Uneversity of Tripoli
Faculty of Engineering
Chemical Engineering Department
Advanced Mathematic for Chemical Engineering
Case Study for Mathematical Modeling in
A Constant Volume Batch Reactor
Name: Zaineb Farag Abdallah
Dr. Hamza Khalaf Allah
Department
Chemical Engineering Department
2023
Abstract
Mathematical modeling describes a process and an object by use of the mathematical
language. A process or an object is presented in a" pure form"
Mathematical modeling of processes is a based on the probability theory. The form of
the solution is one unknown to a system of linear algebraic equation, both ordinary and
partial differential equation.
In this case, we descript the way to model a batch reactor for a constant volume by
using the mole balance for a first order reaction for to concentration A and B.
1. Introduction
A batch reactor is a reactor in which reactants are mix together and the reaction is allow
proceeding for a given time in a closed vessel with no inlet or outlet flows. Provision
for mixing, heating or cooling of the reactants may be required.
A batch reactor is normally use for small-scale operation, testing new processes, the
manufacture of expensive products, and processes difficult to convert to continuous.
Batch reactor is used extensively in industries where only small quantities of product is
made, such as pharmaceuticals.
The advantage is that high conversions is achieved due to leaving the reactants in
reactor.
We often used batch rector in chemical petrochemical or biochemical industry, for the
production of various quality products.
Batch reactor is operate with all the material placed in the reactor prior to the start of
reaction, and all the material is remove after the reaction has been complete.
2. Mole Balance Applied to Batch Reactor
Form the mole balance…
[Rate flow of inlet i mole/time]- [rate of flow of outlet i mole/time] + [rate generation
of i mole/time] = [rate accumulation of i mole/time]
ni (v) − ni (v + ∆v) + ri ∆v =
∆Ni
∆t
For a batch reactor inflow or out flow in a batch reactor so ni (v) ni(v+∆v) = 0
The equation can be limit for Ni = Ni0 t=0
dNi
ri v =
dt
For Ni = Ci v
The equation will be…
ri v = v
dci
dt
For constant volume
dci
Ri =
dt
In this case
k1
k2
A→ B→ C
Figure1: batch reactor sketch
2.1 Objective
Finding a solution of the differential equation describing CB (t) for n=1, m= 1.
2.2 Assumptions
In order to model a batch reactor we need to make the assumption
At any given time, the reactions are well mix so that the composition, temperature and
pressure are the same everywhere in the reactor.
Consider the first order, irreversible reaction .
The reaction rate per unit reaction volume are described by
R A = K 1 CA n , R B = K 1 CA n − k 2 CB m
For n=1, m=1
The linear case is describe by ….
dCA
= −k1 CA → (1)
dt
CA = CA0 exp(−k1 t)
dCB
= k1 CA − k 2 CB → (2)
dt
By dividing the two equation (1, 2) to find…
dCB
k 2 CB
= −1 + ( )
dCA
k1 CA
For a homogeneous form….
𝑉=
𝐶𝐴
𝐶𝐵
𝑑𝐶𝐵
𝑑𝑉
= 𝐶𝐴
+𝑉
𝑑𝐶𝐴
𝑑𝐶𝐴
𝐶𝐴
𝑑𝑉
𝑘2
+ 𝑉 = −1 + (𝑉)
𝑑𝐶𝐴
𝑘1
𝐶𝐴
∫
𝑑𝑉
𝑘2
= −1 + (𝑉) − 𝑉
𝑑𝐶𝐴
𝑘1
𝑑𝑉
𝑑𝐶𝐴
=∫
𝐾
𝐶𝐴
[−1 + ( 2 − 1) 𝑉]
𝐾1
By integration noting V=0 where CA=𝐶𝐴0 as an initial condition
CA
K1
CA
=
[1 − (
)
CB K 2 − K1
CA0
K1
−1
K2
]
Figuer2: a plot to the concentration A, B, C [3]
3. Conclusions
Using mathematical model allows evaluating the differential equation of concentration
of compound B based on the mole balance and assumption that help as to have a simple
model.
4. References
[1] Applied Mathematics and Modeling for Chemical Engineers, Richard G. Rice
Louisiana State University, Duong D. Do University of Queensland St. Lucia,
Queensland, Australia.
[2] Chemical Reaction Engineering 2013, Thomas Rodgers.
[3] Contributed by: Rachael.Baumann (June 2013) University of Colorado Boulder
department of chemical and biological engineering