Class 22

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Pressure Drop in Steady-State PFRs

to be added

Learning Outcomes

After completing Class 22 you should…

  • Know the definition and/or defining equation for open-tube momentum balance, packed-bed PFR, and packed-bed momentum balance
  • Understand that
    • different momentum balances are used when modeling open tube PFRs and packed-bed PFRs
    • when the open-tube momentum balance is used to model a PFR, the differential form of the ideal gas law must be added to the design equations
  • Be able to
    • select, modify, and simplify the mole, energy, and momentum balance equations needed for modeling a given PFR using either the axial position or the cumulative volume as the independent variable
    • use the PFR design equations, together with any other necessary equations, to complete a reaction engineering assignment involving pressure drop
    • use qualitative analysis to assess and explain results from a quantitative analysis

Prepare

  • Review Sections 9.1 through 9.4, and read Examples 9.6.3 and 9.6.4 from REB, The Book.
  • Watch …
  • Write a concise summary of the learning activity assignment below to use in class.

Participate

  • Attend Class 22 virtually using the Class 22 Video, and engage in completing the following Learning Activity.

Class 22 Learning Activity Hill and Root (“Introduction to Chemical Engineering Kinetics and Reactor Design,” 2nd Ed. Wiley, Hoboken, NJ, 2014.) describe the adiabatic oxidation of sulfur dioxide, reaction (1), in a steady-state, packed-bed PFR. A feed containing 8% SO2, 13% O2, and 79% N2 at 1 atm and 370 °C was fed to the first of two reactors at 0.149 lbmol s–1. It was assumed to have an effective viscosity of 0.09 lb h–1 ft–1. Spherical catalyst particles with a diameter of 0.25 in formed a packed bed with a porosity of 0.4 and a density of 0.6 g cm–3. The reactor diameter was 6 ft, and it operated at a conversion of 0.81. The rate expression shown in equation (2) was used with = 1.745 x 105 mol s–1 gcat–1 atm–1.5, = 31 kcal mol–1, = 7.59 x 109 mol s–1 gcat–1 atm, and = 53.6 kcal mol–1. Their analysis assumed no pressure drop, but after completing the analysis they estimated the pressure drop and found it to be significant. They suggested that the pressure drop could be decreased by using a larger reactor diameter.

Repeat their analysis, accounting for pressure drop, for reactors with diameters of 3, 6, and 9 ft and comment on the results. Specifically calculate the required catalyst volume and the pressure drop, and plot the temperature as a function of axial position in the reactor. Selected thermodynamic data are given in Table 1 where the heats of formation are in cal mol–1, and the heat capacities of the reagents in cal mol–1 K–1 can be calculated using the coefficients, \(\alpha_i\), \(\beta_i\), \(\gamma_i\), and \(\delta_i\), with temperatures in K.

\[ SO_2 + \frac{1}{2} O_2 \rightleftarrows SO_3 \tag{1} \]

\[ r = \frac{k_fP_{SO_2}P_{O_2} - k_rP_{SO_3}\sqrt{P_{O_2}}}{\sqrt{P_{SO_2}}} \tag{2} \]

\[ \hat C_{p,i} = \alpha _i + \beta _i T + \gamma _i T^2 + \delta _i T^3 \tag{3} \]

Table 1. Thermodynamic Data.
i \(\Delta H_{f,i}\big\vert_{298\,K}\) \(\alpha_i\) \(\beta_i\) \(\gamma_i\) \(\delta_i\)
SO2 –70950 5.697 0.016 –1.185 x 10–5 3.172 x 10–9
O2 0 6.713 –8.79 x 10–7 4.175 x 10–6 –2.544x 10–9
SO3 –94470 12.13 0.00812
N2 0 7.44 –0.00324 6.4 x 10–6 2.79 X 10–9

Learning Activity 22 Solution

Learning Activity 22 Calculations

Practice

  1. Attempt to complete the assignment below using only the REB Equation Summary, and if you get stuck, each time
    1. note whether you didn’t know what to do or you knew what to do, but not how to do it
    2. refer to just enough of the provided solution to get un-stuck
  2. Check your solution against the provided one and if you find mistakes note whether they were due to
    1. a conceptual misunderstanding
    2. improper simplification of a design equation or substitution into it
    3. computational errors
  3. Make sure you have corrected knowledge gaps and misunderstanding revealed in steps 1 and 2, and think about how you might modify your study methods to avoid similar issues in the future.

Class 22 Practice Assignment A PFR surrounded by a perfectly mixed jacket is going to be used to produce Z according to liquid-phase reaction (1). The feed to the reactor is at 45 psi and flows at 120 ft3 h–1. It contains only A at a concentration of 0.025 lbmol ft–3 and a temperature of 120 °F. Water at 75 °F flows into the shell at 2000 lb h–1. The reactor tubes have a diameter of 1 in, a length of 125 ft, a heat transfer coefficient of 150 BTU ft–2 h–1 °F–1, and a Darcy friction factor of 0.018.

\[ A \rightarrow Z \tag{1} \]

The reacting fluid heat capacity is 8 BTU lbmol–1 °F–1. Its viscosity is 1 lb ft–1 h–1, and its density is 57 lb ft–3. The cooling water heat capacity is 1 BTU lb–1 °F–1. The reaction is exothermic with a heat of reaction of –30,500 BTU lbmol–1. The rate coefficient equals 0.059 s–1 at 120 °F, and the activation energy is 14,000 BTU lbmol–1.

Plot the conversion, reacting fluid temperature and pressure along the length of the PFR.

Class 22 Practice Assignment Solution

Class 22 Practice Assignment Calculations

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