Development of a Rigorous Modeling Framework ... - ACS Publications

Subscriber access provided by - Access paid by the | UCSB Libraries

Process Systems Engineering

Development of a Rigorous Modeling Framework for Solvent-Based CO2 Capture. Part 1: Hydraulic and Mass Transfer Models and Their Uncertainty Quantification Anderson Soares Chinen, Joshua C. Morgan, Benjamin Omell, Debangsu Bhattacharyya, Charles Tong, and David Miller Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b01471 • Publication Date (Web): 12 Jul 2018 Downloaded from http://pubs.acs.org on July 16, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 59 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Industrial & Engineering Chemistry Research

Development of a Rigorous Modeling Framework for Solvent-Based CO2 Capture. Part 1: Hydraulic and Mass Transfer Models and Their Uncertainty Quantification Anderson Soares Chinena, Joshua C. Morgana,b, Benjamin Omellb, Debangsu Bhattacharyyaa,*, Charles Tongc, David C. Millerb a

Department of Chemical and Biomedical Engineering, West Virginia University, Morgantown, WV 26506, USA

b

National Energy Technology Laboratory, 626 Cochrans Mill Rd, Pittsburgh, PA 15236, USA

c

Lawrence Livermore National Laboratory, Livermore, CA 94550, USA

Abstract Rigorous process models are critical for reducing the risk and uncertainty of scaling up a new technology. It is essential to quantify uncertainty in key submodels so that uncertainty in the overall model can be appropriately characterized. In solvent-based post-combustion CO2 capture technologies, mass transfer and column hydraulics are key factors affecting the performance of the absorber. Developing submodels for mass transfer, column hydraulics, and reactions is a challenging multiscale problem since the phenomena are tightly coupled and it is difficult to design experiments to isolate each properly. In particular, simultaneous mass transfer coupled with fast reaction kinetics make it difficult to measure the mass transfer rate and reactions rate individually. The typical approach to solving this issue is to use proxy systems to conduct experiments under mass transfer-limited or reaction-limited conditions. This approach can lead to inaccurate mass transfer submodels. In this paper, a novel simultaneous regression approach is proposed where submodels for mass transfer, diffusivity, interfacial area, and reaction kinetics are optimally identified using experimental data from multiple scales and operating conditions. Since all models have some level of uncertainty, a rigorous uncertainty quantification (UQ) technique is implemented for the hydraulic and mass transfer submodels based on Bayesian inference. Posterior distributions of submodel parameters are propagated through the column model to obtain the uncertainty bounds on critical performance measures. Keywords: uncertainty quantification, hydraulic, mass transfer, MEA, CO2 capture ____________________________________________________________________________ *Corresponding author. Tel:+1-3042939335, Fax: +1-3042934139 E-mail address: [email protected]

ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 59

2

Nomenclature

,

1, 2 ! !"## ℎ ℎ, % , % & ' '( , '( &) , &) * + , , -. / ,

Packing specific area (m2/m3) Wetted packing area (m2/m3) CO2 activity coefficient MEA activity coefficient MEAH activity coefficient MEACOO activity coefficient Column cross sectional area Parameters in Equation (27) Parameter in Equation (15) Parameter in Equation (23) Parameter in Equation (24) Parameter in Equation (6) Parameter in Equation (13) CO2 capture ratio in the absorber Hydraulic diameter in Equation (24) (m) Packing specific diameter (m) Column diameter (m) Diffusivity (m2/s) Parameter in Equation (28) Parameter in Equations (28-29) Energy of activation for reactions 1 and 2 Gas capacity factor F-factor (Pa0.5) Parameter for equation (3) Froude number CO2 flux in the wetted wall column Gravitational acceleration (m/s2) Effective gravity (m/s2) Holdup (v/v) Holdup (v/v) bellow the loading point Parameters in Equation (18) Mass transfer coefficient Parameter in Equation (6) Equilibrium constants for reactions 1 and 2 Reaction constants for reactions 1 and 2 Wetted perimeter (m) Parameter in Equations (28-29) Gas constant Reaction rate for reactions 1 and 2 Channel size (m) Schmidt number Temperature (K) Velocity (m/s) Velocity at the loading point (m/s)


Page 3 of 59 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3

,) ," ,0 10 2 3 Γ 5

Velocity at the flooding point (m/s) Effective velocity (m/s) Reynolds number Weber number Column height

Greek letters 6 7

Corrugation angle Liquid flow based on perimeter (kg/m.s) Void fraction specific mass (kg/m3) viscosity (Pa.s)

Subscripts G L sln exp pred

Gas phase Liquid phase solution Experimental Model prediction



Page 4 of 59

4

Table of Contents Abstract ........................................................................................................................................... 1 1.0 Introduction ............................................................................................................................... 7 2.0 Background ............................................................................................................................. 11 3.0 Deterministic Model Development ......................................................................................... 14 3.1 Models for Column Hydraulics ........................................................................................... 14 3.1.1 Holdup submodel selection .......................................................................................... 18 3.1.2 Pressure drop submodel selection and optimization..................................................... 21 3.2 Mass transfer and kinetic submodels .................................................................................. 24 3.2.1 Mass transfer coefficient submodel .............................................................................. 27 3.2.2 Interfacial area submodel.............................................................................................. 28 3.2.3 Diffusivity submodel .................................................................................................... 28 3.2.4 Reaction Kinetics .......................................................................................................... 29 3.2.5 Integrated model selection and methodology ............................................................... 30 3.2.6 Integrated mass transfer model regression results ........................................................ 30 4.0 Uncertainty Quantification (UQ) ............................................................................................ 36 4.1 Uncertainty quantification methodology............................................................................. 36 4.2 Hydraulics submodel UQ .................................................................................................... 37 4.3 UQ of the integrated mass transfer model ........................................................................... 44 5.0 Conclusion .............................................................................................................................. 48 Acknowledgement ......................................................................... Error! Bookmark not defined. Disclaimer ...................................................................................... Error! Bookmark not defined. Supporting Information ................................................................................................................. 51


Page 5 of 59 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5

References ..................................................................................................................................... 52

List of Figures Captions Figure 1 –Approach to the Development of the Submodels ......................................................... 12 Figure 2 - Submodels for packed column and wetted wall column models ................................. 13 Figure 3 - Comparison of various holdup submodels for MellapakPlus 252Y operating with a low viscosity liquid (1 mPa. s) at FV=1.02 Pa0.5 .......................................................................... 19 Figure 4 - Comparison of various holdup submodels for MellapakPlus 252Y operating with a high viscosity liquid (10 mPa. s) at FV=1.02 Pa0.5 ....................................................................... 20 Figure 5 - Comparison of pressure drop submodels for the low-viscosity case ........................... 22 Figure 6 - Pressure drop candidate submodels validation in a high-viscosity case ...................... 23 Figure 7 - Parity plot of the regressed Billet and Schultes (1999) with regressed Tsai (2010) holdup ........................................................................................................................................... 24 Figure 8 - Parity plot of the CO2 (%) capture in the absorber (Experimental data from Tobiesen et al., 2007) ................................................................................................................................... 32 Figure 9 - Parity plot of the CO2 flux in the WWC (Experimental data from Dugas, 2009) ....... 33 Figure 10 - Zoomed parity plot of the CO2 flux in the WWC (Experimental data from Dugas, 2009) ............................................................................................................................................. 34 Figure 11 - Comparison of the literature model and integrated mass transfer model with data from Notz et al. (2012).................................................................................................................. 36 Figure 12 - Single-parameter prior and posterior marginal probability density functions of the parameters in the hydraulics model .............................................................................................. 39 Figure 13 - Two-parameter prior and posterior marginal posterior distributions of the parameters in the hydraulics model. ................................................................................................................ 40



Page 6 of 59

6

Figure 14 - Stochastic pressure drop estimates for a water-air system in a tower packed with MellapakPlus 252Y. The lines in a and b represent a liquid load of 6 m3/m2·h and the lines in c and d, a liquid load of 18 m/h. Cases ‘a’ and ‘c’ corresponds to the liquid load of 18 m/h. Cases ‘a’ and ‘c’ corresponds to the liquid viscosity of approximately 1 cP and cases ‘b’ and ‘d’ corresponds to the liquid viscosity of approximately 12 cP. ‘*’ represent corresponding experimental data .......................................................................................................................... 42 Figure 15 - Stochastic holdup estimates for a water-air system in a tower packed with MellapakPlus 252Y. The lines in a and b represent an F-factor of 0.71 Pa0.5 and the lines in c and d an F-factor of 1.6 Pa0.5. Cases ‘a’ and ‘c’ regards a viscosity of approximately 1 cP and cases ‘b’ and ‘d’ a viscosity of approximately 12 cP.................................................................... 43 Figure 16 - Sobol analysis of the integrated mass transfer model ................................................ 45 Figure 17- Parameter distributions................................................................................................ 46 Figure 18 - Stochastic response obtained from the uncertainty propagation compared with data from Tobiesen et al. (2007)........................................................................................................... 47 Figure 19 - Probability density function for the fractional CO2 capture for a particular operating condition ....................................................................................................................................... 48


Page 7 of 59 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


7

1.0 Introduction Fossil fuels currently generate most of the world’s electricity. These power plants release over 36 billion metric tons of CO2 a year[1]. Due to the anticipated strong reliance on fossil fuels in the foreseeable future and increased concern over global warming, there has been an increased interest in the development of CO2 capture technologies as an effective means for reducing greenhouse gas emissions from fossil energy plants. Recently, the Carbon Capture Simulation Initiative (CCSI) has developed a suite of computational tools and models to accelerate the development and scale up of CO2 capture technologies. One component of the CCSI Toolset is a “gold-standard” model to serve not only as a definitive reference for benchmarking the performance of solvent-based CO2 capture systems under both dynamic and steady-state conditions over a large operating range[2], but also to provide a framework for developing predictive models of other solvent systems. Given the multiscale nature of phenomena in a CO2 capture process, it is essential that each submodel be rigorously developed and validated while also quantifying its uncertainty. For a “gold-standard” column model, the sub-models for transport properties, thermodynamics, column hydraulics, and mass transfer are critical, and the resulting system model must be validated using pilot-plant data collected over a large range of operating conditions. Our previous publications have focused on the development of rigorous models and their uncertainty quantification (UQ) for the transport[3] and thermodynamic properties[4]. In Part 1 of this paper, we focus on the development of rigorous models for mass transfer and hydraulics while using Bayesian UQ approaches to reduce their uncertainty. In Part 2, the model is validated with the data from the National Carbon Capture Center in Wilsonville, AL coupled with UQ of the integrated model. Mass transfer coefficients and interfacial area depend on the operating system[5]. Sensitivity studies show that the model form has a strong impact on the accuracy of the overall column, especially when operating between 50-85% CO2 capture[6-8]. Most of correlations available in the open literature for mass transfer coefficients, interfacial area, and column hydraulics were not developed for recently developed packings that incorporate significant advances that improve their hydraulic performance by reducing the pressure drop and increasing the operational regime without flooding or weeping. Furthermore, they were not developed for the MEA-H2O-CO2 system. Although it is common practice to apply literature models for mass transfer and column hydraulics directly to the MEA system without any adjustments[6-22], the parameters are likely to



Page 8 of 59

8

be sub-optimal. Tsai[23] presented pioneering work on mass transfer and hydraulics for recently developed packing-types, studying various flow regimes and operating conditions for a H2ONaOH-CO2 system. Describing the mass transfer phenomena in a packed column requires several submodels, including those for diffusivity, interfacial area, and mass transfer coefficients on both the liquid and gas side. When considering reactive solvents in an absorber, the challenge to develop a model is compounded since it is difficult to separate the effects of mass transfer and chemical reactions since they occur simultaneously. Thus, models for both mass transfer and reactivity need to be developed. The typical approach in the literature is to develop these models sequentially by using experiments that try to isolate the effects of specific phenomena. Typically, the diffusivity submodel is first developed assuming no reaction and by using correlations such as the Stokes-Einstein relation. This diffusivity submodel is then applied while developing the submodels for the mass transfer coefficients and reaction kinetics using experimental data from the wetted wall column (WWC)[21,22,24]. Finally, an interfacial area model is developed for a given packing based on experimental data from absorbers/regenerators. Another approach is to obtain the mass transfer coefficient submodel using experimental data from a nonreactive system in the packed tower. Then data for the actual, reactive system are used to develop the interfacial area submodel. There are three issues with this traditional approach. First, it implicitly assumes that the diffusivity and mass transfer coefficient submodels obtained from different equipments (i.e., the WWC column and non-reactive system) are valid for the reactive system with a given packing. However, the hydrodynamics, liquid and gas velocities, loading of the solvent and operating temperatures can be very different. In addition, significant differences in density, viscosity, and surface tension between the reactive and non-reactive systems can affect the wettability and flow characteristics of the fluids, hence the interfacial area. Furthermore, mass transfer for electrolyte systems are affected by the ionic species present in the solution, ion-molecule interactions, and ion mobility. Therefore, a model form that is adequate to capture the physics of the system at a given scale or to capture the physics of a surrogate system may fail to accurately represent the physics of the system at a different scale or to capture the physics of the true system at the same scale. Thus, the error due to the inadequacy of the model form gets propagated to the subsequent


Page 9 of 59 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


9

step of model identification and parameter estimation. The second issue is that the error in the parameter estimation from one step gets propagated to the next step. In other words, the parameter estimates may be locally optimal, but not necessarily globally optimal. The third issue is that instead of using a rigorous rate-based column model, most researchers have used a simple model for the WWC based on the enhancement factor approach so that the parameters for the kinetic and mass transfer coefficient submodels can be easily estimated[25-27]. Recently, a few researchers have considered the rigorous rate-based model while analyzing the WWC experiments[28-30], albeit, following a sequential approach. An alternative approach is to develop a model of the volumetric mass transfer coefficient, where the mass transfer coefficients are multiplied by the interfacial area. While the model of the volumetric mass transfer coefficients can be obtained directly from packing experiments[31,32], the main difficulty is that the mass transfer coefficients and interfacial area become indistinguishable. The range of reported validation data is usually narrow, around 90-100% CO2 capture[8, 11-13,16-19, 21]

. Validation data for lower capture rates, such as 50-85%, is rather limited and typically

consists of only a few data points. Reported models tend to have greater error for low capture rates[7, 9, 10, 14, 15, 20, 22]. In this paper an integrated mass transfer model that can be applied to the packed tower as well as to the WWCs is developed where the submodels for diffusivity, interfacial area, liquid- and gasside mass transfer coefficients, and kinetics are simultaneously identified, as appropriate, and regressed using data from WWCs along with data from packed towers for the MEA-H2O-CO2 system. An accurate submodel for the pressure drop is important for calculating the fluid flowrates, particularly in the gas phase during dynamic simulations especially when the tower approaches flooding conditions. Holdup in the packing affects the extent of reaction as well as pressure drop through the tower. In addition, due to the close coupling between holdup and pressure drop, as will be explained in more details later, it is important to have accurate submodels for both, especially for transient simulation when both of these variables can change significantly, leading to undesired tower operation such as flooding. Holdup also directly affects the rate of change in the transport variables such as the temperature and concentration. While for MEA-H2O-CO2 reaction systems where the reactions are fast, impact of the holdup model on the extent of



Page 10 of 59

10

reaction may not be important, it affects the mass and energy accumulation and therefore also affects the transient response. While this two-part paper focuses on application of the developed submodels from the steady-state perspective, the authors intend to leverage the developed submodels also in the dynamic model. Correlations developed for air-water systems are typically applied to the hydraulic modeling of the MEA-H2O-CO2 systems[6, 7, 15, 33]. Stichlmair et al.[34] and Billet and Schultes[35] proposed pressure drop and holdup submodels that can be applied to both random and structured packings from the loading region up to the flooding point. Billet and Schultes[36] later improved and expanded their submodels by considering a larger database. Rocha et al.[37] proposed a correlation for pressure drop and holdup in the loading region of a structured packing. Other widely used submodels have been reported by Bravo et al.[38,39] ,and Fair and Bravo[40]. These submodels utilize packing-specific parameters to address the effect of geometry on the pressure drop and holdup; therefore, the existence of experimental data for a given packing is critical. Appropriate parameters for a number of recently developed packings with improved mass transfer and hydraulics are not available in the open literature. In this work, updated parameters for the hydraulics of one of the newer structured packings have been determined. Uncertainty in models and their parameters is unavoidable and must be quantified for predictive models. A systematic approach to UQ of not only the mass transfer and hydraulic submodels, but of process models, in general, is rare. Uncertainty in model parameters has been evaluated by a few authors through perturbation methods[41,42] and Monte Carlo analysis[43-45]. A rigorous approach to UQ of the thermodynamic submodels by using a comprehensive Bayesian approach has been reported by Mebane et al.[46], Weber et al.[47] and Sarkar et al.[48]. A deterministic alternative to the Monte Carlo simulations has been proposed by Gong[49] for evaluating the uncertainty of several performance metrics for the gasification technologies. The method has also been applied to quantify the uncertainty of integrated

reforming combined cycle

technologies[50].

Uncertainty quantification of density, surface tension and viscosity submodels using a Bayesian approach has been reported by Morgan et al.[3]. More recently, we have reported a Bayesian inference procedure for UQ of the thermodynamic model of a MEA-H2O-CO2 system where the


Page 11 of 59 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


11

VLE, enthalpy and chemistry submodels were considered together[4]. One significant difference in the UQ of the mass transfer and hydraulic models compared with that of the thermodynamic and transport models is the consideration of the column model. The input space spanned by the priors of the parametric uncertainties and the space of operating conditions must be propagated through the complicated and computationally demanding rate-based column model for Bayesian UQ of the parameter space. This leads to challenging computational issues that are undertaken in this work. In summary, contributions of the current work are as follows: •

An integrated framework for simultaneous parameter estimation of interfacial area submodels, mass transfer coefficient submodels, and kinetic submodels has been developed using multiscale experimental data from the WWCs and packing.

•

Hydraulic submodels are obtained for a new structured packing.

•

Bayesian inference is used to accurately quantify the uncertainty in the mass transfer submodels as well as the hydraulic submodels, by using an initial estimate of the parameters probability distributions.

•

Parametric uncertainty in the mass transfer model is propagated through a process model of the MEA-H2O-CO2 system for evaluating its impact on column performance.

2.0 Background The approach taken to developing the “gold-standard” model is presented in Figure 1. Development of the properties submodels used in this work for density, viscosity, and surface tension have been presented by Morgan et al.[3]. Those submodels were found to have high accuracy over ranges of 0 - 0.5 mol CO2/mol MEA, 0.2 - 0.4 g MEA/g MEA+H2O, and 298.15 353.15 K for the viscosity and density submodels and 303.15 - 333.15 K for the surface tension submodel. Model parameters were optimally estimated by minimizing SSE while the model forms were evaluated using cross validation. The thermodynamic model used in this work has been reported by Morgan et al.[4]. The parameters for the e-NRTL thermodynamic framework, including the heat capacity submodel, were obtained by using the vapor-liquid equilibrium (VLE), heat of absorption, and heat capacity data simultaneously. The large parameter space of



Page 12 of 59

12

the e-NRTL thermodynamic framework was considerably downselected by using the Akaike Information Criterion (AIC). The reaction kinetics were cast in a form that ensures consistency with the thermodynamic model. A parameter selection methodology using the AIC has been applied to determine an optimal set of parameters for representing the thermodynamic model.

Figure 1 –Approach to the Development of the Submodels

The hydraulic and mass transfer submodels are developed as part of this work. The mass transfer model was developed to be valid across multiple scales by considering WWC data and packed column data simultaneously for parameter estimation. An illustration of how each scale depends on the submodels is presented in Figure 2.


Page 13 of 59 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


13

Figure 2 - Submodels for packed column and wetted wall column models

Existing commercial process simulation software such as Aspen Plus cannot be used directly for simultaneous parameter estimation of such an integrated model due to the way the submodels are organized in the software. Diffusivity belongs to the transport model package; reactions belong to a separate callable class while the hydraulics and mass transfer models belong to the column model. Furthermore, large numbers of WWC experiments and tower experiments should be considered simultaneously for parameter estimation of the integrated model. This large-scale optimization problem is computationally expensive and can be difficult to solve in commercial software. The Framework for Optimization, Quantification of Uncertainty and Surrogates (FOQUS)[51], that can read from and write to Aspen Plus models, is utilized in this work for developing the integrated model. More details on the simultaneous regression approach is given in section 3.2.



Page 14 of 59

14

3.0 Deterministic Model Development The methodologies developed in this work are generic and can be readily applied to any solvent system with any packing. However, the performance of the hydraulic and mass transfer submodels strongly depends on the packing in the system. In this work, the models were developed for use at the National Carbon Capture Center (NCCC) in Wilsonville, AL, which uses MellapakPlusTM 252Y, one of the newer packings from Sulzer[52], in both the absorber and stripper. Very little data are available on this packing in the literature. This packing offers a low pressure drop, a wide operating window without flooding or channeling, and a high interfacial area with high mass transfer efficiency[52]. Experimental data used in this work are from MellapakPlus 252Y or packings that are structurally similar to it. 3.1 Models for Column Hydraulics Table 1 presents three leading hydraulic models that have been widely used for calculating pressure drop and holdup. Equations 1-5 represent the Rocha et al.[37] model, which was developed as an update of the previous hydraulic models of Bravo et al.[38,39], and Fair and Bravo[40]. Equations 6-14 represent the Billet and Schultes[36] model, and Equations 15-17 are the Stichlmair[34] model. These models typically consider holdup and pressure drop to be dependent on each other. Equation 18 represents a submodel of holdup which is independent of pressure drop. According to Tsai[23], Equation 18 is more accurate than submodels where holdup is coupled with the pressure drop submodel. This submodel showed an average error of 12% for the entire database considered by Tsai (2010), but the results for MellapakPlus 252Y had an error of above 20% for most cases. The pressure drop calculation ∆P is presented in equations (1 and 2), as a function of the gas

density 6 , the packing channel size S, the packing void fraction ε, the packing corrugation angle

α, the gas velocity and the gas viscosity 7 .

The holdup calculation is presented in Equations 3-5, in which 6 is the liquid density, 7 is the liquid viscosity, is the liquid velocity. The holdup submodel is tied to the pressure drop


Page 15 of 59 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


15

calculation by the effective gravity !"## and the correction factor , which is calculated as a

function of the dimensionless numbers of the liquid phase: ,0 , 10 and .

In the hydraulics submodels given by equations 6-13[36], the pressure drop, ΔP/z, is a function of

the parameter , the gas phase Reynolds number, ,0, the void fraction of the packing, ε, the column holdup, ℎ , the holdup bellow the loading point, ℎ, , the Froude number, , the

specific packing area, a, and the gas capacity factor, . The parameter K is a lumped term that

is a function of the packing void fraction, ε, the packing specific diameter, , and the column diameter, . The holdup, ℎ , is a function of liquid viscosity, 7 , the packing specific area, a, the gravity acceleration, g, the liquid velocity, , the liquid density, 6 . Above the loading

point, the holdup becomes also a function of the gas velocity, , the gas velocity in the flooding

point, ,) , the water viscosity, 79:;"< , the water density, 69:;"< , and the parameter .

Pressure drop and holdup of a packing depend on the flooding point of a given packing, restricting the applicability of a given model. Tsai[23] suggests a value of 1025 Pa/m for the pressure drop h at the flooding point. Billet and Schultes[36] also address this issue. In their model, under the loading point, which is defined as the flow regime in which the liquid flow does not significantly decrease the packing void fraction available for the gas flow, the holdup submodel is given by Equation (8). As the liquid flowrate is increased, the tower goes through a transition region (, !P (above loading point)

(9)

y ℎ = ℎ, + (ℎ,) − ℎ, ) [ \ y,) Y

ℎ,) ℎ,

7 69:;"< = 2.2ℎ, [ \ 79:;"< 6

.X

Z Y

127 =[ \ !6

ZY z {

6 .X ,0 < 5: = U W [ \ 7 !

(10)

.

6 .X ,0 ≥ 5: = 0.85 U W [ \ 7 !

Stichlmair (1989)

.

∆HI

Development of a Rigorous Modeling Framework ... - ACS Publications

Recommend Documents