The Correlation for Effective Distribution Coefficient with Initial Impurity Concentration and Growth Rate for Acrylic Acid in Melt Crystallization

Abstract

The layer growth rates and resulting crystal purity during solid-layer melt crystallization were experimentally measured for acrylic acid (AA) with impurity propionic acid (PA) operated at various cooling temperatures. A power law was adopted to correlate the growth rate with the temperature difference between melt and coolant. The effective distribution coefficient was determined from the resulting crystal purity for each condition. An empirical equation modified from the analytical solution for the mass transfer boundary layer was proposed in this work to relate the effective distribution coefficient to the initial impurity concentration and growth rate.

1. Introduction

Melt crystallization is an important separation technique for the purification of organic compounds in the chemical and pharmaceutical industries [1,2,3,4]. It has many advantages over distillation due to no solvent addition, low energy consumption, low temperature operation, etc. It has been efficiently adopted to replace distillation for heat-sensitive materials. In melt crystallization, a crystal layer of the desired substance growing on a cooled wall is produced from the melt. Both static melt crystallization and falling film melt crystallization have been widely applied in industries. Although melt crystallization is typically a selective operation, some unwanted components can be incorporated into the crystal of the desired substance. The effective distribution coefficient, defined as the ratio of the impurities in the solid phase and the liquid phase, is often used to describe the degree of separation [1]. Numerous studies have investigated the crystal layer growth rate and effective distribution coefficient based on the fundamental energy balance and mass balance equations for various types of melt crystallization [5,6,7,8,9,10,11,12,13,14,15,16].

Acrylic acid (AA) is an important chemical used as a polymer form in many fields, such as diapers, textiles, coatings, and adhesives. Traditional AA synthesis is based on oxidation of propylene or propane obtained by steam cracking in the petrochemical process. Recently, research has focused on the biosynthesis of AA from renewable feed stocks, e.g., fermentable sugars and plant oils [17,18]. However, a high concentration of propionic acid (PA) is usually produced as an impurity in AA biosynthesis. Due to the close boiling points between AA (141 °C) and PA (141.15 °C), it is usually difficult to separate them by distillation. In recent years, melt crystallization has been applied to separate AA from melt with impurity PA [19,20,21]. To elucidate the influence of process parameters on the effective distribution coefficient, the objective of this study is to investigate the dependence of the effective distribution coefficient on the initial impurity concentration and layer growth rate for AA melt with PA as an impurity for industrial applications.

2. Experimental

Solid-layer melt crystallization experiments were performed for the AA melt with impurity PA using a vessel immersed in a water bath with a cylindrical crystallizer internally cooled by the circulating coolant shown in Figure 1 [22]. The melt in the vessel was mixed using a magnetic stirrer with a stirring rate of 130 rpm.

AA (99.5%, ACROS) and PA (99%, ACROS) were used to prepare a 1000 mL binary mixture with each mole fraction of AA, ${\mathrm{C}}_{\mathrm{mt},\mathrm{AA}}$. The chemical structures of AA and PA are illustrated in Figure 2. According to the solid-liquid equilibrium data for the binary mixture of AA and PA reported in the literature [23,24], the relationship between the equilibrium temperature ${\mathrm{T}}_{\mathrm{eq}}$ and the mole fraction of AA ${\mathrm{C}}_{\mathrm{mt},\mathrm{AA}}$ is given by

$${\mathrm{T}}_{\mathrm{eq}}\left(\mathrm{K}\right)=-21.63{\mathrm{C}}_{\mathrm{mt},\mathrm{AA}}{}^2+89.69{\mathrm{C}}_{\mathrm{mt},\mathrm{AA}}+218.07,(\mathrm{for} 0.5<{\mathrm{C}}_{\mathrm{mt},\mathrm{AA}}<1),$$

(1)

As listed in

Table 1. The experimentally measured data of $\mathrm{G}$ and ${\mathrm{C}}_{\mathrm{cry},\mathrm{im}}$ for various ${\mathrm{C}}_{\mathrm{mt},\mathrm{AA}}$ operated at various ${\mathrm{T}}_{\mathrm{C}}$ from the binary AA/PA mixture.

${\mathbf{C}}_{\mathbf{mt},\mathbf{AA}}$ (-)	${\mathbf{C}}_{\mathbf{mt},\mathbf{im}}$ (-)	${\mathbf{T}}_{\mathbf{eq}}$ (K)	${\mathbf{T}}_{\mathbf{C}}$ (K)	$\mathbf{\Delta }\mathbf{T}$ (K)	$\mathbf{G}(\times {\mathbf{10}}^{-\mathbf{6}})$ (m/s)	${\mathbf{C}}_{\mathbf{cry},\mathbf{im}}$ (-)	${\mathbf{k}}_{\mathbf{eff}}$ (-)
0.954	0.046	283.9	273	10.9	3.89	0.023	0.500
			270	13.9	5.39	0.026	0.565
			266.5	17.4	5.89	0.030	0.609
			263	20.9	6.92	0.031	0.674
0.908	0.092	281.7	273	8.68	3.11	0.057	0.620
			270	11.7	4.22	0.063	0.685
			266.5	15.2	5.33	0.067	0.728
			263	18.7	6.33	0.069	0.750
0.861	0.139	279.3	273	6.30	1.78	0.097	0.698
			270	9.30	3.94	0.105	0.755
			266.5	12.8	4.67	0.109	0.784
			263	16.3	5.39	0.114	0.820
0.813	0.187	276.7	273	3.70	1.50	0.138	0.738
			270	6.70	3.25	0.143	0.765
			266.5	10.2	3.67	0.149	0.797
			263	13.7	4.33	0.154	0.824

, ${\mathrm{T}}_{\mathrm{eq}}\left({\mathrm{C}}_{\mathrm{mt},\mathrm{AA}}\right)$ was determined from Equation (1). The melt in the vessel was maintained at the corresponding ${\mathrm{T}}_{\mathrm{eq}}$ for a given ${\mathrm{C}}_{\mathrm{mt},\mathrm{AA}}$ during each experiment. At the beginning of the experiments, the cylindrical crystallizer internally circulated with the coolant at 253 K was submerged into the melt for 5 s to initiate nucleation on the outer surface of the crystallizer. Then, the cylindrical crystallizer was removed from the melt. As the coolant was quickly switched to a specified cooling temperature ${\mathrm{T}}_{\mathrm{C}}$, the cylindrical crystallizer was submerged into the melt again, and the crystal layer gradually grew on the outer surface of the cylindrical crystallizer during the experiments.

The total growth time for each experiment was kept at 600 s. At the end, the overall crystal layer grown on the outer surface of the cylindrical crystallizer ${\mathrm{M}}_{\mathrm{S}}$ was melted and weighed. The impurity mole fraction in the overall crystal layer ${\mathrm{C}}_{\mathrm{cry},\mathrm{im}}$ was determined by GC using a China Chromatograph 2000 with a stainless steel capillary column (Zebron/ZB-FFAP, 30 m × 0.32 mm, Supelco, Bellefonte, PA, USA).

As shown in Figure 3, the bottom part of the cooling finger corresponds to a hemisphere with radius ${\mathrm{R}}_0$. As the height of the overall crystal on the surface of the cooling finger $\mathrm{H}$ was kept the same for each experiment, the overall crystal volume on the surface of the cooling finger was given by

$${\mathrm{M}}_{\mathrm{S}}=[\mathsf{\pi }\left({\mathrm{R}}^2-{\mathrm{R}}_0{}^2\right)\mathrm{H}+\frac{2}{3}\mathsf{\pi }\left({\mathrm{R}}^3-{\mathrm{R}}_0{}^2\right)]{\mathsf{\rho }}_{\mathrm{s}},$$

(2)

where $\mathrm{R}$ represents the outer radius of the overall crystal on the surface of the cooling finger at the end of the experiment and ${\mathsf{\rho }}_{\mathrm{s}}$ is the crystal density of AA. Thus, as ${\mathrm{M}}_{\mathrm{S}}$ was measured at the end of each experiment, $\mathrm{R}$ can be determined from Equation (2) using ${\mathsf{\rho }}_{\mathrm{s}}=1050 \mathrm{kg}/{\mathrm{m}}^3$, ${\mathrm{R}}_0=0.01 \mathrm{m}$ and $\mathrm{H}=0.10 \mathrm{m}$.

Once $\mathrm{R}$ is determined, the growth rate is given by

$$\mathrm{G}=\frac{\mathrm{R}-{\mathrm{R}}_0}{{\mathrm{t}}_{\mathrm{total}}},$$

(3)

where ${\mathrm{t}}_{\mathrm{total}}$ is the total growth time. As the crystal layer $\left(\mathrm{R}-{\mathrm{R}}_0\right)$ was generally smaller than $4.2 \mathrm{mm}$ at ${\mathrm{t}}_{\mathrm{total}}=600 \mathrm{s}$ for all the experiments, ${\mathrm{M}}_{\mathrm{S}}$ was negligible compared to the initial mass of melt. Consequently, ${\mathrm{C}}_{\mathrm{mt},\mathrm{AA}}$ was nearly unchanged during each experiment.

3. Results and Discussion

Table 1 lists the experimentally measured data of $\mathrm{G}$ and ${\mathrm{C}}_{\mathrm{cry},\mathrm{im}}$ for various ${\mathrm{C}}_{\mathrm{mt},\mathrm{AA}}$ operated at various ${\mathrm{T}}_{\mathrm{C}}$. Note that ${\mathrm{C}}_{\mathrm{mt},\mathrm{im}}=1-{\mathrm{C}}_{\mathrm{mt},\mathrm{AA}}$, where ${\mathrm{C}}_{\mathrm{mt},\mathrm{im}}$ represents the mole fraction of PA in the melt. As the coolant was circulated very quickly and the crystal layer grown on the outer surface of the cylindrical crystallizer was thin for each experiment, the crystallization heat released at the crystal–melt interface was assumed to be quickly removed by the coolant. Furthermore, as the melt was maintained at the corresponding ${\mathrm{T}}_{\mathrm{eq}}$ for a given ${\mathrm{C}}_{\mathrm{mt},\mathrm{AA}}$, the heat transfer rate from the crystal layer to the coolant should be related to $\Delta \mathrm{T}={\mathrm{T}}_{\mathrm{eq}}-{\mathrm{T}}_{\mathrm{C}}$. Consequently, the growth rate of the crystal layer should be proportional to the heat transfer rate. Thus, all the data in Figure 4 lead to [6]

$$\mathrm{G}=4.68\times {10}^{-7}\Delta {\mathrm{T}}^{0.89}({\mathrm{R}}^2=0.927),$$

(4)

Note that the line represents the fitted correlation. Figure 4 shows that $\mathrm{G}$ increases with increasing $\Delta \mathrm{T}$.

The effective distribution coefficient between crystal and melt is defined as [1]

$${\mathrm{k}}_{\mathrm{eff}}=\frac{{\mathrm{C}}_{\mathrm{cry},\mathrm{im}}}{{\mathrm{C}}_{\mathrm{mt},\mathrm{im}}},$$

(5)

where ${\mathrm{C}}_{\mathrm{mt},\mathrm{im}}$ is the impurity concentration in the melt and ${\mathrm{C}}_{\mathrm{cry},\mathrm{im}}$ is the impurity concentration in crystal. Low ${\mathrm{k}}_{\mathrm{eff}}$ values indicate high separation efficiency. The value of ${\mathrm{k}}_{\mathrm{eff}}$ ranges from zero if crystals are totally pure to one if no separation occurs. Table 1 also lists the calculated ${\mathrm{k}}_{\mathrm{eff}}$ from the experimentally measured ${\mathrm{C}}_{\mathrm{cry},\mathrm{im}}$ for various ${\mathrm{C}}_{\mathrm{mt},\mathrm{im}}$ operated at various ${\mathrm{T}}_{\mathrm{C}}$. Figure 5 shows ${\mathrm{k}}_{\mathrm{eff}}$ versus $\mathrm{G}$ for each ${\mathrm{C}}_{\mathrm{mt},\mathrm{im}}$. Thus, ${\mathrm{k}}_{\mathrm{eff}}$ decreases with decreasing $\mathrm{G}$ at each ${\mathrm{C}}_{\mathrm{mt},\mathrm{im}}$ while ${\mathrm{k}}_{\mathrm{eff}}$ is generally smaller for a lower ${\mathrm{C}}_{\mathrm{mt},\mathrm{im}}$ at a given $\mathrm{G}$. This trend for the dependence of ${\mathrm{k}}_{\mathrm{eff}}$ on $\mathrm{G}$ and ${\mathrm{C}}_{\mathrm{mt},\mathrm{im}}$ is consistent with the findings reported in various systems [22,25,26].

Figure 6 illustrates the impurity concentration in the mass transfer boundary layer along the $\mathrm{x}$ direction perpendicular to the crystal–melt interface for a melt. As the variations in the impurity concentration in the boundary layer ${\mathrm{C}}_{\mathrm{L},\mathrm{im}}\left(\mathrm{x}\right)$ with time can be neglected for a thin boundary layer [5,6,27,28], one can derive

$$\mathrm{D}\frac{{\mathrm{d}}^2{\mathrm{C}}_{\mathrm{L},\mathrm{im}}\left(\mathrm{x}\right)}{{\mathrm{dx}}^2}+\mathrm{G}\frac{{\mathrm{dC}}_{\mathrm{L},\mathrm{im}}\left(\mathrm{x}\right)}{\mathrm{dx}}=0,$$

(6)

The boundary conditions for Equation (6) are given by

$$\mathrm{at} \mathrm{x}=0, {\mathrm{GC}}_{\mathrm{L},\mathrm{im}}\left(\mathrm{x}=0\right)={\mathrm{GC}}_{\mathrm{cry},\mathrm{im}}-\mathrm{D}{\left[\frac{{\mathrm{dC}}_{\mathrm{L},\mathrm{im}}}{\mathrm{dx}}\right]}_{\mathrm{x}=0},$$

(7)

$$\mathrm{at} \mathrm{x}=\mathsf{\delta }, {\mathrm{C}}_{\mathrm{L},\mathrm{im}}\left(\mathrm{x}=\mathsf{\delta }\right)={\mathrm{C}}_{\mathrm{mt},\mathrm{im}} ,$$

(8)

where ${\mathrm{C}}_{\mathrm{L},\mathrm{im}}\left(\mathrm{x}=\mathsf{\delta }\right)={\mathrm{C}}_{\mathrm{mt},\mathrm{im}}$ is the impurity concentration in the melt and ${\mathrm{C}}_{\mathrm{L},\mathrm{im}}\left(\mathrm{x}=0\right)={\mathrm{C}}_{\mathrm{int},\mathrm{im}}$ is the impurity concentration in the melt at the crystal–melt interface.

The solution of Equation (6) is

$${\mathrm{C}}_{\mathrm{L},\mathrm{im}}\left(\mathrm{x}\right)={\mathrm{C}}_{\mathrm{cry},\mathrm{im}}+\left({\mathrm{C}}_{\mathrm{mt},\mathrm{im}}-{\mathrm{C}}_{\mathrm{cry},\mathrm{im}}\right)\exp [\frac{\mathrm{G}\left(\mathsf{\delta }-\mathrm{x}\right)}{\mathrm{D}}],$$

(9)

Substituting $\mathrm{x}=0$ into Equation (9) yields

$${\mathrm{C}}_{\mathrm{int},\mathrm{im}}={\mathrm{C}}_{\mathrm{cry},\mathrm{im}}+\left({\mathrm{C}}_{\mathrm{mt},\mathrm{im}}-{\mathrm{C}}_{\mathrm{cry},\mathrm{im}}\right)\exp \left(\frac{\mathrm{G}}{{\mathrm{k}}_{\mathrm{d}}}\right),$$

(10)

where the mass transfer coefficient is defined as ${\mathrm{k}}_{\mathrm{d}}=\frac{\mathrm{D}}{\mathsf{\delta }}$ [1]. Combining Equations (5) and (10) yields

$${\mathrm{k}}_{\mathrm{eff}}=[\frac{{\mathrm{C}}_{\mathrm{int},\mathrm{im}}}{{\mathrm{C}}_{\mathrm{mt},\mathrm{im}}}-\exp \left(\frac{\mathrm{G}}{{\mathrm{k}}_{\mathrm{d}}}\right)]/[1-\exp \left(\frac{\mathrm{G}}{{\mathrm{k}}_{\mathrm{d}}}\right)],$$

(11)

Thus, Equation (11) is an analytical solution for ${\mathrm{k}}_{\mathrm{eff}}$ in terms of the dimensionless groups, $\frac{{\mathrm{C}}_{\mathrm{int},\mathrm{im}}}{{\mathrm{C}}_{\mathrm{mt},\mathrm{im}}}$ and $\exp \left(\frac{\mathrm{G}}{{\mathrm{k}}_{\mathrm{d}}}\right)$. However, as ${\mathrm{C}}_{\mathrm{int},\mathrm{im}}$ generally cannot be experimentally measured, it is difficult to determine ${\mathrm{k}}_{\mathrm{eff}}$ from Equation (11). In other words, ${\mathrm{k}}_{\mathrm{eff}}$ is affected by the impurity concentration in melt at the crystal–melt interface, the impurity mass transfer coefficient in the melt, and the layer growth rate.

By comparing the experimental ${\mathrm{k}}_{\mathrm{eff}}$ with Equation (11), some researchers indicated for simplicity that ${\mathrm{k}}_{\mathrm{eff}}$ should be a function of the dimensionless groups, $\frac{{\mathrm{C}}_{\mathrm{mt},\mathrm{im}}}{1-{\mathrm{C}}_{\mathrm{mt},\mathrm{im}}}$ and $\exp \left(\frac{\mathrm{G}}{{\mathrm{k}}_{\mathrm{d}}}\right)$ [29,30,31]. For practical applications, an empirical form of ${\mathrm{k}}_{\mathrm{eff}}$ is proposed in this study as

$${\mathrm{k}}_{\mathrm{eff}}=\mathsf{\alpha }{\left(\frac{{\mathrm{C}}_{\mathrm{mt},\mathrm{im}}}{1-{\mathrm{C}}_{\mathrm{mt},\mathrm{im}}}\right)}^{\mathsf{\beta }}{\left[\exp \left(\frac{\mathrm{G}}{{\mathrm{k}}_{\mathrm{d}}}\right)\right]}^{\mathsf{\gamma }},$$

(12)

where $\mathsf{\alpha }$, $\mathsf{\beta }$ and $\mathsf{\gamma }$ are three coefficients. Equation (12) for $\mathsf{\gamma }=1$ reduces to the correlation adopted by Shiau [22]. However, Equation (12) provides a more generalized equation for applications. Taking the log of both sides, Equation (12) becomes

$${\mathrm{lnk}}_{\mathrm{eff}}=\ln \mathsf{\alpha }+\mathsf{\beta }\ln \left(\frac{{\mathrm{C}}_{\mathrm{mt},\mathrm{im}}}{1-{\mathrm{C}}_{\mathrm{mt},\mathrm{im}}}\right)+\left(\frac{\mathsf{\gamma }}{{\mathrm{k}}_{\mathrm{d}}}\right)\mathrm{G},$$

(13)

Thus, $\mathsf{\alpha }$, $\mathsf{\beta }$ and $\frac{\mathsf{\gamma }}{{\mathrm{k}}_{\mathrm{d}}}$ can be determined by fitting Equation (13) with the experimental data of ${\mathrm{k}}_{\mathrm{eff}}$ versus $\mathrm{G}$ for various ${\mathrm{C}}_{\mathrm{mt},\mathrm{im}}$. It should be noted that, $\mathsf{\gamma }$ cannot be calculated here unless ${\mathrm{k}}_{\mathrm{d}}$ is determined previously.

The data in Figure 5 fitted to Equation (13) lead to

$${\mathrm{k}}_{\mathrm{eff}}=0.99{\left(\frac{{\mathrm{C}}_{\mathrm{mt},\mathrm{im}}}{1-{\mathrm{C}}_{\mathrm{mt},\mathrm{im}}}\right)}^{0.28}\exp \left(59320\mathrm{G}\right) ({\mathrm{R}}^2=0.946),$$

(14)

with ${\mathrm{C}}_{\mathrm{mt},\mathrm{im}}$ in mole fraction and $\mathrm{G}$ in $\mathrm{m}/\mathrm{s}$. Note that each line represents the fitted correlation for each ${\mathrm{C}}_{\mathrm{mt},\mathrm{im}}$.

4. Conclusions

The layer growth kinetics and resulting crystal purity for AA with impurity PA were experimentally measured in a solid-layer cylindrical crystallizer internally cooled by a circulating coolant operated at various cooling temperatures. The growth rate of the crystal layer was correlated well using a power law with the temperature difference between melt and coolant. The effective distribution coefficient was found to increase with the increasing initial impurity concentration and increasing growth rate. The effective distribution coefficient was fitted well to the proposed empirical equation, which provides a simple correlation for the dependence of the effective distribution coefficient on the initial impurity concentration and growth rate.