- Email: [email protected]
Computational Mass Transfer Method for Chemical Process Simulation
Chinese Journal of Chemical Engineering, 16(4) 497ü502 (2008)
PERSPECTIVES
Computational Mass Transfer Method for Chemical Process Simulation* YUAN Xigang (၏ **)رand YU Guocong (ဥڳផ)
State Key Laboratory of Chemical Engineering, Tianjin University, Tianjin 300072, China Abstract The recent works on the development of computational mass transfer (CMT) method and its applications in chemical process simulation are reviewed. Some development strategies and challenges in future research are also discussed. Keywords computational mass transfer, turbulent mass transfer diffusivity, chemical process simulation
1
INTRODUCTION
As a result of the development of computer technology beginning in the sixties of last century, the computational fluid dynamics (CFD) and the subsequent computational heat transfer (CHT) were initiated on the basis of closure of the differential equations of momentum and heat transfer respectively by the fluid dynamic researchers and mechanical engineers. The computational method provides a sound foundation for predicting the velocity and temperature fields in the engineering areas with wide application. Nevertheless, in the chemical engineering work, the prediction of concentration field is equally important, and the task of such development, which may be regarded as computational mass transfer (CMT), is naturally relied on the investigation by the chemical engineers. Recently, works on CMT have been reported covering its basic ground and various applications. Historically, the early research on the numerical simulation of concentration distribution was reported in 1960s, almost at the same time when CFD was developed [1, 2], but the simulation of concentration and temperature distributions depended largely on the pattern of velocity distribution by CFD. The computational mass transfer under investigation now aims to the prediction of concentration distribution of complex fluid systems with simultaneous mass, heat and momentum transports and/or chemical (or biochemical) reactions in chemical processes. More specifically, the CMT method should be used to predict the distributions of concentration, velocity and temperature as well as the transport parameters and operating efficiency simultaneously for the chemical equipment. The key problem of establishing the CMT is to find a method of closure for the mass transfer differential equation, just like those for closing the momentum and the heat transfer differential equations. The early approach of this problem is by the use of empirical turbulent mass transfer diffusivity [3] to calculate the concentration distribution in distillation column. Subsequently, the two equations method for the closure of mass transfer differential equation was developed a few years ago [4, 5]. Since then, the frame work of CMT was established including the funda-
mental equations and its applications to chemical engineering equipment. Undoubtedly, further investigation needs to be done in order that CMT, CFD and CHT are generally to be recognized and used as the three essentials of computational transport. This article addresses the recent work on the CMT method development and its applications in chemical engineering. Some development strategies and the challenges of future research are also discussed. 2 BASIC EQUATIONS OF MASS TRANSFER AND ITS CLOSURE For incompressible fluids, the mass transfer equation of a component species with instantaneous concentration c can be expressed as [6]: wc wc ui wt wxi
D
w 2 c Sc wxi wxi
(1)
where u is instantaneous velocity, D is the molecular diffusivity and Sc is the source term. For the turbulent flow, if substituting c C c and u U u into Eq. (1), in which C and U are the time-average values, we obtain the time-average transport equation for the concentration scalar as follows:
wC wC U j wt wx j
§ wC · (2) ¨ D wx u j c ¸ Sc j © ¹ In the foregoing equation, a new unknown term u j c , the second order covariance of the velocity and concentration appears and may be regarded as the Reynolds mass flux, analogous to Reynolds stress in the CFD. Similar to the Boussinesq postulate made in turbulent modeling for isotropic fluid dynamics, the new variable can be expressed as proportional to the gradient of average concentration [7]: wC (3) u j ci Dt,i i wx j w wx j
where Dt,i is termed as turbulent diffusivity of component i, which is not a constant but related with the velocity, temperature, component concentration and structure of the flow field. Hence, the evaluation of the turbulent diffusivity plays a vital role in solving
Received 2008-05-26, accepted 2008-06-03. * Supported by the National Natural Science Foundation of China (20736005). ** To whom correspondence should be addressed. E-mail: [email protected]
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the equations. Traditionally, the turbulent diffusivity is usually estimated empirically by simple analogy to the turbulent viscosity by assuming that the turbulent mass transfer is simply related to the turbulent momentum transfer. This empirical analogy leads to a simple expression for the turbulent diffusivity as Dt, j P t / Sct , where Pt is a turbulent viscosity which can be acquired by using a CFD model such as k-H model, and Sct is the Schmidt number, usually assuming to be a constant taken between 0.7 and 1.0. Wang et al. [3] used this approach combined with the volume averaged pseudo-single-liquid model to obtain the three-dimensional concentration profile on the trays of an industrial scale distillation column and evaluate the corresponding tray efficiencies. Another empirical approach to estimate Dt,i is from the Peclet Number (Pe), which is equal to the product of a characteristic velocity and characteristic length divided by the turbulent diffusivity. The Peclet number, traditionally applied to represent the back-mixing property of the fluid, is usually determined experimentally by the inert tracer technique in a non-reactive flow and correlated as function of Reynolds number, physical properties and the characteristic dimension of equipment. Such method, however, is still empirical. The two empirical approaches mentioned above for estimating Dt,i need no additional differential equation, and are termed as zero-equation models for the closure of mass transfer equation Eq. (1). The main shortcoming of zero-equation models is that Dt,i is considered only related with the fluid velocity fluctuation. Theoretically, Dt,i is also dependent on the concentration fluctuation. Based on this idea and in reference to the CFD and CHT treatments, Liu [4, 5] suggested that Dt,i is proportional to the product of characteristic velocity and characteristic length, or mathematically: 1
Dt
(4)
Ct k 2 Lm
ables k and İ can be obtained by the k-İ model in CFD. The other two variables c 2 and İc are obtained by solving corresponding equations. Liu et al. [4, 5] derived the exact c 2 and İc equations and the modeling forms, which were further simplified by Sun et al. [8] as follows:
c 2 equation: wc 2 wc 2 Ui wt wxi
2 ª§ Dt · wc º D¸ «¨ » ¹ wxi ¼ ¬© V c wC wC 2 Dt 2H c wxi wxi
w wxi
(6)
İc equation: wH c wH Ui c wt wxi
ª§ Dt · wH c º «¨ V D ¸ wx » «¬© H c ¹ i »¼ H H wC Cc1 c cuic Cc 2 H c 2 k wxi c w wxi
(7)
where D and Dt are the molecular and turbulent diffusivities respectively, with ıc, V H c , C1 and C2 as constants. Eqs. (6), (7) and (2) constitute the c 2 -İc model for the closure of Eq. (3), and are the fundamental part of the computational mass transfer. The foregoing c 2 -İc model involves the variables k and İ which can be solved by the CFD method for turbulent flow consisting of five equations, namely the continuity equation, the moment equation, the Boussinesq’s Dt equation, the k equation and the İ equation. If the process involves heat effect, such as the chemical absorption or exothermic catalytic chemical reaction, the effect of temperature distribution cannot be ignored. In this case, the CHT model, which involves the heat transport equation, the Boussinesq’s
where k is the turbulent kinetic energy, equal to uicuic / 2 , and its square root represents the characteristic velocity, Lm is the characteristic length represented by k1/2/IJm where the mixed time scale IJm is taken as geometric average W PW c in which W P and W c are the
thermal diffusivity Dh equation, the t 2 equation and the İt equation, should be accompanied.
dissipation time scale of the velocity and concentration fluctuations. In CFD, we have W P k / H where İ
Computational mass transfer equation system should be solved by coupling the momentum, heat and mass transport; therefore it consists of three parts: (i) Equation set of mass transfer and its closure, including Eqs. (2), (3), (6) and (7), for predicting the concentration distribution. (ii) Equation set of CFD including five equations as mentioned above for the purpose to obtain the k, İ and velocity distribution. (iii) Equation set for heat transfer: including four equations as mentioned above for the purpose to obtain the temperature distribution. These three parts should be solved simultaneously as the concentration field and temperature field are both affected by the velocity field. It is noted that
is the dissipation rate of concentration fluctuation. Similarly, we may let W c c 2 / H c where c 2 is the variance of concentration fluctuation and İc is the dissipation rate of concentration fluctuation. Then Eq. (4) becomes: 1
Dt
§ kc 2 · 2 ¸¸ Ct k ¨¨ © HH c ¹
(5)
It can be seen that in Eq. (5), there are four variables besides the proportional constant Ct. The vari-
3 COMPUTATIONAL EQUATION SYSTEM
MASS
TRANSFER
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499
Figure 1 Equation system for CMT
under some nearly isothermal conditions, such as in a single distillation tray, the heat transfer equations can be neglected for simplification. It can be seen that CMT methodology involves solving a large number of differential equations and thus the computer load is heavy, but it helps to use more efficient commercial software. The interrelationship of equation system can be represented by the diagram shown in Fig. 1. 4 THE MODELING OF CHEMICAL PROCESS BY COMPUTATIONAL MASS TRANSFER Most chemical processes are under the condition of two flowing phases, such as gas/vapor-liquid flow in the distillation column or the catalytic reactor. In such cases of two-phase flow, the modeling equations of each phase should be established. The number of equations is then double and the computation becomes extremely complicated. Methods of simplification are being sought. Roughly speaking, they can be classified into three categories: pseudo-single-fluid model, mixed-fluid model and two-fluid model. The pseudo-single-fluid model, proposed originally for distillation trays, assumes that the characteristic of two-phase flow can be represented by an equivalent single phase flow with the account for the interacting force by the other. The interaction between phases, represented by a proper constitutive equation, is inserted to the CFD equations as a source term. This model has been used successfully by many investigators [3, 5, 9, 10] for the simulation of two-phase flow in tray/packed distillation column. The advantage of this model is that it is simple and able to simulate counter-current or cross-current two-phase flow, provided the proper expression of source terms. The mixed-fluid model assumes the two-phase mixture be an equivalent mixture of coexisting gas and liquid with weighted average velocity and other parameters. Besides, the interfacial action still need to be accounted for. The advantage of the mixed fluid model is particularly suitable to co-current flows. The two-fluid model formulates the respective equations for each interacting fluid and consequently
the number of equations needed to be solved is doubled, although it is the best method for the numerical simulation. Its advantage is able to obtain the concentration distribution for each phase simultaneously. For the chemical process with solid phase, such as in the case of packed column or catalytic reactor, the simulation can be done with the help of using the Representative Elementary Volume (REV) method [11]. By this approach, the gas-liquid-solid three-phase space is divided into a number of virtual REV, the size of which is large enough to represent the character of three phases configuration, but yet small enough in the flow field as an elementary volume of discretization, as shown in Fig. 2. This approach has been successfully used for simulating the transport processes in packed absorption/distillation column [9, 10] and catalytic reactor [12].
Figure 2
5 5.1
A representative elementary volume
APPLICATIONS Distillation
For the distillation process using tray columns, the temperature of fluid on each tray can be considered nearly constant and the equations in the heat transfer part of CMT methodology may be neglected in order to simplify the computation. However, the temperature of each individual tray is changing from the bottom to the top of the column. With such idea, Sun et al. [5, 8] simulated an industrial scale sieve tray column for cyclohexane and n-heptane separation [13] in FRI (Fractionation Research Institute) by using
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CMT method with c 2 - H c model to obtain the velocity and concentration distributions of each tray and its Murphree tray efficiency. Fig. 3 gives some computational results. It is seen from the figure that the simulated result in outlet concentration of each tray is closely checked by the experiment except on tray No.4 where experimental error is obvious.
(a) Concentration distribution for C6 (a) Velocity distribution for liquid phase
(b) Comparison with experiments for HETP (b) Concentration distribution for cyclohexane
Figure 4 Computation results by CMT with c 2 - H c model for packed distillation column [14]
umns. With the c 2 - H c , k-H and t 2 - H t models to be used in CMT formulation, their simulated results include not only radial and axial concentration, temperature and velocity distributions but also the mass transfer diffusivity, thermal diffusivity and the enhancement factor. All those are in agreement with the experimental data. (c) Comparison with experiments for trays’ outlet concentrations Figure 3 Computation results by CMT with c 2 - H c model for distillation tray [5]
The simulation by Liu et al. [14] using the CMT method to a commercial scale distillation column packed with 50.8 mm pall ring in FRI [15] for C6-nC7 separation is shown in Fig. 4, in which the radial concentration distribution and HETP are clearly seen. 5.2
Chemical absorption
The chemical absorption process usually involves heat effect, and all three parts of CMT equation system should be applied. Liu et al. [9, 10] simulated the absorption of CO2 by using NaOH and MEA (monoethanolamine) separately as absorbent in packed col-
5.3
Catalytic reaction
Liu et al. [9] applied the CMT methodology to a more complicated case, an exothermic fixed bed catalytic reactor with cooling jacket for the synthesis of vinyl acetate from acetic acid and acetylene [16]. Like in the case of chemical absorption, the concentration, temperature and velocity distributions as well as their respective diffusivities are all obtained at once as shown in Fig. 5. It can be seen that there are notable differences among the values of the three kinds of diffusivities in both radial and axial directions. Thereby no simple relationship is seen to be existing between these diffusivities. In other words, the Schmidt number Sc and the Peclet number Pe are not constant throughout. Hence the traditional empirical assumption of assuming constant Sc and Pe is unjustified and may lead to serious error.
Chin. J. Chem. Eng., Vol. 16, No. 4, August 2008
(a) Turbulent viscosity
(b) Turbulent diffusivity of acetylene
501
pirical correlation. The theoretical prediction from microscopic viewpoint is a challenge to the chemical engineers and scientists. Interfacial turbulence The controlling factor of mass transfer rate is usually considered to be the resistance between the surface and the main fluid, and the facial effect is always being neglected. It has been reported that in many cases, interfacial mass-transfer is accompanied by interfacial convection (or interfacial turbulence). When the liquid surface tension gradient on the interface formed by the concentration difference increases to a certain extent, the interface turns out to be unstable, and consequently creates interface convection which is regarded as Marangoni effect [17, 18]. Similarly, the convection induced by density gradient is called as Rayleigh effect [19]. It has been proven that the existence of the interface convection has an obvious effect on interface mass-transfer [20], for instance, it may increase the rate by several folds. Thus, further research on interface convection phenomenon and its impact to the mass-transfer rate is an important fundamental research work. Multi-component mass transfer Many chemical processes deal with multi-component system. The characteristic of such system is different from the binary mixture as the molecular interaction is more complicated. For instance, the mass transfer efficiency of a component in multi-component mixture may be either much higher or much lower than one, which is entirely different from binary mixture. At present, the theoretical ground to solve the multi-component mass transfer is by the application of Maxwell-Stefen equation. Yet the solution of this equation is based on various simplified assumptions and suitable only for some cases. Therefore, further investigation is needed. 6.2 Modeling of the multi-phase flow and relevant transfer parameters
(c) Turbulent thermal diffusivity Figure 5
Transport properties of the fixed bed reactor [14]
Thus, an obvious advantage of CMT methodology is able to predict all the relevant parameters without using empirical assumption or correlation. 6 CHALLENGES AND RESEARCH PERSPECTIVES The success of CMT development reveals the fact that the computational technique could be used more extensively to the chemical engineering area in order to achieve the goal of simulating chemical process and equipment on the basis of more theoretical approach rather than experimental and/or empirical ones. In that direction, we are facing a number of challenges, shown as follows. 6.1
Interface mass-transfer
At present, the evaluation of mass transfer rate between phases is based on the film concept and em-
As stated above, there are three categories of models can be used to simulate multi-phase flow. Although the pseudo-single-phase model is simpler and suitable to many cases as shown in previous sections, more accurate model is still need to investigate. The explicit two-phase flow approach, such as VOF method, can hardly be applied at present for simulation due to the increased heavy computation load. Certainly, the development of more efficient approaches to handle explicitly two-phase flow is of great interest. Furthermore, the current CMT model depends on the reliability of estimating such vital parameters as the inter-phase contact area, the mass transfer rate, etc. The study on these aspects by theoretical or semi-theoretical means is urged. 6.3
Anisotropic problem
In present CMT methodology, the closure of turbulent Navier-Stokes equation is by applying the Boussinesq’s postulate and the isotropic assumption. But in some occasions, such as the boundaries have strong impact on the fluid flow, the transport in porous medium with clear orientations, and the cyclic flow, etc., the
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isotropic assumption is invalid, and may produce considerable error. Therefore it is necessary to develop reliable approaches for the CMT under anisotropic flow, such as those employed in the CFD methodology. 6.4
Multi-scale modeling
As a trend, the depth modeling of chemical process should be considered in multi-scale. For instance, the interface convection occurs on a mesoscopic scale, which affects the macroscopic rate of mass transfer. The Lattice-Boltzmann method to be used in recent years for simulating fluid flow and heat/mass transfer in the complex systems displays significant advantages in describing the transport phenomena from particle distribution to macroscopic scale. More understanding in microscopic and mesoscopic levels of mass transfer will make the modeling on the sound scientific basis. 6.5 Closure of mass transfer equation The use of c 2 - H c model for the closure of mass transfer differential equation has been proven to be successful as a base of CMT methodology. However, this is not the only way to close the mass transfer equation or to solve the unknown parameter u cc . More method of solution is under investigation. Briefly, the perspectives and challenges discussed above are mainly based on CMT investigation that has been made. The exploration on novel strategies for higher level of modeling is still in need and of significance. 7
CONCLUSIONS
Computational mass transfer as a method for chemical process simulation integrates the advances in CFD and transport theories. The recent development
staff in the State Key Laboratories of Chemical Engineering (Tianjin University). REFERENCES 1
2
3
4
5
6 7
8
9
10
11
12
13
2
of c - H c model for closing the mass transfer differential equation constitutes the ground work of CMT methodology for chemical process simulation. The feature of CMT method is able to simulate the concentration, temperature and velocity fields as well as the transport diffusivities and operating efficiency at once without depending on the assumption of constant Sc/Pr number or empirical correlations. The present CMT method has been applied successfully to distillation, chemical absorption and exothermic catalytic reaction processes. However, the CMT method is now still in the developing stage, further investigation is necessary in order to make it relying on the basis of more theoretical and less empirical in order to achieve more reliable and accurate simulation. ACKNOWLEDGEMENTS
The authors acknowledge the assistance from the
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20
McFarlane, R.C., Muller, T.D., Miller, F.G., “Unsteady-state distribution of fluid composition in two-phase oil reservoirs undergoing gas injection”, Soc. Petrol. Eng. J., 7 (1), 6174 (1967). We, C., Farouqali, S.M., Stahl, C.D., “Experimental and numerical simulation of two-phase flow with interface mass transfer in one and two dimensions”, Soc. Petrol. Eng. J., 9 (3), 323337 (1969). Wang, X.L., Liu, C.J., Yuan, X.G., Yu, K.T., “Computational fluid dynamics simulation of three-dimensional liquid flow and mass transfer on distillation column trays”, Ind. Eng. Chem. Res., 43, 25562567 (2004). Liu, B.T., “Study of a new mass transfer model of CFD and its application on distillation tray”, Ph.D. Thesis, Tianjin University, Tianjin, China (2003). (in Chinese) Sun, Z.M., Liu, B.T., Yuan, X.G., Yu, K.T., “New turbulent model for computational mass transfer and its application to a commercial-scale distillation column”, Ind. Eng. Chem. Res., 44 (12), 44274434 (2005). Bird, R.B., Steward, W.E., Lightfoot, E.N., Transport Phenomena, 2nd ed., John Wiley & Sons (2002). Rosén, C., Trägårdh, C., “Prediction of turbulent high Schmidt number mass transfer using a low Reynolds number k-İ turbulent model”, Chem. Eng. J., 59, 153159 (1995). Sun, Z.M., Yu, K.T., Yuan, X.G., Liu, C.J., “A modified model of computational mass transfer for distillation column”, Chem. Eng. Sci., 62, 18391850 (2007). Liu, G.B., Yu, K.T., Yuan, X.G., Liu, C.J., Guo, Q.C., “Simulations of chemical absorption in pilot-scale and industrial-scale packed columns by computational mass transfer”, Chem. Eng. Sci., 61, 65116529 (2006). Liu, G.B., Yu, K.T., Yuan, X.G., Liu, C.J., “New model for turbulent mass transfer and its application to the simulations of a pilot-scale randomly packed column for CO2-NaOH chemical absorption”, Ind. Eng. Chem. Res., 45, 32203229 (2006). Bachmat, Y., Bear, J., “Macroscopic modelling of transport phenomena in porous media (1) The continuum approach”, Transport Porous Media, 1 (3), 213240 (1986). Liu, G.B., Yu, K.T., Yuan, X.G., Liu, C.J., “A computational transport model for wall-cooled catalytic reactor”, Ind. Eng. Chem. Res., 47, 26562665 (2008). Sakata, M., Yanagi, T., “Performance of a commercial scale sieve tray”, In: Institution of Chemical Engineers Symposium Series, 3.2, American Institute of Chemical Engineers, New York, 2134 (1979). Liu, G.B., “Computational transport and its application to mass transfer and reaction processes in packed-bed”, Ph.D. Thesis, Tianjin University, Tianjin, China (2006). (in Chinese) Shariat, A., Kunesh, J.G., “Packing efficiency testing on a commercial scale with good (and not so good) reflux distribution”, Ind. Eng. Chem. Res., 34 (4), 12731279 (1995). Valstar, J.M., van den Berg, P.J., Oyserman, J., “Comparison between two dimensional fixed bed reactor calculations and measurements”, Chem. Eng. Sci., 30 (7), 723728 (1975). Sun, Z.F., Yu., K.T., Wang, S.Y., Miao, Y.Z., “Absorption and desorption of carbon dioxide into and from organic solvents: Effects of Rayleigh and Marangoni instability”, Ind. Eng. Chem. Res., 41, 19051913 (2002). Sha, Y., Cheng, H., Yuan, X.G., Yu, K.T., “Kinetics of mass transfer accompanied by Marangoni effect”, J. Chem. Ind. Eng. (China), 54 (11), 15181523 (2003). (in Chinese) Sha, Y., Yuan, X.G., Cheng, H., Yu, K.T., “Rayleigh-Bénard-Marangoni effect in mass transfer”, Chemical Engineering (China), 31 (5), 813 (2003). (in Chinese) Yu, L.M., Zeng, A.W., Yu, K.T., “Effect of interfacial velocity fluctuations on the enhancement of the mass-transfer process in falling-film flow”, Ind. Eng. Chem. Res., 45, 12011210 (2006).
PERSPECTIVES
Computational Mass Transfer Method for Chemical Process Simulation* YUAN Xigang (၏ **)رand YU Guocong (ဥڳផ)
State Key Laboratory of Chemical Engineering, Tianjin University, Tianjin 300072, China Abstract The recent works on the development of computational mass transfer (CMT) method and its applications in chemical process simulation are reviewed. Some development strategies and challenges in future research are also discussed. Keywords computational mass transfer, turbulent mass transfer diffusivity, chemical process simulation
1
INTRODUCTION
As a result of the development of computer technology beginning in the sixties of last century, the computational fluid dynamics (CFD) and the subsequent computational heat transfer (CHT) were initiated on the basis of closure of the differential equations of momentum and heat transfer respectively by the fluid dynamic researchers and mechanical engineers. The computational method provides a sound foundation for predicting the velocity and temperature fields in the engineering areas with wide application. Nevertheless, in the chemical engineering work, the prediction of concentration field is equally important, and the task of such development, which may be regarded as computational mass transfer (CMT), is naturally relied on the investigation by the chemical engineers. Recently, works on CMT have been reported covering its basic ground and various applications. Historically, the early research on the numerical simulation of concentration distribution was reported in 1960s, almost at the same time when CFD was developed [1, 2], but the simulation of concentration and temperature distributions depended largely on the pattern of velocity distribution by CFD. The computational mass transfer under investigation now aims to the prediction of concentration distribution of complex fluid systems with simultaneous mass, heat and momentum transports and/or chemical (or biochemical) reactions in chemical processes. More specifically, the CMT method should be used to predict the distributions of concentration, velocity and temperature as well as the transport parameters and operating efficiency simultaneously for the chemical equipment. The key problem of establishing the CMT is to find a method of closure for the mass transfer differential equation, just like those for closing the momentum and the heat transfer differential equations. The early approach of this problem is by the use of empirical turbulent mass transfer diffusivity [3] to calculate the concentration distribution in distillation column. Subsequently, the two equations method for the closure of mass transfer differential equation was developed a few years ago [4, 5]. Since then, the frame work of CMT was established including the funda-
mental equations and its applications to chemical engineering equipment. Undoubtedly, further investigation needs to be done in order that CMT, CFD and CHT are generally to be recognized and used as the three essentials of computational transport. This article addresses the recent work on the CMT method development and its applications in chemical engineering. Some development strategies and the challenges of future research are also discussed. 2 BASIC EQUATIONS OF MASS TRANSFER AND ITS CLOSURE For incompressible fluids, the mass transfer equation of a component species with instantaneous concentration c can be expressed as [6]: wc wc ui wt wxi
D
w 2 c Sc wxi wxi
(1)
where u is instantaneous velocity, D is the molecular diffusivity and Sc is the source term. For the turbulent flow, if substituting c C c and u U u into Eq. (1), in which C and U are the time-average values, we obtain the time-average transport equation for the concentration scalar as follows:
wC wC U j wt wx j
§ wC · (2) ¨ D wx u j c ¸ Sc j © ¹ In the foregoing equation, a new unknown term u j c , the second order covariance of the velocity and concentration appears and may be regarded as the Reynolds mass flux, analogous to Reynolds stress in the CFD. Similar to the Boussinesq postulate made in turbulent modeling for isotropic fluid dynamics, the new variable can be expressed as proportional to the gradient of average concentration [7]: wC (3) u j ci Dt,i i wx j w wx j
where Dt,i is termed as turbulent diffusivity of component i, which is not a constant but related with the velocity, temperature, component concentration and structure of the flow field. Hence, the evaluation of the turbulent diffusivity plays a vital role in solving
Received 2008-05-26, accepted 2008-06-03. * Supported by the National Natural Science Foundation of China (20736005). ** To whom correspondence should be addressed. E-mail: [email protected]
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the equations. Traditionally, the turbulent diffusivity is usually estimated empirically by simple analogy to the turbulent viscosity by assuming that the turbulent mass transfer is simply related to the turbulent momentum transfer. This empirical analogy leads to a simple expression for the turbulent diffusivity as Dt, j P t / Sct , where Pt is a turbulent viscosity which can be acquired by using a CFD model such as k-H model, and Sct is the Schmidt number, usually assuming to be a constant taken between 0.7 and 1.0. Wang et al. [3] used this approach combined with the volume averaged pseudo-single-liquid model to obtain the three-dimensional concentration profile on the trays of an industrial scale distillation column and evaluate the corresponding tray efficiencies. Another empirical approach to estimate Dt,i is from the Peclet Number (Pe), which is equal to the product of a characteristic velocity and characteristic length divided by the turbulent diffusivity. The Peclet number, traditionally applied to represent the back-mixing property of the fluid, is usually determined experimentally by the inert tracer technique in a non-reactive flow and correlated as function of Reynolds number, physical properties and the characteristic dimension of equipment. Such method, however, is still empirical. The two empirical approaches mentioned above for estimating Dt,i need no additional differential equation, and are termed as zero-equation models for the closure of mass transfer equation Eq. (1). The main shortcoming of zero-equation models is that Dt,i is considered only related with the fluid velocity fluctuation. Theoretically, Dt,i is also dependent on the concentration fluctuation. Based on this idea and in reference to the CFD and CHT treatments, Liu [4, 5] suggested that Dt,i is proportional to the product of characteristic velocity and characteristic length, or mathematically: 1
Dt
(4)
Ct k 2 Lm
ables k and İ can be obtained by the k-İ model in CFD. The other two variables c 2 and İc are obtained by solving corresponding equations. Liu et al. [4, 5] derived the exact c 2 and İc equations and the modeling forms, which were further simplified by Sun et al. [8] as follows:
c 2 equation: wc 2 wc 2 Ui wt wxi
2 ª§ Dt · wc º D¸ «¨ » ¹ wxi ¼ ¬© V c wC wC 2 Dt 2H c wxi wxi
w wxi
(6)
İc equation: wH c wH Ui c wt wxi
ª§ Dt · wH c º «¨ V D ¸ wx » «¬© H c ¹ i »¼ H H wC Cc1 c cuic Cc 2 H c 2 k wxi c w wxi
(7)
where D and Dt are the molecular and turbulent diffusivities respectively, with ıc, V H c , C1 and C2 as constants. Eqs. (6), (7) and (2) constitute the c 2 -İc model for the closure of Eq. (3), and are the fundamental part of the computational mass transfer. The foregoing c 2 -İc model involves the variables k and İ which can be solved by the CFD method for turbulent flow consisting of five equations, namely the continuity equation, the moment equation, the Boussinesq’s Dt equation, the k equation and the İ equation. If the process involves heat effect, such as the chemical absorption or exothermic catalytic chemical reaction, the effect of temperature distribution cannot be ignored. In this case, the CHT model, which involves the heat transport equation, the Boussinesq’s
where k is the turbulent kinetic energy, equal to uicuic / 2 , and its square root represents the characteristic velocity, Lm is the characteristic length represented by k1/2/IJm where the mixed time scale IJm is taken as geometric average W PW c in which W P and W c are the
thermal diffusivity Dh equation, the t 2 equation and the İt equation, should be accompanied.
dissipation time scale of the velocity and concentration fluctuations. In CFD, we have W P k / H where İ
Computational mass transfer equation system should be solved by coupling the momentum, heat and mass transport; therefore it consists of three parts: (i) Equation set of mass transfer and its closure, including Eqs. (2), (3), (6) and (7), for predicting the concentration distribution. (ii) Equation set of CFD including five equations as mentioned above for the purpose to obtain the k, İ and velocity distribution. (iii) Equation set for heat transfer: including four equations as mentioned above for the purpose to obtain the temperature distribution. These three parts should be solved simultaneously as the concentration field and temperature field are both affected by the velocity field. It is noted that
is the dissipation rate of concentration fluctuation. Similarly, we may let W c c 2 / H c where c 2 is the variance of concentration fluctuation and İc is the dissipation rate of concentration fluctuation. Then Eq. (4) becomes: 1
Dt
§ kc 2 · 2 ¸¸ Ct k ¨¨ © HH c ¹
(5)
It can be seen that in Eq. (5), there are four variables besides the proportional constant Ct. The vari-
3 COMPUTATIONAL EQUATION SYSTEM
MASS
TRANSFER
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Figure 1 Equation system for CMT
under some nearly isothermal conditions, such as in a single distillation tray, the heat transfer equations can be neglected for simplification. It can be seen that CMT methodology involves solving a large number of differential equations and thus the computer load is heavy, but it helps to use more efficient commercial software. The interrelationship of equation system can be represented by the diagram shown in Fig. 1. 4 THE MODELING OF CHEMICAL PROCESS BY COMPUTATIONAL MASS TRANSFER Most chemical processes are under the condition of two flowing phases, such as gas/vapor-liquid flow in the distillation column or the catalytic reactor. In such cases of two-phase flow, the modeling equations of each phase should be established. The number of equations is then double and the computation becomes extremely complicated. Methods of simplification are being sought. Roughly speaking, they can be classified into three categories: pseudo-single-fluid model, mixed-fluid model and two-fluid model. The pseudo-single-fluid model, proposed originally for distillation trays, assumes that the characteristic of two-phase flow can be represented by an equivalent single phase flow with the account for the interacting force by the other. The interaction between phases, represented by a proper constitutive equation, is inserted to the CFD equations as a source term. This model has been used successfully by many investigators [3, 5, 9, 10] for the simulation of two-phase flow in tray/packed distillation column. The advantage of this model is that it is simple and able to simulate counter-current or cross-current two-phase flow, provided the proper expression of source terms. The mixed-fluid model assumes the two-phase mixture be an equivalent mixture of coexisting gas and liquid with weighted average velocity and other parameters. Besides, the interfacial action still need to be accounted for. The advantage of the mixed fluid model is particularly suitable to co-current flows. The two-fluid model formulates the respective equations for each interacting fluid and consequently
the number of equations needed to be solved is doubled, although it is the best method for the numerical simulation. Its advantage is able to obtain the concentration distribution for each phase simultaneously. For the chemical process with solid phase, such as in the case of packed column or catalytic reactor, the simulation can be done with the help of using the Representative Elementary Volume (REV) method [11]. By this approach, the gas-liquid-solid three-phase space is divided into a number of virtual REV, the size of which is large enough to represent the character of three phases configuration, but yet small enough in the flow field as an elementary volume of discretization, as shown in Fig. 2. This approach has been successfully used for simulating the transport processes in packed absorption/distillation column [9, 10] and catalytic reactor [12].
Figure 2
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A representative elementary volume
APPLICATIONS Distillation
For the distillation process using tray columns, the temperature of fluid on each tray can be considered nearly constant and the equations in the heat transfer part of CMT methodology may be neglected in order to simplify the computation. However, the temperature of each individual tray is changing from the bottom to the top of the column. With such idea, Sun et al. [5, 8] simulated an industrial scale sieve tray column for cyclohexane and n-heptane separation [13] in FRI (Fractionation Research Institute) by using
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CMT method with c 2 - H c model to obtain the velocity and concentration distributions of each tray and its Murphree tray efficiency. Fig. 3 gives some computational results. It is seen from the figure that the simulated result in outlet concentration of each tray is closely checked by the experiment except on tray No.4 where experimental error is obvious.
(a) Concentration distribution for C6 (a) Velocity distribution for liquid phase
(b) Comparison with experiments for HETP (b) Concentration distribution for cyclohexane
Figure 4 Computation results by CMT with c 2 - H c model for packed distillation column [14]
umns. With the c 2 - H c , k-H and t 2 - H t models to be used in CMT formulation, their simulated results include not only radial and axial concentration, temperature and velocity distributions but also the mass transfer diffusivity, thermal diffusivity and the enhancement factor. All those are in agreement with the experimental data. (c) Comparison with experiments for trays’ outlet concentrations Figure 3 Computation results by CMT with c 2 - H c model for distillation tray [5]
The simulation by Liu et al. [14] using the CMT method to a commercial scale distillation column packed with 50.8 mm pall ring in FRI [15] for C6-nC7 separation is shown in Fig. 4, in which the radial concentration distribution and HETP are clearly seen. 5.2
Chemical absorption
The chemical absorption process usually involves heat effect, and all three parts of CMT equation system should be applied. Liu et al. [9, 10] simulated the absorption of CO2 by using NaOH and MEA (monoethanolamine) separately as absorbent in packed col-
5.3
Catalytic reaction
Liu et al. [9] applied the CMT methodology to a more complicated case, an exothermic fixed bed catalytic reactor with cooling jacket for the synthesis of vinyl acetate from acetic acid and acetylene [16]. Like in the case of chemical absorption, the concentration, temperature and velocity distributions as well as their respective diffusivities are all obtained at once as shown in Fig. 5. It can be seen that there are notable differences among the values of the three kinds of diffusivities in both radial and axial directions. Thereby no simple relationship is seen to be existing between these diffusivities. In other words, the Schmidt number Sc and the Peclet number Pe are not constant throughout. Hence the traditional empirical assumption of assuming constant Sc and Pe is unjustified and may lead to serious error.
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(a) Turbulent viscosity
(b) Turbulent diffusivity of acetylene
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pirical correlation. The theoretical prediction from microscopic viewpoint is a challenge to the chemical engineers and scientists. Interfacial turbulence The controlling factor of mass transfer rate is usually considered to be the resistance between the surface and the main fluid, and the facial effect is always being neglected. It has been reported that in many cases, interfacial mass-transfer is accompanied by interfacial convection (or interfacial turbulence). When the liquid surface tension gradient on the interface formed by the concentration difference increases to a certain extent, the interface turns out to be unstable, and consequently creates interface convection which is regarded as Marangoni effect [17, 18]. Similarly, the convection induced by density gradient is called as Rayleigh effect [19]. It has been proven that the existence of the interface convection has an obvious effect on interface mass-transfer [20], for instance, it may increase the rate by several folds. Thus, further research on interface convection phenomenon and its impact to the mass-transfer rate is an important fundamental research work. Multi-component mass transfer Many chemical processes deal with multi-component system. The characteristic of such system is different from the binary mixture as the molecular interaction is more complicated. For instance, the mass transfer efficiency of a component in multi-component mixture may be either much higher or much lower than one, which is entirely different from binary mixture. At present, the theoretical ground to solve the multi-component mass transfer is by the application of Maxwell-Stefen equation. Yet the solution of this equation is based on various simplified assumptions and suitable only for some cases. Therefore, further investigation is needed. 6.2 Modeling of the multi-phase flow and relevant transfer parameters
(c) Turbulent thermal diffusivity Figure 5
Transport properties of the fixed bed reactor [14]
Thus, an obvious advantage of CMT methodology is able to predict all the relevant parameters without using empirical assumption or correlation. 6 CHALLENGES AND RESEARCH PERSPECTIVES The success of CMT development reveals the fact that the computational technique could be used more extensively to the chemical engineering area in order to achieve the goal of simulating chemical process and equipment on the basis of more theoretical approach rather than experimental and/or empirical ones. In that direction, we are facing a number of challenges, shown as follows. 6.1
Interface mass-transfer
At present, the evaluation of mass transfer rate between phases is based on the film concept and em-
As stated above, there are three categories of models can be used to simulate multi-phase flow. Although the pseudo-single-phase model is simpler and suitable to many cases as shown in previous sections, more accurate model is still need to investigate. The explicit two-phase flow approach, such as VOF method, can hardly be applied at present for simulation due to the increased heavy computation load. Certainly, the development of more efficient approaches to handle explicitly two-phase flow is of great interest. Furthermore, the current CMT model depends on the reliability of estimating such vital parameters as the inter-phase contact area, the mass transfer rate, etc. The study on these aspects by theoretical or semi-theoretical means is urged. 6.3
Anisotropic problem
In present CMT methodology, the closure of turbulent Navier-Stokes equation is by applying the Boussinesq’s postulate and the isotropic assumption. But in some occasions, such as the boundaries have strong impact on the fluid flow, the transport in porous medium with clear orientations, and the cyclic flow, etc., the
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isotropic assumption is invalid, and may produce considerable error. Therefore it is necessary to develop reliable approaches for the CMT under anisotropic flow, such as those employed in the CFD methodology. 6.4
Multi-scale modeling
As a trend, the depth modeling of chemical process should be considered in multi-scale. For instance, the interface convection occurs on a mesoscopic scale, which affects the macroscopic rate of mass transfer. The Lattice-Boltzmann method to be used in recent years for simulating fluid flow and heat/mass transfer in the complex systems displays significant advantages in describing the transport phenomena from particle distribution to macroscopic scale. More understanding in microscopic and mesoscopic levels of mass transfer will make the modeling on the sound scientific basis. 6.5 Closure of mass transfer equation The use of c 2 - H c model for the closure of mass transfer differential equation has been proven to be successful as a base of CMT methodology. However, this is not the only way to close the mass transfer equation or to solve the unknown parameter u cc . More method of solution is under investigation. Briefly, the perspectives and challenges discussed above are mainly based on CMT investigation that has been made. The exploration on novel strategies for higher level of modeling is still in need and of significance. 7
CONCLUSIONS
Computational mass transfer as a method for chemical process simulation integrates the advances in CFD and transport theories. The recent development
staff in the State Key Laboratories of Chemical Engineering (Tianjin University). REFERENCES 1
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of c - H c model for closing the mass transfer differential equation constitutes the ground work of CMT methodology for chemical process simulation. The feature of CMT method is able to simulate the concentration, temperature and velocity fields as well as the transport diffusivities and operating efficiency at once without depending on the assumption of constant Sc/Pr number or empirical correlations. The present CMT method has been applied successfully to distillation, chemical absorption and exothermic catalytic reaction processes. However, the CMT method is now still in the developing stage, further investigation is necessary in order to make it relying on the basis of more theoretical and less empirical in order to achieve more reliable and accurate simulation. ACKNOWLEDGEMENTS
The authors acknowledge the assistance from the
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McFarlane, R.C., Muller, T.D., Miller, F.G., “Unsteady-state distribution of fluid composition in two-phase oil reservoirs undergoing gas injection”, Soc. Petrol. Eng. J., 7 (1), 6174 (1967). We, C., Farouqali, S.M., Stahl, C.D., “Experimental and numerical simulation of two-phase flow with interface mass transfer in one and two dimensions”, Soc. Petrol. Eng. J., 9 (3), 323337 (1969). Wang, X.L., Liu, C.J., Yuan, X.G., Yu, K.T., “Computational fluid dynamics simulation of three-dimensional liquid flow and mass transfer on distillation column trays”, Ind. Eng. Chem. Res., 43, 25562567 (2004). Liu, B.T., “Study of a new mass transfer model of CFD and its application on distillation tray”, Ph.D. Thesis, Tianjin University, Tianjin, China (2003). (in Chinese) Sun, Z.M., Liu, B.T., Yuan, X.G., Yu, K.T., “New turbulent model for computational mass transfer and its application to a commercial-scale distillation column”, Ind. Eng. Chem. Res., 44 (12), 44274434 (2005). Bird, R.B., Steward, W.E., Lightfoot, E.N., Transport Phenomena, 2nd ed., John Wiley & Sons (2002). Rosén, C., Trägårdh, C., “Prediction of turbulent high Schmidt number mass transfer using a low Reynolds number k-İ turbulent model”, Chem. Eng. J., 59, 153159 (1995). Sun, Z.M., Yu, K.T., Yuan, X.G., Liu, C.J., “A modified model of computational mass transfer for distillation column”, Chem. Eng. Sci., 62, 18391850 (2007). Liu, G.B., Yu, K.T., Yuan, X.G., Liu, C.J., Guo, Q.C., “Simulations of chemical absorption in pilot-scale and industrial-scale packed columns by computational mass transfer”, Chem. Eng. Sci., 61, 65116529 (2006). Liu, G.B., Yu, K.T., Yuan, X.G., Liu, C.J., “New model for turbulent mass transfer and its application to the simulations of a pilot-scale randomly packed column for CO2-NaOH chemical absorption”, Ind. Eng. Chem. Res., 45, 32203229 (2006). Bachmat, Y., Bear, J., “Macroscopic modelling of transport phenomena in porous media (1) The continuum approach”, Transport Porous Media, 1 (3), 213240 (1986). Liu, G.B., Yu, K.T., Yuan, X.G., Liu, C.J., “A computational transport model for wall-cooled catalytic reactor”, Ind. Eng. Chem. Res., 47, 26562665 (2008). Sakata, M., Yanagi, T., “Performance of a commercial scale sieve tray”, In: Institution of Chemical Engineers Symposium Series, 3.2, American Institute of Chemical Engineers, New York, 2134 (1979). Liu, G.B., “Computational transport and its application to mass transfer and reaction processes in packed-bed”, Ph.D. Thesis, Tianjin University, Tianjin, China (2006). (in Chinese) Shariat, A., Kunesh, J.G., “Packing efficiency testing on a commercial scale with good (and not so good) reflux distribution”, Ind. Eng. Chem. Res., 34 (4), 12731279 (1995). Valstar, J.M., van den Berg, P.J., Oyserman, J., “Comparison between two dimensional fixed bed reactor calculations and measurements”, Chem. Eng. Sci., 30 (7), 723728 (1975). Sun, Z.F., Yu., K.T., Wang, S.Y., Miao, Y.Z., “Absorption and desorption of carbon dioxide into and from organic solvents: Effects of Rayleigh and Marangoni instability”, Ind. Eng. Chem. Res., 41, 19051913 (2002). Sha, Y., Cheng, H., Yuan, X.G., Yu, K.T., “Kinetics of mass transfer accompanied by Marangoni effect”, J. Chem. Ind. Eng. (China), 54 (11), 15181523 (2003). (in Chinese) Sha, Y., Yuan, X.G., Cheng, H., Yu, K.T., “Rayleigh-Bénard-Marangoni effect in mass transfer”, Chemical Engineering (China), 31 (5), 813 (2003). (in Chinese) Yu, L.M., Zeng, A.W., Yu, K.T., “Effect of interfacial velocity fluctuations on the enhancement of the mass-transfer process in falling-film flow”, Ind. Eng. Chem. Res., 45, 12011210 (2006).