Dynamic Model and Control Structures for a Hot-Gas Desulfurization

For example, we could consider the use of transport reactor fluidized beds instead of .... The control system is modified to include a controller that...
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PROCESS DESIGN AND CONTROL Dynamic Model and Control Structures for a Hot-Gas Desulfurization Fluidized Process Chang K. Yi Korea Institute of Energy Research, 71-2 Jangdong, Yusung, Taejon 305-343, Korea

William L. Luyben* Department of Chemical Engineering, Lehigh University, Iacocca Hall, Bethlehem, Pennsylvania 18015

A dynamic model is developed of the fluidized-bed desulfurizer/regenerator system for the hotgas desulfurization (HGD) process as part of the integrated gasification combined cycle. The efforts of model development, simulation, control structure exploration, and evaluation of design changes are described in this paper. The desulfurizer and regenerator are modeled as bubbling fluidized beds with perfectly mixed solid phases and plug flow gas phases. Six alternative control structures are tested dynamically for load disturbance rejection. The control schemes in which the H2S concentration is controlled by changing the level of solids in the desulfurizer perform poorly because they give large swings in solid circulation rates. For throughput changes, O2 concentration is controlled very well by the use of a ratio controller with time lag. The dynamic studies indicate that the HGD process can be controlled, but that trade-offs between design and control must be carefully considered. 1. Introduction The hot-gas desulfurization (HGD) process is an integral part of IGCC (integrated gasification combined cycle), which will be able to replace a conventional pulverized-coal-fired power plant. Coal gasified in a high-pressure gasifier must be cleaned before being combusted in the gas turbine because deleterious contaminants (especially hydrogen sulfide, particulate matter, and trace metals) can damage the downstream process equipment and are detrimental to the environment. Development of technologies for efficient removal of hydrogen sulfide (H2S) is now recognized as crucial to the economics in the IGCC process.1,2 The existing conventional wet-type processes require cooling and reheating of the gas stream, resulting in significant reduction in thermal efficiency of the system and large capital investment. The hot-gas desulfurization technology of the dry type offers the potential for improving the overall efficiency and lowering the cost of environmental control. The Korea Institute of Energy Research (KIER) has been developing a hot-gas desulfurization process that has two fluidized bed reactors consisting of a desulfurizer and a regenerator. KIER and Lehigh University are collaborating in a study of the dynamics and control of this process. In the HGD system the concentration of H2S in the product gas leaving the desulfurizer should be maintained below the environmental regulation. Both temperature and pressure must be controlled in both the desulfurizer and the regenerator. The sorbent * To whom correspondence should be addressed. Phone: (610) 758-4256. Fax: (610) 758-5297. E-mail: [email protected].

level should be controlled in one of the vessels, and the solid circulation rate must be controlled. In addition, the amount of gas fed to the regenerator must be sufficient to provide enough oxygen to regenerate the sorbent but not too much so that the process efficiency is reduced. There has been little work published in the open literature on dynamic HGD models. It is important to consider dynamics and control at the early stages of design because of the strong impact of design decisions on the effectiveness of any control system. Since the HGD process is multivariable, there are a large number of alternative control structures. It is also important to explore these alternatives at the design stage so that equipment sizing can be appropriately matched with dynamic control requirements. The objective of this study is to develop an effective control structure for a hot-gas desulfurization system and to explore the impact of design parameters on controllability. 2. Dynamic Model A dynamic model to be used for control studies cannot include all the microscopic details such as gas bubble formation and internal flow of solid inside the bubbles. However, the model must be rich enough to capture the important dynamic features of pressure, temperature, compositions, and holdups. No model is any better than the assumptions made in its development. If they are reasonable and include all important effects, the model can be used to study the dynamics of the system. The model is not intended to study the many important steady-state parameters such as the effect on conversion of the solid particle size distribution or the design of the gas distributor.

10.1021/ie990292g CCC: $18.00 © 1999 American Chemical Society Published on Web 09/30/1999

Ind. Eng. Chem. Res., Vol. 38, No. 11, 1999 4291 Table 1. Physical Parameter Values for Dynamic Simulation Cpg Cps kSo kRo ES ER ∆Hrxn

36 kJ/(kg mol)/K 0.322 kJ/kg/K 7.694 × 106 22.4 × 106 60 190 kJ/(kg mol) 80 710 kJ/(kg mol) 297 000 kJ/(kg mol of O2 reacted)

2.1. Assumptions. The model makes the following basic assumptions. (1) The solid phase is perfectly mixed, and the gas phase flows in plug flow up through the bubbling fluidized bed. This leads to an ordinary differential equation for gas concentrations, with bed height as the independent variable, which can be analytically integrated to yield an algebraic equation for the concentration of the gas phase leaving the top of the bed at each point in time. (2) The temperatures of the gas and the solid phases are equal. (3) The solid phase represents all of the thermal inertia in the system, i.e., there is only one energy equation, which describes how the internal energy of the solid phase varies with time. This assumption is valid because of the much larger mass holdup of the solid phase compared to the gas phase. (4) The rate of reaction of H2S in the desulfurizer depends on the temperature, holdup of solid, concentration of H2S in the gas phase (which varies with height up through the bed), and weight fraction of MeO in the sorbent.

sulfidation reaction:

MeO + H2S f MeS + H2O

(5) The rate of reaction of MeS in the regenerator depends on the temperature, holdup of solid, concentration of O2 in the gas phase (which varies with height), and weight fraction of MeS in the sorbent.

regeneration reaction:

Figure 1. Steady-state design values of a fluidized hot-gas desulfurization system.

MeS + 1.5O2 f MeO + SO2

(6) Solid flowrates can be perfectly measured and controlled; i.e., solid hydraulics are neglected. For the initial study, it was felt that, although the hydraulics are crucial for the operation of the plant, it was more important to gain some insight into the chemical and energy aspects of the process without running the risk of clouding the picture with solid hyraulics phenomena. The essence of the solids hydraulic problem is to design the standpipes to provide sufficient head to overcome the changes in differential pressure between the units. So the first step is to find out how large these pressure swings really are. Solid hydraulics will be discussed in a later paper. (7) Simplified physical properties are used: mean constant heat capacity of the gas phases, solid phases, and ideal gases. Table 1 shows the parameter values. Other base case conditions are shown in Figure 1. The parameters apply for a small pilot-scale HGD plant. (8) Heat transfer rates from the desulfurizer and regenerator are directly adjustable. As discussed in item 6 above, we wanted to concentrate on the most important effects first. Later studies will explore alternative heat-removal/temperature-control systems. The heat of reaction in the desulfurizer is negligible, but heat must still be removed from the desulfurizer

Figure 2. Nomenclature for the dynamic model.

because of the sensible heat of the hot solids coming from the regenerator. Thus, the exothermic heat of reaction in the regenerator is partially removed in the regenerator and partially removed in the desulfurizer. This process is not “heat balanced” as is the catalytic cracking process with which it is quite similar in most other aspects. In catalytic cracking, the exothermic combustion reaction in the regenerator is almost balanced by the endothermic cracking reaction in the reactor. In the HGD process, this balance does not exist. This results in important design issues, most notably the strong interrelationship between intervessel solid circulation and heat removal rates in the two vessels. Higher solids circulation rates require more heat removal in the desulfurizer and less in the regenerator. The impact of this fundamental difference between catalytic cracking and HGD on control structure selection is clearly revealed later in this paper. It also strongly affects the choice of process designs. For example, we could consider the use of transport reactor fluidized beds instead of bubbling beds to reduce vessel sizes and capital investment. However, the solid circulation rate through a transport reactor strongly affects solids holdup, which impacts reaction rates and conversion. But changing the intervessel solids circulation alters the heat balances in the HGD process. This fundamental difference between cat cracking and hot-gas desulfurization should be kept in mind during the design. The superficial similarity between the two processes may tempt HGD designers to simply duplicate many features from the proven technology of cat cracking, which could lead to serious flaws in the process.

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desulfurizer equations gas-phase mass balance (kg/h) d(VSG FSG)/dt ) FSinMSin - FSoutMSout - MH2SRH2S + MH2ORH2O (1) gas-phase component balance ((kg mol)/ h of component j ) H2S) S S S /MSout)/dt ) FSin yin,j - FSout yout,j - RH2S d(VSG FSG yout,j

Figure 3. Control structure CS1 of the HGD system (H2S is not controlled).

(3)

gas-phase component balance ((kg mol)/ h of component j ) H2O) S /MSout)/dt ) d(VSG FSG yout,j S S - FSout yout,j + RH2O (4) FSin yin,j

gas density (kg/m3) FSG ) MSoutPS/(R(TS + 273))

(5)

solid-phase mass balance (kg/h) dWS/dt ) SR - SS + (MMeS - MMeO)RH2S

(6)

solid-phase component balance (kg/ h of component k ) MeO) d(WSxSk )/dt ) SRxRk - SSxSk - MMeORH2S

(7)

solid-phase component balance (kg/ h of component k ) MeS) d(WSxSk )/dt ) SRxRk - SSxSk + MMeSRH2S

(8)

solid-phase component balance (kg/ h of component k ) inert) d(WSxSk )/dt ) SRxRk - SSxSk

(9)

solid-phase energy balance (kJ/h) d(CpsWSTS)/dt ) Cps(SRTR - SSTS) + Cpg(FSin TSin - FSoutTS) - QS (10) kinetics S RH2O ) RH2S ) ConvSFSin yin,H 2S

(11)

The ordinary differential equation describing the quasisteady-state changes in the gas concentration of H2S as it flows up through the solid bed is Figure 4. Load disturbance responses of CS1 for a 20% change of the feed gas (+20% in FSin).

2.2. Modeling. Figure 2 gives a sketch of the process and shows the notation used.

dyS/dw ) -kSySxMeO/FSin

where the limits of integration are yS,in at w ) 0 (the bottom of the bed) and integrating to w ) WS (the top of the bed). At any point in time, all the variables are constant except yS. The solution is Now conversion is

gas-phase component balances ((kg mol)/ h of component j ) H2, CO2, and CO) S S S /MSout)/dt ) FSin yin,j - FSout yout,j d(VSG FSG yout,j

(2)

(12)

ln(yS,top/yS,in) ) -WSkSxMeO/FSin given by

(13)

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ConvS ) (yS,in - yS,top)/yS,in S

S

in

ConvS ) 1 - e(-WSkSxMeOyin,H2S/FS ) kS ) kSoe-ES/(T

S+273)R

of the process. Numerical integration uses the simple first-order explicit Euler with a step size of 0.00005 h.

(14) 3. Results and Discussion

(15)

regenerator equations gas-phase mass balance (kg/h) d(VRG FRG)/dt ) FRinMRin - FRoutMRout - MO2RO2 + MSO2(2/3)RO2 (16) gas-phase component balance ((kg mol)/ h of component j ) N2) R R R /MRout)/dt ) FRin yin,j - FRout yout,j (17) d(VRG FRG yout,j

gas-phase component balance ((kg mol)/ h of component j ) O2) R /MRout)/dt ) d(VRG FRG yout,j R R - FRout yout,j - RO2 (18) FRin yin,j

gas-phase component balance ((kg mol)/ h of component j ) SO2) R /MRout)/dt ) d(VRG FRG yout,j R R - FRout yout,j + RSO2 (19) FRin yin,j

gas density (kg/m3) FRG ) MRoutPR/(R(TR + 273))

(20)

solid-phase mass balance (kg/h) dWR/dt ) SS - SR - (MMeS - MMeO)(2/3)RO2

(21)

solid-phase component balance (kg/ h of component k ) MeO) d(WRxRk )/dt ) SSxSk - SRxRk + MMeO(2/3)RO2 (22) solid-phase component balance (kg/ h of component k ) MeS) d(WRxRk )/dt ) SSxSk - SRxRk - MMeS(2/3)RO2 (23) solid-phase component balance (kg/ h of component k ) inert) d(WRxRk )/dt ) SSxSk - SRxRk

(24)

solid-phase energy balance (kJ/h) d(CpsWRTR)/dt ) Cps(SSTS - SRTR) + Cpg(FRin TRin - FRoutTR) - QR + ∆HRxnRO2 (25) kinetics R RO2 ) (3/2)RSO2 ) ConvRFRin yin,O 2 R

R

R

ConvR ) 1 - e(-WRkRxMeSyin,O2/Fin) kR ) kRoe-ER/(T

R+273)R

(26) (27) (28)

This model is used to simulate the dynamic response

The control structure used in the hot-gas desulfurization process should be safe and simple and give tight control of the H2S concentration of the product gas and reasonably tight control of the O2 concentration of the regenerator exit gas. A number of different alternative control schemes are explored. The central issue in selecting the best control structure is how well the H2S concentration of the desulfurized gas is controlled. The H2S concentration must be maintained below a maximum level to satisfy environmental constraints. As the flowrate of gas from the gasifier increases, the H2S concentration of the desulfurized gas increase unless control action is taken. More sulfur must be removed from the gas in the desulfurizer. In addition, more sulfur must be removed from the solids in the regenerator, which requires that more oxygen be fed to the regenerator. Any increase in the concentration of H2S in the gas from the gasifier will also require changes in both the desulfurizer and the regenerator to remove this additional sulfur. 3.1. Control Structure CS1. The first control structure contains the following simple control loops (see Figure 3). (1) The desulfurizer pressure is controlled by manipulation of the desulfurizer exit gas stream. (2) The regenerator pressure is controlled by manipulation of the regenerator exit gas stream. (3) The desulfurizer temperature is controlled by manipulation of heat removal in the desulfurizer. (4) The regenerator temperature is controlled by manipulation of heat removal in the regenerator. (5) The desulfurizer sorbent bed level is controlled by manipulation of sorbent flow from the desulfurizer. (6) The flowrate of sorbent into the desulfurizer from the regenerator is held constant. (7) The oxygen composition in the regenerator exit gas is controlled by manipulation of the flowrate of gas to the regenerator. Note that there is no control of the composition of the gas from the desulfurizer. All of the controllers are proportional-integra1 (PI) with the exception of the pressure controllers and the level controller, which are proportional (P). The PI controllers are tuned by conducting a relay-feedback test (ATV test) to obtain the ultimate gain Ku and the ultimate period Pu (h) with appropriate lags assumed in the loops. Then the Tyreus-Luyben3 controller settings are calculated and tested. In the regenerator temperature control loop, the controller gain had to be further reduced to obtain nonoscillatory response.

KTL ) Ku/3.2

(29)

τTL ) 2.2Pu

(30)

Table 2 gives the values of the controller tuning constants used. The control of regenerator temperature appears to be the most difficult loop in the process. Figure 4 shows the response of the entire system to a 20% increase in feed gas to the desulfurizer with the CS1 control system in operation. The concentration of H2S in the desulfurized gas leaving the desulfurizer increases rapidly and

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Table 2. Values of Controller Tuning Constants loop

controlled

manipulated

1 2 3 4 5 6 7 4 CS2

WS TS PS TR PR R yout,O 2 S yout,H 2S TR

SS QS FSout QR FRout FRin Wset S QR

Ku

Pu

KTL

τTL

8.5

0.024

2.6

0.053

5

0.038

1.6

0.085

18 141 19.6

0.04 0.11 0.0286

5.7 44 6.1

0.088 0.24 0.063

Kc 4 2.6 1 1 4 5.7 44 3.1

τI 0.053 0.085 0.088 0.24 0.063

Figure 5. Control structure CS2 of the HGD system (cascade control for H2S).

levels out after about 1 h at close to 200 ppm (compared to the desired value of 80 ppm). Therefore, the control system needs to be modified to include a controller of H2S concentration. 3.2. Control Structure CS2. The control system is modified to include a controller that changes the level of sorbent in the desulfurizer to control the concentration of H2S in the desulfurizer exit gas as shown in Figure 5. The composition controller changes the setpoint of the desulfurizer level controller. When the H2S concentration increases, the level in the desulfurizer is increased by cutting back on the flow of sorbent (SS) out of the desulfurizer. This reduces the level of sorbent in the regenerator. The size of the regenerator is increased from the base case conditions to not run out of solid level in the regenerator. The diameter is increased to 0.38 m, and the initial solids holdup in the regenerator is increased from the base case value of 14.3 to 34 kg. This larger holdup results in a lower steadystate MeS concentration in the regenerator: 6.9 wt % drops to 3.0 wt %. The larger regenerator requires retuning of the temperature controller, giving a larger gain and smaller reset time as shown in Table 2. Figure 6 shows the response of this control structure to a 15% increase in throughput. The H2S concentration is returned to 80 ppm by building the sorbent holdup in the desulfurizer to about 70 kg. There is a corresponding drop in sorbent in the regenerator. Note the large changes in O2 concentration in the regenerator exit gas and the corresponding large swings in regenerator inlet and exit gas flow rates. These are undesirable for both the upstream and downstream units. This variability can be reduced by using feedforward control: ratioing the regenerator feed gas flow rate to the desulfurizer feed gas flow rate. 3.3. Control Structure CS3. Figure 7 gives a control structure with feedforward control. The large regenerator is used in this case since manipulation of the desulfurizer level is still employed. Load changes are

Figure 6. Dynamic responses of CS2 for a +15% load change of the feed gas (+15% FSin; larger regenerator).

handled in the regenerator by ratioing the flowrate of the regenerator feed gas to the flowrate of the desulfurizer feed gas. This should account for rate changes, but it would not account for changes in the H2S concentration of the gas fed to the desulfurizer. The performance of this control structure is shown in Figure 8. The disturbance is a 15% increase in feed flowrate. The increase in H2S concentration causes the desulfurizer level control to cut the flowrate of solids from the desulfurizer (SS) to increase the solids holdup in the desulfurizer. The reduced flow of solids to the regenerator causes a large drop in regenerator temperature because less reacting material is being fed to sustain the exothermic reaction. There is a very large increase in the O2 concentration of the regenerator exit gas. This is due to two effects: the decrease in the consumption of oxygen due to the drop in reaction rate, and the increase in the flowrate of the regenerator feed gas due to the ratio controller. Very large swings in the heat removal rate in the regenerator occur. 3.4. Control Structure CS4. Figure 9 shows what

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Figure 7. Control structure CS3 of the HGD system (cascade control for H2S; ratio for O2).

Figure 9. Load disturbance response of CS4 when the H2S is not controlled (+15% in FSin; larger regenerator; FRin ratioed to FSin).

Figure 8. Performance of CS3 (+15% in FSin; larger regenerator; FRin ratioed to FSin).

happens if the H2S controller is put on manual for this same disturbance. The 15% increase in feed gas flowrate causes the H2S concentration in the desulfurizer exit gas to climb to a new steady-state value of about 150 ppm from its design value of 80 ppm. The upset to the regenerator is much less: regenerator temperature, heat removal, and oxygen concentration undergo much smaller swings. These results indicate that the process

is quite sensitive to changes in the circulation rate of the solids. These results clearly show that the use of the solids flowrates as manipulated variables to control concentrations yields poor dynamic performance. This poor performance is a direct result of the interaction between the solids circulation rate and the energy balances in the desulfurizer and the regenerator. 3.5. Control Structure CS5. One approach to the difficult problem of controlling the H2S concentration in the exit gas from the desulfurizer would be to build larger vessels with more solids holdup. That is to say, one way to achieve effective sulfur removal is to overdesign both vessels so that essentially all of the H2S is removed under the worse case conditions, i.e., maximum flowrates and maximum H2S concentration of the feed gas. The desulfurizer and the regenerator could be sized so that, at the maximum throughput and with the maximum amount of H2S in the feed gas from the gasifier, the H2S concentration in the gas leaving the desulfurizer is less than its maximum specification (80 ppm). This would naturally increase capital investment, but it may be well worth the additional cost if it produces a smooth-running HGD unit that filters out the inevitable disturbances coming in from the gasifier due to changes

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Figure 10. Control structure CS5 with no composition control (larger regenerator and desulfurizer; ratio control with lag).

in coal quality and changes in electrical demand. Processes with solid feed streams are usually plagued with frequent and large disturbances. The HGD unit is centrally located in the process, and it should be designed to attenuate disturbances, not amplify them. The larger vessels have the additional advantage of making them less sensitive to gas flowrate disturbances. The changes in pressures are slower, and this permits the pressure controllers to maintain tighter pressure control, which means less variability in the solids flowrates due to pressure imbalances. The idea of overdesigning the HGD unit should not be dismissed without some careful study. The function of the HGD unit is to produce low-sulfur gas to feed to the combustion turbine, and combustion turbines do not like to be disturbed and are probably the most expensive part of the entire process. In many processes it is often more profitable to increase capital investment to reduce product quality variability. To explore this possibility, the volumes of both reactors are almost doubled (WS ) 100 kg and WR ) 60 kg). It is easy to increase bed height twice because the HGD system uses fluidized reactors with ample freeboard sections. The model is converged to the new steady state. The H2S composition in the desulfurizer exit gas drops from 80 to 2.6 ppm. The MeS composition of the solids in the desulfurizer changes slightly from 16.9 to 16.5 wt %. The corresponding change in the regenerator is from 3.23 to 3.0 wt % MeS. The control structure is sketched in Figure 10. Figure 11 gives the responses of this process with a large desulfurizer and regenerator. A very large 50% increase in throughput occurs at time zero. The concentration of H2S in the gas leaving the desulfurizer gradually climbs to about 55 ppm, which is still below specification. This simple control structure gives excellent regulatory control. Building additional capacity into the HGD process makes the control job much easier. This attractive alternative achieves much better dynamic load rejection. Oversizing the desulfurizer and the regenerator increases capital investment but provides much less variability in the H2S concentration of the desulfurized gas and in the flowrate and O2 concentration of the regenerator gas. The economics of additional capital expenditure versus reduced product-quality variability and its impact on the downstream power generation processes must be incorporated into design decisions. 3.6. Control Structure CS6. An alternative method for the control of H2S in the product gas is to change

Figure 11. Response of CS5 for a 50% increase in throughput (+50% FSin; larger regenerator and desulfurizer; no composition control).

Figure 12. Control structure CS6 with H2S controlled by desulfurizer temperature.

the desulfurizer temperature. To react more of the H2S in the feed gas to the desulfurizer, the temperature of the desulfurizer could be raised. The control structure is sketched in Figure 12. Figure 13 shows how the desulfurizer temperature, heat-removal rate, and H2S concentration of the exit gas

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ations and dynamic modeling), (2) incorporating solid flow hydraulics in the model (effect of standpipe height on controllability), and (3) use of feed temperature manipulation to control vessel temperatures instead of direct convective heat removal from the desulfurizer and regenerator vessels (use of heat exchanger bypassing for tight temperature control). Acknowledgment We appreciate the financial support from RaCER (the R&D Management Center for Energy & Resources, Korea) and also in part from KEPRI (Korea Electric Power Research Institute) for this work. Nomenclature

Figure 13. Dynamic response of CS6 for a (20% change in S ). throughput (H2S controller (Kc ) 3, τI ) 0.2) resets Tset

vary for a 20% increase and a 20% decrease in throughput. These results are for the base-case size vessels. Reasonably effective control of H2S is obtained. A 20% increase in throughput requires an increase in desulfurizer temperature from 600 to 626 °C. A 20% decrease results in dropping the desulfurizer temperature to 594 °C. Notice that an increase in throughput results in a lower value of the heat transfer rate. This occurs because the higher temperature of the material leaving the desulfurizer carries more sensible heat from the vessel. 4. Conclusion A dynamic model is developed of the fluidized desulfurizer/regenerator system for the hot-gas desulfurization process for the integrated gasification combined cycle. Six control structures are proposed and tested for load disturbance rejection. The control of H2S concentration by changing the level of solids in the desulfurizer gives poor performance and results in large swings in solid circulation rates. The solid circulation rate appears to be a poor choice to control concentrations because the process is quite sensitive to changes in the circulation rates of solids. The regenerator exit gas O2 concentration is effectively controlled by the ratio controller of FRin to FSin with a time lag for throughput changes. Changes in the sulfur content of the gas fed to the desulfurizer will require a feedback trim from an oxygen composition controller. The impact of vessel size on dynamic controllability is demonstrated in this work. The dynamic studies indicate that the HGD process can be controlled, but that trade-offs between design and control must be carefully considered. Building additional capacity into the process makes the control job much easier. This would increase capital investment, but would provide much less variability in the H2S concentration of the desulfurized gas and in the flowrate and O2 concentration of the regenerator gas. Additional studies of this process will be reported in future papers. Topics covered include (1) use of a transport reactor for the desulfurizer (design consider-

ConvR ) conversion in the regenerator ConvS ) conversion in the desulfurizer Cpg ) mean heat capacity of the gas phase, kJ/(kg mol)/K Cps ) mean heat capacity of the solid phase, kJ/kg/K ER ) activation energy of the regeneration reaction, kJ/ (kg mol) ES ) activation energy of the desulfurization reaction, kJ/ (kg mol) FRin ) total molar flowrate of inlet gases to the regenerator, (kg mol)/h FRout ) total molar flowrate of outlet gases from the regenerator, (kg mol)/h FSin ) total molar flowrate of inlet gases to the desulfurizer, (kg mol)/h FSout ) total molar flowrate of outlet gases from the desulfurizer, (kg mol)/h kR ) regeneration reaction constant, (kg mol)/kg of sorbent/h kS ) desulfurization reaction constant, (kg mol)/kg of sorbent/h kRo ) preexponential factor of the regeneration reaction, (kg mol)/kg of sorbent/h kSo ) preexponential factor of the desulfurization reaction, (kg mol)/kg of sorbent/h Kc ) controller gain of the PI controller KTL ) controller gain of the Tyreus-Luyben setting of the PI controller Ku ) ultimate gain for controller tuning Mk ) molecular weight of component k, kg/(kg mol) MRin ) mean molecular weight of gases to the regenerator, kg/(kg mol) MRout ) mean molecular weight of outlet gases from the regenerator, kg/(kg mol) MSin ) mean molecular weight of inlet gases to the desulfurizer, kg/(kg mol) MSout ) mean molecular weight of outlet gases from the desulfurizer, kg/(kg mol) MeO ) metal oxide MeS ) metal sulfide PR ) pressure in the regenerator, atm PS ) pressure in the desulfurizer, atm Pu ) ultimate period, h QR ) heat removal from the regenerator, kJ/h QS ) heat removal from the desulfurizer, kJ/h R ) gas constant, (atm m3)/(kg mol)/K Rj ) rate of reaction of component j, (kg mol)/h SS ) flowrate of solids from the desulfurizer, kg/h SR ) flowrate of solids from the regenerator, kg/h TR ) temperature in the regenerator, °C TRin ) temperature of inlet gas to the regenerator, °C TS ) temperature in the desulfurizer, °C TSin ) temperature of the inlet gas to the desulfurizer, °C

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VRG ) gas volume in the regenerator, m3 VSG ) gas volume in the desulfurizer, m3 w ) weight of solid, kg WS ) solid inventory in the desulfurizer, kg WR ) solid inventory in the regenerator, kg xRk ) weight fraction of solid component k in the regenerator xSk ) weight fraction of solid component k in the desulfurizer R yin,j ) mole fraction of component j in the inlet gas of the regenerator R yout,j ) mole fraction of component j in the outlet gas of the regenerator S yin,j ) mole fraction of component j in the inlet gas of the desulfurizer S yout,j ) mole fraction of component j in the outlet gas of the desulfurizer yS ) mole fraction of H2S in the gas flowing up through the desulfurizer bed yS,in ) mole fraction of H2S in gas entering desulfurizer bed yS,top ) mole fraction of H2S in the gas at the top of the desulfurizer bed

Greek Symbols FRG ) mean density of gases in the regenerator, kg/m3 FSG ) mean density of gases in the desulfurizer, kg/m3 τI ) integral time of the PI controller, h τTL ) integral time of the Tyreus-Luyben setting of the PI controller, h

Literature Cited (1) Gangwal, S. K.; Harkins, S. M.; Woods, M. C.; Jain, S. C.; Bossart, S. J. Bench-Scale Testing of High-Temperature Desulfurization Sorbents. Environ. Prog. 1989, 8 (4), 265. (2) Woods, M. C.; Gangwal, S. K.; Jothimurugesan, K.; Harrison, D. P. Reaction between H2S and Zinc Oxide-Titanium Oxide Sorbents: 1. Single-Pellet Kinetic Studies. Ind. Eng. Chem. Res. 1990, 29, 1160. (3) Tyreus, B. D.; Luyben, W. L. Tuning of PI Controllers for Integrator/Deadtime Processes. Ind. Eng. Chem. Res. 1992, 31, 2625.

Received for review April 26, 1999 Revised manuscript received July 26, 1999 Accepted August 9, 1999 IE990292G