Melt Crystallization for Purification

Reference ID: MET-4D13 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The melt-crystallization calculation predicts the performance of a single-stage melt crystallizer used to purify a binary fatty-acid mixture (e.g., stearic acid / palmitic acid). By cooling a liquid feed above its melting point, the major component crystallises while the impurity remains in the liquid (mother liquor). The crystals are withdrawn together with a small amount of entrained liquor; the impurity retained in this entrained fraction determines the final product purity. This methodology is essential for designing heat-removal equipment, sizing crystallizer vessels, and estimating achievable purity in the fatty-acid and broader specialty-chemical industries.

Methodology & Formulas

1. Feed composition on a mole basis

Let \(\dot{m}_{F}\) be the total feed mass flow (kg h^-1), \(w_{F}\) the impurity weight fraction in the feed, and \(M_{SA}, M_{PA}\) the molecular weights of stearic and palmitic acid respectively.

\[ n_{SA,F}= \frac{\dot{m}_{F}\,(1-w_{F})}{M_{SA}},\qquad n_{PA,F}= \frac{\dot{m}_{F}\,w_{F}}{M_{PA}} \] \[ n_{tot,F}= n_{SA,F}+n_{PA,F} \] \[ x_{SA,F}= \frac{n_{SA,F}}{n_{tot,F}},\qquad x_{PA,F}=1-x_{SA,F} \]

2. Mother-liquor composition (Schröder-van Laar equation)

The liquidus relationship for the major component (stearic acid) is

\[ \ln\!\bigl(x_{L,SA}\bigr)=\frac{\Delta H_{f}}{R}\left(\frac{1}{T_{m,SA}}-\frac{1}{T_{op}}\right) \]

where \(\Delta H_{f}\) is the molar heat of fusion (J mol^-1), \(R\) the universal gas constant, \(T_{m,SA}\) the melting temperature of stearic acid (K), and \(T_{op}\) the operating temperature (K). The impurity mole fraction follows from the binary constraint:

\[ x_{L,PA}=1-x_{L,SA} \]

Conversion to a weight-basis impurity fraction in the liquid (\(w_{L}\)) uses the molecular weights:

\[ w_{L}=\frac{x_{L,PA}\,M_{PA}}{x_{L,SA}\,M_{SA}+x_{L,PA}\,M_{PA}} \]

3. Mass balances

Stream	Mass flow (kg h^-1)	Impurity wt %	Impurity mass flow (kg h^-1)
Feed	\(\dot{m}_{F}\)	\(w_{F}\)	\(\dot{m}_{F} w_{F}\)
Dry crystals	\(\dot{m}_{C,dry}\)	0	0
Mother liquor	\(\dot{m}_{L}\)	\(w_{L}\)	\(\dot{m}_{L} w_{L}\)

The overall and impurity balances are

\[ \dot{m}_{F} = \dot{m}_{C,dry} + \dot{m}_{L} \] \[ \dot{m}_{F} w_{F} = \dot{m}_{L} w_{L} \]

Solving for the liquor flow and crystal yield:

\[ \dot{m}_{L} = \dot{m}_{F} \frac{w_{F}}{w_{L}},\qquad \dot{m}_{C,dry}= \dot{m}_{F} - \dot{m}_{L} \] \[ \text{Crystal yield}= \frac{\dot{m}_{C,dry}}{\dot{m}_{F}}=1-\frac{w_{F}}{w_{L}} \]

4. Product composition with entrainment

The entrainment ratio \(e\) (kg liquor per kg dry crystal) defines the total product mass flow

\[ \dot{m}_{P} = \dot{m}_{C,dry}\,(1+e) \]

The impurity fraction in the final product is the entrained liquor impurity diluted by the total mass:

\[ w_{P} = \frac{e}{1+e}\;w_{L} \]

5. Separation efficiency

\[ \eta = \frac{w_{F} - w_{P}}{w_{F}} \]

6. Energy balance

Sensible cooling of the feed:

\[ Q_{sens}= \dot{m}_{F}\,C_{p}\,(T_{in}-T_{op}) \]

Latent heat released by crystallisation (per kilogram of dry crystal):

\[ \Delta h_{lat}= \frac{\Delta H_{f}}{M_{SA}} \] \[ Q_{lat}= \dot{m}_{C,dry}\,\Delta h_{lat} \]

Total heat removal (converted to kilowatts):

\[ Q = Q_{sens}+Q_{lat},\qquad Q_{kW}= \frac{Q}{3600} \]

7. Validity checks (empirical ranges)

Criterion	Limit	Expression	Note
Operating temperature vs eutectic	\(T_{op} > T_{eutectic}\)	\(T_{op} - T_{eutectic} > 0\)	Avoid simultaneous crystallisation of impurity.
Feed impurity level	\(w_{F} \le 0.10\)	\(w_{F}\) must satisfy Schröder-van Laar applicability.	Higher impurity requires experimental phase data.
Entrainment ratio	\(0.05 \le e \le 0.15\)	\(e\) within typical filter-cake range.	Adjust based on filtration and washing efficiency.

Melt crystallization is often preferred for heat-sensitive materials because it operates at significantly lower temperatures than distillation, preventing thermal degradation. Key benefits include:

Elimination of solvent recovery systems, reducing overall process complexity.
Higher selectivity for isomers that possess similar boiling points but different freezing points.
Lower energy consumption due to the lower latent heat of fusion compared to the latent heat of vaporization.

The cooling rate is a critical parameter that dictates the crystal growth kinetics and the inclusion of impurities. If the cooling rate is too high, the following issues typically occur:

Rapid crystal growth leads to the entrapment of mother liquor within the crystal lattice.
Increased nucleation rates result in smaller crystals with a higher surface area, which increases surface adsorption of impurities.
Formation of dendritic structures that trap liquid phase contaminants.

Sweating is a controlled partial melting process used to enhance the purity of the crystal cake after the mother liquor has been drained. During this stage:

The temperature of the crystal bed is raised slightly below the melting point of the pure component.
Impurities concentrated at the crystal boundaries or trapped in inclusions melt first and are removed.
The process effectively purifies the crystal surface, ensuring the final product meets stringent quality specifications.

Melt Crystallization for Purification – Worked Example

A single-stage melt crystallizer purifies a binary mixture of stearic acid (main component) and palmitic acid (impurity). Feed is liquid at 80 °C, above the melting point. Operation at 60 °C induces crystallization of stearic acid. The wet cake is separated with an entrainment ratio of 0.10 kg liquor per kg dry crystal.

Known Parameters

Feed mass flow: \(F = 1000.000\ \text{kg/h}\)
Feed impurity (palmitic acid): \(w_{F} = 0.050\ \text{wt fraction}\) (5.0 wt%)
Feed temperature: \(T_{in} = 80.0\ ^\circ\text{C}\)
Operating temperature: \(T_{op} = 60.0\ ^\circ\text{C}\)
Entrainment ratio: \(e = 0.100\)
Heat capacity (liquid): \(C_{p} = 2.100\ \text{kJ/(kg·K)}\)
Stearic acid molecular weight: \(M_{SA} = 284.480\ \text{g/mol}\)
Palmitic acid molecular weight: \(M_{PA} = 256.420\ \text{g/mol}\)
Melting point of stearic acid: \(T_{m,SA} = 342.750\ \text{K}\)
Molar heat of fusion: \(\Delta H_{f} = 47500.0\ \text{J/mol}\)
Universal gas constant: \(R = 8.314\ \text{J/(mol·K)}\)

Step-by-Step Calculation

Feed composition (mole basis)
\(\displaystyle n_{SA,F} = \frac{F\,(1-w_{F})}{M_{SA}} = 3.339\ \text{kmol/h}\)
\(\displaystyle n_{PA,F} = \frac{F\,w_{F}}{M_{PA}} = 0.195\ \text{kmol/h}\)
Total moles: \(n_{tot,F} = 3.534\ \text{kmol/h}\)
Mole fractions: \(x_{SA,F} = 0.945\), \(x_{PA,F} = 0.055\)
Mother liquor composition at 60 °C (Schröder-van Laar equation)
Operating temperature in Kelvin: \(T_{op,K} = 60.0 + 273.15 = 333.15\ \text{K}\)
\(\displaystyle \ln(x_{L,SA}) = \frac{\Delta H_{f}}{R}\left(\frac{1}{T_{m,SA}} - \frac{1}{T_{op,K}}\right) = -0.480\)
\(x_{L,SA} = \exp(-0.480) = 0.619\)
\(x_{L,PA} = 1 - 0.619 = 0.381\) (mole fraction)
Convert to weight basis:
Mass of SA: \(0.619 \times M_{SA} = 175.974\ \text{kg/kmol}\)
Mass of PA: \(0.381 \times M_{PA} = 97.804\ \text{kg/kmol}\)
Total mass: \(273.778\ \text{kg/kmol}\)
Liquor impurity: \(w_{L} = 0.357\ (35.7\ \text{wt%})\)
Crystal yield
Mother liquor flow: \(\displaystyle \dot{m}_{L} = F \cdot \frac{w_{F}}{w_{L}} = 139.963\ \text{kg/h}\)
Dry crystal flow: \(\dot{m}_{C,dry} = F - \dot{m}_{L} = 860.037\ \text{kg/h}\)
Yield: \(\displaystyle \frac{\dot{m}_{C,dry}}{F} = 0.860\ (86.0\%)\)
Product impurity and separation efficiency
Product impurity: \(\displaystyle w_{P} = \frac{e}{1+e} \cdot w_{L} = 0.0325\ (3.3\ \text{wt%})\)
Separation efficiency: \(\displaystyle \eta = \frac{w_{F} - w_{P}}{w_{F}} = 0.350\ (35.0\%)\)
Heat removal
Sensible cooling: \(\displaystyle Q_{sens} = \frac{F \cdot C_{p} \cdot (T_{in} - T_{op})}{3600} = 11.667\ \text{kW}\)
Specific latent heat of crystallization: \(\Delta h_{lat} = \dfrac{\Delta H_{f}}{M_{SA}} = 166.971\ \text{kJ/kg}\)
Latent heat removal: \(\displaystyle Q_{lat} = \frac{\dot{m}_{C,dry} \cdot \Delta h_{lat}}{3600} = 39.889\ \text{kW}\)
Total heat removal: \(Q_{total} = Q_{sens} + Q_{lat} = 51.556\ \text{kW}\)

Final Results

Product purity: 96.8 wt% (3.3 wt% impurity).
Crystal yield: 86.0%.
Total cooling duty: 51.556 kW.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle