Coupled Heat and Mass Transfer during Crystallization of MgSO4 ...

Crystallization of MgSO 4 · 7H 2 O on a Cooled Surface Crystal Growth & Design, Vol. xxx, No. xx, XXXX Edifferent approaches all of which overlap perfectly well witheach other. These approaches were using the Image J programfor measuring the crystal size changes in time, weighing thecrystal amount at the end of the experiment, and also doingconcentration measurement in the beginning and at the end ofthe experiment and extracting the mass flux value from eq 10.The chemical potential difference in eq 15 was calculatedusing Pitzer model. 15 The various equations and the constantsneeded in the calculations are presented in the Appendix.Figure 3. Magnified view (figure is not drawn proportional to its origin)of the surface of Figure 2; k, fraction of the enthalpy of crystallizationcarried out via the cooled surface, see eq 21.The thickness of the salt (δ s ) growing into the liquid directionand the mass flux (J) of the crystals were estimated using fourResults and DiscussionIn Figure 4, single (a) and (c) MgSO 4 · 7H 2 O crystal photoson the TLC surface are presented. The transparency propertyof MgSO 4 · 7H 2 O crystals allows us to follow the temperatureprofiles of the TLC surface both on crystal-free and for thecrystal covered areas. Undistinguished crystal borders on thepost-processing temperature images (b) and (d) support this fact.The part of the heat (due to heat of crystallization) evolved inthe crystal side is transferred both under and near the bordersof the crystal on the TLC.Crystallization is a dynamic process in which molecules orions dissolve and absorb on the crystal surface simultaneouslyduring growth. Therefore warmer or colder temperature readingsunder the crystals compared to the surrounding is possible beforeFigure 4. (a, c) Picture of single crystals (labeled 1 to 7) on the TLC surface; (b, d) crystals post-processing temperature images on the TLCsurface; 1 pixel is 65 µm.

H Crystal Growth & Design, Vol. xxx, No. xx, XXXX Genceli et al.N A ′ )-D AB∆C A∆z(24)The force-flux relation in case of no heat transport can alsobe simplified from eq 15, and mass flux (J) can be defined asa function of mass resistivity (R µµ i,s ) and chemical potential or,in other words, concentration differences.J )- 1R i,s µµ T l∆ l,s µ T (T l ) )- 1R i,s µµ T lRTl∆ l,s (C)(25)C lEquating eqs 24 and 25 and knowing that N A ′ ) J, thediffusion coefficient can be written in terms of mass resistivityas shown in eq 26.D AB ) ∆zR(26)R i,s µµ C lIf we choose the ∆z value as the interface thickness, 0.3µm (δ i ), and use the C l concentration of 2.33 mol/m 3 , theD AB is calculated as 8.5 × 10 -10 m 2 /s. This value iscomparable to the diffusion coefficient of water-MgSO 4system relation deduced from Gmelin and Lobo for the sameconcentration and temperature by having the value of 8.8 ×10 -10 m 2 s -1 . 11,17,18A similar analogy between conductive heat transfer coefficient(λ) and heat resistivity (R qq ) can be derived. Thegeneral heat transfer flux (q′′) relation according to Fourier’slaw is proportional to the temperature (∆T/∆z) gradient inthe direction of the heat transfer and the thermal conductivitycoefficient (λ). 16q ′′ ) λ ∆ l,s (T)∆z(27)The force-flux relation in case of no mass transport canbe simplified from eq 15, and the heat flux (J q ′s ) can bedefined as a function of heat resistivity (R qq i,s ) and temperaturedifference.J ′s q )- 1 ∆ l,s (T)(28)i,sR qq(T l ) 2Equating eqs 27 and 28 knowing that q′′ ) J q ′s , the conductionheat transfer coefficient can be written in terms of heat resistivityas shown in eq 29.λ )∆z(29)i,s(T l ) 2R qqFor comparison, ∆z, as the interface thickness (δ i ) value, ischosen as 0.3 µm. Using bulk side interface temperature andheat resistivity, the thermal conductivity coefficient (λ) iscalculated as 1.5 × 10 -5 Wm -1 K -1 , which is lower than thebulk thermal conductivity value calculated for thewater-MgSO 4 system relation deduced from ref 19 (Melinder)having a value of 0.6 W m -1 K -1 . 11,19 By replacing the bulkthermal conductivity value in eq 29, the interface thickness iscalculated to be 12 mm.Mersmann (eq 3.19) 12 assumes the interface temperature tochange within a very small range and does not define thetemperature as an independent parameter. Rather, the interfacetemperature and concentration were derived from the phasediagram. However, we treat the interface as an open system,where we have heat fluxes in and out of the system (eq 10).The temperature is then an independent variable.ConclusionCrystallization is mostly an exothermic process. According toirreversible thermodynamics, the heat released during crystallizationat the interface is distributed to both the liquid and the solid phaseswhich are in contact with the interface also for isothermalconditions. We present the temperature jump between the solidand the liquid side of the interface and the heat of transfer for bothsides of the surface, using MgSO 4 · 7H 2 O crystal growth on a coldsurface as example. For an epsomite crystal growth rate of 2.33 ×10 -3 mol m -2 s -1 , around 20-30% of the heat of crystallizationis calculated to be transferred back into the liquid side. The interfaceresistivity to mass transfer (R µµ i,s ) is 1.26 × 10 3 Jm 2 sK -1 mol -2 ,while the interface resistivity to heat transfer (R qq i,s )is2.1× 10 -7m 2 K -1 W -1 . This is the first time for which the coupling of theheat and mass transfer equation at the liquid-solid interfacedescription and the heat of distribution calculation during crystallizationis documented.Supporting Information Available: Further details about the PitzerModel. This material is available free of charge via the Internet at http://pubs.acs.org.AppendixTable 2. Equations and ConstantsconstantsunitCp freezium ) 0.00274T(°C) + 2.818 kJ kg -1 K -1F freezium )-0.4T(°C) + 1289 kg m -3λMgSO4[T aq (°C),C ) wt %]) 5.6 × 10 -1 + 2 × 10 -3 (T) -6.9 × 10 -6 (T) 2 - 6.4 × 10 -4 (C)- 2.3 × 10 6 (C)(T) + 7.8 ×10 -9 (C)(T) 2 Wm -1 K -1MgSO4 20 2.427 W m -1 K -1λ saltλ solidλ solid∆H crystmetal+Vaseline ) wall5 W m -1 K -1TLC0.16 W m -1 K -1MgSO4 395 J kmol -1CMgSO4 eq ) 21.3 + 0.206 × T(°C) + wt %0.000833 × T 2 (°C)A cold surface 0.031 m 2δ w 5 mmδ TLC 0.13 mmA cooled 0.0314 m 2F salt 1680 kg m -3(for MgSO 4 · 7H 2 O)measured dataunit and sourceT c in 33.572 °C (PT100)T c out 33.579 °C (PT100)T TLC 33.666 °C (photos-without crystal)T 1 33.933 °C (PT100))T 2 34.274 °C (PT100)T 3 34.805 °C (PT100)F coolant 374.60 l h -1δ s 0.35 mmcalculated valuesunitT c log 33.575 °CCp freezium 3.659 kJ kg -1 K -1T TLC without crystal 33.625 °CT s 33.681 °CT l 33.869 °CJ 2.33 × 10 -3 mol m -2 s -1J′ c q 108.20 W m -2J′ q 52.37 W m -2J∆H cryst. 55.83 W m -2′lJ q 31.02 W m -2 (for no mass flux)a w-l 0.937a Mg-l 0.093a SO4-l 0.093a w-s 0.945a Mg-s 0.085a SO4-s 0.085

Crystallization of MgSO 4 · 7H 2 O on a Cooled Surface Crystal Growth & Design, Vol. xxx, No. xx, XXXX ITable 3. List of Captionslist of symbolsunita activityC concentration mol m -3Cp specific heat kJ kg -1 K -1D AB diffusion coefficient m 2 s -1∆H cryst. heat of crystallization J mol -1h convection heat transfer coefficient W m -2 K -1J mass flux mol m -2 s -1J′ q heat flux J m -2 s -1kenthalpy fraction transferred to thecoolant sideN A ′ convective mass flux mol s -1 m -2R ideal gas constant J K -1 mol -1i,sR qq heat transfer resistivity of the surface m 2 s 1 J -1 K -1to the salt sidei,sR µµ mass transfer resistivity of the surface JK -1 mol -2 m -2 s -1to the salt sideR i,sqµ ) R qµ coupling film resistivities of the m 2 s 1 mol -1 K -1surface to the salt sidei,sR qq heat transfer resistivity of the surface m 2 s 1 J -1 K -1to the liquid sidei,sR µµ mass transfer resistivity of the surface JK -1 mol -2 m -2 s -1to the liquid sideR i,sqµ ) R qµ coupling film resistivities of the m 2 s 1 mol -1 K -1surface to the liquid sideq′′ convective heat flux W m -2q* i,s heat of transfer coefficient ratio to the J mol -1salt side of the surfaceq* i,l heat of transfer coefficient ratio to the J mol -1liquid side of the surfaceT temperature °Cz distance from the salt surface mGreek symbolsUnitδ thickness µm, mmλ thermal conductivity W m -1 K -1µ chemical potential J K -1V measured velocity, salt growth rate m s -1F density at the given condition kmol m -3σ entropy production W m -2 K -1cilsTLCsub-superscriptscoolantinterfaceliquidsaltthermo liquid crystalReferences(1) Kjelstrup, S.; Bedeaux, D. Non-equilibrium thermodynamic of heterogeneoussystems; Series on Advances in Statistical Mechanics, Vol.16; World Scientific: Singapore, 2008.(2) Bedeaux, D.; Albano, A. M.; Mazur, P. Physica A 1976, 82, 438–462.(3) Bedeaux, D. AdV. Chem. Phys. 1986, 64, 47–109.(4) Albano, A. M.; Bedeaux, D. Physica A 1987, 147, 407–435.(5) Ratje, S. K.; Flesland, O. J. Food Eng. 1995, 25, 553–567.(6) Chen, X. C.; Chen, P.; Free, K. W. J. Food Eng. 1997, 31, 395–402.(7) Badam, V. K.; Kumar, V.; Durst, F.; Danov, K. Experimental andtheoretical investigations on interfacial temperature jumps duringevaporation. Exp. Therm. Fluid Sci., 2007, 32(1), 276–292.(8) Kjelstrup Ratkje, S.; Bedeaux, D. J. Electrochem. Soc. 1996, 143 (3),779–789.(9) Gibbs, J. W. The Scientific Papers of J.W. Gibbs; Dover: New York,1961.(10) Bedeaux, D.; Kjelstrup, S. Int. J. Thermodyn. 2005, 8 (1), 25–41.(11) Pronk, P. Fluidized bed heat exchangers to prevent fouling in ice slurrysystems and industrial crystallizers. PhD Dissertation, Delft Universityof Technology, The Netherlands, 2006.(12) Mersmann, A. Crystallization Technology Handbook, 2nd ed.; MarcelDekker Inc.: New York, 2001.(13) Xu, J.; Kjelstrup, S.; Bedeaux, D.; Røsjorde, A.; Rekvig, L. J. ColloidInterface Sci. 2006, 299, 455–463.(14) Delfos, R.; Lagerwaard, R.; Roos, M. Bottom temperature structurein low-conductivity Rayleigh-Benard convection, EUROTHERM-71 Visualization, Imaging and Data Analysis in ConVectiVe Heat andMass Transfer, October 28-30, 2002, Reims, France.(15) Pillay, V.; Gaertner, R. S.; Himawan, C.; Seckler, M. M.; Lewis,A. E.; Witkamp, G. J. J. Chem. Eng. Data 2005, 50 (2), 551–555.(16) Incropera, F. P.; De Witt, D. P. Fundamentals of Heat and MassTransfer, 3rd ed.; John Wiley & Sons: New York, 1990.(17) Gmelins Handbuch der Anorganschen Chemie, 8th ed.; DeutscheChemische Gesellschaft, Verlag Chemie: Weinheim, 1952.(18) Lobo, V. M. M. Handbook of Electrolyte Solutions.; Elsevier:Amsterdam, 1989.(19) Melinder, Å. Thermophysical Properties of Liquid Secondary Refrigerants.Charts and Tables.; International Institute of Refrigeration: Paris,1997.(20) Thermtest. Thermal conductiVity database of materials; http://www.thermtest.com, 2007.CG800377X