No abstract

A free, vertical jet of liquid plunging into a uiescent surface of the same liquid entrains the surrounding gas into the receiving liquid to form bubbles. The entrainment characteristics of such jets of Newtonian liquids of varying physical properties have been investigated by means of high‐speed photography. Although both laminar and turbulent jets entrain gas bubbles, the mechanisms governing the entrainment process of the two types of jets are clearly different. Entrainment by turbulent jets results from the disturbances on the free surface caused by the jet instability; entrainment by laminar jets is accomplished by the formation of a thin shell of gas around the jet at the point of entrance, by the development of oscillations in the shell, and by its subsequent breakup into bubbles. Entrainment occurs only when the average jet velocity exceeds a certain critical value termed minimum entrainment velocity. For a laminar jet having a flat velocity profile at the point of entrance, the following correlation permits prediction of the minimum entrainment velocity: where the dimensionless numbers are based on the liquid properties and the jet diameter at the point where the jet meets the surface of the receiving liquid.

We can see from Table 2 that if both tjl = f l ( T ) and y2 = f 2 ( T) are replaced algebraically in Equation (26) by their respective values as a function of T, the resulting loop wiII then be represented by Figure 12c. This is a simple loop for which there exists a reciprocal path.There are no easy manipulations which one can make in order to get a reciprocal path for loop A. Therefore direct substitution techniques for this problem become useless. A graphical representation ofhas been obtained from a digital computer and is shown in Figure 13. The curve is not monotonic and there is a discontinuity at x30 = 0.699. There are two values R1 and R2 for which ~3~ = xaN. On the forward path C convergence is always at R1. Unfortunately, the solution to the system corresponding to x3 = R1 has no physical meaning.The reciprocal path is shown as C' but cannot be realized via iterative computation since inverse flow is not possible for loop A. A quasilinearization solution of this problem has been carried out. The system has eight nonlinear terms. Depending upon the initial guess for the nonlinear variables, convergence has been obtained on either the real root R2 or the nonreal root R1.As we can see from the study of the last example, a solution by direct substitution loses many of its advantages when nested loops appear. However, the algorithm of Ramirez and Vestal (1972) tends to minimize the number of nested loops required to solve a problem, and there-fore nested loops are fairly rare in actual design computations. If they cannot be avoided, the engineer must be very~careful when he tries to get an actual solution to his problem. For this type of problem quasilinearization is recommended. ACKNOWLEDGMENTSCompany and the authors gratefully acknowledge this support.