Modeling Growth Rates in Static Layer Melt Crystallization - Crystal

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Modeling growth rates in static layer melt crystallization Thorsten Beierling, ramona gorny, and Gabriele Sadowski Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/cg400959a • Publication Date (Web): 14 Oct 2013 Downloaded from http://pubs.acs.org on October 21, 2013

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Crystal Growth & Design

Modeling growth rates in static layer melt crystallization Thorsten Beierling, Ramona Gorny and Gabriele Sadowski*

TU Dortmund University, Department of Chemical and Biochemical Engineering, Laboratory of Thermodynamics Emil-Figge-Str. 70, D-44227 Dortmund, Germany

* To whom correspondence should be addressed. E-mail address: [email protected]; Tel.: ++49 (0)231 755 2635; Fax: ++49 (0)231 755 2572

Keywords: melt crystallization, isomers, modeling, thermal conductivity, growth rate

ABSTRACT The prediction of growth rates as functions of process conditions can reduce experimental efforts to develop crystallization processes. There is a lack of reliable models of static layer melt crystallization in the literature. In this study, a model for the prediction of growth rates for static layer melt crystallization was developed. The essence of this model is the description of heat transport during crystal growth in a naturally convected static melt where, in contrast to other models, the implicit relation between the growth rate and the natural convection is considered. Predicting crystal growth rates requires knowledge of the crystal thermal conductivity, a sensitive physical property that is often estimated or fitted to experimental data. In this study, an approach for measuring the crystal thermal conductivity was developed and successfully validated with literature data. The crystal thermal conductivities of p-xylene, n-hexadecane, n-dodecanal and n-tridecanal were measured. Using these measurements, the crystal growth rates of the binary systems n-dodecanal/iso-dodecanal, n-tridecanal/isotridecanal, and p-xylene/m-xylene from static layer crystallization were predicted as functions of the process conditions. Good agreement with experimental data was achieved without the use of a fitted parameter.

1 INTRODUCTION Layer melt crystallization is a highly selective and technically proven method for separating the components of organic mixtures whereby the purified substance crystallizes on a cooled surface1-3. This method has gained significance over the last several decades, especially for separating azeotropic mixtures or isomers with similar boiling points4-9. Layer melt ACS Paragon Plus Environment


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crystallization can be divided into dynamic and static processes. In the dynamic case, the melt is mixed by forced convection caused by stirring or pumping10 whereas a mixing of the melt in the static case is induced by natural convection10-12. In both cases, crystal purity depends significantly on the crystal growth rate, despite the thermodynamic solid-liquid equilibrium13. Therefore, process development depends upon understanding the crystal growth rate as a function of process conditions, such as concentration, temperature and cooling rates, ideally without conducting experiments. Since the crystal growth in melt crystallization is mainly influenced by transported heat through the melt and crystal layer, as well as the mass transport in the melt near the crystal surface2, modeling requires adequate descriptions of these transport phenomena. In the literature, there are several studies regarding modeling growth rates during dynamic layer melt crystallization14-17. In those studies, heat and mass transport coefficients are constant during the crystallization process and fixed by the intensity of mixing. The same applies to mass transport, which is generally described with boundary layer approaches using mass transport coefficients14, 15. However, these transport coefficients vary during static layer melt crystallization because natural convection, a measure of mixing intensity, is a function of the crystal growth rate, which leads to complex implicit relations. Only several reliable models exist for calculating growth rates from static layer melt crystallization in the literature11, 15, 16, 18, 19. Guardani18, 19 and Scholz15 use a mass transport equation from Smith et al.20 that is valid only for stagnant fluids. Furthermore, these models do not consider implicit relations between natural convection and the growth rate. Other studies14, 18, 19, 21 simplify growth rate models by assuming constant values for physical parameters, crystal layer temperatures, heat transport coefficients and the temperature of the cooled surface. Consequently, these models inadequately describe technical crystallization processes. A further simplification of many models, as well as for the dynamic case, concerns the crystal thermal conductivity. For three reasons, crystal thermal conductivity is often fit to experimental data: 1) crystal thermal conductivities of organic compounds have been rarely reported in the literature; 2) crystal thermal conductivity is a sensitive function of temperature and concentration22; and 3) crystal thermal conductivity is difficult to measure. By fitting the crystal thermal conductivity to experimental data, models for growth rates are not predictive and require experimental validation. Furthermore, it poses the problem that any uncertainty of the model is compensated by the fitted thermal conductivity, wherein it loses its physical meaning. Özoguz16 proposed using the liquid thermal conductivity at the melting temperature, which does not give reliable data. Most experimental methods from the literature rely on calculations of heat flow through a crystal layer and measurements of the resulting layer thickness22, 23. One approach is the use of differential scanning calorimetry, which analyzes the thermal response of a crystal induced by a temperature program24-28. However, this method is not practical for organic compounds, especially for those with melting temperatures below room temperature, because it requires tiny samples with well-defined cylindrical shapes. Holmen23 presented an experimental method for the measurement of crystal thermal conductivities of pure long-chain alkanes. A crystal layer was grown on a plane-cooled surface, and the crystal thermal conductivity was determined from the heat flow and the crystal layer thickness. However, the heat flow was calculated by neglecting the natural convection in the fluid above the crystal layer, which introduced considerable experimental uncertainty. A promising method has been presented by Kim22 who measured the crystal thermal conductivity from a well-mixed water-NaCl mixture growing on a cooling finger in a lab-scale crystallization apparatus. However, he calculated the heat flow through the crystal layer from the difference between the inlet and outlet temperature of the cooling finger. Due to the small apparatus and therefore small temperature differences, considerable measurement uncertainty can be assumed. ACS Paragon Plus Environment

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In the current study, we present a model for the description of heat transport and mass transport during static melt crystallization, which includes 1) measurement of crystal thermal conductivities with high accuracy and 2) prediction of crystal growth rates from static melts as functions of process conditions. We validated our method of measuring the thermal conductivity with known values from the literature, and we measured crystal thermal conductivities of n-dodecanal, n-tridecanal, n-hexadecane and p-xylene as functions of temperature. Furthermore, we modeled crystal growth rates for the static layer melt crystallization of three binary melts: n-dodecanal/iso-dodecanal, n-tridecanal/iso-tridecanal, and p-xylene/m-xylene. Here, our model considers the implicit relations between the growth rate and natural convection, and is capable to handle time-dependent values of the following parameters: temperature of the crystal layer, temperature of the cooled surface, and transport coefficients. Furthermore, physical properties are regarded as functions of concentration and temperature. With knowledge of the crystal thermal conductivities, the model is able to predict growth rates without the use of any fitted parameters. The modeled growth rates are validated with the measured growth rates of n-dodecanal/iso-dodecanal and n-tridecanal/iso-tridecanal from a previous study5. Furthermore, the growth rates of p-xylene/m-xylene were measured in the current study.

2 THEORY 2.1 DISTRIBUTION COEFFICIENT The purity of a crystal is a function of the crystal growth rate and the melt concentration. Since it is not possible to predict the crystal purity, experiments must be conducted2, 13. The distribution coefficient kint is used to quantify the separation efficiency of one crystallization step (Equation 1). The distribution coefficient is defined as the ratio of impurity concentration in the crystal xi,cr and the initial melt xi,m13. Iso-dodecanal, iso-tridecanal and m-xylene are the impurities of the binary systems investigated in this study. , (1) ,

Low kint values indicate high separation efficiency. Because mass ratios between the crystal and the melt are low and growth rates are constant in the experiments, kint is assumed to be constant during each crystallization experiment.

2.2 MODELING CRYSTAL GROWTH RATES Crystal growth during melt crystallization is influenced mainly by heat transport2. The following sections address: 1) the temperature profiles in the static melt, crystal layer and crystallizer construction parts, 2) the calculation of heat transport coefficients, 3) the differential equation for modeling growth rates, and 4) the equations to determine crystal thermal conductivities.

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2.2.1 TEMPERATURE PROFILE Figure 1 shows the assumed qualitative temperature profile in a cross section of the crystallizer. The crystal grows on a cooled cylindrical surface in contact with a static melt. The temperature of the melt is maintained by a double jacket.

Figure 1: Cross section of the used laboratory-scale layer crystallizer with the assumed qualitative temperature profile during crystal growth from a static melt. Arrows indicate natural convection.

The temperatures of the cooling agents in the double jacket Tdj and the cooling finger Tcf are measured. As shown by various authors14-18, 22, 29, the temperature on the crystal surface is assumed to be the equilibrium temperature Teq, which is a function of the mass fraction of the impurity component at the crystal surface wi,eq (Equation 2). Because isomeric compounds have similar molar masses, it is valid to use the mass fraction in this equation. 1 ,

∆ ⋅ 1

(2)

Here, ΔhSLcryst and TSLcryst are the enthalpy of fusion and the melting temperature of the respective crystallizing compound and R is the ideal gas constant. Concerning the investigated melts, the influence of activity coefficients can be neglected because the investigated liquid melts exhibit ideal behavior5, 30. Furthermore, these systems form eutectic solid-liquid equilibria, and the concentration of the crystal is equal to one5, 30-32. Temperature profiles caused by thermal resistance Rth in the crystallizer construction parts are often neglected in the literature. In this study, the temperatures of the outer surface of the cooling finger Tcf,w and the double jacket Tdj,w are calculated from the heat flow from the double jacket to the cooled surface, as well as the thermal resistances (Equation 3). ∆ ∙ "#,$ % #,& '

(3)

The temperature profiles in the crystallizer are parabolic due to its cylindrical geometry. The thermal resistance by heat conduction Rth,λ of a crystallizer part is calculated from its thickness, which is characterized by outer and inner radii r2 and r1, respectively, its length L, and the thermal conductivity λ (Equation 4)33.


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#,$

) ()* , +

2⋅.⋅/⋅0

(4)

The thermal resistance of convective heat transport Rth,α is calculated from the heat transport coefficient α and surface area A (Equation 5)33. #,&

1 1⋅2

(5)

We propose that conductive heat flow in the melt is negligible, and no radial temperature profile exists. The temperature of the melt Tm is calculated using an energy balance of the melt at steady state (Equation 6). 2 ∙ ∙ 1 234,5 ∙ 34,5 ∙ 134,5

(6)

Here, αCr and αdj,w are the heat transport coefficients at the crystal surface and the double jacket wall, respectively. Acr and Adj,w are the surface areas of the crystal and the double jacket, respectively. It says that the convective heat flow from the double jacket to the melt is identical to the convective heat flow from the melt to the crystal surface. The balance scope is defined between the thermal boundary layers in the melt (vertical dashed lines in the melt region in Figure 1). This is valid because the convective heat flows through the crystal and the double jacket are identical to those transported through the boundary layers near the crystal and the double jacket, respectively. The heat of fusion does influence the equilibrium temperature at the crystal surface and therefore the heat flow through the crystal (Equation 2). But since the balance scope does not include the crystal surface and due to the fact that the heat of fusion is only transported through the crystal layer, the heat of fusion does not appear directly in the energy balance. Further, the assumption of a steady state is valid because the temperature of the melt Tm is changing negligibly slow. The heat transfer surface of the double jacket Adj is given by Equation 7. * 134,5 2 ⋅ . ⋅ )34 ⋅ / % . ⋅ )34

(7)

Here, rdj is the inner radius of the double jacket. The assumption of no radial temperature profile in the melt is valid if the ratio J between the specific heat flow by heat conduction q 7 in the melt and by heat transport q 8 is close to zero (Equation 8). 34,5 9 ;& 2 ∙

(8)

Here, λm is the thermal conductivity of the melt and rcf+s is the radius at the crystal surface.

2.2.2 HEAT TRANSPORT COEFFICIENTS The heat transport coefficient in the cooling finger αcf is calculated with the Reynolds number Re and the Nusselt number Nu for turbulent tube flows34, 35. The heat transport coefficient in the double jacket αdj was measured in preliminary experiments as a function of temperature. The heat transport coefficients in the melt at the crystal surface αcr and at the double jacket αdj,w are calculated with the Nusselt number, a function of the natural convection characterized by the Grashof number Gr11, 21 (see Appendix). The required physical properties ACS Paragon Plus Environment


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are considered to be functions of the average temperatures and concentrations in the boundary layers at the crystal surface and at the double jacket. Consequently, the heat transport coefficients αcr and αdj,w are not identical. Natural convection was visually proven by adding small crystalline particles to the melt in preliminary experiments that circulated in the crystallizer.

2.2.3 CALCULATION OF THE GROWTH RATE The integral formulation approach of Chianese and Santilli17 was used to model crystal growth rates from a static melt. The fundamental equation is an energy balance at the moving crystal layer surface, which is assumed to be smooth and two-dimensional14, 15, 17, 18, 36. 2 "?' ∙ "?' "?' % @ ∙ AB "?' ∙ C 0 ∙ D

E "?' F E) GH >

(9)

Here, Tcr, λcr, ρcr and v are the temperature, thermal conductivity, density, and growth rate of the crystal layer, respectively. Furthermore, Δh’ is the amount of heat that must be transported through the crystal layer for crystal growth17 (Equation 10), which includes the heat of fusion and the fact that the solidifying melt has to be cooled down from the temperature of the melt Tm to the temperature at the crystal surface Teq. ∆B "?' ∆I % JK, "?' ⋅ "?' "?'

(10)

A time-dependent parabolic temperature profile in the crystal is assumed to be similar to the approach of Parisi14. "), ?' L"?' ⋅ ) )= % M"?' ⋅ ) )= % J"?' *

(11)

To obtain the coefficient a(t), one has to derive Equation 11 by time and substitute it into the one-dimensional unsteady Fourier equation for cylinders (Equation 12). 1 E E "?' 1 E "?' ⋅ ⋅ N) ⋅ O ⋅ ) E) E) E?

(12)

In this equation, k is the thermal diffusivity of the crystal layer. The coefficients b(t) and c(t) are calculated with the boundary conditions defined by Equations 13 and 14. ) )= =,5

) )=>

(13) (14)

It turned out in preliminary investigations that it is not sufficient to assume a linear temperature profile in the crystal layer wherefore a(t) in Equation 11 cannot be neglected. An example of these findings is given in the results section of this paper. It is important to note that the parameters a, b, and c are calculated from the temperatures and layer thicknesses without fitting to experimental data. It is well known from the literature14, 15 that the impurity concentration increases at the crystal surface during crystal growth. The equilibrium temperature Teq at the crystal surface decreases during crystal growth because it is related directly to the impurity concentration. Scholz15 and Guardini18, 19 calculated the impurity concentration at the crystal surface during static melt ACS Paragon Plus Environment

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crystallization using an analytical expression derived by Smith et al.20, which is only valid for a completely stagnant fluid and a constant growth rate. However, the assumption of a completely stagnant fluid is inconsistent with natural convection. Therefore, in this work, an equation that is used ordinarily for modeling growth rates in dynamic crystallization was applied. Several authors14, 15 showed that the impurity concentration at the crystal surface is a function of the impurity concentration in the bulk phase wi,m, the densities of melt ρm and crystal ρcr, the mass-transport coefficient β and the crystal growth rate (Equation 15). , , ∙ exp N

S ∙ C O S ∙ T

(15)

This equation is valid for the growth of a completely pure crystal. If impurities are embedded by entrapment of liquid inclusions, the impurity concentration at the crystal surface is smaller than in the case of perfect separation and is calculated with Equation 1614. , , ∙ % , , ∙ ∙ exp N

S ∙ C O S ∙ T

(16)

The mass transport coefficient was calculated in analogy to the heat transport coefficient with the Grashof number, the Sherwood number Sh and the Schmidt number Sc. The required diffusion coefficient D was determined using correlations from Vignes, Tyn and Calus33 (see Appendix). As observed in Equations 2 through 16, the crystal layer thickness, the growth rate, the temperatures, the heat transport coefficients, the mass transport coefficient and the physical properties are connected implicitly. Hence, the dependencies of these parameters were initially calculated as functions of the growth rate and crystal layer thickness by iteration using Equations 2 through 7, 15 and 16. With these obtained correlations, the growth rate was modeled as function of the cooling rate and melt concentration using Matlab®. Because our model is based only on physical equations, the calculated growth rates are purely predictive without fitting to experimental data.

2.2.4 CALCULATION OF THERMAL CONDUCTIVITY The thermal conductivity of the crystal layer has a major effect on the crystal growth rate in layer melt crystallization. Therefore, a method was developed which allows the measurement of the crystal thermal conductivity in a laboratory-scale layer crystallizer with high precision and reproducibility. The crystal thermal conductivity was determined at a constant crystal layer thickness and constant temperatures for the cooling finger and double jacket. Equation 17, a short form of Equation 9 with zero growth rate, is the fundamental equation. 2 "?' ∙ "?' "?' 0 ∙ D

E "?' F E) GH>

(17)

The required temperatures Teq and Tm as well as the heat transport coefficients were calculated as described above. The temperature profile of the crystal layer was assumed to be parabolic, and it was calculated with Equations 11, 13, 14 and the Laplace equation (Equation 18). 1 E E ∙ ∙ N) ∙ O0 ) E) E)


(18)


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2.2.5 ACCURACY OF THE MODEL The physical properties of the melt and the crystal were used to determine crystal thermal conductivities and model growth rates. Hence, the experimental uncertainty of each physical property Δpi leads to estimated uncertainties for the crystal thermal conductivity and modeled growth rate. The relation between the crystal thermal conductivity and each physical property pi was determined while fixing all other physical properties pj≠i. The uncertainty of the crystal thermal conductivity Δλcr is derived using the experimental uncertainty of each physical property via Gaussian error propagation (Equation 19). ∆0

E0 VW X N O EY K

Z[\

*

∙ ∆Y ]

(19)

The uncertainty of the modeled growth rates Δv was calculated using Equation 20 in an analogous manner.

EC ∆C VW X N O EY K

Z[\

*

∙ ∆Y ]

(20)

3 EXPERIMENTS 3.1.1 CHEMICALS The chemicals used in this work are given in Table 1. They were used as obtained without further purification. Table 1: Used chemicals, their suppliers, and purities.

Chemical p-xylene m-xylene n-hexadecane

Supplier Merck Merck Merck

Purity [wt.%] ≥ 99 % ≥ 99 % ≥ 99 %

3.1.2 EXPERIMENTAL SETUP Crystallization experiments were carried out in a laboratory-scale static crystallizer with a cooling finger as the cooled surface (Figure 2). This equipment was used in previous studies31, 6, 5 and resembles the experimental equipment used by Özoguz, Kim and Matsuoka16, 22, 37. Therefore, only the main aspects of our crystallizer are mentioned here.


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Figure 2: Schematic drawing of the experimental equipment for static layer melt crystallization experiments. In detail: crystallizer with a double jacket (1), cooling finger (2), double jacket and cooling finger thermostats (3), and camera (4).

The cooling finger, made of stainless steel with an outer diameter of 18 mm, was dipped into the static melt. The melt was tempered by a double jacket made of glass with an inner radius rdj of 30 mm. The wall thicknesses of the cooling finger and the double jacket were 1.5 mm and 2.2 mm, respectively. The average depth of fill L was 70 mm. The temperature of the ethanol cooling agent in the double jacket and the cooling finger was measured with an accuracy of ±0.1 K with pt100 thermometers. To guarantee a uniform cylindrical crystal, the cooling finger had a cap at its bottom and was surrounded by a ring at the point of immersion, both made of chemical-resistant polytetrafluoroethylene. The crystal thickness was recorded online with a camera and analyzed with commercially available Corel PHOTO-PAINT X4 software, with an accuracy of ± 0.5 mm5. For the measurement of crystal thermal conductivity, 150 g of the melt was filled into the crystallizer, and the temperature of the double jacket was set to 2.5 K and 5 K above the liquidus line5, 30. The cooling finger was set to the same temperature and dipped into the melt. To obtain a thin initial crystal layer, the temperature of the cooling finger was decreased rapidly and subsequently increased to 4 K below the liquidus line to form a thin initial crystal layer. Afterwards, the temperature of the cooling finger was decreased in steps of 1 K and held until a constant crystal thickness was obtained. During the measurement of growth rates, the temperature of the double jacket was set to 1 K above the liquidus line. After the formation of a thin initial crystal layer, the temperature of the cooling finger was decreased linearly, as described above. ACS Paragon Plus Environment


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3.1.3 PHYSICAL PROPERTIES OF THE CHEMICALS USED References for the physical properties used to determine the crystal thermal conductivity and modeling growth rates are given in Table 2. These properties were compared to various other literature data38-47 and assumed to be reliable. Table 2: References for the used physical properties of the investigated compounds. 1) Same as n-aldehyde; 2) equimolar mixture of n-tridecanal and iso-tridecanal.

ρm n-dodecanal 48 iso-dodecanal 31 n-tridecanal 5 iso-tridecanal 51) n-hexadecane 48 p-xylene 48 m-xylene

ρcr 31 5

50

cp,m 31 31 5 51) 48 48 48

cp,cr 31 5

48

λm 48 481) 48 481) 53 48 48

de ^_`abc 31

∆fde gh 31

5 5 50 30

5 5 54

ηm 31 51 512) 51 48 48 56

ei ^gh 49 49 5 52

σ 50 50 50 50

55

48

4 RESULTS 4.1 ACCURACY OF THE MODEL The accuracy of the crystal thermal conductivities measured with our method was investigated in this study. The inaccuracy of the modeled growth rates was considered as well, which arose from experimental uncertainties of the used physical properties. Table 3 shows by how many percent the crystal thermal conductivity and modeled growth rate change if a particular physical property has an experimental uncertainty of one percent. Table 3: Deviation (percentage) of crystal thermal conductivity and modeled growth rate for an experimental uncertainty of one percent of the respective physical property.

λcr,n-dodecanal vn-dodecanal/iso-dodecanal

de λm cp,m ηm ρm Tdj ^gh 0.64 0.28 0.25 0.54 77.64 112.33 0.11 0.10 0.06 0.15 45.07 27.75

For example, if the liquid thermal conductivity λm has an experimental uncertainty of one percent, then the deviation of the determined crystal thermal conductivity would be 0.64 percent. Experimental uncertainties in the temperature, particularly the melting temperature TSLcryst, had the largest influence on the measured crystal thermal conductivity and the modeled growth rate. Temperature uncertainty leads to a significant deviation in the heat flow through the crystal layer, which is the main parameter for the determination of the crystal thermal conductivity and the modeled growth rate. Apart from temperatures, the thermal conductivity of the melt and the density of the melt had the largest influence for all investigated systems. In conclusion, it is essential to measure temperature with high accuracy, while physical properties such as the heat capacity of the melt can be estimated with some inaccuracy when they are not known. The crystal thermal conductivity was determined from the crystal layer thickness at fixed temperatures in the double jacket and the cooling finger. Therefore, a measurement uncertainty of the layer thickness also contributed to the experimental ACS Paragon Plus Environment

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uncertainty of the crystal thermal conductivity. The propagated uncertainties of the crystal thermal conductivity and the modeled growth rates via Equations 19 and 20 are shown in sections 4.2 and 4.3.2.

4.2 CRYSTAL THERMAL CONDUCTIVITY The crystal thermal conductivities of n-dodecanal, n-tridecanal, p-xylene, and n-hexadecane were measured as functions of the crystal temperature. The crystal thermal conductivity is determined by measuring the constant thickness of a crystal layer formed due to fixed temperatures in the double jacket and cooling finger. The temperature of the crystal mmmm Tkl is assumed to be the integral average of the cooling finger temperature Tcf,w and equilibrium temperature Teq. Figure 3 shows an example plot of the crystal temperature and the corresponding crystal layer thicknesses for n-hexadecane as functions of time.

Figure 3: Temperature of the cooling agent in the cooling finger Tcf (line) and the corresponding crystal layer thicknesses of n-hexadecane (triangles) as functions of time. The double jacket temperature was held constant at Tdj=296.3 K.

The crystal layer thickness increases and reaches a constant value during each temperature step. The crystal temperatures, the respective constant crystal layer thicknesses and the resulting crystal thermal conductivities are shown in Table 4. The stated uncertainties are the propagated uncertainties calculated via Equation 19.

Table 4: Crystal and double jacket temperatures, crystal layer thicknesses and crystal thermal conductivities for n-hexadecane, p-xylene, n-dodecanal and n-tridecanal.

mmmm ^ _` [K] 288.8 288.3 287.8 287.3

n-hexadecane; Tdj =296.3 K s [mm] λcr [W/m/K] 3.7 0.370 ± 0.052 4.5 0.375 ± 0.052 5.4 0.376 ± 0.052 6.3 0.380 ± 0.052 average 0.375 ± 0.052 n-dodecanal; Tdj =291.5 K mmmm s [mm] λcr [W/m/K] ^ _` [K]

p-xylene; Tdj= 291.6 K mmmm s [mm] λcr [W/m/K] ^_` [K] 284.1 2.0 0.287 ±0.052 283.6 2.5 0.298 ± 0.052 282.6 3.5 0.310 ± 0.052 282.1 4.2 0.320 ± 0.052 average 0.304 ± 0.052 n-tridecanal; Tdj =289.2 K mmmm [mm] λcr [W/m/K] ^ _` [K]



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283.5 282.5 281.5 280.5

2.1 3.3 4.5 5.8 average

0.188 ± 0.039 0.219 ± 0.039 0.234 ± 0.039 0.249 ± 0.039 0.222 ± 0.039

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284.4 283.9 283.4

5.4 6.7 8.4

average

0.247 ± 0.027 0.244 ± 0.027 0.251 ± 0.027 0.248 ± 0.027

For all compounds, a slight increase in the crystal thermal conductivity with decreasing temperature can be observed, which is consistent with the literature57-59. The known crystal thermal conductivities of n-hexadecane are 0.40 ± 0.08 W/m/K23 and 0.35 ± 0.07 W/m/K60, and the known crystal thermal conductivity for p-xylene is 0.29 W/m/K61. These values are in perfect agreement with our measurements, which supports two conclusions: first, the method applied within this work provides reliable values for the crystal thermal conductivity, and second, the proposed model adequately describes heat transport. In particular, the assumption of a constant temperature Tm in the melt is validated by showing that the ratio J between heat conduction and heat transport in the melt is close to zero (Figure 4).

Figure 4: The ratio J of heat conduction and heat transport in the melt as a function of the double jacket temperature for pure p-xylene at constant layer thicknesses (s=1 mm (circles), s=2 mm (squares), s=3 mm (triangles)).

It can be seen that the heat conduction in the melt was more than one order of magnitude less than the heat transport. Therefore, heat flow by heat conduction in the melt was negligible, and the assumption of no radial temperature in the melt is valid. This assumption is more valid for thinner crystal layers and higher double jacket temperatures. Table 4 shows that the crystal thermal conductivities of n-dodecanal and n-tridecanal are in the same range. Nevertheless, it is not possible to conclude that aldehydes with similar chain lengths have similar crystal thermal conductivities. According to the literature, there is no welldefined dependence between crystal thermal conductivity and chemical structure as for melting points62. Crystal thermal conductivities can differ significantly for molecules with different chain lengths in the same chemical group23, 63. It was checked in preliminary investigations whether it was appropriate to assume that the temperature profile in the crystal layer is linear or whether it is necessary to use a parabolic profile. The crystal thermal conductivities were considerably lower for a linear profile than for a parabolic profile. Consequently, the measured values would not agree with the literature ACS Paragon Plus Environment

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data. This deviation increases with crystal layer thicknesses. In the case of p-xylene, the average crystal thermal conductivity would be 11.2 % lower. Thus, we conclude that it is necessary to use a parabolic profile.

4.3 CRYSTAL GROWTH RATES 4.3.1 IMPLICIT RELATIONS IN A NATURALLY CONVECTED MELT In this section, we show that the implicit relations between the growth rate and natural convection are significant in static layer melt crystallization and cannot be neglected for modeling growth rates. Therefore, we present the relations between crystal growth rate, crystal layer thickness, distribution coefficient kint, temperatures and transport coefficients during static melt crystallization. The heat transport coefficient at the crystal surface αcr and the temperature in the melt Tm are chosen for the systems n-tridecanal/iso-tridecanal and p-xylene/m-xylene. Figure 5 illustrates the heat transport coefficient αcr as functions of the crystal layer thickness and the growth rate for the p-xylene/m-xylene system at a p-xylene concentration of wp=0.9.

Figure 5: Heat transport coefficient at the crystal surface αcr as a function of the crystal layer thickness and crystal growth rate for p-xylene/m-xylene during layer crystallization from a static melt (wp=0.9, kint=0).

The heat transport coefficient αcr increases with increasing growth rate and decreasing layer thickness. This behavior can be explained as follows: the faster the growth rate, the higher the concentration of the impurity in front of the crystal layer and therefore the lower the equilibrium temperature Teq. This results in a stronger natural convection due to a higher temperature difference between the crystal surface and the melt. Consequently, the heat transport coefficient increases. Decreasing heat transport coefficients with increasing crystal layer thickness can be explained by the temperature Tm in the melt. Because of cylindrical growth, increasing crystal layer thickness leads to a greater crystal surface area and higher heat flow through the crystal. As illustrated in Figure 6, the result is a lower temperature Tm in the melt. This leads to a smaller temperature difference between the crystal surface and the melt and therefore less natural convection.



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Figure 6: Melt temperature as a function of crystal layer thickness and crystal growth rate for pxylene/m-xylene during layer crystallization from a static melt (wp=0.9, kint=0).

Figure 7 shows the heat transport coefficient αcr as a function of the crystal layer thickness and the growth rate for n-tridecanal/iso-tridecanal for a n-tridecanal concentration of wn=0.9.

Figure 7: Heat transport coefficient at the crystal surface as a function of the crystal layer thickness and the crystal growth rate for n-tridecanal/iso-tridecanal during layer crystallization from a static melt (wn=0.9, kint=0).

Comparing Figures 5 and 7 shows that the heat transport coefficient is lower for n-tridecanal/iso-tridecanal than for p-xylene/m-xylene, with identical crystal layer thicknesses and growth rates. The higher viscosity of n-tridecanal/iso-tridecanal leads to a lower natural convection in the melt and, therefore, a lower heat transport coefficient at the crystal surface. As a result, a smaller amount of heat must be transported through the crystal layer, and the growth rate is faster. Consequently, by assuming identical crystal thermal conductivities and no limiting influences by mass transport, crystals generally grow faster from melts with high viscosity than crystals from melts with low viscosity. However, this statement applies only for the investigated melts and viscosities. For melts with considerably higher viscosities, we assume that mass transport limitations can reduce the growth rates. The previous plots were obtained by assuming the growth of a pure crystal and therefore a distribution coefficient kint equal to zero. However, because the distribution coefficient influences the concentration and temperature at the crystal surface, it can have a significant impact on the crystal growth rate. Therefore, the impurity concentration at the crystal surface ACS Paragon Plus Environment

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is calculated with Equation 16. The distribution coefficient for n-tridecanal/iso-tridecanal was obtained from a previous study5. Figure 8 shows the heat transport coefficient αcr as a function of the crystal layer thickness and growth rate using the distribution coefficient kint for a ntridecanal concentration of wn=0.95.

Figure 8: Heat transport coefficient at the crystal surface as a function of the crystal layer thickness and crystal growth rate for n-tridecanal/iso-tridecanal during layer crystallization from a static melt (wn=0.9, 5 kint=f(v) ).

By considering the distribution coefficient, the heat transport coefficient is lower than with a distribution coefficient equal to zero. The accumulation of impurities at the crystal surface decreases with increasing distribution coefficient. Therefore, the temperature at the crystal surface is higher than when the distribution coefficient is equal to zero, resulting in less natural convection and a lower heat transport coefficient. Consequently, assuming identical physical properties, crystal growth rates are always higher from a melt with lower separation efficiency.

4.3.2 MEASURED AND MODELED GROWTH RATES Figures 9, 10, and 11 show the measured and modeled growth rates from static layer melt crystallization experiments for n-dodecanal/iso-dodecanal, n-tridecanal/iso-tridecanal, and p-xylene/m-xylene as functions of the cooling rate and initial melt concentration.



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Figure 9: Measured (symbols) and predicted (lines) growth rates of n-dodecanal/isododecanal as functions of the cooling rate at four melt concentrations (stars: wn=0.55; triangles: wn=0.8; squares: wn=0.9; circles: wn=1.0). Experimental data 5 from .

Figure 10: Measured (symbols) and predicted (lines) growth rates of n-tridecanal/iso-tridecanal as functions of the cooling rate at two melt concentrations (squares: wn=0.9; circles: wn=1.0) (solid lines: distribution coefficient kint is a function of the growth rate 5; dashed line: kint=0). Experimental data from5.

Figure 11: Measured (symbols) and predicted (lines) growth rates for p-xylene/m-xylene as functions of the cooling rate at three melt concentrations (triangles: wn=0.8; squares: wn =0.9; circles: wn =1.0).

The growth rates of p-xylene/m-xylene were measured in this work and are an average of at least two measurements, with a standard deviation of ±1.9∙10-8 m/s. The growth rates of the aldehyde systems were measured in a previous study5. The average crystal thermal ACS Paragon Plus Environment

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conductivities measured in this work were used to model growth rates. Our model is able to predict growth rates in good agreement with the measured growth rates as a function of the melt concentration and cooling rate. Furthermore, Figures 9 through 11 show that growth rates decrease with decreasing initial melt concentration of the crystallizing substance. This effect can be justified by the equilibrium temperature Teq at the crystal surface, which is connected directly to the impurity concentration wi,eq. Figure 12 shows the ratio between the impurity concentrations at the crystal surface and in the melt as a function of the growth rate and melt concentration for n-dodecanal/iso-dodecanal.

Figure 12: Ratio of the impurity concentrations at the crystal surface and in the melt as a function of the crystal growth rate for different melt concentrations (wn=0.9 (circles), wn=0.8 (squares), wn=0.55 (triangles). System: n-dodecanal/iso-dodecanal).

The accumulation of impurities at the crystal surface increases with increasing growth rate, while the dependence on the melt concentration is negligible, which indicates that the mass transport coefficient does not depend significantly on concentration in the investigated concentration ranges which is in accordance with the literature2. As mentioned above, the equilibrium temperature was connected directly to the impurity concentration by the liquidus line. Because the bending of the liquidus line increased with an increasing impurity concentration, the ratio between the equilibrium temperatures at the crystal surface and in the melt is a function of concentration (Figure 13).



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Figure 13: Ratio of the equilibrium temperature at the crystal surface to the equilibrium temperature in the melt as a function of the crystal growth rate for different melt concentrations (wn=0.9 (circles), wn=0.8 (squares), wn=0.55 (triangles). System: n-dodecanal/iso-dodecanal.

Consequently, the heat transport coefficient and therefore the heat flow through the crystal layer increase with an increasing impurity concentration in the melt (Figure 14), resulting in lower growth rates.

Figure 14: Heat transport coefficient at the crystal surface as a function of the growth rate for different initial melt concentrations for n-dodecanal/iso-dodecanal (wn=0.9 (circles), wn=0.8 (squares), wn=0.55 (triangles)).

Adequate modeling of the growth rates as functions of melt concentration emphasizes the fact that the impurity concentration at the crystal surface is described adequately. This refutes the approach proposed by Scholz15 and Guardani18, 19 who used an equation for stagnant fluids. The equation for stagnant fluids was tested in this work for the n-dodecanal/iso-dodecanal system. That equation led to impurity concentrations at the crystal surface that were 13 % lower than the calculated concentrations. The resulting modeled growth rates were 9 % higher than the growth rates calculated in this work, which is not in agreement with the experimental data. Hence, the equation that we used for the calculation of the impurity concentration (Equation 15) at the crystal surface is more accurate than the equation used by Guardani and Scholz. It is further important to note that it is appropriate to assume an equilibrium temperature at the crystal surface. In contrast, Parisi14 assumed a supercooled temperature at the crystal ACS Paragon Plus Environment

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surface for inducing crystallization and showed that this supercooling decreases with decreasing growth rate. This supercooling is negligible for the growth rates obtained in this study. As illustrated in Figures 9, 10 and 11, the growth rates increased with an increasing cooling rate. This nonlinear trend can be explained by the cylindrical shape of the growing crystal, which leads to superproportional increase in area and volume with increasing crystal layer thickness15. Because, as mentioned above, the heat transport coefficient also increased with an increasing impurity concentration, the nonlinearity increased with an increasing impurity concentration in the melt. Comparing the growth rates of pure n-dodecanal (Figure 9) and n-tridecanal (Figure 10) reveals that the growth rates of n-tridecanal are 29 % faster than for n-dodecanal, although the crystal thermal conductivity is only 12 % higher. This result supports our assumption that crystals grow faster from melts with higher viscosity. As mentioned in the previous section, the distribution coefficient has an impact on the growth rate. This influence is shown for the n-tridecanal/iso-tridecanal system (Figure 10) because it has the highest overall distribution coefficient among the investigated systems. In the case of a distribution coefficient equal to zero, the modeled growth rates are too low to adequately fit the experimental data (Figure 10). By considering the distribution coefficient as a function of the growth rate, the modeled growth rates are higher and in significantly better agreement with measured values. Consequently, it is advisable to consider the distribution coefficient when it is high and a strong function of the growth rate but to ignore it for systems with very high separation efficiencies. Figures 15, 16 and 17 compare measured and modeled growth rates for the investigated systems in parity plots.

Figure 15: Parity plot of the modeled and measured growth rates for n-dodecanal/iso-dodecanal. The solid line is the x=y reference. The dashed lines represent the confidence intervals of the modeled growth rates based on experimental uncertainties of the physical properties.



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Figure 16: Parity plot of the modeled and measured growth rates for n-tridecanal/iso-tridecanal. The solid line is the x=y reference. The dashed lines represent the confidence intervals of the modeled growth rates based on experimental uncertainties of the physical properties.

Figure 17: Parity plot of the modeled and measured growth rates for p-xylene/m-xylene. The solid line is the x=y reference. The dashed lines represent the confidence intervals of the modeled growth rates based on the experimental uncertainties of the physical properties.

Here, the confidence intervals, characterized by the dashed lines, represent the maximum achievable accuracy of the modeling due to the experimental uncertainty of the physical properties: 5.62 %, 5.94 % and 5.91 % for n-dodecanal/iso-dodecanal, n-tridecanal/isotridecanal and p-xylene/m-xylene, respectively. Deviations that are larger than the confidence intervals occurred for mainly two reasons. First, the impurity concentration at the crystal surface requires knowledge of the diffusion coefficient, which was estimated by the equations from Tyn and Calus. Uncertainties in this estimation led to uncertainties in the modeling results. Second, the growth rate calculation is sensitive to the parameters for heat transport through the crystal. Because the crystal was not perfectly cylindrical during these experiments, the area of the crystal and the heat flow were calculated with a small uncertainty. These ACS Paragon Plus Environment

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inaccuracies increased with increasing growth rate. Thus, deviations between experimental and modeled values increased slightly with an increasing cooling rate. Because the investigated melts differ significantly in viscosity, chemical structure and purification efficiency, the good agreement between the measured and modeled growth rates underlines the broad applicability of this model.

5 CONCLUSIONS This study proposes a reliable approach to model growth rates from static layer melt crystallization. In contrast to approaches from the literature, our model considers the thermal resistance of the crystallizer construction parts and non-linear growth rates. Physical properties of the melt and the crystal are regarded as functions of temperature and concentration. Furthermore, implicit relations between growth rate and natural convection are not negligible in static layer melt crystallization. These considerations allow us to model growth rates from static layer melt crystallization processes that meet the needs of industrial applications. Here, the crystal thermal conductivity is an essential physical property for the modeling of growth rates, which is often fitted to experimental data in the literature. This poses the problem that any uncertainty in the model is compensated by the fitted thermal conductivity; thus, these models are no longer predictive. Therefore, we developed a method to measure the crystal thermal conductivity in a laboratory-scale crystallizer. The crystal thermal conductivities of n-dodecanal, n-tridecanal, p-xylene and n-hexadecane were measured as functions of temperature. Our method was validated successfully with known crystal thermal conductivities from the literature. Because this method is highly sensitive to the physical properties of the melt and the crystal, a comprehensive assessment of uncertainties was conducted. Our method provides an experimental uncertainty of 15 %, which is more accurate than many methods from the literature. Additionally, temperatures must be determined with high accuracy, while physical parameters, such as heat capacities or densities, can be estimated with an inaccuracy of up to 10 %. With the knowledge of the crystal thermal conductivity, the crystal growth rates from static layer melt crystallization were predicted as functions of the cooling rate and melt concentration. An assessment of uncertainty showed that, based on the experimental uncertainties of the used physical properties, the modeled growth rates had an inaccuracy of 5.8 %. The deviation between the modeled and predicted growth rates of three binary isomeric mixtures, n-dodecanal/iso-dodecanal, n-tridecanal/iso-tridecanal, and p-xylene/m-xylene, was 9 % on average. This good agreement was achieved without the use of fitted parameters. Furthermore, our model can be used to consider the impact of the distribution coefficient on the growth rate. Because the physical properties of the investigated melts differ significantly, broad applicability of the presented model can be stated.

ACKNOWLEDGEMENTS This work is part of the Collaborative Research Centre project, "Integrated Chemical Processes in Liquid Multiphase Systems". Financial support by the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged. ACS Paragon Plus Environment


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6 APPENDIX Heat transport coefficients in natural convection33

n) 9.81 ⋅

/r |S Suvwx | ⋅ Suvwx s*

(21)

L n) ⋅ y)

z ⋅ JK,

0

y)

(22) (23) ~ +

0.492 {+ "y)' |1 % N O y)

+ ~

+ * / N0.825 % 0.387 ⋅ "L ⋅ {+ "y)'' O % 0.435 ⋅

2⋅/ 0

(24)

(25)

(26)

Mass transport coefficients in natural convection33

/r |S Suvwx | n) 9.81 ⋅ * ⋅ Suvwx s

(27)

L n) ⋅ J J

z

S ∙

(28) (29) ~ +

0.492 {+ "J ' |1 % N O J

+ ~

+ * / N0.825 % 0.387 ⋅ "L ⋅ {+ "J'' O % 0.435 ⋅


(30)

(31)

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T∙/

(32)

Calculation of the diffusion coefficient33 By assuming an ideal mixture, the diffusion coefficient of component A in B is calculated by the Vignes correlation (Equation 33). '5 " '5 " ⋅

(33)

The diffusion coefficients in highly diluted mixtures are calculated by the correlation of Tyn and Calus via equation 34 with an accuracy of ±10 %. +

+

r J* Jr Jr 8.93 ⋅ 10 ⋅ C " ' ⋅ C y I. ⋅ N O ⋅ ⋅ "z yL'+ y

(34)

Here, vA(T ) and vB(T ) are the specific volumes of components A and B at their boiling

points, which were taken from literature48. The parachor of component i is defined by equation 35.

Jr I.* y v ⋅ N D FO

(35)

Mixing rules for physical properties

S4 SI ⋅ % SI4 ⋅ "1 '

JK,4 JK.I ⋅ % JK,I4 ⋅ "1 ' * * 0* 4 ⋅ 0I % "1 ' ⋅ 0I4

z4 Y ⋅ "zI ' % 1 4 ⋅ ln "zI4 '


(36) 31 (37) 31 (38) 16 (39) 16


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NOTATION A

area

[m²]

cp

heat capacity

[kJ/kg/K]

D

diffusion coefficient

[m²/s]

diffusion coefficient of A in B in highly diluted mixtures

[m²/s]

d

diameter

[m]

∆h¨ kl¥¦§

Grashof number

[-]

heat of fusion of crystallizing component

[kJ/kg]

J

ratio between heat conduction and heat transport

[-]

k

thermal diffusivity

[m²/s]

kint

integral distribution coefficient

[-]

L

depth of fill, length

[m]

Nu

Nusselt number

[-]

Pr

Prandtl number

[-]

P

parachor

p

physical property

[-]

q7

heat flow

[W]

q8

specific heat flow by heat conduction

[W/m²]

specific heat flow by heat transport

[W/m²]

R

ideal gas constant

[kJ/kg/K]

rdj

inner radius of the double jacket

[m]

rcf+s

radius of the outer crystal surface

[m]

Ra

Rayleigh number

[-]

D

Gr

Q

[cm³∙mN/mol/m]


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Re

Reynolds number

[-]

Rth

thermal resistance

[K/W]

Rth,α

thermal resistance of conductive heat transport

[K/W]

Rth,λ

thermal resistance of crystallizer construction part

[K/W]

s

crystal layer thickness

[m]

Sc

Schmidt number

[-]

Sh

Sherwood number

[-]

T

temperature

[K]

¨ Tkl¥¦§

boiling temperature of component i

[K]

mmmm T kl

melting temperature of component i

[K]

average temperature of the crystal

[K]

Tcf

temperature of cooling agent in cooling finger

[K]

Tcf,w

temperature of outer surface of cooling finger

[K]

Tdj

temperature of cooling agent in double jacket

[K]

Tdj,w

temperature of outer surface of double jacket

[K]

Teq

equilibrium temperature

[K]

Tm

temperature of the melt

[K]

t

time

[s]

v

crystal growth rate

[m/s]

vi

specific volume of component i

[cm³/mol]

w

mass fraction

[-]

wi,eq

mass fraction of impurity component at the crystal surface

[-]

wi,m

mass fraction of impurity component in the melt (bulk)

[-]

x

mole fraction

[-]

TIª

Greek symbols α

heat transport coefficient

[W/m²/K]

β

mass transport coefficient

[m/s]

η

dynamic viscosity

[kg/m/s]

λ

thermal conductivity

[W/m/K]

ν

kinematic viscosity

[m²/s]

ρ

density

[kg/m³]

σ

surface tension

[kg/s²]

Sub- and superscripts 0i

pure component i ACS Paragon Plus Environment


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cf

cooling finger

cr

crystal

cryst

crystallizing component

dj

double jacket

eq

equilibrium

i

impurity component, component i in mixture

j

component j in mixture

LV

liquid-vapor

m

melt

n

linear aldehyde

p

p-xylene

SL

solid-liquid


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Table of contents use only

Table of content graphic (TOC)

Synopsis

This work provides a detailed description of heat transport in crystal growth during static layer melt crystallization. In contrast to models in the literature, this model includes implicit relations between the growth rate and natural convection. This approach allows predicting growth rates that are in high agreement with experimental data without the use of fitted parameters.



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Modeling Growth Rates in Static Layer Melt Crystallization - Crystal

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