Principles and Industrial Application

Reference ID: MET-B8D5 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

Freeze concentration is a high-fidelity separation process used primarily in the food and pharmaceutical industries to concentrate heat-sensitive aqueous solutions. Unlike thermal evaporation, which can degrade volatile flavor compounds or heat-labile nutrients, freeze concentration operates at sub-zero temperatures, preserving the organoleptic and chemical integrity of the product.

The process relies on the selective crystallization of pure water from a solution, followed by the mechanical separation of the ice crystals from the concentrated mother liquor. It is typically employed for high-value products such as fruit juices, coffee extracts, and biological proteins where maintaining product quality is paramount.

Methodology & Formulas

The engineering design of a freeze concentration system requires balancing thermodynamic cooling requirements with kinetic crystal growth models and fluid mechanical separation limits.

1. Material Balance and Cooling Duty

The mass of ice generated is determined by the concentration ratio required to reach the target solute concentration:

\[ \dot{m}_{ice} = \dot{m}_{feed} \cdot \left( 1 - \frac{C_{feed}}{C_{concentrate}} \right) \]

The total cooling duty required to achieve the phase change and sensible cooling is defined as:

\[ Q = \frac{\dot{m}_{ice} \cdot \lambda + \dot{m}_{feed} \cdot c_{p} \cdot \Delta T}{3600} \]

where \(\Delta T\) is the temperature difference between the feed inlet and the freezing point of the solution.

2. Crystallizer Hydrodynamics

In a scraped-surface heat exchanger, the flow regime is characterized by the rotor Reynolds number, which dictates the heat transfer efficiency:

\[ Re_{rotor} = \frac{\rho \cdot N \cdot D^{2}}{\mu} \]

3. Crystal Growth Kinetics

Crystal size is a function of the growth rate and the residence time within the crystallizer. The average crystal diameter is calculated using the growth law:

\[ G = k_{g} \cdot (\Delta T_{sub})^{n} \]

\[ L_{avg} = G \cdot t_{res} \]

where \(G\) is the linear growth rate, \(k_{g}\) and \(n\) are empirical constants, and \(\Delta T_{sub}\) is the subcooling.

4. Separation Efficiency

Solute loss is primarily driven by the mass of the mother liquor film adhering to the surface of the ice crystals. The mass of mother liquor retained per unit mass of ice is estimated by the specific surface area of the crystals, the film thickness, and the densities of the liquor and ice:

\[ f_{m} = \frac{6 \cdot \delta}{L_{avg}} \cdot \frac{\rho_{liquor}}{\rho_{ice}} \cdot (1 - \eta_{wash}) \]

The corresponding solute loss fraction, relative to the total feed solids, is obtained from the mass balance:

\[ \text{Loss}_{solute} = f_{m} \cdot \frac{\dot{m}_{ice}}{\dot{m}_{feed}} \cdot \frac{C_{concentrate}}{C_{feed}} \]

Operational Regimes and Validity Limits

Parameter	Symbol	Valid Range / Limit
Rotor Reynolds Number	\( Re_{rotor} \)	50 ≤ \( Re_{rotor} \) ≤ 1000
Crystal Diameter	\( L_{avg} \)	50 μm ≤ \( L_{avg} \) ≤ 500 μm
Prandtl Number (liquid phase)	\( Pr \)	3 ≤ \( Pr \) ≤ 100
Concentrate Viscosity	\( \mu \)	< 100 cP (Centrifugal) / > 500 cP (Wash Column)
Solute Loss	\( \text{Loss}_{solute} \)	1% – 5% of total solids

Freeze concentration separates water from a solution by selectively crystallizing pure ice. Because the ice lattice excludes most solutes, the remaining liquid becomes concentrated. This phase-change process preserves heat-sensitive aromas and nutrients far better than thermal evaporation.

The rotor Reynolds number \(Re_{rotor}\) describes the flow regime in a scraped-surface heat exchanger. A value in the range 50–1000 ensures turbulent mixing, which enhances heat transfer from the cold wall to the bulk liquid, promotes uniform subcooling, and controls the nucleation and growth rate of ice crystals.

The ice crystals leaving the crystallizer carry a thin film of concentrated mother liquor on their surface. A wash column or centrifugal rinse removes most of this film. The wash efficiency \(\eta_{wash}\) directly multiplies the adhered liquid mass: even a small improvement (e.g., from 90% to 95%) can halve the solute loss, keeping it within the acceptable 1–5% of total solids.

Worked Example: Freeze Concentration of Orange Juice

Scenario: A process engineer is designing a freeze concentration system to concentrate orange juice from 12 °Brix to 30 °Brix. The feed flow rate is 100 kg/h and enters the crystallizer at 7.0 °C above its freezing point. A swept-surface crystallizer with a rotor diameter of 0.1 m operating at 0.5 rps is used. The juice near freezing has a viscosity of 0.05 Pa·s and density of 1050 kg/m³. Subcooling is maintained at 4.0 °C with a residence time of 10 min. Centrifugal separation is employed with an adhered film thickness of 20 μm and a wash efficiency of 95%. Ice density is taken as 917 kg/m³.

Known Input Parameters

Feed mass rate, \(\dot{m}_{feed} = 100.0\) kg/h
Feed concentration, \(C_{feed} = 12.0\) °Brix
Target concentration, \(C_{target} = 30.0\) °Brix
Sensible cooling range, \(\Delta T = 7.0\) °C
Rotor diameter, \(D = 0.1\) m
Rotational speed, \(N = 0.5\) rps
Viscosity, \(\mu = 0.05\) Pa·s
Liquid density, \(\rho_{liquor} = 1050.0\) kg/m³
Ice density, \(\rho_{ice} = 917.0\) kg/m³
Thermal conductivity, \(k = 0.5\) W/(m·K)
Subcooling, \(\Delta T_{sub} = 4.0\) °C
Residence time, \(t_{res} = 10.0\) min
Adhered film thickness, \(\delta = 20.0\) μm
Wash efficiency, \(\eta_{wash} = 0.95\)
Latent heat of ice, \(\lambda = 334.0\) kJ/kg
Specific heat of juice, \(c_{p} = 4.18\) kJ/(kg·K)
Growth law constants, \(k_{g} = 0.075\) μm/(s·°C\(^{1.5}\)), \(n = 1.5\)

Step-by-Step Calculation

Ice Mass Flow Rate
Using the material balance for solute retention in the liquid phase:
\[ \dot{m}_{ice} = \dot{m}_{feed} \cdot \left(1 - \frac{C_{feed}}{C_{target}}\right) = 100.0 \cdot \left(1 - \frac{12.0}{30.0}\right) = 60.0 \text{ kg/h} \]
Cooling Duty
Heat load includes latent heat of ice formation and sensible cooling of the feed from inlet temperature to freezing point:
\[ Q = \frac{\dot{m}_{ice} \cdot \lambda + \dot{m}_{feed} \cdot c_{p} \cdot \Delta T}{3600} = \frac{60.0 \cdot 334.0 + 100.0 \cdot 4.18 \cdot 7.0}{3600} = 6.379 \text{ kW} \]
Rotor Reynolds Number
\[ Re_{rotor} = \frac{\rho_{liquor} \cdot N \cdot D^{2}}{\mu} = \frac{1050.0 \cdot 0.5 \cdot (0.1)^{2}}{0.05} = 105.0 \]
Crystal Growth Rate
\[ G = k_{g} \cdot (\Delta T_{sub})^{n} = 0.075 \cdot (4.0)^{1.5} = 0.6 \text{ μm/s} \]
Average Crystal Size
\[ L_{avg} = G \cdot (t_{res} \cdot 60) = 0.6 \cdot (10.0 \cdot 60) = 360.0 \text{ μm} \]
Specific Surface Area and Solute Loss Fraction
Specific surface area of spherical crystals (per unit volume of ice): \[ a_{s} = \frac{6}{L_{avg}} = \frac{6}{360.0 \times 10^{-6} \text{ m}} = 1.667 \times 10^{4} \text{ m}^{2}/\text{m}^{3} \] Mass of mother liquor retained per unit mass of ice (after washing): \[ f_{m} = a_{s} \cdot \delta \cdot \frac{\rho_{liquor}}{\rho_{ice}} \cdot (1 - \eta_{wash}) \] \[ f_{m} = 1.667 \times 10^{4} \cdot (20.0 \times 10^{-6}) \cdot \frac{1050.0}{917.0} \cdot (1 - 0.95) = 0.0191 \text{ kg liquor / kg ice} \] Solute loss as a fraction of feed solids: \[ \text{Loss}_{solute} = f_{m} \cdot \frac{\dot{m}_{ice}}{\dot{m}_{feed}} \cdot \frac{C_{target}}{C_{feed}} \] \[ \text{Loss}_{solute} = 0.0191 \cdot \frac{60.0}{100.0} \cdot \frac{30.0}{12.0} = 0.0286 \] i.e., 2.86% of total solids are lost with the ice.

Final Results Summary

Ice production rate: 60.0 kg/h
Concentrate production rate: 40.0 kg/h (calculated as \(\dot{m}_{feed} - \dot{m}_{ice}\))
Cooling duty: 6.379 kW
Rotor Reynolds number: 105.0 (within empirical range 50–1000)
Average crystal size: 360.0 μm (within efficient washing range 50–500 μm)
Solute loss fraction: 0.0286 (2.86% of total solids, within 1–5% industrial target)

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