The unrestricted engineering Bernoulli equation

Abstract

Based on a force analysis, B ird recently derived restricted “engineering Bernoulli equations.” The procedure consisted of forming the scalar product of the velocity with the Navier—Stokes equations of motion and then integrating over the volume of the open system to obtain first a general mechanical-energy balance. For the restricted cases of isothermal and isentropic flow at constant composition (or else at chemical equilibrium), and with each species experiencing the same conservative body forces, B ird converted the mechanical-energy balance into “engineering Bernoulli equations.” In the present paper a Bernoulli equation will be derived by a generalized method; the only condition will be that of identical body forces upon the constituents. 1 1 The treatment is readily extended to the case where the body forces are not identical: In B ird's equations (20) and (21) replace ρ F φ by Σρ i F i = Σρ i [ F iφ + F it ] where F iφ are conservative body forces [equal to − ∇φ i ( x,y,z)], and F it are time-dependent body forces. Then, the integration ▪ may be carried out in a manner entirely analogous to that of B ird (his equation 26). This results in ▪ where φ i is the potential energy of the i-th constituent, the energy resulting from the location of i in the potential field φ F( x,y,z). Also ▪ and e φ ≡ ( 1 ρ )Σρ i φ i ; r i is the rate of production of constituent i, per unit volume. The last four terms of this equation are zero when all the body forces are identical and conservative; inclusion of these terms renders the equations to follow unrestricted. By rewriting equation (f.1) as an expression for dE φ/ dt, each of the other terms may be rationalized into a specific contribution to the total change in E φ.