The Influence of Temperature in the Forward Osmosis Process

Chapter FourMathematical ModelChapter FourTHEORETICAL ANALYSISMA andMathematical ModelingPurpose of the survey is to probe of temperature as a factor that influences the transport of water system across the tissue layer in FO procedure. The steady-state theoretical fibs have been developed to foretell water iux (JouleTungsten) as map of temperature (Thymine) and bulk constriction (C) ( i.e. Draw and Feed absorption ) . It was anyways study the consequence of temperature on some belongingss, such as Solute distribution coefficient (CalciferolSecond) , Mass f are coefficient (K) , Permeability coefficient (A) and Solute electric automobile resistance (Km) .4.1 Osmotic PressureThe osmotic burden per unit area (?) of a root word depends on the assimilation of dissolved ions in solution and the temperature of solution, and fanny be computed by utilizing Va nt Hoff comparisonWhitherNis the vant Hoff factor ( histories for the figure of single atoms of a compound dissolved in the solution ) ,?is the osmotic coefficient,Cis the torpedo parsimony ( molar concentration ) of the solution,Roentgenis the gas invariable andThymineis the absolute temperature of the solution. The vant Hoff factor is introduced to cover divergences from apotheosis solution behaviour that include bounded volume occupied by solute molecules and their common attractive voice suck as in new wave derWaals attractive military force ( Howard, 2003 ) . Table 4.1 show osmotic coefficients (?) for a figure of solutes of physiological importance ( Khudair, 2011 ) . For all solutes?depends on the substance and on its concentration. As the concentration of any solute attacks zero its value of?attacks 1. In ideal solution,?= 1 ( Glass tone, 1974 ) .Table 4.1 Osmotic Coefficients (?) and Vant Hoff Factor ( N ) for a Number of SolutesSubstanceVant Hoff Factor (N)OsmoticCoefficients ( ? )NaCl20.93KCl20.92HCl20.95New hampshire4Chlorine220.92NaHCO320.96CaCl230.86MgCl230.89Sodium2So430.74MgSO4 20.58Glucose11.01Sucrose11.024.2 Concentration Polarization4.2.1 orthogonal Concentration PolarizationConcentration polarisation ( CP ) is the accretion of solutes near the tissue layer out and has inauspicious effects on membrane public presentation. The iux of water supply through the membrane brings feed H2O ( incorporating H2O and solute ) to the membrane surface, and as clean H2O iows through the membrane, the solutes accumulate near the membrane surface. Equations for concentration polarisation run on be derived from i?lm theory and the great unwashed balances. Harmonizing to i?lm theory, a spring bed signifiers at the surface of the membrane. body of water and solutes move through the enclosure bed toward the membrane surface. As H2O base on ballss through the membrane, the solute concentration at the membrane surface additions. The concentration gradient in the boundary bed leads to diffusion of solutes back toward the volume nourishment H2O. During uninterrupted operation, a steady-state status is reached in which the solute concentration at the membrane surface is changeless with regard to clip because the convective iow of solutes toward the membrane is balanced by the diffusing iow of solutes off from the surface.A caboodle balance can be developed at the membrane surface as followsMass accretion = mass in ? mass out ( 4.2 )With no accretion of mass at steady province, the solute iux toward the membrane surface must be balanced by iuxes of solute iowing off from the membrane ( receivable to diffusion ) and through the membrane ( into the fall into place ) as followsWhereMeteris mass of solute,Jouletungstenis the experimental permeate H2O magnetic combine,Tis clip,CalciferolSecondis the diffusion coefficient of the solute,omegathe distance perpendicular to membrane surface,Cpeis the solute concentration in the permeate andEis the surface country of membrane. Equation 4.3 applies non hardly at the membrane surface but besides at any plane in the boundary bed because the net solute iux must be changeless throughout the boundary bed to forestall the accretion of solute anyplace indoors that bed ( the start term in equating 4.3 represents the solute that must go through through the boundary bed and the membrane to stop up in the permeate ) . Rearranging and incorporating equation 4.3 across the thickness of the boundary bed with the boundary conditions C ( 0 ) = CMeterand C ( ?Bacillus) = CF, cell, where CF, cellis the concentration of provender cell solution and CMeteris the concentration at the membrane surface, are done in the lowmentioned equationsIntegration outputsWhereKis the mass conveyance coefficient and?Bacillusthickness of the boundary bed, rearranging the equation 4.6 when utilizing the vant Hoff equation the eventually theoretical account from the concentrative external concentration polarisation at each permeate flux, could be calculated utilizingWhere?F, Bis the osmotic force per unit areas of feed solution at the majority and?F, mis the osmotic force per unit areas of the provender solution at the surface membrane. Note that the advocate is positive, he pointed out that ?F, m& A gt ?F, B.The get in solution in touch with the permeate side of membrane is the being diluted at the permeate membrane interface by the permeating H2O ( gloomy and Kessler, 1976 ) . This is called diluted external CP. Both dilutive external CP phenomena cut down and concentrative the effectual osmotic drive air force. A dilutive external CP modulus be identified as above, merely In the present instance, the concentration of the majority greater than concentration of the draw solution at the membrane surface ( i.e. ?D, B& A gt ?D, m) ( Cath et al. , 2006 ) Where?D, mis the osmotic force per unit areas of the draw solution at the membrane surface and?D, Bis the osmotic force per unit areas of draw solution at the majority. The general equation portraiture H2O conveyance in FO, RO, and PRO i s ( Cath et al. , 2006 ) Where,Athe H2O permeableness invariable of the membrane, ? the contemplation coefficient, and a?P is the applied force per unit area. For FO, a?P is zero for RO, a?P & A gt a?? and for PRO, a?? & A gt a?P ( see figure 4.1 ) .Figure 4.1 Direction and magnitude of H2O as a map of ?P.To pattern the flux public presentation of the forward osmosis procedure in the presence of external concentration polarisation, we start with the flux equation for forward osmosis, given asWe assume that the salt does non traverse membrane, the osmotic contemplation coefficient (?) , assume equal 1. Equation 4.10 predicts Flux as maps of driving force merely in the absence dilutive external concentration polarisation or concentrative, which may to be valid merely if the permeating flux is excessively low. When higher flux rates, must be modified to include this equation both(prenominal) the dilutive external concentration polarisation and concentrativeFigure 4.2 ( a ) show s this phenomenon with a muddy symmetric membrane ( McCutcheon and Elimelech, 2006 ) .4.2.2 Internal Concentration PolarizationIf the holeyness support bed of asymmetric membrane confronting feed solution, as is the instance in force per unit area retarded osmosis ( PRO ) , Polarization bed is established along interior of heavy active bed as H2O and solute propagate the porousness bed ( Figure 4.2 ( B ) ) . This is referred to as concentrative internal concentration polarisation, this phenomenon is similar to concentrative external concentration polarisation, except that it takes topographic point within the porous bed, and therefore, can non be underestimated by cross flow ( Lee et al, 1981 ) Obtained look patterning this phenomenon in force per unit area retarded osmosis ( Loeb et al. 1997 ) . This equation describes internal concentration polarisation ( ICP ) the effects and how it links to H2O flux, salt permeableness coefficient ( B ) and H2O permeableness coefficientWhereK mis the opposition to solute diffusion within the membrane porous support bed,Kmis defined asWhereSecondthe membrane structural parametric quantity,?mis the thickness,?is the tortuousness and?is the porousness of the support bed,Kmis a step how easy it can be dissolved widespread support inside and outside Layer, and hence is a step of the strength of ICP. We maintain the usage of theKmterm due to convention established in old surveies on internal concentration polarisation. Salt permeableness coefficient ( B ) is about negligible compared with the other footings in the equation 4.12. Therefore, we ignore salt flux in the way of H2O flux and any transition of salt from the permeate ( draw solution ) side ( Gray et al. , 2006 ) . Therefore, flux can be solved for implicitly from equation 4.12The exponential term in equation 4.14 is the rectification factor that could be considered the concentrative internal concentration polarisation modulus, defined asWhere ?F, Iis the osmotic forc e per unit area of the feed solution on the interior of the active bed within the porous support. The positive advocate indicates that ?F, I& A gt ?F, B, or that the consequence is concentrative. Substitute Equation 4.8 into 4.14 to obtain an analytical theoretical account for the impact of internal and external concentration polarisation on H2O fluxAll the footings in equation 4.16 are readily determined through computations or experiments. From equation we can cipher the flux of H2O through the membrane where feeding solution is placed against asymmetric support bed and the draw solution on the active bed.In forward osmosis applications for desalinization and H2O intervention, the active bed of the membrane faces the provender solution and the porous support bed faces the draw solution ( Kessler and Moody, 1976 ) . As H2O permeates the active bed, the draw solution within the porous infrastructure becomes diluted. This is referred to as dilutive internal concentration polarisati on ( Figure 4.2 ( degree Celsius ) ) . ( Loeb et al, 1997 ) Descriptions likewise flux behaviour in the development of forward osmosisWhen presuming that B = 0 ( i.e. , the salt permeableness is negligible ) and the equation 4.17 is agreement, are acquiring an inexplicit equation for the flux of H2O permeatingHere, ?D, Bis now corrected by the dilutive internal concentration polarisation modulus, given byWhere ?D, Iis the concentration of the draw solution on the interior of the active bed within the porous support. The controvert advocate because the H2O flux is in the way off from the membrane active bed surface, In other words, the concentration polarisation consequence in our instance is dilutive, intending that ?D, I& A lt ?D, Bby replacing equation 4.7 into 4.18, we getThe footings in equation 4.20 are mensurable system conditions and membrane parametric quantities. Note that here dilutive internal concentration polarisation is coupled with concentrative external concentra tion polarisation, whereas in the equation 4.16, concentrative internal concentration polarisation was coupled with dilutive external concentration polarisation.In each of these instances, the external concentration polarisation and internal concentration polarisation moduli all contribute negatively to the overall osmotic drive force. The negative occasion of each addition with higher flux, which suggests a self-limiting flux behaviour, this implies that increasing osmotic drive force will supply decreasing additions in flux ( hell dust et al. , 2010 ) .Figure 4.2 Illustration of osmotic driving force composes for osmosis through several membrane types and orientations, integrating both internal and external concentration polarisation. ( a ) The profile illustrates concentrative and dilutive external CP. ( B ) PRO manner the profile illustrates concentrative internal CP and dilutive external CP. ( degree Celsius ) FO manner the profile illustrates dilutive internal CP and conc entrative external CP (McCutcheon and Elimelech, 2006 ) .In this be given if taking transmembrane temperature difference into history, the temperature being next to membrane surface will besides differ from that in bulk solution due to the happening of estrus transportation. Hence, utilizing vant Hoff jurisprudence for computation of osmotic force per unit area requires the temperature points to be purely in line with the concentration points asWhereC,TDandTFis the concentration, temperature draw and temperature, with the inferiors F, cell ( feed cell solution ) and D, cell ( draw cell solution ) . The theoretical account to foretell H2O iux can be rewritten to a modii?ed by replacing equation 4.21 and 4.22 in 4.20, we getFigure 4.3 gives the conventional illustration of the concentration and temperature proi?les in FO procedure operated under active bed provender solution ( ALFS ) .Figure 4.3Conventional diagram of mass and heat iux proi?les within boundary bed and membrane duri ng FO procedure under ALFS manner in the presence of temperature difference ( TF, cell& A gt TD, cell) .4.3 Heat FluxHeat transportation from the solution to the membrane surface across the boundary bed in the side of the membrane faculty imposes a opposition to mass reassign The temperature at the membrane surface is lower than the corresponding value at the majority stage. This affects negatively the drive force for mass transportation. Under steady province conditions, derived from the heat balance, the heat transportation in the single compartments of system is represented by the undermentioned equationIn which Q denotes the heat flux, and the inferiors FS BL, m and DS BL represent feed solution boundary bed, membrane and draw solution boundary bed. By stipulating the equation 4.24, we obtainWhereHis the single heat transportation coefi?cient,CPthe specii?c heat of H2O,?tungstenthe H2O denseness. Rearranging the equation 4.25 gives denotative looks of temperature near the m embrane surfaces as ( Zhong et al. , 2012 )It is sensible to dei?ne the temperature at interface of SL and AL by averaging theThymineF, mandThymineD, m4.4 Heat Transfer CoefficientsThe finding of heat transportation coefi?cientHis developed on the footing of the correlativity between Nusselt, Reynolds and Prandtl figure ( Holman, 2009 ) .For the laminar flowFor the disruptive flowWhereNu=hL/? , Pr =CPhosphorus/? ,and.Nu is the Nusselt figure,Rheniumthe Reynolds figure andPraseodymiumthe Prandtl figure. TheCPhosphorusis the specii?c heat,Literlength of the channel,the impulsive viscousness, and ? the thermic conduction of NaCl solution. The valueis obtained harmonizing to = , in which?is the solution denseness, and?the kinematic viscousness. The dependance of?on temperature can be described byWhereAndare the thermic conduction of H2O at temperature T and 298.15 K. The heat transportation coefficientHcalculated byWhere happenNufrom equation 4.29 or 4.30The overall heat transportation coefficientHmof FO membrane embodies the thermic conduction of both liquid-phase H2O go throughing the micro pores and the solid-phase membrane4.5 Mass Transfer CoefficientThe mass transportation coefficient is a map of provender flow rate, cell geometry and solute system. Generalized correlativities of mass transportation, which have been used by several writers ( Sourirajan, 1970 ) , suggest that the Sherwood figure,Sh,is related to the Reynolds figure,Re,and Schmidt figure,Sc,as For the laminar flowFor the disruptive flowWhereand.Shis the Sherwood figure,Scandiumthe Schmidt figure andvitamin DHis the hydraulic diameter, the hydraulic diameter is dei?ned asWhere tungsten and h the channel breadth and channel tallness severally. The parametric quantities,CalciferolSecondand?rely strongly on temperature, which can be quantitatively determined by empirical equations below. For aqueous electrolyte like NaCl,CalciferolSecondvalue of the ions is presented by ( Beijing, 1988 )Where Nis the absolute valley of ions ( i.e. N=1 ) , and ?is the tantamount conduction of Na+and Clions, estimated as( 4.40 )In which( 5.110-3m2/? for Na ions 7.6410-3m2/? for chloride ions ) is the mention tantamount conduction at 298.15 K temperature coefficient,,forSodium+, and,,for, severally.The empirical equations were employed to gauge kinematic viscousness of NaCl solution asWhereis the H2O viscousness at temperature T, expressed asIn whichvitamin E= 0.12,degree Fahrenheit= -0.44,-= -3.713,I=2.792 are the fitting parametric quantities,CSecondthe NaCl molar concentration, andThymineRoentgenthe normalized temperature.There is besides another manner to cipher diffusion coefficient in the liquid stage of a dilute solution can be estimated by the Stokes Einstein equation if the solute radius is clearly larger than the consequence radiusWhereKBacillusis the Boltzmann invariable, T ( K ) is the absolute temperature, is the dynamic viscousness of the liquid and ROis the radius of the sol ute. To cipher diffusion coefficients in aqueous solutions predict that diffusion coefficients really linearly with temperature and reciprocally with viscousness. Indeed, harmonizing to Li and Gregory, ( 1974 ) .In instance of the stokes Einstein relation the diffusion coefficientD ( T )at a temperatureThymineis given asWhere D( TO)is the diffusion coefficient at a mention temperatureThymineOand ( T )and ( TO)are the dynamic viscousnesss at temperaturesThymineandThymineO, severally. Note that temperatures are given in Kelvin.Finally the mass transportation coefficient K calculated byWhereShdiscovery from equation 4.36 or 4.374.6 Water Permeability CoefficientThe equation ciphering pure H2O permeableness coefi?cient A for FO procedure is derived from the theoretical account thereby the H2O iux of rearward osmosis procedure is predicted ( Baker, 2004 )WhereCtungstenis the H2O molar concentration,Volttungstenthe molar volume of H2O,Calciferoleffthe effectual H2O molecule diffusivity within the pores of active bed of the FO membraneWherevitamin DSecond( 4AO) andvitamin DPhosphorus( 7.2AO) are the diameter of H2O molecule and pore, and D the evident diffusivity, which is given asAlong with H2O dynamic viscousness ( w ) predicted byThere is besides another manner to cipher membrane permeableness ( A ) iat-sheet bench-scale RO trial system was used to find the H2O permeableness coefi?cient ( A ) of the CTA membrane. A membrane voucher holding an effectual surface country of 64 centimeter2was the active bed of the membrane confronting the provender solution. Mesh spacers placed in the provender channel enhanced the turbulency of the ultrapure H2O provender watercourse. A hard-hitting positive supplanting pump was used to recirculate the provender solution at 12 L/h.The FO membrane H2O permeableness coefi?cient ( A ) was determined utilizing ( Lee et al. , 1981 ) .Where is the osmotic force per unit area difference across the membrane and ?P is the hydraulic force p er unit area difference across the membrane.Because ultrapure H2O was used as the provender solution, was zero during the experiments. Pressure was increased from 1 saloon to 2 saloon. Pressure was held changeless at each increase for continuance of 3 h. Water iux through the membrane was calculated found on the increasing weight of the permeant H2O on an analytical balance. The temperature was held changeless at 25OC. See figure 4.4Figure 4.4 Flux vs. force per unit area and the swill is representedH2O permeableness coefi?cient ( A ) .4.7 Recovery PercentageThe recovery factor measures how much of the provender is recovered as permeate. It is reported as a per centum ( Al-Alawy, 2000 ) . The recovery of the membrane was calculated by spliting the overall of permeate rate by the provender rate solution. Recovery, or transition, is defined byWhereVoltPhosphorusis the overall permeate volume andVoltFis the provender volume solution.Figure 4.5 the flow chart of patterning FO H2O flux at different temperature matrixes.1