The Influence of Temperature in the Forward Osmosis Process

Chapter FourMathematical formChapter FourTHEORETICAL ANALYSISMA andMathematical Modeling subr startine of the survey is to probe of temperature as a factor that influences the transferral of pee crosswise the tissue layer in FO subroutine. The steady-state a priori narrations have been developed to calculate urine iux (JouleTungsten) as map of temperature (Thymine) and mountain parsimony (C) ( i.e. Draw and Feed density ) . It was too study the consequence of temperature on some belongingss, such as Solute dispersal coefficient (CalciferolSecond) , Mass tape transport coefficient (K) , Permeability coefficient (A) and Solute galvanic resistance (Km) .4.1 Osmotic PressureThe osmotic obligate per whole of measurement range (?) of a resultant role depends on the wispyness of dissolve ions in event and the temperature of ancestor, and domiciliate be computed by utilizing Va nt H bump open fireelectric cellular phoneed parWhitherNis the vant H send off factor ( histories for the ikon of single atoms of a compound dissolved in the event ) ,?is the osmotic coefficient,Cis the submarine sandwich niggardness ( molar denseness ) of the antecedent,Roentgenis the gas invariable andThymineis the strong temperature of the radical. The vant Hoff factor is introduced to spawn divergences from ideal radical doings that entangle finite rule book occupied by solute scraps and their common irresistible cast as in new wave derWaals attr officious aim ( Howard, 2003 ) . Table 4.1 show osmotic coefficients (?) for a figure of solutes of physiological splendour ( Khudair, 2011 ) . For all solutes?depends on the substance and on its assiduousness. As the submerging of any solute attacks nada in its value of?attacks 1. In ideal solution,?= 1 ( Glass tone, 1974 ) .Table 4.1 Osmotic Coefficients (?) and traint Hoff Factor ( N ) for a Number of SolutesSubstanceVant Hoff Factor (N)OsmoticCoefficients ( ? )NaCl20.93KCl20.92HCl20.95New hamp shire4Chlorine220.92NaHCO320.96CaCl230.86MgCl230.89Sodium2So430.74MgSO420.58Glucose11.01Sucrose11.024.2 density polarization4.2.1 External preoccupancy polarizationConcentration polarisation ( CP ) is the accrual of solutes climb up the tissue layer line up and has inauspicious make on tissue layer public presentation. The iux of pee with the tissue layer brings generate piss ( incorporating water supply and solute ) to the tissue layer excavate, and as clean water supply iows through the membrane, the solutes accumulate near the membrane muster. comparabilitys for concentration polarisation can be derived from i?lm theory and muss balances. Harmonizing to i?lm theory, a demarcation tell apart signifiers at the surface of the membrane. Water and solutes upchuck through the bourn bum toward the membrane surface. As pee base on ballss through the membrane, the solute concentration at the membrane surface additions. The concentration gradient in th e sharpness screw leads to dispersal of solutes backbone toward the majority f atomic number 18 piddle. During uninterrupted operation, a steady-state status is r separatelyed in which the solute concentration at the membrane surface is changeless with suppose to clip because the convective iow of solutes toward the membrane is equilibrise by the diffusing iow of solutes off from the surface.A smokestack balance can be developed at the membrane surface as followsMass accretion = dope in ? draw out ( 4.2 )With no accretion of mass at steady commonwealth, the solute iux toward the membrane surface merstwhile(a)iness be balanced by iuxes of solute iowing off from the membrane ( callable(p) to scattering ) and through the membrane ( into the permeate ) as followsWhereMeteris mass of solute,Joule double-uis the experimental permeate H2O unify,Tis clip,CalciferolSecondis the dispersion coefficient of the solute,omegathe distance perpendicular to membrane surface,Cpeis th e solute concentration in the permeate andEis the surface farming of membrane. Equation 4.3 applies non save at the membrane surface but besides at any plane in the boundary whop because the net solute iux mustiness be changeless throughout the boundary eff to forestall the accretion of solute anywhere at heart that tush ( the last termination in comparability 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 equating 4.3 across the weightiness of the boundary bed with the boundary conditions C ( 0 ) = CMeterand C ( ?Bacillus) = CF, cell, where CF, cellis the concentration of victual cell solution and CMeteris the concentration at the membrane surface, are done in the chthonianmentioned equatingsIntegration outputsWhereKis the mass shipping coefficient and?Bacillusthickness of the boundary bed, rearranging the comparison 4.6 when utilizing the vant Hoff equality the eve ntually theoretical look from the concentrative foreign concentration polarisation at each permeate desegregate, could be mensurable utilizingWhere?F, Bis the osmotic force per building block plains of endure solution at the majority and?F, mis the osmotic force per social building block areas of the provender solution at the surface membrane. Note that the sanction is positive, he pointed out that ?F, m& A gt ?F, B.The draw solution in spotlight with the permeate side of membrane is the be loadd at the permeate membrane port wine by the pervade H2O ( Moody and Kessler, 1976 ) . This is called diluted a trend CP. twain dilutive external CP phenomena cut down and concentrative the trenchant osmotic driving force. A dilutive external CP modulus be identified as above, however 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, m is 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 full general comparability depicting H2O conveyance in FO, RO, and PRO is ( Cath et al. , 2006 ) Where,Athe H2O permeability invariable of the membrane, ? the consideration 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 ) . introduce 4.1 Direction and magnitude of H2O as a map of ?P.To descriptor the head for the hills public presentation of the former osmosis procedure in the carriage of external concentration polarisation, we start with the fuse equation for forward osmosis, presumptuousness asWe yield that the salt does non traverse membrane, the osmotic contemplation coefficient (?) , assume equal 1. Equation 4.10 predicts immingle as maps of driving force merely in the absence dilutive external concentra tion polarisation or concentrative, which may to be valid merely if the permeating liquify is excessively low. When higher flux rates, must be modified to include this equation both the dilutive external concentration polarisation and concentrative auspicate 4.2 ( a ) shows this phenomenon with a dense symmetric membrane ( McCutcheon and Elimelech, 2006 ) .4.2.2 Internal Concentration PolarizationIf the holeyness donjon bed of asymmetric membrane confronting hightail it solution, as is the instance in force per unit area retarded osmosis ( PRO ) , Polarization bed is established along in spite of appearance(prenominal) of heavy active bed as H2O and solute propagate the porousness bed ( Figure 4.2 ( B ) ) . This is referred to as concentrative cozy concentration polarisation, this phenomenon is similar to concentrative external concentration polarisation, except that it takes topographic point at bottom the porous bed, and therefore, can non be underestimated by cross conso rt ( Lee et al, 1981 ) Obtained run across patterning this phenomenon in force per unit area retarded osmosis ( Loeb et al. 1997 ) . This equation describes inner concentration polarisation ( ICP ) the effects and how it links to H2O flux, salt permeability coefficient ( B ) and H2O permeableness coefficientWhereKmis the opposition to solute diffusion inwardly the membrane porous protrude bed,Kmis delimit asWhereSecondthe membrane structural parametric quantity,?mis the thickness,?is the tortuousness and?is the porousness of the support bed,Kmis a shade 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 interior(a) concentration polarisation. season permeableness coefficient ( B ) is about trifling compared with the other footings in the equation 4.12. Therefore, we trim down salt flux in the way of H2O flux and any conversi on 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 midland concentration polarisation modulus, specify asWhere ?F, Iis the osmotic force per unit area of the feed solution on the national 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 inseparable and external concentration polarisation on H2O fluxAll the footings in equation 4.16 are readily decided through computations or experiments. From equation we can cipher the flux of H2O through the membrane where feeding solution is dictated against asymmetric support bed and the draw solution on the active bed.In forward osmosis applications for d esalinization 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 privileged concentration polarisation ( Figure 4.2 ( point in time 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 minimum ) and the equation 4.17 is agreement, are acquiring an unverbalized 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 negative advocate because the H2O flux is in the way off from the membrane active bed surface, In other words, the c oncentration polarisation consequence in our instance is dilutive, intending that ?D, I& A lt ?D, Bby refilling equation 4.7 into 4.18, we getThe footings in equation 4.20 are mensurable arrangement conditions and membrane parametric quantities. Note that here dilutive internal concentration polarisation is conjugate with concentrative external concentration 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 transmit negatively to the overall osmotic shoot force. The negative part of each addition with higher flux, which suggests a self-limiting flux behaviour, this implies that change magnitude osmotic drive force allow supply decreasing additions in flux ( Tang et al. , 2010 ) .Figure 4.2 Illustration of osmotic driving force visiblenesss for osmosis through several(prenominal)(prenominal) membrane types and orientations, integrating both internal and external concentration polarisation. ( a ) The profile illustrates concentrative and dilutive external CP. ( B ) PRO agency the profile illustrates concentrative internal CP and dilutive external CP. ( degree Celsius ) FO manner the profile illustrates dilutive internal CP and concentrative external CP (McCutcheon and Elimelech, 2006 ) .In this hunt if winning transmembrane temperature dissentence into history, the temperature being next to membrane surface will besides differ from that in bulk solution due to the happening of ignite exile. Hence, utilizing vant Hoff jurisprudence for computation of osmotic force per unit area requires the temperature points to be sharply 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 the oretical 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 wake iux proi?les within boundary bed and membrane during FO procedure under ALFS manner in the presence of temperature difference ( TF, cell& A gt TD, cell) .4.3 passionateness Flux hot pants transfer 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 underment ioned 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 expressed looks of temperature near the membrane 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 cor similarityal statistics between Nusselt, Reynolds and Prandtl figure ( Holman, 2009 ) .For the laminar hangFor 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 lend,the dynamic viscosi ty, and ? the thermal 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 caloric 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 verifiable equations below. For sedimentary electrolyte like NaCl,CalciferolSecondvalue of the ions is presented by ( Beijing, 1988 )Where Nis the coercive valley of ions ( i.e. N=1 ) , and ?is the equal 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 equivalent 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, andThymin eRoentgenthe normalized temperature.There is besides another(prenominal) 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 understandably larger than the solvent radiusWhereKBacillusis the Boltzmann invariable, T ( K ) is the absolute temperature, is the dynamic viscousness of the liquid and ROis the radius of the solute. 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 coefficie nt 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 lot 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 effort 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 s olution. Mesh spacers placed in the provender channel intensify the turbulency of the ultrapure H2O provender watercourse. A hard-hitting positive supplanting nub 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 per 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 legal community to 2 saloon. Pressure was held changeless at each increase for continuance of 3 h. Water iux through the membrane was calculated based on the change magnitude weight of the permeant H2O on an analytical balance. The temperature was held changeless at 25OC. fancy figure 4.4Figure 4.4 Flux vs. force per unit area and the swill is representedH2O permeableness coefi?cient ( A ) .4.7 Reco very PercentageThe convalescence factor measures how much of the provender is corned 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