12. Steady-State Nonisothermal Reactor Design—Flow Reactors with Heat Exchange

Research is to see what everybody else sees, and to think what nobody else has thought.

—Albert Szent-Gyorgyi

¹ Radial gradients are discussed in Chapters 17 and 18.

The heat flow to the reactor, , is given in terms of the overall heat-transfer coefficient, U, the heat exchange area, ΔA, and the difference between the ambient temperature, T_a, and the reactor temperature T

where a is the heat exchange area per unit volume of reactor. For a tubular reactor

where D is the reactor diameter. Substituting for in Equation (12-1), dividing by ΔV, and then taking the limit as ΔV → 0, we get

Expanding

From a mole balance on species i, we have

Differentiating the enthalpy Equation (11-19) with respect to V

Substituting Equations (12-3) and (12-4) into Equation (12-2), we obtain

Rearranging, we arrive at

This form of the energy balance will also be applied to multiple reactions.

and

where

Q_g = r_A ΔH_Rx

Q_r = Ua(T − T_a)

which is Equation (T11-1G) in Table 11-1 on pages 498–500.

For exothermic reactions, Q_g will be a positive number. We note that when the heat “generated,” Q_g, is greater than the heat “removed,” Q_r (i.e., Q_g > Q_r), the temperature will increase down the reactor. When Q_r > Q_g, the temperature will decrease down the reactor.

For endothermic reactions, Q_g will be a negative number and Q_r will also be a negative number because T_a > T. The temperature will decrease if (–Q_g) > (–Q_r) and increase if (–Q_r) > (–Q_g).

Equation (12-5) is coupled with the mole balances on each species, Equation (12-3). Next, we express r_A as a function of either the concentrations for liquid systems or molar flow rates for gas systems, as described in Chapter 4. We will use the molar flow rate form of the energy balance for membrane reactors and also extend this form to multiple reactions.

We could also write Equation (12-5) in terms of conversion by recalling F_i = F_A0(Θ_i + v_iX) and substituting this expression into the denominator of Equation (12-5).

PFR energy balance

For a packed-bed reactor dW = ρ_b dV where ρ_b is the bulk density

PBR energy balance

Equations (12-6) and (12-7) are also given in Table 11-1 as Equations (T11-1D) and (T11-1F). As noted earlier, having gone through the derivation to these equations, it will be easier to apply them accurately to CRE problems with heat effects.

Energy balance

must be coupled with the mole balance

Mole balance

and with the pressure drop equation

Pressure drop

and solved simultaneously. If the temperature of the heat-exchange fluid, T_a, varies down the reactor, we must add the energy balance on the heat-exchange fluid. In the next section, we will derive the following equation for co-current heat transfer

Heat exchanger

along with the equation for countercurrent heat transfer. A variety of numerical schemes can be used (e.g., Polymath) to solve these coupled differential equations: (A), (B), (C), and (D).

Numerical integration of the coupled differential equations (A) to (D) is required.

Consequently, we need to only solve equations (A), (D), and (E) simultaneously.

We now carry out an energy balance on the coolant in the annulus between R₁ and R₂, and between V and V + ΔV, as shown in Figure 12-2. The mass flow rate of the heat-exchange fluid (e.g., coolant) is . We will consider the case when the reactor is cooled and the outer radius of the coolant channel R₂ is insulated. Recall that by convention is the heat added to the system.

For co-current flow, the reactant and the coolant flow in the same direction.

The energy balance on the coolant in the volume between V and (V + ΔV) is

where T_a is the temperature of the heat transfer fluid, i.e., coolant, and T is the temperature of the reacting mixture in the inner tube.

Dividing by ΔV and taking limit as ΔV → 0

Analogous to Equation (12-4), the change in enthalpy of the coolant can be written as

The variation of coolant temperature T_a down the length of reactor is

The equation is valid whether the heat-transfer fluid is a coolant or a heating medium.

Typical heat-transfer fluid temperature profiles are shown here for both exothermic and endothermic reactions when the heat transfer fluid enters at T_a0.

Again, we write an energy balance over a differential reactor volume to arrive at

At the entrance, V = 0 ∴ X = 0 and T_a = T_a2.

At the exit, V = V_f ∴ T_a = T_a0.

We note that the only difference between Equations (12-10) and (12-11) is a minus sign, i.e., (T – T_a) vs. (T_a – T).

The solution to a countercurrent flow problem to find the exit conversion and temperature requires a trial-and-error procedure.

TABLE 12-1 PROCEDURE TO SOLVE FOR THE EXIT CONDITIONS FOR PFRs WITH COUNTERCURRENT HEAT EXCHANGE

Table 12-2 gives the algorithm for the design of PFRs and PBRs with heat exchange: In Case A Conversion is the reaction variable and in Case B Molar Flow Rates are the reaction variables. The procedure in Case B must be used to analyze multiple reactions with heat effects.

Be sure you can explain why these curves look the way they do.

In Chapter 11 the steady-state energy balance was derived as

Recall that is the shaft work, i.e., the work done by the stirrer or mixer in the CSTR on the reacting fluid inside the CSTR. Consequently, because the convention that done by the system on the surroundings is positive, the CSTR stirrer work will be a negative number, e.g., = –1,000 J/s. (See problem P12-6_B, a California Professional Engineers’ Exam Problem.)

These are the forms of the steady-state balance we will use.

Note: In many calculations the CSTR mole balance derived in Chapter 2

(F_A0X = –r_AV)

will be used to replace the term following the brackets in Equation (11-28), that is, (F_A0X) will be replaced by (–r_AV) to arrive at Equation (12-12).

Rearranging yields the steady-state energy balance

Although the CSTR is well mixed and the temperature is uniform throughout the reaction vessel, these conditions do not mean that the reaction is carried out isothermally. Isothermal operation occurs when the feed temperature is identical to the temperature of the fluid inside the CSTR.

³ Information on the overall heat-transfer coefficient may be found in C. J. Geankoplis, Transport Processes and Unit Operations, 4th ed. (Englewood Cliffs, NJ: Prentice Hall, 2003), p. 300.

For exothermic reactions (T>T_a2>T_a1)

The following derivations, based on a coolant (exothermic reaction), apply also to heating mediums (endothermic reaction). As a first approximation, we assume a quasi-steady state for the coolant flow and neglect the accumulation

For endothermic reactions (T_1a>T_2a>T)

term (i.e., dT_a/dt = 0). An energy balance on the heat-exchanger fluid entering and leaving the exchanger is

Energy balance on heat-exchanger fluid

where C_Pc is the heat capacity of the heat exchanger fluid and T_R is the reference temperature. Simplifying gives us

Solving Equation (12-16) for the exit temperature of the heat-exchanger fluid yields

From Equation (12-16)

Substituting for T_a2 in Equation (12-18), we obtain

Heat transfer to a CSTR

For large values of the heat-exchanger fluid flow rate, , the exponent will be small and can be expanded in a Taylor series (e^–x = 1 – x + . . .) where second-order terms are neglected in order to give

Then

where T_a1 ≅ T_a2 = T_a.

Valid only for large heat-transfer fluid flow rates!!

With the exception of processes involving highly viscous materials such as Problem P12-6_B, a California Professional Engineers’ Exam Problem, the work done by the stirrer can usually be neglected. Setting in Equation (11-27) to zero, neglecting ΔC_P, substituting for , and rearranging, we have the following relationship between conversion and temperature in a CSTR

Solving for X

Equation (12-22) is coupled with the mole balance equation

to design CSTRs.

We now will further rearrange Equation (12-21) after letting

then

Let κ and T_C be non-adiabatic parameters defined by

Non-adiabatic CSTR heat exchange parameters: κ and T_c

Then

The parameters κ and T_c are used to simplify the equations for non-adiabatic operation. Solving Equation (12-24) for conversion

Solving Equation (12-24) for the reactor temperature

Table 12-3 shows three ways to specify the design of a CSTR. This procedure for nonisothermal CSTR design can be illustrated by considering a first-order irreversible liquid-phase reaction. To solve CSTR problems of this type we have three variables X, T, and V and we can only specify two and then solve for the third. The algorithm for working through either cases A (X specified), B (T specified), or C (V specified) is shown in Table 12-3. Its application is illustrated in Example 12-3.

Forms of the energy balance for a CSTR with heat exchange

We will see in the next section that there may be multiple exit values of conversion and temperature (Multiple Steady States, MSS) that satisfy the parameter values and entrance conditions.

⁶ S.A. Vejtasa and R. A. Schmitz, AIChE J., 16 (3), 415 (1970).

2Na₂S₂O₃ + 4H₂O₂ → Na₂S₃O₆ + Na₂SO₄ + 4H₂O

in a CSTR operated adiabatically. The multiple steady-state temperatures were examined by varying the flow rate over a range of space times, τ.

Reconsider the X_MB(T) curve, Equation (E12-3.10), shown in Figure E12-3.2, which has been redrawn and shown as dashed lines in Figure E12-3.2A. Now consider what would happen if the volumetric flow rate υ₀ is increased (τ decreased) just a little. The energy balance line, X_EB(T), remains unchanged, but the mole balance line, X_MB, moves to the right, as shown by the curved, solid line in Figure E12-3.2A. This shift of X_MB(T) to the right results in the X_EB(T) and X_MB(T) intersecting three lines, indicating three possible conditions at which the reactor can operate.

When more than one intersection occurs, there is more than one set of conditions that satisfy both the energy balance and mole balance (i.e., X_EB = X_MB); consequently, there will be multiple steady states at which the reactor may operate. These three steady states are easily determined from a graphical solution, but only one could show up in the Polymath solution. Thus, when using the Polymath nonlinear equation solver, we need to either choose different initial guesses to find if there are other solutions, or plot X_MB and X_EB versus T, as in Example 12-3.

We begin by recalling Equation (12-24), which applies when one neglects shaft work and ΔC_P (i.e., ΔC_P = 0 and therefore )

where

and

Using the CSTR mole balance , Equation (12-24) may be rewritten as

G(T) = Heat-generated term

The left-hand side is referred to as the heat-generated term

The right-hand side of Equation (12-28) is referred to as the heat-removed term (by flow and heat exchange) R(T)

R(T) = Heat-removed term

To study the multiplicity of steady states, we shall plot both R(T) and G(T) as a function of temperature on the same graph and analyze the circumstances under which we will obtain multiple intersections of R(T) and G(T).

Heat-removed curve R(T)

Vary Non-adiabatic Parameter κ. If one increases κ by either decreasing the molar flow rate, F_A0, or increasing the heat-exchange area, A, the slope increases and for the case of T_a < T₀ the ordinate intercept moves to the left, as shown in Figure 12-8.

On the other hand, if T_a > T₀, the intercept will move to the right as κ increases.

To obtain a plot of heat generated, G(T), as a function of temperature, we must solve for X as a function of T using the CSTR mole balance, the rate law, and stoichiometry. For example, for a first-order liquid-phase reaction, the CSTR mole balance becomes

Solving for X yields

First-order reaction

Substituting for X in Equation (12-31), we obtain

Finally, substituting for k in terms of the Arrhenius equation, we obtain

Note that equations analogous to Equation (12-33) for G(T) can be derived for other reaction orders and for reversible reactions simply by solving the CSTR mole balance for X. For example, for the second-order liquid-phase reaction

Second-order reaction

the corresponding heat generated term is

Let’s now examine the behavior of the G(T) curve. At very low temperatures, the second term in the denominator of Equation (12-33) for the first-order reaction can be neglected, so that G(T) varies as

Low T

(Recall that means that the standard heat of reaction is evaluated at T_R.)

At very high temperatures, the second term in the denominator dominates, and G(T) is reduced to

High T

G(T) is shown as a function of T for two different activation energies, E, in Figure 12-9. If the flow rate is decreased or the reactor volume increased so as to increase τ, the heat-generated term, G(T), changes, as shown in Figure 12-10.

Can you combine Figures 12-10 and 12-8 to explain why a Bunsen burner goes out when you turn up the gas flow to a very high rate?

Heat-generated curves, G(T)

If one were now to increase the entering temperature to T₀₂, the G(T) curve, y, would remain unchanged, but the R(T) curve would move to the right, as shown by line b in Figure 12-11, and will now intersect the G(T) at point 2 and be tangent at point 3. Consequently, we see from Figure 12-11 that there are two steady-state temperatures, T_s2 and T_s3, that can be realized in the CSTR for an entering temperature T₀₂. If the entering temperature is increased to T₀₃, the R(T) curve, line c (Figure 12-12), intersects the G(T) curve three times and there are three steady-state temperatures, T_s4, T_s5, and T_s6. As we continue to increase T₀, we finally reach line e, in which there are only two steady-state temperatures, a point of tangency at T_s10 and an intersection at T_s11. By further increasing T₀, we reach line f, corresponding to T₀₆, in which we have only one reactor temperature that will satisfy both the mole and energy balances, T_s12. For the six entering temperatures, we can form Table 12-4, relating the entering temperature to the possible reactor operating temperatures.

Both the mole and energy balances are satisfied at the points of intersection or tangency.

By plotting T_s as a function of T₀, we obtain the well-known ignition-extinction curve shown in Figure 12-13. From this figure, we see that as the entering temperature T₀ is increased, the steady-state temperature T_s increases along the bottom line until T₀₅ is reached. Any fraction-of-a-degree increase in temperature beyond T₀₅ and the steady-state reactor temperature T_s will jump up from T_s10 to T_s11, as shown in Figure 12-13. The temperature at which this jump occurs is called the ignition temperature. That is, we must exceed a certain feed temperature, T₀₅, to operate at the upper steady state where the temperature and conversion are higher.

If a reactor were operating at T_s12 and we began to cool the entering temperature down from T₀₆, the steady-state reactor temperature, T_s3, would eventually be reached, corresponding to an entering temperature, T₀₂. Any slight decrease below T₀₂ would drop the steady-state reactor temperature to the lower steady-state value T_s2. Consequently, T₀₂ is called the extinction temperature.

The middle points 5 and 8 in Figures 12-12 and 12-13 represent unstable steady-state temperatures. Consider the heat-removed line d in Figure 12-12, along with the heat-generated curve, which is replotted in Figure 12-14. If we were operating at the middle steady-state temperature T_s8, for example, and a pulse increase in reactor temperature occurred, we would find ourselves at the temperature shown by vertical line , between points 8 and 9. We see that along this vertical line , the heat-generated curve, y ≡ G(T), is greater than the heat-removed line d ≡ R(T) i.e., (G > R). Consequently, the temperature in the reactor would continue to increase until point 9 is reached at the upper steady state. On the other hand, if we had a pulse decrease in temperature from point 8, we would find ourselves on a vertical line between points 7 and 8. Here, we see that the heat-removed curve d is greater than the heat-generated curve y (R > G), so the temperature will continue to decrease until the lower steady state is reached. That is, a small change in temperature either above or below the middle steady-state temperature, T_s8, will cause the reactor temperature to move away from this middle steady state. Steady states that behave in this manner are said to be unstable.

In contrast to these unstable operating points, there are stable operating points. Consider what would happen if a reactor operating at T_s9 were subjected to a pulse increase in reactor temperature indicated by line in Figure 12-14. We see that the heat-removed line d is greater than the heat-generated curve y (R > G), so that the reactor temperature will decrease and return to T_s9. On the other hand, if there is a sudden drop in temperature below T_s9, as indicated by line , we see the heat-generated curve y is greater than the heat-removed line d (G > R), and the reactor temperature will increase and return to the upper steady state at T_s9. Consequently, T_s9 is a stable steady state.

Next, let’s look at what happens when the lower steady-state temperature at T_s7 is subjected to pulse increase to the temperature shown as line in Figure 12-14. Here, we again see that the heat removed, R, is greater than the heat generated, G, so that the reactor temperature will drop and return to T_s7. If there is a sudden decrease in temperature below T_s7 to the temperature indicated by line , we see that the heat generated is greater than the heat removed (G > R), and that the reactor temperature will increase until it returns to T_s7. Consequently, T_s7 is a stable steady state. A similar analysis could be carried out for temperatures T_s1, T_s2, T_s4, T_s6, T_s11, and T_s12, and one would find that reactor temperatures would always return to locally stable steady-state values when subjected to both positive and negative fluctuations.

While these points are locally stable, they are not necessarily globally stable. That is, a large perturbation in temperature or concentration, while small, may be sufficient to cause the reactor to fall from the upper steady state (corresponding to high conversion and temperature, such as point 9 in Figure 12-14), to the lower steady state (corresponding to low temperature and conversion, point 7).

When we have multiple reactions occurring, we have to account for, and sum up, all the heats of reaction in the reactor for each and every reaction. For q multiple reactions taking place in the PFR where there are m species, it is easily shown that Equation (12-5) can be generalized to

---

Energy balance for multiple reactions

---

---

i = Reaction number
j = Species

---

Note that we now have two subscripts on the heat of reaction. The heat of reaction for reaction i must be referenced to the same species in the rate, r_ij, by which ΔH_Rxij is multiplied, which is

where the subscript j refers to the species, the subscript i refers to the particular reaction, q is the number of independent reactions, and m is the number of species. We are going to let

and

Then Equation (12-35) becomes

Equation (12-37) represents a nice compact form of the energy balance for multiple reactions.

Consider the following reaction sequence carried out in a PFR:

One of the major goals of this text is that the reader will be able to solve multiple reactions with heat effects, and this section shows how!

The PFR energy balance becomes

Again, we must account for the “heat generated” by all the reactions in the reactor. For q multiple reactions and m species, the CSTR energy balance becomes

---

Energy balance for multiple reactions in a CSTR

---

Substituting Equation (12-20) for , neglecting the work term, and assuming constant heat capacities and large coolant flow rates , Equation (12-39) becomes

For the two parallel reactions described in Example 12-5, the CSTR energy balance is

Major goal of CRE

One of the major goals of this text is to have the reader solve problems involving multiple reactions with heat effects (cf. Problems P12-23_C, P12-24_C, P12-25_C, and P12-26_B). That’s exactly what we are doing in the next two examples!

⁷ An introductory webinar on COMSOL can be found on the AIChE webinar Web site: http://www.aiche.org/resources/chemeondemand/webinars/modeling-non-ideal-reactors-and-mixers.

COMSOL application

We are going to carry out differential mole and energy balances on the differential cylindrical annulus shown in Figure 12-15.

F_iz = W_izA_cz

where W_iz is the molar flux in the z direction (mol/m²/s), and A_cz (m²) is the cross-sectional area of the tubular reactor.

In Chapter 14 we discuss the molar fluxes in some detail, but for now let us just say they consist of a diffusional component, –D_e(∂C_i/∂z), and a convective flow component, U_zC_i

where D_e is the effective diffusivity (or dispersion coefficient) (m²/s), and U_z is the axial molar average velocity (m/s). Similarly, the flux in the radial direction is

Radial direction

where U_r (m/s) is the average velocity in the radial direction. For now, we will neglect the velocity in the radial direction, i.e., U_r = 0. A mole balance on species A (i.e., i = A) in a cylindrical system volume of length Δz and thickness Δr as shown in Figure 12-15 gives

Mole Balances on Species A

Dividing by 2πrΔrΔz and taking the limit as Δr and Δz → 0

Similarly, for any species i and steady-state conditions

Using Equations (12-42) and (12-43) to substitute for W_iz and W_ir in Equation (12-44) and then setting the radial velocity to zero, U_r = 0, we obtain

For steady-state conditions and assuming U_z does not vary in the axial direction

This equation will also be discussed further in Chapter 17.

In terms of the molar fluxes and the cross-sectional area, and (q = /A_c)

The q term is the heat added to the system and almost always includes a conduction component of some form. We now define an energy flux vector, e, (J/m² · s), to include both the conduction and convection of energy.

e = energy flux vector J/s · m²

where the conduction term q (kJ/m² · s) is given by Fourier’s law. For axial and radial conduction, Fourier’s laws are

where k_e is the thermal conductivity (J/m·s·K). The energy transfer (flow) is the flux vector times the cross-sectional area, A_c, normal to the energy flux

Energy flow = e · A_c

(e_r2πrΔz)|_r – (e_r2πrΔz)|_{r+ Δr} + e_z 2πrΔr|_z – e_z 2πrΔr|_{z+ Δz} = 0

Dividing by 2πrΔrΔz and taking the limit as Δr and Δz → 0

The radial and axial energy fluxes are

e_r = q_r + ΣW_irH_i

e_z = q_z + ΣW_izH_i

Substituting for the energy fluxes, e_r and e_z

and expanding the convective energy fluxes, ΣW_i H_i,

Substituting Equations (12-51) and (12-52) into Equation (12-50), we obtain upon rearrangement

Recognizing that the term in brackets is related to Equation (12-44) and is just the rate of formation of species i, r_i, for steady-state conditions we have

Recalling

and

r_i = v_i(–r_A)

Σr_iH_i = Σv_iH_i(–r_A) = –ΔH_Rxr_A

we have the energy in the form

where W_iz is given by Equation (12-42). Equation (12-54) would be coupled with the mole balance, rate law, and stoichiometric equations to solve for the radial and axial concentration at gradients. However, a great amount of computing time would be required. Let’s see if we can make some approximations to simplify the solution.

Some Initial Approximations

Assumption 1. Neglect the diffusive term, wrt, the convective term in Equation (12-42) in the expression involving heat capacities

ΣC_PiW_iz = ΣC_{Pi(0 + UzCi}) = ΣC_PiC_iU_z

With this assumption, Equation (12-54) becomes

For laminar flow, the velocity profile is

where U₀ is the average velocity inside the reactor.

Assumption 2. Assume that the sum C_Pm = ΣC_PiC_i = C_A0ΣΘ_iC_Pi is constant. The energy balance now becomes

Energy balance with radial and axial gradients

Equation (12-56) is the form we will use in our COMSOL problem. In many instances, the term C_Pm is just the product of the solution density (kg/m³) and the heat capacity of the solution (kJ/kg · K).

Coolant Balance

We also recall that a balance on the coolant gives the variation of coolant temperature with axial distance where U_ht is the overall heat transfer coefficient and R is the reactor wall radius

Boundary and Initial Conditions

A. Initial conditions if other than steady state (not considered here) t = 0, C_i = 0,   T = T₀,   for z > 0 all r

B. Boundary conditions

1) Radial

(a) At r = 0, we have symmetry ∂_T/∂_r = 0 and ∂C_i/∂r = 0.

(b) At the tube wall, r = R, the temperature flux to the wall on the reaction side equals the convective flux out of the reactor into the shell side of the heat exchanger.

(c) There is no mass flow through the tube walls ∂C_i/∂r = 0 at r = R.

2) Axial

(a) At the entrance to the reactor z = 0

T = T₀ and C_i = C_i0

(b) At the exit of the reactor z = L

The preceding equations were used to describe and analyze flow and reaction in a tubular reactor with heat exchange as described in the following example, which can be found in the Expanded Material on the CRE Web site. What follows is only a brief outline of that example with a few results from the output of the COMSOL program.

COMSOL Web Site

Examples can be found on the COMSOL Web site (www.comsol.com/ecre), including the radial effects in tubular reactor examples: Isothermal Reactor, Nonisothermal Adiabatic Reactor, Nonisothermal Reactor with Isothermal Cooling Jacket, and Nonisothermal Reactor with Variable Coolant Temperature. You can also access the Tubular Reactor app directly from the Web site. This app solves the equations defined in the Nonisothermal Reactor with Variable Coolant Temperature exercise, which is embedded in the app. In these examples, you are also in the Living Example Program format (LEP). In accessing these LEPs, make sure that you either have COMSOL installed on your laptop or that it is available on your university computer. On the COMSOL Web site, www.comsol.com/ecre, you can also find how to locate the LEPs in the installed COMSOL Application Libraries.

Also, other applications based on exercises from this book are found at the COMSOL Web site (www.comsol.com/ecre). These include the following applications: activation_energy based on the example Determination of Activation Energy in this book; cstr_startup from the exercise Startup of a CSTR; non_ideal_cstr based on the paragraph Real CSTR Modeled as Two CSTRs with Interchange; and non_isothermal_plug_flow based on the example Production of Acetic Anhydride.

⁸ Courtesy of J. Singh, Chemical Engineering, 92 (1997) and B. Venugopal, Chemical Engineering, 54 (2002).

Runaway reactions are the most dangerous in reactor operation, and a thorough understanding of how and when they could occur is part of the chemical reaction engineer’s responsibility. The reaction in the last example in this chapter could be thought of as running away. Recall that as we moved down the length of the reactor, none of the cooling arrangements could keep the reactor from reaching an extremely high temperature (e.g., 800 K). In the next chapter, we study case histories of two runaway reactions. One is the nitroaniline explosion discussed in Example E13-2 and the other is Example E13-6, concerning the recent (2007) explosion at T2 Laboratories. (http://www.chemsafety.gov/videoroom/detail.aspx?VID=32).

There are many resources available for additional information on reactor safety and the management of chemical reactivity hazards. Guidelines for managing chemical reactivity hazards and other fire, explosion, and toxic release hazards are developed and published by the Center for Chemical Process Safety (CCPS) of the American Institute of Chemical Engineers. CCPS books and other resources are available at www.aiche.org/ccps. For example, the book Essential Practices for Managing Chemical Reactivity Hazards, written by a team of industry experts, is also provided free of charge by CCPS on the site www.info.knovel.com/ccps. A concise and easy-to-use software program that can be used to determine the reactivity of substances or mixtures of substances, the Chemical Reactivity Worksheet, is provided by the National Oceanic and Atmospheric Administration (NOAA) for free on its Web site, www.noaa.gov.

The Safety and Chemical Engineering Education (SAChE) program was formed in 1992 as a cooperative effort between the AIChE, CCPS, and engineering schools to provide teaching materials and programs that bring elements of process safety into the education of undergraduate and graduate students studying chemical and biochemical products and processes. The SAChE Web site (www.sache.org) has a great discussion of reactor safety with examples as well as information on reactive materials. These materials are also suitable for training purposes in an industrial setting.

The following instruction modules are available on the SAChE Web site (www.sache.org).

1. Chemical Reactivity Hazards: This Web-based instructional module contains about 100 Web pages with extensive links, graphics, videos, and supplemental slides. It can be used either for classroom presentation or as a self-paced tutorial. The module is designed to supplement a junior or senior chemical engineering course by showing how uncontrolled chemical reactions in industry can lead to serious harm, and by introducing key concepts for avoiding unintended reactions and controlling intended reactions.

2. Runaway Reactions: Experimental Characterization and Vent Sizing: This instruction module describes the ARSST and its operation, and illustrates how this instrument can easily be used to experimentally determine the transient characteristics of runaway reactions, and how the resulting data can be analyzed and used to size the relief vent for such systems.

3. Rupture of a Nitroaniline Reactor: This case study demonstrates the concept of runaway reactions and how they are characterized and controlled to prevent major losses.

4. Seveso Accidental Release Case History: This presentation describes a widely discussed case history that illustrates how minor engineering errors can cause significant problems; problems that should not be repeated. The accident was in Seveso, Italy, in 1976. It was a small release of a dioxin that caused many serious injuries.

Membership in SAChE is required to view these materials. Virtually all U.S. universities and many non-U.S universities are members of SAChE—contact your university SAChE representative, listed on the SAChE Web site, or your instructor or department chair to learn your university’s username and password. Companies can also become members—see the SAChE Web site for details.

Note: Access is restricted to SAChE members and universities

Certificate Program

SAChE also offers several certificate programs that are available to all chemical engineering students. Students can study the material, take an online test, and receive a certificate of completion. The following two certificate programs are of value for reaction engineering:

1. Runaway Reactions: This certificate focuses on managing chemical reaction hazards, particularly runaway reactions.

2. Chemical Reactivity Hazards: This is a Web-based certificate that provides an overview of the basic understanding of chemical reactivity hazards.

Many students are taking the certificate test online and put the fact that they successfully obtained the certificate on their résumés.

More information on safety is given in the Summary Notes and Professional Reference Shelf on the Web. Particularly study the use of the ARSST to detect potential problems. These will be discussed in Chapter 13 Professional Reference Shelf R13.1 on the CRE Web site.

In terms of conversion

2. The temperature dependence of the specific reaction rate is given in the form

3. The temperature dependence of the equilibrium constant is given by van’t Hoff’s equation for ΔC_P = 0

4. Neglecting changes in potential energy, kinetic energy, and viscous dissipation, and for the case of no work done on or by the system, large coolant flow rates (), and all species entering at the same temperature, the steady-state CSTR energy balance for single reactions is

5. Multiple steady states

where and

6. When q multiple reactions are taking place and there are m species

7. Axial or radial temperature and concentration gradients. The following coupled, partial differential equations were solved using COMSOL:

and

1. COSMOL Example W12-8

2. Problems

COMSOL LEP W12-8

WP12-18_C

WP12-21_B

WP12-24_B

WP12-31_B

• Learning Resources

1. Summary Notes

2. Web Module COMSOL Radial and Axial Gradients

3. Interactive Computer Games (ICGs)

A. Heat Effects I

B. Heat Effects II

4. Solved Problems

A. Example 12-2          Formulated in AspenTech: Download AspenTech directly from the CRE Web site.

A step-by-step AspenTech tutorial is given on the CRE Web site.

B. Example CD12-1     ΔH_Rx(T) for Heat Capacities Expressed as Quadratic Functions of Temperature

C. Example CD12-2     Second-Order Reaction Carried Out Adiabatically in a CSTR

5. PFR/PBR Solution Procedure for a Reversible Gas-Phase Reaction

• Living Example Problems

1. Example 12-1 Isomerization of Normal Butane with Heat Exchange

2. Example 12-2 Production of Acetic Anhydride

3. Example 12-2_Asp AspenTech Formulation

4. Example 12-4 CSTR with Cooling Coil

5. Example 12-5 Parallel Reaction in a PFR with Heat Effects

6. Example 12-6 Multiple Reactions in a CSTR

7. Example 12-7 Complex Reactions

8. Example W12-8 COSMOL Radial Effects in a Tubular Reactor

9. Example R12-1 Industrial Oxidation of SO₂

10. Example 12-T12-3 PBR with Variable Coolant Temperature, T_a

• Professional Reference Shelf

R12.1. Runaway in CSTRs and Plug-Flow Reactors

Phase Plane Plots. We transform the temperature and concentration profiles into a phase plane.

The trajectory going through the maximum of the “maxima curve” is considered to be critical and therefore is the locus of the critical inlet conditions for C_A and T corresponding to a given wall temperature.

R12.2. Steady-State Bifurcation Analysis. In reactor dynamics, it is particularly important to find out if multiple stationary points exist or if sustained oscillations can arise.

R12.3. Heat Capacity as a Function of Temperature. Combining the heat of reaction with the quadratic form of the heat capacity

C_Pi = α_i + β_iT + γ_iT²

we find that

Example 12-2 is reworked on the PRS for the case of variable heat capacities.

R12.4. Manufacture of Sulfuric Acid. The details of the industrial oxidation of SO₂ are described. Here, the catalyst quantities, the reactor configuration, and operating conditions are discussed, along with a model to predict the conversion and temperature profiles.

In each of the questions and problems, rather than just drawing a box around your answer, write a sentence or two describing how you solved the problem, the assumptions you made, the reasonableness of your answer, what you learned, and any other facts that you want to include. See Preface Section I for additional generic parts (x), (y), and (z) to the home problems.

Before solving the problems, state or sketch qualitatively the expected results or trends.

Q12-2_A Read over the problems at the end of this chapter. Make up an original problem that uses the concepts presented in this chapter. To obtain a solution:

(a) Make up your data and reaction.

(b) Use a real reaction and real data. See Problem P5-1_A for guidelines.

(c) Prepare a list of safety considerations for designing and operating chemical reactors. (See www.sache.org and www.siri.org/graphics.) The August 1985 issue of Chemical Engineering Progress may be useful for part (c).

• Get a more intuitive feel reactor system.

• Gain insight about the most sensitive parameters (e.g., E, KC) and how they affect outlet conditions.

• Learn how reactors are affected by different operating conditions.

• Simulate dangerous situations such as potential runaway reactions.

• Compare the model and parameters with experimental data.

• Optimize the reaction system.

P12-1_A Download the following Polymath programs (i.e., LEPs) from the CRE Web site where appropriate. In addition to the LEPs having Polymath and MATLAB codes, some LEPs have Wolfram. Previous students suggest using Wolfram to save time in working through the parameter variation problems to understand how the reactor behaves when parameters are varied.

(a) Example 12-1. Safety. Suppose the value of the equilibrium constant and heat of reaction were measured incorrectly and were found to be K_C = 1,000 mol/dm³ at 330 K and ΔH_Rx = –20,000 kJ/mol. (1) Redo Example 12-1 using these values. (2) Let Q_g = r_AΔH_Rx and Q_r = Ua (T – T_a), and then plot Q_g and Q_r on the same figure as a function of V. (3) Vary the coolant flow rate (0 < < 2,000 kg/h) and the entering temperature 273 (K < T₀ < 315 K), and describe what you find. (4) Vary some of the other parameters and see if you can find unsafe operating conditions. (5) Plot Q_r and T_a as a function of V necessary to maintain isothermal operation.

(b) Example 12-2 (Use the LEP). (1) Let Q_g = r_AΔH_Rx and Q_r = Ua (T – T_a), and then plot Q_g and Q_r on the same figure as a function of V. (2) Fix the reactor volume at 0.5 m³ and the entrance conditions at (T₀ = 1050 K, T_a0 = 1250 K), and then make a table, X_e, X, T_a, and T for each of the heat exchange cases. Change the inlet conditions and determine which heat exchanger case gives the greatest differences in the conversion. (3) Repeat (2) for V = 5 m³. (4) Plot Q_g, Q_r, and –r_A versus V for all four cases on the same figure and describe what you find. (5) For each of the four heat exchanger cases, investigate the addition of an inert I with a heat capacity of 50 J/mol · K, keeping F_A0 constant and letting the other inlet conditions adjust accordingly (e.g., ε). (6) Vary the inert molar flow rate (i.e., Θ_I, 0.0 < Θ_I < 3.0 mol/s). Plot X and analyze versus Θ_I. (7) Finally, vary the heat-exchange fluid temperature T_a0 (1,000°F < T_a0 < 1350°F). Write a paragraph describing what you find, noting interesting profiles or results.

(c) Example 12-2. AspenTech Formulation. Repeat P12-2(b) using AspenTech.

(d) Problem 12-3_B Solution. Load the LEP for this problem and answer the questions in P12-3_B. Students have pointed out that using Wolfram can greatly speed the solutions to the various parts of this problem.

(e) Example 12-3 (Use the LEP). Describe how your answers would change if the molar flow rate of methanol were increased by a factor of 4.

(f) Example 12-4 (Use the LEP). (1) Use Figure E12-3.2 and Equation (E12-4.4) to plot X versus T to find the new exit conversion and temperature. (2) Other data show and C_PA = 29 Btu/lb-mol/°F. How would these values change your results? (3) Make a plot of conversion as a function of heat exchanger area. [0 < A < 200 ft²].

(g) Example 12-5 (Use the LEP). (1) Why is there a maximum in temperature? How would your results change if there is (2) a pressure drop with α = 1.05 dm^–3? (3) Reaction (1) is reversible with K_C = 10 at 450 K? (4) How would the selectivity change if Ua is increased? Decreased?

(h) Example 12-6 (Use the LEP). (1) Vary T₀ to make a plot of the reactor temperature, T, as a function of T₀. What are the extinction and ignition temperatures? (2) Vary τ between 0.1 min and 0.001 min and describe what you find. (3) Vary Ua between 4,000 and 400,000 J/min/K and describe what you find.

(i) Example 12-7 (Use the LEP). (1) Safety. Plot Q_g and Q_r as a function of V. How can you keep the maximum temperature below 700 K? Would adding inerts help and if so what should the flow rate be if C_PI = 10 cal/mol/K? (2) Look at the Figures. What happened to species D? What conditions would you suggest to make more species D? (3) Make a table of the temperature (e.g., maximum T, T_a) and molar flow rates at two or three volumes, comparing the different heat-exchanger operations. (4) Why do you think the molar flow rate of C does not go through a maximum? Vary some of the parameters to learn if there are conditions where it goes through a maximum. Start by increasing F_A0 by a factor of 5. (5) Include pressure in this problem. Vary the pressure-drop parameter (0 < αρ_b < 0.0999 dm^–3) and describe what you find.

(j) CRE Web site SO₂ Example PRS-R12.4-1. Download the SO₂ oxidation LEP R12-1. How would your results change if (1) the catalyst particle diameter were cut in half? (2) The pressure were doubled? At what particle size does pressure drop become important for the same catalyst weight, assuming the porosity doesn’t change? (3) You vary the initial temperature and the coolant temperature? Write a paragraph describing what you find.

(k) SAChE. Go to the SAChE Web site, www.sache.org. Your instructor or department chair should have the username and password to enter the SAChE Web site in order to obtain the modules with the problems. On the left-hand menu, select “SAChE Products.” Select the “All” tab and go to the module entitled, “Safety, Health and the Environment” (S, H & E). The problems are for KINETICS (i.e., CRE). There are some example problems marked “K” and explanations in each of the above S, H & E selections. Solutions to the problems are in a different section of the site. Specifically look at: Loss of Cooling Water (K-1), Runaway Reactions (HT-1), Design of Relief Values (D-2), Temperature Control and Runaway (K-4) and (K-5), and Runaway and the Critical Temperature Region (K-7). Go through the K problems and write a paragraph on what you have learned.

(l) COMSOL Homework Problem (LEP). There is a COMSOL problem in the Expanded Material on the CRE Web site (www.comsol.com/ecre).

P12-2_A Use Wolfram LEP to find the value of Θ_B that would be required to not allow reactor temperature to rise above 360 K?

P12-3_B Load Polymath or Wolfram or MATLAB for LEP 12-T12-3 for Table T12-2 from the CRE Web site for this exothermic reversible reaction with a variable coolant flow rate.

has the following parameter values for the base case:

Vary the following parameters in the ranges shown in parts (a) through (i). Write a paragraph describing the trends you find for each parameter variation and why they look the way they do. Use the base case for parameters not varied. Hint: See and at the very end of the Chapter 12 Summary Notes on the CRE Web site. The feedback from students on this problem is that one should use Wolfram in the LEP 12-3 on the Web to carry out the parameter variations. Note: Using the Wolfram LEP on the CRE Web site could save a lot of time on this homework problem.

(a) F_A0: 1 ≤ F_A0 ≤ 8 mol/s

(b) Θ_I: 0.5 ≤ Θ_I ≤ 4

*Note: The program gives Θ_I = 1.0. Therefore, when you vary Θ_I, you will need to account for the corresponding increase or decrease of C_A0 because the total concentration, C_T0, is constant.

(c)

(d) T₀:      310 K ≤ T₀ ≤ 350 K

(e) T_a:      300 K ≤ T_a ≤ 340 K

(f) :     1 ≤ m_c ≤ 1000 g/s

(g) Repeat (f) for countercurrent coolant flow.

(h) Determine the conversion in a 5,000-kg fluidized CSTR where UA = 500 cal/s·K with T_a = 320 K and ρ_b <= 2 kg/m³.

(i) Repeat (a), (b), and (d) if the reaction were endothermic with K_c = 0.01 at 303 K and .

A partial solution is given in Living Example Problems on the CRE Web site.

P12-4_A Download the Interactive Computer Games (ICG) from the CRE Web site. Play the game, and then record your performance number for the module, which indicates your mastery of the material. Note: For simulation (b), only do the first three reactors, as reactors 4 and above do not work because of the technician tinkering with them.

(a) ICG Heat Effects Basketball 1 Performance # ___________.

(b) ICG Heat Effects Simulation 2 Performance # ___________.

P12-5_C Safety Problem. The following is an excerpt from The Morning News, Wilmington, Delaware (August 3, 1977): “Investigators sift through the debris from blast in quest for the cause [that destroyed the new nitrous oxide plant]. A company spokesman said it appears more likely that the [fatal] blast was caused by another gas—ammonium nitrate—used to produce nitrous oxide.” An 83% (wt) ammonium nitrate and 17% water solution is fed at 200°F to the CSTR operated at a temperature of about 510°F. Molten ammonium nitrate decomposes directly to produce gaseous nitrous oxide and steam. It is believed that pressure fluctuations were observed in the system and, as a result, the molten ammonium nitrate feed to the reactor may have been shut off approximately 4 min prior to the explosion.

Assume that at the time the feed to the CSTR stopped, there was 500 lb_m of ammonium nitrate in the reactor. The conversion in the reactor is believed to be virtually complete at about 99.99%.

Additional information (approximate but close to the real case):

where M is the mass of ammonium nitrate in the CSTR (lb_m) and k is given by the relationship below.

The enthalpies of water and steam are

H_w (200°F) = 168 Btu/lb_m

H_g (500°F) = 1202 Btu/lb_m

(a) Can you explain the cause of the blast? Hint: See Problem P13-3_B.

(b) If the feed rate to the reactor just before shutoff was 310 lb_m of solution per hour, what was the exact temperature in the reactor just prior to shutdown? Hint: Plot Q_r and Q_g as a function of temperature on the same plot.

(c) How would you start up or shut down and control such a reaction? (Hint: See Problem P13-2_B.)

(d) Explore this problem and describe what you find. For example, add a heat exchanger UA (T – T_a), choose values of UA and T_a, and then plot R(T) versus G(T)?

(e) Discuss what you believe to be the point of the problem. The idea for this problem originated from an article by Ben Horowitz.

P12-6_B The endothermic liquid-phase elementary reaction

A + B → 2C

proceeds, substantially, to completion in a single steam-jacketed, continuous-stirred reactor (Table P12-6_B). From the following data, calculate the steady-state reactor temperature:

Reactor volume: 125 gal

Steam jacket area: 10 ft²

Jacket steam: 150 psig (365.9°F saturation temperature)

Overall heat-transfer coefficient of jacket, U: 150 Btu/h · ft² · °F

Agitator shaft horsepower: 25 hp

Heat of reaction, = + 20,000

Btu/lb-mol of A (independent of temperature)

P12-7_B Use the data in Problem P11-4_A for the following reaction.

Th elementary, irreversible, organic liquid-phase reaction

is carried out in a flow reactor. An equal molar feed in A and B enters at 27°C, and the volumetric flow rate is 2 dm³/s and C_A0 = 0.1 kmol/m³.

Additional information:

(a) Calculate the conversion when the reaction is carried out adiabatically in one 500-dm³ CSTR and then compare the results with the two adiabatic 250-dm³ CSTRs in series.

The reversible reaction (part (d) of P11-4_A) is now carried out in a PFR with a heat exchanger. Plot and then analyze X, X_e, T, T_a, Q_r, Q_g, and the rate, –r_A, for the following cases:

(b) Constant heat-exchanger temperature T_a

(c) Co-current heat exchanger T_a (Ans.: At V = 10 m³ then X = 0.36 and T = 442 K)

(d) Countercurrent heat exchanger T_a (Ans.: At V = 10 m³ then X = 0.364 and T = 450 K)

(e) Adiabatic operation

(f) Make a table comparing all your results (e.g., X, X_e, T, T_a). Write a paragraph describing what you find.

(g) Plot Q_r and T_a as a function of V necessary to maintain isothermal operation.

P12-8_A The gas-phase reversible reaction as discussed in P11-7_B

A B

is now carried out under high pressure in a packed-bed reactor with pressure drop. The feed consists of both inerts I and species A with the ratio of inerts to the species A being 2 to 1. The entering molar flow rate of A is 5 mol/min at a temperature of 300 K and a concentration of 2 mol/dm³. Work this problem in terms of volume. Hint: V = W/p_B,

Additional information:

Plot and then analyze X, X_e, T, T_a, and the rate (–r_A) profiles in a PFR for the following cases. In each case, explain why the curves look the way they do.

(a) Co-current heat exchange

(b) Countercurrent heat exchange (Ans.: When V = 20 dm³ then X = 0.86 and X_e = 0.94)

(c) Constant heat-exchanger temperature T_a

(d) Compare and contrast each of the above results and the results for adiabatic operation (e.g., make a plot or a table of X and X_e obtained in each case).

(e) Vary some of the parameters, e.g., (0 < Θ_I < 10) and describe what you find.

(f) Plot Q_r and T_a as a function of V necessary to maintain isothermal operation.

P12-9_A Algorithm for reaction in a PBR with heat effects and pressure drop

The elementary gas-phase reaction

in P11-8_B is now continued and carried out in packed-bed reactor. The entering molar flow rates are F_A0 = 5 mol/s, F_B0 = 2F_A0, and F_I = 2F_A0 with C_A0 = 0.2 mol/dm³. The entering temperature is 325 K and a coolant fluid is available at 300 K.

Additional information:

Plot X, X_e, T, T_a, and –r_A down the length of the PFR for the following cases:

(a) Co-current heat exchange

(b) Countercurrent heat exchange

(c) Constant heat-exchanger temperature T_a

(d) Compare and contrast your results for (a), (b), and (c) along with those for adiabatic operation and write a paragraph describing what you find.

P12-10_B Use the data and reaction in Problems P11-4_A and P12-7_A for the following:

(a) Plot and then analyze the conversion, Q_r, Q_g, and temperature profiles up to a PFR reactor volume of 10 dm³ for the case when the reaction is reversible with K_C = 10 m3/kmol at 450 K. Plot and then analyze the equilibrium conversion profile.

(b) Repeat (a) when a heat exchanger is added, Ua = 20 cal/m³/s/K, and the coolant temperature is constant at T_a = 450 K.

(c) Repeat (b) for both a co-current and a countercurrent heat exchanger. The coolant flow rate is 50 g/s, C_Pc = 1 cal/g · K, and the inlet coolant temperature is T_a0 = 450 K. Vary the coolant rate (10 < < 1,000 g/s).

(d) Plot Q_r and T_a as a function of V necessary to maintain isothermal operation.

(e) Compare your answers to (a) through (d) and describe what you find. What generalizations can you make?

(f) Repeat (c) and (d) when the reaction is irreversible but endothermic with = 6,000 cal/mol. Choose T_a0 = 450 K.

P12-11_B Use the reaction data in Problem P11-4_A and P12-7_A for the case when heat is removed by a heat exchanger jacketing the reactor. The flow rate of coolant through the jacket is sufficiently high that the ambient exchanger temperature is constant at T_a = 50°C.

(a) (1) Plot and then analyze the temperature conversion, Q_r, and Q_g profiles for a PBR with

where

p_b = bulk density of the catalyst (kg/m³)

a = heat-exchange area per unit volume of reactor (m²/m³)

U = overall heat-transfer coefficient (J/s · m² · K)

(2) How would the profiles change if Ua/p_b were increased by a factor of 3000?

(3) If there is a pressure drop with α = 0.019 kg^–1?

(b) Repeat part (a) for co-current and countercurrent flow and adiabatic operation with = 0.2 kg/s, C_Pc = 5,000 J/kg K, and an entering coolant temperature of 50°C.

(c) Find X and T for a “fluidized” CSTR with 80 kg of catalyst.

(d) Repeat parts (a) and (b) for W = 80.0 kg, assuming a reversible reaction with a reverse specific reaction rate of

Vary the entering temperature, T₀, and describe what you find.

(e) Use or modify the data in this problem to suggest another question or calculation. Explain why your question requires either critical thinking or creative thinking. See Preface Section I and http://www.umich.edu/scps.

P12-12_C Derive the energy balance for a packed bed membrane reactor. Apply the balance to the reaction in Problem P11-5_B

A B + C

for the case when it is reversible with K_C = 1.0 mol/dm³ at 300 K. Species C diffuses out of the membrane with k_C = 1.5 s^–1.

(a) Plot and then analyze the concentration profiles for different values of K_C when the reaction is carried out adiabatically.

(b) Repeat part (a) when the heat transfer coefficient is Ua = 30 J/s·kg-cat·K with T_a = 50°C.

P12-13_B Circle the correct answer.

(a) The elementary reversible isomerization of A to B was carried out in a packed-bed reactor. The following profiles were obtained:

If the total entering volumetric flow rate remains constant, the addition of inerts to the feed stream will most likely

A) Increase conversion.

B) Decrease conversion.

C) Have no effect.

D) Insufficient information to tell

(b)

Which of the following statements are true?

A) The above reaction could be adiabatic.

B) The above reaction could be exothermic with constant cooling temperature.

C) The above reaction could be endothermic with constant heating temperature.

D) The above reaction could be second order.

P12-14_A The irreversible reaction

is carried out adiabatically in a CSTR. The “heat generated” G(T) and the “heat removed” R(T) curves are shown in Figure P12-14_A.

(a) What is the ΔH_Rx of the reaction?

(b) What are the inlet ignition and extinction temperatures?

(c) What are all the temperatures in the reactor corresponding to the inlet ignition and extinction temperatures?

(d) What are the conversions at the ignition and extinction temperatures?

P12-15_B The first-order, irreversible, exothermic liquid-phase reaction

A → B

is to be carried out in a jacketed CSTR. Species A and an inert I are fed to the reactor in equimolar amounts. The molar feed rate of A is 80 mol/min.

Additional information

(a) What is the reactor temperature for a feed temperature of 450 K?

(b) Plot and then analyze the reactor temperature as a function of the feed temperature.

(c) To what inlet temperature must the fluid be preheated for the reactor to operate at a high conversion? What are the corresponding temperature and conversion of the fluid in the CSTR at this inlet temperature?

(d) Suppose that the fluid inlet temperature is now heated 5°C above the reactor temperature in part

(c) and then cooled 20°C, where it remains. What will be the conversion?

(e) What is the inlet extinction temperature for this reaction system? (Ans.: T₀ = 87°C.)

P12-16_B The elementary reversible liquid-phase reaction

takes place in a CSTR with a heat exchanger. Pure A enters the reactor.

(a) Derive an expression (or set of expressions) to calculate G (T) as a function of the heat of reaction, equilibrium constant, temperature, and so on. Show a sample calculation for G (T) at T = 400 K.

(b) What are the steady-state temperatures? (Ans.: 310, 377, 418 K.)

(c) Which steady states are locally stable?

(d) What is the conversion corresponding to the upper steady state?

(e) Vary the ambient temperature T_a and make a plot of the reactor temperature as a function of T_a, identifying the ignition and extinction temperatures.

(f) If the heat exchanger in the reactor suddenly fails (i.e., UA = 0), what would be the conversion and the reactor temperature when the new upper steady state is reached? (Ans.: 431 K.)

(g) What heat exchanger product, UA, will give the maximum conversion?

(h) Write a question that requires critical thinking and then explain why your question requires critical thinking. Hint: See Preface Section I.

(i) What is the adiabatic blowout flow rate, υ₀ ?

(j) Suppose that you want to operate at the lower steady state. What parameter values would you suggest to prevent runaway, e.g., the upper SS?

Additional information

P12-17_B The reversible liquid phase reaction

A B

is carried out in a 12-dm³ CSTR with heat exchange. Both the entering temperature, T₀, and the heat exchange fluid, T_a, are at 330 K. An equal molar mixture of inerts and A enter the reactor.

(a) Choose a temperature, T, and carry out a calculation to find G(T) to show that your calculation agrees with the corresponding G(T) value on the curve shown below at the temperature you choose.

(b) Find the exit conversion and temperature from the CSTR. X = _______ T = _____.

(c) What entering temperature T₀ would give you the maximum conversion? T₀ = _______ X = ______

(d) What would the exit conversion and temperature be if the heat-exchange system failed (i.e., U = 0)?

(e) Can you find the inlet ignition and extinction temperatures? If yes, what are they? If not, go on to the next problem.

(f) Use Preface Section I to ask another question.

Additional information

The G(T) curve for this reaction is shown below

P12-18_C The elementary gas-phase reaction

2A C

is carried out in a packed-bed reactor. Pure A enters the reactor at a 450-K flow rate of 10 mol/s, and a concentration of 0.25 mol/dm³. The PBR contains 90 kg of catalyst and is surrounded by a heat exchanger for which cooling fluid is available at 500 K. Compare the conversion achieved for the four types of heat exchanger operation: adiabatic, constant T_a, co-current flow, and countercurrent flow.

Additional information

P12-19_C A reaction is to be carried out in the packed-bed reactor shown in Figure P12-19_C.

The reactants enter the annular space between an outer insulated tube and an inner tube containing the catalyst. No reaction takes place in the annular region. Heat transfer between the gas in this packed-bed reactor and the gas flowing countercurrently in the annular space occurs along the length of the reactor. The overall heat-transfer coefficient is 5 W/m² · K. Plot the conversion and temperature as a function of reactor length for the data given in reactor length for the data given in Problem P12-7_B

P12-20_B The reaction

A + B 2C

is carried out in a packed-bed reactor. Match the following temperature and conversion profiles for the four different heat-exchange cases: adiabatic, constant T_a, co-current exchange, and countercurrent exchange.

(a) Figure 1 matches Figure ________

(b) Figure 2 matches Figure ________

(c) Figure 3 matches Figure ________

(d) Figure 4 matches Figure ________

P12-21_B The irreversible liquid-phase reactions

are carried out in a PFR with heat exchange. The following temperature profiles were obtained for the reactor and the coolant stream:

The concentrations of A, B, C, and D were measured at the point down the reactor where the liquid temperature, T, reached a maximum, and they were found to be C_A = 0.1, C_B = 0.2, C_C = 0.5, and C_D = 1.5, all in mol/dm³. The product of the overall heat-transfer coefficient and the heat-exchanger area per unit volume, Ua, is 10 cal/s · dm³ · K. The entering molar flow rate of A is 10 mol/s.

Additional information

(a) What is the activation energy for Reaction (1)?

P12-22_B The following elementary reactions are to be carried out in a PFR with a heat exchange with constant T_a:

The reactants all enter at 400 K. Only A and B enter the reactor. The entering concentration of A and B are 3 molar and 1 molar at a volumetric flow rate of 10 dm³/s.

Additional information

What constant coolant temperature, T_a, is necessary such that at the reactor entrance, i.e., V = 0, ?

P12-23_B The complex gas-phase reactions are elementary

and carried out in a PFR with a heat exchanger. Pure A enters at a rate of 5 mol/min, a concentration of 0.2 mol/dm³, and temperature 300 K. The entering temperature of an available coolant is 320 K.

Additional information

k_1A1 = 50 dm³/mol • min@305 K with E₁ = 8,000 J/mol

k_2C2 = 4000 dm⁹/mol³ • min@310 K with E₂ = 4,000 J/mol

K_CA = 10 dm³/mol@315 K

Note: This is the equilibrium constant with respect to A in reaction 1 when

using van Hoff’s equation, i.e., (S12-4),

The reactor volume is 10 dm³.

(a) Plot (F_A, F_B, F_C) on one graph and (T and T_a) on another along the length of the reactor for adiabatic operation, heat exchange with constant T_a, and co-current and countercurrent heat exchange with variable T_a. Only turn in a copy of your code and output for co-current exchange.

Adiabatic operation

(b) What is the maximum temperature and at what reactor volume is it reached?

(c) At what reactor volume is the flow rate of B a maximum, and what is F_Bmax at this value?

Constant T_a

(d) What is the maximum temperature and at what reactor volume is it reached?

(e) At which reactor volume is the flow rate of B a maximum, and what is F_Bmax at this volume?

Co-current exchange

(f) At what reactor volume does T_a become greater than T? Why does it become greater

Countercurrent exchange

(g) At what reactor volume does T_a become greater than T?

Hint: Guess T_a at entrance around 350 K.

P12-24_B The elementary liquid-phase reactions

are carried out adiabatically in a 10 dm³ PFR. After streams A and B mix, species A enters the reactor at a concentration of C_A0 = 2 mol/dm³ and species B at a concentration of 4 mol/dm³. The entering volumetric flow rate is 10 dm³/s.

Assuming you could vary the entering temperature between 300 K and 600 K, what entering temperture would you recommend to maximize the concentration of species C exiting the reactor? (±25˚K). Assume all species have the same density.

Additional information

P12-25_C (Multiple reactions with heat effects) Xylene has three major isomers, m-xylene (A), o-xylene (B), and p-xylene (C). When m-xylene (A) is passed over a Cryotite catalyst, the following elementary reactions are observed. The reaction to form p-xylene is irreversible:

The feed to the reactor is pure m-xylene (A). For a total feed rate of 2 mol/min and the reaction conditions below, plot the temperature and the molar flow rates of each species as a function of catalyst weight up to a weight of 100 kg.

(a) Plot the concentrations of each of xylenes down the length (i.e., V) of a PBR.

(b) Find the lowest concentration of o-xylene achieved in the reactor.

(c) Find the maximum concentration of o-xylene in the reactor.

(d) Repeat part (a) for a pure feed of o-xylene (B). What is the maximum concentration of meta xylene and where does it occur in the reactor?

(e) Vary some of the system parameters and describe what you learn.

(f) What do you believe to be the point of this problem?

Additional information⁹

⁹ Obtained from inviscid pericosity measurements.

All heat capacities are virtually the same at 100 J/mol ·K.

P12-26_C (Comprehensive problem on multiple reactions with heat effects) Styrene can be produced from ethylbenzene by the following reaction:

However, several irreversible side reactions also occur:

[J. Snyder and B. Subramaniam, Chem. Eng. Sci., 49, 5585 (1994)]. Ethylbenzene is fed at a rate of 0.00344 kmol/s to a 10.0-m³ PFR (PBR), along with inert steam at a total pressure of 2.4 atm. The steam/ethylbenzene molar ratio is initially, i.e., parts (a) to (c), 14.5 : 1 but can be varied.

Given the following data, find the exiting molar flow rates of styrene, benzene, and toluene along with for the following inlet temperatures when the reactor is operated adiabatically:

(a) T₀ = 800 K

(b) T₀ = 930 K

(c) T₀ = 1100 K

(d) Find the ideal inlet temperature for the production of styrene for a steam/ethylbenzene ratio of 58 : 1. Hint: Plot the molar flow rate of styrene versus T₀ . Explain why your curve looks the way it does.

(e) Find the ideal steam/ethylbenzene ratio for the production of styrene at 900 K. Hint: See part (d).

(f) It is proposed to add a countercurrent heat exchanger with Ua = 100 kJ/m³/min/K, where T_a is virtually constant at 1000 K. For an entering stream to ethylbenzene ratio of 20, what would you suggest as an entering temperature? Plot the molar flow rates and .

(g) What do you believe to be the major points of this problem?

(h) Ask another question or suggest another calculation that can be made for this problem.

Additional information

The kinetic rate laws for the formation of styrene (St), benzene (B), and toluene (T), respectively, are as follows (EB = ethylbenzene).

The temperature T is in Kelvin and P_i is in atm.

P12-27_B The liquid-phase, dimer-quadmer series addition reaction

4A → 2A₂ → A₄

can be written as

and is carried out in a 10-dm³ PFR. The mass flow rate through the heat exchanger surrounding the reactor is sufficiently large so that the ambient temperature of the exchanger is constant at Ta = 315 K. The reactants enter at a temperature T₀, of 300 K. Pure A is fed to the rector at a volumetric flow rate of 50 dm³/s and a concentration of 2 mol/dm³.

(a) Plot, compare, and analyze the profiles F_A, F_A2, and F_A4 down the length of the reactor up to 10 dm³.

(b) The desired product is A₂ and it has been suggested that the current reactor may be too large. What reactor volume would you recommend to maximize F_A2?

(c) What operating variables (e.g., T₀, T_a) would you change and how would you change them to make the reactor volume as small as possible and to still maximize F_A2? Note any opposing factors in maximum production of A₂. The ambient temperature and the inlet temperature must be kept between 0˚C and 177˚C.

Additional information

Turn in your recommendation of reactor volume to maximize F_A2 and the molar flow rate at this maximum.

ARIS, R., Elementary Chemical Reactor Analysis. Upper Saddle River, NJ: Prentice Hall, 1969, Chaps. 3 and 6.

FOGLER, JOSEPH, A Reaction Engineer’s Handbook of Thermochemical Data, Jofostan Press, Riça, Jofostan (2018).

2. Safety

CENTER FOR CHEMICAL PROCESS SAFETY (CCPS), Guidelines for Chemical Reactivity Evaluation and Application to Process Design. New York: American Institute of Chemical Engineers (AIChE) 1995.

CROWL, DANIEL A., and JOSEPH F. LOUVAR, Chemical Process Safety: Fundamentals with Applications, 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2011.

MELHEM, G. A., and H. G. FISHER, International Symposium on Runaway Reactions and Pressure Relief Design. New York: Center for Chemical Process Safety (CCPS) of the American Institute of Chemical Engineers (AIChE) and The Institution of Chemical Engineers, 1995.

See the Center for Chemical Process Safety (CCPS) Web site, www.aiche.org/ccps.

3. A number of example problems dealing with nonisothermal reactors may or may not be found in

BURGESS, THORNTON W., The Adventures of Jerry Muskrat. New York: Dover Publications, Inc., 1914.

FROMENT, G. F., and K. B. BISCHOFF, Chemical Reactor Analysis and Design, 3rd ed. New York: Wiley, 2010.

WALAS, S. M., Chemical Reaction Engineering Handbook of Solved Problems. Amsterdam: Gordon and Breach, 1995. See the following solved problems: Problem 4.10.1, page 444; Problem 4.10.08, page 450; Problem 4.10.09, page 451; Problem 4.10.13, page 454; Problem 4.11.02, page 456; Problem 4.11.09, page 462; Problem 4.11.03, page 459; Problem 4.10.11, page 463.

4. A review of the multiplicity of the steady state and reactor stability is discussed by

PERLMUTTER, D. D., Stability of Chemical Reactors. Upper Saddle River, NJ: Prentice Hall, 1972.

5. The heats of formation, H_i (T), Gibbs free energies, G_i (T_R), and the heat capacities of various compounds can be found in

GREEN, DON W., and ROBERT H. PERRY, Perry’s Chemical Engineers’ Handbook, 8th ed. (Chemical Engineers Handbook). New York: McGraw-Hill, 2008.

LIDE DAVID R., CRC Handbook of Chemistry and Physics, 90th ed. Boca Raton, FL: CRC Press, 2009.