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Freezing Times for Food Using Plank’s Equation

Heat Transfer and Engineering
HVAC Systems Components and Engineering

Freezing Times for Food Using Plank’s Equation

One of the most widely known simple methods for estimating freezing times of foods and beverages was developed by Plank (1913, 1941). Convective heat transfer is assumed to occur between the food and the surrounding cooling medium. The temperature of the food is assumed to be at its initial freezing temperature, which is constant throughout the freezing process. Furthermore, constant thermal conductivity for the frozen region is assumed. Plank’s freezing time estimation is as follows:

θ = L_f / ( T_f - T_m ) [ PD / h + RD² / k_s ]

Where

L_f = volumetric latent heat of fusion
T_f = initial freezing temperature of the food
T_m = freezing medium temperature
D = thickness of the slab or diameter of the sphere or infinite cylinder
h = convective heat transfer coefficient
k_s = thermal conductivity of the fully frozen food,
P and R = geometric factors.

For an infinite slab, P = 1/2 and R = 1/8. For a sphere, P = 1/6 and R = 1/24; for an infinite cylinder, P = 1/4 and R = 1/16.

Plank’s geometric factors indicate that an infinite slab of thickness D, an infinite cylinder of diameter D, and a sphere of diameter D, if exposed to the same conditions, would have freezing times in the ratio of 6:3:2. Hence, a cylinder freezes in half the time of a slab and a sphere freezes in one-third the time of a slab.

For more information see ASHRAE 2018 Refrigeration, SI Edition

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