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New Formulation for Semi-Empirical Correlations for Penetration Jets

Correlations for the extension of a water vapor jet injected in a liquid pool were historically proposed considering the mass flux (kg/m/s) as a constant. The results were satisfactory, however adjusting the values by linear regression. Although, it presents the following drawbacks: 1) the formulation is only valid for the specific range of data for what it was created; 2) it does not allow the analytical evaluation of the heat transfer coefficient from the extension equation. This paper proposes a new formulation for the calculation of the mass flux, in such a way to remove both of these drawbacks.


Introduction
The phenomenon of Direct Contact Condensation (DCC) has been discussed in the literature since Kerney [1], due to its importance as a solution of engineering, where large values of heat transfer coefficient are needed. Through the DCC, several different sets of equipment can be designed, when, for example, condensation is in a small frame of time, or when reduced space is required. The present work proposes a new formulation for the flux of mass in the currently presented correlations in the literature, in such a way that their validity would extend to any set of data, and which would allow a direct deduction of the heat transfer coefficient from the adjusted correlation of the dimensionless extension. This proposition grounds itself in an analytical procedure considering the 1 st and the 2 nd law of the thermodynamics.

Literature Review
The first proposal to evaluate the extension of the jet, as in [1], is a development of the mass conservation, applied to a simple model, in which a superheated vapor jet produces a cavity full of vapor, discharged in an atmospheric water bath, as presented in Figure 1. In Equation (1), the vapor flow rate (" m  ") along the injection axis "x" is related to the jet radius "r" and the rate of condensation "R".
The dimensions r' and dx' are depicted in Figure 1. According to this proposal, there is an outflow of water along the lateral vapor-liquid interface of the jet.
This amount of water, crossing the jet interface, promptly freezes, assuming the properties of the surrounding water bath. This condensation is governed by R, which takes the following form: The mass flow (Equation (3)) also takes part in this development. Initial conditions are applied in Equation (4).
where B is defined as the driving potential for condensation (Equation (6)), and S as the dimensionless transport modulus, which is analogous to the Stanton Number (Equation (7)), although it presents some unorthodoxy, since the heat transfer coefficient (h) and the specific heat (C P ) are related to the liquid phase, and the mass flow rate (G) is related to the vapor phase. X is defined as the dimensionless jet length, Equation (8). For sake of clarity, Y replaces the term presented Equation (9).
( ) At this point, namely Equation (5), [1] comes to a crossroad, since G and S depend on Y, and for this, Equation (5) cannot be integrated. A pure analytic result is not achievable. However, assuming G M and S M as constant, this author proposed the correlation, Equation (10): And S M is described by Equation (11): Considering the results of their experiment, treated by linear regression, the numeric format of Equation (12) was proposed, within 13.6% accuracy.
In [3], a method to extend the semi-empirical correlation scope, in order to present results considering fluids other than water, and pressures other than ambient in the pool was presented. To achieve it, the proposed correlation considers the influence of the fluid density.

R. R. Pacheco et al. World Journal of Nuclear Science and Technology
The development is not analogous to that in [3] since it uses a full set of equation (continuity of mass, linear momentum, and energy) while [1] used only the continuity of mass. Another theoretical remark is that the former considered that there is a cross flow of vapor to the bath, which is condensed within it, while the latter considered the opposite entrainment of water into the vapor jet, which is partially evaporated, and creates a two-phase flow zone.
This development also presented a point where the integration of the continuity of the mass is not possible since the integrand is not a defined function of the mass cross flow. An approximated correlation is considered, and this further development, considering experimental data, yields: where G W is not an arbitrary fixed value since it was considered the mass flux in the point where the jet finishes an isentropic expansion. After that, the jet allows entrained water in. The "W" properties are calculated considering the laws of the Thermodynamics, for the depth in which the jet starts the two-phase flow. The average absolute deviation found was 21.9%, higher than the value found in the precedent work.

A slightly better average absolute deviation is found both by Kerney [1] and
Weimer [3], when they depart of the condition to have a priori fixed value exponents, and consider them a free product of the regression. No further development of the latter was found in the literature.
Based on previous work, Chun [4], consider that the characteristics of the jet are mainly dependent on 1) the degree of subcooling of the bath, 2) the steam mass flux, 3) the nozzle direction and 4) the depth of the nozzle. The efficiency of the Direct Contact Condensation as a mechanism of heat transfer is also praised, although attention is called to the fact that no reliable correlation to determine the length of the jet exists. According to this paper, much of the disagreement is related to the fact that, experimentally, the length of the jet is obtained by a visual method, what raises issues related to the geometric limits of the jet. In this paper, the end of the jet was considered as the interface between pure vapor and two-phase flow regions. Theoretical development was not presented, and this paper focused on the development of new values for old parameters. The theoretical expression has its roots in Kerney [1], as shown in Equation (14).
Empirically, through visual method, the values of X are determined, which allows Chun [4] propose Equation (15) The value of G M is also not mentioned throughout the work, for what is supposed that in [4], Chun considered the fixed value of 275 kg/m 2 /s as formerly proposed.
Proceeding a similar experiment (vapor injection in a subcooled water pool, atmospheric pool), reference Kim, et al. (2001) produced the correlation as presented in Equation (17) and Equation (18): The value of G M was again considered 275 kg/m 2 /s, as originally proposed. It is worth to notice that Equation (10), Equation (15) and Equation (17)  Gulawani, in [5] and Kang, in [6] present CFD as a new tool to perform the geometric analysis of the jet, and its heat transfer coefficient. Shah, in [7], also by CFD analysis, found heat transfer coefficients ranging between 0, 6 and 08 MW/m 2 /K and a dimensionless length ranging between 3.8 and 8.
In [8], it was experimentally proposed a different form of correlation. For instance, this model includes a pressure correction factor, as shown in Equation This correlation is within a 40% band of error.
This development leads to the proposal of another heat transfer correlation, as shown in Equation (22): As a summary, Table 1

Development
The experimental works under this scope are related to the development of a correlation for the non-dimensional length and heat transfer coefficient, through the propositions of variations of the originally proposed development from which results in the length correlation (Equation (10)). The extension of the works studied shows concern related to increasing the accuracy in a variety of experimental settings and parameters, in order to reduce the band of adjustment numerical error, since no fully analytical model is yet available.
The band of the error band is determined by fluctuations, which were neglected when Equation (5) was approximated. Sonin, in [12] experimentally investigated this phenomenon related to pressure waves propagating along the pool, while Youn [13] focused the particular case in which pressure waves are Table 1. Extension correlation selected in the literature. In order to achieve a more realistic value of G M , this paper proposes a new correlation. This value is developed through the application of the 1 st and 2 nd law of the thermodynamics in the jet, considering it an isentropic discharge, and a function of the conditions in the pressure chamber, as displayed from Equation (24) to Equation (30):

Dimensionless Extension Equation Number
Within the scope of this paper, the proposed correlations of G M (Equation (24)) substituted the formerly constant value of G M = 275 kg/m 2 /s in those correlations presented in Table 1. The numerical analysis that follows considered the experimental data obtained from Kerney [1]. Table 2 presents some important data related to the numerical procedure.

1) The Validity of G M as Product of an Isentropic Process
Item (i) presents the least square adjustment error when this procedure is applied to the correlation in the respective literature. All the correlations in item (i) were applied to the experimental data presented in Kerney [1]. Item (ii) presents the adjustment error found when the least square procedure is applied to the correspondent correlation when using the value of G M as proposed in Equation (24), as suggested by this paper. The experimental data comes also from Kerney [1]. The percentage difference between (i) and (ii) is presented in item (iii). Item   Kerney [1], experimental data.     belong to the same magnitude order, which indicates that this procedure to obtain h may be valid.
In Figure 9, difference decreases as long as the pressure in the chamber increases. On the other hand, Figure 10 and Figure 11 present the value of the heat transfer coefficient, found through the application of the least square procedure in an extension correlation in order to determine M with data from Kerney [1]. These values are plotted vs G 0 . When "h" is represented as a function of "G 0 ", its smoothly dependence on "G 0 " is easily observed. Once more, both results belong to the same magnitude order.
R. R. Pacheco et al.

Conclusions
The decision to maintain S M as a constant is numerically satisfactory since the chosen value is adjusted by the linear regression to fit the found data. Through this premise, any value proposed for G M would generate the same adjustment error. This decision, proposed originally by Kerney [1], has been assumed up to works in 2015, although it does not depict the physics of the problem.  (24) vs. mass flux ratio. h from correlation of Chun [4] vs. mass flux ratio. Data from Kerney [1]. Figure 11. Heat Transfer Coefficient considering G M from Equation (24) vs. mass flux ratio. h from the correlation of Kim [11] vs. mass flux ratio. Data from Kerney [1].
A full analytic function is not possible to be achieved since the area and the variation of major parameters as a function of the flux of mass in Equation (5) are unknown. These gaps have been fulfilled with experimental data, allowing the propositions of correlations, which present some degree of adjustment error, and are suitable only for the range of the experimented data. Considering the scope of correlations presented in this work, the proposed correlation of G M as an isentropic function of the pressure chamber stands as reasonable and would apply to any extension of thermodynamical conditions.
Besides, the isentropic formulation of G M allows the direct deduction of the heat transfer coefficient from the formulation of the extension, which reduces the dependence on experimental data. The present analysis indicates the possibility to reduce the dependence on experimental data to determine h, and points to a direction where more experimental efforts could be expended.
Further works could focus on the reinforcement of the presented correlation through the experimental analysis, considering a large range of parameters.