Freezing Time by Plank's Equation

The classic refrigeration estimate that splits the answer into the two resistances that actually control it. The surface term P a / h is how hard it is to get heat off the product, which is airflow; the internal term R a^2 / k is how hard it is to get heat out of the middle, which is thickness, and it enters SQUARED. Halving the thickness cuts the internal term by four, while doubling the air velocity only helps the surface term. The shape constants are P = 1/2 and R = 1/8 for a slab, 1/4 and 1/16 for a cylinder, 1/6 and 1/24 for a sphere. Latent heat is the product's water, not pure water's 143.4 BTU/lb, so a 74% moisture beef slab carries about 106 BTU/lb, and the freezing point sits a few degrees below 32 F because of dissolved solids. A 3 in (0.25 ft) beef slab at 65 lb/ft3, freezing point 28 F, blast freezer at -10 F, h = 3.0 and frozen k = 0.9 gives a driving term of 6890 / 38 = 181.3, a surface term of 0.0417 and an internal term of 0.0087, so the time is about 9.1 hr and the product is airflow-limited. Double the slab to 6 in and the time more than doubles, because the internal term grew fourfold while the surface term only doubled. An engineering estimate; Plank neglects sensible heat above and below the freezing point and assumes a single freezing temperature, so it runs optimistic, and a measured product-temperature log governs a food-safety decision.

freezing_time_hr = (density_pcf x latent_heat_btu_lb / (freezing_point_f - medium_temp_f)) x (P x a_ft / h_coeff + R x a_ft^2 / k_frozen), with P and R the shape constants (slab 1/2 and 1/8; cylinder 1/4 and 1/16; sphere 1/6 and 1/24).

Plank's equation for freezing time with the standard shape constants as published in the ASHRAE Handbook Refrigeration chapter on food freezing times, cited by name and not reproduced; Plank neglects sensible heat above and below the freezing point, so it runs optimistic, and a measured product-temperature log governs a food-safety decision.

Plank's equation is public physics from 1913 and the shape constants follow from the geometry. The product's density, latent heat, freezing point, surface coefficient, and frozen conductivity are the user's values.

Estimate. AHJ and licensed professional govern.

Field names used by the API: a_ft, shape, density_pcf, latent_heat_btu_lb, freezing_point_f, medium_temp_f, h_coeff, k_frozen, freezing_time_hr, surface_term, internal_term

• Shape constants slab P = 1/2, R = 1/8; cylinder P = 1/4, R = 1/16; sphere P = 1/6, R = 1/24Plank's equation as published in ASHRAE Handbook Refrigeration
• Latent heat the product's water, not pure water's 143.4 BTU/lb; a 74% moisture product carries about 106 BTU/lbfood refrigeration practice
• Known limitation Plank neglects sensible heat above and below the freezing point and assumes one freezing temperature, so it runs optimisticASHRAE Handbook Refrigeration