Lecture 4

Mole balance: calculation of membrane reactors and unsteady state in tank

reactors. Analysis of rate data

Mole Balance in terms of Concentration and Molar Flow Rates

• Working in terms of number of moles (NA, NB,..) or molar flow rates (FA, FB etc) rather than conversion could be more convenient at some instances

• The difference in calculation: we will write mole balance for each and every species in the reactor

Isothermal reaction design algorithm

Membrane reactors• used to increase

conversion when the reaction is thermodynamically limited (e.g. with small K)

• or to increase selectivityin when multiple reactions are occurring

6 12 2 6 63C H H C H⎯⎯→ +←⎯⎯

inert membrane reactor with catalyst pellet on the feed side (IMRCF)

catalyst membrane reactor (CMR)

Membrane reactors

• Mole balances:

6 12 2 6 63C H H C H⎯⎯→ +←⎯⎯

3A B C⎯⎯→ +←⎯⎯

AA

dF rdV

= CC

dF rdV

=

BB B

dF r RdV

= −| | 0V V VB B B BF F R V r V

+Δ− − Δ + Δ =

IN by flow

OUT by flow

OUT by diffusion

generation

no accumulation

Membrane reactors

• Assuming CBS=0 and introducing

BB B

dF r RdV

= −

( )B B c B BSR W a k a C C′= = −

Diffusion flux

area per volume

concentration in the sweep gas channel

mass transfer coefficient

2

4/ 4

DLaLD Dπ

π= =

c ck k a′= B c BR k C=

Example: Dehydrogenation reaction• Typical reactions:

– dehydrogenation of ethylbenzene to styrene;

– dehydrogenation of butane to butene– dehydrogenation of propane to propene

• Problem: for a reaction of typewhere an equilibrium constant Kc=0.05 mol/dm3; temperature 227ºC, pure A enters chamber at 8.2 atm and 227ºC at a rate of 10 mol/min– write differential mole balance for A, B, C– plot the molar flow rate as a function of space and time– calculate conversion at V=400 dm3.

• Assume that the membrane is permeable for B only, catalyst density is ρb=1.5 g/cm3, tube inside diameter 2cm, reaction rate k=0.7 min-1 and transport coefficient kc=0.2 min-1.

A B C⎯⎯→ +←⎯⎯

Example• Mole balance:

AA

dF rdV

= BB B

dF r RdV

= − CC

dF rdV

=

• Rate lawB C

A AC

C Cr k CK

⎛ ⎞− = −⎜ ⎟

⎝ ⎠

• Transport out of the reactor

B c BR k C=

• Stoichiometry

0A

A TT

FC CF

= 0B

B TT

FC CF

= 0C

C TT

FC CF

=

300 0

0

0.2 mol/dmA TPC C

RT= = =

B C Ar r r= = −

Example• POLYMATH solution

Use of Membrane reactors to enhance selectivity

• B is fed uniformly through the membrane

BB B

dF r RdV

= +

A B C D+ → +

Unsteady state operation of stirred reactors

• during the start up of a reactor:• slow addition of component B to a large quantity of A

e.g. when reaction is highly exothermic or unwanted side reaction can occur at high concentration of B

• one of the products (C) is vaporized and withdrawn continuously.

reactor start-up semibatchw. cooling

reactive distillation

A B C D+ → +

Startup of CSTR• Conversion doesn’t have any meaning in startup so we have

to use concentrations

0A

A A AdNF F r Vdt

− + =

00

0

,AA A A

VdCC C rdt v

τ τ τ− + = =• For liquid phase with constant overflow

• For the 1st order reactions

01, AAA A A

CdC kr kC Cdt

ττ τ+

− = + =

( )0 1 exp 11

AA

C tC kk

ττ τ

⎧ ⎫⎡ ⎤= − − +⎨ ⎬⎢ ⎥+ ⎣ ⎦⎩ ⎭

• e.g. to reach 99% steady state concentration

0 , 4.61 1

AAS s

CC tk k

ττ τ

= =+ +

4.6st τ=

4.6st k=

for small k:

for large k:

0 0;v v V V= =

Semibatch reactors• semibatch reactors could be used e.g. to improve selectivity

DkA B D+ ⎯⎯→

UkA B U+ ⎯⎯→

2D A Br kC C=

2U A Br kC C=

• selectivity: /D D A

D UU U B

r k CSr k C

= =Selectivity can be increased by keeping concentration A high and concentration B low

Semibatch equations• For component A: no flow in/out

( )AA A AA

d C VdN VdC C dVr Vdt dt dt dt

= = = +

0 0V V v t= +

0A

A AVdCv C r V

dt− + =

• For component B: 0

0 0

BB B

B BB B

dN r V Fdt

VdC C dV r V v Cdt dt

= +

+ = +

( )00

BB B B

vdC r C Cdt V

= + −

0AA A

vdC r Cdt V

= −

Example 4.9• Production of methyl bromide is carried as an irreversible

liquid phase reaction in isothermal semibatch reactor. Consider reaction to be elementary.

3 2 3 2CNBr CH NH CH Br NCNH+ → +3

03

03

03

0 3 23

5

( ) 0.05 /

0.05 /

( ) 0.025 /

2.2 /

V dm

C CNBr mol dm

v dm s

C CH NH mol dm

k dm s mol

=

=

=

=

= ⋅

initial volume of fluid

initial concentration CNBr

flow of CH3NH2 solutionconcentration CH3NH2

rate constant

Solution

• mole balance:

• rate law:

• combining

0 0 0( ); B BA BA A B

v v C CdC dCr C rdt V dt V

−= − = +

A A Br kC C= −

0 0 0

0 0

0 0

0 0 0

0 0 0

( );

; ;

B BA BA B A A B

C DA B C A B D

A A A A

A A

v v C CdC dCkC C C kC Cdt V dt V

dC v vdCkC C C kC C Cdt V dt V

V V v tN N C V C VX

N C V

−= − − = − +

= − − = − −

= +

− −= =

Polymath calculation

Analysis of rate data

• Main question: How to collect rate data and deduce the reaction rate law?

• The Algorithm:1. Postulate a rate law2. Select appropriate reactor type and mole balance3. Determine the variables and process the

experimental data. Apply simplification and create a model

4. Fit model to the data and find the coefficients. 5. Analyze your rate model for ”goodness of fit”.

Batch reactor data

• For irreversible reactions, it is possible to determine reaction order α and ratee constant k by either non-linear regression or differentiating concentration vs time data

• Differential method is most applicable when:– rate is function of one concentration only:

– method of excess is used

Products

A A A b

A Br k C Cα β

+ →

− =

Products

A A A

Ar k C α

→

− =

Batch reactor: differential method• Combing mole balance and rate law:

ln ln ln

AA A

AA A

dC k Cdt

dC k Cdt

α

α

− =

⎡ ⎤− = +⎢ ⎥⎣ ⎦

( )( / )A

A

Ap

dC dtkC

α

−=

Batch reactor: differential method• Techniques to obtain dC/dt from the data:

– Graphical– Numerical– Differentiation of polinomial fit.

• Graphical:– plotted vs. t,

– smooth curve plotted to appoximate the area under the histogramm

Ct

ΔΔ

Batch reactor: differential method

• Numerical method:– three-point differentiation formulas

( )

0 1 2

0

( 1)1

( 2) ( 1)

3 42

24 32

A A AA

t

A iA iA

ti

A f A f AfA

tf

C C CdCdt t

C CdCdt t

C C CdCdt t

−+

− −

− + −⎛ ⎞ =⎜ ⎟ Δ⎝ ⎠−⎛ ⎞ =⎜ ⎟ Δ⎝ ⎠− +⎛ ⎞ =⎜ ⎟ Δ⎝ ⎠

first point

interior point

last point

Batch reactor: differential method• Polynomial fit:

– fit CA with a polynomial and differentiate2

0 1 2

2 11 2 3

...

2 3 ...

nA n

nAn

C a a t a t a tdC a a t a t na tdt

−

= + + + +

⎛ ⎞ = + + + +⎜ ⎟⎝ ⎠

3rd order polynomial 5th order polynomial

significant error in derivative

Batch reactor: integral method• Used when reaction order is known to evaluate the rate

constant. The procedure:– assume the reaction order– integrate the model equation– plot the data in appropriate coordinates and fit the data

• Example ProductsA→

0

;AA

A

A A

dC rdt

dC kdt

C C kt

=

= −

= − 0

;

ln

AA

AA

A

A

dC rdt

dC kCdtC ktC

=

= −

=

0th order 1st order

2

0

;

1 1

AA

AA

A A

dC rdt

dC kCdt

ktC C

=

= −

− =

2nd order

Batch reactor: integral method

Batch reactor: non-linear regression• parameters in the non-linear equation (e.g. reaction

order and rate constant) are varied to minimize the sum of squares:

22 21 ( )

N

im ici

s r rN K N K

σ = = −− − ∑

number of parameters to be determined

number of runs

measured value

experimental value

steepest gradient descent trajectory to find α and k

Example 5-3

• The reaction of trityl and methanol is carried out in a solution of benzene and pyridine. Pyridine reacts with HCl and precipitate so the reaction is irreversible. Initial concentration of methanol 0.5 mol/dm3. and trityl 0.05 mol/dm3. Determine reaction order with trityl.

( ) ( )6 5 3 6 5 33 3C H CCl CH OH C H COCH HCl

A B C D

+ → +

+ → +

1 101

1

AA

A A

dC k Cdt

C Ctk

α

α α

α

− −

′− =

−=

′ −

Method of initial rates• Differential method might be problematic if

backward reaction rate is significant. In this case method of initial rates should be used instead

• Series of experiments are carried out at different initial concentrations CA0 and initial rate –rA0 is determined

• rate dependence is plotted to determine the reaction order, e.g.

0 0A Ar kC α− =

Method of half-lives• Half-life time of the reaction (time for the concentration of the

reactant to fall by half) is determined as function of initial concentration

1 10

1

1/2 10

Products

1 1 1( 1)

2 1 1( 1)

AA A

A A

A

AdC r kCdt

tk C C

tk C

α

α α

α

α

α

α

− −

−

−

→

= − =

⎡ ⎤= −⎢ ⎥− ⎣ ⎦

−=

−1/2 012A At t at C C= =

Differential reactor• consist of a tube with a thin catalyst disk (wafer)• conversion of the reactants, concentration and

temperature should be small• most commonly used catalytic reactor to obtain

experimental data

0

0( )

A AeA

A A A

F FrW

r r C

−′− =Δ

′ ′− = − bed concentration is equal to the inlet

Problems• P4-26: A large component in the processing train for fuel cell

technology is the water gas shift membrane reactor, where H2can diffuse out the sides of the membrane while the other gases cannot.

2 2 2CO H O CO H⎯⎯→+ +←⎯⎯

• Based on the following information plot the concentration and molar flow rates of each of the reacting species down the length of the membrane reactor. Assume: the volumetric feed is 10dm3/min at 10atm; equil molar feed of CO and water vapour with CT0=0.4mol/dm3, equilibrium constant Ke=1.44, reaction rate k=1.37 dm6/mol·kg cat·min, mass transfer coefficient for H2 kc=0.1dm3/mol·kg cat·min.Compare with PFR.

• P5-13