Engineering & Fluid Mechanics
Plank’s Freezing Time Equation: Blast-Freezer & Biopharma Cryopreservation Slab Formula
Calculate phase-change freezing time for infinite slabs, cylinders, and spheres using Rudolf Plank’s moving-boundary equation t_f = (rho_f * L_f / (T_f - T_inf)) * (P * a / h + R * a² / k_f) alongside the freezing Biot number.
Separating external convective resistance (P*a/h) from internal frozen-crust conduction (R*a²/k_f)!
Derived by German refrigeration pioneer Rudolf Plank (1913, Zeitschrift für die gesamte Kälte-Industrie), Plank’s quasi-steady moving-front equation predicts latent phase-change time from initial freezing point T_f across a product of full thickness/diameter a cooled by medium at T_inf (driving difference deltaT = T_f - T_inf): t_f = ((rho_f * L_f) / deltaT) * ((P * a) / h + (R * a^2) / k_f)!
Shape constants (P, R) equal (1/2, 1/8) = (0.500, 0.125) for an infinite slab cooled from both faces, (1/4, 1/16) = (0.250, 0.0625) for an infinite cylinder, and (1/6, 1/24) for a sphere! For a 40 mm biopharma/food tray slab (a = 0.040 m, rho_f = 980 kg/m^3, effective latent heat L_f = 250,000 J/kg, k_f = 1.65 W/(m*K), deltaT = 26 K) in an air-blast tunnel (h = 45 W/(m^2*K), Bi = h*a/k_f = 1.091), t_f = 1.481 hours—slashing to 0.754 hours in an impingement/plate freezer at h = 120 W/(m^2*K)!
| Freezer Heat Transfer Coefficient h | Freezing Biot Number Bi = h*a/k_f | Plank Freezing Time t_f (Hours) | Controlling Thermal Bottleneck |
|---|---|---|---|
| Air-Blast Tunnel h = 45 W/(m²·K) | 1.091 | 1.481 hours (88.9 min) | Mixed Surface Convection + Ice Conduction |
| Plate / Impingement h = 120 W/(m²·K) | 2.909 | 0.754 hours (45.2 min) | Internal Frozen-Shell Conduction Dominates |
| Ambient T_inf >= T_f (deltaT <= 0) | No Freezing Drive | "Invalid Plank Freezing Inputs" | Sub-Freezing Medium Guard |
Plank Phase-Change Freezing Time in Hours t_f = (rho_f*L_f / deltaT) * (P*a/h + R*a²/k_f) / 3600
Computes latent freezing duration in hours from frozen density rho_f (B2), latent heat L_f (B3), driving temperature difference deltaT = T_f - T_inf (B4), thickness/diameter a (B5), convective h (B6), frozen thermal conductivity k_f (B7), and Plank geometry constants P (B8) and R (B9).
=IF(OR(MIN(B2:B9) <= 0), "Invalid Plank Freezing Inputs", ROUND(((B2 * B3) / B4) * ((B8 * B5) / B6 + (B9 * (B5 ^ 2)) / B7) / 3600, 3))Freezing Biot Number Bi = (h * a) / k_f
Evaluates the ratio of internal ice-layer conduction resistance to external convective film resistance.
=IF(OR(MIN(B2:B9) <= 0), "Invalid Plank Freezing Inputs", ROUND((B6 * B5) / B7, 3))Try the example
Use the original values or change them to test your understanding.