Mole Balance Equations

MM 305: Kinetics of Materials

Ajay Singh Panwar and N Venkataramani

Fall 2020

MEMS, IIT Bombay

Cis-trans isomerization

Isomerization is possible only if the double bond is temporarily broken – allowing the two ends to rotate freely
 Low Temperature Diamond CVD
Effect of temperature using CO2/CH4 gas mixtures

𝐸𝑎
𝑅𝑎𝑡𝑒 ∝ 𝑒𝑥𝑝 −
𝑘𝑇

CVD diamond lab, Chemistry, Univ of Bristol

Haber’s process ammonia

Proposed reaction mechanism and energy profile for ammonia synthesis. Reaction mechanism and energy profile for ammonia synthesis over (a)
conventional catalyst and (b) Ru/C12A7:e-. (a) N2 and H2 react on the catalyst surface through a Langmuir–Hinshelwood mechanism to form NH3 in
which N2 dissociation is the RDS. The energy barrier (Edis) for this step corresponds to the apparent activation energy (Ea) for ammonia synthesis. As
for Ru/C12A7:e- (b), the rate-limiting step is not N2 dissociation but the formation of N–Hn species. NH3 is formed through the Langmuir–Hinshelwood
mechanism (route 1) and the direct reaction of N adatoms with H radicals (nascent hydrogen) derived from cage H_ anions (route 2). Ea is determined by
the difference between the top of the barrier for N–Hn formation and the energy level of reactant molecules (N2 and H2).

Read more at: https://phys.org/news/2015-03-catalyst-redefines-limitations-ammonia-production.html#jCp

Rate of reaction
➢ The rate of reaction for a species “A”, 𝒓𝑨 : The number of molecules of A that lose their chemical identity
per unit time per unit volume through the breaking and re-forming of chemical bonds during the reaction

➢ Conservation of mass: The overall mass is conserved (mass balance) in a reaction

➢ Individual masses of the species involved in a reaction disappear and form with particular rates

➢ A species may lose its chemical identity by: decomposition, combination or isomerization

➢ The reaction rate law for 𝒓𝑨 is,

▪ The rate of formation of A
Can vary if conditions
▪ An algebraic equation
vary at different points
▪ Independent of the reactor type
in the same reactor
▪ Solely a property of reacting materials or reacting conditions

Chapter 1 - H. Scott Fogler, “Elements of Chemical Reaction Engineering” 5th Ed. Prentice-Hall (2016)
Mole Balances

In Generation Out

Mole balance on a species “j”

in a volume V

𝑵𝒋 : Number of moles of species “j” at a time t The rate of formation is independent of

𝒓𝒋 : Rate of formation of species “j” position in the system volume, V …

𝑮𝒋 = 𝒓𝒋 𝑽 … if all system variables are uniform

throughout the volume,
• Temperature
• Species concentration
• Catalytic activity

Chapter 1 - H. Scott Fogler, “Elements of Chemical Reaction Engineering” 5th Ed. Prentice-Hall (2016)
Mole Balances
Convert to an integral – consider an infinitesimal volume ∆𝑽 → 𝟎 and 𝑴 → ∞

Replace in the
conservation equation

If the rate of formation varies with position in V,

then 𝑟𝑗 will be different at different locations

▪ Need to determine rate of generation locally! General Mole Balance Equation

▪ The total rate of generation would be a sum of

local rates Develop design equations for
various types of reactors

Chapter 1 - H. Scott Fogler, “Elements of Chemical Reaction Engineering” 5th Ed. Prentice-Hall (2016)
Reactor types
Batch reactors Continuous-flow reactors

Batch reactor Continuous Stirred Plug Flow Packed Bed

Tank Reactor (CSTR) Reactor (PFR) Reactor (PBR)

Chapter 1 - H. Scott Fogler, “Elements of Chemical Reaction Engineering” 5th Ed. Prentice-Hall (2016)
Visual Encyclopedia of Chemical Engineering at the University of Michigan

Lots of interesting stuff here!

http://encyclopedia.che.engin.umich.edu/Pages/Reactors/menu.html
Mole Balance in a Batch Reactor
➢ Small scale operations: Testing new processes, expensive products, continuous operation is difficult
➢ Advantage: High conversion corresponding to long charging times
➢ Disadvantages: Expensive, batch-to-batch variability, small-scales

➢ Charge from the top: No inflow, no outflow → 𝑭𝒋𝟎 = 𝑭𝒋 = 𝟎

0 0

For a well-mixed reaction mixture, the reaction rate is

independent of position, 𝒓𝒋 = 𝒄𝒐𝒏𝒔𝒕

• This is the mole balance for a batch reactor

• The equation can be integrated to obtain the
conversion time for a species

http://encyclopedia.che.engin.umich.edu/
Chapter 1 - H. Scott Fogler, “Elements of Chemical Reaction Engineering” 5th Ed. Prentice-Hall (2016)
Mole balance in a batch reactor: Integral form
What is the time required to reduce the number of moles 𝑵𝑨𝟎 (initial value) to a 𝑵𝑨𝟏 (desired number)?

𝐴 →𝐵

• Integral form of the mole balance equation

• Time required to reduce the number of moles of A to a desired amount

Chapter 1 - H. Scott Fogler, “Elements of Chemical Reaction Engineering” 5th Ed. Prentice-Hall (2016)
Mole Balance in a CSTR
➢ Liquid phase reactions: Stirred tank operated continuously
➢ Operated at steady state and perfectly mixed
➢ No time-dependence and position-dependence – uniform concentration, temperature and reaction rate!
➢ Conditions in the exit stream are identical to those in the tank
➢ Steady-state and uniform reaction rate →

0
𝑟𝑗 𝑉

• This is the design equation for a CSTR

• Determines the reactor volume required to reduce the exit flow rate
to 𝐹𝑗 when species are disappearing at a rate 𝑟𝑗

CSTR mole balance equation is

algebraic not differential

Product of concentration and

volumetric flow rate (more accessible)

http://encyclopedia.che.engin.umich.edu/
Chapter 1 - H. Scott Fogler, “Elements of Chemical Reaction Engineering” 5th Ed. Prentice-Hall (2016)
Mole Balance in a PFR
➢ Tubular reactor: Mostly used for gas phase separations
➢ Consists of a cylindrical pipe and operated at steady state
➢ Reactants consumed continuously down the length of the reactor – concentration varies axially

➢ Reaction rate is not uniform – a function of axial position!

Reaction rate varies along

the axial direction
0

http://encyclopedia.che.engin.umich.edu/
Chapter 1 - H. Scott Fogler, “Elements of Chemical Reaction Engineering” 5th Ed. Prentice-Hall (2016)
Mole Balance in a PFR: Differential form

Mole balance over an

infinitesimal volume

The degree of completion of a reaction in an ideal PFR does not depend on its
shape – only on its total volume

In the limit ∆𝑉 → 0
𝐴 →𝐵
Reactor volume over
which the molar flow rate
of A reduces to 𝐹𝐴1

http://encyclopedia.che.engin.umich.edu/
Chapter 1 - H. Scott Fogler, “Elements of Chemical Reaction Engineering” 5th Ed. Prentice-Hall (2016)
Mole Balance in a PBR
➢ Used for catalytic processes: fluid-solid heterogeneous reactions Homogeneous
➢ Reaction rate depends on the mass of the catalyst reaction rate
➢ Catalyst mass (W) plays the role of volume (V) – reactor volume
of secondary significance Bulk density of
the catalyst
mol of A reacting
➢ Heterogeneous reaction rate of A, 𝒓′𝑨 = time ×mass of catalyst

• Design equations for the batch reactor, CSTR and PFR are based
on reactor volume, V

• The formulation of the design equations is similar to PFR

• Replace volume coordinate, V with catalyst mass coordinate, W

http://encyclopedia.che.engin.umich.edu/
Chapter 1 - H. Scott Fogler, “Elements of Chemical Reaction Engineering” 5th Ed. Prentice-Hall (2016)
Mole Balance in a PBR: Differential form

Mole balance over an

infinitesimal catalyst weight

In the limit ∆𝑊 → 0

Catalyst weight required

to reduce the molar flow
rate of A to 𝐹𝐴1

http://encyclopedia.che.engin.umich.edu/
Chapter 1 - H. Scott Fogler, “Elements of Chemical Reaction Engineering” 5th Ed. Prentice-Hall (2016)
Summary of reactor mole balances

Chapter 1 - H. Scott Fogler, “Elements of Chemical Reaction Engineering” 5th Ed. Prentice-Hall (2016)