Workshop 02 Solution: Conversion and reactor sizing

Lecture notes for chemical reaction engineering

Problem 1

P2-3: You have two CSTRs and two PFRs, each with a volume of 1.6 m^3 . Use Figure 1 to calculate the conversion for each of the reactors in the following arrangements.

1. Two CSTRs in series.
2. Two PFRs in series.
3. Two CSTRs in parallel with the feed, F_{A0}, divided equally between the two reactors.
4. Two PFRs in parallel with the feed divided equally between the two reactors.
5. A CSTR and a PFR in parallel with the flow equally divided. Calculate the overall conversion, X_{ov}

X_{ov} = \frac{F_{A0}-F_{A,CSTR} - F_{A,PFR}}{F_{A0}} with F_{A,CSTR} = \frac{F_{A0}}{2} - \frac{F_{A0}}{2} X_{CSTR} , \text{and } F_{A,PFR} = \frac{F_{A0}}{2} (1 - X_{PFR})

6. A PFR followed by a CSTR.
7. A CSTR followed by a PFR.
8. A PFR followed by two CSTRs. Is this arrangement a good arrangement or is there a better one?

Figure 1: Figure-2-2b

Solution:

To read the CSV file use the genfromtxt function from numpy

import numpy as np

p1_expt_file = './workshop-02-problem-1-data.csv'
p1_expt_data = np.genfromtxt(p1_expt_file, 
                             delimiter=',', 
                             dtype=[('x', float), 
                                    ('fa0_by_ra', float)], 
                             skip_header=1)

To interpolate the data, use CubicSpline function from scipy.interpolate.

import scipy.interpolate as interpolate

p1_interp = interpolate.CubicSpline(p1_expt_data['x'], 
                                    p1_expt_data_data['fa0_by_ra'])

Data plotting using matplotlib.pyplot

import matplotlib.pyplot as plt

fig,ax = plt.subplots()

ax.scatter(p1_expt_data['x'], 
           p1_expt_data['fa0_by_ra'], 
           marker='s', 
           color='red')

ax.set_xlabel('Conversion X')
ax.set_ylabel('$\\frac{F_{A0}}{-r_A} (m^3)$')

# Setting x and y axis limits
ax.set_xlim(0, 1)
ax.set_ylim(0, 12)

plt.show()

Add fit line to the plot

x_interp =np.linspace(0,1,100)
ax.plot(x_interp, p1_interp(x_interp), color='grey')

Add rectangle to the plot

# Adds rectangle from (0,0) with a width of x1 and height of y1
rectangle = plt.Rectangle((0, 0), x1, y1, color='skyblue', alpha=0.4)
ax.add_patch(rectangle)

Color area under the curve

 Fill the area under the curve between x1 and x2
x_fill = np.linspace(x1, x2, 100)
y_fill = p1_interp(x_fill)
ax.fill_between(x_fill, y_fill, color='skyblue', alpha=0.4)

Figure 2: Levenspiel plot of processed data 2 for problem 2-2b

1. Two CSTRs in series (Figure 3).

Figure 3: Conversion from two CSTR in series

2. Two PFRs in series (Figure 4).

Figure 4: Conversion from two PFR in series

3. Two CSTRs in parallel with the feed, F_{A0}, divided equally between the two reactors (Figure 5).

Figure 5: Conversion from two CSTR in parallel

4. Two PFRs in parallel with the feed divided equally between the two reactors (Figure 6).

Figure 6: Conversion from two PFR in parallel

5. A CSTR and a PFR in parallel with the flow equally divided. Calculate the overall conversion, X_{ov} (Figure 7)

Figure 7: Conversion from a CSTR and a PFR in parallel

6. A PFR followed by a CSTR (Figure 8).

Figure 8: Conversion from a PFR followed by CSTR

7. A CSTR followed by a PFR (Figure 9).

Figure 9: Conversion from a CSTR followed by PFR

8. A PFR followed by two CSTRs (Figure 10). Is this arrangement a good arrangement or is there a better one?

Figure 10: Conversion from a PFR followed by two CSTRs

Figure 11: Conversion from two CSTRs followed by a PFR

Two CSTRs followed by a PFR (Figure 11) yield final conversion of X_3 = 0.92.

Figure 12: Conversion from two PFRs followed by a CSTR

Two PFRs followed by a CSTR (Figure 12) yield final conversion of X_3 = 0.97.

Problem 2

P2-4: The exothermic reaction of stillbene (A) to form the economically important trospophene (B) and methane (C), i.e.,

\ce{A -> B + C}

was carried out adiabatically and the following data recorded:

Table 1: Problem 2.4 rate data

X	-r_A (mol/dm^3 min)
0	1
0.2	1.67
0.4	5
0.45	5
0.5	5
0.6	5
0.8	1.25
0.9	0.91

The entering molar flow rate of A was 300 mol/min.

1. What are the PFR and CSTR volumes necessary to achieve 40% conversion?
2. Over what range of conversions would the CSTR and PFR reactor volumes be identical?
3. What is the maximum conversion that can be achieved in a 105 dm^3 CSTR?
4. What conversion can be achieved if a 72 dm^3 PFR is followed in series by a 24 dm^3 CSTR?
5. What conversion can be achieved if a 24 dm^3 CSTR is followed in a series by a 72 dm^3 PFR?
6. Plot the conversion and rate of reaction as a function of PFR reactor volume up to a volume of 100 dm^3.

Solution:

The rate data (-r_A) is given. We need F_{A0}/-r_A. Dividing F_{A0} = 300 by -r_A we get:

Table 2: Processed data for problem 2

X	-r_A	F_{A0}/-r_A
0	1	300
0.2	1.67	179.641
0.4	5	60
0.45	5	60
0.5	5	60
0.6	5	60
0.8	1.25	240
0.9	0.91	329.67

Figure 13: Levenspiel plot of processed data for problem 2-4

Trying to fit a single cubic spline or a polynomial doesn’t work well due to the nature of the data (Figure 13). The data consists of three liniear segments. Therefore, we fit a piecewise linear function using numpy.piecewise. We also use lambda functions to define the linear segments.

def piecewise_linear_fit(x, x0, y0, k1, k2, k3):
    """
    Piecewise linear function defined by slopes and a constant part.
    x0, y0: Coordinates of the piecewise function's bending points.
    k1, k2, k3: Slopes of the first, second, and third parts.

    Note that in this problem, 
    k1 = -600.0
    k2 = 0 
    k3 = 898.9010989010986 

    x0 = [0.4, 0.6]
    y0 = [60.0, 60.0]

    We can call this function as
    piecewise_linear_fit(x, *args)
    
    args = ([0.4, 0.6], [60.0, 60.0], -600.0, 0, 898.9010989010986)
    """

    return np.piecewise(x, 
                        [x < x0[0], (x >= x0[0]) & (x <= x0[1]), x > x0[1]], 
                        [lambda x: k1*x + y0[0] - k1*x0[0], 
                         lambda x: y0[1], 
                         lambda x: k3*x + y0[1] - k3*x0[1]])

Figure 14: Piecewise linear fit processed data for problem 2-4

1. What are the PFR and CSTR volumes necessary to achieve 40% conversion?

Figure 15: Reactor volume for CSTR to achieve X = 0.4

From @fig-problem-2a, $V_{CSTR}$ = 24\.00 $dm^3$.

2. Over what range of conversions would the CSTR and PFR reactor volumes be identical?

   As the slope of F_{A0}/-r_A vs.\ X line is 0 between 0.4 and 0.6, the CSTR and PFR volumes over this range would be identical.

3. What is the maximum conversion that can be achieved in a 105 dm^3 CSTR?

Figure 16: Reactor conversion for CSTR with volume = 105 dm^3

From Figure 16, X = 0.70.

4. What conversion can be achieved if a 72 dm^3 PFR is followed in series by a 24 dm^3 CSTR?

Figure 17: Reactor conversion for PFR followed by CSTR

From Figure 17, X_{PFR} = 0.40 and X_{CSTR} = 0.64.

5. What conversion can be achieved if a 24 dm^3 CSTR is followed in a series by a 72 dm^3 PFR?

Figure 18: Reactor conversion for CSTR followed by PFR

From Figure 18, X_{CSTR} = 0.40 and X_{PFR} = 0.91.

6. Plot the conversion and rate of reaction as a function of PFR reactor volume up to a volume of 100 dm^3.

To create this plot (Figure 19), we will need to calculate the volume first for all X as

V = \int_0^x \frac{F_{A0}{-r_A}} dX

This data is given in Table 3

Table 3: Reactor conversion and rate as function fo volume

V (dm^3)	X	-r_A
0	0	1
48	0.2	1.67
72	0.4	5
75	0.45	5
78	0.5	5
84	0.6	5
113.978	0.8	1.25
142.451	0.9	0.91

Figure 19: Reactor conversion and rate as function fo volume

Problem 3

P2-7: The adiabatic exothermic irreversible gas-phase reaction

\ce{2A + B -> 2C}

is to be carried out in a flow reactor for an equimolar feed of A and B. A Levenspiel plot for this reaction is shown in Figure 20 .

1. What PFR volume is necessary to achieve 50% conversion?
2. What CSTR volume is necessary to achieve 50% conversion?
3. What is the volume of a second CSTR added in series to the first CSTR (Part b) necessary to achieve an overall conversion of 80%?
4. What PFR volume must be added to the first CSTR (Part b) to raise the conversion to 80%?
5. What conversion can be achieved in a 6 \times 10^4 m^3 CSTR? In a 6 \times 10^4 m^3 PFR?
6. Think critically to critique the answers (numbers) to this problem.

Figure 20: fig-p2-7b

Solution:

Problem 4

P2.10: The curve shown in Figure 21 is typical of a gas-solid catalytic exothermic reaction carried out adiabatically.

1. Assuming that you have a fluidized CSTR and a PBR containing equal weights of catalyst, how should they be arranged for this adiabatic reaction? Use the smallest amount of catalyst weight to achieve 80% conversion of A.
2. What is the catalyst weight necessary to achieve 80% conversion in a fluidized CSTR?
3. What fluidized CSTR weight is necessary to achieve 40% conversion?
4. What PBR weight is necessary to achieve 80% conversion?
5. What PBR weight is necessary to achieve 40% conversion?
6. Plot the rate of reaction and conversion as a function of PBR catalyst weight, W.

Additional information: FA0 = 2 mol/s.

Figure 21: Figure P2-10b

Solution:

Digitized graph: (Figure 22)

Figure 22: Levenspiel plot for an adiabatic exothermic heterogeneous reaction.

1. Assuming that you have a fluidized CSTR and a PBR containing equal weights of catalyst, how should they be arranged for this adiabatic reaction? Use the smallest amount of catalyst weight to achieve 80% conversion of A. (Figure 23)

Figure 23: Levenspiel plot for an adiabatic exothermic heterogeneous reaction.

2. What is the catalyst weight necessary to achieve 80% conversion in a fluidized CSTR? (Figure 24)

Figure 24: Catalyst weight for 80% conversion in CSTR

3. What fluidized CSTR weight is necessary to achieve 40% conversion? (Figure 25)

Figure 25: Catalyst weight for 40% conversion in CSTR

4. What PBR weight is necessary to achieve 80% conversion? (Figure 26)

Figure 26: Catalyst weight for 80% conversion in PBR

5. What PBR weight is necessary to achieve 40% conversion? (Figure 27)

Figure 27: Catalyst weight for 40% conversion in PBR

6. Plot the rate of reaction and conversion as a function of PBR catalyst weight, W. (Data table: Table 4; Plots: Figure 28)

Table 4: Reactor conversion and rate as function fo volume

V (dm^3)	X	-r_A
0.0730885	0.00122266	0.0334255
1.597	0.02795	0.0368764
3.97751	0.0756876	0.0436557
6.4248	0.134583	0.0531844
8.88462	0.207988	0.0670574
11.6107	0.313744	0.089225
14.0622	0.436315	0.108972
16.2747	0.55899	0.109126
18.6969	0.681754	0.0921045
21.4901	0.794571	0.0699448
23.7561	0.865033	0.0549146
25.4624	0.907618	0.0450597
27.1073	0.941305	0.037535

Figure 28: Reactor conversion and rate as function fo volume