Reference ID: MET-9F44 | 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, and a detailed heat transfer analysis for convective freezing further explains how the model accounts for both convective heat transfer at the surface and conductive heat transfer through the frozen layer of the product.
This calculation is critical in the design and operation of industrial freezing equipment, such as air blast freezers, tunnel freezers, and contact plate freezers. By determining the freezing time, engineers can optimize conveyor speeds, freezer residence times, and energy consumption, ensuring that products reach the required core temperature while maintaining quality and safety standards.
Methodology & Formulas
The freezing time is calculated by evaluating the thermal resistance of the system, which is divided into a convective component and a conductive component. The governing equation, known as Plank’s equation for freezing time, with the temperature driving force expressed as a positive difference \(\Delta T = T_{f} - T_{a}\) (valid for normal freezing where the coolant is colder than the product’s initial freezing point), is:
Where the effective latent heat of fusion is determined by the water mass fraction of the product:
\[ \lambda = w \cdot \lambda_{0} \]
The geometry factors \(P\) and \(R\) are dimensionless constants derived from analytical solutions of the one-dimensional phase-change heat transfer problem for regular shapes. They represent the resistance contributions from surface convection and internal conduction through the frozen layer, respectively.
Geometry
Characteristic Length (\(d\))
\(P\)
\(R\)
Infinite Slab
Thickness
1/2
1/8
Infinite Cylinder
Diameter
1/4
1/16
Sphere
Diameter
1/6
1/24
Variable Definitions:
\(t_{f}\): Freezing time (s)
\(\rho\): Product density (kg/m³)
\(\lambda\): Effective latent heat of fusion (J/kg)
\(\lambda_{0}\): Latent heat of pure water (≈ 334 × 10³ J/kg)
\(w\): Water mass fraction (dimensionless)
\(T_{f}\): Initial freezing point of the product (°C or K)
\(T_{a}\): Cooling medium temperature (°C or K)
\(\Delta T = T_{f} - T_{a}\): Temperature driving force (positive when \(T_{f} > T_{a}\))
\(h\): Convective heat transfer coefficient (W/m²·K)
\(k\): Thermal conductivity of the frozen product (W/m·K)
\(d\): Characteristic length (m) – see table for definition per geometry
The dimensionless factors \(P\) and \(R\) arise from the analytical solution of the one-dimensional phase-change heat conduction problem for regular geometries. Physically:
\(P\) (convective factor): Accounts for the shape's influence on the external convective resistance at the product surface. The term \(P d / h\) represents the effective thermal resistance due to the cooling medium boundary layer.
\(R\) (conductive factor): Accounts for the shape's influence on the internal conductive resistance through the growing frozen layer. The term \(R d^{2} / k\) represents the effective resistance to heat conduction from the freezing front to the surface.
These factors are not adjustable; they are fixed by geometry (Infinite Slab: \(P=1/2,\ R=1/8\); Infinite Cylinder: \(P=1/4,\ R=1/16\); Sphere: \(P=1/6,\ R=1/24\)).
For a fixed mass, the slab geometry (large surface area, small thickness) provides a shorter maximum distance that heat must travel from the thermal centre to the surface. This is captured mathematically by the characteristic length \(d\) and the factors \(P\) and \(R\):
The slab's convective term \(P d / h\) is smaller because \(P_{\text{slab}}=1/2\) gives a larger denominator effect compared to \(P_{\text{sphere}}=1/6\) when the geometry is "flattened".
Critically, the conductive term scales with \(d^{2}\); the slab's much smaller characteristic thickness dominates the resistance reduction.
The combined effect is that the total thermal resistance for the slab is significantly lower, leading to a faster freezing time.
Plank's equation is an invaluable first approximation, but process engineers must be aware of its simplifying assumptions:
Constant thermal properties: The frozen-layer thermal conductivity \(k\) and density \(\rho\) are assumed constant, whereas real food materials exhibit temperature-dependent properties.
Sharp freezing point: The model assumes a distinct phase-change interface at a single temperature \(T_{f}\). Real foods (especially with high water content and solutes) freeze progressively over a range of temperatures.
Uniform initial temperature: The entire product is assumed to be at the freezing point at the start of the phase-change period; any sensible cooling above \(T_{f}\) is neglected.
Regular geometries only: The tabulated \(P\) and \(R\) values are valid strictly for infinite slabs, infinite cylinders, and spheres. Irregular shapes require numerical methods (e.g., finite differences or CFD) or empirical shape-factor correlations.
Constant surface heat transfer coefficient: \(h\) is assumed uniform and constant, which may not hold in high-velocity air blast freezing where local turbulent effects vary.
For industrial design, Plank's equation is typically used for preliminary sizing, with final validation performed using numerical simulations or pilot-plant trials.
Worked Example: Freezing of a Spherical Meatball vs. Flat Patty Using Plank's Equation
Scenario: A process engineer needs to compare the freezing time of a 50 g spherical meatball and a 50 g flat patty (infinite-slab approximation) in an air blast freezer. Both products are ground meat with 65% water content. The freezer operates at −30 °C with a convection coefficient of 25 W/(m²·K).
Knowns
Mass, \(m\) = 0.050 kg
Density, \(\rho\) = 1050 kg/m³
Water fraction, \(w\) = 0.65
Initial freezing point, \(T_{f}\) = −2.0 °C
Coolant temperature, \(T_{a}\) = −30.0 °C
Temperature driving force, \(\Delta T = |T_{f} - T_{a}|\) = 28.0 K
Convection coefficient, \(h\) = 25.0 W/(m²·K)
Frozen thermal conductivity, \(k\) = 1.5 W/(m·K)
Slab thickness, \(d_{\text{slab}}\) = 0.010 m
Latent heat of water, \(\lambda_{0}\) = 334 × 10³ J/kg
Spherical meatball: \(t_{f}\) ≈ 2 900 s (≈ 48 min)
Flat patty (slab): \(t_{f}\) ≈ 1 700 s (≈ 28 min)
The sphere freezes approximately 1.7 times slower than the slab of the same mass, driven by the less favourable surface-area-to-volume ratio and longer internal conduction path.
"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
