Accepted Manuscript. Title: Linearized description of the non-isothermal flow of a saturated vapor through a micro-porous membrane

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1 Title: Linearized description of the non-isothermal flow of a saturated vapor through a micro-porous membrane Author: Thomas Loimer PII: S (07) DOI: doi: /j.memsci Reference: MEMSCI 7912 To appear in: Journal of Membrane Science Received date: Revised date: Accepted date: Please cite this article as: T. Loimer, Linearized description of the non-isothermal flow of a saturated vapor through a micro-porous membrane, Journal of Membrane Science (2007), doi: /j.memsci This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

2 Manuscript Linearized description of the non-isothermal flow of a saturated vapor through a micro-porous membrane Thomas Loimer Institut für Strömungsmechanik und Wärmeübertragung, TU Wien, Resselg. 3, 1040 Wien, Austria. Abstract The one-dimensional flow of a fluid near saturation through a micro-porous membrane is considered. Upstream of the porous membrane the fluid is in a state of saturated vapor. Downstream, the fluid is in a state of unsaturated vapor. Due to the Joule-Thomson effect, the fluid is cooler at the downstream side of the membrane than at the upstream side. Due to the temperature difference and the heat conduction in downstream direction, the saturated vapor condenses fully or partially and the fluid re-evaporates further downstream within the membrane. The flow process is described taking into account (i) the temperature difference due to the Joule-Thomson effect, (ii) the capillary pressure across a curved meniscus in the porous medium and (iii) the vapor pressure reduction at a curved meniscus given by Kelvin s equation. Depending on the material properties of the fluid and the membrane and on the pressure difference, different types of flow occur. These flow types are analyzed, expressions for the mass flux are given for each case and a flow map is drawn to show the conditions under which the different types of flow occur. A comparison is made with isothermal models and with experimental data from the Preprint submitted to Elsevier Science 30th May 2007 Page 1 of 38

3 literature. Key words: flow through porous media, transport models, Joule-Thomson effect, phase change 1 Introduction When a vapor near saturation flows through a porous membrane, it may condense. Due to the flow of condensate through a part of the membrane and due to the large pressure difference across curved menisci that form at fronts of phase change in the porous medium, the mass flux is quite different from the value that would be expected from the flow of a vapor that does not undergo phase changes. The vapor may condense due to capillary condensation, i.e., vapor pressure reduction at a curved meniscus, and due to the Joule-Thomson effect. Here, the flow process is described accounting for both these reasons of condensation. The influence of the Joule-Thomson effect on the flow process under investigation was studied by Schneider [1] and later by Loimer [2 4]. Due to the Joule-Thomson effect, the vapor is cooler at the downstream side of the membrane than at the upstream side. In general, the temperature of a fluid may increase in a Joule-Thomson process, but on the dew line, all fluids have a positive Joule-Thomson coefficient and cool down. Due to the temperature difference, heat is conducted in downstream direction through the membrane. However, the saturated vapor at the upstream side of the membrane can only Tel: ; fax: address: thomas.loimer@tuwien.ac.at (Thomas Loimer). 2 Page 2 of 38

4 release heat by condensation, hence it condenses fully or partially. Equating the flux of enthalpy convected to the front surface of the membrane to the heat flux conducted downstream, Schneider [1] found that a critical permeability exists, which is a function of the material properties of the fluid and the membrane material only. For a permeability of the membrane larger than the critical permeability, the fluid condenses partially at the upstream surface of the membrane and a two-phase mixture flows through a part of the membrane. For a permeability of the membrane smaller than the critical permeability, the fluid condenses fully, a liquid film may form in front of the membrane, and liquid flows through a part of the membrane. Here, permeability is a quantity defined by Darcy s law. It is essentially a property of the pore space geometry only and it is given in units of length squared. For commonly used fluids and membrane materials, e.g., iso-butane and a polypropylene membrane, the critical permeability is of the order of m 2, which corresponds to a pore diameter of about 60 nm. The critical permeability can be much larger at elevated temperatures or for membrane materials with high thermal conductivities, e.g., ceramics or sintered metals. Note that condensation due to the Joule-Thomson effect does not depend on the wetting properties between the fluid and the membrane material. Capillary condensation is well known. It occurs due to the reduction of the equilibrium pressure between the liquid and the vapor phase at a curved interface with respect to its value at a plane interface, the latter being the saturation pressure. The change of the equilibrium pressure is described by Kelvin s equation. The equilibrium pressure is reduced for a fluid whose liquid phase wets the solid material of the membrane, but for a non-wetting liquid phase of the fluid the equilibrium pressure is increased. The change of the 3 Page 3 of 38

5 equilibrium pressure must be mainly considered for pore sizes of the order of 100 nm and smaller. For larger pores, the change of the equilibrium pressure is small. For instance, in perfectly wetted pores with a diameter of 200 nm, at 20 C the vapor pressure of iso-butane is reduced by 25 mbar from 3.02 bar to bar. The flow of vapors through porous media with small pore sizes was experimentally studied by a number of researchers [5 12]. Often, at the upstream side unsaturated vapor was applied, but some data for a saturated vapor at the upstream side was also reported [7 9, 11]. Several models were proposed to account for condensation [5 7, 9, 11, 13, 14]. The porous medium was usually modeled as an equivalent bundle of capillaries [5 7, 9], but also as a network of pores [13]. The flow was described accounting for capillary condensation and for surface flow [5 7, 13, 14]. Surface flow refers to the transport of fluid by adsorption to the pore walls at the upstream side of the membrane, transport in a thin liquid layer and desorption at the downstream side of the membrane [15, 16]. Rhim & Hwang [5] mentioned that the enthalpy of vaporization released at the condensation front must be conducted to the evaporation front and implemented an energy balance, but they assumed that the upstream and the downstream temperatures are the same. Most other authors assumed isothermal flow conditions throughout the whole membrane [6, 7, 9, 13, 14]. Complex behavior was observed. For instance, in a plot of permeance vs. mean reduced pressure p red,with p red =(p 1 + p 2 )/(2p sat ), where p 1 and p 2 are the upstream and the downstream pressure, respectively, and p sat refers to the saturation pressure at the upstream temperature, some authors observed a maximum of the permeance, e.g., around p red =0.8, and a strong decrease of the permeance when further increasing p red [5, 8, 10]. Other authors reported 4 Page 4 of 38

6 a monotonically increasing permeance [7 9, 11, 12]. Here, permeance refers to the mass flux through the membrane divided by the applied pressure difference. It seems that those authors who measured at small pressure differences, p 1 p 2 < 0.1p sat, observed a maximum [8, 10], while those who applied larger pressure differences reported monotonically increasing permeances. This view is supported by an observation of Tzevelekos et al. [8] who noticed that a maximum appeared in their plot of permeance versus mean reduced pressure when the pressure difference was increased. The increase and also the reduction of the mass flow rate are both believed to be caused by condensation. The capillary pressure across a meniscus in a porous membrane is usually estimated with the Young-Laplace equation [5, 7, 9]. If a liquid plug forms inside the membrane with one end of the plug located at the upstream surface of the porous medium where the curvature of the meniscus can be small, then the large capillary pressure at the other end of the plug within the porous medium drags the liquid downstream and thus enhances the mass flow rate [5, 7, 9]. The reduction of the mass flux was qualitatively explained by clogging of pores with condensate [15, 16]. Pore clogging can occur if the porous medium consists of a network of pores, in which the topology allows menisci with different curvatures at the end of a liquid plug, hence the plug can withstand a certain pressure difference without flow to occur [13]. In a straight capillary, a liquid plug could withstand a certain pressure difference if the contact angle can change, e.g., by contact angle hysteresis. In the flow model proposed here, the porous medium is modeled as a bundle of capillaries. A unique contact angle without hysteresis is assumed, hence, pore clogging is not taken into account. Different to previous works [1 4], here 5 Page 5 of 38

7 the vapor pressure reduction at a curved meniscus is taken into account by applying Kelvin s equation. Also, a correction is applied to the vapor viscosity to account for Knudsen flow. The flow is described for a saturated vapor at the upstream side of the membrane. The energy balance is satisfied and account is taken of the temperature variation across the membrane. Surface flow is not considered because for a vapor close to saturation the film of adsorbate that forms at the pore walls is believed to be quite immobile and contribute little to the total mass flow [10, 16]. However, at fronts of phase change where curved menisci form, a liquid film effectively reduces the pore size and increases the curvature of the menisci. The thickness of the liquid film was estimated to be less than one nm [8]. Therefore, the description of the flow presented here becomes probably inaccurate for wetted membranes with pore sizes less than approx. 10 nm. For larger pore sizes and for any pore size for non-wetted membranes the present description should be accurate. 2 Analysis 2.1 Governing equations The one-dimensional, steady flow through a porous membrane is described by solving the balances of mass, momentum and energy. The membrane extends from z = 0 to z = L, where z is the coordinate in downstream direction. The fluid is in a state of saturated vapor upstream of the porous membrane, p 1 = p sat (T 1 )atz,wheret is the absolute temperature. At the downstream side, the pressure is given, p = p 2 at z = L. The fluid is assumed to be in local thermodynamic equilibrium everywhere. The mass balance is 6 Page 6 of 38

8 trivially satisfied by having a constant mass flow rate per unit area, ṁ = const. (1) The momentum balance is given by Darcy s law, ṁ = κ dp ν dz, (2) where κ is the permeability and ν is the kinematic viscosity of the fluid. Darcy s law is assumed to hold for all states of the fluid, i.e., for a vapor, for a liquid, or for a two-phase mixture. To account for Knudsen flow in the vapor phase the kinematic viscosity of the vapor, ν g, is replaced with an apparent viscosity, ν app = ν g (1 + βkn) 1, (3) where Kn is the Knudsen number and β is a number that depends on the topology of the porous medium. This result is obtained by assuming that the mass flux is the sum of the contributions due to Knudsen flow and due to viscous flow. For a bundle of parallel, straight capillaries, the Knudsen number is Kn = λ/(2a) and the mass flux due to Knudsen flow is given by ṁ = κ(dp/dz)/(3a πrt) [17, p. 172]. Here, λ denotes the mean free path, a istheporeradiusandr is the specific gas constant. Substituting κ = ɛa 2 /8, where ɛ refers to the porosity, estimating the mean free path by λ =3ν π/(8rt ) [18, p. 20] and substituting a by the Knudsen number eq. (3) follows with β =8.1. In practice, the value of β for a porous medium with an irregular pore space is best determined from measurements of the mass flow rate of a gas through the porous medium. 7 Page 7 of 38

9 The energy balance is given by ṁh + q = const., (4) where h denotes the specific enthalpy of the fluid and q is the heat flux. Equation (4) is valid for small fluid velocities, because the contribution of the kinetic energy is ignored. Hence, eq. (4) is applicable for small Eckert numbers, u 2 /[c p (T 1 T 2 )] 1, where u is the fluid velocity and c p is the specific heat capacity at constant pressure. The heat flux is expressed by Fourier s law of heat conduction, q = k dt dz, (5) where k refers to the thermal conductivity. At locations of complete phase change, curved menisci form. Recalling that the porous medium is modeled as a bundle of capillaries, these menisci are located in parallel in all capillaries. All these menisci have the same curvature and the same location with respect to the z-coordinate. For cylindrical capillaries the radius of curvature is given by r = a/ cos θ, whereθ is the contact angle. The pressure difference across a meniscus is given by the Young-Laplace equation, p =2σ/r, (6) where σ is the surface tension. The pressure at the vapor side of a meniscus, p K, is given by Kelvin s equation, ln (p K /p sat )= (2σ/r)v l /RT, (7) 8 Page 8 of 38

10 where v l refers to the specific volume of the liquid phase of the fluid. The location of the menisci and, hence, of fronts of phase change, is found from the condition that in the interior of the membrane the pressure at the vapor side of the menisci is equal to p K (T ). The pressure of a two-phase mixture within the porous medium is also equal to p K (T ). Menisci at the membrane surfaces can have a curvature that varies between zero and the value within the membrane. The pressure at the vapor side of these menisci varies accordingly between p sat and p K. It is assumed that at fronts of incomplete phase change there is no pressure difference between a vapor region and a region with a two-phase mixture. The pressure difference at a front between a two-phase mixture and the liquid phase is the same as between the vapor phase and the liquid phase. This choice is justified by findings from numerical solutions to a similar, non-linear problem [3] and from the results presented later in Section 4. It was found that the vapor volume fraction in the two-phase mixture is mostly close to one. The governing equations are simplified by linearizing the dependencies of the saturation pressure p sat and of p K on temperature around the respective values at the upstream temperature T 1 [1]. The downstream temperature is calculated from the Joule-Thomson coefficient, T 2 = T 1 ( T/ p) h (p 1 p 2 ). (8) With the exception of the kinematic viscosity of the vapor, all other material properties are taken to be constant and equal to their respective values at T 1. For instance, the energy balance, eq. (4), yields, with ṁh = ṁ(h l +ẋ h vap ), 9 Page 9 of 38

11 where h l is the specific enthalpy of the liquid, h vap is the specific enthalpy of vaporization and ẋ refers to the vapor mass flow fraction, ṁẋ h vap + q = const. (9) Equations (9) and (5) state that the enthalpy of vaporization released by condensation is conducted downstream and consumed by evaporation. In previous works a similar problem was investigated in which, different to the present work, the change of the equilibrium pressure at a curved meniscus was neglected [2, 3]. Solutions to the linearized and numerical solutions to the nonlinear problem that results from accounting for the temperature dependence of the material properties were presented. Some conclusions can be carried over to the present work. Little difference was found between the solutions to the linearized and the non-linear problem. The major difference stems from the use of a constant Joule-Thomson coefficient in the linearized description. Hence, the downstream temperatures in the solutions to the non-linear and the linearized problem differ slightly. Also, it was found that it is crucial for the accuracy of the solution to the linearized problem to choose an appropriate kinematic viscosity of the vapor. The kinematic viscosity should be taken as the geometric mean of the viscosities at the beginning and the end of the vapor flow section. This result is obtained from assuming the ideal gas law and a constant dynamic viscosity for the vapor, and integrating Darcy s law. An equivalent and well known formulation is to substitute the kinematic viscosity with the dynamic viscosity times the specific volume of the vapor at the mean pressure in the vapor flow section. 10 Page 10 of 38

12 In the present work, the description of the flow is restricted to cases in which the downstream state of the fluid is an unsaturated vapor. State diagrams such as a Mollier, a temperature-entropy or, most clearly, a pressure-enthalpy diagram show that an isenthalpic change of state leads for decreasing pressure from the dew line into the vapor region, as long as the initial state is a certain distance apart from the critical point. In a region around the critical point, the downstream state can be a two-phase mixture. Hence, restricting the investigation to cases with the downstream state of an unsaturated vapor means to stay apart from the critical point, in a region where the linearized governing equations very well approximate the non-linear equations. 2.2 Flow map Nine different types of flow can occur. In anticipation of results obtained in Section 2.3, in Fig. 1 a flow map is presented to show the conditions under which these types of flow occur. On the abscissa, the permeability of the membrane is plotted, made dimensionless with the critical permeability κ K, ν l k m,l κ K = h vap (dp K /dt ). (10) Here, k m,l refers to the effective thermal conductivity of the liquid-filled membrane and ν l denotes the kinematic viscosity of the liquid phase of the fluid. From eq. (7) we obtain dp K dt = p K dp sat p sat dt + p 2σ K r v l RT ( 1 T 1 dσ σ dt 1 ) dv l. (11) v l dt The second term on the right-hand side is usually small. The critical permeability was presented in Ref. [1], denoted there as κ c. However, the definitions 11 Page 11 of 38

13 of κ K and κ c differ slightly, κ c = κ K (dp K /dt )/(dp sat /dt ). Both definitions yield the same value for a contact angle of 90. The six different flow types for wetting systems are shown on the positive side of the ordinate in Fig. 1. On the negative side, the three flow types for nonwetting systems are shown. Different scalings are used on the positive and the negative side. On the positive side, the inverse of the group C cc is shown, C cc =(p 1 p 2 )(1 n)/(p sat p K ). (12) Here, n is a characteristic quantity for the Joule-Thomson process of a saturated vapor. It is defined by n =( T/ p) h (dp K /dt ), (13) evaluated at the upstream temperature T 1.Forn>1 the downstream state can be a two-phase mixture, for n<1the downstream state is an unsaturated vapor. For brevity, where not noted otherwise, p sat and p K refer to their respective values at T 1, p sat (T 1 )andp K (T 1 ). The group C cc is always positive. With (p 1 p 2 )n = p K (T 1 ) p K (T 2 ) we obtain that, for C cc > 1, the downstream pressure is smaller than the vapor pressure at a meniscus within the membrane, p 2 <p K (T 2 ), hence the fluid evaporates in the interior of the membrane and vapor flows downstream through the remaining part of the membrane. For C cc < 1 the fluid evaporates at a meniscus located at the downstream front of the membrane. For non-wetting systems the group C cap is used. It is defined by C cap = n(p 1 p 2 ), (14) 2σ/r 12 Page 12 of 38

14 which is always negative because the the radius of curvature r is negative for a non-wetting fluid. C cap approximately relates the difference between p 1 and the pressure in the process for which the fluid remains in its liquid state to the capillary pressure. Hence, for C cap > 1 liquid does not penetrate into the membrane and a liquid film is formed in front of the membrane. The different types of flow are distinguished as follows: For κ>κ K the fluid condenses partially and a two-phase mixture flows through a part of the membrane. For κ κ K the fluid condenses fully and liquid flows through a part or all of the membrane. Furthermore, for a wetting system a liquid film is formed in front of the membrane if κ<κ f,withκ f κ K.ForC cc 1 the entire membrane is filled with liquid or a two-phase mixture and a meniscus is located at the downstream end of the membrane. Hence, for C cc 1, which corresponds to a small pressure difference, the phenomenon of capillary condensation is reproduced. Regarding non-wetting systems, liquid cannot penetrate into the membrane for C cap > 1 (p sat p K )/(2σ/r), where the second term on the right-hand side is a small correction to ( 1). The curves κ f /κ K (C cc ), C cc,2ph (κ/κ K ) and the correction to C cap = 1 for the line at C cap 1 depend on the properties of the fluid, the rest of the flow map stays the same for all fluids. 2.3 Solutions In the following, the solutions to the different types of flow are presented in detail. The governing equations for each type of flow are a subset of the equations listed in Table 1. The key states in the process are defined in Table Page 13 of 38

15 First, the six solutions for wetted membranes, then the three solutions for non-wetted membranes are presented, in the order from large to small pressure differences and from small to large permeabilities. For a sufficiently large pressure differences, C cc > 1, and small permeabilities, κ<κ f,whereκ f is given below by eq. (17), a liquid film forms in front of the membrane, liquid flows through a part of the membrane and evaporates completely at a front of phase change within the membrane. Vapor flows through the remaining part of the membrane. Such a flow configuration was described previously by several authors [1, 5, 6, 19]. The locations of the surface of the liquid film and of the front of phase change in the interior of the membrane are at z = δ f and z = δ m, respectively. This type of flow is governed by eqs. (34), (35, with z = δ m ), (37, z = L δ m ), (38) and (45), see Table 1. The solution is given by ṁ = κ ( pk p 2 n(p 1 p 2 ) + p ) 1 p K + n(p 1 p 2 )+2σ/r, (15) L ν g ν l ( ) δ f = Lk (p l 1 p K ) n(p 1 p 2 ) 1 κ Kκ 2σ/r k m,l [p K p 2 n(p 1 p 2 )] ν l ν g + p 1 p K + n(p 1 p 2 )+2σ/r, (16) where k l is the thermal conductivity of the saturated liquid. The maximum permeability at which this type of flow occurs, κ f, is calculated from eq. (16) by demanding that the thickness of the liquid film becomes zero, δ f =0,which yields κ f n(p 1 p 2 ) = κ K p 1 p K + n(p 1 p 2 )+2σ/r for C cc 1. (17) For κ f κ κ K and C cc > 1 a meniscus is located at the upstream front of the membrane, liquid flows through a part of the membrane and evaporates within the membrane. Such a flow configuration was described by a number 14 Page 14 of 38

16 of authors [1, 5 7, 9, 12, 14, 19]. In front of the membrane, the temperature of the vapor increases. The curvature of the meniscus at the upstream front increases with increasing permeability from zero at κ = κ f to its maximum at κ = κ K. In Fig. 2 a sketch of the flow through a single pore, the pressure and the temperature distribution and the states of the fluid in a p-t diagram are shown for isobutane and a polypropylene membrane. The governing equations are eqs. (35, z = δ m ), (37, z = L δ m ), (39) and (45). The mass flow is given by ṁ = κ p K p 2 n(p 1 p 2 ) + p ( ) 1 p K + n(p 1 p 2 ) 1+ 2σ/r p 1 p K +2σ/r ( (18) L ν g 1+ κ 2σ/r ) νl κ K p 1 p K For κ>κ K and C cc > 1 the fluid condenses partially at the upstream surface of the membrane, a two-phase mixture flows through a part of the membrane and evaporates completely at an evaporation front within the membrane, cf. Ref. [1]. For κ/κ K = 2 the flow field is plotted in Fig. 3 for the flow of isobutane through a polypropylene membrane. The governing eqs. (36), (37), (40) and (42) apply. Eliminating p 7 p 8 and T 7 T 8 from eqs. (36) yields an implicit equation for the vapor mass flow fraction ẋ, k m,2ph ν 2ph 1 ẋ = κ h vap (dp K /dt ). (19) Here, k m,2ph is the effective thermal conductivity of the membrane filled with a two-phase mixture and ν 2ph is the effective kinematic viscosity of the twophase mixture. Both k m,2ph and ν 2ph are functions of ẋ. The form of these functions depends on the chosen two-phase model. Assuming homogeneous flow, we have ẋ = x = αv 2ph /v g, (20) 15 Page 15 of 38

17 ν 2ph =[αµ g +(1 α)µ l ]v 2ph, (21) k m,2ph =(1 ɛ)k m + ɛ[αk g +(1 α)k l ]. (22) Here, x and α stand for the mass fraction and the volume fraction of the vapor, respectively, and µ g and µ l denote the dynamic viscosities of the vapor and the liquid, respectively. The specific volume of the two-phase mixture is given by v 2ph =[(1 α)/v l + α/v g ] 1. With the solution of eqs. (19) to (22) the mass flux is given by ṁ = κ [ pk p 2 n(p 1 p 2 ) + p ] 1 p K + n(p 1 p 2 ). (23) L ν g Equation (19) yields the expression for the critical permeability, eq. (10), by taking the values of ẋ, k m,2ph and ν 2ph to the limit of full condensation, ẋ 0, k m,2ph k m,l and ν 2ph ν l. The next three types of flow occur for small pressure differences, C cc 1. With respect to membrane processes, these flow types are presumably of little interest because the pressure differences are too small to cause an appreciable mass transport through the membrane. However, the solutions are briefly reported. The three types of flow have the common feature that the entire membrane is filled with liquid or a two-phase mixture and the fluid evaporates at a meniscus located at the downstream end of the membrane. ν 2ph For κ<κ f and C cc 1 a liquid film forms in front of the membrane and the entire membrane is filled with liquid, see also Refs. [5, 6]. Equations (34), (35, z = L), (38) and (46) apply. The solution for the mass flux is given by ṁ = κ(p 1 p 2 ) ν l L ( 1+ 2σ/r ) p K np 1 p 1 p K p K n(p 1 p 2 ). (24) 16 Page 16 of 38

18 For the thickness of the liquid film we obtain δ f = L m,l 1+ 2σ/r p K np 1 p 1 p K p K n(p 1 p 2 ) k l k κ K κ n 1 (25) and the permeability κ f at which δ f =0isgivenby ( κ f = n 1+ 2σ/r ) 1 p K np 1 for C cc 1. (26) κ K p 1 p K p K n(p 1 p 2 ) For κ f κ κ K and C cc 1orforκ>κ K and C cc C cc,2ph,wherec cc,2ph is a function of κ/κ K, the fluid condenses at a meniscus at the upstream front of the membrane, liquid flows through the entire membrane and evaporates at a meniscus at the downstream front of the membrane. Such a flow configuration was also described in Refs. [5, 6, 19, 20]. The flow is governed by eqs. (35, z = L), (39) and (46), which yields for the mass flux ṁ = (p 1 p 2 )κ ν l L p 1 p K 2σ/r + n + p K np 1 p K n(p 1 p 2 ) κ κ K + p 1 p K 2σ/r. (27) The expression for C cc,2ph is found from the condition that, for C cc = C cc,2ph, the pressure difference across the upstream meniscus becomes the possible maximum, p 1 p 5 =2σ/r. Using this and solving eqs. (35, z = L), (39) and (46) yields, together with eq. (12), κ 1 κ K 1 n C cc,2ph 2σ/r 1 C cc,2ph ( p 1 p K 1 n p1 1 n p K 1 ) =1+ n C cc,2ph 1 n C cc,2ph.(28) ( Simplifying eq. (28) by considering that 1 n p1 1 n p K 1 ) C cc,2ph 1weobtain C cc,2ph = ( κ + p )/[ 1 p K κ + p 1 p K κ K 2σ/r κ K 2σ/r ] κ/κ K n. (29) 1 n For κ>κ K and C cc,2ph >C cc > 1 the fluid condenses partially at the upstream surface of the membrane, a two-phase mixture flows through a part 17 Page 17 of 38

19 of the membrane and condenses fully at a second condensation front within the membrane. Liquid flows through the remaining part of the membrane. Equations (35, z = L δ m ), (36), (40), (41) and (46) apply. The mass flux is given by ṁ = (p [( 1 p 2 )κ ν l L ( κ κ K 1 ) 1 ν )( l n + p ) 1 p K ν 2ph p 1 p 2 ( 1 κ )( ν l 1+ 2σ/r p K np 1 κ K ν 2ph p 1 p K p K n(p 1 p 2 ) 2σ/r )]. (30) p 1 p 2 For membranes which are not wetted by the liquid phase of the fluid three types of flow are possible. These flow types have the common feature that a liquid film of finite thickness forms in front of the membrane. If a large pressure difference is applied to a membrane with a small permeability, C cap < 1 (p 1 p K )/(2σ/r) andκ κ K, a liquid film forms in front of the membrane, liquid flows through a part of the membrane and evaporates at a front within the membrane. The flow is governed by the same equations as the type of flow for a wetting liquid with C cc > 1andκ<κ f, eqs. (34), (35, z = δ m ), (37, z = L δ m ), (38) and (45). The solution is given by eqs. (15) and (16). In contrast to the case for a wetting liquid, here always a liquid film of finite thickness forms. The value of C cap below of which this type of flow occurs is derived from the observation that liquid penetrates into the membrane. Hence, the inequality p 1 p 9 > 2σ/r holds. Substituting for p 9 from eq. (45) and using the definition for C cap,eq.(14),yields C cap < 1 p 1 p K 2σ/r. (31) For a pressure difference which is too small to overcome the capillary pressure, 18 Page 18 of 38

20 C cap 1 (p 1 p K )/(2σ/r), a liquid film forms in front of the membrane and the liquid evaporates at a meniscus located at the upstream surface of the membrane, see Fig. 4. The flow is governed by eqs. (34), (37, z = δ m ), (38) and (44). The mass flux is given by ṁ = (p 1 p 2 )κ ν g L p 1 p 1 K n ( ). (32) 1+ p 1 p K 2σ/r 1 n p 1 p 2 p K This type of flow is similar to the flow configuration of membrane distillation [21, 22]. The difference is that in membrane distillation there is liquid throughout at the upstream side, not only a liquid film. Hence, in membrane distillation the enthalpy of vaporization must be supplied from the outside and conducted to the evaporation front. Here, the enthalpy of vaporization is supplied by condensation of the vapor at the surface of the liquid film and conducted only a short distance through the liquid film. For κ>κ K and C cap < 1 (p 1 p K )/(2σ/r) a liquid film forms in front of the membrane, the liquid evaporates partially at the upstream front of the membrane and a two-phase mixture flows through a part of the membrane. The fluid evaporates completely at a front within the membrane and vapor flows through the remaining part of the membrane. The mass flux is given by ṁ = κ { ( 1 [n(p 1 p 2 )+p 1 p K ] 1 ) + p 1 p 2 + 2σ/r }. (33) L ν 2ph ν g ν g ν 2ph 3 Experiments Schneider [1] reported experimental data for the flow of isobutane through a Celgard c 2500 polypropylene membrane. At the upstream side, the isobutane was in a state of saturated vapor at 20 C. The downstream side was kept at 19 Page 19 of 38

21 atmospheric pressure. A mass flux of 0.3 kg/m 2 s was measured. At 20 C, the saturation pressure of isobutane is p sat =3.02 bar, the specific volumes of the saturated liquid and the saturated vapor are v l = m 3 /kg and v g = m 3 /kg, respectively, and the surface tension is σ =10.7 mn/m. The specific enthalpy of vaporization is h vap = kj/kg and the Joule-Thomson coefficient is (dt/dp) h =2.703 K/bar. The dynamic viscosities are given by µ l = Pas [23] and µ g = Pas, respectively, and the thermal conductivity is k l = W/mK. Except where otherwise noted, the above values were taken from Ref. [24]. The permeability of Celgard c 2500, κ = m 2, was obtained by measuring the mass flux of liquid isobutane through the membrane [1]. The porosity, ɛ =0.45, was taken from the product data sheet. An image taken with transmission electron microscopy shows that the pores are approximately 400 nm long and between 20 and 150 nm wide [25]. The majority of the pores seem to be about 50 nm wide [25, Fig. 1(C)], hence the membrane is modeled to consist of parallel, tortuous slits with a width b =50nm.With the permeability related to the width b of the slits by κ =(ɛ/τ)b 2 /12, we obtain a tortuosity of τ =5.8, which is in agreement with values reported elsewhere [26, 27] (see also [28]). The thermal conductivity of polypropylene is k m =0.22 W/mK [29]. The coefficient β is determined from measurements of the air flow rate given in the product data sheet, which yields β = Page 20 of 38

22 4 Results For the experiment reported above, the mass flux is calculated as follows: From the material properties we obtain n =0.243 and κ K = m 2. From the applied pressure drop, together with the assumption of ideal wetting, we have C cc = 28. Hence, because κ<κ K and C cc > 1atypeofflow occurs which is shown in Fig. 2. First, the apparent vapor viscosity is calculated. The pressures at the boundaries of the vapor flow section are p 9 and p 2.Withp 9 given from eq. (45) the geometric mean of the kinematic viscosities is ν =[(1/2)(ν ν 1 2 )] 1 = m 2 /s. Estimating the mean free path from ν yields Kn = 0.3. The apparent vapor viscosity becomes ν app = m 2 /s. Substituting ν app for ν g in eq. (18) yields the mass flux ṁ =0.25 kg/m 2 s. A posteriori, the assumption of isenthalpic flow and the omission of the kinetic energy in the energy balance, eq. (4), can now be confirmed. With c p = 2418 J/kgK and T 1 T 2 =5.4 Kfor2barpressure difference the Eckert number is Ec = For comparison, calculating the mass flux for the isothermal flow of a vapor that does not undergo phase changes, a flow configuration which is equivalent to the isothermal flow of a non-wetting fluid, yields ṁ =0.2 kg/m 2 swhen accounting for viscous and molecular flow contributions. Applying a model of the isothermal flow of an ideally wetting fluid gives a flow configuration with a flat meniscus at the upstream surface of the membrane, liquid flow through a part of the membrane, a meniscus with a radius of curvature r = b within the membrane and vapor flow through the remaining part of the membrane[7,9,12].amassfluxofṁ =1.12 kg/m 2 s is obtained, which is much larger than according to the non-isothermal description presented in 21 Page 21 of 38

23 this work because the flat meniscus at the upstream surface of the membrane does not cause a pressure difference which partially would balance the large pressure difference caused by the meniscus within the membrane. In Fig. 5 the mass flux vs. the permeability is plotted for the flow of isobutane at 20 C through a membrane that, apart from the varying permeability, has the properties of a Celgard c polypropylene film. Experimental data is compared with theoretical predictions according to the present analysis and with calculations assuming isothermal flow. To show the influence of the wetting properties of the fluid, the mass flux was calculated for the two cases of an ideally wetting and an ideally nonwetting liquid phase of the fluid, respectively. In the range of permeabilities presented here, the wetting properties have a significant influence on the mass flux. Obviously, a small contact angle enhances the mass flux. The description of the isothermal flow of an ideally wetting fluid [7, 9, 12], the dot-dashed line, yields a mass flux which is nearly an order of magnitude larger than according to the present, non-isothermal description. Assuming an isothermal flow of a non-wetting liquid, the dotted line in Fig. 5, yields a mass flux which is nearly identical to the non-isothermal description of a wetting fluid, the solid line, for permeabilities a few times larger than the critical permeability. The isothermal flow of a non-wetting liquid is equivalent to the flow of a vapor that does not undergo phase changes, hence the mass flux is readily calculated by applying Darcy s law and possibly accounting for Knudsen flow. The difference to the non-isothermal description is small because the two-phase mixture predicted by the latter has a vapor volume fraction very close to one, see Fig. 6. Therefore, the flow resistance of the 22 Page 22 of 38

24 two-phase mixture is dominated by the vapor viscosity if a homogeneous twophase model is used. The effect of using different two-phase models is discussed elsewhere, see Fig. 10 in Ref. [3]. The non-isothermal description for a wetting fluid has three solutions in a small region at κ/κ K 1. One solution exists for a flow type with full condensation as depicted in Fig. 2, which is indicated by the dot-dot-dashed line in Fig. 5, and two solutions exist for a flow with partial condensation and two-phase flow as depicted in Fig. 3, which is indicated by the solid line in Fig. 5. The additional solutions are owed to the fact that the equation for the mass flow fraction, eq. (19), together with the equations for the utilized two-phase model, eqs. (20) to (22), have two solutions in a small range of permeabilities. In Fig. 6 the left-hand side and the right-hand side of eq. (19), using the definitions of eqs. (20) to (22), is plotted vs. α for four different permeabilities. The plot shows that two solutions exist if κ is slightly smaller than κ K. Multiple solutions may not exist for two-phase models other than eqs. (20) to (22). The existence of multiple solutions was not addressed in the discussion of the flow map depicted in Fig. 1. Fig. 6 also shows that the vapor volume fraction rapidly approaches one for κ>κ K and increasing the permeability, as was mentioned above. 5 Conclusions When a saturated vapor flows through a membrane with a small permeability, the fluid condenses partially or fully and re-evaporates. When the downstream state is an unsaturated vapor, nine different types of flow are possible. It depends on the permeability of the membrane with respect to the critical 23 Page 23 of 38

25 permeability, on the applied pressure drop and on the wetting properties of the fluid, which type of flow occurs. To describe the different types of flow it is necessary to include the momentum and the energy balance, the Joule- Thomson effect, the pressure difference across fronts of phase change and the change of the equilibrium pressure in the interior of the porous medium. However, the mass flux of a wetting fluid that flows through a membrane with a permeability a few times larger than the critical permeability is very well approximated by assuming isothermal flow, e.g., by applying Darcy s law. Such an approximation would be much in error for a permeability smaller than the critical permeability. For a vanishing pressure difference across the membrane, the description presented in this work reproduces the effect of capillary condensation. The flow of a non-wetting fluid must be described as a non-isothermal process for small and for large permeabilities. For not too large permeabilities, a flow configuration similar to that of membrane distillation occurs. The advantage of the process described here is that the enthalpy of vaporization does not need to be supplied from the outside to the flow. List of Symbols a radius of a capillary (m) b width of slits (m) C cc dimensionless pressure difference, cf. eq. (12) C cap dimensionless pressure difference, cf. eq. (14) c p specific heat capacity at constant pressure (J/kgK) Ec Eckert number, Ec = u 2 /[c p (T 1 T 2 )] 24 Page 24 of 38

26 h h vap Kn specific enthalpy (J/kg) specific enthalpy of vaporization (J/kg) Knudsen number, Kn = λ/(2a) k thermal conductivity (W/m K) L membrane thickness (m) ṁ mass flux (kg/m 2 s) n dimensionless coefficient, cf. eq. (13) p pressure (Pa) p K equilibrium pressure at a curved meniscus (Pa), cf. eq. (7) p sat saturation pressure (Pa) q heat flux (W/m 2 ) R specific gas constant (J/kg K) r radius of curvature (m) T absolute temperature (K) u velocity (m/s) v specific volume (m 3 /kg) x vapor mass fraction ẋ vapor mass flow fraction z coordinate normal to membrane surface (m) Greek letters α vapor volume fraction β correction factor for free molecular flow, cf. eq. (3) δ m δ f location of front of phase change (m) thickness of liquid film (m) 25 Page 25 of 38

27 ɛ θ porosity contact angle κ permeability (m 2 ) κ f permeability at which a liquid film forms (m 2 ), cf. eqs. (17) and (26) κ K critical permeability (m 2 ), cf. eq. (10) λ mean free path (m) µ dynamic viscosity (Pa s) ν kinematic viscosity (m 2 /s) ν app apparent kinematic viscosity of the vapor (m 2 /s), cf. eq. (3) σ surface tension (N/m) τ tortuosity Subscripts 1 upstream state 2 downstream state 3 9 states of the fluid, cf. Table 2 2ph two-phase mixture g vapor phase l liquid phase m membrane Acknowledgements Prof. Schneider brought the flow process investigated here to my attention. His support and his interest in the progress of this work are greatly acknowledged. 26 Page 26 of 38

28 References [1] W. Schneider, Vapor flow through a porous membrane a throttling process with condensation and evaporation, Acta Mech. 47 (1983) [2] T. Loimer, A Joule-Thomson process with condensation and evaporation, Proc. Appl. Math. Mech. 3 (2003) [3] T. Loimer, The flow of saturated vapor with positive Joule-Thomson coefficient through a porous membrane, in: Proc. Fourth Europ. Thermal Sci. Conf., 29th 31st Mar. 2004, Birmingham, UK, 2004, paper POM1 S14. [4] T. Loimer, Effects of capillarity on a Joule-Thomson process of a fluid near saturation, in: Y. Wang, K. Hutter (Eds.), Trends in Applications of Mathematics to Mechanics. Proc. 14th STAMM, Aug , 2004, Seeheim, Germany, 2005, pp [5] H. Rhim, S.-T. Hwang, Transport of capillary condensate, J. Colloid Interface Sci. 52 (1975) [6] K.-H. Lee, S.-T. Hwang, The transport of condensible vapors through a microporous Vycor glass membrane, J. Colloid Interface Sci. 110 (1986) [7] B. Abeles, L.-F. Chen, J. W. Johnson, J. M. Drake, Capillary condensation and surface flow in microporous Vycor glass, Israel J. Chem. 31 (1991) [8] K. P. Tzevelekos, G. E. Romanos, E. S. Kikkinides, N. K. Kanellopoulos, V. Kaselouri, Experimental investigation on separations of condensable from non-condensable vapors using mesoporous membranes, Micropor. Mesopor. Mat. 31 (1999) [9] P. S. Sidhu, E. L. Cussler, Diffusion and capillary flow in track-etched membranes, J. Membr. Sci. 182 (2001) Page 27 of 38

29 [10] J.-S. Bae, D. D. Do, Study on diffusion and flow of benzene, n-hexane and CCl 4 in activated carbon by a differential permeation method, Chem. Eng. Sci. 57 (2002) [11] P. Uchytil, R. Petrickovic, S. Thomas, A. Seidel-Morgenstern, Influence of capillary condensation effects on mass transport through porous membranes, Sep. Purif. Technol. 33 (2003) [12] P. Uchytil, R. Petrickovic, A. Seidel-Morgenstern, Study of capillary condensation of butane in a Vycor glass membrane, J. Membr. Sci. 264 (2005) [13] K. P. Tzevelekos, E. S. Kikkinides, M. E. Kainourgiakis, A. K. Stubos, N. K. Kanellopoulos, V. Kaselouri, Adsorption-desorption flow of condensable vapors through mesoporous media: network modeling and percolation theory, J. Colloid Interface Sci. 223 (2000) [14] J.-S. Bae, D. D. Do, Permeability of subcritical hydrocarbons in activated carbon, AIChE J. 51 (2005) [15] K. Kammermeyer, Gas and vapor separations by means of membranes, in: E. S. Perry (Ed.), Progress in Separation and Purification, Vol. 1, Interscience, 1968, pp [16] J.-G. Choi, D. D. Do, H. D. Do, Surface diffusion of adsorbed molecules in porous media: monolayer, multilayer, and capillary condensation regimes, Ind. Eng. Chem. Res. 40 (2001) [17] A. E. Scheidegger, The Physics of Flow through Porous Media, 3rd Edition, University of Toronto Press, [18] B. R. Bird, W. E. Stewart, E. N. Lightfoot, Transport Phenomena, Wiley, [19] R. J. R. Uhlhorn, K. Keizer, A. J. Burggraaf, Gas transport and separation with ceramic membranes. Part I. Multilayer diffusion and capillary 28 Page 28 of 38

30 condensation, J. Membr. Sci. 66 (1992) [20] K. P. Tzevelekos, E. S. Kikkinides, A. K. Stubos, M. E. Kainourgiakis, N. K. Kanellopoulos, On the possibility of characterising mesoporous materials by permeability measurements of condensable vapours: theory and experiments, Adv. Colloid Interface Sci (1998) [21] K. W. Lawson, D. R. Lloyd, Membrane distillation, J. Membr. Sci. 124 (1997) [22] E. Curcio, E. Drioli, Membrane distillation and related operations A review, Sep. Purif. Rev. 34 (2005) [23] C. Wohlfarth, B. Wohlfarth, Pure organic liquids, in: M. D. Lechner (Ed.), Viscosity of Pure Organic Liquids and Binary Liquid Mixtures, Vol. 18B of Landolt-Börnstein. New Series, Group IV: Physical Chemistry, Springer, [24] E. W. Lemmon, M. O. McLinden, D. G. Friend, Thermophysical properties of fluid systems, in: P. Linstrom, W. Mallard (Eds.), NIST Chemistry WebBook, NIST Standard Reference Database Number 69, National Institute of Standards and Technology, Jun. 2005, ( [25] T. Sarada, L. C. Sawyer, M. I. Ostler, Three dimensional structure of Celgard r microporous membranes, J. Membr. Sci. 15 (1983) [26] R. Prasad, K. K. Sirkar, Dispersion-free solvent extraction with microporous hollow-fiber modules, AIChE J. 34 (1988) [27] A. Kiani, R. R. Bhave, K. K. Sirkar, Solvent extraction with immobilized interfaces in a microporous hydrophobic membrane, J. Membr. Sci. 20 (1984) [28] G. T. Vladisavljevic, M. Shimizu, T. Nakashima, Permeability of hydrophilic and hydrophobic Shirasu-porous-glass (SPG) membranes to 29 Page 29 of 38

31 pure liquids and its microstructure, J. Membr. Sci. 250 (2005) [29] VDI-Gesellschaft Verfahrenstechnik und Chemieingenieurwesen (Ed.), VDI-Wärmeatlas, 9th Edition, VDI-Verlag, Düsseldorf, Page 30 of 38

32 1 C cc C cap κ f/κk C cc,2ph κ/κ K Figure 1. Flow map. The nine different types of flow that can occur are shown in a map of Ccc 1 and Ccap 1 vs. the dimensionless permeability κ/κ K. The fluid is isobutane at 80 C. Wetting systems are shown on the positive side of the ordinate, non-wetting systems on the negative side. Note the different scaling for wetting and non-wetting systems. The various flow types are indicated by sketches of the flow in a single pore. Gray shaded areas denote liquid, gray striped areas stand for a two-phase mixture. 31 Page 31 of 38

33 a) b) c) p [bar] p [bar] δ m L 0 L z δ m p sat pk T [K] Figure 2. Flow of isobutane under wetting conditions, θ = 0, κ/κ K = 0.5, κ f < κ and C cc = The membrane has the thermal conductivity of a Celgard c polypropylene film, k m = 0.22 W/mK and slit-like pores, hence κ K = m 2. The width of the slits is b = 36 nm. The pressure difference is p 1 p 2 = 2 bar. (a) Sketch of the flow in a single pore. Numbers refer to different states of the fluid, cf. Table 2. (b) Pressure (solid line, left ordinate) and temperature (dashed line, right ordinate) distribution. (c) Path of the process in a p-t diagram. The saturation pressure, p sat, is indicated by a solid line, p K by a dashed line z T [K] 32 Page 32 of 38

34 a) b) c) p [bar] p [bar] δ m L 0 δ m L z p sat p K 8, T [K] Figure 3. Flow of isobutane under wetting conditions, θ = 0, κ/κ K = 2, and C cc = The permeability corresponds to a pore width of b = 71 nm, the pressure difference is p 1 p 2 = 2 bar. For remaining caption, see Fig. 2. (a) Sketch of the flow in a single pore. The gray striped area stands for two-phase flow. 1 3,7 2 z T [K] 33 Page 33 of 38

35 a) b) c) p [bar] p [bar] δ f δ f L 0 L z p K T [K] Figure 4. Flow of isobutane under non-wetting conditions, θ = 180, κ/κ K =2,and C cap = The permeability corresponds to a pore width of b =71nm,the pressure difference is p 1 p 2 = 2 bar. For remaining caption, see Fig. 2. p sat 1,4 z T [K] 34 Page 34 of 38

36 m [kg/m 2 s] κ/κ K a) b) m [kg/m 2 s] κ/κ K Figure 5. Mass flux ṁ vs. κ/κ K for the flow of isobutane at 20 C through a Celgard c membrane, p 1 p 2 = 2 bar. Figure 5a is an enlarged view of Fig. 5b. Shown are the experimental value from Ref. [1] (diamond symbol) and calculations of the mass flux according to the present analysis of a non-isothermal flow and assuming perfect wetting, θ =0 (dot-dot-dashed and solid line) and non-wetting, θ = 180 (dashed line). The mass flux is also plotted assuming isothermal flow and θ =0 (dot-dashed line) and θ = 180 (dotted line). The flow configurations are sketched in Fig. 5b. For non-isothermal flow and θ =0, the flow type changes at κ/κ K from liquid-vapor to 2ph-vapor flow. 35 Page 35 of 38

37 κ/κ K = 0.95 κ/κ K = 2 κ/κ K = 1 κ/κ K = α Figure 6. Left-hand side (solid line) and right-hand side (dotted lines) of eq. (19) vs. vapor volume fraction α for four different permeabilities. 36 Page 36 of 38

38 states description equations 4 5 liquid film p 5 = p 4, T 4 T 5 = ṁ h vap δ f /k l (34) 5 6 liquid flow p 5 p 6 = ṁ zν l /κ a, T 5 T 6 = ṁ h vap z/k m,l a (35) 7 8 2ph. flow p 7 p 8 = ṁδ m ν 2ph /κ, T 7 T 8 = ṁ(1 ẋ) h vap δ m /k m,2ph p 7 p 8 = dp K dt (T 7 T 8 ) (36) 9 2 vapor flow p 9 p 2 = ṁ zν app /κ b, T 9 = T 2 (37) 1 4 vap. liq., z = δ f p 1 = p 4, T 1 = T 4 (38) 3 5 vap. liq., z =0 p 1 p 5 = dp k T 3 T 1, T 5 = T 3 2σ/r dt p 1 p K (39) 3 7 vap. 2ph, z =0 p 7 = p 1, T 7 = T 3 ; T 3 T 1 = p 1 p K dp K /dt (40) 8 5 2ph liq., z = δ m p 8 p 5 =2σ/r, T 5 = T 8 (41) 8 9 2ph vap., z = δ m p 9 = p 8, T 9 = T 8 (42) 5 7 liq. 2ph, z = δ m p 5 p 7 = 2σ/r, T 7 = T 5 (43) 5 9 liq. vap., z =0 p 1 p 5 p 9 p = K n(p 1 p 2 ) ( ), 2σ/r 2σ/r + p 1 p K n(p 1 p 2 ) p1 p K 1 T 9 = T 5 (44) 6 9 liq. vap., z = δ m p 6 p 9 = 2σ/r, T 9 = T 6, p 9 = p K n(p 1 p 2 ) (45) ( ) p 2 p 6 (p 1 p 2 ) 1 p 1 p 6 2 liq. vap., z = L = K n ( ), T 2σ/r p 1 p K n(p 1 p 2 ) p1 2 = T 6 (46) p K 1 a z = δ m, L or (L δ m ) b z = L δ m or L Table 1 Governing, linearized equations. Equations (34) to (37) are the momentum and energy balances for the different flow regions. Eq. (36), in addition, contains the condition that a 2ph-mixture is always in thermodynamic equilibrium, p = p K (T ). Equations (38) to (46) are the conditions that apply across fronts of phase change. The states of the fluid are defined in Table Page 37 of 38