Reactor Design

Lecture 4

Fall 2007

ChEE ABE 481a/581a

Mass Balance on Reactive System• In - out + gen - cons = accumulation

• A mass balance for the system is

• NA is the mass of “A” inside the system.

GARate of generation/consumption

FA0Rate of flow in

FARate of flow out

System

GARate of generation/consumption

FA0Rate of flow in

FARate of flow out

System

dt

dNGFF A

AAA 0

• The reaction term can be written in more familiar terms,

GA = rA V

• V is volume of the system. • Note that the units for this relation are consistent:

• If GA (and hence rA) varies with position in the system volume, we can take this into account by evaluating this term at several locations. Then GA1 = rA1 V1,

volumetimevolume

mass

time

mass

• Summing the reactions over the entire volume yields:

• As (that is, as we decrease the size of these cubes and

increase their number)

• which gives

k

iiAi

k

iAiA VrGG

11

k

V

AA dVrG

0V

Generalized Design Equation for Reactors

• In - out + gen - cons = accumulation

dt

dNdVrFF A

V

AAA 0

Types of Reactors• Batch

– No flow of material in or out of reactor

– Changes with time

• Fed- Batch– Either an inflow or an outflow of material but not both

– Changes with time

• Continuous– Flow in and out of reactor

– Continuous Stirred Tank Reactor (CSTR)

– Plug Flow Reactor (PFR)

– Steady State Operation

Batch Reactor• Generalized Design Equation for

Reactors

• No flow into or out of the reactor, then, FA = FA0 = 0

• Good mixing, constant volume

dt

dNdVrFF A

V

AA0A

V

AA dVr

dt

dN

Vrdt

dNA

A A

AA rdt

dC

dt

VNdor

Enzyme Batch Reactor(constant volume, well mixed)

• integrate from t = 0 to t = t, we obtain

Kmln (S0/S) + (S0 -S) = vmax t

• Batch reactors are often used in the early stage of development due to their ease of operation and analysis

SK

Sv

dt

dSr

M max

Fed Batch Reactor

• Reactor Design Equation

• No outflow FA = 0

• Good Mixing rA dV term out of the integral

dt

dNdVrFF A

V

AAA 0

dt

VCd

dt

dNVrF AA

AA

0

Fed Batch Continued• Convert the mass (NA) to concentration. Applying

integration by parts yields

• Since

• Then

• Rearranging

dt

dVC

dt

dCVVrF A

AAA 0

0AFdt

dV

00 AAA

AA FCdt

dCVVrF

V

FCr

V

F

dt

dC AAA

AA 00

Fed Batch Continued

• Or

• Used when there is substrate inhibition and for bioreactors with cells.

AAAA rC

V

F

dt

dC 10

Assumptions for a fed batch reactor include

Only

a fe

ed in

Eith

er a

feed

in o

r a...

Ste

ady

stat

e

2 a

nd 3

All

of the

above

0%

64%

7%

29%

0%

1. Only a feed in

2. Either a feed in or a removal stream

3. Steady state

4. 2 and 3

5. All of the above

Continuous Stirred Tank Reactor

• Assume rate of flow in = rate of flow out

• FA = v CA and FA0

= v CA0

• v = volumetric flow rate (volume/time)

CSTR - continued• General Reactor Design Equation

• Assume Steady State

• Well Mixed

• So or

dt

dNdVrFF A

V

AAA 0

0dt

dN A

A

V

A VrdVr

00 AAA VrFFA

AA

r

FFV

0

CSTR for Enzymes(Enzyme remains inside)

• Input - output + generation - consump = accumulation

• F - flow rate l/hr• S - substrate conc. • V- reactor volume • r - reaction rate• at Steady State dS/dt = 0

dt

dSvrVFSFS 0

CSTR - enzymes

rV = F(S0 - S)

or r = F/V(S0 - S) = D(S0 - S)

• D= dilution Rate (hr-1) = residence time (hr)

If

ThenSK

Svr

M max

SS

SvKS M

0

max

Plug Flow Reactor (PFR)

• Tubular Reactor• Pipe through which fluid flows and reacts. • Poor mixing • Difficult to control temperature variations. • An advantage is the simplicity of construction.

PFR Design Equation• Design Equation

• Examine a small volume element (V) with length y and the same radius as the entire pipe.

• If the element is small, then spatial variations in rA are negligible, and

dt

dNdVrFF A

V

AAA 0

VrdVr A

V

A

Flow of A into Element

Flow of A out of Element

Assumption of “good mixing” applies only to the small volume element

• If volume element is very small, then assume steady state with no changes in the concentration of A.

• Simplify design equation to:

• rA is a function of position y, down the length of the pipe and reactant concentration

• The volume of an element is the product of the length and cross-sectional area,

V = A y • Design Equation becomes:

0dt

dN A

0 VryyFyF AAA

A

AA Ary

yFyyF

• take the limit where the size of a volume element becomes infinitesimally small

• or because y A = V,

• This is the Design Equation for a PFR

• Bioapplications - Sometimes hollow fiber reactor analysis is simplified to a PFR

AA

y

Ardy

dF

lim

0

AA r

dV

dF