Investigation of Simplifying Assumptions in Mathematical Modeling of

838

Ind. Eng. Chem. Res. 2010, 49, 838–846

Investigation of Simplifying Assumptions in Mathematical Modeling of Natural Gas Dehydration Using Adsorption Process and Introduction of a New Accurate LDF Model M. Gholami and M. R. Talaie* Department of Chemical Engineering, College of Engineering, UniVersity of Isfahan, P.O. Box 81746-73441, Isfahan, Iran

In this study, a comprehensive mathematical model was developed to investigate the simplifying assumptions in modeling of natural gas dehydration by the adsorption process. In the developed model the variations of pressure, velocity, and temperature along the bed and the temperature changes inside the particles were considered. Convective heat and mass transfer were considered outside the particles and a diffusion mechanism was taken into account for the heat and mass transfer inside the particles. A dual site Langmuir model was selected to predict adsorption equilibrium and the Peng-Robinson equation of state was used to calculate the gas compressibility factor. The experimental data of Mohamadinejad et al. [Mohamadinejad, H.; et al. Sep. Sci. Technol. 2000, 35, 1] was used to verify the model. Good agreement was observed between the predictions of the comprehensive model and the experimental data. The results showed that applying the assumptions of uniform temperature distribution inside the pellet (lump method), thermal equilibrium between gas and particles, and isothermal and isobaric conditions had no significant effects on the predictions. It was also concluded that external mass transfer resistance was negligible at the industrial operating conditions of gas dehydration processes and common linear driving force models (LDFs) were not able to predict the performance of the dehydration bed well. Also, a new accurate correlation was introduced for an LDF proportionality coefficient applicable in gas dehydration systems. The predictions based on the proposed correlation were in good agreement with the results of the comprehensive model. Using this new accurate LDF model instead of diffusion model saves a large amount of CPU time without a loss of accuracy. 1. Introduction Natural gas is the one of the main sources of energy for industry, transportation, and even homes. Natural gas fed to a gas refinery is usually saturated with water vapor and needs special treatments to be dehydrated. The presence of water vapor in natural gas leads to corrosion and the deposition of gas hydrate. The adsorption process has been used widely in new installations for natural gas dehydration. The commercial molecular sieves of type A are used as the sorbent material. This type of zeolite removes water vapor from the mixed gas stream selectively. Such an adsorption process is normally performed in a tower packed with zeolite pellets under unsteady-state conditions. This process can be designed based upon data generated in a pilot plant. However, an interesting alternative approach is to use an accurate mathematical model. The mathematical model can be also employed for optimizing the operation of working units. These are the main reasons why the mathematical modeling of adsorption processes has attracted a great attention among many researchers. The large number of equations involved in a comprehensive model of the adsorption process, and hence the large amount of CPU time needed to solve them, is the main obstacle to using such a model for applied purposes. Figure 1 shows the equations involved in a comprehensive mathematical model of the adsorption process in a packed bed. Thus the simplifying assumptions must be used to increase the practical applicability of the model without reduction of accuracy. The most simplifying assumption is that internal heat and mass transfer resistances (particle inside) is negligible compared to the external ones * To whom correspondence should be addressed. Tel.: +98 311 793 4011. Fax: +98 311 793 2679. E-mail: [email protected].

(particle outside). According to this assumption local equilibrium between the gas stream flowing through the bed and adsorbed phases exists. For this case, particles are considered just as lumped sinks and sources in mass and energy conservation energy equations for fluid flowing through the bed. The next level of simplification can be obtained by using the linear driving force (LDF) model to evaluate mass transfer rate into the particles. The LDF approximation which was first proposed by Glueckauf and Coates2 transformed the partial differential equation expressing mass conservation for gas penetrating pores into an ordinary differential equation. By now, many attempts have been made to develop new correlations for the LDF proportionality coefficient. Table 1 summarizes some of these works (Nakao and Suzuki,7 Farooq and Ruthven,10 Alpay and Scott,11 Carta,12 Serbezov and Sotirchos,13 Malek and Farooq,6 Serbezov and Sotirchos,14 and Magalhaes and Mendes15). Several researchers have made efforts to evaluate the accuracy of simplified models. Table 2 summarizes some of these works (Cavenati et al.,3 Delgado et al.,4 Jee et al.,5 Malek and Farooq,6 Nakao and Suzuki,7 Park et al.,8 Sankararao and Gupta9). The main objective of this study was to develop a comprehensive mathematical model for the prediction of natural gas dehydration performance. This thorough model was then used to examine the simplifying assumptions such as thermal equilibrium and the LDF model. Because using the LDF model could reduce the computational time significantly, a new accurate LDF model was developed for the adsorption of water vapor from natural gas into 5A zeolite particles.

10.1021/ie901183q  2010 American Chemical Society Published on Web 12/09/2009

Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010

839

Figure 1. Equations used in the comprehensive model and their applications. Table 1. Research Performed To Modify the LDF Model reference

objectives 7

Nakao and Suzuki

Farooq and Ruthven10 Alpay and Scott11 Carta12 Serbezov and Sotirchos13 Malek and Farooq6

Serbezov and Sotirchos14

Magalhaes and Mendes15

They compared the LDF approximation with the numerical solution of the diffusion equation for cycle time smaller than 0.1 and proposed a graphic correlation from which the values of the LDF coefficient are determined as a function of the dimensionless time (single component adsorption). They improved an LDF model based on mass transfer resistances which considers macropore, micropore, and film resistances to mass transfer, adapted to cylindrical pellets. They developed an analytical expression based on Fourier series for the graphic correlation of Nakao and Suzuki7 (single component adsorption). He also developed an analytical expression based on Laplace transforms for the graphic correlation of Nakao and Suzuki7 (single component adsorption). They formulated a general methodology for the development of LDF approximations for multicomponent mixtures. They improved an LDF model based on mass transfer resistances. This model was satisfactorily capable of predicting breakthrough curves of methane, ethane, and propane in a bed of activated carbon and silica gel. Based on the dusty-gas model, they formulated a generalized LDF approximation. This can satisfactorily predict the overshoot in the uptake of the less adsorbable/faster moving species in a multicomponent mixture. They developed a generalized LDF approximation for adsorption of multicomponent mixtures. The orthogonal collocation method was proposed as the basis for developing generalized LDF approximations for adsorption and desorption of multicomponent mixtures in a single particle, independently of the mass transport model adopted.

2. Mathematical Model 2.1. Assumptions and Limitations. The porous volume of zeolite-type adsorbents can be separated into two parts of micropore and macropore based on their pore sizes. The pellets of the adsorbent have a relatively large pore network (macropores) for the transport of molecules to the interior where zeolite crystals (micropores) are dispersed to capture them. Thus the

mass transfer from gas phase into the micropore volume occurs via the following paths: 1. mass transfer to the surface of pellet 2. mass transfer into the macropore volume 3. mass transfer into the micropore volume The comprehensive mathematical model was developed based on the following assumptions:

840


Table 2. Research Performed To Evaluate the Accuracy of Simplifying Assumptions in Mathematical Modeling of Adsorption Processes reference

mass transfer rate model

adsorbent

Cavenati et al.3 13X and CMS3K Delgado et al.4 silicalite Park et al.8 activated carbon and 5A zeolite 5 Jeong Jee et al. activated carbon and 5A zeolite Gorbach et al.16 zeolite 4A Sankararao and 5A zeolite Gupta9

components adsorbed

LDF LDF LDF

CH4, CO2, N2 CH4, CO2 CH4, CO2, CO, H2

LDF

CH4, CO2, CO, N2, H2

)

∫

RP 2

0

r

∂qci dr ∂t

(1)

(2)

where qci is the mass of component i adsorbed into the micropore volume at r position per unit volume of pellet. The following equation expresses the component material balance over a spherical shell in the adsorbent particles:

(

)

∂Cpi 1 - εp ∂qci ∂Cpi 1 ∂ ) Deffir2 2 ∂r ∂r ε ∂t ∂t r p

(3)

This equation was used to find the composition of the gas penetrating macropore volume at each radial position. qci, which varies radially inside the particles, was obtained by using mass balance on micropore volumes existing in a spherical shell of the pellets. Using the LDF model for the mass transfer rate through micropore volumes, this mass balance equation was reduced to the following equation: 15Dci ∂qci ) (q*ci - qci) ∂t Rc2

∑b

1+

j1pj

(5)

n

∑b

j2pj

j)1

in which the constants bj (j ) 1, 2) and qsj (j ) 1, 2) depend on the temperature in the following way:

This equation was used to find the distribution of gas composition along the bed. In the above equation qpi is the mass of component i adsorbed into the pellet per unit volume of the pellet. It can be calculated by integrating mass adsorbed into the micropore volume over the spherical pellets: ∂qpi 3 ) 3 ∂t RP

bi2pi

+ qsi,2

n

j)1

1. The profiles of temperature and concentration are uniform across the cross section of the bed (bed diameter is larger than particle diameter). 2. The flow pattern is described by the plug-flow model (bed diameter is larger than particle diameter). 3. The multicomponent diffusion model in the macropore volume and a linear driving force in the micropore volume can be used to evaluate the intraparticle mass transfer rate. 4. The wall of the bed is insulated and the heat capacity of the wall is negligible. 5. The natural gas merely contains methane, carbon dioxide, nitrogen, and water vapor. 2.2. Governing Equations. Writing the gas-phase component material balance for a differential control volume of the adsorption column yields

(

bi1pi

q*ci ) qsi,1 1+

LDF water vapor pore diffusion and N2, O2 tuned LDF C2H6, CO2, N2

1 - εb ∂qpi ∂Cbi ∂(UCbi) ∂Cbi ∂ ) D ∂Z ax ∂Z ∂Z εb ∂t ∂t

zeolites. This result has been attained by comparing the predictions of these four models with more than 1000 data points of adsorption equilibrium for pure and binary and ternary mixtures of light gases on zeolites type A and X. Thus the extended dual site Langmuir isotherm for multicomponent adsorption was applied to find equilibrium concentrations (Keller17):

(4)

where q*ci is the equilibrium concentration of component i in the micropore volume. Gholami et al.18 have shown that the dual site Langmuir model was more accurate than the Langmuir, LangmuirFreundlich, and Toth models for predicting adsorption equilibrium for binary and ternary mixtures of light gases on A and X

( )

bj ) b0j exp

Ej RT

j ) 1, 2

(6)

Aij + Aij j ) 1, 2; i ) 1, 2 (7) T The Clausius-Clapeyron relation can be used to find the relationships between the heat of adsorption and the isotherm model. Based on the Clausius-Clapeyron relation, the following equation was derived for the heat of adsorption: qsj )

∆Hs )

E1qs1b1(1 + b2P)2 + E2qs2b2(1 + b1P)2 qs1b1(1 + b2P)2 + qs2b2(1 + b1P)2

(8)

Because adsorption is an exothermic process, heat is generated and hence temperature may vary radially inside the spherical adsorbent particle. The generated heat is conducted to the surface of particles and then is transferred to the gas phase by a convection mechanism. The following equations were obtained by writing the differential energy balance in the bed and inside the spherical adsorbent particles:

(

)

6(1 - εb)hb ∂(FgUCp,gTb) ∂Tb ∂ k (TP - Tb) ) ∂Z eff ∂Z ∂Z εbdp ∂ (F C T ) (9) ∂t g p,g b

(

)

∂TP ∂ kSr2 + 2 ∂r r ∂r

n

∑ ∆H

ads i

i)1

∂qci ∂ ) (FCp,STP) ∂t ∂t

(10)

The above equations were used to find the distributions of temperature both inside the particles and along the bed. As a result of pressure and temperature variations, gas density and hence gas velocity vary along the bed. The following equation expressing the overall material balance for a differential control volume of the bed was used to calculate gas velocity: -

1 - εb ∂(UCbt) ∂Z εb

n

∑ i)1

∂Cbt ∂qpi ) ∂t ∂t

(11)

Cbt, which is the gas density, was correlated to the temperature and pressure by using the Peng-Robinson equation of state. Ergun’s equation was applied to predict the pressure drop across the bed by means of the following momentum balance equation: dPbed 150µ(1 - ε)2 1.75(1 - ε) 2 )u-F u dZ dp2ε3 dpε3

(12)

2.3. Initial and Boundary Conditions. The initial and boundary conditions for the equations are summarized in Table 3. 2.4. Parameters. The parameters A11, A12, A21, A22, b01, b02, E1, and E2 appearing in eqs 6 and 7 were obtained by fitting to


841

Table 3. Initial and Boundary Equations for Governing Equations governing equation

initial and boundary conditions

mass conservation of component i in bed {gas phase, eq 1}

mass conservation of component i in pellet {gas phase, eq 3}

0 , Cpi | r,t)0 ) Cbi

∂Cpi ) 0, ∂r r)0,t

Tb | z,t)0 ) Tin,

energy conservation in pellet {solid phase, eq 10}

εpDeffi

Cbi | z)0,t ) Cini

∂Cpi ) kfi(Cbi - Csi) ∂r r)RP

0 qci | t)0 ) q* ci (Cbi)

mass conservation of component i in crystal {solid phase, eq 4}

energy conservation in bed {gas phase, eq 9}

∂Cbi ) 0, ∂Z z)L,t

0 , Cbi | z,t)0 ) Cbi

TP | z,t)0 ) Tin,

Tb | z)0,t ) Tin,

∂TP ) 0, ∂r r)0,t

-kS

∂Tb )0 ∂Z z)L,t

∂TP ) hb(TP | r)RP - Tb) ∂r r)RP

U| z)0 ) Uin

overall material balance {gas phase, eq 11}

Pbed | z)0 ) Pin

momentum balance equation {gas phase, eq 12}

Table 4. Adsorption Equilibrium Parameters of Methane, Carbon Dioxide, and Nitrogen in Zeolite component

A11 (mol K/kg)

A12 (mol/kg)

b01 (1/kPa) -7

E1 (J/mol)

reference

H2O CO2 CH4 N2

-3799.940 516.743 348.971 605.423

18.711 -0.794 0.542 -0.582

3.580 × 10 3.320 × 10-7 6.770 × 10-6 3.730 × 10-5

44 140.040 41 077.100 13 672.210 7 528.091

Mihajlo et al.19 Mulloth20 Pakseresht et al.21 Nam et al.22

component

A21 (mol K/kg)

A22 (mol/kg)

b02 (1/kPa)

E2 (J/mol)

reference

45 199.990 29 812.290 20 307.220 7 941.248

Mihajlo et al.19 Mulloth20 Pakseresht et al.21 Nam et al.22

H2O CO2 CH4 N2

3684.491 -932.131 348.971 605.423

-4.450 6.083 0.542 -0.582

Table 5. Kinetic Parameters of Water Vapor, Methane, Carbon Dioxide, and Nitrogen in Zeolite 5A Crystal (Micropore Volume) component H2O CO2 CH4 N2

D0 (m2/s) -8

2.39 × 10 5.9 × 10-11 7.20 × 10-12 5.20 × 10-13

E (J/mol)

reference

17 288.47 26 334 12 551.94 6 275.97

Paoli et al.23 Yucel and Ruthven24 Chen et al.25 Chen et al.25

experimental equilibrium data of pure methane, carbon dioxide, nitrogen, and water vapor on zeolite 5A. Table 4 summarizes the values of these parameters and the references of the experimental data. The parameters D0 and E which appear in an equation for crystal diffusivity were evaluated by fitting to experimental data. The values of these two parameters and the references of the experimental data used to obtain them are listed in Table 5. The other gas properties needed in the model were calculated based on the relations listed in Table 6 (Dongsheng and Ding,26 Poling et al.,27 and Ruthven28), where Dmi, DKi, and kf were calculated by using suitable equations and correlations. The values of these variables depend on the condition. The orders of magnitude of Dmi, DKi, and kfi are about 10-7 m2/s, 10-4 m2/ s, and 10-1 m/s, respectively. 3. Method of Solution The governing equations were solved by using the finitevolume method, power-law scheme. Uniform grid spacing was

-5

1.620 × 10 6.430 × 10-7 6.130 × 10-7 3.180 × 10-5

used to discretize the differential equations.29 The system of algebraic equations obtained by discretizing the differential equations was solved using Newton’s method. 4. Results and Discussion 1. The results obtained from the present study are discussed in terms of the following items: 2. examination of the uniform temperature assumption inside the pellet 3. examination of the thermal equilibrium assumption between pellet and gas 4. examination of isothermal and isobaric conditions in the bed 5. investigation on mass transfer resistance inside and outside particles 6. examination of common linear driving force models 4.1. Verification of the Mathematical Model. In order to validate the developed model, the calculated results were compared with the experimental data of Mohamadinejad et al.1 In this experiment an atmospheric gas stream containing H2O, CO2, and N2 has been dehydrated on a bed of 5A zeolite at ambient temperature. The ternary breakthrough curves of H2O, CO2, and N2 have been obtained in this study. Figure 2 shows the comparison of calculated results with the experimental breakthrough curves of CO2 at the midpoint of the bed and H2O at both the midpoint and the outlet of the bed. As can be seen

842


Table 6. Correlations Used To Calculate the Mass and Heat Transport Properties effective transport coefficients parameter

equation Dax ) γ1Dmi + γ2dpu

Daxi

0 Deff,i

0 Deffi

keax

(

1 1 ) + Dmi DKi

)

-1

keax ke0 ) + 0.5(Re)p(Pr) kg kg

( )

Ei Dci ) D0i exp RT

Dci

(Nu)kg hb ) dp

(Sh)Dmi dp

Dongsheng and Ding26

Ruthven28

Re ) 5000-10300

( )

kfidp Re ) 1.15 Dmi εb

Poling et al.27

Re e 4000

Nu 20.4 (Re)-0.815 ) 0.95ε (Re)(Pr)1/3

Sh ≡

Ruthven28

Ruthven28

Nu ) 2 + 1.1(Pr)1/3(Re)0.6

jH )

kfi )

reference

0.5

Ruthven28 (Sc)0.33

kfi ) ShDmi/dp

from Figure 2, there is a good agreement between the calculated results and the experimental data. In order to examine the assumptions mentioned in the first paragraph of section 4, the model was used to simulate the dehydration of a gas stream containing CH4, CO2, H2O, and N2 in a bed of zeolite 5A particles. The calculations were performed for the conditions of constant mass of sorbent and constant gas flow rate. Natural gas fed to the dehydration process is usually saturated with water vapor at high pressure. Water vapor content cannot exceed the natural gas dew point, which is a maximum 200 ppm (0.2 mol %) at industrial operating conditions. Thus the maximum value of 200 ppm for water content of wet natural gas was applied in these calculations. The computations were stopped when the

Figure 2. Comparison of comprehensive model prediction with experimental data of Mohamadinejad et al.1

concentration of water vapor in the gas stream leaving the bed rose to 5 ppm. The characteristics of the adsorption bed, sorbent, and gas flow are shown in Table 7. In the next sections the sensitivity of the computational results to the several mentioned assumptions is investigated. 4.2. Assumption of Uniform Temperature Distribution Inside the Particle (Lump Approach). In order to examine the assumption of uniform temperature distribution inside the particles, the maximum temperature difference between the center and the surface of the particles during dehydration process was considered. The maximum temperature difference was obtained equal to 0.118 °C. This small temperature difference can validate this assumption. In Figure 3 the variations of temperatures at the center and the surface of the particles along the bed are compared in the second minute after starting up the process. The second minute was selected because the maximum temperature difference between the center and the surface of the particles occurred at this time. 4.3. Local Thermal Equilibrium Assumption. In order to examine the assumption of local thermal equilibrium between gas and particles, the maximum temperature difference between the bulk of gas and the surface of the particles during dehydration process was considered. The maximum temperature difference was obtained less than 0.5 °C. Because the effect of this small temperature difference on transport properties is unimportant, the assumption of local thermal equilibrium can be truly acceptable. Figure 4 shows the axial temperature distribution for the gas and solid surface at the second minute of process. The second minute was selected because the maximum temperature difference was observed at this time.


843

Table 7. Characteristics of Adsorption Bed, Sorbent Pellet, and Gas Flow component

mole fraction

Inlet Gas Composition H2O CO2 CH4 N2

0.00184 0.00998 0.953 0.03518 Initial Values of Gas Composition in the Bed

H2O CO2 CH4 N2

0 0 0.9644 0.0356 Inlet Gas Characteristics

molar gas flow rate (kmol/h) inlet gas temperature (K) inlet gas pressure (bar)

23929 295.5 64

Bed Characteristics bed diameter (m) bed length (m) bed porosity

3.5 5.5 0.34

Figure 4. Variations of temperatures at the pellet surface and gas bulk along the bed at second minute. Table 8. Comparison between Calculated Break Times and Bed Saturation Percents Using Comprehensive Model and Based on Isothermal and Isobaric Assumptions

Pellet Characteristics pellet porosity pellet-solid density (kg/m3) pellet diameter (m) pellet heat capacity (J/kg · K) pellet thermal conductivity (W/m2 · K) mean crystal diameter (m) mean macropore diameter (m)

0.36 1812.5 0.026 1000 0.5 1 × 10-6 1.7 × 10-7

4.4. Assumptions of Isothermal and Isobaric Conditions in the Bed. In order to validate these assumptions, the break time and bed saturation percent calculated using the comprehensive model were compared with those obtained using isothermal and isobaric assumptions. The time at which the concentration of water vapor in the gas leaving the bed rose to 5 ppm is considered as the break time. Bed saturation percent is the percent of the bed which is saturated with adsorbed water at the break time. Break time and bed saturation percent were selected because these two parameters are important in industrial design of adsorption beds. The values of break time and bed saturation percent obtained using the comprehensive model and those obtained based on isothermal and isobaric assumptions are compared in Table 8. As seen in this table, these assumptions are valid for this adsorption system.

comprehensive model isothermal model isobaric model

break time (min)

saturation percent (%)

610 611 610

82.2 82.3 82.2

4.5. Investigation on Mass Transfer Resistance Inside and Outside Particles. Mass transfer from a gas stream into a sorbent pellet occurs in two steps, namely external mass transfer by convection and internal mass transfer by diffusion. The mass transfer rate model can be simplified by neglecting external resistance with respect to internal resistance. In order to examine this assumption, the break time and bed saturation percent predicted using the comprehensive model were compared with those obtained by the simplified model. The results showed that the average relative differences between the predictions of the two models for break time and bed saturation percent are about 0.82% in a wide range of industrial operational parameters. This result shows that the external mass transfer is not an important step and its resistance can be ignored. 4.6. Examination of Common Linear Driving Force Models. The LDF model which was introduced for the first time by Glueckauf and Coates2 can be obtained by simplifying the diffusion model. By now many attempts have been made to develop correlations for accurate prediction of the LDF mass transfer coefficient. However, these equations have not been justified for use for a wide range of different systems. The degree of agreement between the results of the adsorption model based on the LDF model and those of the comprehensive model can be used as a measure of LDF model accuracy. Therefore, the ability of the LDF models for predicting the performance of adsorption beds can be evaluated by comparing the results of them with the results of the diffusion model. The following four famous relations introduced in the literature were selected to examine the LDF model for the system of natural gas dehydration:

Figure 3. Variations of temperatures at the surface and center of the pellets along the bed at the second minute.

standard form of LDF model developed by Glueckauf and Coates2 15Deff KLDF ) (13) RP2

844


Table 9. Comparison of the Results of the Model Based on the Existing LDF Relations with Comprehensive Model

break time (min) saturation percent of bed (%) unused length (m)

comprehensive

Malek and Farooq6

Glueckauf and Coates2

Glueckauf30

Farooq and Ruthven10

fitted on a pellet

610 82.2 0.98

419 56.2 2.41

726 98.8 0.07

721 98.6 0.08

161 26.1 4.06

693 94.3 0.31

modified LDF model of Malek and Farooq6 RP q*i RP2 q*i Rc2 -1 KLDF ) + + 3kf Cbi 15εpDeff Cbi 15Dc

(14)

LDF model of Glueckauf30 KLDF )

5Deff

LDF model of Farooq and Ruthven10 rP2q*(Cini)τ rc2 rPq*(Cini) KLDFi ) + + 2kfiPyini 8εpDi-HePyini 15Dc

[

(15)

RP2

]

-1

(16)

The LDF model whose coefficient,KLDF, was fitted to the results of the diffusion model for a single particle, is not shown here. The parameters Deffi, kfi, and Dc were calculated using correlations listed in Table 6. In Table 9 the results of the adsorption model based on the LDF models are compared with those of the comprehensive model for natural gas dehydration. The results indicate that none of these models is capable of providing a satisfactory result. Table 9 also shows that using KLDF obtained by fitting the diffusion model for a single particle cannot predict the performance of the water vapor adsorption process accurately. 4.7. Development of a New Correlation for LDF Proportionality Coefficient for Water Vapor Adsorption. The advantage of the LDF model (much less computational time compared with the diffusion model) caused many researchers to modify and develop this model for different systems. However, the results shown in Table 9 showed that the existing LDF models are not able to predict the water vapor adsorption performance accurately. This problem can be attributed to the high polarity of water vapor, which leads to a heterogeneous adsorption. This heterogeneity affects the dynamics of adsorption and causes strong nonlinearity of the equilibrium isotherm. Figure 5 shows the variations of KLDF at different positions of the bed and times. The results shown in this figure were obtained using the comprehensive model. As shown in Figure 5, the LDF proportionality coefficient cannot be considered constant. The LDF proportionality coefficient may be related to transport coefficients, gas bulk concentration, equilibrium isotherm, and

particle size. The similar shapes of KLDF profiles at different times are due to the similar conditions of mass transfer zone which moves to the end of the bed continuously as the process proceeds. This fact leads to the idea that all profiles can coincide if they are drawn versus a suitable parameter. Figure 6 shows the plots of these profiles versus the parameter of Cwater vapor/ q*(Cwater vapor). As shown in this figure, all plots match each other. This coincidence makes it possible to correlate the LDF proportionality coefficient based on the ratio of Cwater vapor/ q*(Cwater vapor). According to this conclusion, the following dimensionless parameter was used as the correlating parameter: x)

Cwater vapor/q*(Cwater vapor) Cwater vapor inlet/q*(Cwater vapor inlet)

The calculated LDF coefficients can be satisfactorily correlated by the use of the following equation: KLDF-1 )

RP2Tin c (a + bx + + d exp(-x)) DeffyinPin x

(17)

As can be seen in eq 17, the LDF proportionality coefficient depends on the equilibrium concentration of component i in the particle, the diameter of particles, the effective diffusivity, the inlet pressure, the inlet temperature, and the inlet water mole fraction. The values of the coefficients of eq 17 are listed in Table 10. The dimensions of all parameters are Pa/K. These constants were obtained by fitting the results of the model based on the LDF model on the results of the comprehensive model for the conditions of case 1 shown in Table 11. In order to validate the proposed correlation, 17 other cases (cases 2-18 shown in Table 11) were considered. The conditions of all cases are listed in Table 11. The accurate predictions of break time and bed saturation percent could serve as a measure of proposed model accuracy. The values of break time and bed saturation percent obtained using the proposed LDF model are compared with those obtained by the comprehensive model in Table 12. The mean relative errors of 0.69% and 0.65% for break time and saturation percent, respectively, show

Figure 6. Variation of LDF coefficients versus ratio of Cwater vapor/ q*(Cwater vapor). Table 10. Coefficients of eq 17

Figure 5. Variation of LDF coefficient along the bed at different times.

a

b

c

d

327 570.286 8

-205 895.896

6 260.348 102

-343 454.455


845

Table 11. Operating Conditions for 18 Case Studies diffusion inlet coeff case inlet inlet bed particle water mole ratio Deffi/ 0 study temp (K) press. (bar) diam (m) diam (mm) fraction Deffi ) 1/τ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

295.5 295.5 295.5 288 288 288 308 308 308 295.5 295.5 295.5 295.5 295.5 295.5 295.5 295.5 295.5

64 45 80 64 45 80 64 45 80 64 64 64 64 64 64 64 64 64

3.5 3 4 3 4 3.5 4 3.5 3 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5

2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 1 1.5 2 2.5 3 2.5 2.5 2.6 2.6

0.001 84 0.001 84 0.001 84 0.001 84 0.001 84 0.001 84 0.001 84 0.001 84 0.001 84 0.001 84 0.001 84 0.001 84 0.001 84 0.001 84 0.001 84 0.001 84 0.000 92 0.003 68

1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.5 2 1 1

Table 12. Comparison of the Results of the Model Based on New LDF Relation with Comprehensive Model break time (min) case study 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 av error (%)

saturation percent (%)

comprehensive LDF rel comprehensive LDF rel model model error (%) model model error (%) 610 614 603 613 606 607 601 606 609 699 680 652 618 575 519 668 1173 309

612 631 601 619 611 603 604 614 612 707 686 657 620 575 520 671 1179 309

0.33 2.77 0.33 0.98 0.83 0.66 0.50 1.32 0.49 1.14 0.88 0.77 0.32 0.00 0.19 0.45 0.51 0.00 0.69

82.2 83.1 81.1 82.3 81.0 81.2 82.3 83.2 83.8 94.3 91.7 87.9 83.3 77.4 69.8 90.1 82.2 81.5

82.4 85.3 80.7 83.0 81.5 80.6 82.5 84.3 84.1 95.4 92.5 88.6 83.5 77.4 69.9 90.5 82.6 81.5

0.34 2.61 0.61 0.80 0.59 0.67 0.24 1.30 0.29 1.11 0.86 0.75 0.31 0.01 0.18 0.44 0.50 0.01 0.65

that this model is able to predict the results of the diffusion model for water vapor adsorption systems very well. The CPU computation time for running the comprehensive model for each case study was about 24 h, while this time when the LDF model was used reduced to about 45 min. These computation times were obtained for a Visual.Net program using a computer with a 4400 DualCore AMD Athlon X2 processor. Figure 7 shows the comparison between the results of the new LDF model with the experimental data of Mohamadinejad et al.1 As seen, there is a good agreement between the calculated results and the experimental data. 5. Conclusion In the present study, a comprehensive mathematical model was developed to investigate the simplifying assumptions in the modeling of natural gas dehydration by adsorption. The good agreement between the results of the comprehensive model and the experimental data confirms the accuracy of the model. From the results obtained, the following conclusions were drawn: 1. The temperature distribution inside the pellet is uniform. 2. The gas and pellets are in thermal equilibrium. 3. The effects of isothermal and isobaric conditions on computation results are negligible. 4. The external mass transfer resistance is negligible under industrial operating conditions.

Figure 7. Comparison of LDF model prediction with experimental data of Mohamadinejad et al.1

The results also revealed that the LDF model using the existing correlations for KLDF could not predict the mass transfer rate accurately. Using the LDF model can reduce the computation time significantly. Thus, in order to use the LDF model for accurate calculation of mass transfer rate, a new correlation was developed to predict KLDF. The predictions based on the LDF model using the proposed correlation for KLDF were in good agreement with the results of the comprehensive model. Acknowledgment This research was supported financially by Petropars Company. Nomenclature A11, A21 ) parameters in isotherm model (mol K/kg) A12, A22 ) parameters in isotherm model (mol/kg) bi ) adsorption affinity parameter of component i (1/kPa) Cbi ) concentration of component i in gas phase (bed) (mol/m3) 0 Cbi ) initial concentration of component i in gas phase (bed) (mol/ m3) Cpi ) concentration of component i in gas phase (voidage of particle) (mol/m3) Cini ) inlet concentration of component i in gas phase (mol/m3) Csi ) concentration of component i in gas phase at surface of pellet (mol/m3) Cbt ) total concentration in gas phase (mol/m3) Cp,g ) gas heat capacity (J/kg K) Cp,S ) particle heat capacity (J/kg K) Dax ) axial dispersion (m2/s) Deffi ) effective diffusivity in particle applied in eq 3 (m2/s) 0 Deffi ) effective diffusivity calculated from equations in Table 5 (m2/s) DKi ) Knudsen diffusivity in particle (m2/s) Dci ) crystalline diffusivity (m2/s) D0 ) corrected crystalline diffusivity (m2/s) db ) bed diameter (m) dp ) particle diameter (m) Ei,j ) affinity constant activation energy of component i on site j (J/mol), j ) 1, 2 hb ) gas-particle heat transfer coefficient (W/m2 K) ∆Hiads ) heat of adsorption of component i (J/mol) kzeff ) effective thermal conductivity in gas phase (W/m K) kS ) particle thermal conductivity (W/m K)

846


KLDF ) LDF model mass transfer coefficient (1/s) kfi ) external film mass transfer coefficient of component i (m/s) L ) bed length (m) Pbed ) pressure in bed (Pa) Pin ) inlet pressure (Pa) qc,i ) concentration of component i in particle crystal (mol/m3) qjci ) average concentration of component i in particle crystal (mol/ m3) qjpi ) average concentration of component i in particle (mol/m3) q*i ) equilibrium concentration of component i in particle (mol/ m3) RP ) sorbent particle radius (m) Rc ) sorbent crystal radius (m) R ) gas constant (8.314 J/mol K) t ) time (s) Tb ) gas phase temperature (K) Tin ) inlet temperature (K) TP ) particle temperature (K) u ) interstitial velocity (m/s) Uin ) inlet interstitial velocity (m/s) yin ) inlet water mole fraction Z ) bed length (m) Greek Symbols µ ) gas viscosity (Pa · s) Fg ) gas density (kg/m3) Fs ) particle skeleton density (kg/m3) εb ) bed porosity εp ) particle porosity τ ) tortuosity

Literature Cited (1) Mohamadinejad, H.; Knox, J. C.; Smith, J. E. Experimental and Numerical Investigation of Adsorption/Desorption in Packed Sorption Beds under Ideal and Nonideal Flows. Sep. Sci. Technol. 2000, 35, 1. (2) Glueckauf, E.; Coates, J. I. Theory of chromatography. Part IV. The influence of incomplete equilibrium on the front boundary of chromatograms and on the effectiveness of separation. J. Chem. Soc. 1947, 1315. (3) Cavenati, S.; Grande, C. A.; Rodrigues, A. E. Separation of CH4/ CO2/N2 mixtures by layered pressure swing adsorption for upgrade of natural gas. Chem. Eng. Sci. 2006, 61, 3893. (4) Delgado, J. A.; Uguina, M. A.; Sotelo, J. L.; Ruiz, B.; Gomez, J. M. Fixed-bed adsorption of carbon dioxide/methane mixtures on silicalite pellets. Adsorption 2006, 12, 5. (5) Jee, J. G.; Kim, M. B.; Lee, C. H. Adsorption Characteristics of Hydrogen Mixtures in a Layered Bed: Binary, Ternary, and Five-Component Mixtures. Ind. Eng. Chem. Res. 2001, 40, 868. (6) Malek, A.; Farooq, S. Comparison of isotherm models for hydrocarbon adsorption on activated carbon. AIChE J. 1996, 42, 3191. (7) Nakao, S. I.; Suzuki, M. Mass transfer coefficient in cyclic adsorption and desorption. J. Chem. Eng. Jpn. 1983, 16, 114. (8) Park, J. H.; Kim, J. N.; Cho, S. H.; Kim, J. D.; Yang, R. T. Adsorber dynamics and optimal design of layered beds for multi-component gas adsorption. Chem. Eng. Sci. 1998, 53, 3951.

(9) Sankararao, B.; Gupta, S. K. Modeling and simulation of fixed bed adsorbers (FBAs) for multi-component gaseous separations. Comput. Chem. Eng. 2007, 31, 1282. (10) Farooq, S.; Ruthven, D. M. Heat Effects in Adsorption Column Dynamics. 1. Comparison of One- and Two-Dimensional Models. Ind. Eng. Chem. Res. 1990, 29, 1076. (11) Alpay, E.; Scott, D. M. The linear driving force model for shortcycle adsorption and desorption in a spherical particle. Chem. Eng. Sci. 1992, 47, 499. (12) Carta, G. Exact solution and linear driving force approximation for cyclic mass transfer in bidisperse sorbent. Chem. Eng. Sci. 1993, 48, 1613. (13) Serbezov, A.; Sotirchos, S. V. Generalized linear driving force approximation for multicomponent pressure swing adsorption simulations. Presented at the AIChE Annual Meeting, Chicago, 1996. (14) Serbezov, A.; Sotirchos, S. V. On the formulation of linear driving force approximations for adsorption and desorption of multicomponent gaseous mixtures in sorbent particle. Sep. Purif. Technol. 2001, 24, 343. (15) Cruz, P.; Magalha˜es, F. D.; Mendes, A. Generalized linear driving force approximation for adsorption of multicomponent mixtures. Chem. Eng. Sci. 2006, 61, 3519. (16) Gorbach, A.; Stegmaier, M.; Eigenberger, G. Measurement and Modeling of Water Vapor Adsorption on Zeolite 4A-Equilibria and Kinetics. Adsorption 2004, 10, 29. (17) Keller, J. U.; Staudt, R. Gas Adsorption Equilibria: Experimental Methods and AdsorptiVe Isotherms; Springer: Boston, 2005. (18) Gholami, M.; Talaie, M. R.; Roodpeyma, Sh. Evaluation of several common adsorption models predicting natural gas component adsorption on type A and X zeolite. Farayandno 2009, 19, 2. (19) Golubovic, M. N.; Hettiarachchi, H. D. M.; Worek, W. M. Sorption properties for different types of molecular sieve and their influence on optimum dehumidification performance of desiccant wheels. Int. J. Heat Mass Transfer 2006, 49, 2802–2809. (20) Mulloth, L. M. Carbon Dioxide Adsorption on a 5A Zeolite Designed for CO2 RemoVal in Spacecraft Cabins; NASA/TM 208752; 1998. (21) Pakseresht, S.; Kazemeini, M.; Akbarnejad, M. M. Equilibrium isotherms for CO, CO2, CH4 and C2H4 on the 5A molecular sieve by a simple volumetric apparatus. Sep. Purif. Technol. 2002, 28, 53. (22) Nam, G. M.; Jeong, B. M.; Kang, S. H.; Lee, B. K.; Choi, D. K. Equilibrium Isotherms of CH4, C2H6, C2H4, N2, and H2 on Zeolite 5A Using a Static Volumetric Method. J. Chem. Eng. Data 2005, 50, 72. (23) Paoli, H. A.; Ethivier, M. H.; Krause, C.; Pfeifer, H.; Stallmach, F.; Karger, J. Comparative QENS and PFG NMR diffusion studies of water in zeolite NaCaA. Microporous Mesoporous Mater. 2002, 55, 147. (24) Yucel, H.; Ruthven, D. M. Diffusion of CO2 in 4A and 5A Zeolite. Crystals. J. Colloid Interface Sci. 1980, 74, 186. (25) Chen, N. Y.; Degnan, T. F.; Morris Smith, C., Jr. Molecular Transport and Reaction in Zeolites; VCH: New York, 1994. (26) Dongsheng, W.; Ding, Y. Heat transfer of gas flow through a packed bed. Chem. Eng. Sci. 2006, 61, 3532. (27) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids; McGraw-Hill: New York, 2000. (28) Ruthven, D. M. Principles of adsorption and adsorption processes; Wiley: New York, 1984. (29) Patankar, S. V. Numerical Heat Transfer and Fluid Flow; Hemisphere: New York, 1980. (30) Glueckauf, E. Theory of Chromatography: Part 10sFormulae for diffusion into spheres and their application to chromatography. Trans. Faraday Soc. 1955, 51, 1540.

ReceiVed for reView July 26, 2009 ReVised manuscript receiVed October 9, 2009 Accepted November 16, 2009 IE901183Q

Investigation of Simplifying Assumptions in Mathematical Modeling of

Recommend Documents