CHAPTER 5 Batch and Semi-Batch Operations

Transcription

1 CHAPTER 5 Batch and Semi-Batch Operations PROCESS CYCLE TIME We must know the process cycle time for a batch or semi-batch process in order to estimate the number of reactors required for a given plant capacity and to estimate the staffing requirement for that plant. By process cycle time, we mean the time between initiating one reaction and the next, subsequent reaction. One process cycle for a batch or semi-batch process involves charging the reactor with feed; heating the reactor contents to a specified reaction temperature; completing the reaction; cooling the reactor contents to a specified discharge temperature; discharging product from the reactor; flushing, i.e., cleaning, the reactor; drying the reactor; preparing the reactor for the next charge. We must estimate the time required for each of the above steps before finalizing our plant design. We can easily calculate the time to charge a batch or semi-batch reactor by initially specifying a reactor size and a feed volumetric flow rate. Therefore, if we specify a 3000 gallon (11.4 m 3 ) batch reactor and a volumetric flow rate of 100 gallons/min (0.379 m 3 /min), then the time required to fill the reactor 80% full with reactant is t 5 V Liquid Q Feed where t is the fill time. 5 0:8 ð3000 gallonsþ 100 gallons=min 5 24 min After charging the feed to the batch or semi-batch reactor, we begin heating it to reaction temperature. Consider a batch reactor containing V Liquid with a heat capacity of c P/Liq at an initial temperature T Initial. Batch and Semi-batch Reactors. Copyright 2015 Elsevier Inc. All rights reserved.

2 86 Batch and Semi-batch Reactors Our batch reactor contains a heat exchanger in the form of coiled loops through which an isothermal condensing fluid flows, thereby heating the liquid contents of the reactor. The heat accumulating in the reactor s contents equals the rate of heat transfer from the coiled loops; that is dt V Liquid ρ Liq c P=Liq 5 UAðT HeatFluid 2 TÞ dt where V Liquid is the liquid volume in the reactor (m 3 ); ρ Liq is the density of the liquid in the reactor (kg/m 3 ); c P/Liq is the specific heat capacity of the liquid in the reactor (m 2 /K s 2 ); T is the temperature of the liquid in the reactor (K); t is the time (s); U is the overall heat transfer coefficient for the coiled loops in the reactor (kg/k s 3 ); A is the heat transfer surface area of the coiled loops (m 2 ); and T HeatFluid is the temperature of the heating medium flowing through the coiled loops (K). Rearranging gives dt dt 5! UA ðt HeatFluid 2 TÞ V Liquid ρ Liq c P=Liq Then, separating variables and integrating yields ð TReaction! ðt dt T Initial ðt HeatFluid 2 TÞ 5 UA dt V Liquid ρ Liq c P=Liq 0 " #! 2 ln T HeatFluid 2 T Reaction UA 5 t T HeatFluid 2 T Initial V Liquid ρ Liq c P=Liq Finally, moving the negative sign inside the logarithm function gives ln T! HeatFluid 2 T Initial UA 5 t T HeatFluid 2 T Reaction V Liquid ρ Liq c P=Liq Solving for time t yields V Liquid ρ Liq c P=Liq ln T HeatFluid 2 T Initial 5 t UA T HeatFluid 2 T Reaction This equation gives the time required to heat the liquid volume in the batch reactor from its initial temperature to the specified reaction

3 Batch and Semi-Batch Operations 87 temperature. 1,2 The derivation is the same for a jacketed batch reactor; therefore, the above equation is valid for a batch reactor with coiled loops or with a jacket. Other heating options are available. For example, what if we replace the isothermal heating fluid with a non-isothermal heating fluid? For the same equipment as above, the heating fluid enters the coiled loop at temperature T Inlet, but exits the coiled loop at T Exit. The heat lost by the heating fluid while passing through the coiled loop equals the heat transferred to the batch reactor; that is Q HeatFluid 5 Q HeatTransfer w HeatFluid c P=HeatFluid ðt Inlet 2 T Exit Þ 5 UAðLMTDÞ where w HeatFluid is the weight velocity of heating fluid through the coiled loop (kg/s); c P/HeatFluid is the specific heat capacity of the heating fluid (m 2 /K s); U and A have been previously defined; and LMTD is the log mean temperature difference, defined as LMTD 5 T Inlet 2 T Exit ln½t Inlet 2 T=T Exit 2 TŠ where T is the temperature of the liquid inside the batch reactor. The heat balance equation is, therefore T Inlet 2 T Exit w HeatFluid c P=HeatFluid ðt Inlet 2 T Exit Þ 5 UA ln½t Inlet 2 T=T Exit 2 TŠ Note, however, that the temperature of the liquid inside the batch or semi-batch reactor changes with time. The energy balance for the liquid inside the batch or semi-batch reactor is V Liq ρ Liq c P=Liq dt dt 5 w HeatFluid c p=heatfluid ðt Inlet 2 T Exit Þ We can solve this differential equation by solving the second equation above for T Exit, then substituting T Exit into the differential equation. Doing so yields T Exit 5T 1 T Inlet 2 T e ðua=ðwc PÞ HeatFluid Þ and gives dt V Liq ρ Liq c P=Liq 5w HeatFluid c p=heatfluid T Inlet 2 T 1 T Inlet 2T dt e ðua=ðwc PÞ HeatFluid Þ

4 88 Batch and Semi-batch Reactors Expanding, simplifying, rearranging, then integrating yields ð TReaction dt T Inlet 2 T 5 w! HeatFluidc p=heatfluid e ðua=ðwc PÞ Þ HeatFluid ð 2 1 t dt V Liq ρ Liq c P=Liq e ðua=ðwc PÞ HeatFluid Þ 0 T Initial The integration gives ln T Inlet 2 T Initial 5 w! HeatFluidc p=heatfluid e ðua=ðwc PÞ Þ HeatFluid 2 1 t T Inlet 2 T Final V Liq ρ Liq c P=Liq ð Þ e UA=ðwc PÞ HeatFluid where we have moved the negative sign from integrating into the logarithm term. The time to heat the liquid inside the batch reactor is then 1,2 V Liq ρ Liq c P=Liq e ðua=ðwc P Þ HeatFluid Þ! w HeatFluid c p=heatfluid e ðua=ðwc PÞ HeatFluidÞ 2 1 ln T Inlet 2 T Initial T Inlet 2 T Final 5 t It is not uncommon for a batch or semi-batch reactor to possess neither internal coiled loops nor a jacket for heat transfer. For reactors without internal coiled loops or a jacket, we pump the liquid inside the reactor through an external heat exchanger, generally a shell and tube exchanger. We call these external heat exchangers pump-around loops. We withdraw the liquid inside the reactor from the bottom of the reactor, pump it through a shell and tube heat exchanger, then return it to the top of the reactor, either spraying it into the reactor s headspace or distributing it in the liquid inside the reactor via a submerged pipe and nozzle. For a pump-around loop using an isothermal heating fluid, the energy balance is V Liq ρ Liq c P=Liq dt dt 5 w Liq c p=liq ðt Xger 2 TÞ where V Liquid is the liquid volume in the reactor (m 3 ); ρ Liq is the density of the liquid in the reactor (kg/m 3 ); c P/Liq is the specific heat capacity of the liquid in the reactor (m 2 /K s 2 ); T is the temperature of the liquid in the reactor (K); t is the time (s); w Liq is the weight flow rate (kg/s); and T Xger is the temperature of the reactor liquid exiting the heat exchanger (K). 1(p628) The energy balance around the heat exchanger is T Xger 2 T w Liq c p=liq ðt Xger 2 TÞ 5 UA lnðt HeatFluid 2 T=T HeatFluid 2 T Xger Þ

5 Batch and Semi-Batch Operations 89 where T HeatFluid is the temperature of the heating fluid entering and exiting the shell and tube heat exchanger; all other variables are defined above. Solving the above equation for T Xger gives T Xger 5 T HeatFluid 2 T HeatFluid 2 T e ðua=w Liqc p=liq Þ Substituting T Xger into the above differential equation yields dt V Liq ρ Liq c P=Liq 5 w Liq c p=liq T HeatFluid 2 T HeatFluid 2 T 2 T dt e ðua=w Liqc p=liq Þ which gives, upon rearranging and integrating ln T! HeatFluid 2 T Initial w Liq e ðua=w Liqc p=liq Þ t T HeatFluid 2 T Final V Liq ρ Liq e ðua=w Liqc p=liq Þ The time to heat the liquid inside the batch reactor is then V Liq ρ Liq e ðua=w Liqc p=liq Þ ln T HeatFluid 2 T Initial 5 t w Liq e ðua=w Liqc p=liq Þ 2 1 T HeatFluid 2 T Final For analysis of more complex heating systems, see the open literature, which is rife with 1(p628, chp 18) them. We commence the chemical reaction once the liquid inside a batch or semi-batch reactor reaches reaction temperature. The time required to complete the chemical reaction depends upon the chemical kinetics of the reaction mechanism and the desired final product concentration or reactant concentration. For a first-order reaction, the time required to reach a given product concentration is t 5 1 k ln P Final P Initial or to reduce a given reactant to a specified concentration is t 52 1 ln ½RŠ Final 5 1 ln ½RŠ Initial k ½RŠ Initial k ½RŠ Final For a second-order chemical reaction, the time to reach a specified final reactant concentration is t 5 1 R Initial 2 R Final k R Final R initial

6 90 Batch and Semi-batch Reactors If the kinetics for the chemical reaction are too complex to be reduced to a manageable equation, then we can plot product concentration or reactant concentration as a function of time and read the time required to reach a specified concentration from the plot (see Chapter 2). After completing the chemical reaction, we may need to cool the liquid contents of the batch or semi-batch reactor before discharging it. For a batch or semi-batch reactor equipped with internal coiled loops or a jacket containing finished product at temperature T Reaction, the time to cool the liquid volume using an isothermal cooling fluid at T CoolFluid is 1 t 5 V Productρ Product c P=Product UA ln T Reaction 2 T CoolFluid T Final 2 T CoolFluid where all the other variables have been previously defined. For a similar process using a non-isothermal cooling fluid, the time to cool the liquid volume inside the reactor is t5 eðua=w CoolFluidc P=CoolFluid Þ V Product ρ Product ln T CoolFluid 2T Initial e ðua=w CoolFluidc P=CoolFluid Þ 21 w CoolFluid c P=CoolFluid T CoolFluid 2T Final For analysis of more complex cooling systems, see the open 1(chp 18) literature. After cooling the product in the batch or semi-batch reactor, we pump it from the reactor. The time required to discharge product from the reactor depends upon the volume of product V Product (m 3 ) and the capacity of the pump, defined as product volumetric flow rate Q Product (m 3 /s). The time to discharge product from the reactor is t 5 V Product Q Product To clean the batch or semi-batch reactor after discharging product from it, we flush it with solvent, either water or an organic liquid. Flushing a reactor involves multiple time steps, which are charge the reactor with flush solvent; agitate the solvent inside the reactor to solubilize any product or reactant remaining in the reactor; discharge the flush solvent; repeat the above steps if necessary.

7 Batch and Semi-Batch Operations 91 The time required to charge and discharge solvent are t 5 V FlushSolvent Q FlushSolvent where V FlushSolvent is the volume of flush solvent charge to the reactor (m 3 ) and Q FlushSolvent is the volumetric discharge flow rate of flush solvent (m 3 /s). The time required to solubilize any product or reactant remaining in the reactor requires measurement either in a pilot plant reactor or a commercial-sized reactor because scaling such information from laboratory data is difficult. The batch or semi-batch reactor requires drying after flushing its interior. If our reactor has a jacket, we dry it by applying steam to the jacket and streaming either ambient or hot gas, generally air, through the reactor s interior. If the reactor does not possess a jacket, then we stream hot gas through it. The drying rate for a batch or semi-batch reactor is R Drying 52 1 dm A dt where R Drying is the drying rate (kg/m 2 s); A is the interior surface area covered with liquid (m 2 ); m is the liquid mass (kg); and t is the time (s). When we plot R Drying as a function of time, we see it is a composite function: at early time, R Drying is constant; at later times, R Drying decreases linearly with time. Thus R Drying 5 R ConstantRate 1 R FallRate where R ConstantRate occurs when R Drying is constant and R FallRate occurs when R Drying decreases with increasing time. Therefore, the total drying time will be t T 5 t CR 1 t FR where t T is the total drying time (s); t CR is the drying time during the constant drying rate period (s); and t FR is the drying time during the falling period rate (s). When R Drying is constant, the liquid inside the reactor is adjusting itself to the drying conditions. 3 During this time period, R Drying is independent of the amount of liquid inside the reactor, implying the

8 92 Batch and Semi-batch Reactors presence of a continuous liquid film on the interior surface of the reactor. When this liquid film is no longer continuous, R Drying demonstrates a decreasing rate as time increases. R Drying becomes mass dependent subsequent to this point in time. We describe the constant drying rate period as R ConstantRate 52 1 dm A dt Rearranging and integrating yields ð tcr 1 dt 52 dm 0 AR ConstantRate m Initial where t C is the time to reach the point when the liquid film inside the reactor no longer continuously covers the area being dried (s); m Initial is the starting weight of liquid inside the reactor (kg); and m C is the weight of liquid inside the reactor when the liquid film becomes discontinuous (kg). Performing the above integration gives t CR 52 ðm C 2 m Initial Þ 5 ðm Initial 2 m C Þ AR ConstantRate AR ConstantRate Drying rate becomes dependent upon the mass of liquid inside the reactor once the liquid surface becomes discontinuous, at which time R Drying begins decreasing as time increases. Thus R FallRate 5 k Drying m where R FallRate is the drying rate during this time period (kg/m 2 s); k Drying is the mass transfer rate constant for drying (1/m 2 s); and m is the mass of liquid being dried (kg). But R FallRate 52 1 dm A dt Substituting the penultimate equation into the above equation gives k Drying m 52 1 dm A dt Rearranging and integrating yields k Drying A ð tfr 0 dt 52 ð mc ð mfinal m C dm m

9 Batch and Semi-Batch Operations 93 where t FR is the drying time for the falling rate period (s); m C is the weight of liquid inside the reactor when the liquid film becomes discontinuous (kg); and m Final is the final weight of liquid inside the reactor. Performing the above integration gives k Drying At FR 52ln m Final m C Rearranging the last equation yields 1 t FR 5 ln k Drying A 5 ln m C m Final The total drying time is then t T 5 ðm Initial 2 m C Þ 1 1 ln AR ConstantRate k Drying A m C m Final m C m Final Rearranging provides us with 1 t T 5 ðm Initial 2 m C Þ 1 R ConstantRate ln AR ConstantRate k Drying m C m Final The final step in a batch or semi-batch reactor cycle is preparing the reactor for use. This step involves inspecting the reactor for cleanliness, reassembling the reactor if piping was decoupled or hatches and manways opened, then pressure testing the reassembled reactor for leaks. We establish this time duration by observing the operating personnel doing it. If the operating personnel undertaking this task are highly trained and skilled at it, then the duration will be minimized. If the operating personnel are new and still learning this task, then the duration will be longer than desired but will decrease as their skill level increases. We identify this time duration as t Prep. The ultimate time we must record is down time. We classify that time when there is no activity occurring at the reactor as down time t Down. Down time occurs during holidays and other periods when the production facility is closed. It also occurs when t Prep ends an hour before shift change during 24/7 operation or ends an hour before quitting time if the production facility staffs only one shift. We can now calculate the possible number of batches per reactor for a given production facility. If the production facility operates 24/7,

10 94 Batch and Semi-batch Reactors 365 days/year, then the number of batches per batch or semi-batch reactor is ð365 days=yearþð24 h=dayþ Batches 5 t Charge 1 t Heat 1 t Reaction 1 t Cool 1 t Discharge 1 t Flush 1 t Dry 1 t Prep 1 t Down where each t has units of h/batch. The above equation determines the capacity of a particular batch or semi-batch reactor at a given production facility. To optimize the reaction time t Rxn for a batch or semi-batch reactor, we must determine the net profit (NP) Process for the process or reaction conducted in the reactor. The total operating cost C Total for one batch or semi-batch reactor is C Total 5 C Charge 1 C Heat 1 C Rxn 1 C Cool 1 C Discharge 1 C Flush 1 C Dry 1 C Prep 1 C Down For a simple, generalized reaction, such as the cost of reaction is A 1 B-P C Rxn 5 w A ða t 2 A t50 Þ 1 w B ðb t 2 B t50 Þ where w A is the unit price per mole of reactant A; w B is the unit price per mole of reactant B; A t is the moles of reactant A in the reactor at time t; A t50 is the initial number of moles of reactant A in the reactor; and B t is the moles of reactant B in the reactor at time t; B t50 is the initial number of moles of reactant B in the reactor. The revenue from the product P in the reactor is w P P, where w P is the unit market value per mole for product P. The net profit for the reaction is ðnpþ Rxn 5 w P P 2 ½w A ða t 2 A t50 Þ 1 w B ðb t 2 B t50 ÞŠ Substituting C Rxn into the above equation yields ðnpþ Rxn 5 w P P 2 C Rxn

11 Batch and Semi-Batch Operations 95 Subtracting all the other costs from (NP) Rxn gives the net profit for the process; thus ðnpþ Process 5 ðnpþ Rxn 2 C Charge 2 C Heat 2 C Cool 2 C Discharge 2 C Flush 2 C Dry 2 C Prep 2 C Down 5 w P P 2 C Rxn 2 C Charge 2 C Heat 2 C Cool 2 C Discharge 2 C Flush 2 C Dry 2 C Prep 2 C Down Substituting for C Total yields ðnpþ Process 5 w P P 2 C Total Differentiating with respect to t Rxn gives dðnpþ Process 5 w P dp Note that we assumed C Total to be independent of t Rxn. Multiplying the above equation by one, i.e., V/V, gives dðnpþ Process 5 w P V dðp=vþ 5 w P V d½pš Fitting a polynomial equation to a plot of [P] versus time provides (see Chapter 2) ½PŠ 5 α 1 βt 2 γt 2 1 δt 3 1? which becomes upon differentiation d½pš 5 β 2 2 γt 1 3 δt 2 1? Substituting this last differential equation into the differential equation for d(np) Process / gives dðnpþ Process 5 w P V d½pš 5 w P Vðβ 2 2 γt 1 3 δt 2 1?Þ Setting the above differential equation to zero optimizes (NP) Process, namely dðnpþ Process 5 w P Vðβ 2 2 γt 1 3 δt 2 1?Þ 5 0

12 96 Batch and Semi-batch Reactors which gives β 2 2 γt 1 3 δt 2 1? 5 0 Solving the above equation for t provides the optimum t Rxn for the process. 4 PROCESS OPERATING TEMPERATURE When operating a batch or semi-batch reactor, we must determine the best operating temperature; the best temperature progression for the reaction to maximize production, in the case of single reactions, or to maximize yield, in the case of multiple reactions, i.e., in the case where byproducts form in the reactor. 5 For irreversible reactions involving a single product, i.e., for a reaction that does not produce by-products, determining the best operating temperature is relatively simple: it is the highest possible temperature at which reactor materials and solvent, reactant, and product physical properties are compatible. 6 In this case, we choose the highest possible temperature to obtain maximum conversion or to achieve the shortest possible time for reaching a specified reactant conversion. For exothermic reversible, single reactions, an optimum operating temperature exists because, as the reaction approaches its equilibrium conversion, increasing operating temperature shifts the equilibrium toward higher reactant concentrations. 6 The reverse reaction will be slow at low reactant conversion; however, it increases as reactant conversion, i.e., product formation, increases. In addition, the activation energy of the reverse reaction, E R, is greater than the activation energy of the forward reaction, E F ; otherwise, an economic amount of product does not accumulate in the reactor. Because E R. E F, the rate of the reverse reaction increases faster with increasing temperature than does the rate of the forward reaction. Therefore, we should use a temperature sequence when operating a reversible, exothermic reaction. Initially, we use a high temperature to produce product at an economically acceptable rate, but, as conversion increases, we decrease the operating temperature to reduce the reverse reaction rate relative to the forward reaction rate, which preserves the product already formed in the reactor. 6(p380),7

13 Batch and Semi-Batch Operations 97 We can quantify our reasoning as follows: consider the reversible reaction A3B For a batch reactor, the component balance for reactant A, in terms of moles, is 2 dn A dt 5 R A where R A is R A 5 k F n A 2 k R n B with k F being the forward rate constant (1/s); n A, the moles of reactant A; k R, the reverse rate constant (1/s); and n B, the moles of product B. At any given time n B 5 n B=t50 1 n A=t50 2 n A where n A/t50 and n B/t50 are the initial moles of reactant A and product B in the batch reactor. Of course, at t 5 0, n B/t50 is zero; therefore, the above equation becomes n B 5 n A=t50 2 n A Substituting this last equation into the equation for R A gives Defining conversion as R A 5 k F n A 2 k R ðn A=t50 2 n A Þ x 5 n A=t50 2 n A n A=t50 5 n B n A=t50 then rearranging that definition in terms of n A and n B gives n B n A 5 xn A=t50 5 ð1 2 xþn A=t50 Substituting the above definitions into the last equation for R A, then rearranging yields R A 5 n A=t50 ½ð1 2 xþk F 2 xk R Š

14 98 Batch and Semi-batch Reactors Differentiating R A with respect to operating temperature T at a specified conversion x, then setting the differential equal to zero to determine the optimum operating temperature, yields dr A dt 5 n d A=t50 ½ dt ð1 2 xþk F 2 xk R Š5 0 or d ½ dt ð1 2 xþk F 2 xk R Š5 0 The Arrhenius equation relates the rate constants to operating temperature as and k F 5 A F e 2ðE F=R g TÞ k R 5 A R e 2ðE R=R g TÞ where the subscript F references the forward reaction and the subscript R references the reverse reaction; A is the frequency factor or collision parameter for each respective reaction; E is the activation energy for each respective reaction; R g is the gas constant; and T is the operating temperature. Differentiating each rate constant with respect to T yields dk F dt 52A F dk R dt 52A R E F R g E R R g!! e 2ðE F=R g TÞ e 2ðE R=R g TÞ Substituting the above differential equations into the optimization equation yields E F ð1 2 xþ 2A F e 2ðE F=R g TÞ E R 2 x 2A R e 2ðE R=R g TÞ 5 0 R g R g Rearranging the above equation, then simplifying gives x 1 2 x 5 A FE F e ðe R2E F Þ=R g T A R E R Solving this last equation for T yields the optimum operating temperature for a specified conversion x; thus

15 Batch and Semi-Batch Operations 99 E R 2 E F T Optimum 5 R g ln½ða R E R =A F E F Þðx=ð1 2 xþþš This equation yields the optimum temperature at each conversion for a reversible, exothermic reaction. Note that we generally conduct reversible, exothermic reactions in tubular reactors since they respond faster to a desired temperature change than do batch reactors. However, such reactions can be performed in batch reactors equipped with external heat exchangers for removing heat. The circulation flow rate must be high to achieve the desired operating temperature profile. For a reversible, endothermic reaction, E F. E R ; thus, the forward reaction rate increases relative to the reverse reaction rate at any operating temperature, independent of conversion. Therefore, the optimum temperature sequence is the maximum allowable operating temperature for the process. 7 For most multiple reaction processes in which by-products form, mathematical analysis can be daunting and requires numerical solution. However, we can use deductive reasoning to obtain insight into the optimum temperature profile required for an economically viable process. Consider the parallel reaction scheme X 1 Y-P X 1 Y-BP where P identifies desired product and BP identifies undesired byproduct. We can optimize this reaction scheme for product formation or for product yield. If the energy of activation for by-product BP (E BP ) is greater than the energy of activation for product P (E P ), then a low operating temperature favors P formation. However, a low operating temperature extends the time required to complete the reaction, thereby necessitating a large reactor. Thus, to maximize product formation, we would use a large reactor and operate the process at a constant low temperature. To maximize product yield, we utilize a low operating temperature initially, then increase the operating temperature as the concentrations of X and Y decrease. Consider the consecutive reaction scheme X 1 Y-P-BP

16 100 Batch and Semi-batch Reactors where the symbols have been defined above. If E BP. E P,thenalow,continuous operating temperature maximizes product yield, but at a slow rate. 5(p129) To maximize product formation rate, specifying a high initial operating temperature permits the rapid accumulation of P; to conserve that accumulated P, we decrease the operating temperature while continuing to consume X and Y, which requires a reactor with an external cooler capable of a high circulation rate. 6(p382),8 If E P. E BP, then a high, continuous operating temperature maximizes product yield. 5(p129) The chemical engineering literature abounds with many other such examples. 9,10 SUMMARY This chapter discussed process cycle time, which is the total time from charging feed to a batch or semi-batch reactor to the next feed charge of the same reactor, and its optimization. This chapter also discussed the reaction time required to optimize the profitability of a given batch or semi-batch reactor. It finished by presenting scenarios for maximizing product formation and quality through process temperature optimization. REFERENCES 1. Kern D. Process heat transfer. New York, NY: McGraw-Hill Book Company; p Coker A. Modeling of chemical kinetics and reactor design. Boston, MA: Gulf Professional Publishing; p McCabe W, Smith J. Unit operations of chemical engineering. 3rd ed. McGraw-Hill Book Company; p Aris R. The optimal design of chemical reactors. New York, NY: Academic Press; Denbigh K. Chemical reactor theory: an introduction. Cambridge, UK: Cambridge University Press; p Froment G, Bischoff K. Chemical reactor analysis and design. New York, NY: John Wiley & Sons, Inc.; p Smith J. Chemical engineering kinetics. 3rd ed. New York, NY: McGraw-Hill Book Company; p Rase H. Chemical reactor design for process plants: volume 1 principles and techniques. New York, NY: John Wiley & Sons, Inc.; p Fournier C, Groves F. Isothermal temperatures for reversible reactions. Chem Eng 1970;77 (3): Fournier C, Groves F. Rapid method for calculating reactor temperature profiles. Chem Eng 1970;77(13):157 9.