Mole balance for chemical reaction engineering

(Design Equations for reactors)Lec 3 week 3

The General Mole Balance Equation

• To perform a mole balance on any system, the system boundaries must first be specified. The volume enclosed by these boundaries is referred to as the system volume.

where Nj , represents the number of moles of species j in the system at time t.

• The rate of generation of species j is expressed as The product of the reaction term and 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) then

which gives

k

iiAi

k

iAiA VrGG

11

k

V

AA dVrG

0V

we now replace G in equation (1) to get :

• From this general mole balance equation we can develop the design equations for the various types of industrial reactors: batch, semi-batch. and continuous- flow.

Types of Reactors• Batch

– No flow of material in or out of reactor– Changes with time

• Fed- Batch (semi 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 ( no change with time)

General Mole Balance in terms of number of moles

Batch reactor mole balance• Generalized Design Equation

for Reactors

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

• Good mixing, constant volume

Vrdt

dNA

A

Batch Reactor

Batch Reactor

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

dNVrF A

AA 0

CSTR• 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

Continuous Stirred Tank Reactor

CSTR

Tubular Reactor (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

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

0dt

dN A

0 VryyFyF AAA

AA

v

rdV

dF

lim

0

This is the Design Equation for a PFR

• 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

v

rdV

dF

lim

0

AA r

dV

dF

Plug Flow Reactor Mole Balance

PFR:

The integral form is:

V dFAr

AFA 0

FA

This is the volume necessary to reduce the entering molar flow rate (mol/s) from FA0 to the exit molar flow rate of FA.

Packed Bed Reactor

PBR

The integral form to find the catalyst weight is:

W dFA

r AFA 0

FA

FA0 FA r AdW dNA

dt

Reactor Mole Balance Summary