Sequential-Based Process Modeling of Natural Gas Combustion in a

Article pubs.acs.org/EF

Sequential-Based Process Modeling of Natural Gas Combustion in a Fluidized Bed Reactor Abolhasan Hashemi Sohi,† Ali Eslami,† Amir Sheikhi,†,‡ and Rahmat Sotudeh-Gharebagh* Process Design and Simulation Research Centre, Oil and Gas Processing Centre of Excellence, School of Chemical Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran ABSTRACT: A model for the simulation of a fluidized bed combustion reactor was developed using sequential modular approach in which two submodels for describing physical and chemical phenomena were integrated simultaneously. The adopted submodels were hydrodynamic and kinetic submodels, respectively. Dynamic two-phase model was used as the hydrodynamic submodel, and the latter submodel was extracted from literatures. The combustion reactor was divided into two regions, namely dense bed and freeboard. The dense bed was divided into several sections, determined by a newly introduced dimensionless number, in which gaseous phase was considered as a plug flow through the bubble phase, and a mixed flow through the emulsion phase. Moreover, energy balance and the effect of bubble growth were taken into account, and a new correlation for obtaining the critical temperature of natural gas combustion was proposed. Various sets of experimental data were derived from the literature and used to validate the proposed model, and a close agreement was observed between them. The sequential modular approach, which is developed in this research, can be used for the simulation of nonideal fluidized bed combustors inside the industrial process simulators such as Aspen Plus and Aspen HYSYS.

1. INTRODUCTION The combustion of natural gas in fluidized bed reactors provides a great potential for waste-to-energy applications by using low calorific fuel upgrading as well as cofiring, gas reburn and direct combustion.1 Among the natural gas burners, fluidized beds offer a wide range of advantages, including smaller sizes, simplicity of setting up, flexibility in solid, liquid, and/or gas processing, and also large combustion efficiencies at eminently low temperature ranges, while minimizing thermal NOx generation.1 Moreover, natural gas fluidized bed combustors can be employed in a wide range of industrial plants, offering remarkable advantages such as low emission, high conversion and energy efficiencies, easy-to-prepare fuel requirements, and economical operating conditions.1 Furthermore, natural gas-based fluidized beds are offered as the best technology for residential applications due to their bed-heat exchanger large heat transfer capabilities.1,2 Combustion processes are concerned by mass and energy balances as well as chemical reactions.3 Designing virtual physical and chemical platforms to predict the behavior of combustion reactors is of the most important challenges in the industry, since the real experimental studies are not only time and energy consuming but also difficult to carry out. Combustion processes are divided into three major categories with respect to the needed fuel, namely, gaseous, liquid, and solid combustions. Most of the time, complete combustion is ended in gaseous phase;1 hence, studying these kinds of multiphase systems and having the ability of analyzing them, from either conversion or emission point of views, without spending too much, is of the most vital concerns for process engineers. The operation of fluidized bed reactors for combustion purposes is conducted by gas and air feeding upward through a distributor located at the bottom of the bed. By the gas flowing © 2012 American Chemical Society

upward, liquid-like behavior of particles is started, leading the process to have a high heat-transfer rate. This results in almostisothermal reactions without hot spots occurring throughout the bed.4−7 Study on the combustion of gaseous mixtures, mostly methane, in bubbling fluidized bed reactors is done by several researchers.8−13 Although some comprehensive investigations on premixed fluidized bed fuel-combustion reactors can be found in the literature (e.g., Wu et al.,12 Srinivasan et al.,14 and Friedman and Li15),16 easy-to-conduct modeling methods for fluidized bed combustors, as highly nonideal systems, are required to gain the ability of analyzing process key parameters such as temperature, concentration gradient, the conversion of each reactant as a function of process operating condition, and emission level, especially for further pursuing optimization and scale up purposes. The modeling is commonly based on two important submodels; the hydrodynamic submodel, which introduces the physical phenomena, and the reaction submodel, which involves the chemical reactions occurring in the fluidized bed combustor. Two general types of models have been reported in the literature: empirical and mechanistic models.17 Empirical models ignore the hydrodynamic complexity and therefore are less valuable than mechanistic models, which are based on pseudo-homogeneous or two-phase approaches.18 Since the homogeneous model describes the hydrodynamics using a single-phase approach (completely mixed, plug flow, or dispersion models), it usually represents a poor prediction of fluidized bed combustor behavior. The two-phase approach provides a satisfactory technique by considering the combustor as two one-dimensional paths (bubble as a high-voidage phase Received: February 3, 2012 Revised: March 9, 2012 Published: March 9, 2012 2058

dx.doi.org/10.1021/ef300204j | Energy Fuels 2012, 26, 2058−2067

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and emulsion as a high-solid-fraction phase) connected together based on a mass transfer mechanism. Two-phase models are investigated in several studies.19−21 Yanata et al.22 considered a premixed combustor as a porous media and developed a simple model to explain the behavior of such fluidized beds of inert particles. In their model, a plug-flow gas was assumed to exchange heat with the particles through an energy-balance mechanism.3 Bubble-phase, as a descent reaction medium compared to the dense phase, was suggested by Dennis et al.,23 who neglected the reactions in the dense phase. Also, van der Vaart24 modeled the homogeneous fluidized-bed combustion of stoichiometric mixture of methane using sand particles as a solid phase and considering three complicated reactor models including the following main characteristics: (a) constant bubble size and two interacting phases, (b) three phases including constant bubble size, and (c) three phases including bubble growth. Moreover, to explain the behavior of the freeboard region, a basic plug flow model was taken into account, separately.24−26 In this study, a sequential modular simulation (SMS) approach was developed. Relevant calculations start using known feed composition and flow and continue through a unitby-unit analysis pathway to obtain all the unknowns in any unit operation in the flowsheet. To simulate the behavior of a fluidized bed combustor using SMS approach, several ideal reactors, with their well-understood hydrodynamic behavior and available reaction kinetic models, are linked to each other with a wise logic presenting the conversion of the fuel (e.g., natural gas) in the multiphase reactor. Moreover, to reach temperature profile inside the bed, leading to more accurate conversion rates of the present species, energy balance and the effect of bubble diameter variation should be taken into account.1,24

Figure 2. Interactions in the dense bed of a fluidized bed combustion reactor.

As shown in Figure 3, to model a fluidized bed reactor, it is axially divided into a series of CSTRs and PFRs. While a PFR represent the gas flow through the bubbles, a CSTR considers the gas flow through the emulsion phase. The assumptions made in developing the equations of the model are summarized as follows: (1) The hydrodynamics of both phases are characterized by generalized steady-state mass balance equations for either bubble or emulsion phases in each stage (similar to Jafari et al.28). (2) Reactions occur in both bubble and emulsion phases (similar to Kiashemshaki et al.29). (3) Bubble growth was considered along the bed height (based on the suggestion of van der Vaart24). (4) Temperature variation was taken into account inside the bed. 2.1. Hydrodynamic Submodel. The required hydrodynamic parameters are listed in Table 1.20,24,30−35 As it is mentioned earlier, bubble growth occurring simultaneously with severe exothermic reactions of gas phase is considered in the model.24 It is assumed that the main reason of bubble diameter variation is the bubble temperature alteration, which can be presented as the following equation for rc, the bubble radius at a favored temperature (Tb).24

2. REACTOR MODEL Figure 1 shows a schematic diagram of a typical fluidized bed combustion reactor. Methane and air are passed through the

⎛ Tb ⎞1/3 rc = ⎜ ⎟ rc, ∞ ⎝ T∞ ⎠ Figure 1. Schematic diagram of a fluidized bed combustor.

(1)

In this model, it is considered that the gas temperature is constant in the emulsion phase (therefore, the volume of emulsion phase will remain unchanged due to the constant density of the gas inside it)24 while it is changed only in the bubble phase.24 The molar balance in the bubble phase can be written as eq 2:

distributor to ensure a uniform gas distribution and temperature profile, and the reactor is usually surrounded by a heater. Also, a burner can be used as a reactor base to provide the energy needed to heat up the bed materials. The combustion reactor is divided into two regions: dense bed and freeboard. The dense bed of a fluidized bed combustor consists of two phases, namely, emulsion and bubble phases, as shown in Figure 2. A fluidized bed rector is a nonideal multiphase system, so it cannot be modeled as a plug flow reactor (PFR) or a completely stirred tank reactor (CSTR).27

CAb(i − 1)UbAb − Abε b

∫z

zi i−1

rA(i) dz

− Kbe(CAb(i) − CAe(i))Vb(i) − CAb(i)UbAb = 0 (2) 2059


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equation is solved and the bubble growth is taken into account, simultaneously. 2.2. Kinetic Submodel. Different complex reaction-network-based studies on methane combustion are available in the literature and are formed from a vast range of elementary reactions.3,18,24 In a simple, yet accurate, way of kinetic analysis, combustion of methane is assumed to lead to CO production, and CO was then assumed to be burned to CO2 according to the following reactions: CH 4 + CO +

3 O2 → CO + 2H2O 2

(5)

1 O2 → CO2 2

(6)

36

Kozlov investigated methane combustion in small tubes and developed the following global expressions for the reaction rates: −

d[CH 4] [CH 4]−0.5 [O2 ]1.5 ⎛ −6000 ⎞ ⎟ = 7.0 × 108 exp⎜ ⎝ RT ⎠ dt T (7)

−

d[CO] = 1.04 × 1012 dt ×

[CO][O2 ]0.25 [H2O]0.5 T 2.5

⎛ −32000 ⎞ ⎟ exp⎜ ⎝ RT ⎠ (8)

Equations 7 and 8 predict that the overall reaction rate is inversely proportional to the methane concentration.

3. RESULTS AND DISCUSSION The most important part of an SMS approach, after combining the ideal reactors in a logical manner, is to define the best number of stages to obtain accurate predictions. The number of stages depends on several operating parameters such as superficial gas velocity, bed temperature, particle size, difference between gas and solid density, and air factor defined as the ratio of air volume actually fed into the reactor to the volume corresponding to the stoichiometric mixture, etc. Based on the several sets of experimental data obtained from fluidized bed reactors, it was found that hydrodynamics, heat transfer, and reaction kinetics affect the number of stages the most, which are required to be taken into account to simulate the fluidized bed combustors.28,29,37 Three most important parameters affecting the hydrodynamic-related part of the model are the ratio of superficial gas velocity to minimum fluidization gasvelocity, difference between the solid and gas densities, and aspect ratio (AR) (the ratio of bed height to its diameter). Moreover, bubble to emulsion heat-transfer coefficient, specific heat capacity of gas and solids, and critical temperature, which will be discussed in details later, should be considered to evaluate the impact of energy transfer. It should be noted that bed temperature is also a crucial parameter in determining the combustion reaction kinetics. The effect of increasing superficial gas velocity is obvious in increasing the solid and gas heat transfer due to vigorous mixing in the emulsion phase. Also, increasing aspect ratio will increase the bubble phase volume fraction inside the bed. The reactions are mainly occurring in the bubble phase. Therefore, by the increase of AR, the number of stages should be decreased to achieve a particular conversion. Van der Vaart38

Figure 3. Schematic diagram of sequential modular simulation of a dense bed.

while the energy balance over the bubble phase is presented as eq 3: n

n

∑ Cj b,(i)CPj ,(i)UbAbTb,(i) − ∑ Cj b,(i − 1)CPj ,(i − 1)Ub j=1

× AbTb,(i − 1) − ΔHAbε b

j=1 zi

∫z

i−1

rA(i) dz

− 4πhbe(Tb,(i) − Tb,(i − 1))rc 2 = 0

(3)

and the molar balance in the emulsion phase is as follows (eq 4): CAe(i − 1)UeA e − rA(i)VCSTR(i) + Kbe(CAb(i) − CAe(i)) ⎛ δ ⎞ ⎟ − C × Ve(i)⎜ Ae(i)UeA e = 0 ⎝1 − δ ⎠

(4)

The mass transfer among the phases is calculated as illustrated in Figure 4. As the mass transfer calculations at an imaginary section i are completed, the corresponding flow distribution is properly evaluated to move to the next section (i + 1). The calculation is continued upward until the top of the bed is reached. Also, after each section, the energy balance 2060


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Table 1. Required Hydrodynamic Parameters param.

correlation

Archimedes number

ref. 30

3

Ar =

ρgd p (ρp − ρg)g μg 2

bubble diameter

D b = 0.21Hf 0.8(U0 − Umf )0.42 exp[− 0.25(U0 − Umf )2 − 0.1(U0 − Umf )]

31

minimum fluidization velocity

Umf ρgd p

32

μg

=

27.22 + 0.0408Ar − 27.2

bubble rise velocity

Ubr = 0.711 gD b

bubble velocity

Ub = U0 − Ue + Ubr

emulsion velocity bubble to emulsion mass transfer coefficient

20

U0 − δUb 1−δ 1 1 1 = + Kbe Kbc Kce

Ue =

⎛ D 1/2g 1/4 ⎞ ⎛U ⎞ 1 ⎟ = 4.5⎜ e ⎟ + 5.85⎜⎜ AB 5/4 ⎟ Kbc ⎝ Db ⎠ ⎝ Db ⎠

33

⎛ D ε U ⎞1/2 1 = 6.77⎜⎜ AB e3 br ⎟⎟ K ce ⎠ ⎝ Db gas-to-particle heat transfer coefficient

⎛ kg ⎞ hgp,br = ⎜⎜ ⎟⎟(2 + 0.6Re p1/2Prp1/3) ⎝ dp ⎠

bubble-to-emulsion heat transfer coefficient

⎧ ⎡ ⎞⎤⎫ ⎛ h ⎪ gp,br k g ⎟⎥⎪ ⎬ hbe = 0.25UbrC p,pρpε b⎨1 − exp⎢− 6τ⎜⎜ ⎟⎥ ⎢ ⎪ ⎝ k sC p,pρpd p ⎠⎦⎪ ⎣ ⎩ ⎭

voidage in bubble phase

⎛ U −U ⎞ mf ⎟ ε b = A void − b(1) + A void − b(2) exp⎜⎜− 0 ⎟ ⎝ A void − b(3) ⎠

voidage in emulsion phase

⎛ U −U ⎞ mf ⎟ εe = A void − e(1) + A void − e(2) exp⎜⎜− 0 ⎟ ⎝ A void − e(3) ⎠

emulsion fraction

⎛ U −U ⎞ mf ⎟ δ = 1 − A f(1) − A f(2)exp⎜⎜− 0 A f(3) ⎟⎠ ⎝

residence time of particle in bubble

τ=

34,35

24

2.3rc/g

agreement with the maximum bed temperature reported in experimental combustion literature. It is noteworthy that by increasing the loaded-mass of sand particles into a fluidized bed reactor (and consequently, increasing the bed height), the residence time of gas inside the bed will increase, which results in a gently occurring combustion in the bed and a lower critical temperature. In such condition, less gas bursting is observed in the freeboard section of reactors. In this work, five series of experimental data were employed in a wide range of bed temperature, superficial gas velocity, and reactor type. Properties and operating conditions of each case are given in Table 2.10,13,24,41,42 Predicted results from the model presented in this study were compared to the experimental data at different operating conditions in terms of methane conversion, effluent carbon monoxide and carbon dioxide, and bubble temperature. Figures 5 and 6 illustrate methane conversion in the combustion process as a function of bed temperature for various numbers of sections. As shown in these figures, increasing the bed temperature increases the final conversion. This observation is in full agreement with the Srinivasan et al.14 demonstration of the higher methane

demonstrated that the ignition front moves closer to distributor by the increase of particle size and bed temperature and the decrease of excess gas velocity. It is showed that there a critical temperature exists in combustion processes, which varies according to the type of fluidized beds. Below the so-called critical temperature, the combustion may shift toward the freeboard region. It is important to note that Hesketh and Davidson39 have shown that above the combustion critical temperature, the emulsion phase exhibits homogeneous combustion behavior in a fluidized bed reactor.40 To determine the critical temperature for further simulations, the following correlation is derived using Zukowski,10 van der Vaart,24 and Dounit et al.41 experimental data: ⎡ (U − U )(d )2 ⎤ mf p ⎥ Tcr − T0 = C ln⎢ ⎥ ⎢ Lsνg ⎦ ⎣

24

(9)

where T0 and C are constants, which are found to be 1326, and 11.94, respectively, and Ls is the static bed height of sand loaded in the bed, which represents the bed height. T0 is obtained from a least-squared curve fitting and is in a close 2061


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Figure 5. Methane conversion as a function of bed temperature at various number of sections: comparison between the model and experiment,42 the values of NS for the experimental data (T = 780− 950 °C) are 10.28, 10.01, 9.72, 9.49, 9.21, and 8.78, respectively.

Figure 4. Schematic diagram of mass and energy transfer in a dense bed.

stages required to simulate fluidized bed natural gas combustors:

conversions achieved by the increase of the reactor temperature due to the front-movement of ignition section closer to the distributor. In addition, the methane conversion versus the ratio of superficial gas velocity to minimum fluidization gas-velocity is presented for various divisions of combustion reactor in Figure 7. It is noteworthy that superficial gas velocity leads to the increase of the methane conversion as a result of improved mixing in the emulsion phase. As can be seen in Figures 5−7, using one section in the simulation, the calculated conversion overpredicts the experimental data. In contrast, increasing the number of stages causes the model to fall below the experimental conversion. This can be explained by the fact that increasing the number of stages in a tank-in-series model turns it into a plug flow reactor, and according to eqs 7 and 8, which present the reaction kinetics, a CSTR offers more conversion than a plug flow reactor. Therefore, there is an optimum number of stages (for each operating condition) to simulate the combustion fluidized bed reactors appropriately. The same procedure was repeated for all of the experimental data listed in Table 2, and the optimum number of sections was determined in each case. Based on the previous discussion on the influencing hydrodynamic, heat transfer, and reaction parameters, the following dimensionless number (NS) is introduced in order to predict the generalized number of

NS = 16.2 ×

⎡ ⎤ ⎛ Re ⎞5⎛ (ρp − ρg)C pg,mU0 ⎞⎟⎥ 103⎢⎜ m ⎟ ⎜ ⎢⎝ Re ⎠ ⎜⎝ ⎣

hbe

0.4

⎟⎥ ⎠⎦

⎡ 1 ⎤Tbed / Tcr [AR]−3 ×⎢ ⎥ ⎣α⎦

(10)

in which Rem is the particle Reynolds number at minimum fluidization velocity, hbe is bubble-to-emulsion heat transfer coefficient, and alpha (α) is the air factor. The relationship between the number of stages and dimensionless number, NS, in various operating conditions of fluidized bed combustors is presented in Table 3. In each case, all affecting parameters, except one, for example bed temperature, were considered to be constant, and then, the desired parameter was manipulated to calculate the proper effect on the proposed dimensionless number. The performance of the model was evaluated quantitatively using error statistics such as BIAS and scatter index (SI), defined as N

BIAS =

∑ i=1

1 (Yi − Xi) N

(11)

Table 2. Experimental Operating Conditions

a

param.

unit

ref 10

ref 13

ref 24

ref 41

ref 42

H D dp ρs Tbed T0 P αa U

m m μm kg/m3 °C °C Pa

0.4 0.096 385−430 2643 550−1050 20 101300 1.4 300

0.3 0.122 250−600 2643 730−930

0.2 0.07 327 2650 850−975 177 101300 2 241

1.4 0.36 350 2650 600−850 20

0.3 0.18 350 2650 500−950 20 101300 1.875−2.25 136

mm/s

101300 2.57 290

1−1.5 160−320

The volume ratio of oxygen to methane in the feed. 2062


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mean value of experimental data. Also, correlation coefficient (CC) is calculated for each case based on eq 13: ∑iN= 1 (Xi − X̅ )(Yi − Y ̅ )

CC =

∑iN= 1 (Xi − X̅ )2 ∑iN= 1 (Yi − Y ̅ )2

(13)

in which Y̅ is the mean value of the predicted data. Measured errors, presented in Table 4, are remarkably low, so that they confirm the validity of the proposed model. Table 4. BIAS, SI, and CC of Model Results for Various Number of Sections reactor result ref ref ref ref

Figure 6. Methane conversion as a function of bed temperature at various number of sections: comparison between the model and experiment,24 the values of NS for the experimental data (T = 850− 980 °C) are 13.46, 12.63, and 12.37, respectively.

SI =

n 1 2 3 4 5

CC

0.07 0.01 0.01 0.03

0.99 0.97 0.95 0.98

place closer to the distributor at higher temperatures because inhibition has less effect at higher temperatures. The inhibition of methane combustion by sand particles is caused by reduced free-radical concentration due to the fact that, on solid surfaces, reactive radicals such as CH3 recombine together shortly and a lot of inert byproduct, such as CO, is produced.40,43 The parity plot of predicted against experimental conversions24,42 is also presented in Figure 9. It could be seen in this figure that the model predictions are in a very close agreement with the experimental results. Figure 10 shows the effluent CO concentration as a function of bed temperature. As it is shown, the CO concentration rises as bed temperature increases up to a critical temperature. As discussed earlier, the bubble explosion occurs in freeboard regime at temperatures relatively lower than the critical temperatures. The sand particles act as an inhibitor and cool the gases in the bed, and consequently, the increase of bed temperature leads to a rise in CO concentration. On the other hand, increase of temperature above the critical temperature shifts the reactions zone deeper into the bed which leads to

1 N ∑ (Y − Xi)2 N i=1 i

X̅i

SI

0.03 0.08 −0.06 −0.09

Figure 8. Methane conversion along the fluidized bed combustor at two different bed temperatures.

Table 3. Number of Stages (n) Defined by NS Dimensionless Number NS

BIAS

Figure 8 illustrates methane conversion along the reactor bed at 850 and 975 °C. Based on this figure, the combustion takes

Figure 7. Change of effluent CO2 with respect to normalized feed velocity, the values of NS for the experimental data (U/Umf = 3−7) are 18.78, 16.69, 15.05, 14.13, and 13.69, respectively.

NS > 24.3 18.8 < NS < 24.3 13.5 < NS < 18.8 8.4 < NS < 13.5 NS < 8.4

42 24 13 10

(12)

where Xi and Yi denote the experimental and predicted values, respectively, and N is the number of observations. X̅ is the 2063


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Figure 9. Comparison between the experimental data and the model prediction in terms of methane conversion percentage.

Figure 11. CO and CO2 yields along the fluidized bed reactor: comparison between the model and experiments.42

Figure 10. CO concentration versus bed temperature: comparison between the model and experiments.10

Figure 12. Bubble temperature and CO yield as functions of bed height.

bubble explosion in the early stages of bed and provides enough time to convert CO to CO2 completely. In Figure 11, CO and CO2 yields, which are defined as the ratio of their concentration to methane concentration in the feed stream, are illustrated along the reactor. According to the kinetics of combustion reactions, in the first reaction (eq 7), methane is converted to CO and then, CO is burned to CO2. Therefore, there is a maximum in CO yield, which has a considerable importance in environmental emission protocols. Further environmental analysis is achievable using our modeling method for emission predictions. The comparison between the model and the experimental data are quite satisfactory. In Figure 12, the bubble temperature and CO yield profiles are illustrated along the fluidized bed reactor at a bed temperature of 700 °C, simultaneously. As can be seen, CO production rises continuously to about the middle of the bed and decreases rapidly at the location where maximum temperature occurs. As discussed earlier, the increase of temperature confronts the inhibition effects of sand particles in bed, which leads to a continuous growth of CO concentration. High specific heat capacity of inert particles and the well-mixed behavior of emulsion phase make the temperature of this phase

Figure 13. Methane conversion along the fluidized bed reactor at two different temperatures, without considering the bubbles growth.

constant. On the other hand, there is a maximum in bubble temperature near the end of the reactor bed. This can be explained by the fact that combustion of methane is a highly 2064


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Table 5. BIAS, SI, and CC of Model Results for Bubble Diameter Variations BIAS

SI

T = 850 °C

T = 975 °C

T = 850 °C

T = 975 °C

T = 850 °C

T = 975 °C

constant, d = 0.024 m with bubble growth

−0.21 −0.14

−0.55 −0.23

0.20 0.15

0.17 0.10

0.99 0.99

0.97 0.98

Present Address

exothermic reaction, and at high temperatures, the reaction occurs near the distributor. Therefore, the heat of reaction cannot be transferred to emulsion phase because of the extreme rate of its generation, and consequently, the bubble temperature increases rapidly. However, below the critical temperature, the heat generation is damped by sand particles, which shift the temperature peak toward the freeboard section. Variation of bubble diameter is a crucial parameter in the hydrodynamics of fluidized bed combustors, which has been studied in this research. This important effect is demonstrated in Figure 13, which presents methane conversion along the fluidized bed reactor at two different temperatures, considering bubbles to be of a constant size. Neglecting the alteration of bubble diameter causes model prediction to underestimate experimental data. Model calculations are compared with experimental data for two bed temperatures and error statistics are presented in Table 5. One of the effective parameters on the variation of bubble diameter is bubble temperatures, which are inter-related by eq 1. Therefore, the assumption of having unchanged bubble size results in higher errors, especially at higher temperatures, which has been confirmed in Table 5.

‡

Department of Chemical Engineering, McGill University, Montreal, Quebec H3A 2B2, Canada Author Contributions †

These authors contributed equally to this work.

Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS Authors would like to thank Ms. Mojgan Abbasi for her comments at the beginning of this work.

4. CONCLUSION A model for fluidized bed natural gas combustion reactors was developed using several ideal reactors combined in an intelligent manner. This model consists of two submodels, namely, hydrodynamic and chemical reaction submodels, which were solved simultaneously. A typical fluidized bed reactor was divided into two sections of dense bed and freeboard. Using dynamic two-phase approach for describing the hydrodynamic behavior of the reactor, the dense bed was divided into two phases: bubble phase and emulsion phase. In each section, the gas flow was assumed to be plug flow through bubbles and completely mixed through the emulsion phase. To generalize the simulation package, a new dimensionless number (NS) for optimum stage-number identification was developed based on the logical interpretation of various affecting operating parameters, which is offered along with a generalized correlation for combustion critical temperature predictions. The predicted results by the model were compared to various experimental data sets, and a great agreement was observed between them. By means of our proposed model in this study, the prediction of the behavior of nonideal fluidized bed natural gas combustion reactors at different operating conditions becomes possible. We also suggest our outcomes to be integrated with the industrial process simulators such as Aspen Plus and Aspen HYSYS for nonlinear analyses. Finally, the whole results of this study could be used for bringing nonideal fluidized bed combustors into process simulators for further purposes such as optimization and scale up.

■

CC

bubble diam.

AUTHOR INFORMATION

Corresponding Author

*Tel.: +98 21 6697-6863. Fax: +98 21 6646-1024. E-mail: [email protected]. 2065

NOMENCLATURE A = cross-sectional area (m2) Af(1) = constant of the dynamic two-phase model Af(2) = constant of the dynamic two-phase model Af(3) = constant of the dynamic two-phase model Avoid‑b (1) = constant of the dynamic two-phase model Avoid‑b (2) = constant of the dynamic two-phase model Avoid‑b (3) = constant of the dynamic two-phase model Avoid‑e (1) = constant of the dynamic two-phase model Avoid‑e (2) = constant of the dynamic two-phase model Avoid‑e (3) = constant of the dynamic two-phase model Ar = Archimedes number, ((dp3ρg(ρp − ρg)g)/(μg2)) AR = aspect ratio BIAS = bias of data CA = concentration of component A (mol/m3) Cp = molar specific heat capacity (J/(mol.K)) Cp,m = specific heat capacity (J/(kg·K)) CC = correlation coefficient dp = particle diameter (m) D = bed diameter (m) DAB = diffusion coefficient (m2/s) Db = bubble mean diameter (m) Dr = reactor diameter (m) f = emulsion phase fraction Fb,in = flow entering the bubble phase (mol/s) Fb,out = flow exiting the bubble phase (mol/s) Fe,in = flow entering the emulsion phase (mol/s) Fe,out = flow exiting the bubble phase (mol/s) g = acceleration of gravity (m/s2) H = reactor height (m) ΔH = heat of combustion of methane (J/mol) hbe = bubble-emulsion heat-transfer coefficient (W/(m2·K)) hgp,br = gas-to-particle heat transfer coefficient based on Ubr (W/(m2·K)) Hf = bubbling bed height (m) k = thermal conductivity (W/(m·K)) Kbc = bubble-to-cloud mass transfer coefficient (1/s) Kbe = bubble-to-emulsion mass transfer coefficient (1/s) Kce = cloud-to-emulsion mass transfer coefficient (1/s) Ls = static bed height (m) n = number of stages N = number of observations NS = proposed combustion dimensionless number dx.doi.org/10.1021/ef300204j | Energy Fuels 2012, 26, 2058−2067

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Pr = Prandtl number, (Cpμ/k) R = universal gas constant (kJ mol−1 K−1) r A = reaction rate with respect to component A (mol/(m3 s)) rc = bubble radius (m) rc,∞ = bubble radius at T∞ in the absence of reaction (m) T = temperature (K) T∞ = bed temperature (K) Tb = favored temperature to calcuate bubble radius at (K) t = time (s) Ub = bubble velocity (m/s) Ubr = bubble rise velocity (m/s) Ue = emulsion gas velocity (m/s) Umf = minimum fluidization velocity (m/s) U0 = superficial gas velocity (m/s) Vb = bubble-phase volume (m3) VCSTR = emulsion-phase volume (m3) Xi = experimental values X̅ = mean value of experimental data Yi = predicted values Y̅ = mean value of predicted values z = distance from distributor (m) Greek Symbols

α = air factor δ = bubble phase fraction ε = average bed porosity εb = bubble phase porosity εe = emulsion phase porosity μg = gas viscosity (Pa.s) ρg = gas density (kg/m3) ρp = particle density (kg/m3) τ = residence time of particle in bubble (s) Subscripts

b = bubble phase c = in the absence of chemical reaction e = emulsion phase g = gas mf = evaluated at minimum fluidizing velocity p = particle

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REFERENCES

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