Reference ID: MET-293D | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Plank's Equation is a fundamental analytical model in food process engineering used to estimate the time required to freeze a product of a specific geometry, and the contact freezing time calculation provides a practical tool for applying this model to real‑world scenarios.
This calculation is critical in the design and operation of industrial freezing equipment, such as plate freezers, air‑blast tunnels, and cryogenic immersion systems. It allows engineers to determine throughput capacities, optimize cooling rates, and ensure product quality by predicting the time required for the freezing front to penetrate the product core, as detailed in our critical freezing rate determination guide.
Methodology & Formulas
The freezing time is determined by calculating the resistance to heat transfer, which consists of both convective and conductive components. The total freezing time t is defined by the following relationship, which is closely linked to the impact of freezing rate on ice crystal size.
To assess the validity of the model and the dominant heat transfer regime, the following dimensionless numbers are utilized, and a detailed IQF freezing time calculation is provided for further reference.
Parameter
Formula
Description
Biot Number (\(Bi\))
\( Bi = \dfrac{h \cdot (d/2)}{k} \)
Ratio of internal conductive resistance to external convective resistance. A large Bi (\(>0.5\)) indicates that conduction resistance controls the freezing rate, making Plank's Equation particularly appropriate.
Stefan Number (\(Ste\))
\( Ste = \dfrac{c_{p} \cdot \Delta T}{\lambda} \)
Ratio of sensible heat to latent heat; used to justify the exclusion of sensible heat. Values below 0.5 support the phase‑change‑dominance assumption.
The geometric constants P and R are determined by the shape of the product:
Geometry
P
R
Infinite Slab
0.5
0.125
Infinite Cylinder
0.25
0.0625
Sphere
0.1667
0.0417
Note: For the slab geometry, the characteristic dimension d represents the full thickness of the product when cooled from both sides. If cooling occurs from only one side, the half‑thickness should be used as the characteristic dimension; see the geometry factors in Plank’s Equation for more detail.
Plank's Equation estimates the freezing time of a product by balancing heat removal through conduction (in the frozen layer) and convection (at the surface). It applies to simple, regular geometries—infinite slab, infinite cylinder, and sphere—for which the geometric constants \(P\) and \(R\) have been derived.
The key assumptions of the model are:
The material is initially at its freezing temperature (no pre‑cooling).
Phase change occurs at a single, constant temperature.
Thermal properties (density, latent heat, thermal conductivity) are constant.
Heat flow is one‑dimensional and quasi‑steady through the frozen layer.
The convective heat‑transfer coefficient at the surface is constant.
The model is particularly useful when internal conduction resistance is significant (\(Bi > 0.5\)); for very low Biot numbers simpler lumped‑capacitance methods may suffice, though Plank's Equation remains mathematically valid.
The calculation follows these steps:
Identify the characteristic dimension \(d\) (m). For a slab cooled from both sides, \(d\) is the full thickness.
Determine the temperature difference: \(\Delta T = |T_{f} - T_{a}|\), where \(T_{f}\) is the freezing point and \(T_{a}\) is the coolant temperature (°C or K).
Obtain the material properties: density \(\rho\) (kg/m³), latent heat of fusion \(\lambda\) (J/kg), and thermal conductivity \(k\) (W/(m·K)).
Measure or estimate the convective heat‑transfer coefficient \(h\) (W/(m²·K)) at the cooling surface.
Select the geometry constants for a slab: \(P = 0.5\), \(R = 0.125\).
Compute the Biot number to confirm the regime: \(Bi = h \cdot (d/2) \,/\, k\). Plank's Equation is applicable for all Biot values and is especially important when \(Bi > 0.5\).
Apply Plank's formula:
\[ t = \frac{\rho \cdot \lambda}{\Delta T} \left( \frac{P \cdot d}{h} + \frac{R \cdot d^{2}}{k} \right) \]
where \(t\) is the freezing time (s).
Accurate use of Plank's Equation requires the following properties, evaluated at the freezing temperature range:
Density \(\rho\) (kg/m³)
Thermal conductivity \(k\) (W/(m·K))
Latent heat of fusion \(\lambda\) (J/kg)
Freezing point \(T_{f}\) (°C)
Convective heat‑transfer coefficient \(h\) (W/(m²·K)) for the cooling medium
Specific heat capacity \(c_{p}\) is not directly used in the basic Plank equation but is needed to calculate the Stefan number, which verifies that sensible heat can be neglected. If any property varies significantly with temperature, use an average value over the freezing interval or perform a segmented analysis.
Plank's Equation assumes constant surface temperature and simple geometry. To handle more complex cases:
Divide the product into smaller elements that each approximate a slab and apply Plank's Equation to each element.
Use a time‑dependent surface temperature \(T_{s}(t)\) in the driving force, integrating numerically over the freezing period.
For cylindrical or spherical shapes, use the appropriate \(P\) and \(R\) constants (see the Methodology table) or employ analytical solutions (e.g., Carslaw–Jaeger solutions) for more detailed transient analysis.
Incorporate variable \(h(t)\) if the cooling medium changes temperature or flow conditions during the process.
These approaches retain the physical insight of Plank's method while extending its applicability to real‑world process‑engineering scenarios.
Worked Example: Freezing Time of a Fish Slab Using Plank's Equation
Product: Fish block (slab geometry, both sides cooled)
Density (\(\rho\)): 1000.0 kg/m3
Latent heat (\(\lambda\)): 300000.0 J/kg
Specific heat capacity (\(c_{p}\)): 2000 J/(kg·K)
Thermal conductivity (\(k\)): 1.5 W/(m·K)
Freezing temperature (\(T_{f}\)): –2.0 °C
Coolant temperature (\(T_{a}\)): –28.0 °C
Heat transfer coefficient (\(h\)): 500.0 W/(m2·K)
Slab thickness (\(d\)): 0.05 m
Geometry constants: \(P = 0.5\), \(R = 0.125\)
Compute temperature difference:
\(\Delta T = |T_{f} - T_{a}| = |(-2.0) - (-28.0)| = 26.0\ \text{K}\).
Stefan number: \(Ste = \dfrac{c_{p} \cdot \Delta T}{\lambda}
= \dfrac{2000 \times 26.0}{300000} \approx 0.173\).
Since \(Ste < 0.5\), the sensible heat contribution is small, supporting the assumption of phase‑change dominance.
"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
