

Whereas the right hand, we have heat removed from the reactor (by heat

transfer and mass flow), R(T):

() )

=

1 + 0

. ( )


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Heat Generation term

Investigating the function:

To plot heat generated G(T) as a function of reaction temperature, solve for X

as a f(T) using mole balance, rate law and stoichiometry. 1st order reaction:

. .

(1) ( 1 )

. (1)


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Solving for X gives:

Which then yields:

Heat generation is a function of residence time in the CSTR

With:

. 1 +

.

. .

1 +

. 1 + .

1 +


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At low temperatures k is small and so k1, which gives an approximation:

.

1 + .

1 +

.


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G(T) can be derived for other reaction orders and for reversible reactions. For

example, for second-order liquid phase reaction:

: 2 + 1 4 + 12

: () [ 2 + 1 4 + 1]2


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Heat Removal term

Analysing the heat removal function:

R(T) increases linearly with temperature, with gradient varying with K.

As entering temperature TiO is increased, the line retains the same slope but

shifts to the right:

() )

=1 + 0


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Heat Removal term

Varying the non-adiabatic parameter, K. If K increases either by decreasing

the molar flow rate or increasing heat-exchange area, the slope of R(T)increases and intercept moves to the left

)

=1 + 0 .=


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Ignition-Extinction Curve

By equating the heat generation term G(T) with the heat removal term (G(T),

the steady-state operation in a CSTR can be found. The points of intersection of R(T) and G(T) give the temperature the reactor

can operate at steady state.

Increasing the inlet temperature, TiO:


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Ignition-Extinction Curve

Entering

temperature

(T0)

Steady State Reactor

Temperatures (TS)

T01 (a) TS1

T02 (b) TS2 TS3

T03 (c) TS4 TS5 TS6

T04 (d) TS7 TS8 TS9

T05 (e) TS10 TS11

T06 (f) TS12


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Ignition Temperature

Start from T01, increase entering

temperature.

Follow bottom line until T05 is

reached.

Any fraction of temperatureincrease beyond T05 the steady

state T will jump to TS11.

The temperature at which jump

occurs is called the ignition

temperature.


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Extinction Temperature

Start from T12, decrease entering

temperature.

Follow top line until T02 is

reached.

Any fraction of temperaturedecrease below T02 the steady

state T will drop to TS2.

The temperature at which jump

occurs is called the extinction

temperature.


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Perturbation to steady states

If operating at TS9 or TS7. Small

perturbation up in T results in

R(T) > G(T), hence T move back

to TS9 eventually.

Vice versa for perturbation down

in T. Locally stable steady states.

For middle point 8 represent

unstable steady state

temperatures.

If steady state at TS8 and sudden

increase in T, G(T) > R(T), T

continue to increase until TS9.

Vice versa for sudden decrease.


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Runaway Reactions In many reacting systems, the temperature of the upper steady state may be

sufficiently high that it is undesirable or dangerous to operate on.

Runaway is said to occur when at the ignition temperature is exceeded. At

this point, the slope of G(T) and R(T) are equal. dG(T)/dT = dR(T)/dT.

From the heat generation term:

Assuming heat of reaction ( ) is temperature independent:

For irreversible reaction:

( )

( ) ()

.

.()


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Runaway Reactions

Substituting back into tangent of heat generation curve:

From the heat removal term:

( )

()

2 ()

2

()

2

.

2 ()

() )

=1 + 0

() )=

1 + ( )

0 + 1 +


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Runaway Reactions The tangent of the heat removal is given by:

At the point where the tangents are equal, we have:

Also at this point, G(T)=R(T):

()

)= 1 + ( )

() )

=

1 + ()

()

() () () 2 ()

2


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Runaway Reactions If the difference between reaction temperature and Tc is exceeded

Transition to the upper steady state will occur! Runaway!

For many industrial reactions E/RT is typically between 14 and 24. Reaction

temperatures, T may be 300 to 500 K.

Consequently this critical temperature difference T-Tc will be somewhere

around 15 to 30 K.

> 2

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