Suppose that a sphere of diameter D moves at a velocity U relative to a liquid, and that the sphere temperature T_S exceeds the liquid temperature T_L far from the sphere. The rate of heat transfer from sphere to liquid (energy per unit time) may be expressed as

(a) It is customary to embed h in a Nusselt number,

Confirm that this is dimensionless.

(b) If N₁ = Nu, U appears only in N₂, and μ appears only in N₃, what are suitable choices for the dimensionless groups?

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Q=hA(Ts - Tt) (P1.4-1) where A is the surface area of the sphere and h is the heat transfer coefficient. The value of h depends on the thermal conductivity of the liquid (k, SI unit W m -¹ K-¹) and its heat capacity per unit mass at constant pressure (CP, SI unit J kg -¹ K¯¹), in addition to D, U, p, and µ. Temperature adds to (M, L, T) a fourth primary dimension, denoted as Ⓒ.