Friction factor for a bubble in a clean liquid, when a gas bubble moves through a liquid, the bulk of the liquid behaves as if it were in potential flow; that is, the flow field in the liquid phase is very nearly given by Eqs.4B.5-2 and 3. The drag force is closely related to the energy dissipation in the liquid phase (see Eq. 4.2-18) F_kv_∞ = Ev, show that for irrotational flow the general expression for the energy dissipation can be transformed into the following surface integral: E_v = µf'(n ∙ ∆v²) dS next show that insertion of the potential flow velocity profiles into Eq. 6D.1-2, and use of Eq. 6D.l-1 leads to f = 48/Re A somewhat improved calculation that takes into account the dissipation in the boundary layer and in the turbulent wake leads to the following result: This result seems to hold rather well up to a Reynolds number of about 200.

2.2 48 (1 f = Re VRe.