MEMO

To: Dr. Michael Cunningham, Xin Su

From: Tanis Worthy 10045353, Sagesse Deane 10049303, Miguel Granero 10163458

Date: Nov 6-2014

Subject: CHEE 321 – Design Assignment

Part A

Ethylbenzene takes part in a reversible dehydrogenation reaction to produce styrene with a side product of hydrogen. This can be simplified to:

A↔B+C

The dehydrogenation of ethylbenzene to produce styrene follows elementary rate laws in both the forward and reverse reactions. Therefore the rate law can be written so that all concentration components have an exponent of 1.

−r A=K cCA=K f C A−K rCBCC

Since the reaction follows the ideal gas law, the equilibrium constant with regards to concentration is equivalent to the equilibrium constant with regards to pressure divided by the gas constant and temperature.

KC=K p

RT

Similarly ,

K f=K f , p

RT

Theequilibrium constant canbe defined by the quotient of the forward rate constant∧the reverse rate constant .

This canberearranges ¿ substitute for the reverse rate∈equations .

K r=K f

K c

Subbing all of these values in gives a rate law expression of:

−r A=K fC A−K rCBCC=K f CA−K f

K c

CBCC

A stoichiometric table can be used to solve for the output flowrates of each component. These can then be converted into concentrations and subbed into the rate law.

Table 1: Stoicheometric Table

Species Input Change OutputA F A0 −X A F A0 (1−X ¿¿ A )F A0=F A¿

B 0 X A F A0 X A F A0 = FB

C 0 X A F A0 X A F A0=FC

I F I 0 ¿S F A0 0 S FA 0=F I

Total F0=(1+S)FA 0 F t=(1+S+X A)F A0

Using the ideal gas law with the ratio of initial conditions to final conditions, the volumetric flowrate can be found.

VV 0

=F t

F0

The initial and final flowrates can be subbed into this equation to give an expression for the final volumetric flowrate.

V=V 0

F t

F0=V 0

(1+S+X A )F A 0

(1+S)F A 0

=V 0

(1+S+X A )(1+S)

Now that the volumetric flowrate is solved for, this can be used with the final flowrates found in the stoichiometric table to find the final concentrations of each species.

C A=F A

V=

(1−X ¿¿ A )F A0

V 0

(1+S+X A )(1+S)

=(1−X ¿¿ A)(1+S )F A0

(1+S+X A )V 0

=(1−X ¿¿A )(1+S )C A0

(1+S+X A )¿¿¿

CB=FB

V=

X AF A 0

V 0

(1+S+X A )(1+S)

=X ACA 0(1+S)

(1+S+X A )

CC=CB=X AC A0(1+S)

(1+S+X A )

Each of the concentrations can now be subbed in for a completed rate law.

−r A=K f

(1−X ¿¿ A) (1+S )C A0

(1+S+X A )−K f

K c ( X ACA 0 (1+S )

(1+S+X A ) )2

¿

At steady state:

dC A

dt=0

Which implies that ,

0=K f

(1−X ¿¿ A) (1+S )C A0

(1+S+X A )−

K f

K c ( X AC A0 (1+S )

(1+S+X A ) )2

=K cC A ¿

As

K f=K r=K f

K c

More detailed calculations are in Page 1 of the appendix.

Part B.

Figure 1 Comparing the effect on conversion of varying temperature (a and b), varying pressure (c and d) and varying steam ratio (e and f)

The conversion plots in Figure 1 were created by solving for Xa using the quadratic equation. The equation from which the a, b, and c values were obtained was as follows:

X A2 (CA 0 (1+S )+KC )+X A (S KC )− (KC (1+S ) )=0

In examining Figure 1, the effects of temperature, pressure, and steam ratio can be compared. Plots a) and b) show that increased temperature increases the conversion significantly past approximately 550K. These plots also show that conversion is higher with a lower pressure and higher steam ratio, but these changes are not as significant as the temperature changes. Plots c) and d) show that with lower pressures, the conversion rate is higher, especially between 1atm and approximately 3atm. Plot c) shows that higher steam ratios also contribute to higher conversions. Plot d) shows that there is even more of a contribution from temperature, seeing as how at very high temperatures and very low temperature, the conversion is very high and very low (respectively), and the pressure is almost insignificant to the conversion. Plots e) and f) show that as the steam ratio increases, so does the conversion, and plot e) shows that at more extreme temperatures, the steam ratio has less of an impact. With varying steam ratio, as seen in plot f), the conversion is not affected greatly at different pressures, except for at 1 atm. The conclusions to be drawn are that higher steam ratios and low pressures increase conversion slightly, but temperature is the main factor in conversion rate.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10000

20000

30000

40000

50000

60000

70000

c)

S=0 P=1atm T=953.90KS=0 P=2atm T=953.90K

Conversion, X[-]

Fao/

-ra

[L]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100000200000300000400000500000600000

a)

S=0 P=1atm T=953.90KS=3 P=1atm T=953.9K

Conversion, X [-]

Fao/

-ra

[L]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10000200003000040000500006000070000

b)

S=0 P=1atm T=953.90Ks=0 P=1atm T=1000K

Conversion, X [-]

Fao/

-ra

[L]

Figure 2 Levenspiel plots for varying S (plot a), varying T (plot b), and varying S (plot c)

In Figure 2, the Levenspiel plot a) shows that with a higher S value, the volume increased for both the CSTR and the PBR, which is not ideal for a reactor because with increased volume comes increased cost. With lower pressure and higher temperature (plots a and b), the volume of both the PBR and CSTR are minimized. From the design equations for a PBR vs. a CSTR (PBR volume is area under the curve and CSTR volume is area above the curve), the volume of a CSTR is always larger than the volume of a PBR. Therefore, using a PBR would minimize volume thereby decreased cost for a more optimized process.

Part c. Optimisation of product per volume of reactor.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10000

20000

30000

40000

50000

60000

70000

S=0 P=1atm T=953.90K

Conversion, X [-]

Fao/

-ra

[ L]

Figure 3 Levenspiel plot for optimized reactor conditions

The Levenspiel plot for the optimized conditions shows an exponential trend so that higher conversions always results in a much higher reactor volume. This is especially true when the conversion is increased above 0.7. Therefore, if the minimum required conversion is 0.7, the system needs to be optimized for that value.

According to our Conversion vs T, P and S plots for section b), the conversion is maximized when pressure is minimized and the temperature and steam ratio are maximized.

However, according to our Levenspiel plots, in figure 2. It is possible to conclude that a small increase in steam ratio results in a very high increase in volume. This means that to maximize the product per L of reactor we would need to diminish S. This is because the steam (inert) that enters the reactor also occupies a volume, therefore the reactor needs to be bigger.

Therefore, it is possible to state that there is trade-off between conversion and reactor volume for a change in S. Analysing the X vs S plot and the Levenspiel plot with a change in S it is possible to determine that the change in conversion can be neglected as a lower steam ratio does not diminish the conversion coefficient nearly as much as a higher steam ratio increases the reactor volume.

As a consequence, our final theoretical optimization would be a steam ratio of 0, a pressure of 1 atm and a temperature of 953.90K, which at steady state results in a conversion of 0.7. The reactor volume for this system is 14752 L for CSTR and 7663L for the PFR. This is expected as the PFR

minimizes the volume as it can be observed from the Levenspiel plot. The volume under the curve is the volume corresponding to the PFR and the area of y-axis (1/-Ra) times x-axis(Xa) is the volume for the CSTR. To calculate the volume from the PFR in Excel, the sum of rectangles methodwas used as excel does not have a built-in function.

The inlet volumetric flowrate was calculated by V ¿=¿ = 0.07827 cubic meters per s

The residence time was calculated using the formula: t=V PFR

V ¿ = 1m 37s. This means that a particle

entering the optimized reactor will take 1m 37s to exit the reactor.

The maximum flowrate of Fb per L of reactor volume was found to be Fb/Vpfr = 9.17*10^-5 g/L/s.

This is just a theoretical approximation and it is important to take into account other factors like to what extent temperature needs to be minimized as heating the mixture requires energy and implies higher running costs. In this case the steam ratio was set to zero to maximize the flowrate of product per L of reactor but in an actual set up steam is required to heat up the gas mixture. If we account for a reasonable steam ratio such as 10 [1] and want to diminish the volume of the reactor, the pressure could be increased to 200Kpa[1] so that the temperature would need to be at 885.5 K according to our optimization algorithm for the conversion. This temperature is in the actual Temperature range that an actual reactor that produces ST from EB would be at [1]. This would result in a volume of 243000L for the CSTR and 126100L for the PFR, which give a residence time of 311 seconds. With these conditions, Fb/Vpfr =5.55 * 10^-6 g/L/s.

It is clear from these results that the production rate of product B is not maximised under these conditions as Fb/Vpfr is lower than it was with S=0, but the volume is still reasonable, about 20 times less than an Olympic swimming pool (2,500,000L).

Operating conditions are also going to be affected by costs of reactants, volume available as well as the materials for the reactor. Another important factor would be maximum temperature and pressure that the reactor can operate and the conditions that the chemical compounds can withstand.

Appendix:

Table A 1 Optimization for Fb/Vpfr

Optimization minimizing S

S P T Kc CA0 Xa -ra 1/-ra Fb vPFR vCSTR Fb/V v0Kpa K mol/L mol/L/s L L mol/L m^3/s

0101.3

3953.9

00.0124

0.012776271

0.05

0.000150256

6.66E+03

0.05

332.8 332.77

0.000150256

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.10

0.000142348

7.03E+03

0.10

674.8 702.50

0.000148198

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.15

0.00013444

7.44E+03

0.15

1036.4

1115.74

0.000144738

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.20

0.000126531

7.90E+03

0.20

1419.9

1580.64

0.000140856

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.25

0.000118623

8.43E+03

0.25

1828.2

2107.51

0.000136745

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.30

0.000110715

9.03E+03

0.30

2264.8

2709.66

0.000132463

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.35

0.000102807

9.73E+03

0.35

2733.8

3404.45

0.000128029

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.40

9.48985E-05

1.05E+04

0.40

3240.4

4215.03

0.000123442

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.45

8.69903E-05

1.15E+04

0.45

3791.2

5172.99

0.000118696

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.50

7.90821E-05

1.26E+04

0.50

4394.7

6322.54

0.000113773

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.55

7.11739E-05

1.41E+04

0.55

5062.1

7727.55

0.000108651

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.60

6.32657E-05

1.58E+04

0.60

5808.5

9483.82

0.000103297

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.65

5.53575E-05

1.81E+04

0.65

6655.3

11741.87

9.76668E-05

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.70

4.74493E-05

2.11E+04

0.70

7633.8

14752.60

9.16978E-05

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.75

3.9541E-05

2.53E+04

0.75

8792.9

18967.63

8.52961E-05

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.80

3.16328E-05

3.16E+04

0.80

10215.5

25290.18

7.83126E-05

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.85

2.37246E-05

4.22E+04

0.85

12059.6

35827.75

7.04836E-05

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.90

1.58164E-05

6.32E+04

0.90

14693.9

56902.90

6.12497E-05

0.078270101

0101.3

3953.9

00.0124

0.012776271

0.95

7.90821E-06

1.26E+05

0.95

19435.9

120128.35

4.88787E-05

0.078270101

Table A 2 Optimal conditions from data from Reference 1 (Optimal conditions for a real life reactor)

Actual optimization accounting for the steamrequirement to heat up the mixture

S P T Kc CA0 Xa minus ra #NAME?Kpa K mol/L mol/L/s

10 200 885.25 0.003875880.00247036

6 0.05 9.0961E-06 109937.2154

10 200 885.25 0.003875880.00247036

6 0.1 8.61736E-06 116044.8384

10 200 885.25 0.003875880.00247036

6 0.15 8.13862E-06 122871.0054

10 200 885.25 0.003875880.00247036

6 0.2 7.65987E-06 130550.4432

10 200 885.25 0.003875880.00247036

6 0.25 7.18113E-06 139253.8061

10 200 885.25 0.003875880.00247036

6 0.3 6.70239E-06 149200.5066

10 200 885.25 0.003875880.00247036

6 0.35 6.22365E-06 160677.4686

10 200 885.25 0.003875880.00247036

6 0.4 5.74491E-06 174067.2576

10 200 885.25 0.003875880.00247036

6 0.45 5.26616E-06 189891.5538

10 200 885.25 0.003875880.00247036

6 0.5 4.78742E-06 208880.7092

10 200 885.25 0.003875880.00247036

6 0.55 4.30868E-06 232089.6769

10 200 885.25 0.003875880.00247036

6 0.6 3.82994E-06 261100.8865

10 200 885.25 0.003875880.00247036

6 0.65 3.3512E-06 298401.0131

10 200 885.25 0.003875880.00247036

6 0.7 2.87245E-06 348134.5153

10 200 885.25 0.003875880.00247036

6 0.75 2.39371E-06 417761.4184

10 200 885.25 0.003875880.00247036

6 0.8 1.91497E-06 522201.7729

10 200 885.25 0.003875880.00247036

6 0.85 1.43623E-06 696269.0306

10 200 885.25 0.003875880.00247036

6 0.9 9.57484E-07 1044403.546

10 200 885.25 0.003875880.00247036

6 0.95 4.78742E-07 2088807.092

10 200 885.25 0.003875880.00247036

6 0.9 9.57484E-07 1044403.546

References:

(1) Material Balances Project Styrene Manufacture, West Virginia University. http://www.che.cemr.wvu.edu/publications/projects/styrene/styrene-a.pdf (accessed: Nov 6, 2014)