Microsoft Word - 46gerke.docx CHEMICAL ENGINEERING TRANSACTIONS VOL. 69, 2018 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Guest Editors: Elisabetta Brunazzi, Eva Sorensen Copyright © 2018, AIDIC Servizi S.r.l. ISBN 978-88-95608-66-2; ISSN 2283-9216 Investigation of the Separation Performance of a Stripping Microchannel Device: a Model-Based Approach Chafika Adiche Technische Universität Darmstadt, Institute for Thermal Process Engineering, Department of Mechanical Engineering, Otto- Berndt-Straße 2, 64287 Darmstadt, Germany. adiche@tvt.tu-darmstadt.de In this work, a mathematical model describing the transport mechanisms in a recently developed microstripper (Adiche, 2018) was established and analytically solved. Based on this model, the overall volumetric mass transfer capacity coefficient could be estimated according to a three-resistance mass transfer model and used subsequently in combination with the HTU-NTU concept for the theoretical prediction of the separation performance of the microstripper in terms of stripping degree (Adiche, 2018). Accordingly, a sensitivity analysis of the separation performance of the device to the relevant process parameters including the liquid channel depth and the volumetric flow rates of the feed solution and carrier gas nitrogen was conducted. The comparison of the estimated stripping degrees determined in this work with those obtained experimentally from a previous work (Adiche, 2018) indicates that the developed model-based approach allows a correct prediction of the separation performance of the microstripper and constitutes therefore a useful tool towards the design optimization of the device. 1. Introduction A liquid–vapor / gas separation microchannel device has been recently developed and successfully tested as microstripper for the removal of acetone from a dilute aqueous solution with a concentration of 5 wt % acetone using dry nitrogen as carrier gas (Adiche, 2018). In this work, a model-based approach is presented for the theoretical investigation of the separation performance of the microstripper in terms of stripping degree. For this purpose, a mathematical model describing the transport mechanisms in the separation device including the fluid channels and the membrane contactor is established and analytically solved. Subsequently, a three- resistance mass transfer model is applied to estimate the overall volumetric mass transfer capacity coefficient. The latter is then used within a HTU-NTU approach for the prediction of the stripping degree for given liquid channel depth and volumetric flow rates of feed solution and carrier gas nitrogen. Finally, the ability of the model as predictive design tool for the microstripper is examined by comparing the estimated values of stripping degree in this work with those obtained experimentally in a previous work (Adiche, 2018). 2. Prediction of mass transfer properties in the microstripper The overall mass transfer coefficient KL (based on the liquid phase) in the microstripper is determined according to a three-resistance model including the resistance to mass transfer in the liquid channel, in the membrane pores and in the gas channel (Adiche, 2018) so that GcMcLL kH 1 kH 1 k 1 K 1 ++= , (1) where Hc is the Henry constant for acetone in nitrogen and kL, kM and kG are the individual mass transfer coefficients based on concentration difference as driving force in the liquid phase, through the membrane pores and in the gas phase, respectively.The individual mass transfer coefficient kL/G for the liquid / carrier gas stream can be obtained by DOI: 10.3303/CET1869113 Please cite this article as: Adiche C., 2018, Investigation of the separation performance of a stripping microchannel device: a model-based approach, Chemical Engineering Transactions, 69, 673-678 DOI: 10.3303/CET1869113 673 dh D hS k G/L G/LL/G G/L = , (2) with ShL/G is the Sherwood number in the liquid / gas channel, DL/G the diffusion coefficient of acetone in water / nitrogen and dhL/G is the hydraulic diameter of the liquid / gas channel given by G/L L/G G/L HW H W 2 dh + = . (3) In a previous work (Adiche, 2018) it was shown that for the microstripper considered in this study, the molecular diffusion is the prevailing mass transfer mechanism in the membrane pores so that the mass transfer coefficient in the membrane may be expressed as D k G M δχ ε = , (4) with ε is the membrane porosity , δ the membrane thickness and χ is the membrane tortuosity. To determine the Sherwood number in the liquid / gas channel, a two-dimensional mathematical model describing the transport mechanisms i.e. continuity, momentum, and species conservation equations, for each fluid channel of the microstripper was established [Figure 1]. The latter is based on the following assumptions: i. the microstripper is operating under steady state, isothermal condition (ca. 23 °C) and at atmospheric pressure, ii. axial dispersion along the fluid flow direction and free convection at the interface between the membrane and fluid channels are neglected, iii. laminar fully developed velocity and concentration profiles in both liquid and gas channels (Adiche, 2018), which are considered here as straight instead of meandering channels, iv. the acetone diffusion flux N through the membrane pores is constant along the fluid flow direction and v. the interface between liquid and gas phases at the membrane pore entrance is planar. Accordingly, the Sherwood number (in analogy with Nusselt number) is expected to be constant for each fluid channel (here for a rectangular channel with constant diffusion flux through one porous side: membrane pores) (Shah and London, 1978). Figure 1: Model of the microstripper (exploded view) with the fluid channels are represented as straight channels instead of meandering channels (the membrane module is here omitted). In reason of the similarity between the transport mechanisms in both fluid channels, only the equations for the liquid channel are presented in this section as follows: The Navier-Stokes equation writes 674 dz dp1 y v x v L L 2 z,L 2 2 z,L 2 ⋅−= ∂ ∂ + ∂ ∂ μ . (5) The species conservation equation for acetone is given by ( ) z c y,xv D 1 = y c x c L z,L L 2 L 2 2 L 2 ∂ ∂⋅⋅        ∂ ∂+ ∂ ∂ , (6) with the mass balance equation along the flow direction writes z,L L v Na z c = ∂ ∂ , (7) where a is the specific interfacial area per unit volume of the wetted device estimated by LHa ε= , (8) and z,Lv is the average velocity over the channel cross section area W×HL defined as     = L L H 0 W 0 H 0 W 0 z,L z,L dxdy dxdyv v , (9) and finally the acetone diffusion flux N is given by ( )M L B LL cckN −= , (10) with B Lc and M Lc are the average concentrations of acetone in the liquid phase over the channel cross section area W×HL and at the membrane pore entrance along the length W, respectively. The boundary conditions for the velocity field are given by 0v 0xz,L = = , 0v Wxz,L = = , 0v 0yz,L = = , 0v LHyz,L = = , (11) and those for the concentration field by 0 x c 0x L = ∂ ∂ = , 0 x c Wx L = ∂ ∂ = , 0 y c 0y L = ∂ ∂ = and LHy L D N y c L −= ∂ ∂ = , with Wx0 ≤≤ and LHy0 ≤≤ . (12) The dimensionless variables are defined as follows W x=ξ , W y=η , with 10 ≤≤ ξ and W H 0 L L =≤≤ βη , with βL is the aspect ratio of the liquid channel, (13.a) ( ) ( ) Z p . W y,xv ,u L L 2 z,L ,L Δ μ ηξς = , (13.b) and LL in,LL L DNdh cc − =ω , with cL,in is the inlet molar concentration of acetone in the feed solution. (13.c) Accordingly, the dimensionless Navier-Stokes equation and the related boundary conditions write ( ) ( ) 1 ,u,u 2 ,L 2 2 ,L 2 −= ∂ ∂ + ∂ ∂ η ηξ ξ ηξ ςς , 0u 0,L = =ξς , 0u 1,L = =ξς , 0u 0,L = =ης , 0u L ,L = =βης . (14) The solution of Eqs. (14) is obtained analytically (Spiga and Morini, 1994) and is given by ( ) ( ) ( )  ∞ ∞ +       = odd n odd m 222 L L 4 2 L ,L mnnm msinnsin 16 ,u β β ηππξ π βηξς . (15) Combining Eqs. (9) with (13.b) and (15) gives rise to L L L 2 6 2 L z,L A Z PW64 v μ Δ π β ⋅= , with ( )  ∞ ∞ + = odd n odd m 222 L 22L mnmn 1 A β . (16) 675 Furthermore, the dimensionless species conservation equation for acetone with the related boundary conditions are given by ( ) ( )ηξ μ Δ β βε η ω ξ ω ς ,u Z PW v 1 2 1 ,L L L 2 z,L 2 L L 2 L 2 2 L 2 ⋅⋅+= ∂ ∂+ ∂ ∂ (17.a) and 0 0 L = ∂ ∂ =ξξ ω , 0 1 L = ∂ ∂ =ξξ ω , 0 0 L = ∂ ∂ =ηη ω , ( ) L LL 2 1 L β β η ω βη + −= ∂ ∂ = . (17.b) Combining Eqs. (17.a) with (15) and (16) results in ( ) ( ) ( )  ∞ ∞ +       ⋅⋅+= ∂ ∂+ ∂ ∂ odd n odd m 222 L L L 2 2 L L 2 L 2 2 L 2 mnnm msinnsin A42 1 β β ηππξ π β βε η ω ξ ω . (18) The concentration profile is then determined (Matlab®) by solving the mass balance equation (Eq. (18)) according to the analytical method developed in the work of Spiga and Morini (1996) for the determination of the temperature profile and the Nusselt number in rectangular ducts for H2 boundary conditions. Subsequently, the Sherwood number is obtained by combining Eq. (2) with Eqs. (10) and (13.c) such that M L B L L 1 Sh ωω − = , (19) with the average dimensionless acetone concentration in the liquid channel B Lω is given by     = L L 0 1 0 ,L 0 1 0 L,L B L d d u d d u β ς β ς ηξ ηξω ω , (20) and the average dimensionless liquid acetone concentration at the membrane pore entrance M Lω writes ( ) ξβξωω d , 1 0 LL M L = . (21) Once the Sherwood numbers for the liquid and carrier gas streams have been determined, the respective individual mass transfer coefficients kL/G can be obtained using Eq. (2). Subsequently, combining Eqs. (1) and (4) provides the overall mass transfer coefficient KL. The latter is then multiplied by the estimated specific interfacial area for mass transfer a (Eq. (8)); providing thereby the predicted overall volumetric mass transfer capacity coefficient (KLa)predic . 3. Results and discussion The estimated overall volumetric mass transfer capacity coefficient (KLa)predic in the microstripper was determined for each tested liquid channel depth (100 and 300 μm) according to the model-based approach presented in Section 2. For this purpose, the relevant thermodynamic and transport properties of acetone were used [Table 1]. In addition, a membrane tortuosity χ equal to 2 was considered. Table 1: Thermodynamic and transport properties of acetone in the microstripper (Adiche, 2018) Property Value Hc (kmol/m3/kmol/m3) 1.3 10-3 DL (m 2/sec) 1.2 10-9 DG (m 2/sec) 1.3 10-5 Table 2 shows that for both liquid channel depths, the predicted overall volumetric mass transfer capacity coefficients underestimate the corresponding values determined experimentally in the previous work (Adiche, 2018) with however a higher extent for the greater liquid channel depth (aspect ratio). This discrepancy may be attributed to the restriction of the elaborated model to fully developed velocity and concentration profiles in straight fluid channels with constant acetone diffusion flux along the fluid flow direction. So that the entrance- length effects as well as the developing regions, which may occur in the meandering fluid channels of the microstripper are here not considered. Consequently, the developed model provides a minimum Sherwood number in each fluid channel (Clark and Kays, 1952); resulting therefore in a minimum overall mass transfer 676 coefficient in the microstripper. A further possible cause of this discrepancy is the actually curved interface between the liquid and vapor / gas phases at the membrane contactor entrance due to the prevailing surface tension between the feed solution and the hydrophobic membrane contactor (Probstein, 2003). In this work however the interface between the liquid and vapor / gas phases is assumed to be planar. That results in underestimating the specific interfacial area a because of a smaller surface area for the wetted device as compared to that expected for a curved interface; leading consequently to lower values of predicted overall volumetric mass transfer capacity coefficients as compared to those obtained experimentally. Table 2: Comparison between the experimental (Adiche, 2018) and predicted values of overall mass transfer coefficients in the microstripper HL (μm) (KLa)exp (s -1)(Adiche, 2018) (KL,a)predic (s -1) (this work) [(KL,a)predic – (KLa)exp]/ (KLa)exp (%) 100 0.091 0.062 -31.9 300 0.018 0.011 -38.9 Further investigations were carried out to examine the impact of varying alternatively the liquid and carrier gas flow rate on the stripping degree of the microstripper for each tested liquid channel depth by using the corresponding predicted value of overall mass transfer coefficient determined in this work [Table 2]. The predicted stripping degree is therefore determined according to the HTU-NTU concept (Adiche, 2018) using the following relationship ( ) ( ) 1 Q S 1S Z H W aK Exp S 1S 1 L LpredicL predic −        − −−=σ , (22) where QL and S are the liquid volumetric flow rate and the stripping factor, respectively. Figure 2 shows that for the two tested liquid channel depths, similar trends were found for the variation of both experimental (σexp) and predicted (σpredic) stripping degrees with the liquid residence time. For HL = 100 μm, the relative deviation of σpredic with respect to σexp varies between -2.0 % and -21.4 %, increasing with the decrease of the liquid residence time. Similarly for HL = 300 μm, the relative deviation of σpredic to σexp increases with decreasing the liquid residence time, but in this case in a range between -10.0 % and -29.3 %. This discrepancy can be well explained by the lower predicted overall volumetric mass transfer capacity coefficients as compared to the respective experimental values with however a higher extent for the greater liquid channel depth. Figure 2: Influence of the liquid residence time on the experimental and predicted stripping degrees for a carrier gas flow rate of 163 ml /min. Figure 3 illustrates the variation of both experimental and predicted stripping degrees with the carrier gas flow rate. Here also, it is shown that for each tested HL both curves present similar trends with higher values for σexp as compared to the respective σpredict. However, it is worth noticing that contrary to the results presented in the previous section [see also Figure 2], the relative deviation of σpredic with respect to σexp is here hardly affected by the variation of the carrier gas flow rate, with average values of -6.5 % and -25.9 % for HL equal to 100 μm and 300 μm, respectively. 677 Figure 3: Influence of the carrier gas flow rate on the experimental and predicted stripping degrees for a liquid residence time of 34 s. 4. Conclusion The comparison between the estimated stripping degrees determined in this work using a model-based approach and those obtained experimentally from a previous work (Adiche, 2018) reveals that the developed model allows a qualitative prediction of the impact of both miniaturization (with decreasing the liquid channel depth) and variation of the liquid and carrier gas flow rates on the separation performance of the microstripper. The accuracy of the model in predicting the stripping degree was found strongly sensitive to the liquid flow rate (liquid residence time) and to the liquid channel depth while barely affected by the carrier gas flow rate, with the best value achieved for a liquid channel depth of 100 μm and a liquid residence time of 68 s corresponding to a relative deviation of σpredic with respect to σexp of -2.0 %. References Adiche C., 2018, Stripping of acetone from water in a microchannel device, Separation and Purification Technology, 199, 105-113. Clark S. H. and Kays W. M., 1952, Laminar-Flow Forced Convection in Rectangular Tubes, Transactions of the ASME, 859-866. Curcio E. and Drioli E., 2005, Membrane distillation and related operations-A review, Separation and Purification Reviews, 34, 1, 35-86. McCabe W. L., Smith J. C., Harriott P., 2005, Unit Operations of chemical engineering, seventh Edition, McGraw Hill International Edition. Probstein R. F., 2003, Physicochemical Hydrodynamics, An Introduction, Second Edition, John Wiley & Sons, Inc., Hoboken, New Jersey. Shah R.K and London A.L, 1978, Laminar Flow Forced Convection in Ducts, Advanced Heat Transfer, Academic Press, New York. Spiga M. and Morini G. L., 1994, A symmetric solution for velocity profile in laminar flow through rectangular ducts, International Communications in Heat and Mass Transfer, 21, 4, 469-475. Spiga M. and Morini G. L., 1996, Nusselt numbers in laminar flow for H2 boundary conditions, Int. J. Heat Mass Transfer, 39, 6, 1165-1174. 678