2013 Heat and mass transfer effects of ice growth mechanisms in pure water and aqueous solutions

Interactions between heat and mass diffusion determine growth mechanisms during ice crystallization. The effects of heat and mass transfer on ice growth in pure water and magnesium sulfate solution were investigated by studying the evolution of the
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  Heat and Mass Transfer E ff  ects on Ice Growth Mechanisms in PureWater and Aqueous Solutions Michael Kapembwa, Marcos Rodr í guez-Pascual, and Alison E. Lewis * Crystallization and Precipitation Research Unit, Chemical Engineering Department, University of Cape Town, Rondebosch 7701,Cape Town, South Africa * S  Supporting Information  ABSTRACT:  Interactions between heat and mass di ff  usiondetermine growth mechanisms during ice crystallization. Thee ff  ects of heat and mass transfer on ice growth in pure waterand magnesium sulfate solution were investigated by studyingthe evolution of the gradient of the refractive index using colorSchlieren de fl ectometry. For pure water, the gradient of therefractive index of water was used to calculate the temperatureand therefore the local supersaturation. Its e ff  ect on the icecrystal growth rate and morphology was studied. It was foundthat, for local supersaturations greater than 2.8, themorphology was dendritic ice, with a growth rate 2 orders of magnitude higher than that for layered growth. Duringdendritic growth, 3 − 16% of the heat of crystallization di ff  usedto the liquid side, which is counter to current understanding. Atthe transition (between the time of partial melting of the dendritic ice and the beginning of the layered ice growth), a highersupersaturation than that responsible for layered growth was observed. For ice growth from an aqueous salt solution, a mass andthermal di ff  usion boundary layer in front of the growing ice was created by di ff  usion of the solutes from the ice and by the releaseof heat of crystallization. ■  INTRODUCTION The desire to improve product quality and e ffi ciency of processes in volving ice crystallization in industries such as freeze-dr ying, 1 freeze concentration, 2  water desalination by freezing, 3 ,4 and eutectic freeze crystallization 5 requires knowl-edge of and insight into ice growth mechanisms. Because of this, research on ice crystallization processes remains animportant area of study. During ice crystallization, ice growthmechanisms play an important role in determining thestructure, size, and morphology of ice, all of which have ane ff  ect in the separation 6 and the purity of the product.Therefore, understanding the phenomena involved in icecrystallization is useful in providing information on parametersand processes that in fl uence the growth and quality of the icecrystals.Ice crystallization is an exothermic process in which heat of crystallization is released to the surroundings. In classicaltheories, it is assumed that all the heat of crystallization istransferred to the colder side, that is, the solid phase. 7,8 However, later experimental studies 9 showed that the heatreleased di ff  uses to both the solid and liquid sides of thedeveloping interface. This was predicted by using heat and masstransfer equations based on the irreversible thermodynamics of Onsager relations presented by Kjelstrup and Bedeaux. 10 Following this point of view, part of the heat released di ff  usesto the liquid side and is coupled with the rejection of solutes when ice is growing in an aqueous salt solution. This couplede ff  ect results in the formation of a mass and thermal di ff  usion boundary layer in front of the growing ice layer. 11 Because of this coupled heat and mass di ff  usion, temperature andconcentration gradients in the liquid phase in front of thegrowing ice are created. These determine the onset of instabilities and thus the type of growth 12  by a ff  ecting thelocal supersaturation and therefore, the crystal growth rate, themorphology, and consequently the amount of impurities andliquid inclusions. 13 Figure 1 illustrates the growth process of anice crystal layer from an aqueous MgSO 4  solution on a cooledheat exchanger surface.This paper analyses the evolution of the boundary layer inpure water and in an aqueous MgSO 4  solution by measuringthe evolution of the refractive index in front of the growing icelayer. To achieve this, in situ color Schlieren de fl ectometry measurements were performed to obtain the refractive index di ff  erences caused by temperature and concentration gradientsin front of the growing ice crystal. Measurements of ice growingin pure water were used to quantify the amount of heattransferred to the liquid side during crystallization and tomeasure the local supersaturation responsible for di ff  erent typesof morphology. Received:  September 25, 2013 Published:  December 4, 2013 Reviewpubs.acs.org/crystal © 2013 American Chemical Society  389  dx.doi.org/10.1021/cg401428v  |  Cryst. Growth Des.  2014, 14, 389 − 395  ■  MATERIALS AND METHODS The experimental setup in Figure 2 is similar to the one described by  Gupta and co-workers, 14 and details of the operating principles of color Schlieren de fl ectometry have been presented extensively in theliterature. 14 − 16 The setup consisted of an light-emitting diode (LED)cold white light source (REVOX SLG-50S) connected to a 400  μ mpinhole with a  fi  ber optical cable for obtaining a point light source.The pinhole was placed at the focal length of the  fi rst lens to collimatethe light going through the test cell where crystallization occurred. Thetest cell was placed between the  fi rst and second lens. The second lensfocused the light onto a two-dimensional (2-D) axisymmetric color fi lter of 2.32 mm in radius fabricated on a photographic  fi lm. The color fi lter was generated as recommended by Greenberg and co-workers. 17 The  fi lter was positioned in front of the second lens at its focal point.The images covering a section of the cell were captured using a digitalcamera with a resolution of 640  ×  480 pixels connected to a computeracquisition system to record the optical data. The optical equipment was placed on translation microstages, and experiments were carriedout in a dark room. The positions of the test section, the decollimatinglens, and the digital camera satis fi ed the relation: + =  p q f  1 1 1 2  (1) where  p  is the distance (mm) between the test cell and thedecollimating lens,  q  is the distance (mm) between the decollimatinglens and the camera, and  f  2  is the focal length (mm) of decollimatinglens. The fraction  q /  p  is the magni fi cation of the image that wasprojected on the lens of the camera.Deionized water was used for the experiments of ice growth in pure water. An MgSO 4  solution of 8.4 wt % was prepared from 99 wt %anhydrous MgSO 4  and was used for ice growth experiments in theaqueous salt solution. Both experiments were carried out in the testcell shown in Figure 3. The cell was made of Plexi-glass and consistedof the crystallizing and cooling chamber separated by a 1 mm thick stainless steel plate which acted as a heat exchanger. Deionized waterand an ambient temperature 8.4 wt % MgSO 4  solution were injectedinto the crystallizing chamber. Cooling was achieved by circulating thecoolant (Kryo 40) into the cooling chamber. Condensation on the walls of the cell was avoided by continuously blowing dry air on bothsides of the cell. Data Analysis.  In order to obtain quantitative data, that is, convertthe color of the image to the angle of refraction, the color  fi lter wascalibrated. The refraction angle  θ   (in radians) as a function of positionin the  x −  y  plane (which is normal to the beam) was quanti fi ed by  fi nding the displacement distance of the light using color changes onthe image. This was done by capturing the full image of the  fi lter from which a transmissivity function was generated and then used forcalculation of the refractive index. For a 2-D Schlieren of extent  L along the optical axis in the  y  direction, the de fl ected angle  θ   is given by  θ   = ∂∂  Lnn y  y 0  (2) where  L  (mm) is the length of the test section and  n 0  is the refractiveindex of the ambient air (approximately 1). Therefore θ  ∂∂= n y L  y (3)The beam displacement Δ d   (mm) of the light beam at the positionof the color  fi lter due to the de fl ection of light beam passing throughthe test section by an angle  θ   y  is given as θ θ  Δ = ≈ d f f  tan  y y 2 2  (4)Making  θ   y  in eq  4 the subject of the formula and substituting it intoeq  3 gives Figure 1.  Crystal layer growing on a cooled plate in salt solution. 11 Figure 2.  The schematic view of the color Schlieren de fl ectometry setup. Figure 3.  Test cell. Crystal Growth & Design  Review dx.doi.org/10.1021/cg401428v  |  Cryst. Growth Des.  2014, 14, 389 − 395 390  ∂∂= Δ n yd f L 2  (5)Image processing was used to convert the hue value to atransmissivity function, while computational codes based on eqs 2 − 5 were used to calculate the gradient of the refractive index of thesystems under study. Equation 5 represents the di ff  erential index of refraction. When integrated, it gives the gradient of the refractiveindex. The gradient of the refractive index was then related to thetemperature of the deionized water using refractive index-temperaturecurves.Figure 4 shows the processes involved in calculating thetemperature of the pure water (deionized water) in front of thegrowing ice at di ff  erent stages of crystal growth. The location at whichthe temperature was mapped is represented by the blue vertical colorstrip. The black color in images A and B during ice growth representsthe block of ice growing from the heat exchanger, and the whitesection in image B during dendritic growth shows a mushy layer of dendrites. The uniform color distribution in image A before cooling was due to the uniform temperature distribution, and the subsequentcolor di ff  erences in images A during cooling and crystallization weredue to the variations in refractive index caused by the temperaturegradients. The color scale in image B represents the values of thedi ff  erential index of refraction d n/  d  y  shown in eq  5. The values of d n/  d  y  were then related to temperature. The growth rate was measured by analyzing images of the advancing ice front by means of a cameraoperating at a speed of 15 frames per second (fps). Thus, the changein the thickness of the ice layer with time was measured by calculatingthe change in the thickness of the ice layer with time, the distance being calculated using pixels, where the distance between two pixels inthe direction of the layer growth was equivalent to 0.009 mm. All thesemeasurements were done in triplicate. ■  RESULTS AND DISCUSSION E ff  ect of Local Supersaturation on Growth Rate andMorphology of Ice in Pure Water.  The spatial mapping of temperature made it possible to measure temperatures of supercooled water at locations less than 30  μ m away from thesolid − liquid interface of ice during growth. Therefore, it waspossible to measure the local supersaturation, expressed as  Δ T  =  T  *  −  T   , where  T  *  is the equilibrium melting temperature of ice, i.e., 0  ° C, and  T   is the temperature of the water below 0  ° C.Figure 5 shows the e ff  ect of local supersaturation on the growthrate of ice.It can be seen in Figure 5 that, not only did higher localsupersaturation result in a faster growth rate, but also that thereis an order of magnitude di ff  erence between the growth rate of  Figure 4.  Image of deionized water during ice growth: (A) Unprocessed image; (B) map of di ff  erential index of refraction; (C) graph of gradient of index along the highlighted area in image B; (D) graph of temperature along the highlighted area in B. Figure 5.  E ff  ect of local supersaturation on growth rate. Crystal Growth & Design  Review dx.doi.org/10.1021/cg401428v  |  Cryst. Growth Des.  2014, 14, 389 − 395 391  the layer growth and that of the dendritic growth. From Figure5 , it can be seen that, when the supersaturation was greater than2.8, the growth rate was 2 orders of magnitude higher than thatfor layer growth and that the ice growth morphology wasdendritic.The growth rates in melt crystallization are  quite often  in therange of about 10 − 6 to 10 − 4 m/s. 11 However, these growth ratesdepend on the supersaturation due to the rate of heat removedfrom the system (heat  fl ux). Therefore, the order of magnitudeof the growth rates varies depending on the heat removal fromthe supersaturated system. For example, for a supercooling of 0 to 1  ° C for ice growing in water 18 and NaCl solution, 19 thegrowth rates were between 10 − 6 to 10 − 3 m/s. In theexperiments presented here, the supercooling for ice growingin pure water ranged from 0.2 to 3.8  ° C, and consequently theorder of magnitude of the measured growth rates is 10 − 5 to10 − 2 m/s, 10 times higher. The high growth rates of magnitude10 − 2 m/s were observed just after nucleation when the system was highly supersaturated.Dendritic ice, which forms due to high supersaturation andlow temperature gradients, was observed at early stages of iceformation. The growth of ice dendrites in pure water is adi ff  usion-limited process where the transport of heat dictatesthe behavior of the solid − liquid interface. Dendrites grew intoregions of high supersaturation, and the subsequent release of the heat of crystallization resulted in limiting the extent of growth as supersaturation was consumed. Because the super-saturation was simultaneously increasing while the dendrites were dissolving (due to continuous cooling of the heatexchanger surface), the dissolution rate of the dendritesdecreased with time. The heat that partly di ff  used into thesolution and that was partly conducted by the dendritescontrolled the growth rate, the extent of growth as well as thedissolution rate.Figure 6 shows the two di ff  erent morphologies of iceobserved during crystallization. It can be seen in Figure 6 that,at supersaturations lower than 2.8, the ice surface was smoothat the macroscopic scale and grew in layers.It can also be observed in Figure 6 that the supersaturation was comparatively higher during the transition stage, which wasobserved after the dendrites had partially melted but before theinitiation of layered growth. This higher supersaturation was asa result of two aspects: (1) continuous cooling of the systemand (2) the reduction of the thermal resistivity due to thedecrease in the thickness of the ice. Thus, it can be concludedthat a high supersaturation was needed to prevent a completemelting of the ice and to initiate the layered growth. However,during layered growth, the supersaturation was comparatively low and gradually decreased with time. The reduction in thesupersaturation and thus the increase in temperature duringlayered growth were due to the consumption of thesupersaturation by the release of the heat of crystallizationcoupled with the lower heat extraction by the heat exchanger asa result of increased thermal resistivity by the growing ice layer. Quanti fi cation of Heat Transferred to the Liquid Sideduring Dendritic Growth in Pure Water.  Figure 7 showsthe heat  fl ux as a function of growth rate during dendritic andlayered growth. Figure 6.  Surface morphology of ice growing in pure water at di ff  erent supersaturations as a function of time after nucleation. Figure 7.  E ff  ect of heat  fl ux on growth rate: (A) dendritic growth; (B) layered growth. Crystal Growth & Design  Review dx.doi.org/10.1021/cg401428v  |  Cryst. Growth Des.  2014, 14, 389 − 395 392  Heat  fl ux is the rate of heat energy that is transferred througha surface per unit area. In this case, it was the rate of heatenergy that was transferred from the liquid through thecrystalline ice and the heat exchanger to the coolant (thermalliquid).In Figure 7 A, before the formation of the ice (dendrites), thesupersaturation was high and the deionized water was in ametastable state. Therefore, when the metastability was broken,the growth of the dendrites was not necessarily a result of theheat removal by the coolant but because of the already createdhigh supersaturation. Hence it can be concluded that, duringdendritic growth, the magnitude of the heat  fl ux was as a resultof the growth of the dendrites. Since the temperature of theinlet coolant was maintained at  − 25  ° C, as the thickness of theice increased, the thermal resistivity also increased and thus thegrowth rates decreased. At higher growth rates, the rate of heatgeneration was higher than at slower growth rates. Con-sequently, the amount of heat removed at the higher growthrates was more than the amount removed at the slower growthrates. Thus, during dendritic growth, the heat  fl ux decreased with an increase in the growth rate.In Figure 7B, during layered growth, the system was lesssupersaturated and the growth of the ice was due to theremoval of the heat from the growing front. This implies that,at a low growth rate when the thermal resistivity was high, theamount of heat removed by the coolant was less than at ahigher growth rate.Using irreversible thermodynamics and the Onsagerequations, the fraction  k   can be used to quantify the amountof heat that di ff  used to the liq uid side compared to the solidside during dendritic growth. 9 The  k   is de fi ned as the heattransferred to the solid side divided by the enthalpy of crystallization and is given by  =∗Δ kqH  i  ,scryst  (6)  where  q * i  , s (J/mol) is the heat transfer coe ffi cient ratio for thesolid side of the surface.The  k   value for the heat transferred to the liquid side wasobtained by substituting  q * i  ,s in eq  6 with  q * i  , l . Figure 8 presentsthe change in the value of   k   during dendritic growth and therelationship between  k   and the growth rates. From Figure 8 , itcan be seen that about 3 − 16% of the heat of crystallization wastransferred to the liquid side and that highest value of   k   (0.16) was when the dendrites reached maximum thickness beforethey started to melt. Both the 3 − 16% of the heat transferred tothe liquid side and the 84 − 97% of the heat transferred to theice side were responsible for controlling the growth dynamicsof the dendrites and the eventual partial dissolution of the iceduring dendritic growth and dissolution.The results in Figure 8 demonstrate that heat of crystallization is not only transferred to the ice side but alsoto the liquid side. It therefore agrees with current theories based on irreversible thermodynamics coupled with Onsagerrelations. The heat of crystallization transferred to the liquidside during layered growth was not quanti fi ed due toexperimental limitations. This was because the detection of the amount of heat released during the ice layered growth, andtherefore the temperature gradient change, in the growing frontsolution did not create measurable variations in the refractiveindex. Compared to the literature, which gives a  k   value relatedto 20 − 30% of MgSO 4  of the heat transferred to the liquid side, 9 the  k   values for ice were less. One of the reasons for thedi ff  erence could be the di ff  erence in thermal conductivities between ice and water. The conductivity of ice at 0  ° C is 0.0214 W  cm − 1 ° C − 1 and for water at the same temperature of 0  ° C is0.00561 W  cm − 1 ° C − 1 . 20 This represents about a 3.8 timesincrease, and hence it was expected that a much larger fractionof the heat would be conducted by the ice. Comparison of Ice Growth Layer in Pure Water and inAqueous MgSO 4  solution.  Figure 9 shows the evolution of the di ff  erential index of refraction (d n/  d  y ) along the verticaldirection, as shown in Figure 4 , of the MgSO 4  solution anddeionized water before and during the crystallization of ice. Italso shows the corresponding temperatures for the deionized water. The refractive index of water is a ff  ected by temperature, whereas that of an salt solution is a ff  ected by temperature andconcentration. The refractive index of water increasesexponentially with the decrease in temperature and is maximumat 0  ° C. After this, it decreases exponentially with the decreasein temperature when supercooled. 20 However, the refractiveindex of MgSO 4  solution decreases linearly with an increase intemperature ( T   ≥  0  ° C) at constant concentration 20 andincreases linearly with an increase in concentration 21 atconstant temperature. This means that, in an MgSO 4  systemin which both temperature and concentration are changing, therefractive index will change according to the coupled e ff  ect of the temperature and the concentration. The temperatures forthe two systems shown in Figure 9 decreased from pixel 0toward pixel 590 where the heat exchanger was placed.Therefore the refractive index for the deionized water wasexpected to increase toward the heat exchanger for  T   ≥  0  ° Cand decrease for  T   < 0  ° C, whereas that of the MgSO 4  wasexpected to generally increase toward the heat exchanger.In image A of Figure 9 , d n/  d  y  for the deionized water wasobserved to be almost constant (at 1.9  ×  10 − 4 ) between pixels0 and 380. This implies that the refractive index of the waterconstantly increased with decreasing temperature. The d n/  d  y dramatically decreased between pixels 380 and 490 to 0 andincreased between pixels 490 and 590 to 2  ×  10 − 4 . Since themaximum refractive index of water is at 0  ° C where the d n/  d T  is 0, the temperature at pixel 490 was 0  ° C. Therefore, thedeionized water between pixels 490 and 590 was supercooledand was in a metastable state. This pro fi le of d n/  d  y  was due tothe corresponding temperature pro fi les shown in image A  where the temperature decreased from the bulk toward the heatexchanger surface at pixel 590. A   “  jump ”  in the growth of thedendritic ice was observed just after nucleation due to the highsupersaturation, which was created just before the onset of  Figure 8.  Fraction  k   during dendritic ice growth. Crystal Growth & Design  Review dx.doi.org/10.1021/cg401428v  |  Cryst. Growth Des.  2014, 14, 389 − 395 393
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