Engineering
Biot (Bi), Prandtl (Pr) & Nusselt (Nu) Heat Transfer Dimensionless Numbers
Calculate Biot, Prandtl and Nusselt numbers with distinct solid/fluid conductivities and an illustrative small-Biot screening check.
Never confuse Biot Number (k_solid!) with Nusselt Number (k_fluid!)!
Bi = h*Lc/k_solid compares internal conduction resistance with external convection resistance, not the inverse. A small Biot number supports a nearly uniform body-temperature approximation; Bi<0.1 is a common screening rule, not a guarantee of exact uniformity or universal accuracy.
Nu = h*D/k_fluid uses fluid conductivity and the length convention of the applicable flow correlation. Pr = Cp*mu/k_fluid compares momentum and thermal diffusivity. Use dynamic viscosity in Pa·s, heat capacity in J/(kg·K), conductivity in W/(m·K), h in W/(m²·K), and lengths in meters.
The example uses supplied positive material properties and h, rather than predicting h from a flow correlation. Check property temperature, geometry, boundary conditions and other heat-transfer mechanisms before applying a lumped transient model. A higher Biot number alone does not quantify the actual gradients.
| Thermal Dimensionless Group | Formula | Water Cooling + Aluminum Part | Calculated Value |
|---|---|---|---|
| Biot Number (Bi) | (B2 * B3) / B4 (uses k_solid!) | h=80, Lc=0.01m, k_Al=200 | 0.0040 (below screening threshold) |
| Ceramic Part Check | k_solid set to 1.6 W/(m*K) | (80 * 0.01) / 1.6 | 0.5000 (not below screening threshold) |
| Fluid Prandtl Number (Pr) | (B5 * B6) / B7 | Cp=4180, mu=8.9E-4, k_w=0.60 | 6.20 (Water at ~25°C) |
| Pipe Nusselt Number (Nu) | (B2 * B8) / B7 (uses k_fluid!) | h=80, D=0.05m, k_w=0.60 | 6.67 |
Biot Number Bi = (h * L_c) / k_solid & Lumped Capacitance (< 0.1) Test
Use volume divided by the surface area actually exchanging heat for Lc. Thickness/2 applies to a broad wall with both faces exchanging heat, and radius/3 to a fully exposed sphere; other exposed surfaces change Lc.
=IF(COUNT(B2:B4)<>3,"Invalid inputs",IF(OR(B2<0,B3<=0,B4<=0),"Invalid inputs",ROUND((B2 * B3) / B4, 4)))Prandtl Number Pr = (Cp * mu) / k_fluid and Nusselt Number Nu = (h * D) / k_fluid
Both Pr and Nu use the surrounding FLUID's thermal conductivity k_fluid (B7), never the solid wall's conductivity.
=IF(COUNT(B5:B7)<>3,"Invalid inputs",IF(OR(B5<=0,B6<=0,B7<=0),"Invalid inputs",ROUND((B5 * B6) / B7, 2)))Nusselt number using fluid conductivity
Use the pipe diameter supplied in B8 for this example. A different geometry may need a different characteristic length.
=IF(OR(COUNT(B2)<>1,COUNT(B7:B8)<>2),"Invalid inputs",IF(OR(B2<0,B8<=0,B7<=0),"Invalid inputs",ROUND((B2 * B8) / B7, 2)))Small-Biot screening comparison
Compare the unrounded Bi with 0.1. Equality is not below the threshold; this is not a validation of an entire heat-transfer model.
=IF(COUNT(B2:B4)<>3,"Invalid inputs",IF(OR(B2<0,B3<=0,B4<=0),"Invalid inputs",IF((B2 * B3) / B4 < 0.1, "Below small-Biot screen", "Not below small-Biot screen")))Try the example
Use the original values or change them to test your understanding.