Churchill-Bernstein Cylinder Crossflow Nusselt Number Formula

Choose the geometry and property basis

This worked model concerns forced crossflow over a circular cylinder with a uniform surface temperature. Use fluid properties at the film temperature, the average of the surface and free-stream temperatures. It is not an internal-pipe, tube-bank, natural-convection or radiation calculation.

Re_D = rho × U × D / mu and Pr = cp × mu / k must refer to the same fluid state. D is the outer diameter; k is fluid thermal conductivity, not the conductivity of the cylinder material. The example supplies Re and Pr directly. Without density and viscosity there is no unique conversion from Re to wind speed.

The cited EES documentation describes tested use with Re × Pr > 0.2. This worksheet adopts that strict screen and flags equality too. Passing this arithmetic test does not guarantee applicability for every geometry, flow disturbance, property variation or extreme Reynolds number.

Compute raw average Nu in B7

Enter numeric positive Re in B2 and Pr in B3. This guarded formula keeps the correlation result unrounded. At Re=20000 and Pr=0.71, Nu displays as 79.31 when formatted to two decimals. No dimensionless ratio should carry temperature or velocity units.

=IF(COUNT(B2:B3)<>2,"Enter numeric Re and Pr",IF(OR(B2<=0,B3<=0),"Re and Pr must be positive",IF(B2*B3<=0.2,"Outside adopted RePr > 0.2 range",0.3+(0.62*B2^0.5*B3^(1/3)/(1+(0.4/B3)^(2/3))^0.25)*(1+(B2/282000)^(5/8))^(4/5))))

Compute h from the validated Nu in B8

Enter fluid k in W/(m K) in B4 and diameter D in metres in B5. The output h is in W/(m² K). The formula propagates any Nu validation message and checks k and D before division, so invalid flow inputs cannot produce a plausible-looking h.

For the supplied k=0.026 and D=0.04, h displays as 51.55 W/(m² K). At Re=100000, the corresponding values display as Nu=215.35 and h=139.98. Display formatting should not replace the raw Nu used to calculate h.

=IF(ISNUMBER(B7),IF(COUNT(B4:B5)<>2,"Enter numeric k and D",IF(OR(B4<=0,B5<=0),"k and D must be positive",B7*B4/B5)),B7)

Interpret the changes carefully

At fixed supplied Re and Pr, doubling k doubles h and doubling D halves h. Those are algebra checks, not a physical prediction at fixed velocity: changing diameter at fixed velocity also changes Re, and changing fluid state can change both Pr and k.

This example estimates an average convection coefficient. It does not compute drag, radiation, wall temperature, allowable electrical current or a complete heat-loss balance. Additional inputs and a suitable model are needed for those outputs.

Further reading

• F-Chart EES — External_Flow_Cylinder_ND
• Churchill and Bernstein (1977), bibliographic reference