Chapter 6- Numerical simulations of chemical engineering systems
Classification of ODEs and PDEs
Book: “Mathematical Modeling In Chemical Engineering, Anders Rasmuson, Bengt Andersson,
Louise Olsson” Page-80 Chapter-6
Newton Raphson, Bisection, Euler, RK4, Adam Bashforth, Backward difference, Forward
difference, Central Difference, Finite Difference Method
Numerical: Also see class notes and book
(Bisection, Newton Raphson, False-position method)
Example questions
1. A cylindrical tank with a cross-sectional area A=2 m2 has an outflow governed by a
nonlinear valve equation. The liquid height h(in meters) at steady state satisfies the mass
balance:𝑄𝑖𝑛 = 𝑄𝑜𝑢𝑡 = 𝑘√ℎ where Qin=0.5 m3/min is the inlet flow rate, and
k=0.4 m2.5/min is the valve constant. Find the steady-state height h using numerical
approach.
2. In a chemical process, a gas is stored in a rigid container at a fixed temperature and
pressure. The volume V (in liters) of the gas is described by the vander Waals equation, a
nonlinear equation accounting for non-ideal behavior:
(P+a/V2) (V−b) =RT
where:
P=10 atm(pressure),
a=1.36 atm⋅L2/mol2 (van der Waals constant for attraction),
b=0.0318 L/mol (van der Waals constant for volume exclusion),
R=0.08206 L⋅atm/(mol⋅K) (gas constant),
T=300 K (temperature).
Find the molar volume V for 1 mole of gas by using numerical method.
(Euler, RK4, Adam Bashforth)
1. A batch reactor has a first-order reaction: dCA/dt = -0.2 C_A (mol/L min), with initial
concentration CA(0) = 1 mol/L. Use the Euler method with step size h = 1 min to estimate
C_A at t = 3 min.
2. Consider a first-order irreversible reaction A->B in a CSTR, where the concentration of
species A, denoted C_A, changes over time due to reaction and flow. The governing ODE
for the concentration in a transient CSTR is:
𝑑𝐶𝐴 𝑄
= (𝐶𝐴𝑜 − 𝐶𝐴 ) − 𝑘𝐶𝐴
𝑑𝑡
𝑉
Where: Q/V = 0.1s-1, k = 0.2s-1, CA0= 1.0mol/L, Initial condition: CA(0) = 0.5mol/L. Find CA at
time t=1s using all numerical methods.
Heat transfer problems
1. Derive a relation on how a 1-D heat transfer relation can be solved for a homogeneous
material with constant thermal properties using numerical approach.
2. A rod of length L = 0.8m has thermal diffusivity 0.005m2/s. Boundary conditions: u (0, t)
= 80°C, u (0.8, t) = 20°C, Initial condition: u(x,0) = 0°C for 0 < x < 0.8. Compute the
temperature at t = 4s using FDM with 4 spatial steps and 3-time steps.
3. A rod of length L = 0.6m has thermal diffusivity alpha = 0.008m^2/s. Both ends (x = 0, 0.6)
𝑥
are insulated. Initial condition: 𝑢(𝑥, 0) = 50(1 − 0.6)°C for (0 < x < 0.6). Compute the
temperature at t = 3, s using FDM with 4 spatial steps and 3-time steps.
4. A rod of length L=1.2 m has thermal diffusivity α=0.012 m2/s. Boundary conditions: u (0,
t) =60∘C u (0, t) = 60°C, u (0, t) =60∘C, insulated at x=1.2. Initial condition: u(x,0) =30°C for
0≤x<1.2. Compute the temperature at t=6s using FDM with 4 spatial steps and 3-time
steps.
5. A rod of length L=0.4 m has a thermal diffusivity α=0.006 m2/s. The rod has fixed
temperatures at the boundaries: u (0, t) =50°C, u (0.4, t) =10°C. The initial temperature is
a piecewise linear profile:
𝑥
40 (1 −
) 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 0.2
𝑢(𝑥, 0) = {
0.2
0 𝑓𝑜𝑟 0.2 < 𝑥 < 0.4
Compute the temperature at t=2s using FDM ensuring the stability factor F=0.2.
Mass transfer problems
1. Derive a relation on how the concentration of a species evolves over time t and space x in
a medium using numerical approach. Also, derive an expression for diffusive flux of the
species due to a concentration gradient in steady-state condition.
2. A thin membrane of thickness L=0.02 m has a diffusion coefficient D=1×10 −5 m2/s. The
concentration on one side (x=0) is C(0) =5 kg/m3 and on the other side (x=0.02) is C(0.02)
=1 kg/m3. Compute the steady-state concentration profile and diffusive flux using FDM.
3. A liquid-filled tube of length L=0.5 m has a diffusion coefficient D=0.01 m2/s. The left end
(x=0) is maintained at C(0,t)=20 kg/m3, and the right end (x=0.5) is at C(0.5,t)=0 kg/m3.
The initial concentration is C(x,0)=0 kg/m3 for 0<x<0.5. Use FDM to compute the
concentration at t=0.5 s, and calculate the diffusive flux J at x=0.2 m at t=0.5s.
4. A liquid-filled tube of length L=0.4 m has a diffusion coefficient D=0.008 m2/s. Both ends
(x=0 and x=0.4) are insulated. The initial concentration is:
𝐶(𝑥, 0) =
𝜋𝑥
10 sin (0.4) 𝑘𝑔
𝑓𝑜𝑟0 ≤ 𝑥 ≤ 0.4
𝑚3
representing a sinusoidal distribution (e.g., a localized concentration peak). Use FDM to
compute the concentration at t=0.2s, and calculate the flux J at x=0.2m.
5. A liquid-filled tube of length L=0.4 m has a diffusion coefficient D=0.008 m2/s. Boundary
conditions: C(0, t) =40 kg/m3, C(0.4, t) =10kg/m3. Initial condition: C(x,0) = 20kg/m3 for
0<x<0.4. Compute the concentration at t=1s using FDM with 3 spatial steps (4 points) and
2-time steps.
6. A tube of length L=0.5 m has D=0.01 m2/s with both ends insulated. Initial condition:
C(x,0) =30(1−x/0.5) kg/m3 for 0≤x≤0.5. Compute at t=1.5 s using FDM with 3 spatial steps
(4 points) and 2-time steps.
7. A liquid-filled tube of length L=0.3 m has a diffusion coefficient D=0.007 m2/s. Boundary
conditions: C(0, t) =50kg/m3, C(0.3, t) =0 kg/m3. Initial condition: C(x,0) =25kg/m3 for
0<x<0.3. Compute the concentration at t=0.6s using FDM with 3 spatial steps (4 points)
and 2-time steps. Calculate the diffusive flux J at x=0.15 m at t=0.6s.
Sensitivity analysis problems- See class notes
Chapter-7 Artificial Intelligence-based models
Book: Neural Networks in Bioprocessing and Chemical Engineering (D. R. Baughman and Y. A. Liu
1. What is a machine learning model?
2. Explain how deep neural networks fit into the concept of "Deep Learning."
3. Describe the structure of an Artificial Neural Network (ANN), including input layers,
hidden layers, and output layers.
4. Why are ANN-based models considered as a "black-box approach," and what does this
imply for understanding underlying phenomena?
5. What is the role of nodes in an ANN, and how do they perform computations using
activation functions?
6. Define the learning process in ANNs, including how weights are adjusted to minimize
errors.
7. What is a feedforward multilayered ANN, and why is it commonly used in chemical and
biochemical applications?
8. Differentiate between feedforward neural networks and backpropagation neural
networks (BPNNs).
9. How does information processing occur in a single neuron, including the use of weighted
inputs and activation functions?
10. What are activation functions, and name some common types used in ANNs?
11. Describe a univariate regression problem and the type of neural network typically used
for it.
12. What is graph regression, and what kind of neural network is suitable for multivariate
regression problems?
13. Explain binary classification and provide an example of a neural network used for
classification.
14. How does multiclass classification differ from binary classification, and give examples.
15. List and explain the three main sources of data for training ANN models: experimental
data from literature, planned experiments, and simulated data.
16. Why is data normalization important in ANN model development, and what range is
typically used?
17. Explain the training process of ANNs, including how weights and biases are determined
through error minimization.
18. Define overfitting in ANNs, and how can it be identified and avoided during training?
19. What is underfitting, and how does it differ from overfitting?
20. What is an epoch in machine learning, and why is it an important hyperparameter?
21. How is the network error calculated in ANNs, and what optimization scheme is used to
minimize it?
22. What are the limitations of ANN-based models?