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Correlations for the extension of a water vapor jet injected in a liquid pool were historically proposed considering the mass flux (kg/m
^{2}/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.

The phenomenon of Direct Contact Condensation (DCC) has been discussed in the literature since Kerney [

DCC is the natural solution for applications where superheated vapor must be discharged in the atmosphere. Within this scope, DCC offers a safe alternative to reduce the heat and pressure up to the threshold where the discharge to the atmosphere is plausible, as it occurs in relief tanks.

For the nuclear industry, it emerges as a solution for containing the discharge of the high energetic water contained in a primary loop of a Pressure Water Reactor (PWR), and also, to direct it to the chemical and radiological plant treatment facilities.

For Boiling Water Reactors (BWR), DCC emerges as a solution to recollect the water vaporized within the reactor vessel. For this, BWR plants prescribe Suppression Pools in their projects, where the DCC phenomenon takes place.

The DCC phenomenon reached well-established economic importance and has become an object of research since it is related to the safety of nuclear power plants, where it occurs as a high energetic vapor injection in a tank containing liquid compressed water. The heat transfer coefficient is increased due to the turbulent character of the interface of the jet and the bath, and this is the key to the efficiency of thermal exchange [

The design of devices where the DCC may occur determined the interest in the research of the phenomenon. The research focused mainly on the determination of a model for the extension of the jet, and another model for the determination of the heat transfer coefficient.

Nevertheless, the highly complex physics of this phenomenon restrains the development of a full analytical model for the extension of the jet and for its heat transfer coefficient. Experimental data are supplied to fulfill the gaps in the analytical development and semi-empirical correlations have been proposed, observing the following restrictions: 1) these correlations are valid only for the specific range of experimental data; 2) the analytical evaluation of the heat transfer directly from the length correlation was not possible, considering the available formulation.

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.

The first proposal to evaluate the extension of the jet, as in [

d m ˙ d x ′ = − 2 π r ′ R (1)

_{C}) supplies superheated vapor. After the restriction by a valve, the vapor is injected in the water pool. The setting of the experiment allows a choked flow injection since it would reduce fluctuations in the jet.

The dimensions r' and dx' are depicted in

R = h ( T S − T ∞ ) h g f (2)

The mass flow (Equation (3)) also takes part in this development. Initial conditions are applied in Equation (4).

m ˙ = π r ′ 2 G (3)

m ˙ 0 = π r ′ 0 2 G 0 (4)

After some algebraic effort, Equation (1) and Equation (4) produce Equation (5):

X = ∫ 1 0 [ − ( G 0 G ) 1 / 2 ⋅ B − 1 ⋅ S − 1 ] d Y (5)

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).

B = C P ( T S − T ∞ ) h g f (6)

S = h C P ⋅ G (7)

X = L / r 0 (8)

Y = ( m ˙ / m 0 ) 1 / 2 (9)

At this point, namely Equation (5), [_{M} and S_{M} as constant, this author proposed the correlation, Equation (10):

X = ( G 0 G M ) 1 / 2 ⋅ S M − 1 ⋅ B − 1 (10)

And S_{M} is described by Equation (11):

S M = h C P ⋅ G M (11)

Considering the results of their experiment, treated by linear regression, the numeric format of Equation (12) was proposed, within 13.6% accuracy.

G_{M} was arbitrarily chosen as 275 kg/m^{2}/s, constant value, related to the critical vapor mass rate of the nozzle, as a representative value of the order of the magnitude of the mass rate. The value of 1.932 for S_{M} in Equation (12), is partially originated from the arbitrarily chosen value of 275 kg/m/s^{2} and the conditions of the experiment (h, and C_{P}), which Kerney [

X = 1 1.932 ( G 0 G M ) 1 / 2 ⋅ B − 1 (12)

In [

The development is not analogous to that in [

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:

X = 35.5 ( G 0 G W ) 1 / 2 ⋅ B − 1 ⋅ ( ρ ∞ ρ w ) − 1 / 2 (13)

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 [

Based on previous work, Chun [

X = F ( G 0 G M , B , S M ) (14)

Empirically, through visual method, the values of X are determined, which allows Chun [

X = 0.5923 ⋅ ( G 0 G M ) 0.3444 ⋅ B − 0.66 (15)

In sequence, as product of a regression, it is also proposed a correlation for the heat transfer coefficient, Equation (16):

h C P ⋅ G M = 0.8012 ⋅ ( G 0 G M ) 0.3444 ⋅ X − 1.0079 B − 0.6247 (16)

The value of G_{M} is also not mentioned throughout the work, for what is supposed that in [^{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):

X = 0.503 ⋅ ( G 0 G M ) 0.47688 ⋅ B − 0.70127 (17)

h C P ⋅ G M = 1.4453 ⋅ ( G 0 G M ) 0.13315 ⋅ B 0.03587 (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) are reasonably similar in their forms. Both are directly obtained from the original mathematical development.

Gulawani, in [^{2}/K and a dimensionless length ranging between 3.8 and 8.

In [

X = 0.868 ( P 0 P a ) 0.2 ⋅ ( G 0 G M ) 0.5 ⋅ B − 0.60 (21)

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):

S M = h C P ⋅ G M = 0.576 ( P 0 P a ) 0.2 ⋅ ( G 0 G M ) 0.5 ⋅ B − 0.4 (22)

Xu, in [^{2} K and 11.36 MW/m^{2} K.

Chong, in [^{2}, which represents the expansion ratio between a straight-line nozzle and actual orifice nozzle (for straight-line nozzle, (ε/ε')^{2} = 1), as presented in Equation (21).

X = 0.3866 ( ε ϵ ) 2 ⋅ ( G 0 G M ) 0.78 ⋅ B − 0.80 (23)

As a summary,

One can notice that the constant value of G_{M} = 275 kg/s is broadly found (up to 2015) in the literature ( [_{M} takes a different constant value, valid only for the respective correlation and for the respective range of data. A more realistic formulation of G_{M} would be of great interest since it directly impacts the formulation of X, S_{M}, and h.

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 [

Dimensionless Extension | Equation Number |
---|---|

X = 0.5200 ( G 0 G M ) 1 / 2 ⋅ B − 1 | (12) |

X = 35.5 ( G 0 G w ) 1 / 2 ⋅ B − 1 ⋅ ( ρ ∞ ρ w ) − 1 / 2 | (13) |

X = 0.5923 ⋅ ( G 0 G M ) 0.3444 ⋅ B − 0.66 | (15) |

X = 0.503 ⋅ ( G 0 G M ) 0 , 4768878 ⋅ B − 0.70127 | (17) |

X = 0.868 ( P S P a ) 0.2 ⋅ ( G 0 G M ) 0.5 ⋅ B − 0.60 | (21) |

X = 0.3866 ( ε ϵ ) 2 ⋅ ( G 0 G M ) 0.78 ⋅ B − 0.80 | (23) |

created by a low mass flux of injection. The case where multiple jets are simultaneously created by different holes was explored by Cho [

The experimental works follow a standard: the length of the jet is observed, what allows the definition of a correlation of length and the determination of the heat transfer coefficient. The experimental data collected supplies complementary information in order to complement the development of Equation (5).

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):

G M = ρ x ⋅ V C r i t (24)

V C r i t = 2 ⋅ ( h 0 − h c r i t ) (25)

h 0 = h ( P 0 , T 0 ) (26)

h c r i t = h ( P c r i t , s c r i t ) (27)

P c r i t = 0.577 ⋅ P 0 (28)

ρ x = ρ ( 101.3 kPa , S C r i t ) (29)

s c r i t = s 0 (30)

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

1) The Validity of G_{M} as Product of an Isentropic Process

_{M} as proposed in Equation (24), as suggested by this paper. The experimental data comes also from Kerney [_{M} as found in the respective literature, while (v) presents S_{M} recalculated, where the data originally used by each respective author is swapped by the data presented in Kerney [_{M}, and the exponents of G_{0}/G_{M}, B). This is evidenced with the small difference for the column of Equation (12), but greater for the remaining columns. Last item, (vi), evaluates the value of S_{M} as a product of the isentropic relation in Equation (24). The difference between correspondent values of references (v) and (vi) is small, indicating that the isentropic correlation stands for a broad range of parameters.

For the analysis of _{M} on the range of parameters employed in the experiment, since items (iv) and (v) present a large difference. When the isentropic formulation of G_{M} (Equation (24)) is used in item (vi), this difference is vastly reduced (see items (v) and (vi)).

The currently proposed distribution of G_{M} as an isentropic relation (Equation (24)), applied to the experimental data within Kerney [_{C}), as presented in

G M = 127.6367 + 26.4510 ⋅ ln ( P C ) (30)

The graphics in Figures 4-7 present comparisons between the correlations proposed by the respective authors, considering G_{M} = 275 kg/m^{2}/s constant, as originally proposed by Kerney [_{M}, within the scope of this paper, Equation (24). The respective results were plotted against the experimental data.

As noticed, the correlations considering both propositions of G_{M} found a narrow agreement, what validates the use of G_{M} as the product of an isentropic assumption (Equation 24) as proposed by the present paper.

2) Heat Transfer Coefficient

Once the value of G_{M} is specified by Equation (24), the heat transfer coefficient (h) can be analytically derived from Equation (11), in conjunction with the adjusted value of S_{M}, which is obtained from the linear coefficient of the least square adjustment in the correlations of

Item | Description | Equation (12) | Equation (15) | Equation (17) | Equation (21) |
---|---|---|---|---|---|

(i) | Adjustment Error (G_{M} = 275 kg/m^{2}/s) | 2.268 | 2.762 | 2.241 | 2.949 |

(ii) | Adjustment Error (G_{M} as in Equation 24) | 2.409 | 2.907 | 2.407 | 2.458 |

(iii) | Difference in Percentage (i) and (ii) | 6.255% | 5.273% | 7.403% | 16.65% |

(iv) | S_{M} literature, calculate by the respective author, and with data from the respective author | 1.923077 | 1.688334 | 1.988072 | 1.152074 |

(v) | S_{M} (G_{M} = 275 kg/m^{2}/s) recalculated with Kerney [ | 1.918281 | 0.59453 | 0.776398 | 0.534474 |

(vi) | S_{M} (G_{M} as in Equation 24), calculate with data from Kerney [ | 1.850481 | 0.581395 | 0.749625 | 0.514933 |

through the application of correlations in the literature (Equation (16) and Equation (18)).

_{C}. Both sets of data belong to the same magnitude order, which indicates that this procedure to obtain h may be valid.

In _{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.

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 [

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.

This work presented satisfactory arguments to question the original formulation, where the mass flow rate (G_{M}) is constant. This original formulation stands as reasonable when a formulation for the extension of the jet is proposed since any constant value proposed fits when applied the 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, Equation (10).

This way, the isentropic formulation of G_{M} adds flexibility for the extension jet equation, once it is less dependent of experimental data, and allows the analytical evaluation of the heat transfer coefficient from the extension equation.

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.

The authors would like to acknowledge Nuclear and Energy Research Institute, IPEN-CNEN/SP and CTMSP/SP for the infrastructure, particularly the computer laboratory.

The authors declare no conflicts of interest regarding the publication of this paper.

Pacheco, R.R., Freire, L.O., Rocha, M.S., Scuro, N.L., Menezes, M.O. and Andrade, D.A. (2019) New Formulation for Semi-Empirical Correlations for Penetration Jets. World Journal of Nuclear Science and Technology, 9, 96-111. https://doi.org/10.4236/wjnst.2019.92007

A Heat Transfer Area (m^{2})

B Condensation Driving Potential

C_{p }_{ }Water Specific Heat (J/kg/˚C)

G Steam Mass Flux (kg/m^{2}/s)

h Average Heat Transfer Coefficient (W/m^{2}˚C)

h_{fg} Condensation Enthalpy (kJ/kg)

L Steam Jet Length (m)

m ˙ Vapor Flow Rate (kg/s)

P Pressure (kPa)

r Jet Radius (m) R Rate of Condensation (kg/m/s^{2})

S Dimensionless Transport Modulus

T Temperature (˚C)

x Axial Coordinate (m)

X Non-Dimensional Jet Length

ρ Density (kg/m³)

A Atmospheric Conditions

C Conditions in the Pressure Chamber

f Conditions of Saturated Liquid

S Conditions in the Vapor-Bath Interface

g Conditions of Saturated Vapor

∞ Average Conditions in the Pool, Far Away from the Jet

0 Conditions in the Nozzle

M Average Conditions over the Interface Surface

W Developed as [