CHE 354 Chemical Reactor Design Rate Laws. PFR Steady state This is the integral form. Often the differential form is more useful. Take the derivative.

CHE 354CHE 354Chemical Reactor Chemical Reactor

DesignDesignRate LawsRate Laws

PFRPFR

dt

dNdVrFF AA

V

AA 0

Steady state

This is the integral form. Often the differential form is more useful. Take the derivative with respect to volume of each term.

PFRPFR

00

dV

dVrd

dV

dF

dV

dF A

V

AA

Derivative of a constant is

just 0 rA

PFRPFR

AA r

dV

dF

)1(0 AA FF

A

A

r

F

dV

d

0

Rate of ReactionRate of Reaction

rate = amount/(time x “volume”)rate = amount/(time x “volume”)

amount [=] mol, gamount [=] mol, g

time [=] s, min, htime [=] s, min, h

““volume” [=] L (dmvolume” [=] L (dm33), kg cat, m), kg cat, m22 cat cat

rrAA = mol/s/L = mol/s/L

rrAA’ = mol/s/g’ = mol/s/g

rrAA” = mol/s/m” = mol/s/m22

Generic ReactionGeneric ReactionaA + bB aA + bB cC + dD cC + dD

Elementary (irreversible) Elementary (irreversible)

-r-rAA = k C = k CAAaa C CBB

bb

Elementary (reversible) Elementary (reversible)

-r-rAA = k = kff C CAAaa C CBB

b b – k– krr C CCCcc C CDD

dd

Generic ReactionGeneric ReactionaA + bB aA + bB cC + dD cC + dD

11stst Order -r Order -rA A = k C= k CAA

22ndnd Order -r Order -rAA = k C = k CAA22

Power Law -rPower Law -rAA = C = CAA C CBB

Complex -rComplex -rA A = k= k11 C CAA/(1+k/(1+k2 2 CCAA))

So how do we know? So how do we know?

Experiment!Experiment!

Generic ReactionGeneric ReactionaA + bB aA + bB cC + dD cC + dD

Assume experiment shows reaction is Assume experiment shows reaction is elementary and irreversible: elementary and irreversible:

-r-rAA = k C = k CAAaa C CBB

bb

Now, rewrite the stoichiometric equation:Now, rewrite the stoichiometric equation:

A + (b/a)B A + (b/a)B (c/a)C + (d/a)D (c/a)C + (d/a)D

What is rate law?What is rate law?

So what about the rate So what about the rate constant?constant?

Arrhenius EquationArrhenius Equation

k = A exp(-Ea/RT)k = A exp(-Ea/RT)

ln k = ln A – ln k = ln A – Ea/RTEa/RT

m = -Ea/RT

Y = b + mX

Y = ln k

X = 1/T

OK, what is this activation OK, what is this activation energy?energy?

OK, what is this activation OK, what is this activation energy?energy?

A + B

ABP

Firefly Flashing Firefly Flashing FrequencyFrequency

T (T (ooC)C) 2121 2525 3030

FlasheFlashes/ mins/ min

9.09.0 12.1612.16 16.216.2

Evidently, an activated Evidently, an activated process!process!

Firefly Flashes

y = -5.8014x + 21.933

R2 = 0.9952

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.40 3.41

1000/T (K)

ln r

ate

(Fla

shes

/min

)