Modeling and simulation of a top-fired reformer

1832 Ind. Eng. Chem. Res. 1988,27, 1832-1840

Modeling and Simulation of a Top-Fired Reformer?

C. V. S. Murty* Chemical Engineering Division, Regional Research Laboratory, Hyderabad 500 007, India

M. V. Krishna Murthys Department of Mechanical Engineering, Indian Institute of Technology, Madras 600 036, India

With the advent of high-speed computers and innovative numerical methods, mathematical models have been gaining considerable importance in recent years. The application of mathematical modeling in process engineering has been confined so far to model validation studies only, and simulation, in the strictest sense, has remained virtually a neglected area. Studies related to the numerical simulation of process equipment, particularly on a commercial scale, are therefore called for. One such investigation is reported in the present work on a primary reformer, a vital equipment in the ammonia fertilizer industry. The study deals with the development of a complete mathematical model of the reformer and its validation using the data obtained on an industrial reformer. It is demonstrated through the subsequent simulation program how important design information could be derived from the mathematical model.

1. Introduction Some of the popular approaches employed for designing

new process equipment or optimizing the performance of an existing unit are (i) combining intuition with some empiricism to treat the whole process as a work of art, (ii) conducting experiments on prototype models and extrap- olating the information so gathered for scale-up studies, and (iii) numerical experimentation on mathematical models. Of these, the usage of mathematical models has been gaining considerable ground in recent years, mostly because progressively more sophisticated computers and innovative numerical methods are becoming available, both of which are crucial to the development and application of mathematical models. Mathematical models prove especially useful in yielding the desired results in a very short time. However, the development of a robust model could itself be sometimes a time-consuming and costly affair.

While mathematical modeling aims at the mathematical approximation of a physical process in terms of models, simulation serves to illustrate the practical utility of the models in several ways. Simulation is a natural corrollary to modeling, whether physical or numerical. As opposed to actual experimentation, numerical simulation is fast and risk-free. Besides, it could be carried out a t a fraction of the cost of the former. It could be employed, for instance, to find out the effect of change of variables on the process, or it could determine the best operating conditions for an existing equipment in a given situation to improve its efficiency or productivity. Because of these and other advantages, mathematical modeling, together with simulation, forms an important research activity in many areas.

In spite of the fact that great strides have been made over the years in the area of combustion engineering, mathematical modeling of practical and industrial heaters like reformers, rotary kilns, tube-still furnaces, etc., has not been commensurate with this progress. Even in those few instances where it has been attempted, the investi- gators have mostly confiied themselves to model validation studies only. Numerical simulation has remained by and large a neglected area. The present study, therefore, at- tempts to make some useful contributions in this direction, through the mathematical modeling and simulation of a top-fired reformer, which is an important equipment in

RRLH Communication 2085. 8 Present address: Department of Mechanical Engineering,

Indian Institute of Science, Bangalore 560 012, India.

the ammonia fertilizer industry. In the modeling phase, a complete mathematical model of the reformer is developed, and it is validated with the help of operating data collected on an industrial reformer. In simulation, which follows next, numerical experimentation is carried out on the model to obtain data that will be helpful not only in establishing the best operating conditions in any given situation but also in facilitating the design of new reformers.

2. Earlier Work Roesler (1967) reported a pioneering work on the ap-

plication of a two-flux method for analyzing radiative heat transfer in top-fired cocurrent reformer furnaces. He has restricted his study to the furnace side of the reformer and thus excluded the tube-side processes from his treatment. Roesler’s model is stated to have been employed for the successful modeling of many industrial reformers. McGreavy and Newman (1969) made use of Roesler’s model for steady-state and dynamic modeling of a steam reformer. They have, however, validated their model for the steady-state operation of the reformer only. Roesler’s model has been modified later by Filla (1984) by including in the model the effects of diffuse reflection at the side- walls. The modifications led ostensibly to some im- provement in the model predictions.

Exclusive modeling of tube-side processes in reformers has been reported in a few cases. In one such study by Hyman (1968), the effect of parameters like feed pressure, tube wall temperature, etc., on the conversion of methane has been investigated, while in another by Davies and Lihou (1971) simulations have been performed for establishing the optimal values of tube thickness and temperature levels in the reformer. Modeling of the kinetics of the steam-methane reforming reaction has been carried out by several others, although not in the context of reformer modeling, e.g., Grover (1970), De Deken et al. (1982), Murray and Snyder (1985), and others. This list is not exhaustive, however.

Singh and Saraf (1979) considered both the tube-side and furnace-side phenomena in their modeling studies on side-fired reformers. The total heat transfer from the flame to the reformer tubes has been treated simultane- ously with the heat transfer to the reacting gases. They have used their model to check the performance of some side-fired steam hydrocarbon reformers.

The classical Zone method has been used in some instances for radiative exchange calculations inside the

0888-5885/88/2627-1832$01.50/0 0 1988 American Chemical Society

Ind. Eng. Chem. Res., Vol. 27, No. 10, 1988 1833

flowing along the outside of the tubes. The process gas becomes increasingly richer along the tubes in carbon monoxide and hydrogen, which are the products of the reforming reactions. At the end of reforming, it flows out to the secondary reformer for subsequent processing.

4. Mathematical Model For a given set of operating conditions, like fuel flow

rate, the flow rate, temperature, and composition of the process gas at the tube inlet, the reformer model predicts the heat duty of the reformer, temperature profile of the reformed gas, concentration profiles of the constituents of the process gas, temperature profiles of the flue gas and tube skin, etc. The complete model is comprised of two submodels, one for the furnace side and the other for the tube side.

4.1. Furnace-Side Model. The furnace-side model is solely a heat-transfer model, with both the convection and radiation accounted for. The combustion products are assumed to travel down the furnace in a plug flow fashion, and the convective heat transfer is taken care of by means of an empirical heat-transfer coefficient. Radiation, on the other hand, is handled in a rigorous manner. The radiation model is based on the Schuster-Schwarzschild flux method. Roesler (1967) used this method in his work on reformer modeling.

4.1.1. Model Equations. Allowance for the radiative behavior of the combustion products has been made by Roesler (1967) by dividing the wavelength spectrum of radiation into window and band portions. Consequently, four radiative fluxes have been used to represent the radiation model, two each in the band and window regions. This is equivalent to using a one-clear-one-gray gas model (Hottel and Sarofim, 1967) for the emissivity of the species.

In the case of luminous flames, the absorbinglemitting medium may be approximated by a gray gas (Murty, 1987). This is because soot, which is an important constituent of the luminous flames, emits continuously over all the wavelengths of thermal radiation and covers effectively the window regions of the gas (COz + HzO) radiation spectrum. The accuracy of predictions does not suffer much because of the gray gas assumption, and on the other hand, computation time for the radiative exchange calculations is reduced considerably. This treatment is extended to the present case also, and the medium is represented by a gray gas. Through the application of this model, Roesler’s analysis is greatly simplified, and the number of flux equations is reduced to two, one for the upward flux, q-, and the other for the downward flux, 4’. The modified Roesler’s model is called the “gray-gas model” for ease of reference later. Concentration variations are not taken into account, and a uniform absorption coefficient is used throughout the reformer. The resulting differential equations for the two fluxes for a differential section of height dz (Figure 2 ) are

Process gas

I I

ox ox ox ox ox I x x 0 x O x Px O x Ox 7 Burners

x x x x x

1 Tubes

Figure 1. Reformer configuration.

primary reformer. Shen and Yu (1979) and Yu et al. (1980) coupled the Zone method with a kinetic model for the reforming reactions and validated this complete reformer model with data obtained from a commercial unit. The works published subsequently by Stehlik et al. (1986a-c) dealt solely with the modeling of radiative heat transfer inside a reformer using the Zone method. They too have validated their model with plant data.

3. Physical Model The primary reformer is a top-fired box-type furnace

with the structure of a straight-flow, cocurrent-type heat exchanger (Figure 1). Circular tubes made of special quality steel are suspended in the reformer vertically in several lanes, each lane holding a specific number of tubes. The reformer is fired with a fuel in burners arranged in several rows, interrupted by alternating rows of tubes, such that there is a sandwich type of arrangement for the tubes and the burners. The tubes are filled with a special reforming catalyst.

The primary reformer is used for carrying out steam reforming of light hydrocarbons inside the tubes, and the reforming reactions, which are catalytic in nature, are aided by the catalyst in the tubes. The hydrocarbons are assumed to be converted to methane a t the entrance to the reformer in a “hydrocracking” reaction, and the methane so produced reacts with steam according to the steam- methane reforming reaction:

(1) CHI + HzO + CO + 3Hz

AH(298 K) = +2.062 X lo5 J/mol

The carbon monoxide produced in the above reaction reacts with steam according to the Shift reaction:

CO + H20 COZ + Hz ( 2 )

AH(298 K) = -4.12 X lo4 J/mol

The reaction in eq 1 is endothermic, while the reaction in eq 2 is exothermic, but the heats of the reactions (AH) of the two indicate that the endothermicity of the methane-steam reaction is much more than the exothermicity of the other reaction, and consequently, heat has to be supplied for sustaining the reforming reactions. This is done by burning fuel outside the tubes, and as the tube- side gases flow down, the heat necessary for the reforming reactions is absorbed from the hot combustion products

(3)

- dq- dz

-- -

The change in the radiative fluxes in the axial direction is related by eq 3 and 4 to the emission/attenuation of the radiation fromlby the gas, refractory, and tubes. The

1834 Ind. Eng. Chem. Res., Vol. 27, No. 10, 1988

r*

Figure 2. Reformer tube.

flame-side model is complete with the equation for the energy flux of the combustion products. The axial variation of this is determined by an energy balance on the furnace gas and is given by dHg/dz =

-4KE, + 2K(q+ + 4-1 - Bh,,At(T, - TJ + q c - q L (5)

The above equation does not consider conduction effects. The heat released along the length of the flame (9,) through the combustion of fuel is obtained from the heat distribution function (Roesler, 1967). The furnace gas- to-tube heat-transfer coefficient, h,,, is calculated by using a Dittus-Boelter-type correlation for turbulent flows (Kern, 1950).

The tube skin temperature, T,, occurring in eq 3,4, and 5 is calculated by making an energy balance on the tubes:

Methods used for evaluating the overall coefficient, U, are discussed in a later section dealing with the tube-side model.

The flux equations (3) and (4) may be further simplified by making some assumptions regarding the emission and reflection of radiation from the refractory surface. If the adiabatic refractory is assumed to emit/reflect whatever radiation it has absorbed from an upward-bound beam of radiation back into it and similarly for the downward- bound beam, the flux equations (3) and (4) are reduced to the form

dq+/dz = 2KE, - (2K + ttAt)q+ + ttA&

-dq-/dz = 2KE, - (2K + ttAJq- + t&Et

( 7 )

(8)

The effect of making such an assumption as above will be discussed later when the solution of the flux equations is presented. This model is called t,he “extended gray-gas model”.

4.1.2. Boundary Conditions. (a) Radiative Fluxes. For radiatively adiabatic end walls, the radiative fluxes should satisfy

q+ = q- at z = 0, L (9)

(b) Furnace Gas Energy Flux. The enthalpy of the furance gas a t the top of the reformer is related to the

furnace gas temperature through the relation

4.2. Tube-Side Model. With the data available from the flame-side model in the form of net heat fluxes on the tubes, the tube-side model predicts the temperature and concentration profiles for the process gas. The model equations comprise four differential equations, one each for the concentration of methane, concentration of carbon monoxide, process gas temperature, and process gas pressure.

4.2.1. Model Equations. The variation of the species concentration depends upon the rates of reactions in eq 1 and 2. For a differential section of height dz (Figure 2), the differential equations for the concentrations assume the form (Singh and Saraf, 1979)

dCCH, P c

dz GP, = rl-Mav --

Heat transfer from the tube wall to the gases flowing within the tubes is influenced mainly by two resistances, namely, resistance a t the wall and resistance within the catalyst bed, and care has to be exercised in estimating them. Beek (1962) observes that, in the one-dimensional approximation, an overall heat-transfer coefficient con- taining the effective thermal conductivity can be used to describe the heat-transfer process in a packed bed. The effective thermal conductivity is introduced into the overall coefficient, U , through the use of the Biot number ( N B ~ ) ; thus,

hi (1 + 0.25N~i)

U = (13)

The overall coefficient for tube-side heat transfer may thus be obtained by applying a correction to the wall heat-transfer coefficient, through a correction term con- taining the effective thermal conductivity. This, in effect, means that the total resistance in the bed is lumped with the resistance at the wall in order to approximately account for the radial variation of the temperature. This also affords a means of doing away with the estimation of the effective thermal conductivity. In the present study, the overall heat-transfer coefficient has been computed from the wall heat-transfer coefficient through an arbitrary correction factor, whose value is estimated during the model validation. It is interesting to note that Hyman (1968) recommends that the wall coefficient may be cor- rected by a factor of 0.4 for ring-shaped catalyst particles, whereas the correction factor in the present study is found to take the value of 0.38. In a one-dimensional heat- transfer model, the tube-side overall heat-transfer coefficient based on the tube outside surface area can, therefore, be calculated by

_ - - ( - I n - + - - Dto ””) (14) Cfhi Dti U 2k,

where the wall heat-transfer coefficient, hi, may be obtained from (Beek, 1962)

NNu = 2.58(NR,)1/3(Np,)1/3 + 0 .094(N~)0~8(Np, )0~4 (15)

The heat transferred to the tubes from the flames is utilized to meet the heat requirements of the endothermic reforming reactions and also to raise the temperature of


700

Table I. Reformer Operating Data

natural gas (fuel) flow rate, m3/h combustion air flow rate, m3/h

temp of combustion air, K

flue gas temp a t reformer outlet, K

Process Side natural gas (feed) flow rate, m3/h 4818 naphtha feed rate, kg/h 3950 steam flow rate, kg/h 39 350 steam pressure, kPa 2400 steam temp, K 770 feed temp, K 718 process gas temp at reformer exit, K

size of catalyst, mm net calorific value of natural gas, kJ/kg net calorific value of naphtha, kJ/kg

Furnace Side 3687 54 894 99.5 592 3.95 1232

pressure of combustion air, kPa

oxygen in flue gas, mol 70

1018 1100 17 X 17 X 6 2707 2519

bulk density of catalyst, kg/m3

the process gas. The variation of process gas temperature with length may be accordingly given as

- dTPg = ( - 2 r 1 - - r 2 AH )loo:r - + UA,,(T, - Tpg)

dz CP AfiCpGp,

(16) The pressure drop suffered by the process gas while flowing through the catalyst bed is modeled by using Ergun’s (1952) equation. The auxiliary parameters, like the rates of reactions, equilibrium constants, and heats of reactions occurring in the tube-side model equations, are obtained from appropriate sources (Hyman, 1968; Haldor Topsoe, 1965; Davies and Lihou, 1971).

4.2.2. Boundary Conditions. The boundary conditions for the process gas temperature, pressure, and composition are a t z = 0

Tpg = Tpg,i (17)

P = Pi (18) C C H ~ = CCH4,i (19)

CCO = CC0,i (20) While the boundary conditions for temperature and pressure are specified, the concentrations of CHI and CO are obtained from the equilibrium composition of the reaction mixture formed at the reformer inlet in the hydrocracking reaction between the hydrocarbon and steam (Subramaniam, 1967).

5. Solution of the Model Equations The tube-side and the furnace-side differential equa-

tions, together with the respective auxiliary equations, constitute the complete model. These are integrated si- multaneously to obtain the profiles of the desired variables. The numerical integration is carried out by using the fourth-order Runge-Kutta method, with a step length suitable for both the tube-side and furnace-side models. The basic computation scheme is iterative because the value of the furnace gas temperature a t the top of the reformer is not known to start with and it is refined in successive iterations. Convergence is assumed to have been achieved whenever the maximum of the fractional residues of all the variables a t all the points is less than the convergence criterion.

6. Model Validation The mathematical model setup for the reformer is tested

against the operating data of an industrial reformer for the

- I I I

Table 11. Comparison of Model Predictions with Measured Data

measd calcd flue gas temp at reformer exit, K 1232 1216 process gas temp at reformer exit, K 1018 1040 tube skin temp (3.7 m from the top), K 1173 1173 process gas composition a t reformer exit, mol %

CHI 8.80 8.32 co 11.13 10.09 COZ 13.80 14.06 HZ 66.04 67.53

Table 111. Parameters Used/Derived dur ing Model Validation ammonia production rate, TPD 480 steam-to-carbon ratio 3.65 heat input to reformer through the fuel, MW 50 flame length, m 4.2 % excess air used for combustion 18.14 emissivity of the tubes 0.80 emissivity of the refractory 0.75 absorption coefficient, l / m 0.25 heat lost to surroundings as a fraction of the total heat 0.02

correction factor to inside wall heat-transfer coeff 0.38 step size for integration, m 0.01 convergence criterion 0.001

generated

1100 I - Cokulotcd 0 Mcorurcd

- P ti

t - Colculotcd 0 Mcorurcd

8ot co., 80 -

r I 0 0.2 0.4 0.6 0.8 1.0

X , Dimensionless oxiol distoncc

Figure 4. Concentration profiles in the reformer.

purpose of validating the model. Details regarding the geometrical configuration of the reformer, dimensions of the tubes, etc., are not furnished here, as the concerned information is proprietary in nature. However, operating data obtained from the reformer while it was in operation are presented in Table I.

The results of the model validation are shown in Table 11. Profiles for some of the variables like temperatures and concentrations are presented in Figures 3 and 4.


Table IV. Exit Composition of Process Gas (Mole Percent) equilibrium values

plant values (measd) a t 1018 K at 1040 K CH, 8.8 10.37 8.25 co ~ 11.13 8.88 10.10 co2 13.80 14.80 14.06 H2 66.04 65.94 67.59

Table V. Comparison of Predictions of Different Models Roesler, extended Filla,

variable 1967 gray gas gray gas 1984 flue gas temp at 1220 1225 1216 1215

process gas temp at 1043 1040 1040 1042

tube skin temp 1182 1166 1173 1180

reformer exit, K

reformer exit, K

(max), K reformer duty, kW 29 982 29 733 29 773 29 989 process gas

composition at reformer exit, mol % CH, co COP Hz

8.10 8.38 8.32 8.17 10.21 10.05 10.09 10.17 13.99 14.09 14.06 14.01 67.70 67.49 67.53 67.65

These calculations are performed by using the gray-gas model on the furnace side. Some of the important parameters used or derived in the course of the model validation are given in Table 111. A comparison between the calculated and measured values shows that the model formulated makes good predictions of the flue gas temperature and process gas composition. The agreement between the calculated process gas outlet temperature (1040 K) and the measured value (1018 K) is not as good. The process gas composition at the reformer exit can be assumed to be the same as the equilibrium composition a t the prevailing exit temperature. But the equilibrium composition corresponding to the measured temperature does not agree well with the measured exit composition, as may be seen from Table IV. On the other hand, the equilibrium composition at 1040 K is much closer to the measured composition.

One may, therefore, conclude that the actual temperature prevailing at the reformer exit is higher than the measured value and that the model prediction of the process gas temperature is better than what is apparent. The discrepancy between the actual and measured temperatures may be attributed to some heat losses taking place from the duct carrying the process gas.

Model validation is also carried out by using Roesler's model, the extended gray-gas model, and the model of Filla (1984) for furnace-side calculations. A comparison of the predictions made by the different models on the furnace side is shown in Table V. It is clear that both the gray-gas model and its extended version make predictions that are as good as Roesler's and Filla's predictions, thus confirming the validity of the assumptions made. Both of the models are found to be about 20% faster computationally than Roesler's model. In the simulation work, which follows, the gray-gas model is used throughout.

7. Simulation of the Reformer During the model validation, some of the model param-

eters are fine-tuned, so that the model as a whole conforms to the specific features characteristic to the equipment. The response of the equipment to various hypothetical input conditions could then be simulated on the computer with the help of the validated model. A full-scale simu-

a.

f 4

1301 /

E 11.0 c

.P e

8 mo f

L - 9 0

E - 80 5.

- 7 0 Y

0. U I

\ 16.0 I I

2'0 2 5 30 35 L o 45 50

S C R , Stem-to-carbon ratio

Figure 5. Effect of steam-to-carbon ratio.

lation is preceded by a sensitivity analysis to identify the important operating variables. Simulation investigations are carried out later to generate data useful for the prediction of the performance of the general class of top-fired reformers.

7.1. Sensitivity Analysis. Sensitivity analysis helps in identifying the variables to which the reformer performance is sensitive, so that only the important variables are used in simulation in their useful ranges of operation. The performance of the reformer is measured by means of certain indicators like the peak tube skin temperature and the concentration of methane a t the reformer outlet. Four variables are used in the sensitivity analysis, three on the tube side (steam-to-carbon ratio, feed pressure, and feed temperature) and one on the furnace side (flame length). The effect of the variables is discussed with reference to the same reformer that has been used in model validation. The production rate is kept constant throughout, and the fuel flow rate is adjusted such that the peak tube skin temperature is maintained around plant operating value for ease in comparison.

7.1.1. Steam-to-Carbon Ratio. Steam-to-carbon ratio (S/C ratio) at the inlet of the primary reformer is the single most important parameter affecting the reformer performance. This ratio must fall within a rather fixed range of values for a trouble-free operation of the reformer (Craig and Burklow, 1980). The base case S/C ratio is 3.65 (here, base case refers to the operating conditions of the reformer a t the time of data collection). The effect of a variation in the SIC ratio on the reformer performance is shown in Figure 5. As the SIC ratio is increased, the fuel consumption increases, while the exit methane concentration falls. Fuel requirements are high at the higher S/C ratios because the endothermic steam-methane reaction proceeds to greater extents and more heat is required to sustain the reaction temperature.

At lower S/C ratios, the primary reformer exit methane concentration increases. The unconverted methane coming from the primary reformer further reacts with steam in the secondary reformer. The heat required for sustaining this reaction is supplied by the combustion reactions taking place there. Fuel consumption in the primary reformer goes down with decreasing S/C ratios (Figure 5), but the temperature of the outgoing process gas from secondary


vol \ 5

-11.0 g

g

P

Y

- 10.0 c- .P e c

9.0 f

I65 - f - f .I

i 5 160-

9 1 6.0 5

1600 2000 2400 2800 3200 3600

P, Pressure, k Pa

Figure 6. Effect of tube-side inlet pressure.

reformer also decreases, affecting thereby the steam generation in the downstream waste heat boilers. Thus, the fuel savings likely to occur in the primary reformer at low S/C ratios may be offset to some extent by the loss in steam generation. Another important aspect is that, if the methane concentration at the secondary reformer exit increases beyond a certain value, the purge rate in the ammonia synthesis loop is affected.

Proper evaluation of all the above factors is, therefore, necessary for the selection of a suitable S/C ratio. I t should not be so high that excessive fuel consumption occurs, and on the other hand, a very low value may place unduly high methane loads on the secondary reformer. The lower limit is fixed, however, by considerations of risk of carbon formation in the reformer tubes at low concentrations of steam during transient conditions.

7.1.2. Tube-Side Pressure at the Inlet. The plant value for the inlet pressure for the base case is 2400 kPa. The pressure is varied from 1600 to 3600 kPa, and its effect on the reformer performance is studied. The variations of the fuel flow rate and the exit methane concentration with the pressure are shown in Figure 6. As the pressure is increased, the conversion of methane falls and, along with it, the fuel consumption also falls. This is under- standable because lowering the pressure should favor the forward reaction, as the number of moles formed in the reforming reactions is more than the number of moles reacted. Operation of the reformer at very low pressures is not advisable because subsequent operations like shift conversion and COz absorption are adversely affected by lowering the pressure. The upper limit is set, however, by the design pressure of the tubes, which is about 3500 kPa for the tubes employed in reformers. Hence, the process-side pressure at the reformer inlet may be varied within a narrow range of, say, 2000-2800 kPa. The reformer performance is not very sensitive to such a type of operation (Figure 6).

7.1.3. Tube-Side Temperature at the Inlet. The feed is sent a t a temperature of 718 K for the base case in the plant. The results obtained in the sensitivity analysis involving the feed temperature are shown in Figure 7. With an increase in feed temperature, the fuel flow rate

I 1

970 seo sso io00 io10 1020 1030 T I inlet tempnature on the th-s ide. 'K

Figure 7. Effect of tube-side inlet temperature.

- 11.0 *

3 i

E E

9

;.

- 10.0 1

.P

u

- .- x w - 9.0 0. *

130 1 / \ I

I / I a I 8.0

0.2 0.4 0.6 0.8 1.0 1.7.

L f /Lfb ,Flame length as traction of base case value

Figure 8. Effect of flame length.

is seen to fall, indicating that part of the heat required on the tube side is met by the increased sensible heat of the feed. The change in the exit methane concentration is almost negligible. From this, it may be concluded that the reformer performance is insensitive to changes in process feed temperature.

7.1.4. Flame Length. Flame length is an important characteristic of turbulent diffusion flames employed in industrial furnaces. Fuel combustion takes place along the length of the flame, the rate of combustion being controlled by the mixing of fuel and air. Depending upon the requirements of the process, high/low heat fluxes are im- parted by employing a short/long flame through a careful selection of the burner and its operation.

The simulation results for a variation in the flame length are shown in Figure 8. With a decrease in flame length, fuel flow decreases and exit methane concentration increases. A t shorter flame lengths with higher combustion intensities, more heat is generated than can be taken up by the process fluid in the tubes, and consequently max-


Ts, ,,, .Peak tube skin temperature. K CCH4,0,Exit methane conccntratii,mok percent

6 8 10 12 14

Hi ,Heat input.MW

200 400 600 800

1050 1100 1150 1200 1250 - I

SCR .Steam-to-carbon ratio

-I Flame length

- - -

I

// I Rp,Ammnia production rate,TPD

Figure 9. Sizing nomogram for peak tube skin temperature.

imum tube skin temperature is higher in the flame region, the shorter its length is. As the maximum tube skin temperature is fixed at a constant value in the present case, it is logical that fuel consumption should decrease with decreasing flame length. The conversion of methane suffers at shorter flame lengths because of the reduced fuel flow, and methane concentration at the reformer exit increases with decreasing flame length. On the basis of the foregoing discussion, it may be said that it is preferable to operate the reformer with shorter flame lengths, if the peak tube skin temperature is within tolerable limits and the increased methane load can be taken care of by the secondary reformer. This information will prove especially useful in new reformer designs.

7.2. Sizing a Reformer. Sensitivity analysis performed in the last section established the important variables to be considered for the assessment of the performance of a primary reformer. By use of the information about these variables and their ranges of operation, an extensive simulation program is undertaken to generate data useful for the design of the general class of top-fired reformers.

Out of the four variables tested in the sensitivity analysis, two are selected for inclusion in the simulation program. They are S/C ratio and the flame length. The excess air for the combustion of fuel is fixed at 1270, the tube-side inlet pressure is fixed at 2400 kPa, and the tube-side inlet temperature is fixed at 718 K. Five other parameters are considered for the simulation, two opera- tional (ammonia production rate and heat input through fuel) and three geometrical (number of tubes, tube inside

S C R , Steam-to<arbon ratio

/ -&Flame length

I

I 10 ,20 ,30 40 50

I , 1 1 X K ) 400 600 800

Rp,Amnorjo prodKtion ratr.TPD

Figure 10. Sizing nomogram for exit methane concentration.

diameter, and tube length). The ranges in which the variables studied are

steam-to-carbon ratio 3.0-4.0 ammonia production rate 300-600 tons/day heat input through fuel no. of tubes 160-300 tube inside diameter 80-120 mm tube length flame length/tube length 0.3-0.64

15000-60000 kW

6.5-8.5 m

The computational effort involved in generating the simulation data is huge, considering the fact that the information to be obtained about the reformer operation should reflect the effect of as many as seven variables. The resulting simulation data are subjected to multiple re- gression analysis to deduce correlations for the reformer performance characteristics Tapn, CCH.,,~, Qh, Irpgp, Tg,o, Qco,~, and QCH4,+ in terms of the simulation Variables:

Nomograms are prepared on the basis of the correlations obtained above, and some of the important ones are presented in Figures 9-11. The reformer performance characteristics, M , refer solely to the primary reformer. Qh, for instance, is calculated on the basis of fuel consumed in the primary reformer and hence does not take into account subsequent operations in secondary reformer, etc.

The nomograms together with the design correlations facilitate the task of the design of a new reformer. If, for example, a reformer of 400 tonslday is to be designed, Figure 9 gives, for a peak tube skin temperature of 1173 K, a reformer configuration of 256 tubes of 6-m length and 100-mm i.d. for a steam-to-carbon ratio of 3.0, heat input of 35 MW, and a flame length of 60% of the tube length.


certainly have made the design information catalyst-in- dependent and hence more general.

Notwithstanding the shortcomings in the modeling and simulation procedures adapted in the present work, it may be credited with having demonstrated how mathematical modeling could be put to use to generate design information for a process equipment. More studies of this kind are needed with respect to other practical combustors like rotary kilns, petroleum refinery heaters, etc., to evolve better design procedures for these equipment than are available now.

Nomenclature An = tube inside flow area, m2 A, = half refractory area per unit free volume, m2/m3 A,, = tube outside surface area per unit length, m2/m A, = half tube surface area per unit free volume, m2/m3 C = concentration, mol % c f = correction factor to heat-transfer coefficient cp = specific heat at constant pressure, J/(K kg) Dti = tube inside diameter, m D,, = tube outside diameter, m E = black emissive power, W/m2 G = mass velocity, kg/(m2 s) Hg = energy flux of furnace gas, W/m2 Hi = heat input to the reformer, kW AH = heat of reaction, J/mol h = heat-transfer coefficient, W/(m2 K) K = absorption coefficient, l / m k = thermal conductivity, W/(m K) L = heated length of the tube, m Lf = flame length, m Lfi = base case flame length, m M = reformer performance characteristics Ma, = average molecular weight NBi = Biot number (=Dtihi/2k,) NPr = Prandtl number NNu, yb = Nusselt and Reynolds numbers based on catalyst

Nt = total number of tubes P = tube-side fluid pressure, kPa Q = flow rate, kmol/h Qf, = specific consumption of fuel, kg of fuel/kg of NH3

q+ = radiation flux in the positive axial direction, W/m2 q- = radiation flux in the negative axial direction, W/m2 qc = heat released across the flame, W/m3 qL = heat lost to the surroundings, W/m3 RP = ammonia production rate, tons/day r = rate of reaction, kmol/(kg of catalyst s) SCR = steam-to-carbon ratio, mol/atom T = temperature, K U = overall heat-transfer coefficient, W/ (m2 K) X = dimensionless length (=z /L ) z = axial distance, m Greek Symbols c = emissivity pc = bulk density of catalyst, kg/m3 Subscripts e = effective f = fuel g = furnace gases i = inside; inlet m = maximum o = outside; exit pg = process gas r = refractory ref = reference s = tube skin t = tubes

particle

produced

Of,,Specific f w l conwmtion.kg fuel/kg m i a

Y SCR ,Steam.to-carbon ratio I/, I concentration, molc DcTcent

Dt,.Tube inside diameterpm

Rp,Ammon~lproduction 4 / 40 rate,TPD 800

Figure 11. Sizing nomogram for specific fuel consumption.

The corresponding value of the methane concentration for these conditions can be read from Figure 10. If the value obtained is not acceptable from the point of view of secondary reformer operation, a different set of parameters is called for. The ultimate set of parameters selected should satisfy both of the aspects of peak tube skin temperature and exit methane concentration. The nomogram for the specific fuel consumption (Figure 11) is particularly useful in establishing an optimum set of operating parameters, which results in the minimum consumption of fuel for a particular ammonia production rate.

8. Conclusions A mathematical model has been developed for predicting

the performance of a top-fired primary reformer of an ammonia fertilizer plant. Both the process-side and furnace-side phenomena have been treated in an integrated model. The overall model has been tested against the data collected on an industrial reformer, and the model predictions have been found to be reasonably good. A par- ametric study has been conducted on the model to determine the important simulation variables for using them in the subsequent simulation investigations. The data generated during the simulation have been used to develop correlations and nomograms for the reformer performance characteristics in terms of several operating and con- structional parameters of the reformer.

The development of the mathematical model and the subsequent simulation has been handicapped by a lack of adequate information regarding the characteristics of the catalyst employed in the reformer tubes. Although it is doubtful that the results of the simulation are seriously affected because of this, more details in this regard, particularly concerning the activity of the catalyst, would

1840 Ind. Eng. Chem. Res. 1988,27, 1840-1848

w = wall 1,2 = reactions in eq 1 and 2

Literature Cited

Beek, J. Advances in Chemical Engineering; Academic: New York,

Craig, D. F.; Burklow, B. W. Ammonia Plant Saf. 1980, 22, 131. Davies, J.; Lihou, D. A. Chem. Process Eng. 1971,52, 71. De Deken, J. C.; Devos, E. F.; Froment, G. F. ACS Symp. Ser. 1982,

Ergun, S. Chem. Eng. Prog. 1952,48,89. Filla, M. Chem. Eng. Sci. 1984, 39, 159. Grover, S. S. Hydrocarbon Process. 1970,49(4), 109. Haldor Topsoe, Communication 698, The Institution of Gas Engi-

Hottel, H. C.; Sarofim, A. F. Radiatiue Transfer; McGraw-Hik New

Hyman, M. H. Hydrocarbon Process. 1968, 47(7), 131. Kern, D. Q. Process Heat Transfer; McGraw-Hill: Kogakusha, To-

1962; Vol. 111, p 234.

196, 181-197.

neers, 31st Autumn Research Meeting, 1965.

York, 1967.

kyo, 1950.

McGreavy, C.; Newman, M. W. Paper presented at the Institution of Electrical Engineers Conference on the Industrial Applications of Dynamic Modeling, Durham, 1969.

Murray, A. P.; Snyder, T. S. Ind. Eng. Chem. Process Des. Dew. 1985, 24, 286.

Murty, C. V. S. Ph.D. Thesis, Indian Institute of Technology, Ma- dras, 1987.

Roesler, F. C. Chem. Eng. Sci. 1967, 22, 1325. Shen, Cai-Da; Yu, Zun-Hong Shang-hai Hua Kung Hsueh Yuan

Singh, C. P. P.; Saraf, D. N. Ind. Eng. Chem. Process Des. Dew. 1979,

Stehlik, P.; Sika, J.; Bebar, L. Chem. Prum. 1986a, 36(61), 342. Stehlik, P.; Sika, J.; Bebar, L. Chem. Prum. 1986b, 36(61), 454. Stehlik, P.; Sika, J.; Bebar, L. Chem. Prum. 1986c, 36(61), 512. Subramaniam, T. K. Hydrocarbon Process. 1967,46(9), 169. Yu, Zun-Hong; Shen, Cai-Da; Pan, Hui-Qin; Cai, Guo-Qiang; Sun,

Received for review November 30, 1987 Revised manuscript received May 18, 1988

Accepted June 6, 1988

Hsueh Pao 1979, 1-2,51.

18(1), 1.

Xing-Yuan Huagong Xuebao 1980,2,143.

Scheduling in Serial Multiproduct Batch Processes with Finite Interstage Storage: A Mixed Integer Linear Program Formulation

Hong-ming Ku and I f t e k h a r A. Karimi* Department of Chemical Engineering, Northwestern University, Evanston, Illinois 60208

Production scheduling of N products in an M-unit serial multiproduct batch process under the minimum makespan criterion is studied. A finite number of storage units may be present between consecutive processing units. Recurrence relations requiring O(NM) simple steps are presented for computing operation times of all products in a given production sequence. An optimal MILP formulation is developed for scheduling a set of batches with stable intermediates and is then extended to allow for unstable intermediates. A heuristic strategy is also proposed to reduce the number of binary variables in the above formulation by assigning products to specific neighborhoods of a sequence. This approximate formulation yields solutions within 1% of the optima when applied to a set of test problems.

For years, continuous operations have been the most prevalent and sought-after mode in chemical processing. In recent years, however, there has been a renewed interest in batch processes for processing situations involved in the production of complex, high-value chemicals or the production of multiple products by sharing the use of process equipment. An important problem that arises in such operations is the scheduling of products to make efficient use of a multiprocessor production facility with limited resources. A general short-term scheduling problem in multiproduct batch processes consists of scheduling a set of product batches on a network of processors so as to optimize a suitable cost or performance measure.

Normally, this problem requires specifications of a matrix of batch processing times for various operations, matrices of resource utilization requirements for each operation, a matrix of changeover times and costs for pairs of products for each operation, and a set of rules governing the transfer of batches between processing stages. Then scheduling involves the determination of a sequence in which the batches should be processed and of a corresponding time table for all operations to optimize a cost or system performance criterion.

Because of the staged nature of process equipment in batch chemical plants, interstage storage is a vital com-

* Author t o whom correspondence should be addressed.

0888-5885/88/ 2627-1840$01.50/0

ponent of batch processes. Due to the unsteady-state nature of batch operations, the equipment utilization and the productivity are usually low in these processes. Storage installed between processing stages can help reduce idle times in these stages by freeing them to process other batches and thus increase the equipment utilization and the productivity of a multiproduct batch process. Inter- mediate storage can also be used to decouple periodic operation of adjacent batch or semicontinuous units, to minimize the effects of variation in process parameters, to moderate the effeds of equipment failures, and to isolate intermediates associated with different products. In ad- dition, it may be useful to have interstage storage for various other purposes. Hence, various forms of storage are used in batch plants, and its absencelpresence sig- nificantly affects the scheduling of operations.

The rules governing the transfer of batches between processing stages can be classified into four storage policies, depending upon the number of batches that can be stored in the interstage storage. These policies are (1) unlimited intermediate storage (UIS), (2) finite intermediate storage (FIS), (3) no intermediate storage (NIS), and (4) zero wait or no wait (ZW or NW).

The interstage storage capacity is measured in terms of the number of units and not the physical size of storage since it is usually assumed that each storage unit can temporarily hold any product batch. In the UIS and FIS

0 1988 American Chemical Society

Documents

Modeling and simulation of a top-fired reformer