Chemical Reactions Engineering
Fundamentals of Materials Balances
1-Mass Balances:
A mass balance, also called a material balance, is an application of mass conservation laws to analyze physical
systems, by accounting for material entering (input) and leaving a system (output).
Material balances are important first step when designing a new process or analyzing an existing one. They are
always prerequisite to all other calculations in the solution of process engineering problems. Material balances
are nothing more than the application of the law of conservation of mass, which states that mass can neither be
created nor destroyed.
Thus, we cannot, for example, specify an input to a reactor of one ton of naphtha and an output of two tons of
gasoline or gases or anything else. One ton of total material input will only give one ton of total output, that is to
say total mass of input equals total mass of output.
A material balance is an accounting for material. Thus, material balances are often compared to the balancing of
current accounts. They are used in industry to calculate mass flow rates of different streams entering or leaving
chemical or physical processes.
1-1- Aims of Mass Balances:
Mass balances are used widely in engineering and environmental analysis:
-
to design chemical reactors and obtain process with high efficiency
-
to understand process
-
to analyze alternative processes to produce chemicals
-
to model the dispersion of pollution
-
To estimate cost evaluation of process (by establishing energy balances)
2- Open and Closed Systems
2-1- System
By system we mean any arbitrary portion of process that you want to consider for analysis. You can define a
system such as a reactor, a section of a pipe. We can define the limits of the system by drawing the system
boundary, namely a line that encloses the portion of the process that you want to analyze.
2-2-Closed System
Figure below shows view of a bowl containing 1000 kg of H2O. Note that material neither enters or leaves
the bowl, that is to say, no material crosses the system boundary. Changes can take place inside the system,
but for a closed system, no mass exchange occurs with the surroundings. Only heat can be exchanged with
environment. For an isolated system neither material and heat are exchanged
System boundary
1000 Kg
H2O
Closed System
2-3-Open System
Figure below is an example of an open system (also called a flow system) because material crosses the
system boundary.
Open steady state System
3-Reactors & Chemical Reactions
A chemical reactor is an enclosed volume in which a chemical reaction takes place. In chemical engineering,
it is generally understood to be a process vessel used to carry out a chemical reaction, which is one of the
classic unit operations in chemical process analysis. The design of a chemical reactor deals with multiple
aspects of chemical engineering. Chemical engineers design reactors to maximize net present value for the
given reaction. Designers ensure that the reaction proceeds with the highest efficiency towards the desired
output product, producing the highest yield of product while requiring the least amount of money to
purchase and operate. Normal operating expenses include energy input, energy removal, raw material costs,
labor, etc. Energy changes can come in the form of heating or cooling, pumping to increase pressure,
frictional pressure loss or agitation.
Chemical reaction engineering is the branch of chemical engineering which deals with chemical
reactors and their design, especially by application of chemical kinetics to industrial systems.
Mixer
Reactor
Products
Raw Materials
Raw Materials
Energy
Byproducts
Catalysts
Energy
Catalysts
Open reactor system
In any of chemical processes such as grinding of solids, crystallization, filtration, extraction, etc., chemical
engineer is concerned with the process based on four basic principles:
• Mass conservation law
the law of mass conservation or principle of mass conservation states that for any closed system to all
transfers of matter and energy, the mass of the system must remain constant over time, as the system's
mass cannot change, so quantity can neither be added nor be removed. Therefore, the quantity of mass is
conserved over time.
The law implies that mass can neither be created nor destroyed, but it may be rearranged in space, or the
entities associated with it may be changed in form. For example, in chemical reactions, the mass of the
chemical components before the reaction (reactants) is equal to the mass of the components after the
reaction (products).
This law implies the balancing of material incoming and outgoing the process, accumulated in the unit and
transformed during the reaction.
• Energy conservation law:
The law of conservation of energy states that energy can neither be created nor destroyed, only converted
from one form of energy to another. This means that a system always has the same amount of energy,
unless it's added from the outside. This law implies the balancing of the energy consumed in the process
and resulted.
For Example, water can produce electricity: When water falls from the sky, it converts potential energy to
kinetic energy. This energy is then used to rotate turbine of a generator to produce electricity. The potential
energy of water in a dam can be turned into kinetic energy.
Or conversion of chemical energy to a calorific energy in case of combustion reactions for example.
Chemical energy is the energy that will be released or absorbed when chemicals react with one another.
For example, Bunsen burners (and gas applicances) use the reaction between CH4 and O2 to produce heat:
CH4(g)+2O2(g)→CO2(g)+2H2O(l)
If you mix some CH4 and some oxygen at room temperature, the mixture will remain unchanged indefinitely.
However, if you strike a spark, the two gases react violently, producing a great deal of thermal energy. This
thermal energy does not “come from nowhere”; it was originally stored in the CH4 and O2 molecules
• Chemical equilibrium law:
defines balancing or chemical equilibrium when a chemical reaction is at equilibrium. At equilibrium, the
concentration of reactants and products remains the same over time. In other words, the forward and
reverse chemical reaction are the same. Note: this does not mean the concentration of reactants and
products is the same.
The Law of Chemical Equilibrium is a relation stating that in a reaction mixture at equilibrium, there is a
condition (given by the equilibrium constant, Kc) relating the concentrations of reactants and products. For
the reaction:
a A(g) + b B(g) ↔ c C(g) + d D(g)
The equilibrium constant is calculated by the formula:
𝐶
𝐾𝐶 =
𝐴
𝐶
𝑎
𝐷
𝐵
𝑑
𝑏
• The principle of chemical reactions
According to the modern view of chemical reactions, bonds between atoms in the reactants must be broken,
and the atoms are reassembled into products by forming new bonds. Energy is absorbed to break bonds, and
energy is given up as bonds are made.
It is therefore a cracking bonds in the reactants to produce new bonds in the resulting material, leading to the
formation of various new materials having proper chemical and physical characteristics.
4-Mass conservation law
The law of matter conservation or law of mass conservation known as Lavoisier-Lomosov law is a law stating
the following:
• The weight of two chemicals separated is equivalent to the weight of the new material resulting from
their union
• Nothing loses, nothing creates, everything turns
• During a chemical reaction, we do not waste material and do not create new material, but we convert the
material we have
Examples:
1-A closed jar filled with ice and weighed again when the ice has melted. It will be noted that the total mass
will not have changed:
2- Burning a piece of wood:
C6H12O6 + 6O2 -> 6CO2 + 6H2O
Initially, we have a piece of wood mainly composed of cellulose (material made of carbon) and oxygen. At the
end of combustion, we had formation of gas (CO2), H2O and ashes (incomplete oxidation case).
In short, the molecules are transformed, but no atom has disappeared.
3- chemical transformation between silver nitrate and copper:
Reacted at temperature T = 20°C and at pressure P=1Bar (ambient pressure and temperature), an aqueous
solution of silver nitrate (which contains silver ions (Ag +)and nitrate ions (NO3-) with a piece of Copper (Cu)
Initial matter quantity (t0):
n (Ag+) = n (NO3-) = 0,1 mol
n(Cu) = 1 mol
At the end of the reaction, temperature and pressure have not changed, and we can observe:
a metallic silver deposit with an amount of n
(Ag)
= 0.1 mol on the copper metal. The amount of remaining
copper material is n (Cu) final = 0.95 mol.
Piece of
copper
silver nitrate solution
metallic silver deposit
Solution turned
blue due to the
presence of copper
ions (Cu2+)
the silver nitrate solution turned blue, that proves the presence of copper ion Cu 2 +
Final matter quantity (tf):
n (Cu2+) final = 0,05 mol
n (Ag+) final = 0 mol
n (NO3-) final = 0,1 mol
Chemical conversion can be represented by the following scheme
Initial state
Final state
P = 1 bar ; T = 20°C
n (Cu (s)) = 1 mol
n [Ag+(aq)] = 0,1 mol
n[NO3-(aq)]= 0,1 mol
Water molecules H2O(l)
P = 1 bar ; T = 20°C
products made
Ag (s) : nAg(s) = 0,1 mol
n(Cu2+)aq = 0,05 mol
Viewer ion
n(NO3-aq) = 0,1 mol
remaining reactive
n(Cu(s)) = 0,95 mol
Water H2O
4-1-Mass Balance in a Reactor
A mass balance is the starting point of every study in Chemical Engineering. The following scheme is explaining
how to perform a mass balance in a chemical reactor. The general considerations (Input, Output, Generation
and Accumulation) defined below can then be refined when applied to specific situations (closed reactor,
steady state...).
Reactor
Output Reactants FA
)C(
Input Reactants FA0
(A+B)
A+B
Generation GA
(Production – Consumption)
C
Reactants Accumulated in the
reactor NA
Consider a component A that enters a chemical reactor, the component can have different fates depending on
what happens in the system :
•It can just leave the system in the same state it entered
•It can disappear, or appear, for example through a chemical reaction
•It can stay in the reactor, what we will call accumulation in the reactor
Those different quantities, when summed, should be equal to the amount of material introduced in the
reactor (nothing has disappeared, all material entered must have gone somewhere).
The following general mass balance relation can therefore be written :
Input + (Production – Consumption) = Output + Accumulation
Generation (G)
𝐹𝐴0 +
𝑟𝐴 𝑑𝑉
FA0 : input mole flow rate [mol.s-1]
FA : output mole flow rate [mol.s-1]
rA : volume mole flow rate or reaction rate[mol.m-3.s-1]
NA : mole accumulation [mol]
=
𝐹𝐴
𝑑𝑁𝐴
+
𝑑𝑡
4-2-Steady state
In systems theory, a system or a process is in a steady state if the variables (called state variables) which define
the behavior of the system or the process, are unchanging in time.
That means for those properties (p) of the system, the partial derivative over time is zero and remains so:
𝜕𝑝
= 0 𝐹𝑜𝑟 𝑎𝑙𝑙 𝑡
𝜕𝑡
for example, in case of chemical engineering, these properties correspond to the temperature, pressure,
concentration of reactants and products or to the volume.
•For non-reacting systems, Generation = 0
•For systems operated at steady state, Accumulation = 0
Then, Mass and Energy Balances can be reduced to
Input = Output
4-3-Examples of Reactors & Chemical Reactions
1-Saponification:
Chemical reaction:
Ester
+
Base
ethyl acetate + NaOH
O
CH3-C-O-CH2-CH3 + NaOH
+ alcohol
Soap
sodium acetate
+ ethanol
O
CH3-C-O-Na
Soap
+ CH3-CH2-OH
Ethanol
Process of Saponification:
Hot water
Oil +
Ethanol +
Sodium
hydroxide
Heater
Cold
water
stirrer
Soap
Buchner filter
filter paper
water soak
Example of provisional calculation for proccess end : predicting
The saponification reaction of ethyl propanoate (C2H5CO2C2H5) is studied with sodium hydroxide (sodium
hydroxide). 8.10-3 mol of the ester are dissolved in water, sodium hydroxide is added there to to adjust the pH to
12 at 25 ° C.
CH3-CH2-COO-CH2-CH3 + OH-
CH3-CH2-COO- + CH3-CH2-OH
The amount of alcohol formed is equal to the amount of reacted OH- ion :
When pH = 12
[OH-]0 = 0,01 mol/L ?
Knowing the following relations:
[OH-]t*[H3O+]t = 10-14
[H3O+]t = 10-pH
[alcohol] t = 0,01 -10-14+pH mol/L
Predecting of pH end
The reaction is complete when at least one reactant has disappeared: the soda is in excess.
0.008 mol of sodium hydroxide was consumed by 0.008 mol of ester; it remains :
0,01-0,008 = 0,002 mole of sodium hydroxide remaining in one litre
[OH-]f = 0,002
[OH-]f*[H3O+]f = 10-14
2-Example of separations Calculation
n moles
100 moles
10% C2H5OH
90% H2O
40% C2H5OH
Magic
Separating
Machine
80 moles
x % C2H5OH
Conservation of total moles number: 100 = n+80  n= 20
Conservation of C2H5OH moles : (100*0,1) = (20*0,4) + (x*80)
x = 0,025 (2,5%)
3-Example of preparing dilution
If you are asked to prepare 2000 kg of dough (or slurry) consisting of 5% calcium hydroxide (CaOH2) in water by
diluting a dough at 20% , calculate the quantities required for that? Knowing that all proportions by weight.
Let the unknown X, the quantity of slurry before dilution:
The mass balance on CaOH2 ,is written as:
CaOH2 (input) = CaOH2 (output)
𝑋. 0,2 = 0,05 ∗ 2000
𝑋 = 500 𝐾𝑔 𝑜𝑓 𝑠𝑙𝑢𝑟𝑟𝑦 (𝐶𝑎𝑂𝐻2 + 𝑊𝑎𝑡𝑒𝑟)
Let the unknown Y, the quantity of water added for dilution:
The mass balance on water ,is written as:
H2O (input) = H2 O(output)
𝑋. 1 − 0,2 + 𝑌 = (1 − 0,05) ∗ 2000
𝑌=
1 − 0,05 ∗ 2000 − 500 ∗ 0,8 = 1500𝐾𝑔 𝑤𝑎𝑡𝑒𝑟
We can check the material balance on the total quantity:
𝑋 + 𝑌 = 2000
4-Example of distillation column
Estimate the distillate (D) and the bottom (W) for the distillation colomn shown in the figure below:
It is necessary to make a material balance balance to determine the top and bottom product flow rates
Assuming that acetone loss in bottom is neglected, the material balance on acetone is as following:
1000 ∗ 0,1 = 𝐷 ∗ 0,99
𝐷 = 101 𝑘𝑔/ℎ 𝑜𝑓 𝑑𝑖𝑠𝑖𝑙𝑙𝑎𝑡𝑒
The bottom (W):
The total material balance gives:
𝐹 =𝐷+𝑊
W= 1000 – 101 = 899 kg/h
5-Conversion of weight (mass) percentage to molar fraction
Technical grade of hydrochloric acid has a strength of 28% (w/w). Express this as mole fraction
28% (w/w)
For 100 Kg solution, we have 28 Kg of HCl and 72 Kg of water
Mass of acid:
𝑀𝐻𝐶𝑙 = 0,28 ∗ 100 = 28 𝑘𝑔
Mass of water:
𝑀𝐻2𝑂 = 0,72 ∗ 100 = 72 𝑘𝑔
Mol number of acid:
𝑛𝐻𝐶𝑙
𝑀𝐻𝐶𝑙
28
=
=
= 767,12 𝑚𝑜𝑙
𝑀𝑤 𝐻𝐶𝑙 36,5 ∗ 0,001
Mol number of water:
𝑛𝐻2𝑂
𝑀𝐻2𝑂
72
=
=
= 4000 𝑚𝑜𝑙
𝑀𝑤 𝐻2𝑂 18 ∗ 0,001
Total mol number :
𝑛𝑡𝑜𝑡𝑎𝑙 = 𝑛𝐻𝐶𝑙 + 𝑛𝐻2𝑂 = 767,12 + 4000 = 4767,12 𝑚𝑜𝑙
From where, the mol fracion of HCl:
𝑥𝐻𝐶𝑙
𝑛𝐻𝐶𝑙
767,12
=
=
= 0,16
𝑛𝑡𝑜𝑡𝑎𝑙 4767,12
and, the mol fracion of water:
𝑥𝐻2𝑂
𝑛𝐻2𝑂
4000
=
=
= 0,84
𝑛𝑡𝑜𝑡𝑎𝑙 4767,12
Check total:
0,84 + 0,16 = 1
Reminder of Fundamental Concepts and
Definitions
1- Evolution of quantities of matter
During a chemical transformation, reactants are consumed and products are formed. The corresponding
quantities of matter evolve until the end of the reaction: the quantities of the reactants decrease, those of
the products increase. The overall mass of the system is conserved.
At the macroscopic scale, the chemical system and its evolution are described by a model represented by the
chemical reaction and the equation of the reaction associated with it.
Mass of
reactants
Mass of
products
2- Definition of quantities associated with stoichiometry
2-1- Stoichiometric Relations
For a reacting chemical system, the principle of mass conservation is implicitly stated when writing the
stoichiometric relations that define the chemical transformation.
A chemical transformation is modeled by a balance equation taking into account the proportions in which the
reactants react and the products are formed. The coefficients that adjust a balance equation are called
stoichiometric numbers (coefficient).
Consider the general form of any chemical reaction with the following stoichiometric relation :
a A +b B
cC+dD
a: the stoichiometric cofficient of species A
b:the stoichiometric cofficient of species B
c:the stoichiometric cofficient of species C
d:the stoichiometric cofficient of species D
When the reactants are all exhausted at the same time, they are said to have reacted in stoichiometric
proportions.
These proportions depend on the stoichiometric numbers in the balance equation. In the example presented
above, this would imply that:
𝑛0 (𝐴) 𝑛0 (𝐵)
=
𝑎
𝑏
𝑛0 𝐴 =
𝑎
. 𝑛0 (𝐵)
𝑏
At the end of the reaction, there would be neither of the two reactants left.
By writing convention, the species disappearing during a reaction described by a stoichiometric relation will
be assigned a negative stoichiometric coefficient and, in the same way, the species appearing will be assigned
a positive stoichiometric coefficient.
3- Concept of progress
It is a quantity allowing the reaction to be monitored; it is noted (X) and is expressed in mole (mol). The
progress corresponds to the quantity of matter consumed for a reactant having a single stoichiometric
number. In the initial state, the progress is zero and it reaches its final value when the reaction is finished
Time
Example : S2O82−​(aq) + 2I−(aq)→2SO42−​(aq) + I2(aq)
If we note X the quantity of Iodine (I2) formed, the reaction equation allows us to deduce that:
the quantity of sulfate (SO42− ) formed is 2x mole;
the quantity of Peroxodisulfate (S2O82-) consumed is x mol;
the quantity of iodide (I−) consumed is 2x mol.
3-1- Table of progress
The progress table is a tool to understand the evolution of a chemical system.
A progress table always takes the form:
A progress table always takes the form:
Progress
Initial state
intermediate
state
Final state
3-2- Determination of the final composition of the system
The one of the two reactants (A and B) which runs out first is said to be the limiting reactant. To identify the
reactant in question, it is necessary to compare the two values xmax1 and xmax2 verifying the equations:
𝑛0 𝐴 − 3𝑥𝑚𝑎𝑥1 = 0
and
𝑛0 𝐵 − 𝑥𝑚𝑎𝑥2 = 0
Only the smaller of these two values is retained to complete the last row (line) of the table
4- Some Basic Concepts of Chemical Kinetics
4-1- Definition of reaction rate
the reaction rate (speed) of a species i (ri) is a term used to describe the rate (speed) of formation or
disappearance of a substance.
For a chemical reaction occurs, it is obviously necessary that the various reactants meet.
Thus, in the following chemical reaction:
H2O2 + 2 HI = 2 H2O + I2
Hydrogen peroxide Hydrogen iodide
Iodine
to form a molecule of I2, three molecules must meet: a molecule of H2O2 and two molecules HI
However, the probability of meeting molecule depends on its concentration. More the molecule is rare (low
concentration), the less chance of colliding with another molecule
The rate (speed) of I2 formation is therefore directly proportional to the product of the reactants
concentrations. For the current example, the rate is proportional to:
[H2O2].[HI].[HI] or[H2O2].[HI]2
We say that the rate of a chemical reaction in a homogeneous solution is proportional to the product of the
reactant concentrations. Each concentration is raised to a power equal to the corresponding stoichiometric
coefficient .
r = k.[H2O2].[HI]2
r:reaction rate (speed) [mol.m-3.s-1]
K : rate constant
We can also set the rate of the chemical reaction (r) in several ways:
H2O2 + 2 HI  2 H2O + I2
- the rate of disappearance of H2O2
- the rate of disappearance of HI(double of H2O2)
- the rate of appearance of H2O (opposite sign but same value as HI)
- the rate of appearance of I2 (half of H2O).
Arbitrarily choose the last definition. The rate of the chemical reaction is defined as the change of I2
concentration as a function of time
[𝐼2 ]𝑡2 −[𝐼2 ]𝑡1 𝑑 𝐼2
𝑑[𝐻2 𝑂]
𝑑 𝐻2 𝑂2
𝑑 𝐻𝐼
𝑟=
=
=
=−
=−
𝑡2 − 𝑡1
𝑑𝑡
2𝑑𝑡
𝑑𝑡
2𝑑𝑡
by convention, the positive sign is assigned to the formation of a substance and the negative sign to the
disappearance of a substance
Iodine concentration (mol/l)
Time (s)
Starting from the derivative of I2 concentration versus time, instantaneous reaction rate can be easily be
determined by drawing two tangents at two points of the curve
4-2- General Reaction Rate
the expression of the general reaction rate is the variation of the concentration versus time of a given
substance divided by its stoichiometric coefficient in the balanced chemical equation.
For the reaction:
2HBr (g)
H2(g) + Br2(g)
the expression of the general reaction rate is defined by:
Δ[𝐵𝑟2 ] Δ[𝐻2 ]
Δ[𝐻𝐵𝑟]
𝑟=
=
=−
Δ𝑡
Δ𝑡
2Δ𝑡
Consider the general form of any chemical reaction:
a A +b B
𝑟=
cC+dD
Δ[𝐶] Δ[𝐷]
Δ[𝐴]
Δ[𝐵]
=
=−
=−
𝑐. Δ𝑡 𝑑. Δ𝑡
𝑎. Δ𝑡
𝑏. Δ𝑡
The instantaneous reaction rate is the limit of the variation of the reactant in a very short time interval:
Δ[𝑟𝑒𝑎𝑐𝑡𝑖𝑓]
𝑟 = lim −
Δ𝑡
Δ𝑡→0
𝑑[𝑟𝑒𝑎𝑐𝑡𝑖𝑓])
=−
𝑑𝑡
4-3- Reaction order
Reaction order is a concept related to chemical kinetics and refers to the effect of the concentration of
reactants on the kinetics of a reaction
The order reaction is defined by the exponent appearing in the concentration of a reactant in the expression of
the reaction rate
Overall order of reaction is the sum of all orders
In our example, the reaction is first order for hydrogen peroxide (H2O2) and second order in hydrogen iodide
(HI). The overall order of the reaction is equal to three (H2O2 + 2 HI = 2 H2O + I2).
r = k.[H2O2].[HI]2
Consider the reaction:
A+B→C
The terms of the rate law for different orders are then:
- zero order : r = k (independent of the concentration constant)
- first global order: r = k.[A] or k.[B]
- Second global order: r = k.[A].[B] or k.[A]2 or k.[B]2
The variation of the reactant concentration as a function of time can be obtained by integrating the differential
rate law equation.
Zero order :
𝑑𝐴
𝑟=−
= 𝑘. 𝐴
𝑑𝑡
0
=𝑘
𝑑 𝐴 = −𝑘. 𝑑𝑡
Integration gives:
[𝐴]𝑡
𝑡
𝑑 𝐴 = −𝑘.
[𝐴]0
𝑑𝑡
0
[𝐴𝑡 ] − 𝐴0 = −𝑘. 𝑡
[𝐴𝑡 ] = −𝑘. 𝑡 + 𝐴0
[A]0 is the concentration of A at time t = 0
[A]t is the concentration of A at time t
A zero-order reaction is characterized by a linear dependence of the concentration: [A] as a function of time.
The constant k is measured in [mol.l-1.s-1].
First order reaction:
𝑑𝐴
𝑟=−
= 𝑘. [𝐴]
𝑑𝑡
𝑑[𝐴]
= −𝑘. 𝑑𝑡
[𝐴]
Integration gives:
[𝐴]𝑡
[𝐴]0
𝑑𝐴
= −𝑘.
[𝐴]
𝑡
𝑑𝑡
𝑙𝑛[𝐴𝑡 ] = −𝑘. 𝑡 + 𝑙𝑛 𝐴0
0
Thus, a first-order reaction is characterized by a linear dependence of ln [A] as a function of time.
The constant k is measured in [s-1]
Second order reaction:
𝑑𝐴
𝑟=−
= 𝑘. 𝐴
𝑑𝑡
−
2
𝑑𝐴
= 𝑘. 𝑑𝑡
2
𝐴
Integration gives:
𝐴𝑡
𝐴0
𝑑𝐴
−
=
2
𝐴
1
[𝐴]
𝑡
𝑘. 𝑑𝑡
0
𝐴𝑡
= 𝑘. 𝑡
𝐴0
1
1
−
= 𝑘𝑡
𝐴𝑡
𝐴0
1
1
= 𝑘𝑡 +
𝐴𝑡
𝐴0
Thus, a second-order reaction is characterized by a linear dependence of 1/[A] as a function of time.
The constant k is measured in [l.mol-1.s-1]
Thus, a second-order reaction is characterized by a linear dependence of 1/[A] as a function of time.
The constant k is measured in [l.mol-1.s-1]
• For a first order reaction: rA = k.CA
• For a second order reaction: rA = k.CA2
In our case:
• For a first order reaction: rA = k.[H2O2]=k.[HI]
• For a second order reaction: rA = k.[H2O2]2=k.[HI]2
In summary:
If the concentration of one of the reactants is doubled, the order of the reaction is:
-zero if no effect on rate reaction is observed
-First order if the rate reaction is double
- Second if the rate is doubled twice (the rate reaction is increased four times)
The kinetic rate law for rA is:
• The rate of formation of species A per unit volume (mol/s⋅m3)
• Solely a function of the properties of reacting materials and reaction conditions (e.g., concentration,
temperature, pressure, catalyst, or solvent [if any]) and does not depend on reactor type
• An intensive quantity (i.e., it does not depend on the total amount)
• An algebraic equation, not a differential equation (e.g., –rA = kCA, –rA = kCA2)
By convention, –rA is the rate of disappearance of species A and rA is the rate of formation of species A.
General Mole Balance in a reactor
1- General mole balance on a system volume V
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. We shall perform a mole balance
on species A in a system volume, where species A represents the particular chemical species of
interest.
System volume V
FA0
FA
V
GA
NA
The general mole balance on a system volume V is as following:
Input + (Production – Consumption) = Output + Accumulation
Generation (G)
𝐹𝐴0 +
𝑟𝐴 𝑑𝑉
=
𝐹𝐴
𝑑𝑁𝐴
+
𝑑𝑡
FA0 : input mole flow rate [mol.s-1]
FA : output mole flow rate [mol.s-1]
rA : volume mole flow rate or reaction rate [mol.m-3.s-1]
NA : mole accumulation [mol]
The variable rA shall represent the rate of formation of species A per unit volume. In other words, rA has units of
moles per unit volume per unit time (i.e. concentration per time mol.m-3.t-1)
The reaction rate is defined as rA. It is is a negative number for a for a reactant being consumed and a positive
number for product being produced.
The rate equation is a function of the properties of the reacting materials and reaction conditions (not the
type of reactor)
The general mole balance is given as the following for species A:
Input + (Production – Consumption) = Output + Accumulation
Generation (G)
𝐹𝐴0 +
𝑟𝐴 𝑑𝑉
=
𝐹𝐴
𝑑𝑁𝐴
+
𝑑𝑡
If the system variables are uniform throughout the system volume, then:
𝐺=
𝑟𝐴 𝑑𝑉 = 𝑟𝐴
𝑑𝑉 = 𝑟𝐴 . 𝑉
2- Diffrent types of reactors
2-1- Introduction
The heart of a chemical plant is the chemical reactor. The term chemical reactor refers to any device for
carrying out a chemical reaction. That is to say, to transform chemical reactants into other, more useful
chemicals.
The basic objective of a reactor design is to achieve the production of product with specified quality, at a given
flow rate, from known reactants. That is to say, the optimal implementation of processes for the physicochemical transformation of raw materials into more useful products.
To deal with these multiple tasks we must:
➢ Define the optimal operating conditions (working temperature, pressure, reactor volume, conversion rate,
physical state of reactants, …)
➢ Imagine, design and sizing the devices based on experiments carried out in laboratory.
➢ Extrapolate to an industrial scale.
➢ Optimize the operation of the installation.
2-2- Definition
A chemical reactor is any device in which a chemical transformation takes place, either to study it or to use it
for the production of new substances. It is therefore the heart of a chemical plant:
Exchange of
reactants
Reactor
Recycling
Separation
Non Reacted
Reactant
Simplified diagram of an industrial unit
Products
2-3 Classification of reactors
According to the operating mode, we distinguish:
2-3-1 Batch Reactor
A batch reactor is used for small-scale operation, for testing new processes that have not been fully developed,
for the manufacture of expensive products, and for processes that are difficult to convert to continuous
operations. The batch reactor has the advantage of high conversions that can be obtained by leaving the
reactant in the reactor for long periods of time, but it also has the disadvantages of high labor costs per batch,
the variability of products from batch to batch, and the difficulty of large-scale production.
It is the simplest device that can be envisaged for carrying out a chemical transformation. It consists of a
container into which the reactants are introduced at the start of the operation. The reaction takes place until
the desired conversion rate, the device is then drained.
Batch processes are organized for successive batches production of homogeneous products. The quantity
produced is limited by the equipment size (mixing tanks, reactors…).
In batch mode, we have to wait the end of the process step (depletion of the Batch introduced initially), to run
the following step process.
Batch reactor model is therefore a device operating in batch mode, initially charged with the reactants and
then closed to any entry or exit during operation. In the ideal model, is assumed perfectly stirred reactor, the
concentration of each species is uniform in the device.
After a certain time of process, the reactor is opened, and the products are removed. Batch reactor is no a
steady state process.
stirrer
Batch mixer reactor
The essential elements of this device are a container capable of containing a volume V of reaction fluid and a
stirring system for mixing reactants and products.
There are two types of batch reactors:
• Closed: no reactant addition
Charge
Stirrer
Closed Batch Mixing Reactor
• Open: reactants can be added without removing the products.
Adding
reactants
Stirrer
Open Batch Mixing Reactor
This reactor is characterized by the spatial uniformity and the non-permanence of the concentrations: it is
assumed that the composition of the reaction medium is identical at all points of the reactor but varies with
time. The batch reactor is operated at non-steady state.
A batch reactor has neither inflow nor outflow of reactants or products while the reaction is being carried out:
FA0 = FA = 0. The resulting general mole balance on species A is:
𝑑𝑁𝐴
𝑟𝐴 𝑑𝑉 =
𝑑𝑡
If the reaction mixture is perfectly mixed so that there is no variation in the rate of reaction throughout the
reactor volume, we can take rA out of the integral, integrate, and write the mole balance in the form:
𝑑𝑁𝐴
= 𝑟𝐴 𝑉
𝑑𝑡
Let’s consider the isomerization of species A in a batch reactor:
A B
As the reaction proceeds, the number of moles of A decreases and the number of moles of B increases, as
shown in following Figure:
Mole-time trajectories
We might ask what time, t1, is necessary to reduce the initial number of moles from NA0 to a final desired
number NA1. Applying the above Equation to the isomerization:
𝑑𝑁𝐴
= 𝑟𝐴 𝑉
𝑑𝑡
rearranging,
𝑑𝑁𝐴
𝑑𝑡 =
𝑟𝐴 𝑉
and integrating with limits that at t = 0, then NA = NA0, and at t = t1, then NA= NA1, we obtain:
𝑡=𝑡1
𝑁𝐴1
𝑑𝑡 =
𝑡=0
𝑁𝐴0
𝑁𝐴0
𝑡1 =
𝑁𝐴1
𝑑𝑁𝐴
𝑟𝐴 𝑉
𝑑𝑁𝐴
−𝑟𝐴 𝑉
This equation is the integral form of the mole balance on a batch reactor. It gives the time, t1, necessary to
reduce the number of moles from NA0 to NA1 and also to form NB1 moles of B.
2-3-2 Continuous Reactor
Continuous-flow reactors are almost always operated at steady state. We will consider three types: the
continuous-stirred tank reactor (CSTR), the plug-flow reactor (PFR), and the packed-bed reactor (PBR).
2-3-2-1 Continuous Stirred Tank Reactors (CSTR): Uniform concentration and temperature in the volume.
A type of reactor commonly used in industrial processing is the stirred tank operated continuously (Figure
below). It is referred to as the continuous-stirred tank reactor (CSTR) or vat, or backmix reactor, and is primarily
used for liquid-phase reactions.
Stirrer
Input
Output
Continuous Stirred Tank Reactor
It is normally operated at steady state and is assumed to be perfectly mixed; consequently, there is no time
dependence or position dependence of the temperature, concentration, or reaction rate inside the CSTR. That
is, every variable is the same at every point inside the reactor.
Because the temperature and concentration are identical everywhere within the reaction vessel, they are the
same at the exit point as they are elsewhere in the tank. Thus, the temperature and concentration in the exit
stream are modeled as being the same as those inside the reactor.
We consider a device similar to the previous one, except that it is provided with an inlet tube and an outlet
tube. A continuous flow passes through the device.
If the reaction does not affect the density of the mixture, the outlet volume flow is identical to inlet volume
flow entering the device.
Most industrial reactors operate in continuous mode, which allows a greater production.
The CSTR (Continuous Stirred tank reactor) is characterized by the spatial uniformity of concentrations. It is
considered that the reactants are diluted as they enter the reactor, and the mixture is perfect (the concentration
is the same everywhere in the reactor).
When the general mole balance equation
𝐹𝐴0 +
𝑑𝑁𝐴
𝑟𝐴 𝑑𝑉 = 𝐹𝐴 +
𝑑𝑡
is applied to a CSTR operated at steady state (i.e., conditions do not change with time),
𝑑𝑁𝐴
=0
𝑑𝑡
in which there are no spatial variations in the rate of reaction (i.e., perfect mixing):
𝑟𝐴 𝑑𝑉 = 𝑉. 𝑟𝐴
it takes the familiar form known as the design equation for a CSTR:
𝑭𝑨𝟎 − 𝑭𝑨
𝑽=
−𝒓𝑨
The CSTR design equation gives the reactor volume V necessary to reduce the entering flow rate of species A
from FA0 to the exit flow rate FA, when species A is disappearing at a rate of –rA. We note that the CSTR is
modeled such that the conditions in the exit stream (e.g., concentration and temperature) are
identical to those in the tank.
The molar flow rate FA is just the product of the concentration of species A and the volumetric flow rate V’:
𝐹𝐴 = 𝐶𝐴 . 𝑉′
𝑚𝑜𝑙
𝑚𝑜𝑙 𝑉𝑜𝑙𝑢𝑚𝑒
=
.
𝑡𝑖𝑚𝑒 𝑣𝑜𝑙𝑢𝑚𝑒 𝑡𝑖𝑚𝑒
Similarly, for the entrance molar flow rate we have FA0 = CA0 · V’0. Consequently, we can substitute for FA0 and FA
into volume Equation to write a balance on species A as:
𝑪𝑨𝟎 𝑽′𝟎 − 𝑪𝑨 𝑽′
𝑽=
−𝒓𝑨
The ideal CSTR mole balance equation is an algebraic equation, not a differential equation.
2-3-2-2 Continuous Tubular Reactors: Plug-Flow Reactor (PFR)
In addition to the CSTR and batch reactors, another type of reactor commonly used in industry is the tubular
reactor. It consists of a cylindrical pipe and is normally operated at steady state, as is the CSTR. Tubular
reactors are used most often for gas-phase reactions. A schematic and a photograph of industrial tubular
reactors are shown in the following Figure.
Continuous tubular reactors
Tubular reactor schematic.
Longitudinal tubular reactor.
In the tubular reactor, the reactants are continually consumed as they flow down the length of the reactor. In
modeling the tubular reactor, we assume that the concentration varies continuously in the axial direction
through the reactor. Consequently, the reaction rate, which is a function of concentration, will also vary axially
(except for zero order reactions).
We will consider a flow field modeled as plug-flow profile (for example, uniform velocity as in turbulent flow),
as shown in following Figure. That is, there is no radial variation in reaction rate, and the reactor is referred to
as a plug-flow reactor (PFR).
Plug-flow tubular reactors
The tubular reactor is operated at steady state. Concentration and temperature vary continuously
down the tube, and therefore, so does the reaction rate (except for zero order reactions).
The tubular reactor consists, in its basic form, of a tube inside which the reaction medium circulates. It is the
simplest structure that we can imagine for a continuous operating system.
Continuous tubular reactors
It is characterized by the permanence and non-uniformity of concentrations: the composition of the reaction
mixture depends only on the distance z in the direction of flow, from the inlet.
At a given moment
It is a device particularly well suited to carry out rapid reactions, especially in the gas phase (cracking, etc.).
The general mole balance equation is given by Equation:
𝐹𝐴0 +
𝑑𝑁𝐴
𝑟𝐴 𝑑𝑉 = 𝐹𝐴 +
𝑑𝑡
The equation we will use to design PFRs at steady state can be developed in two ways:
• directly from the above Equation by differentiating with respect to volume V, and then rearranging the
result
• from a mole balance on species j in a differential segment of the reactor volume ΔV
Let’s choose the second way to arrive at the differential form of the PFR mole balance. The differential volume,
shown in Figure below, will be chosen sufficiently small such that there are no spatial variations in reaction
rate within this volume.
FA/V
ΔGA
FA/V+ΔV
FA0
Mole balance on species A in volume ΔV
Thus the generation term,ΔGA , is:
∆𝐺𝐴 =
𝑟𝐴 𝑑𝑉 = 𝑟𝐴 ∆𝑉
𝑰𝒏𝒍𝒆𝒕 𝒎𝒐𝒍𝒆 𝒇𝒍𝒐𝒘 𝒓𝒂𝒕𝒆 𝒐𝒇 𝒔𝒑𝒆𝒄𝒊𝒆𝒔 𝑨 𝒂𝒕 𝒗𝒐𝒍𝒖𝒎𝒆 (𝑽) + 𝒎𝒐𝒍𝒆 𝒇𝒐𝒍𝒘 𝒓𝒂𝒕𝒆 𝒐𝒇 𝒈𝒆𝒏𝒆𝒓𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒔𝒑𝒆𝒄𝒊𝒆𝒔 𝑨 𝒘𝒊𝒕𝒉𝒊𝒏 𝒗𝒐𝒍𝒖𝒎𝒆 (∆𝑽)
=
𝒐𝒖𝒕𝒍𝒆𝒕 𝒎𝒐𝒍𝒆 𝒇𝒍𝒐𝒘 𝒓𝒂𝒕𝒆 𝒐𝒇 𝒔𝒑𝒆𝒄𝒊𝒆𝒔 𝑨 𝒂𝒕 𝒗𝒐𝒍𝒖𝒎𝒆 𝑽 + ∆𝑽 + 𝒎𝒐𝒍𝒆 𝒇𝒍𝒐𝒘 𝒓𝒂𝒕𝒆 𝒐𝒇 𝒂𝒄𝒄𝒖𝒎𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒔𝒑𝒆𝒄𝒊𝒆𝒔 𝑨 𝒘𝒊𝒕𝒉𝒊𝒏 (∆𝑽)
zero
𝐹𝐴/𝑉 + 𝑟𝐴 ∆𝑉 =FA/V+ΔV
Dividing by ΔV and rearranging:
FA/V+ΔV − 𝐹𝐴/𝑉
= 𝑟𝐴
∆𝑉
the term in brackets resembles the definition of a derivative
lim
∆𝑥→0
𝑓 𝑥 + ∆𝑥 − 𝑓(𝑥)
𝑑𝑓
=
∆𝑥
𝑑𝑥
Taking the limit as approaches zero, we obtain the differential form of steady state mole balance on a PFR
𝑑𝐹𝐴
= 𝑟𝐴
𝑑𝑉
Again, consider the isomerization A → B, this time in a PFR. As the reactants proceed down the reactor, A is
consumed by chemical reaction and B is produced. Consequently, the mole flow rate FA decreases, while FB
increases as the reactor volume V increases, as shown in Figure below.
Profiles of molar flow rates in a PFR
We now ask what is the reactor volume V1 necessary to reduce the entering molar flow rate of A from FA0 to
FA1.
Rearranging the mole balance Equation in the form:
𝑑𝐹𝐴
𝑑𝑉 =
𝑟𝐴
and integrating with limits at V = 0, then FA = FA0, and at V = V1, then FA = FA1
𝐹𝐴1
𝑉1 =
𝐹𝐴0
𝑑𝐹𝐴
=
𝑟𝐴
𝐹𝐴0
𝐹𝐴1
𝑑𝐹𝐴
−𝑟𝐴
V1 is the volume necessary to reduce the entering molar flow rate FA0 to some specified value FA1 and also the
volume necessary to produce a molar flow rate FB1 of B.
2-3-2-3 Packed-Bed Reactor (PBR)
The principal difference between reactor design calculations involving homogeneous reactions and those
involving fluid solid heterogeneous reactions is that for the latter, the reaction takes place on the surface of the
catalyst. The greater the mass of catalyst, the greater the reactive surface area.
Consequently, the reaction rate is based on mass of solid catalyst, W, rather than on reactor volume, V. For a
fluid–solid heterogeneous system, the rate of reaction of a species A is defined as:
−𝑟𝐴′
𝑚𝑜𝑙 𝐴 𝑟𝑒𝑎𝑐𝑡𝑖𝑛𝑔
=
(mol. 𝐾𝑔−1 . 𝑠 −1 )
𝑡𝑖𝑚𝑒 ∗ 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑐𝑎𝑡𝑎𝑙𝑦𝑠𝑡
The mass of solid catalyst is used because the amount of catalyst is what is important to the rate of product
formation. We note that by multiplying the heterogeneous reaction rate, –rA , by the bulk catalyst density, ρb
(mass/volume), we can obtain the homogeneous reaction rate, –rA:
𝑟𝐴 = 𝜌. 𝑟𝐴 ′
mol.m-3.s-1 =
Kg.m-3 .mol.Kg-1.s-1
The reactor volume that contains the catalyst is of secondary significance.
Longitudinal catalytic packed-bed reactor
In the three idealized types of reactors just discussed (the perfectly mixed batch reactor, the plug-flow tubular
reactor [PFR]), and the perfectly mixed continuous-stirred tank reactor [CSTR]), the design equations (i.e.,
mole balances) were developed based on reactor volume.
The derivation of the design equation for a packed-bed catalytic reactor (PBR) will be carried out in a manner
analogous to the development of the tubular design equation. To accomplish this derivation, we simply
replace the volume coordinate in the general mole Equation by the catalyst mass (i.e., weight) coordinate W.
Packed-bed reactor schematic
As with the PFR, the PBR is assumed to have no radial gradients in concentration, temperature, or reaction
rate. The generalized mole balance on species A over catalyst weight ΔW results in the following equation:
𝑰𝒏𝒍𝒆𝒕 𝒎𝒐𝒍𝒂𝒓 𝒇𝒍𝒐𝒘 𝒓𝒂𝒕𝒆 𝒐𝒇 𝒔𝒑𝒆𝒄𝒊𝒆𝒔 𝑨 𝒂𝒕 𝒘𝒆𝒊𝒈𝒉𝒕 (𝑾) + 𝒎𝒐𝒍𝒂𝒓 𝒇𝒐𝒍𝒘 𝒓𝒂𝒕𝒆 𝒐𝒇 𝒈𝒆𝒏𝒆𝒓𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒔𝒑𝒆𝒄𝒊𝒆𝒔 𝑨 𝒘𝒊𝒕𝒉𝒊𝒏 𝒘𝒆𝒊𝒈𝒉𝒕 (∆𝑾)
=
𝒐𝒖𝒕𝒍𝒆𝒕 𝒎𝒐𝒍𝒂𝒓 𝒇𝒍𝒐𝒘 𝒓𝒂𝒕𝒆 𝒐𝒇 𝒔𝒑𝒆𝒄𝒊𝒆𝒔 𝑨 𝒂𝒕 𝒘𝒆𝒊𝒈𝒉𝒕 𝑾 + ∆𝑾 + 𝒎𝒐𝒍𝒂𝒓 𝒇𝒍𝒐𝒘 𝒓𝒂𝒕𝒆 𝒐𝒇 𝒂𝒄𝒄𝒖𝒎𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒔𝒑𝒆𝒄𝒊𝒆𝒔 𝑨 𝒘𝒊𝒕𝒉𝒊𝒏 (∆𝑾)
zero
𝐹𝐴/𝑊 + 𝑟𝐴 ′∆𝑊 =FA/V+ΔW
The dimensions of the generation term in the above Equation are:
rA’.ΔW= mol.Kg-1.s-1.Kg (mol.s-1)
which are, as expected, the same dimensions of the molar flow rate FA (mol/time). After dividing by ΔW and
taking the limit as ΔW → 0, we arrive at the differential form of the mole balance for a packed-bed reactor:
𝑑𝐹𝐴
= 𝑟𝐴 ′
𝑑𝑊
When pressure drop through the reactor and catalyst decay are neglected, the integral form of the packedcatalyst bed design equation can be used to calculate the catalyst weight
𝐹𝐴1
𝑊=
𝐹𝐴0
𝑑𝐹𝐴
=
𝑟′𝐴
𝐹𝐴0
𝐹𝐴1
𝑑𝐹𝐴
−𝑟′𝐴
W is the catalyst weight necessary to reduce the entering molar flow rate of species A, FA0, down to a flow rate
FA .
Summary
Application 1
Consider the liquid phase cis – trans isomerization of 2–butene
which we will write symbolically as:
The reaction is first order in A (–rA = kCA) and is carried out in a tubular reactor in which the volumetric flow rate
,V’, is constant, i.e., V’=V’0 .
V’
V’
1.
Sketch the concentration profile.
2.
Derive an equation relating the reactor volume to the entering and exiting concentrations of A, the rate
constant k, and the volumetric flow rate V’ .
3.
Determine the reactor volume, V1, necessary to reduce the exiting concentration to 10% of the entering
concentration, i.e., CA = 0.1CA0, when the volumetric flow rate is 10 dm3/min (i.e., liters/min) and the
specific reaction rate, k, is 0.23 min-1 .
Solution:
1.
Sketch the concentration profile.
Species A is consumed as we move down the reactor, and as a result, both the molar flow rate of A and the
concentration of A will decrease as we move. Because the volumetric flow rate is constant, V’=V’0 , one can use
Equation FA = CAV’ to obtain the concentration of A, CA = FA/V’ , and then by comparison with the Profiles plot of
molar flow rates plot in a PFR, we obtain the concentration of A as function of reactor volume, as shown in
Figure below.
Concentration profile
2. Derive an equation relating V, V’, k, CA0, and CA.
For a tubular reactor, the mole balance on species A is given by the following Equation:
𝑑𝐹𝐴
= 𝑟𝐴
𝑑𝑉
For a first-order reaction, the rate law:
–rA = kCA
Because the volumetric flow rate, V’, is constant ( V’=V’0), as it is for most all liquid-phase reactions:
𝑑𝐹𝐴 𝑑(𝐶𝐴 𝑉 ′ ) 𝑑(𝐶𝐴 𝑉0′ )
𝑑𝐶𝐴
′
=
=
= 𝑉0
= 𝑟𝐴
𝑑𝑉
𝑑𝑉
𝑑𝑉
𝑑𝑉
Multiplying both sides of Equation above by minus one and then substituting yields:
−𝑉0′
𝑑𝐶𝐴
= −𝑟𝐴 = kCA
𝑑𝑉
𝑉0′ 𝑑𝐶𝐴
−
= 𝑑𝑉
𝑘 𝐶𝐴
Using the conditions at the entrance of the reactor that when V = 0, then CA = CA0
−
𝑉0′
𝑘
𝐶𝐴
𝐶𝐴0
𝑑𝐶𝐴
=
𝐶𝐴
𝑉
𝑑𝑉
0
Carrying out the integration:
𝑉0′
𝐶𝐴
𝑉0′
𝐶𝐴0
− 𝑙𝑛
= 𝑉 = 𝑙𝑛
𝑘
𝐶𝐴0
𝑘
𝐶𝐴
We can also rearrange the above Equation to obtain the concentration of A as a function of reactor volume :
𝑉0′
𝐶𝐴0
𝑉 = 𝑙𝑛
𝑘
𝐶𝐴
𝑘𝑉
𝐶𝐴0
′
= 𝑒 𝑉0
𝐶𝐴
𝐶𝐴0
𝑘𝑉
𝑙𝑛
= ′
𝐶𝐴
𝑉0
CA0
Concentration Profile
𝑘𝑉
𝐶𝐴 = 𝐶𝐴0 𝑒
− ′
𝑉
0
3. Calculate V. We want to find the volume, V1, at which CA= 0,1.CA0 for k = 0,23 min–1 and V’0 = 10 dm3/min
Substituting CA0, CA, V0’, and k in the above Equation we obtain:
𝑉0′
𝐶𝐴0
10
𝑉 = 𝑙𝑛
=
ln 10 = 100 𝑑𝑚3 (100 mL or 0,1 𝑚3 )
𝑘
𝐶𝐴
0,23
Let’s calculate the volume (V) to reduce the entering concentration to CA = 0,01.CA0. Again, using the same
Equation:
𝑉0′
𝐶𝐴0
10
𝑉 = 𝑙𝑛
=
ln 100 = 200 𝑑𝑚3 (200 mL or 0,2 𝑚3 )
𝑘
𝐶𝐴
0,23
Note: We see that a larger reactor (200 dm3) is needed to reduce the outlet (exit) concentration to a smaller
fraction of the (inlet) entering concentration (e.g., CA = 0,01 CA0).
We see that a reactor volume of 0,1 m3 is necessary to convert 90% of species A entering into product B for the
parameters given.
Conclusion: For this irreversible liquid-phase first order reaction (i.e., –rA = kCA) being carried out in a PFR, the
concentration of the reactant decreases exponentially down the length (i.e., volume V) of the reactor. The
more species A consumed and converted to product B, the larger must be the reactor volume V. The purpose
of the example was to give a vision of the types of calculations we will be carrying out as we study chemical
reaction engineering (CRE).
Application 2
The reaction:
is to be carried out isothermally in a continuous-flow reactor. The entering volumetric flow rate V0’ is 10 dm3/h.
(Note: FA=CA.V’. For a constant volumetric flow rate, V’=V0’, then FA=CAV0’ . Also, CA0 = FA0 ⁄V’0 = 5 mol.h-1 ⁄10
dm3.h-1 =0,5 mol.dm-3 (CA0= 0,5 mol.dm-3)
1- Calculate both the CSTR and PFR reactor volumes necessary to consume 99% of A (i.e., CA =0,01CA0) when
the entering molar flow rate (FA0) is 5 mol/h, assuming the reaction rate –rA is:
a- –rA = k with k = 0,05 mol.h-1.dm-3
b- –rA = kCA with k = 0.0001 s–1
c- –rA =kCA2 with k = 300 dm3.mol-1.h-1
2- Calculate the time necessary to consume 99,9% of species A in a1000 dm3 constant-volume batch reactor
with CA0 = 0.5 mol/dm3.
Solution:
We want to find the volume, V, at which CA= 0,01.CA0 , V’0 = 10 dm3/min, FA0=5mol/h
a- –rA = k with k = 0,05 mol.h-1.dm-3
• For CSTR:
Derive an equation relating V, V’, k, CA0, and CA.
𝐹𝐴0 − 𝐹𝐴
𝑉=
−𝑟𝐴
Similarly, for the entrance molar flow rate we have FA0 = CA0 · V’0. Consequently, we can substitute for FA0 and FA
into volume Equation to write a balance on species A as:
𝐶𝐴0 𝑉0′ − 𝐶𝐴 𝑉0′
𝑉=
−𝑟𝐴
On the hand:
–rA = k with k = 0,05 mol.h-1.dm-3
By substituting -rA by k and CA by 0,01 CA0 we obtain:
𝐶𝐴0 𝑉0′ − 0,01𝐶𝐴0 𝑉0′
0,5 ∗ 10 − (0,01 ∗ 0,5 ∗ 10)
𝑉=
=
= 99 𝑑𝑚3
𝑘
0,05
• For PFR:
𝑑𝐹𝐴
𝑑𝑉 =
𝑟𝐴
and integrating with limits at V = 0, then FA = FA0, and at V = V1, then FA = FA1
𝐹𝐴1
𝑉1 =
𝐹𝐴0
𝑑𝐹𝐴
=
𝑟𝐴
𝐹𝐴0
𝐹𝐴1
𝑑𝐹𝐴
=
−𝑟𝐴
𝐶𝐴0
𝐶𝐴1
𝑑𝐶𝐴 𝑉 ′
𝑉′
10
𝑑𝐶𝐴 = −
𝐶𝐴0 − 𝐶𝐴1 =
0,5 − 0,005 = 99𝑑𝑚3
−𝑟𝐴
𝑟𝐴
0,05
b- –rA = kCA with k = 0,0001 s–1 = 0,0001 / h/3600 =0,36 h-1
• For CSTR:
Derive an equation relating V, V’, k, CA0, and CA.
𝑉=
𝐹𝐴0 − 𝐹𝐴
−𝑟𝐴
Similarly, for the entrance molar flow rate we have FA0 = CA0 · V’0. Consequently, we can substitute for FA0 and FA
into volume Equation to write a balance on species A as:
On the hand:
𝐶𝐴0 𝑉0′ − 𝐶𝐴 𝑉0′
𝑉=
−𝑟𝐴
–rA = kCA
𝐶𝐴0 𝑉0′ − 𝐶𝐴 𝑉0′
𝑉=
𝑘. 𝐶𝐴
(0,5 ∗ 10) − (0,5 ∗ 0,01 ∗ 10)
𝑉=
0,36 ∗ 0,005
𝑉 = 2750 𝑑𝑚3
• For PFR:
the differential form of steady state mole balance on a PFR
𝑑𝐹𝐴
= 𝑟𝐴
𝑑𝑉
Substitute:
𝑑𝐹𝐴 = 𝑑 𝐶𝐴 𝑉 ′ = 𝑉 ′ 𝑑𝐶𝐴
𝑟𝐴 = −𝑘𝐶𝐴
𝑉 ′ 𝑑𝐶𝐴
= −𝑘𝐶𝐴
𝑑𝑉
𝑑𝐶𝐴
𝑑𝑉 = −𝑉′
𝑘𝐶𝐴
′
V
V=−
𝑘
𝐶𝐴
𝐶𝐴0
𝑑𝐶𝐴
V′
𝐶𝐴
V′
𝐶𝐴0
= − 𝑙𝑛
= 𝑙𝑛
= 128 𝑑𝑚3
𝐶𝐴
𝑘
𝐶𝐴0
𝑘
𝐶𝐴
c- –rA =kCA2 with k = 300 dm3.mol-1.h-1
• For CSTR:
Derive an equation relating V, V’, k, CA0, and CA.
𝐹𝐴0 − 𝐹𝐴
𝑉=
−𝑟𝐴
Similarly, for the entrance molar flow rate we have FA0 = CA0 · V’0. Consequently, we can substitute for FA0 and FA
into volume Equation to write a balance on species A as:
𝐶𝐴0 𝑉0′ − 𝐶𝐴 𝑉0′
𝑉=
−𝑟𝐴
On the hand:
–rA = kC2A
𝐶𝐴0 𝑉0′ − 𝐶𝐴 𝑉0′
𝑉=
−kC2A
(0,5 ∗ 10) − (0,5 ∗ 0,01 ∗ 10)
𝑉=
300 ∗ 0,005 ∗ 0,005
𝑉 = 660 𝑑𝑚3
• For PFR:
the differential form of steady state mole balance on a PFR
𝑑𝐹𝐴
= 𝑟𝐴
𝑑𝑉
Substitute:
𝑑𝐹𝐴 = 𝑑 𝐶𝐴 𝑉 ′ = 𝑉 ′ 𝑑𝐶𝐴
𝑟𝐴 = −𝑘𝐶𝐴 2
𝑉 ′ 𝑑𝐶𝐴
= −𝑘𝐶𝐴 2
𝑑𝑉
𝑑𝐶𝐴
𝑑𝑉 = −𝑉′
𝑘𝐶𝐴 2
V=−
V′
𝑘
𝐶𝐴
𝐶𝐴0
𝑑𝐶𝐴 V ′ 1
1
10
3
=
−
=
200
−
2
=
6,6
𝑑𝑚
𝐶𝐴 2
𝑘 𝐶𝐴 𝐶𝐴0
300
2- Calculate the time necessary to consume 99,9% of species A in a1000 dm3 constant-volume batch reactor
with CA0 = 0,5 mol/dm3 in cases (a) and (b).
- In case (a):
–rA = k with k = 0,05 mol.h-1.dm-3
The resulting general mole balance on species A is:
𝑑𝑁𝐴
𝑟𝐴 𝑑𝑉 =
𝑑𝑡
If the reaction mixture is perfectly mixed so that there is no variation in the rate of reaction throughout the
reactor volume, we can take rA out of the integral, integrate, and write the mole balance in the form:
𝑑𝑁𝐴
= 𝑟𝐴 𝑉
𝑑𝑡
rearranging,
𝑑𝑁𝐴
𝑑𝑡 =
𝑟𝐴 𝑉
and integrating with limits that at t = 0, then NA = NA0, and at t = t1, then NA= NA1, we obtain:
𝑡=𝑡1
𝑁𝐴1
𝑑𝑡 =
𝑡=0
𝑁𝐴0
𝑁𝐴0
𝑡1 =
𝑁𝐴1
𝑑𝑁𝐴
𝑟𝐴 𝑉
𝑑𝑁𝐴
−𝑟𝐴 𝑉
Substituting –rA by k and NA by CA.V, and assuming that volume is constant:
𝐶𝐴0
𝑡1 =
𝐶𝐴1
𝑑(𝐶𝐴 𝑉) 1
1
= 𝐶𝐴0 − 𝐶𝐴 =
0,5 − 0,5 ∗ 0,001
𝑘𝑉
𝑘
0,05
= 9,99 ℎ
- In case (b):
–rA = kCA with k = 0,0001 s–1 = 0,0001 / h/3600 =0,36 h-1
The resulting general mole balance on species A is:
𝑑𝑁𝐴
𝑟𝐴 𝑑𝑉 =
𝑑𝑡
If the reaction mixture is perfectly mixed so that there is no variation in the rate of reaction throughout the
reactor volume, we can take rA out of the integral, integrate, and write the mole balance in the form:
𝑑𝑁𝐴
= 𝑟𝐴 𝑉
𝑑𝑡
rearranging,
𝑑𝑁𝐴
𝑑𝑡 =
𝑟𝐴 𝑉
Substituting –rA by kCA and NA by CA.V:
𝑑(𝐶𝐴 𝑉)
𝑑𝐶𝐴
𝑑𝑡 =
=
−𝑘𝐶𝐴 𝑉 −𝑘𝐶𝐴
1
𝐶𝐴
1
0,0005
𝑡=
𝑙𝑛
=−
𝑙𝑛
= 19,18 ℎ
−𝑘
𝐶𝐴0
0,36
0,5
Application 3
The irreversible liquid phase second order reaction (-rA =kCA2)
is carried out in a CSTR. The entering concentration of A, CA0, is 2 molar. and the exit concentration of A, CA is
0,1 molar. The volumetric flow rate, V’, is constant at 3 dm3/s.
What is the corresponding reactor volume?
Solution
CSTR Mole Balance:
𝐹𝐴0 − 𝐹𝐴
𝑉=
−𝑟𝐴
Rate Law (2nd order):
-rA =kCA2
Combine:
𝐹𝐴0 = 𝐶𝐴0 𝑉 ′ = 2 ∗ 3 = 6 𝑚𝑜𝑙/𝑠
𝐹𝐴 = 𝐶𝐴 𝑉 ′ = 0,1 ∗ 3 = 0,3 𝑚𝑜𝑙/𝑠
𝐹𝐴0 − 𝐹𝐴
6 − 0,3
𝑉=
=
= 19000 𝑑𝑚3
2
kCA
0,1 ∗ 0,1 ∗ 0,03
𝐹𝐴0 − 𝐹𝐴
𝑉=
kCA2
Conversion and Reactor Sizing
1- Definition of Conversion
In defining conversion, we choose one of the reactants as the basis of calculation and then relate the other
species involved in the reaction to this basis . In almost all cases, we have to choose the limiting reactant as the
basis for the calculation. We develop the stoichiometric relations and design equations by considering the
general reaction.
(1)
The uppercase letters represent chemical species, and the lowercase letters represent stoichiometric
coefficients. We shall choose species A as our limiting reactant and, thus, our basis of calculation. The limiting
reactant is the reactant that will be completely consumed first after the reactants have been mixed.
Next, we divide the reaction expression through by the stoichiometric coefficient of species A, in order to
arrange the reaction expression in the form:
(2)
Now we ask such questions as “How can we quantify how far a reaction proceeds to the right?” or “How many
moles of C are formed for every mole of A consumed?”
A convenient way to answer these questions is to define a parameter called conversion. The conversion XA is the
number of moles of A that have reacted per mole of A fed to the system:
𝑀𝑜𝑙𝑒𝑠 𝑜𝑓 𝐴 𝑟𝑒𝑎𝑐𝑡𝑒𝑑
𝑋𝐴 =
𝑀𝑜𝑙𝑒𝑠 𝑜𝑓 𝐴 𝑓𝑒𝑑
For irreversible reactions, the maximum conversion is 1 (that is, complete conversion). For reversible reactions,
the maximum conversion is the equilibrium conversion Xe (that is, Xmax = Xe).
2- Batch Reactor Design Equations
In most batch reactors, the longer a reactant stays in the reactor, the more the reactant is converted to product
until either equilibrium is reached or the reactant is exhausted. Consequently, in batch systems the conversion
X is a function of the time the reactants spend in reactor. If NA0 is the number of moles of A initially present in
the reactor (that is, t = 0), then the total number of moles of A that have reacted (that is, have been consumed)
after a time t is [NA0X].
𝑚𝑜𝑙𝑒𝑠 𝑜𝑓 𝐴 𝑟𝑒𝑎𝑐𝑡𝑒𝑑
𝑚𝑜𝑙𝑒𝑠 𝑜𝑓 𝐴 𝑟𝑒𝑎𝑐𝑡𝑒𝑑 (𝑐𝑜𝑛𝑠𝑢𝑚𝑒𝑑) = 𝑚𝑜𝑙𝑒𝑠 𝑜𝑓 𝐴 𝑓𝑒𝑑 .
𝑚𝑜𝑙𝑒𝑠 𝑜𝑓 𝐴 𝑓𝑒𝑑
𝑚𝑜𝑙𝑒𝑠 𝑜𝑓 𝐴 𝑟𝑒𝑎𝑐𝑡𝑒𝑑 (𝑐𝑜𝑛𝑠𝑢𝑚𝑒𝑑) = 𝑁𝐴0 . 𝑋𝐴
(3)
Now, the number of moles of A that remain in the reactor after a time t, NA, can be expressed in terms of NA0
and X:
The number of moles of A in the reactor after a conversion X has been achieved is:
NA = NA0 -NA0 . X = NA0.(1- X )
(4)
When no spatial variations in reaction rate (rA) exist, the mole balance on species A for a batch system is given by
the following equation:
𝑑𝑁𝐴
= 𝑟𝐴 𝑉
𝑑𝑡
This equation is valid whether or not the reactor volume is constant.
(5)
In the general chemical reaction, reactant A is disappearing; therefore, we multiply both sides of balance
Equation by –1 to obtain the mole balance for the batch reactor in the form:
𝑑𝑁𝐴
−
= (−𝑟𝐴 )𝑉
𝑑𝑡
For batch reactors, we are interested in determining how long to leave the reactants in the reactor to achieve a
certain conversion X. To determine this length of time, we write the mole balance, Equation (5), in terms of
conversion by differentiating Equation (4) with respect to time, remembering that NA0 is the number of moles of
A initially present in the reactor and is therefore a constant with respect to time.
𝑑𝑁𝐴 𝑑𝑁𝐴0
𝑑𝑋
𝑑𝑋
=
− 𝑁𝐴0
= −𝑁𝐴0
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
0
Combining the above Equation with equation (5):
𝑑𝑋
−𝑁𝐴0
= (𝑟𝐴 )𝑉
𝑑𝑡
For a batch reactor, the design equation in differential form is:
𝑑𝑋
𝑁𝐴0
= (−𝑟𝐴 )𝑉
𝑑𝑡
(6)
We call Equation (6) the differential form of the design equation for a batch reactor because we have written
the mole balance in terms of conversion. Batch reactors are frequently used in industry for both gas-phase and
liquid-phase reactions. The laboratory bomb calorimeter reactor is widely used for obtaining reaction rate data.
Liquid-phase reactions are frequently carried out in batch reactors when small-scale production is desired or
operating difficulties do not allow the use of continuous-flow systems.
To determine the time to achieve a specified conversion X, we first separate the variables in Equation (6) as
follows:
𝑑𝑡 = 𝑁𝐴0
𝑑𝑋
−𝑟𝐴 𝑉
This equation is now integrated with the limits that the reaction begins at time equals zero where there is no
conversion initially (when t = 0, X = 0) and ends at time t when a conversion X is achieved (thats is to say, when t
= t, then X = X). Carrying out the integration, we obtain the time t necessary to achieve a conversion X in a batch
reactor.
𝑋
𝑡 = 𝑁𝐴0
0
𝑑𝑋
−𝑟𝐴 𝑉
(7)
Batch time, t, required to achieve a conversion X.
The longer the reactants are left in the reactor, the greater the conversion will be. Equation (6) is the differential
form of the design equation, and Equation (7) is the integral form of the design equation for a batch reactor.
3- Design Equations for Flow Reactors
For a batch reactor, we saw that conversion increases with time spent in the reactor. For continuous-flow
systems, this time usually increases with increasing reactor volume, thats is to say, the bigger/longer the
reactor, the more time it will take the reactants to flow completely through the reactor and thus, the more time
to react. Consequently, the conversion X is a function of reactor volume V. If FA0 is the molar flow rate of species
A fed to a system operated at steady state, the molar rate at which species A is reacting within the entire system
will be FA0.X
𝑚𝑜𝑙𝑒𝑠 𝑜𝑓 𝐴 𝑓𝑒𝑑 𝑚𝑜𝑙𝑒𝑠 𝑜𝑓 𝐴 𝑟𝑒𝑎𝑐𝑡𝑒𝑑 𝑚𝑜𝑙𝑒𝑠 𝑜𝑓 𝐴 𝑟𝑒𝑎𝑐𝑡𝑒𝑑
𝐹𝐴0 . 𝑋 =
.
=
= 𝐹𝐴0 − 𝐹𝐴
𝑡𝑖𝑚𝑒
𝑚𝑜𝑙𝑒𝑠 𝑜𝑓 𝐴 𝑓𝑒𝑑
𝑡𝑖𝑚𝑒
𝑁𝐴0
𝑡
𝑁𝐴0 − 𝑁𝐴
𝑁𝐴0
𝑁𝐴0 − 𝑁𝐴
𝑡
𝑁𝐴0
𝑡
𝑁𝐴
𝑡
We obtain the following relation:
𝐹𝐴0 − 𝑋. 𝐹𝐴0 = 𝐹𝐴
Molar flow rate at which A is fed to
the system
Molar rate at which A is consumed within
the system
Molar flow rate at which A leaves
the system
Rearranging gives:
𝐹𝐴 = 𝐹𝐴0 (1 − 𝑋)
(8)
The entering molar flow rate of species A, FA0 (mol/s), is just the product of the entering concentration, CA0
(mol/m3), and the entering volumetric flow rate, (m3/s):
𝐹𝐴0 = 𝐶𝐴0 𝑉′
(9)
For liquid systems, the volume flow rate, , is constant and equal to V’, and CA0 is commonly given in terms of
molarity, for example, CA0 = 2 mol/m3.
For gas systems, CA0 can be calculated from the entering mole fraction, yA0, the temperature, T0, and pressure,
P0, using the ideal gas law or some other gas law. For an ideal gas:
𝑃𝐴0 𝑉 = 𝑁𝐴0 𝑅𝑇
𝑁𝐴0
𝑃𝐴0 =
𝑅𝑇 = 𝐶𝐴0 𝑅𝑇
𝑉
𝑁𝐴0
𝑃𝐴0 𝑦𝐴0 =
𝑅𝑇 = 𝐶𝐴0 𝑅𝑇
𝑉
𝐶𝐴0
𝑃𝐴0 𝑦𝐴0
=
𝑅𝑇
(10)
3-1 CSTR (Also Known as a Backmix Reactor or a Vat)
Recall that the CSTR is modeled as being well mixed such that there are no spatial variations in the reactor.
For the general reaction:
(2)
the CSTR mole balance Equation can be arranged to:
𝐹𝐴0 − 𝐹𝐴
𝑉=
−𝑟𝐴
(11)
We substitute for FA in terms of FA0 and X:
𝐹𝐴 = 𝐹𝐴0 (1 − 𝑋)
and then substitute Equation (12) into (11):
𝑉=
𝐹𝐴0 − 𝐹𝐴0 (1 − 𝑋)
−𝑟𝐴
(12)
Simplifying, we see that the CSTR volume necessary to achieve a specified conversion X is:
𝑉=
𝐹𝐴0 𝑋
(−𝑟𝐴 )
(13)
CSTR Equation design
Because the reactor is perfectly mixed, the exit composition from the reactor is identical to the composition
inside the reactor, and, therefore, the rate of reaction, –rA, is evaluated at the exit conditions.
3-2 Tubular Flow Reactor (PFR)
We model the tubular reactor as having the fluid flowing in plug flow, that’s is to say no radial gradients in
concentration, temperature, or reaction rate. As the reactants enter and flow axially down the reactor, they
are consumed and the conversion increases along the length of the reactor. To develop the PFR
design equation, we first multiply both sides of the tubular reactor design equation by –1. We then express
the mole balance equation for species A in the reaction as:
−
𝑑𝐹𝐴
= −𝑟𝐴
𝑑𝑉
(14)
For a flow system, FA has previously been given in terms of the entering molar flow rate FA0 and the
conversion X:
𝐹𝐴 = 𝐹𝐴0 − 𝐹𝐴0 𝑋
Differentiating:
𝑑𝐹𝐴 = 𝑑𝐹𝐴0 − 𝑑(𝐹𝐴0 𝑋)
𝑑𝐹𝐴 = −𝐹𝐴0 𝑑𝑋
and substituting into Equation (14), gives the differential form of the design equation for a plug-flow reactor
(PFR):
𝐹𝐴0 𝑑𝑋
(15)
= −𝑟𝐴
𝑑𝑉
PFR Equation design
We separate now the variables and integrate with the limits V=0 when X=0 to obtain the plug-flow reactor
volume necessary to achieve a specified conversion X:
𝑑𝑋
𝑑𝑉 = −𝐹𝐴0
𝑟𝐴
𝑋
𝑉 = 𝐹𝐴0
0
𝑑𝑋
(−𝑟𝐴 )
(16)
3-3 Packed-Bed Reactor (PBR)
Packed-bed reactors are tubular reactors filled with catalyst particles. In PBRs it is the weight of catalyst W that
is important, rather than the reactor volume.
The derivation of the differential and integral forms of the design equations for packed-bed reactors are
analogous to those for a PFR[cf. Equations (15) and (16)].
That is to say, substituting Equation (12) for FA in the mole balance Equation (differential form) related to PBRs
reactor:
𝑑𝑋
𝐹𝐴0
= −𝑟𝐴′
𝑑𝑊
(17)
PBR Equation design
The differential form of the design equation [i.e., Equation (17)] must be used when analyzing reactors that
have a pressure drop along the length of the reactor.
In the absence of pressure drop, that is to say, ΔP = 0, we can integrate Equation (17) with limits X = 0 at W = 0,
and when W = W then X = X to obtain:
𝑑𝑋
𝑑𝑊 = 𝐹𝐴0
(−𝑟𝐴′ )
𝑋
𝑊 = 𝐹𝐴0
0
𝑑𝑋
(−𝑟𝐴 )
(18)
Equation (18) can be used to determine the catalyst weight W (mass) necessary to achieve a conversion X when
the total pressure remains constant.
4- Sizing Continuous-Flow Reactors
In this section, we are going to show how we can size CSTRs and PFRs (that’s is to say how to determine their
reactor volumes) from knowledge of the rate of reaction, –rA, as a function of conversion, X [thahs is to say, –rA
= f(X)]. The rate of disappearance of A, –rA, is almost always a function of the concentrations of the various
species present.
When only one reaction is occurring, each of the concentrations can be expressed as a function of the
conversion X ; consequently, –rA can be expressed as a function of X.
Starting from the relations:
𝐹𝐴 = 𝐹𝐴0 − 𝐹𝐴0 𝑋
𝐹𝐴0 = 𝐶𝐴0 𝑉′
𝐹𝐴 = 𝐶𝐴 𝑉′
𝐶𝐴 = 𝐶𝐴0 − 𝐶𝐴0 𝑋
𝐶𝐴 = 𝐶𝐴0 (1 − 𝑋)
In case first-order reaction:
−𝑟𝐴 = 𝑘𝐶𝐴 = 𝑘𝐶𝐴0 (1 − 𝑋)
Where:
K: the specific (constant) reaction rate and is a function only of temperature
CA0 :the entering concentration of A
We note in Equations (13) and (16) that the reactor volume is a function of the reciprocal of –rA. For this firstorder dependence, a plot of the reciprocal rate of reaction (1/–rA) as a function of conversion X, gives a curve
similar to the one shown in the following Figure
1
1
1
=
−𝑟𝐴 𝑘𝐶𝐴0 1 − 𝑋
Typical curve of the reciprocal rate as a
function of conversion .
We can use the Figure above to size CSTRs and PFRs for different entering flow rates. By sizing we mean either
determine the reactor volume for a specified conversion or determine the conversion for a specified reactor
volume.
Before sizing flow reactors, let’s consider some insights. If a reaction is carried out isothermally, the rate is
usually greatest at the start of the reaction when the concentration of reactant is greatest (that’s when there is
negligible conversion [X ≅ 0])
Hence, the reciprocal rate (1/–rA) will be small. Near the end of the reaction, when the reactant has been
mostly used up and thus the concentration of A is small (That’s, the conversion is large), the reaction rate will
be small. Consequently, the reciprocal rate (1/–rA) is large.
For all irreversible reactions of greater than zero order, as we approach complete conversion where all the
limiting reactant is used up, that’s to say X = 1, the reciprocal rate approaches infinity as does the reactor
volume.
A
B+C
Consequently, we see that an infinite reactor volume is necessary to reach complete conversion, X = 1
For reversible reactions (A
B), the maximum conversion is the equilibrium conversion Xe. At equilibrium, the
reaction rate is zero (rA = 0).
A
B+C
Therefore:
and we see that an infinite reactor volume would also be necessary to obtain the exact equilibrium conversion,
X = Xe.
4-1 Examples of Reactor Design and Staging Given rA = f(X)
To illustrate the design of continuous-flow reactors (i.e., CSTRs and PFRs), we consider the isothermal gasphase isomerization.
A
B
We are going to the laboratory to determine the rate of chemical reaction as a function of the conversion of
reactant A. The laboratory measurements given in the following Table show the chemical reaction rate as a
function of conversion. The temperature was 500 K (440°F), the total pressure was 830 kPa (8.2 atm), and the
initial charge to the reactor was pure A. The entering molar flow of A rate is F A0 = 0.4 mol/s.
Recalling the CSTR and PFR design equations, (13) and (16), we see that the reactor volume varies directly with
𝐹 𝑋
the molar flow rate FA0 and with the reciprocal of –rA (1/-rA), for example 𝑉 = 𝐴0
. Consequently, to size
(−𝑟𝐴 )
reactors, we first convert the raw data in Table below, which gives –rA as a function of X first to (1/-rA) as a
function of X. Next, we multiply by the entering molar flow rate, FA0, to obtain (FA0/-rA) as a function of X as
shown in the following Table of the processed data for FA0 = 0.4 mol/s.
To size reactors for different entering molar flow rates, FA0, we would plot (FA0/-rA) as a function of X to
construct the following figure, called Levenspiel plot:
Levenspiel plot
We are now going to use the Levenspiel plot of the processed data (Figure above) to size a CSTR and a PFR.
4-1-1 Sizing a CSTR
The reaction described by the data in Table above:
A
B
is to be carried out in a CSTR. Species A enters the reactor at a molar flow rate FA0 = 0,4 mol.s-1, which is the
flow rate used to construct the Levenspiel plot.
1- Using the data in either the Table or Levenspiel plot, calculate the volume necessary to achieve 80%
conversion in a CSTR.
2- Shade the area in Levenspiel plot that would give the CSTR volume necessary to achieve 80% conversion
Solution
1- Using the data in either the Table or Levenspiel plot, calculate the volume necessary to achieve 80%
conversion in a CSTR.
Equation (13) gives the volume of a CSTR as a function of FA0, X, and –rA
𝐹𝐴0 𝑋
𝑉=
(−𝑟𝐴 )
(13)
In a CSTR, the composition, temperature, and conversion of the effluent stream are identical to that of the fluid
within the reactor, because perfect mixing is assumed. Therefore, we need to find the value of –rA (or
reciprocal) at X = 0,8.
From either Table above or Figure Levenspiel plot, we see that when X = 0,8,then:
1
)
=
−𝑟𝐴 𝑋=0,8
(
20 m3.s.mol-1
Substitution into Equation (13) for an entering molar flow rate, FA0 = 0.4 mol/s and X = 0,8 gives:
𝑉=
𝐹𝐴0 𝑋
(−𝑟𝐴 )
= 0,4 ∗ 0,8 ∗ 20 = 6,4 m3
2- Shade the area in Levenspiel plot that would give the CSTR volume necessary to achieve 80% conversion
Rearranging Equation (13) gives:
𝐹𝐴0
𝑉=
𝑋
−𝑟𝐴
In the following figure, the volume is equal to the area of a rectangle with a height (FA0/–rA = 8 m3) and a base (X
= 0,8). This rectangle is shaded in the figure.
V = Levenspiel rectangle area = height × width
Levenspiel CSTR plot.
The CSTR volume necessary to achieve 80% conversion is 6,4 m3 when operated at 500 K, 830 kPa (8,2 atm), and
with an entering molar flow rate of A of 0,4 mol/s. This volume corresponds to a reactor about 1,5 m in
diameter and 3,6 m high. It’s a large CSTR, but this is a gas-phase reaction, and CSTRs are normally not used for
gas-phase reactions. CSTRs are mainly used for liquid phase reactions.
4-1-2 Sizing a PFR
The reaction described by the data in previous Tables is to be carried out in a PFR. The entering molar flow rate
of A is again 0,4 mol/s.
1- First, determine the PFR reactor volume necessary to achieve 80% conversion.
2- Next, shade the area in in Levenspiel plot that would give the PFR volume necessary to achieve 80%
conversion.
3- Finally, make a qualitative sketch of the conversion, X, and the rate of reaction,–rA, down the length (volume)
of the reactor.
Solution
We start by repeating rows 1 and 4 of the previous Table to produce the results shown in the following Table.
1- determine the PFR reactor volume necessary to achieve 80% conversion.
For the PFR, the differential form of the mole balance is:
Rearranging and integrating gives:
𝐹𝐴0 𝑑𝑋
= −𝑟𝐴
𝑑𝑉
𝐹𝐴0 𝑑𝑋
𝑑𝑉 =
−𝑟𝐴
𝑋=0,8
𝑉 = 𝐹𝐴0
𝑋=0
𝑑𝑋
=
−𝑟𝐴
0,8
0
𝐹𝐴0
𝑑𝑋
−𝑟𝐴
We shall use the five-point quadrature formula to numerically evaluate Equation above (16)
𝑋4
𝑋0
∆𝑋
𝑓 𝑋 𝑑𝑋 =
(𝑓 + 4𝑓1 + 2𝑓2 + 4𝑓3 + 𝑓4 )
3 0
The five-point formula with a final conversion of 0.8 gives four equal segments between X = 0 and X = 0.8, with a
segment length of ΔX= 0,8/4 =0,2.
The function inside the integral is evaluated at X = 0, X = 0.2, X = 0.4, X = 0.6, and X = 0.8.
Using values of [FA0/(–rA)] corresponding to the different conversions in Table above, we obtain:
𝑉=
0,2
3
0,89 + 4 ∗ 1,33 + 2 ∗ 2,05 + 4 ∗ 3,54 + 8 =2,164 m3
The PFR reactor volume necessary to achieve 80% conversion is 2165 dm3. This volume could result from a bank
of 100 PFRs that are each 0.1 m in diameter with a length of 2.8 m
2- Shade the area in in Levenspiel plot that would give the PFR volume necessary to achieve 80% conversion.
The PFR volume, that’s is to say, the integral in Equation (16), can also be evaluated from the area under the
curve of a plot of (FA0/–rA) versus X.
0,8
𝑉=
0
𝐹𝐴0
𝑑𝑋 = Area under the curve between X = 0 and X = 0.8
−𝑟𝐴
Levenspiel PFR plot.
The area under the curve will give the tubular reactor volume necessary to achieve the specified conversion of
A. For 80% conversion, the shaded area is roughly equal to 2165 dm3 (2.165 m3).
3- Sketch the profiles of rA and X down the length of the reactor [that’s rA=f(V) and X=f(V)].
We know that as we proceed down the reactor, the conversion increases as more and more reactant is
converted to product. Consequently, as the reactant is consumed, the concentration of reactant decreases, as
does the rate of disappearance of A for isothermal reactions.
- For X = 0.2, we calculate the corresponding reactor volume using Simpson’s rule with increment X = 0.1 and
the data in rows 1 and 4 in the table:
One-third rule (three-point) formula with a final conversion of 0.2 gives two equal segments between X = 0 and X
= 0.2, with a segment length of ΔX= 0,2/2 =0,1.
𝑋2
𝑋0
∆𝑋
∆𝑋
∆𝑋
𝑓 𝑋 𝑑𝑋 =
(𝑓 + 4𝑓1 + 𝑓2 )
3 0
Using values of [FA0/(–rA)] corresponding to the different conversions in Table above, we obtain:
𝑉=
0,1
3
0,89 + (4 ∗ 1,08) + 1,33 =0,218 m3
This volume (218 dm3) is the volume at which X = 0.2.
Therefore, at X = 0.2, then –rA = 0,3 mol.m-3.s-1 and V= 0,218 m3
- For X = 0.4, we calculate the corresponding reactor volume using again the Simpson’s rule with increment X =
0.2 and the data in rows 1 and 4 in the table:
One-third rule (three-point) formula with a final conversion of 0.4 gives two equal segments between X = 0 and
X = 0.4, with a segment length of ΔX= 0,4/2 =0,2.
Using values of [FA0/(–rA)] corresponding to the different conversions in Table above, we obtain:
𝑉=
0,2
3
0,89 + (4 ∗ 1,33) + 2,05 =0,55 m3
This volume (550 dm3) is the volume at which X = 0.4.
Therefore, at X = 0.4, then –rA = 0,195 mol.m-3.s-1 and V= 0,55 m3
- For X = 0.6, we calculate the corresponding reactor volume using again the Simpson’s rule with increment X =
0.2 and the data in rows 1 and 4 in the table:
We shall use Simpson’s three-eighths rule (four-point) to numerically evaluate Equation above (16)
𝑋3
𝑋0
3∆𝑋
𝑓 𝑋 𝑑𝑋 =
(𝑓0 + 3𝑓1 + 3𝑓2 + 𝑓3 )
8
The five-point formula with a final conversion of 0.6 gives three equal segments between X = 0 and X = 0.6, with
a segment length of ΔX= 0,6/3 =0,2.
Using values of [FA0/(–rA)] corresponding to the different conversions in Table above, we obtain:
𝑉=
0,2∗3
8
0,89 + 3 ∗ 1,33 + 3 ∗ 2,05 + 3,54 =1,093 m3
This volume (1093 dm3) is the volume at which X = 0.6.
Therefore, at X = 0.6, then –rA = 0,113 mol.m-3.s-1 and V= 1,093 m3
We gather all the results in the following table:
Conversion profile
Reaction rate profile
We observe that the reaction rate, –rA, decreases as we move down the reactor while the conversion increases.
These plots are typical for reactors operated isothermally.
5- Reactors in Series
Many times, reactors are connected in series so that the exit stream of one reactor is the feed stream for another
reactor. When this arrangement is used, it is often possible to speed calculations by defining conversion in terms
of location at a point downstream rather than with respect to any single reactor. That is, the conversion X is the
total number of moles of A that have reacted up to that point per mole of A fed to the first reactor.
For reactors in series:
𝑇𝑜𝑡𝑎𝑙 𝑜𝑓 𝑚𝑜𝑙𝑒𝑠 𝑜𝑓 𝐴 𝑟𝑒𝑎𝑐𝑡𝑒𝑑 𝑢𝑝 𝑡𝑜 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑖
𝑋𝑖 =
𝑀𝑜𝑙𝑒𝑠 𝑜𝑓 𝐴 𝑓𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑟𝑒𝑎𝑐𝑡𝑜𝑟
However, this definition can only be used when the feed stream only enters the first reactor in the series and
there are no side streams either fed or withdrawn. The molar flow rate of A at point i is equal to the moles of A
fed to the first reactor, minus all the moles of A reacted up to point i.
𝐹𝐴𝑖 = 𝐹𝐴0 − 𝐹𝐴0 𝑋𝑖
For the reactors shown in Figure below, X1 at point i = 1 is the conversion achieved in the PFR, X2 at point i = 2 is
the total conversion achieved at this point in the PFR and the CSTR, and X3 is the total conversion achieved by all
three reactors.
Reactors in series
To demonstrate these ideas, let us consider three different schemes of reactors in series: two CSTRs, two PFRs,
and then a combination of PFRs and CSTRs in series. To size these reactors, we shall use laboratory data that
give the reaction rate at different conversions.
5-1 CSTRs in Series
The first scheme to be considered is the two CSTRs in series shown in Figure below:
Two CSTRs in series
For the first reactor, the rate of disappearance of A is –rA1 at conversion X1.
A mole balance on reactor 1 gives:
𝐼𝑛𝑝𝑢𝑡 − 𝑂𝑢𝑡𝑝𝑢𝑡 + 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 0
Reactor 1:
𝐹𝐴0 − 𝐹𝐴1 + 𝑟𝐴1 𝑉1 = 0
(19)
The molar flow rate of A at point 1 is:
𝐹𝐴1 = 𝐹𝐴0 − 𝐹𝐴0 𝑋1
(20)
Combining Equations (19) and (20), or rearranging:
𝐹𝐴0 𝑋1
𝑉1 =
−𝑟𝐴1
(21)
In the second reactor, the rate of disappearance of A, –rA2, is evaluated at the conversion of the exit stream of
reactor 2, X2. A steady-state mole balance on the second reactor is:
𝐼𝑛𝑝𝑢𝑡 − 𝑂𝑢𝑡𝑝𝑢𝑡 + 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 0
Reactor 2:
𝐹𝐴1 − 𝐹𝐴2 + 𝑟𝐴2 𝑉2 = 0
(22)
The molar flow rate of A at point 2 is:
𝐹𝐴2 = 𝐹𝐴0 − 𝐹𝐴0 𝑋2
(23)
Combining and rearranging:
𝐹𝐴1 − 𝐹𝐴2 𝐹𝐴0 − 𝐹𝐴0 𝑋1 − 𝐹𝐴0 + 𝐹𝐴0 𝑋2
𝑉2 =
=
−𝑟𝐴2
−𝑟𝐴2
𝐹𝐴1
𝑉2 =
(𝑋 − 𝑋1 )
−𝑟𝐴2 2
(24)
5-1-1 Comparing Volumes for CSTRs in Series
For the two CSTRs in series, 40% conversion is achieved in the first reactor.
What is the volume of each of the two reactors necessary to achieve 80% overall conversion of the entering
species A?
In this example, we shall again use the molar flow rate of A FA0 = 0.4 mol /s and the reaction conditions given in
the table below:
For Reactor 1, we observe from either Table or Levenspiel plot that when X = 0.4, then:
𝐹𝐴0 𝑋
𝑉1 =
= 2,05 ∗ 0,4 = 0,82 𝑚3
(−𝑟𝐴 )
For Reactor 2, when X2 =0.8, then
𝐹𝐴0
(−𝑟𝐴 )
= 8:
𝑉2 =
𝐹𝐴1
(𝑋2 − 𝑋1 ) = 8 ∗ (0,8 − 0,4) = 3,2 𝑚3
−𝑟𝐴2
Two CSTRs in series
The shaded areas in Figure above can also be used to determine volumes of CSTR 1 and CSTR 2.
Note again that for CSTRs in series, the rate rA1 is evaluated at a conversion of 0.4 and rate rA2 is evaluated at a
conversion of 0.8. The total volume for these two reactors in series is:
V = V1 + V2 = 0,82 + 3,2 = 4,02 m3
By comparison, the volume necessary to achieve 80% conversion in one CSTR is:
𝑉=
𝐹𝐴0 𝑋
(−𝑟𝐴 )
= 0,4 ∗ 0,8 ∗ 20 = 6,4 m3
Notice in this example that the sum of the two CSTR reactor volumes (4.02 m3) in series is less than the volume
of one CSTR (6.4 m3) to achieve the same overall conversion.
5-2 PFRs in Series
We saw that two CSTRs in series gave a smaller total volume than a single CSTR to achieve the same conversion.
This case does not hold true for the two plug-flow reactors connected in series shown in Figure below:
Two PFRs in series
We can see from Figure below and from the following equation:
𝑋2
𝑉=
𝐹𝐴0
0
𝑑𝑋
=
−𝑟𝐴
𝑋1
𝐹𝐴0
0
𝑑𝑋
+
−𝑟𝐴
𝑋2
𝐹𝐴0
𝑋1
𝑑𝑋
−𝑟𝐴
that it is immaterial whether you place two plug-flow reactors in series or have one continuous plug-flow
reactor; the total reactor volume required to achieve the same conversion is identical.
Levenspiel plot for two PFRs in series
5.3 Combinations of CSTRs and PFRs in Series
The final sequences we shall consider are combinations of CSTRs and PFRs in series. An industrial example of
reactors in series is shown in Figure below. This sequence is used to dimerize propylene (A) into olefins (B)
according to the following reaction:
Schematic of a real system
For the sake of illustration, let’s assume that the reaction carried out in the reactors in Figure above follows the
same FA0/(-rA) versur X curve given previous table.
The volumes V1 and V2 of the first two CSTRs in series are unchanged:
Reactor 1:
𝑉1 =
𝐹𝐴0 𝑋1
−𝑟𝐴1
Reactor 2:
𝐹𝐴1
𝑉2 =
(𝑋 − 𝑋1 )
−𝑟𝐴2 2
Starting with the differential form of the PFR design equation
𝐹𝐴0 𝑑𝑋
= −𝑟𝐴
𝑑𝑉
rearranging and integrating between limits, when V = 0, then X = X2, and when V = V3, then X = X3 we obtain:
Reactor 3:
𝑋3
𝑉3 =
𝑋2
𝐹𝐴0
𝑑𝑋
−𝑟𝐴
The corresponding reactor volumes for each of the three reactors can be found from the shaded areas in Figure
below:
Levenspiel plot to determine the reactor volumes V1, V2, and V3
5-3-1 Application for Adiabatic Liquid-Phase Isomerization
The isomerization of butane:
was carried out adiabatically in the liquid phase. The data for this reversible reaction are given in Table below.
The reaction is carried out adiabatically, and the reactor used is schematized in in Figure below.
1- Calculate the volume of each of the reactors for an entering molar flow rate of n-butane of 50 kmol/hr.
Solution:
Taking the reciprocal of –rA and multiplying by FA0, we obtain the following Table:
(a) For the first CSTR:
when X = 0.2, then
𝐹𝐴0
−𝑟𝐴
= 0,94
𝐹𝐴0 𝑋1
𝑉1 =
= 0,94 ∗ 0,2 = 0,188𝑚3
−𝑟𝐴1
(b) For the PFR:
0,6
𝑉2 =
0,2
𝐹𝐴0
𝑑𝑋
−𝑟𝐴
Using Simpson’s three-point formula with ΔX = (0.6 – 0.2)/2 = 0.2, and X1 = 0.2, X2= 0.4, and X3 = 0.6
𝑉2 =
0,2
0,94 + 0,85 ∗ 4 + 1,32 = 0,37777𝑚3
3
For the last reactor, the mole balance on A for the CSTR:
𝐼𝑛𝑝𝑢𝑡 − 𝑂𝑢𝑡𝑝𝑢𝑡 + 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 0
𝐹𝐴2 − 𝐹𝐴3 + 𝑟𝐴3 𝑉3 = 0
Rearranging:
𝑉3 =
By another hand:
𝐹𝐴2 − 𝐹𝐴3
−𝑟𝐴
𝐹𝐴2 = 𝐹𝐴0 − 𝐹𝐴0 𝑋2
𝐹𝐴3 = 𝐹𝐴0 − 𝐹𝐴0 𝑋3
From where:
𝐹𝐴2 − 𝐹𝐴3 = 𝐹𝐴0 (𝑋3 − 𝑋2 )
𝐹𝐴0
𝑉3 = (
)(𝑋3 − 𝑋2 )
−𝑟𝐴
when X = 0.65, then
𝐹𝐴0
−𝑟𝐴
=2
𝑉3 = 2 ∗ 0,65 − 0,6 = 0,1𝑚3
A Levenspiel plot of (FA0/–rA) vs. X is shown in Figure below:
Levenspiel plot for adiabatic reactors in series.
6- Some Further Definitions
Some terms and equations commonly used in reaction engineering need to be defined. We also consider the
special case of the plug-flow design equation when the volumetric flow rate is constant.
6.1 Space Time or Holding Time
The space time tau, τ, is obtained by dividing the reactor volume (V) by the volume flow rate entering the
reactor (V’)
the volume V [m3] of the reactor is crossed by a volume flow V‘ [m3.s-1] of product
The ratio
𝑉
𝑉′
defines the time of stay τ (s). It represents a measure of reactor performance
For example, a 2 minute space time means that every 2 minutes the reactor treats a feed volume equal to the
reactor volume
The space time is the time necessary to process one reactor volume of fluid based on entrance conditions. For
example, consider the tubular reactor shown in Figure below, which is 20 m long and 0.2 m3 in volume. The
dashed line represents 0.2 m3 of fluid directly upstream of the reactor. The time it takes for this fluid to enter
the reactor completely is called the space time tau. It is also called the holding time or mean residence time.
For example, if the reactor volume is 0.2 m3 and the inlet volumetric flow rate is 0.01 m3/s, it would take the
upstream equivalent reactor volume (V = 0.2 m3), shown by the dashed lines, a time τ equal to:
𝜏=
0,2
= 20 𝑠
0,01
to enter the reactor (V = 0.2 m3). In other words, it would take 20 s for the fluid molecules at point a to move to
point b, which corresponds to a space time of 20 s.
We can substitute for FA0=CA0.V’ in Equations (13) and (16) and then divide both sides by V’ to write our mole
balance in the following forms:
For a PFR:
𝑉𝑝
𝜏𝑝 =
= 𝐶𝐴0
𝑉′
For a CSTR:
𝑋
0
𝑑𝑋
−𝑟𝐴
𝑉
𝑋
𝜏=
= 𝐶𝐴0
𝑉′
−𝑟𝐴
6-3 Space Velocity
The space velocity (SV), which is defined as:
𝑉′ 1
𝑆𝑉 = =
𝑉 𝜏
6-3 Examples: Reactor Space Times and Space Velocities
Calculate the space time, 𝜏, and space velocities for the reactor in previous Example for an entering volume
flow rate of 2 dm3/s.
We consider the isothermal gas-phase isomerization.
A
B
From this Example, the CSTR volume was 6.4 m3 and the corresponding space time, τ, and space velocity, SV
are:
𝑉
6,4
𝜏=
=
= 3200 𝑠 = 0,89 ℎ
𝑉′
0,002
𝑆𝑉 =
𝑉′ 1
−
= = 1,12 ℎ 1
𝑉 𝜏
7- Calculation of Some Particular Chemical Reactors
The study of a chemical reaction consists in making calculations called balances which inform us about the
profitability of the operation:
➢ Conversion rate
➢ Working temperature
➢ Required volume of the reactor
It is necessary to choose the reactor requiring the smallest volume and giving the best conversion.
7-1 Batch Reactor
They are used in liquid phase reactions  no significant volume change during the reaction.
The input and output terms of general material balance, are null (no material exchange with the outside in
batch reactor).
7-1-1 Isothermal Closed Reactor
One of the jobs in which chemical engineers are involved is the scale-up of laboratory experiments to pilotplant operation or to full-scale production.
In modeling a batch reactor, we assume there is no inflow or outflow of material, and that the reactor is well
mixed. For most liquid-phase reactions, the density change with reaction is usually small and can be neglected
(i.e., V = V0). In addition, for gas-phase reactions in which the batch reactor volume remains constant, we also
have V = V0.
7-1-1-1 Batch Reaction Times
The time necessary to achieve a specific conversion depends upon how fast the reaction takes place, which in
turn is dependent on the rate constant and the reactant concentration. To get a feel of how long it takes to
carry a batch reaction, we shall calculate the batch reaction times for different values of the reaction rate
constant, k, for a first- and for a second-order reaction. First, let’s solve for the time to achieve a conversion X
for the second-order reaction.
2A → B + C
The Chemical Reaction Engineering Algorithm (CRE Algorithm) is as following:
1. The mole balance on a constant-volume, V = V0, batch reactor is:
𝑑𝑋
𝑁𝐴0
= (−𝑟𝐴 )𝑉
𝑑𝑡
Dividing by NA0 and recognizing CA0=NA0/V0, we obtain:
𝑑𝑋 (−𝑟𝐴 )
=
𝑑𝑡
𝐶𝐴0
2. The rate law is:
−𝑟𝐴 = 𝑘2 𝐶𝐴2
3. From stoichiometry for a constant-volume batch reactor, we obtain:
𝐶𝐴 = 𝐶𝐴0 (1 − 𝑋)
4. Combining the mole balance, rate law, and stoichiometry we obtain:
2
−𝑟𝐴 = 𝑘2 𝐶𝐴0
(1 − 𝑋)2
Next, we substitute for –rA for in the mole balance Equation:
2
𝑑𝑋 (𝑘2 𝐶𝐴0
(1 − 𝑋)2 )
=
𝑑𝑡
𝐶𝐴0
𝑑𝑋
= 𝑘2 𝐶𝐴0 (1 − 𝑋)2
𝑑𝑡
5. To evaluate, we separate the variables and integrate
𝑑𝑋
= 𝑘2 𝐶𝐴0 𝑑𝑡
(1 − 𝑋)2
Initially, if t = 0, then X = 0. If the reaction is carried out isothermally, k will be constant; we can integrate this
equation to obtain:
𝑡
𝑑𝑡 =
0
𝑡=
1
𝑘2 𝐶𝐴0
𝑋
0
𝑑𝑋
(1 − 𝑋)2
1
𝑋
(
)
𝑘2 𝐶𝐴0 1 − 𝑋
This time is the reaction time t (i.e., tR) needed to achieve a conversion X for a second-order reaction in a batch
reactor. In a similar fashion, we can apply the CRE algorithm to a first-order reaction to obtain the reaction time,
tR, needed to achieve a conversion X.
1. The mole balance on a constant-volume, V = V0, batch reactor is:
𝑁𝐴0
𝑑𝑋
= (−𝑟𝐴 )𝑉
𝑑𝑡
Dividing by NA0 and recognizing CA0=NA0/V0, we obtain:
𝑑𝑋 (−𝑟𝐴 )
=
𝑑𝑡
𝐶𝐴0
2. The rate law is:
−𝑟𝐴 = 𝑘1 𝐶𝐴
3. From stoichiometry for a constant-volume batch reactor, we obtain:
𝐶𝐴 = 𝐶𝐴0 (1 − 𝑋)
4. Combining the mole balance, rate law, and stoichiometry we obtain:
−𝑟𝐴 = 𝑘1 𝐶𝐴0 (1 − 𝑋)
Next, we substitute for –rA for in the mole balance Equation:
𝑑𝑋 𝑘1 𝐶𝐴0 (1 − 𝑋)
=
𝑑𝑡
𝐶𝐴0
𝑑𝑋
= 𝑘1 (1 − 𝑋)
𝑑𝑡
5. To evaluate, we separate the variables and integrate
𝑑𝑋
= 𝑘1 𝑑𝑡
(1 − 𝑋)
Initially, if t = 0, then X = 0. If the reaction is carried out isothermally, k will be constant; we can integrate this
equation to obtain:
𝑋
0
𝑑𝑋
= 𝑘1
(1 − 𝑋)
1
𝑡𝑅 =
−ln(1 − 𝑋)
𝑘1
1
0
𝑡
𝑑𝑡
0
1
1
= ln
𝑘1
1−𝑋
1
1
𝑡𝑅 = ln
𝑘1
1−𝑋
It is important to understand the order of magnitudes of batch reaction times, tR, to achieve a given conversion,
say 90%, for different values of the product of specific reaction rate, k, and initial concentration, CA0. Table below
shows the algorithm to find the batch reaction times, tR, for both first- and second-order reactions carried out
isothermally. We can obtain these estimates of tR by considering the first- and second-order irreversible
reactions of the form:
2A → B + C
ALGORITHM TO ESTIMATE REACTION TIMES
For first-order reactions, the reaction time to reach 90% conversion (i.e., X 0.9) in a constant-volume batch
reactor scales as:
1
1
1
1
2,3
𝑡𝑅 = ln
= ln
=
𝑘1
1−𝑋
𝑘1
1 − 0,9
𝑘1
If k = 10-4 s-1
tR = 6,4 h
For second-order reactions, the reaction time to reach 90% conversion (i.e., X 0.9) in a constant-volume batch
reactor scales as:
If k2CA0 = 10-3 s-1
1
𝑋
9
𝑡𝑅 =
=
𝑘2 𝐶𝐴0 1 − 𝑋
𝑘2 𝐶𝐴0
𝑡𝑅 = 2,5 ℎ
Example :Determining k from Batch Data
It is desired to design a CSTR to produce 200 million pounds of ethylene glycol per year by hydrolyzing ethylene
oxide. However, before the design can be carried out, it is necessary to perform and analyze a batch-reactor
experiment to determine the specific reaction-rate constant, k. Given that the reaction will be carried out
isothermally, the specific reaction rate will need to be determined only at the reaction temperature of the
CSTR. At temperatures above 80°C, there is a significant by-product formation, while at temperatures below
40°C, the reaction does not proceed at a significant rate; consequently, a temperature of 55°C has been chosen.
Because water is present in excess, its concentration (55.5 mol/dm3) may be considered constant during the
course of the reaction. The reaction is first-order in ethylene oxide.
In the laboratory experiment, 500 mL of a 2 M solution (2 kmol/m3) of ethylene oxide (A) in water was mixed
with 500 mL of water (B) containing 0.9 wt % sulfuric acid, which is a catalyst. The temperature was
maintained at 55C. The concentration of ethylene glycol (C) was recorded as a function of time (Table below).
(a) Derive an equation for the concentration of ethylene glycol as a function of time.
(b) Rearrange the equation derived in (a) to obtain a linear plot of a function concentration versus time.
(c) Using the data in Table , determine the specific reaction rate at 55°C.
CONCENTRATION-TIME DATA
Solution:
(a) Derive an equation for the concentration of ethylene glycol as a function of time.
1. The mole balance on ethylene oxide (A) for a constant volume, V0, well-mixed batch reactor can be written
as:
𝑑𝑁𝐴
= 𝑟𝐴 𝑉0
𝑑𝑡
Taking V0 inside the differential and recalling that the concentration is;
𝑁𝐴
𝐶𝐴 =
𝑉0
then the differential mole balance becomes:
𝑑𝐶𝐴
= 𝑟𝐴
𝑑𝑡
2. The rate law for the ethylene oxide hydrolysis is:
−𝑟𝐴 = 𝑘. 𝐶𝐴
Because water is present in such excess, the concentration of water at any time t is virtually the same as the
initial concentration, and the rate law is independent of the concentration of H2O
3. Stoichiometry. Liquid phase, no volume change, V = V0
STOICHIOMETRIC TABLE
Recall that ΘB is the ratio of the initial number of moles of B to A (i.e.,ΘB=NB0/NA0).
For species B, that’s to say, water, the quantity remaining:
𝑁𝐵 = 𝜃𝐵 𝑁𝐴0 − 𝑁𝐴0 𝑋
𝐶𝐵 = 𝜃𝐵 𝐶𝐴0 − 𝐶𝐴0 𝑋 = 𝐶𝐴0 𝜃𝐵 = 𝐶𝐵0
We quickly see that water is in excess, as the molarity of water is 55 moles per liter. The initial concentration of
A after mixing the two volumes together is 1 molar. Therefore:
𝑁𝐵0 55
𝜃𝐵 =
=
= 55
𝑁𝐴0
1
The maximum value of X is 1, and 𝜃𝐵 ≫ 1, therefore CB is virtually constant:
𝐶𝐵 = 𝜃𝐵 𝐶𝐴0 − 𝐶𝐴0 𝑋 = 𝐶𝐴0 𝜃𝐵 = 𝐶𝐵0
For species C, that’s to say ethylene glycol, the concentration is
𝑁𝐶 𝑁𝐴0 𝑋 𝑁𝐴0 − 𝑁𝐴
𝐶𝐶 =
=
=
= 𝐶𝐴0 − 𝐶𝐴
𝑉0
𝑉0
𝑉0
4. Combining the rate law and the mole balance, we have:
𝑑𝐶𝐴
= −𝑘. 𝐶𝐴
𝑑𝑡
𝑑𝐶𝐴
= −𝑘𝑑𝑡
𝐶𝐴
5. Evaluate. For isothermal operation, k is constant, so we can integrate the equation above:
𝐶𝐴
𝐶𝐴0
𝑑𝐶𝐴
= −𝑘
𝐶𝐴
𝑡
𝑑𝑡
0
using the initial condition that when t = 0, then CA = CA0 = 1 mol/dm3 =1 kmol/m3.
𝐶𝐴0
𝑙𝑛
= 𝑘𝑡
𝐶𝐴
The concentration of ethylene oxide (A) at any time t is:
𝐶𝐴 = 𝐶𝐴0 𝑒 −𝑘𝑡
The concentration of ethylene glycol (C) at any time t can be obtained from the reaction stoichiometry:
𝐶𝐶 = 𝐶𝐴0 − 𝐶𝐴 = 𝐶𝐴0 − 𝐶𝐴0 𝑒 −𝑘𝑡
(b) Rearrange the equation derived in (a) to obtain a linear plot of a function concentration versus time.
We are now going to rearrange the Equation above for the concentration of ethylene glycol in such a manner
that we can easily use the data in the Table to determine the rate constant k
𝐶𝐶 = 𝐶𝐴0 − 𝐶𝐴 = 𝐶𝐴0 − 𝐶𝐴0 𝑒 −𝑘𝑡
𝐶𝐶 − 𝐶𝐴0
= −𝑒 −𝑘𝑡
𝐶𝐴0
𝐶𝐶 − 𝐶𝐴0
𝑙𝑛
𝐶𝐴0
= 𝑘𝑡
(c) Using the data in Table , determine the specific reaction rate at 55°C.
We see that a plot of ln[(CA0 - CC)/CA0] as a function of t will be a straight line with a slope k. Using Table, we can
construct the followingTable and use Excel to plot ln(CA0 - CC)/CA0 as a function of t.
PROCESSED DATA
From the slope of a plot of ln[(CA0 - CC)/CA0] versus t, we can find k, as shown in the Excel plot in the following
Figure.
Slope = –k = –0.311 min–1
The rate law becomes:
−𝑟𝐴 = 0,311. 𝐶𝐴
7-1-1-2 Batch Reaction Thermal Balance
T = cst
 Thermal Balance= 0
7-1-2 Adiabatic Closed Reactor
7-1-2-1 Thermal Balance
𝐻𝑒𝑎𝑡 𝑓𝑙𝑜𝑤 𝑜𝑓 𝑅𝑒𝑎𝑐𝑡𝑖𝑜𝑛 = 𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑒𝑑 𝐻𝑒𝑎𝑡 𝑓𝑙𝑜𝑤
𝑑𝑇
𝑄𝑀 . 𝑟𝐴 . 𝑉 = 𝐶𝑣
[ 𝐽/𝑠 𝑜𝑟 𝑊𝑎𝑡𝑡]
𝑑𝑡
And: 𝐶𝑣 =
𝑁𝑖 𝐶𝑖
𝑖
Mol number reacted by second
[mol.s-1]
Where:
Q: molar heat of species (A) [J.mol-1]
rA: reaction rate [mol.m-3.s-1]
V:reactor volume [m3]
Cv: total heat capacity of fluid (total mixture) [J.K-1]
Ci: molar heat capacity of species (i) [J.mol-1.K-1]
Ni:moles number of species (i) [mol]
The accumulation of energy in a body corresponds to an increase in its internal energy over time, that is to say
the variation of its temperature during the time interval dt: dT/dt.
𝜑𝑎𝑐𝑐
𝜕𝑇
= 𝐶𝑖 .
𝜕𝑡
7-1-2-2 Material Balance for A (reactif)
The same as if the reactor was isothermal:
𝑑𝐶𝐴
𝑑𝐶𝐴
𝑟𝐴 =
=−
𝑑𝑡
𝑑𝑡
Thermal and material balance equations give:
𝑄. 𝑟𝐴 . 𝑉 = 𝐶𝑣
𝑑𝑇
𝑑𝑡
𝑑𝐶𝐴
𝑟𝐴 = −
𝑑𝑡
𝑑𝑇
𝑄. 𝑉
−
=
𝑑𝐶𝐴
𝐶𝑣
𝑑𝑇 = −
𝑄. 𝑉
𝑑𝐶𝐴
𝐶𝑣
∆𝑇 = −
𝑄. 𝑉
𝑄. 𝑉
. 𝐶𝐴 − 𝐶𝐴0 =
. 𝐶𝐴0 − 𝐶𝐴
𝐶𝑣
𝐶𝑣
𝑄. 𝑉
∆𝑇 =
. 𝐶𝐴0 − 𝐶𝐴
𝐶𝑣
A maximum variation of temperature (T), T = TF corresponds to the end of the reaction ([A]F= 0)
𝑄. 𝑉
∆𝑇𝐹 = 𝑇𝐹 − 𝑇0 =
𝐶
𝐶𝑣 𝐴0
∆𝑇
𝐶𝐴0 − 𝐶𝐴
=
∆𝑇𝐹
𝐶𝐴0
𝐶𝐴0 − 𝐶𝐴
= 𝑋 (𝐶𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝐴)
𝐶𝐴0
7-1-3 Closed reactor with thermostatic bath
There is a heat exchange between the fluid inside the reactor at the temperature T and the fluid outside at
temperature To.
q [w or J/s]: heat flow by conduction and convection through the reactor wall
𝑞 = 𝑘𝑆(𝑇 − 𝑇0 )
q: heat flow by conduction and convection through the reactor wall [w]
k: global exchange coeficient including convection and conduction [w.m-2.K-1]
S: exchange surface [m2]
 Thermal Balance
𝑅𝑒𝑎𝑐𝑡𝑖𝑜𝑛 𝑓𝑙𝑜𝑤 𝑜𝑓 𝑅𝑒𝑎𝑐𝑡𝑖𝑜𝑛 = 𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑒𝑑 𝐻𝑒𝑎𝑡 𝑓𝑙𝑜𝑤 + 𝑂𝑢𝑡𝑝𝑢𝑡 𝐻𝑒𝑎𝑡 𝑓𝑙𝑜𝑤
𝑄. 𝑟𝐴 . 𝑉
=
𝑑𝑇
𝐶𝑣
𝑑𝑡
 Material Balance for A (reactif)
The same as if the reactor was isothermal or adiabatic:
𝑑𝐶𝐴
𝑑𝐶𝐴
𝑟𝐴 =
=−
𝑑𝑡
𝑑𝑡
+
𝐾𝑆(𝑇 − 𝑇0 )
7-1-4 Isothermal open reactor
Generally used in the liquid phase: with excess solvent or it is necessary to work under reflux  T is constant
steam
inlet and outlet of
the exchange fluid
steam condensation
system (reflux)
 Thermal Balance
T is constant -----> the thermal balance is nul
 Material Balance for A (reactif)
Input + (Production – Consumption) = Accumulation
Generation (G)
𝐹𝐴0 +
𝑟𝐴 𝑑𝑉
𝑑𝑁𝐴
=
𝑑𝑡
if the reaction mixture is perfectly mixed, then rA is independent of position:
𝑑𝑁𝐴
𝐹𝐴0 + 𝑟𝐴 𝑉 =
𝑑𝑡
𝐹𝐴0
𝑑𝑁𝐴
= 𝑟𝐴 +
𝑉
𝑑𝑡. 𝑉
𝐹𝐴0
𝑑𝐶𝐴
= 𝑟𝐴 +
𝑉
𝑑𝑡
Example :
First order reaction:
𝑑𝐴
𝑟=−
= 𝑘𝐶𝐴
𝑑𝑡
𝑑𝐶𝐴
= −𝑘. 𝑑𝑡
𝐶𝐴
Integration gives:
[𝐴]𝑡
[𝐴]0
𝑑𝐶𝐴
= −𝑘.
𝐶𝐴
[A0]
𝑡
𝑑𝑡
𝑙𝑛𝐶𝐴 𝑡 = −𝑘. 𝑡 + 𝑙𝑛𝐶𝐴0
0
Reactant concentration
(mol/L)
𝐶𝐴 𝐶𝐴0 𝑒 (−𝑘𝑡)
[A0]/2
0
𝐶𝐴 = 𝑒 (−𝑘𝑡+ln 𝐶𝐴0 )
t1/2
t (s)
7-2- Isothermal Continuous Stirred Tank Reactor in steady state (SCTR)
Continuous-stirred tank reactors (CSTRs), such as the one shown here schematically, are typically used for liquidphase reactions.
we derived the following design equation for a CSTR:
𝐹𝐴0 𝑋
𝑉=
(−𝑟𝐴 )𝑒𝑥𝑖𝑡
(13)
which gives the volume V necessary to achieve a conversion X. As we saw, the space time, τ, is a characteristic
time of a reactor. To obtain the space time, τ, as a function of conversion, we first substitute for FA0 = V’CA0 in
Equation (13)
𝑉=
𝑉′𝐶𝐴0 𝑋
(−𝑟𝐴 )𝑒𝑥𝑖𝑡
and then divide by V’ to obtain the space time, τ, to achieve a conversion X in a CSTR:
𝑉
𝐶𝐴0 𝑋
𝜏= =
𝑉′ (−𝑟𝐴 )𝑒𝑥𝑖𝑡
This equation applies to a single CSTR or to the first reactor of CSTRs connected in series.
7-2-1 Single SCTR
7.2.1.1 First-Order Reaction
Let’s consider a first-order irreversible reaction for which the rate law is:
−𝑟𝐴 = 𝑘𝐶𝐴
For liquid-phase reactions, there is no volume change during the course of the reaction, so we can use Equation
(12) to relate concentration and conversion (Stoichiometry):
𝐹𝐴 = 𝐹𝐴0 (1 − 𝑋)
𝐶𝐴 𝑉′ = 𝐶𝐴0 𝑉′(1 − 𝑋)
𝐶𝐴 𝑉′ = 𝐶𝐴0 𝑉′(1 − 𝑋)
𝐶𝐴 = 𝐶𝐴0 (1 − 𝑋)
(12)
𝑋
𝑑𝑋
),the rate law, and the concentration equation
We can combine the mole balance equation (7), (𝜏 = 𝑁𝐴0
−𝑟𝐴 𝑉
(12) to obtain:
0
𝑋
𝜏 = 𝑁𝐴0
0
𝑑𝑋
𝑁𝐴0
=
−𝑟𝐴 𝑉 −𝑟𝐴 𝑉
𝑋
𝑑𝑋 =
0
𝑁𝐴0 𝑋 𝐶𝐴0 𝑋
𝜏=
=
𝑘𝐶𝐴 𝑉
𝑘𝐶𝐴
𝜏=
𝐶𝐴0 𝑋
𝑘𝐶𝐴0 (1 − 𝑋)
𝑁𝐴0 𝑋
−𝑟𝐴 𝑉
𝑋
𝜏=
𝑘 (1 − 𝑋)
Rearranging:
𝑋=
𝜏𝑘
1 + 𝜏𝑘
A plot of conversion as a function of τk using Equation above is shown in following Figure.
First-order reaction in a CSTR
We can increase τk by either increasing the temperature to increase k or increasing the space time τ by
increasing the volume V or decreasing the volume flow rate V’.
7.2.1.2 A Second-Order Reaction in a CSTR
For a second-order liquid-phase reaction being carried out in a CSTR, the combination of the rate law and the
design equation yields:
𝑋𝐹𝐴0 𝑋𝐹𝐴0
𝑉=
=
−𝑟𝐴
𝑘𝐶𝐴2
Using our stoichiometric table for constant volume , CA = CA0(1 – X), and FA0X=V’CA0X, then
𝑉=
𝑋𝐹𝐴0
𝑉′𝑋
=
−𝑟𝐴
𝑘𝐶𝐴0 (1 − 𝑋)2
𝑉
𝑋
𝜏= =
𝑉′ 𝑘𝐶𝐴0 (1 − 𝑋)2
Rearranging, and assuming 𝜏𝑘𝐶𝐴0 = D, we obtain:
DX2 –X(2D+1) +D=0
We solve a second-degree equation in X:
𝑋=
2𝐷 + 1 − 4𝐷 + 1
2𝐷
The minus sign must be chosen in the quadratic equation because X cannot be greater than 1. Conversion is
plotted as a function of the Damköhler number (D= 𝜏𝑘𝐶𝐴0 ) for a second-order reaction, in Figure below.
Conversion as a function of the Damköhler (𝝉𝒌𝑪𝑨𝟎 ) number for a
second-order reaction in a CSTR.
We observe from this figure that at high conversions (say 67%), a 10-fold increase in the reactor volume (or
increase in the specific reaction rate by raising the temperature) will only increase the conversion up to 88%.
This observation is a consequence of the fact that the CSTR operates under the condition of the lowest reactant
concentration (i.e., the exit concentration), and consequently the smallest value of the rate of reaction.
7-2-2 SCTRs in series
A first-order reaction with no change in the volume flow rate (V0’= V’) is to be carried out in two CSTRs placed
in series.
Two CSTRs in series
The effluent concentration of reactant A from Reactor 1 can be found using the following Equation:
𝜏1 𝑘1
𝑋1 =
1 + 𝜏1 𝑘1
Since:
𝐹𝐴1 = 𝐹𝐴0 (1 − 𝑋1 )
𝐶𝐴1 𝑉′ = 𝐶𝐴0 𝑉′(1 − 𝑋1 )
𝐶𝐴1 𝑉′ = 𝐶𝐴0 𝑉′(1 − 𝑋1 )
𝐶𝐴1 = 𝐶𝐴0 (1 − 𝑋1 )
𝐶𝐴1
𝑋1 = 1 −
𝐶𝐴0
𝜏1 𝑘1
𝐶𝐴1
=1−
1 + 𝜏1 𝑘1
𝐶𝐴0
𝐶𝐴1
𝐶𝐴0
=
1 + 𝜏1 𝑘1
From a mole balance on reactor 2:
𝐹𝐴1 + 𝑟𝐴2 𝑉2 = 𝐹𝐴2
𝐹𝐴1 − 𝐹𝐴2 𝑉 ′ 𝐶𝐴1 − 𝑉′𝐶𝐴2
𝑉2 =
=
−𝑟𝐴2
−𝑟𝐴2
𝑉 ′ 𝐶𝐴1 − 𝑉′𝐶𝐴2
𝑉2 =
𝑘𝐶𝐴2
Solving for CA2, the concentration exiting the second reactor, we obtain:
𝐶𝐴2 = 𝐶𝐴1 − 𝑋2 𝐶𝐴1
𝐶𝐴2
𝑋2 = 1 −
𝐶𝐴1
By the other hand:
𝑋2 =
𝜏2 𝑘2
𝐶𝐴2
=1−
1 + 𝜏2 𝑘2
𝐶𝐴1
𝜏2 𝑘2
1 + 𝜏2 𝑘2
𝐶𝐴2
𝐶𝐴1
𝐶𝐴0
=
=
1 + 𝜏2 𝑘2 (1 + 𝜏2 𝑘2 )(1 + 𝜏1 𝑘1 )
If both reactors are of equal size (τ1 = τ2 = τ) and operate at the same temperature (k1 = k2 = k), then:
𝐶𝐴2
𝐶𝐴0
=
(1 + 𝜏𝑘)2
Example:Producing 200 Million Pounds per Year in a CSTR
About one-half of the ethylene glycol is used for antifreeze, while the other half is used in the manufacture of
polyesters. In the polyester category, 88% was used for fibers and 12% for the manufacture of bottles and films.
It is desired to produce 200 million pounds per year of EG. The reactor is to be operated isothermally. A 16.1
mol/dm3 solution of ethylene oxide (EO) in water is mixed (see Figure below) with an equal volumetric solution
of water containing 0.9 wt % of the catalyst H2SO4 and fed to a CSTR. The specific reaction-rate constant is
0.311 min–1, as determined in the previoius Example.
a- If 80% conversion is to be achieved, determine the necessary CSTR volume.
b- If two 800-gal reactors were arranged in parallel with the feed equally divided, what would be the
corresponding conversion?
c- If two 800-gal reactors were arranged in series, what would be the corresponding conversion?
Single CSTR.
Solution
The specified ethylene glycol (EG) production rate in mol/s is:
2108 ∗ 0,454
𝐹𝐶 =
= 2,88 𝐾𝑔/𝑠
365 ∗ 24 ∗ 3600
2,88
2,88
𝐹𝐶 =
=
= 46,45 𝑚𝑜𝑙/𝑠
𝑀𝑤
0,062
From the reaction stoichiometry:
𝐹𝐶 = 𝐹𝐴0 𝑋
we find the required molar flow rate of ethylene oxide for 80% conversion to be:
𝐹𝐴0 =
𝐹𝐶 46,45
=
= 58,06 𝑚𝑜𝑙/𝑠
𝑋
0,8
(a) We now calculate the single CSTR volume to achieve 80% conversion using the CRE algorithm.
1. CSTR Mole Balance:
2. Rate Law:
𝐹𝐴0 𝑋
𝑉=
−𝑟𝐴
𝑟𝐴 = −𝑘𝐶𝐴
3. Stoichiometry: Liquid phase (V0’=V’)
4. Combining:
𝐹𝐴 𝐹𝐴 𝐹𝐴0 (1 − 𝑋)
𝐶𝐴 =
= ′=
= 𝐶𝐴0 (1 − 𝑋)
′
𝑉′ 𝑉0
𝑉0
𝐹𝐴0 𝑋
𝐶𝐴0 𝑉0′ 𝑋
𝑉0′ 𝑋
𝑉=
=
=
𝑘𝐶𝐴0 (1 − 𝑋) 𝑘𝐶𝐴0 (1 − 𝑋) 𝑘 (1 − 𝑋)
5. Evaluate:
The entering volume flow rate of stream A, with CA01 16.1 mol/dm3 before mixing, is:
′
𝑉𝐴0
𝐹𝐴0
58
=
=
= 3,6 𝑑𝑚3 /𝑠
𝐶𝐴01 16,1
From the problem statement:
′
′
𝑉𝐴0
= 𝑉𝐵0
′
′
𝐹𝐵0 = 𝐶𝐵0 𝑉𝐵0
= 𝑉𝐴0
𝐶𝐵0
𝐹𝐵0 = 3,6
1000
= 200 𝑚𝑜𝑙/𝑠
18
The total entering volumetric flow rate of liquid is:
′
′
𝑉0′ = 𝑉𝐴0
+ 𝑉𝐵0
= 7,2 𝑑𝑚3 /𝑠
Substituting in volume Equation , recalling that k 0.311 min-1, yields:
𝑘1 = 0,311 𝑚𝑖𝑛−1 =
0,311
= 0,00518 𝑠 −1
60
𝑉0′ 𝑋
𝑉=
𝑘 (1 − 𝑋)
7,2 ∗ 0,8
𝑉=
= 5539 𝑑𝑚3
0,00518 ∗ (1 − 08)
(b) CSTRs in parallel. What would happen if two 3,028-m3 (3028 dm3) CSTRs arranged in parallel as shown in
following Figure had a volume flow rate V’ = 3.62 dm3/s fed to each reactor? Would the conversion increase,
decrease, or remain the same?
CSTRs in parallel.
To find out, let’s begin by rearranging Equation:
𝑉0′ 𝑋
𝑉=
𝑘 (1 − 𝑋)
𝑉𝑘
𝑋
=
𝑉′ 1 − 𝑋
to obtain:
𝜏𝑘 =
𝑋
1−𝑋
Where:
𝜏=
𝑉 3028
=
= 836,4 𝑠
𝑉′ 1,81
The Damköhler number for a first-order reaction is:
𝐷 = 𝑘𝜏 = 836,4 ∗0,00518 = 4,33
Substituting into Equation gives us:
𝜏𝑘
4,33
𝑋=
=
= 0,81
1 + 𝜏𝑘 1 + 4,33
The conversion exiting both of the CSTRs in parallel is 81%.
Problem asks you to generalize the result for n equal-size reactors Vi in parallel with equal feed rates (FA0/n) and
show that the conversion would also be the same if everything were fed to one big reactor of volume V = nVi
(c) CSTRs in series. What will happen to the conversion in Part (a) if the reactors are placed in series? Will it
increase or decrease?
CSTRs in series
If the 3028-m3 reactors are arranged in series, the conversion in the first reactor is:
𝜏1 𝑘1
𝑋1 =
1 + 𝜏1 𝑘1
Where:
𝑉 3028
𝜏1 = =
= 418,2 𝑠
𝑉′ 7,24
𝜏1 𝑘1
𝑋1 =
= 0,684
1 + 𝜏1 𝑘1
To calculate the conversion exiting the second reactor, we recall that V1=V2, V1’=V2’=V’, then:
𝜏2 = 𝜏1 = 𝜏 =
𝑉 3028
=
= 418,2 𝑠
𝑉′ 7,24
A mole balance on the second reactor is:
𝐹𝐴1 − 𝐹𝐴2 + 𝑟𝐴2 𝑉2 = 0
Basing the conversion on the total number of moles reacted up to a point per mole of A fed to the first reactor:
𝐹𝐴1 = 𝐹𝐴0 (1 − 𝑋1 )
𝐹𝐴2 = 𝐹𝐴0 (1 − 𝑋2 )
Rearranging:
𝐹𝐴1 − 𝐹𝐴2 𝐹𝐴0 (𝑋2 − 𝑋1 )
𝑉=
=
−𝑟𝐴2
−𝑟𝐴2
−𝑟𝐴2 = 𝑘𝐶𝐴2
𝐹𝐴2
𝐹𝐴0 1 − 𝑋2
=𝑘 ′ =𝑘
𝑉
𝑉′
Substituting and Rearranging:
𝑉𝑘 1 − 𝑋2 = 𝑉 ′ (𝑋2 − 𝑋1 )
c
𝑉𝑘 + 𝑉 ′ 𝑋1 = 𝑋2 (𝑉 ′ + 𝑉𝑘)
𝑉𝑘 + 𝑉′𝑋1
3028 ∗ 0,00518 + (7,28 ∗ 0,684)
𝑋2 = ′
=
= 0,9
𝑉 + 𝑉𝑘
7,24 + (3028 ∗ 0,00518)
7-2- Isothermal Tubular Reactors
Gas-phase reactions are carried out primarily in tubular reactors where the flow is generally turbulent. By
assuming that there is no dispersion and there are no radial gradients in either temperature, velocity,
concentration, or reaction rate, we can model the flow in the reactor as plug flow.
tubular reactor
The differential form of the PFR design equation such as:
𝑑𝐹
= 𝑟𝐴
𝑑𝑉
c
By other hand:
𝐹𝐴 = 𝐹𝐴0 − 𝐹𝐴0 𝑋
𝑑𝐹𝐴0 − 𝐹𝐴0 𝑑𝑋
= 𝑟𝐴
𝑑𝑉
c
𝑑𝑋
𝐹𝐴0
= −𝑟𝐴
𝑑𝑉
This differential form is used when there is a pressure drop in the reactor or heat exchange between the PFR
and the surroundings.
In the absence of pressure drop or heat exchange, the integral form of the plug-flow design equation can be
used:
𝑋
𝑉 = 𝐹𝐴0
0
𝑑𝑋
−𝑟𝐴
As an example, consider the elementary reaction:
for which the rate law is:
−𝑟𝐴 = 𝑘𝐶𝐴2
We shall first consider the reaction to take place as a liquid-phase reaction and then as a gas-phase reaction.
- Liquid Phase : V’ = V0’
The combined PFR mole balance and rate law is
𝑑𝑋
𝑑𝑋
𝑑𝑉 = 𝐹𝐴0
= 𝐹𝐴0 2
−𝑟𝐴
𝑘𝐶𝐴
If the reaction is carried out in the liquid phase, and according to the stoichiometry, the concentration of A is:
𝐶𝐴 = 𝐶𝐴0 (1 − 𝑋)
and for isothermal operation, we can bring k outside the integral
𝑑𝑋
𝑉′𝐶𝐴0 𝑑𝑋
𝑑𝑉 = 𝐹𝐴0 2
=
2
𝑘𝐶𝐴0 (1 − 𝑋)2 𝑘𝐶𝐴0
(1 − 𝑋)2
𝑉′
𝑉=
𝑘𝐶𝐴0
𝑉=
𝑋
0
𝑑𝑋
(1 − 𝑋)2
𝑉′
𝑋
𝑘𝐶𝐴0 1 − 𝑋
This equation gives the reactor volume to achieve a conversion X. Dividing by V’ (τ = V/V’) and solving for
conversion, we find:
𝜏=
1
𝑋
𝑘𝐶𝐴0 1 − 𝑋
𝑋=
- Gas Phase: 𝑽′ = 𝑽′𝟎 𝟏 + 𝜺𝑿 .
𝜏𝑘𝐶𝐴0
1 + 𝜏𝑘𝐶𝐴0
𝑻 𝑷
.
𝑻𝟎 𝑷𝟎
For gas-phase flow systems, the process becomes more complicated, as the volume flow rate for gases can vary
with conversion, temperature, and pressure, and we need to develop the relations relating V’ and X. Thas is to
say:
𝑉 ′ = 𝑉0′ 1 + 𝜀𝑋 .
𝑇 𝑃
.
𝑇0 𝑃0
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑜𝑙𝑒𝑠 𝑓𝑜𝑟 𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛
𝜀=
𝑇𝑜𝑡𝑎𝑙 𝑚𝑜𝑙𝑒𝑠 𝑓𝑒𝑑
FA non constant (FA ≠ FAo)
𝐹𝐴0 = 𝐶𝐴0 𝑉0′
𝐹𝐴 = 𝐶𝐴 𝑉′
𝐹𝐴 𝐹𝐴0 (1 − 𝑋)
𝐶𝐴0 𝑉0′ (1 − 𝑋)
𝐶𝐴 =
=
=
𝑇 𝑃
𝑉′
𝑉′
𝑉0′ 1 + 𝜀𝑋 . .
𝑇0 𝑃0
𝐶𝐴 =
𝐶𝐴0 (1 − 𝑋)
𝑇 𝑃
1 + 𝜀𝑋 . .
𝑇0 𝑃0
For constant-temperature (T = T0) and constant-pressure (P = P0) gas-phase reactions, the concentration is
expressed as a function of conversion:
𝑉 ′ = 𝑉0′ 1 + 𝜀𝑋 .
𝐶𝐴0 (1 − 𝑋)
𝐶𝐴 =
1 + 𝜀𝑋 .
and then combining the PFR mole balance, rate law, and stoichiometry
𝑉′𝑑𝑋
𝑑𝑉 =
𝑘𝐶𝐴0 (1 − 𝑋)2
𝑉0′ 1 + 𝜀𝑋 𝑑𝑋
𝑑𝑉 =
𝑘𝐶𝐴0 (1 − 𝑋)2
𝑑𝑉 =
𝐹𝐴0 1 + 𝜀𝑋 𝑑𝑋
2
𝑘𝐶𝐴0
(1 − 𝑋)2
𝐹𝐴0
𝑉=
2
𝑘𝐶𝐴0
𝑋
0
1 + 𝜀𝑋
𝑑𝑋
(1 − 𝑋)2
𝑉0′
𝑋 1+𝜀
𝑉=
2𝜀 1 + 𝜀 ln 1 − 𝑋 + 𝜀 2 𝑋 +
𝑘𝐶𝐴0
1−𝑋
2
Effect of ε on Conversion
We now look at the effect of the change in the total number of moles in the gas phase on the relation between
conversion and volume. For constant temperature and pressure, we have:
𝑉 ′ = 𝑉0′ 1 + 𝜀𝑋 .
Let’s now consider three types of reactions, one in which ε = 0, one in which ε < 0, and one in which ε > 0. When
there is no change in the number of moles with reaction, (e.g., A → B) ε = 0, then the fluid moves through the
reactor at a constant volume flow rate (V’ = V’0) as the conversion increases.
When there is a decrease in the number of moles (ε < 0) in the gas phase, the volume gas flow rate decreases
and the conversion increases. For example, when pure A enters for the reaction 2A → B, and taking A as the basis
of calculation, then A → B/2 and we have =(1-2)/2 = - 0,5.
Then:
𝑉 ′ = 𝑉0′ 1 − 0,5. 𝑋 .
Consequently, the gas molecules will spend more time in the reactor than they would if the flow rate were
constant, V’ = V0’. As a result, this longer residence time would result in a higher conversion than if the flow
were constant at V0’.
On the other hand, if there is an increase in the total number of moles (ε > 0) in the gas phase, then the
volume flow rate will increase as the conversion increases. For example, for the reaction A → 2B, then,
2−1
𝜀=
=1
1
𝑉 ′ = 𝑉0′ 1 + 𝑋 .
and the molecules will spend less time in the reactor than they would if the volume flow rate were constant. As
a result of this smaller residence time in the reactor, the conversion will be less than what would result if the
volume flow rate were constant at V0’.
V’
Change in gas-phase volume flow rate down the length of the reactor.
Figure above shows the volume flow rate profiles for the three cases just discussed. We note that, at the end of
the reactor, virtually complete conversion has been achieved.
Example:Producing 300 Million Pounds per Year of Ethylene in a Plug-Flow Reactor
Design of a Full-Scale Tubular Reactor
Ethylene ranks first in the United States in total tons of organic chemicals produced each year, and it is the
number-one organic chemical produced each year. Over 27,2 milliard tons were produced in 2010, and it sold
for $0.75 per kg. Sixty-five percent of the ethylene produced is used in the manufacture of fabricated plastics,
20% for ethylene oxide, 16% for ethylene dichloride and ethylene glycol, 5% for fibers, and 5% for solvents.
Determine the plug-flow reactor volume necessary to produce 136 million kg (300 million pound) of
ethylene a year by cracking a feed stream of pure ethane. The reaction is irreversible and follows an
elementary rate law (1st order reaction). We want to achieve 80% conversion of ethane, operating the reactor
isothermally at 1100 K and at a pressure of 6 atm. The specific reaction rate at 1,000 K is 0.072 s–1 and the
activation energy is 82,000 cal/mol.
Solution
The molar flow rate of ethylene exiting the reactor is
300 ∗ 0,4536 ∗ 106
𝐹𝐵 =
= 154,1 𝑚𝑜𝑙/𝑠
365 ∗ 24 ∗ 3600 ∗ 0,028
Next, calculate the molar feed rate of ethane, FA0, when 80% conversion is achieved,
𝐹𝐵 = 𝑋𝐹𝐴0
𝐹𝐴0 =
𝐹𝐵
𝑋
=
154,1
0,8
= 192,62 mol/s
1. Plug-Flow Mole Balance:
𝑑𝑋
𝐹𝐴0
= −𝑟𝐴
𝑑𝑉
Rearranging and integrating for the case of no pressure drop and isothermal operation yields
𝑋
𝑉 = 𝐹𝐴0
0
𝑑𝑋
−𝑟𝐴
2. Rate Law:
−𝑟𝐴 = 𝑘𝐶𝐴
with k = 0.072 s–1 at 1000 K
The activation energy is 82 kcal/g mol.
3. Stoichiometry: For isothermal operation and negligible pressure drop, the concentration of ethane is
calculated as follows:
Gas phase, constant T and P:
𝑉 ′ = 𝑉0′ 1 + 𝜀𝑋 .
𝐹𝐴 𝐹𝐴0 (1 − 𝑋) 𝐶𝐴0 𝑉0′ (1 − 𝑋)
𝐶𝐴 =
=
= ′
𝑉′
𝑉′
𝑉0 1 + 𝜀𝑋 .
𝐶𝐴0 (1 − 𝑋)
𝐶𝐴 =
1 + 𝜀𝑋 .
4. Combine volume and concentration Equations, we obtain
𝑋
𝑉 = 𝐹𝐴0
0
𝑋
𝑘𝐶𝐴0
𝑑𝑋
𝐹𝐴0
1 + 𝜀𝑋
=
𝑑𝑋
1−𝑋
𝑘𝐶𝐴0 1 − 𝑋
0
1 + 𝜀𝑋
5. Evaluate:
Since the reaction is carried out isothermally, we can take k outside the integral sign and to carry out our
integration.
𝑉=
𝐹𝐴0
1
1 + 𝜀 𝑙𝑛
− 𝜀𝑋
𝑘𝐶𝐴0
1−𝑋
6. Parameter evaluation:
For gas systems, CA0 can be calculated from the entering mole fraction, yA0, the temperature, T0, and pressure,
P0, using the ideal gas law or some other gas law
𝐶𝐴0 =
𝐶𝐴0
𝑃𝐴0 𝑦𝐴0 𝑃0
=
𝑅𝑇0
𝑅𝑇0
𝑦𝐴0 𝑃0
1 ∗ 6 ∗ 105
=
=
= 65,56 𝑚𝑜𝑙/𝑚3
𝑅𝑇0
8,32 ∗ 1100
CA0 = 0,0655 mol/l
𝟐−𝟏
𝜺=
=𝟏
𝟏
The rate constant k is given at 1000 K, and we need to calculate k at reaction conditions, which is 1100 K.
𝑘 𝑇2 =
𝐸 1 1
−
𝑘(𝑇1 )𝑒𝑥𝑝𝑅 𝑇1 𝑇2
E= 82 kcal/g mol = 342,76 kJ/mol.
𝑘 1100 =
342760
0,001−0,000909
0,072𝑒𝑥𝑝 8,32
𝑘 1100 = 3,04 𝑠 −1
Substituting into Volume Equation :
𝐹𝐴0
1
𝑉=
1 + 𝜀 𝑙𝑛
− 𝜀𝑋
𝑘𝐶𝐴0
1−𝑋
192,62
1
𝑉=
1 + 1 𝑙𝑛
− 0,8
3,04 ∗ 65,56
1 − 0,8
𝑉 = 2,3 𝑚3