cylinders with their axes vertical, the expressions for plane surfaces can be used provided the curvature effect is not too significant. This represents the limit where boundary layer thickness is small relative to cylinder diameter . The correlations for vertical plane walls can be used when where is the Grashof number. External flow, horizontal plates W. H. McAdams suggested the following correlations for horizontal plates. The induced buoyancy will be different depending upon whether the hot surface is facing up or down. For a hot surface facing up, or a cold surface facing down, for laminar flow: and for turbulent flow: For a hot surface facing down, or a cold surface facing up, for laminar flow: The characteristic length is the ratio of the plate surface area to perimeter. If the surface is inclined at an angle θ with the vertical then the equations for a vertical plate by Churchill and Chu may be used for θ up to 60°; if the boundary layer flow is laminar, the gravitational constant g is replaced with g cos θ when calculating the Ra term. External flow, horizontal cylinder For cylinders of sufficient length and negligible end effects, Churchill and Chu has the following correlation for . External flow, spheres For spheres, T. Yuge has the following correlation for Pr≃1 and . Vertical rectangular enclosure For heat flow between two opposing vertical plates of rectangular enclosures, Catton recommends the following two correlations for smaller aspect ratios. The correlations are valid for any value of Prandtl number. For : where H is the internal height of the enclosure and L is the horizontal distance between the two sides of different temperatures. For : For vertical enclosures with larger aspect ratios, the following two correlations can be used. For 10 < H/L < 40: For : For all four correlations, fluid properties are evaluated at the average temperature—as opposed to film temperature—, where and are the temperatures of the vertical surfaces and . Forced convection Internal flow, laminar flow Sieder and Tate give the following correlation to account for entrance effects in laminar flow in tubes where is the internal diameter, is the fluid viscosity at the bulk mean temperature, is the viscosity at the tube wall surface temperature. For fully developed laminar flow, the Nusselt number is constant and equal to 3.66. Mills combines the entrance effects and fully developed flow into one equation