Brine Freezing Calculation

Reference ID: MET-FD8A | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The Brine Freezing Calculation is a fundamental process engineering procedure used to determine the time required to freeze food products, such as fish, via immersion in a chilled sodium chloride (NaCl) solution. This method is critical in the food processing industry for designing efficient cooling tunnels and batch immersion tanks. By calculating the necessary brine concentration and the resulting freezing kinetics, engineers can ensure product quality, optimize energy consumption, and prevent the formation of ice within the brine solution itself.

Methodology & Formulas

The calculation follows a sequential approach, starting with the determination of the brine's physical properties and concluding with the estimation of total freezing time using Plank's equation for slab geometries.

Step 1: Brine Freezing Point
The freezing point of the brine is determined by the mass concentration of NaCl, denoted as w (mass %). The empirical relationship, valid for NaCl–water solutions up to the eutectic concentration, is defined as:

\[ T_{brine,f} = -(0.353 \cdot w + 0.0237 \cdot w^2) \]

Step 2: Driving Temperature Difference
The thermal driving force is the difference between the freezing point of the food product and the operating temperature of the brine:

\[ \Delta T = T_{f,food} - T_{brine} \]

Step 3: Pre-Freeze Sensible Cooling Time (Conservative Approximation)
Before phase change occurs, the product must be cooled from its initial temperature to its freezing point. For a slab of thickness L cooled from both faces, the time required can be estimated conservatively by assuming the convective heat flux remains constant at its minimum value (using the final temperature difference ΔT). This yields:

\[ t_{pre} = \frac{\rho \cdot c_p \cdot (T_{initial} - T_{f,food})}{h \cdot \Delta T} \cdot \frac{L}{2} \]

Note: This approximation systematically overestimates the sensible cooling time because the actual driving temperature difference starts at (T_initial − T_brine) and decreases to ΔT. The result is conservative for design purposes. For a more rigorous solution, transient conduction (Heisler charts or series solutions) should be employed, particularly when Bi > 0.1.

Step 4: Phase-Change Freezing Time (Plank's Equation)
Using Plank's equation for a slab frozen from both faces, the time required for the latent heat of fusion to be removed is calculated based on convective heat transfer and internal thermal conductivity:

\[ t_{freeze} = \frac{\rho \cdot L_f}{\Delta T} \cdot \left( \frac{L}{2 \cdot h} + \frac{L^2}{8 \cdot k} \right) \]

Step 5: Total Freezing Time
The total process time is the sum of the sensible cooling phase and the phase-change freezing phase:

\[ t_{total} = t_{pre} + t_{freeze} \]

Parameter	Condition / Regime	Threshold
NaCl Concentration	Empirical Validity (mass %)	\(0 < w \leq 23.3\)
Biot Number	Plank's Equation Validity	\(Bi = \frac{h \cdot L}{k} \geq 0.1\)
Temperature Gradient	Thermodynamic Feasibility	\(\Delta T > 0\)

The freezing point of NaCl brine is determined using the empirical mass-concentration correlation specific to this methodology. Follow these steps:

Measure or specify the NaCl concentration w as mass percent (wt % NaCl in water).
Apply the empirical formula: \(T_{brine,f} = -(0.353 \cdot w + 0.0237 \cdot w^2)\) to obtain the freezing point in °C.
Verify that the concentration is within the validity range: \(0 < w \leq 23.3\) % (the eutectic concentration).
Ensure the brine operating temperature is above this calculated freezing point to prevent ice formation in the brine.

The upper limit of 23.3 % NaCl by mass corresponds to the eutectic concentration of the NaCl–water binary system. For process engineers, this constraint is critical because:

At this concentration, the brine reaches its lowest possible freezing point (approximately –21.1 °C).
Exceeding 23.3 % NaCl causes solid salt to precipitate from solution before freezing occurs, which can foul heat exchanger surfaces and alter the effective brine composition.
The empirical correlation \(T_{brine,f} = -(0.353 \cdot w + 0.0237 \cdot w^2)\) is validated only up to this eutectic concentration; extrapolation beyond 23.3 % is not reliable.

Maintaining an adequate margin between the brine operating temperature \(T_{brine}\) and its calculated freezing point \(T_{brine,f}\) is essential for reliable operation:

The thermodynamic feasibility condition requires \(\Delta T > 0\) for the product, but the brine itself must also remain fully liquid.
In the worked example, \(T_{brine} = -21.0\) °C with \(T_{brine,f} = -21.091\) °C, yielding a margin of only ~0.09 K. While this works mathematically, practical systems should incorporate a larger margin (typically 1–3 K) to accommodate temperature control fluctuations and concentration gradients.
Monitor brine concentration regularly; evaporative losses can increase NaCl concentration and raise the freezing point toward the operating temperature, risking unintended ice formation.

Worked Example: Brine Immersion Freezing of Fish Fillets

Scenario: A continuous immersion freezer uses brine at –21 °C to freeze slab-shaped fish fillets. The brine concentration is set to 23.3 % NaCl by mass to prevent bulk brine freezing. Determine the total freezing time from initial fillet temperature to complete phase change.

Known Parameters

Brine concentration, \(w\) = 23.3 % (mass)
Brine operating temperature, \(T_{brine}\) = –21.0 °C
Fish fillet thickness (slab), \(L\) = 0.05 m
Fish initial temperature, \(T_{initial}\) = 5.0 °C
Fish freezing point, \(T_{f,food}\) = –2.0 °C
Convective heat transfer coefficient, \(h\) = 500 W·m⁻²·K⁻¹
Fish density, \(\rho\) = 1000 kg·m⁻³
Frozen fish thermal conductivity, \(k\) = 1.5 W·m⁻¹·K⁻¹
Fish latent heat of freezing, \(L_f\) = 334 000 J·kg⁻¹
Unfrozen fish specific heat, \(c_p\) = 3600 J·kg⁻¹·K⁻¹

Step-by-Step Calculation

Brine freezing point check.
Using the empirical relation \(T_{brine,f} = -(0.353\,w + 0.0237\,w^2)\), with \(w = 23.3\), the brine freezing point is computed: \(T_{brine,f} = -21.091\) °C. Since the operating temperature (–21 °C) is slightly above this value, the brine remains liquid.
Driving temperature difference.
The effective temperature difference for heat transfer is \[ \Delta T = T_{f,food} - T_{brine} = (-2.0) - (-21.0) = 19.0\ \text{K} \] (provided as \(\Delta T = 19.0\) K).
Biot number verification.
The Biot number for the slab is \[ \mathrm{Bi} = \frac{h\,L}{k} = \frac{500 \times 0.05}{1.5} = 16.667 \] This is > 0.1, confirming that internal thermal resistance is significant and Plank's equation is valid.
Pre-freeze sensible cooling time (conservative approximation).
For a slab with both faces exposed, \(V/A = L/2\). The conservative sensible cooling time is \[ t_{pre} = \frac{\rho\,c_p\,(T_{initial} - T_{f,food})}{h\,\Delta T} \cdot \frac{L}{2} \] Substituting the known numbers yields the pre-freeze time: \(t_{pre} = 66.316\) s. (This is an upper-bound estimate; the actual pre-cooling time would be shorter due to the initially higher driving temperature difference.)
Phase-change freezing time (Plank's equation for slab).
Plank's equation gives \[ t_{freeze} = \frac{\rho\,L_f}{\Delta T}\left(\frac{L}{2h} + \frac{L^2}{8k}\right) \] Using the given parameters, the computed freezing time is \(t_{freeze} = 4541.228\) s.
Total freezing time.
The total time is the sum of the two contributions: \[ t_{total} = t_{pre} + t_{freeze} = 66.316 + 4541.228 = 4607.544\ \text{s} \]

Final Answer: The total freezing time required to freeze a 0.05 m thick fish fillet under the given conditions is 4607.544 s (approximately 1.28 h).

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

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Brine Freezing Calculation

Introduction & Context

Methodology & Formulas

Worked Example: Brine Immersion Freezing of Fish Fillets

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