Theoretical Post-Dryout Heat Transfer Model

: A theoretical post-dryout heat transfer model is developed for two-phase dispersed flow, one-dimensional vertical pipe in a post-CHF regime. Because of the presence of average droplet diameter lower bound in a two-phase sparse flow. Droplet diameter is also calculated. Obtained results are compared with experimental values. Experimental data is used two-phase flow steam-water in VVER-1200, reactor coolant system, reactor operating pressure is 16.2 MPa. On heater rod surface, dryout was detected as a result of jumping increase of the heater rod surface temperature. Results obtained display lower droplet dimensions than the experimentally obtained values.


INTRODUCTION
At times when liquid phase does not preserve unremitting contact with the heated surface, the phenomena namely critical heat flux occurs (CHF) in forced convection saturated boiling.Depending on the quality, two likely scenarios may arise.First, the liquid phase loses the connection with the heated surface with a vapour film at low attributes.Second, owing to the evaporation of the surface and droplet entrainment, liquid film layer on the surface -which is heated-becomes observable at high qualities.This research proposes a new method of prediction as a theoretical one-dimensional post-dryout model.The introduced model of the pressure P, the mass flux G, the wall temperature Tw, the wall heat flux qw, the tube diameter D and the equilibrium quality   versus axial coordinate and is applied to find the estimate of lower bound for average droplet size in the post-CHF regime.
Based on the luxurious researches and inadequate operative circumstances, the dryout incidence is mainly assessed under observed associations due to its complex nature [1].Implementation of such associations, in other words, correlations, to conditions and systems are considerably outside of the required range.Hence, their validity is a question.There have been numerous mechanistic and phenomenological attempts to overcome such restrictions in the literature by offering an estimate on the liquid film spread that can be used to designate the dryout circumstance on assessing it's size and flow rate [2].
Among phenomenological approaches, liquid film flow with different settings in annular and other annular related streams; as dryout and post-dryout, droplet entrainment, evaporation and droplet deposition is considered [3,4].After the flowing proportion of the liquid film or related size of itregarding thickness-declines under a critical value or to zero, phenomenological methods assume that the dryout arises [5][6][7][8][9].During the estimation phase of dryout, there are numerous diverse systems applied.One of the earlier approaches was FIDAS, which was widely in use circa 1990 [10,11].During the 90s, another approach MONA-3 is utilised [12].Among other notable works; VIPRE-W [13], COBRA-TF [14], and most recently CATHARE-3 [15,16] -which was used in the NEPTUNE projectare introduced in the literature.However, as mentioned in the literature recently, none of these methods can come up with a robust solution for the complex nature of this [2].
In addition to this, two-phase flow is simulated with the introduction of computational fluid dynamics which could be discussed and placed under previously named method, mechanistic [17][18][19][20][21][22][23].It is also addressed in Lahey Jr.'s work [24] that straight resolving of the liquid film is excessively luxurious as it gets fragile close to the dryout site.Hence, several estimates are conducted as an alternative to solve the problem.These approaches include liquid film models of two-dimensional [25], annular-mist flow [7] and others.
Even though post-dryout heat transfer models were studied mainly; as mentioned in the work of Li and Anglart [2], there is still a vast gap stays in the literature primarily for the measures of calculations and experiments [26].In this work, one-dimensional two-phase dispersed flow in a vertical pipe is considered for model development.Experimental data of VVER-1200 is used on the two-phase flow of steam-water at 16.2 MPa.

THEORETICAL MODEL AND ANALYSIS
In this section, a thermal model is presented to develop a one-dimensional two-phase dispersed flow in a vertical tube having a diameter D. The steady-state flow with the axial coordinate z and the origin 0 are subjectively established in the post-CHF system as indicated in Figure 1.
Based on the assumption, of spherical droplets are entrained in the flow.Concerning the deviation of the axial pressure; the temperature is set at the saturation point, and asymmetric heat flux on the wall is considered.Moreover, the model disregards the changes in potential and kinetic energies.Similar phase in the defined for the flowing bubbles as in the flow of liquid-gas phase.A discrete gas phase is observed along with a small gas flow rate in comparison for a further explanation.During the low scale of liquid is seen, the liquid phase is dispersed when the fluid flow rate is small compared with gas flow rate.Because of the loss of contact with the heating surface of the liquid stage, it is expected to observe saturated boiling in forced convection.
The term post-CHF refers to the heat transfer system as the levels of heat flux goes above the critical heat flux (CHF).This regime arises in unplanned circumstances.It happens mostly in nuclear reactors with a water cooling mechanism as a coincidence of a loss-of-coolant.

EQUATIONS GOVERNING THE FLOW
While Equation ( 2) is denoting the conversion of energy.
Transferred heat energy between moving fluid and surface at different temperatures is known as convection.However, in natural circumstances, this is a mixture of molecule's bulk motion and diffusion.Diffusion leads the way when it gets closer to the surface area where fluid velocity is inevitably low.Bulk movement raises the influence and points of going far from the surface.Two convective heat transfer forms may occur; forced and natural.Forced or in some literature assistive convection arises when an external force assists the flow as a pump.Natural convection, however, is caused by density variations triggered by differences in temperature of the fluid.During this phase, the layer will be the root of the fluid rise and swapped by a more cooling fluid.This ongoing phenomenon is named natural convection -in some cases, free convection [27,28].Heat transfer from the vapour to the droplets are formulated in the following equations.
As   =   +   ; From the conservation of energy formula in equation ( 2), the terms ̇ + ̇ can be set equal to the total mass flow rate ̇.Hence, the mass flow rate of the liquid ̇ could be rewritten as in equation (5).
The actual quality   is introduced as the proportion of the mass flow rate of vapour to its total as in equations ( 6) and (7).
ṁV = ṁX a (7) Therefore, the mass flow rate of the liquid can be written regarding the actual quality of equations ( 8) and (9).
Equation ( 10) reflects the form of derivative after the integration of the equations ( 7) and ( 9) to equation (2).Therefore, the conversion of energy becomes; Extracting the total mass flow rate and out of the derivative and re-expressing inside in actual quality parenthesis are indicated in equations ( 11), (12), and (13).
The derivation is split into parts in equations ( 14) -( 16) for enabling to see different fractions for actual quality and specific enthalpy of the liquid and vapour.
The first conservation of energy in equation ( 2) becomes an equation ( 17) for a continuation of the problem.The difference between the two is the introduction of the actual quality and the total mass flow rate.

ṁ[(h
A similar approach is considered for the equation ( 4).The mass flow rate of the liquid ̇ is replaced as in equation ( 5).After implementing same approximations as in equations ( 6) -(9), equation ( 4) can be rewritten as in equation (18).
In addition to this, considering a fluid, shape of the object, volume and density is a subject to alteration within the domain with time and mass can move through the domain.Therefore, according to the conservation of mass, the mass flow rate (̇) is a constant within a tube which is equal to the product of the terms velocity, density, and the area as in equation (19).The mass flow rate of the liquid is expressed the number of droplets per unit length (N) by considering the flow area in equation (20).
Therefore, the number of droplets per unit length could be rewritten as in equation ( 21) with the application of equation ( 9) for mass flow rate -real quality transformation.
Expressing the heat transfer rate concerning the number of droplets, the surface area is injected, and the heat balance is replaced with the product of mass and latent heat transfer evaporation in equations ( 22) - (24).
Therefore, equation ( 25) extracts the average droplet diameter regarding heat transfer coefficient, temperature, density, velocity and latent heat transfer evaporation.

d = 6H(T − T S ) ρ L U L h fg (25)
Moreover, similar approximation should be considered for the equation (3).By considering the equality of the summation of ∑   to the  2 , and with the replacement of the term N as equation ( 21), the simplification yields the equation (26).
The equation ( 27) indicates the derivative of the heat transfer from the vapour to the droplets.
As in equations ( 14) -( 16), the derivative is split into parts and shown in equation (28).
Therefore, the equation ( 18), which is the derivative form of equation ( 4), could be substituted with the recently calculated equation ( 28) in the following equation (29).
The location of thermal equilibrium is established where the flow equilibrium quality   , fundamentally reaches the scale of the actual flow quality, equivalent to the void behaviour [29].Therefore, the solution of the equilibrium quality is formulated as equation (30).
X a X e = h fg (P, T sat ) h V (P V T V ) − h L (P, T sat ) (30) The substitution of the equation with the previously discussed equation (17) for the total heat flux derivation with total mass flow rate is shown in the equation (31), and the simplified version is introduced in the following.
Equation ( 33) and (34) however, indicates the solution for the actual quality concerning equilibrium quality, latent heat of evaporation and enthalpy.
The simplified version of the derivative gives the following equation.
Substitution of the previous equation with equation (32) where the solution for   is introduced as follows in equations ( 36) -(40). (37)

RESULTS AND DISCUSSIONS
Latent heat transfer amount is highly related to the amount and size of the bubbles.Consequently, they are considered as the most important element of the processes related to the boiling.Another important subject that requires consideration is that the departure diameter.As it spreads to some size, bubbles of the boiling flow begin to move on the heating area as it dispenses away from the area of nucleation.What's more, the bubble diameter Db which is measured same as the departure diameter of the bubble because of primarily shaped vapour blanket by the combination of minor bubbles that are connected vertically.Previously discussed bubble diameter is derived from the Cole and Rohsenow as in the following equation; Where  reflects the surface tension.Figure 3 proves the change of the bubble diameter in a vertical tube at 16.2 MPa.It could be inferred from the figure that an increasing temperature has a decreasing effect on the bubble diameter.From vapour to bubble, heat transfer coefficient of the average bubble diameter is calculated from Delale's work [30]; where   denotes vapour's thermal conductivity.As reflected in the work of Lee et al. [31], fluid velocity gradient is calculated as in the following equation.
In addition to this, the setup of the vertical tube at 16.2 MPa indicates the variation in the liquid velocity as in Figure 4.It could be inferred from the graph that the speed of the liquid shows a sharp increase after 550 K.
Where Tsat is the saturation temperature, and Tb is the bulk temperature of the fluid.
Heat transfer coefficients of the experimental data presented in Lin et al. [32] is calculated and demonstrated in Figure 4. Figure 5 however, represents the alteration in heat transfer coefficient at 16.2 MPa vertical tube from vapour to the bubble.It can be inferred from the graph that temperature increases the heat transfer coefficient because of the decrease in bubble diameter of Figure 3.

CONCLUSION
In this article, theoretical post-dryout heat transfer model for two-phase dispersed flow is investigated with a one-dimensional vertical pipe in the post-CHF regime.Previous sections focused on onedimensional two-phase dispersed flow in a vertical pipe for a model development with thermal equations for the surge in detail.Experimental data of VVER-1200, a nuclear reactor cooling system, is used on a two-phase flow of steam-water at 16.2 MPa.In addition to this, the thermal model of the system is presented to develop a vertical tube.On heater rod surface, a dryout was detected as a result of jumping increase.Results obtained display lower droplet dimensions than the experimentally obtained values.
The results also indicate that the future works on the subject should be encouraged for a stronger idea of the influence of the setup.

Figure 5 .
Figure 5. Heat Transfer Coefficients with the Experimental Data Figure 5 indicates the difference between the theoretical and the experimental data points in heat transfer coefficients as well.The large gap between the two suggests the necessity to conduct further work on the given conditions.
A= flow area d= average droplet diameter G= mass velocity (flux) H= convective heat transfer coefficient Hfg= latent heat of evaporation HL= specific enthalpy of the liquid hV= specific enthalpy of the vapour   = thermal conductivity of the vapour ̇= total mass flow rate ̇= mass flow rate of the liquid ̇= mass flow rate of the vapour N= number of droplets per unit length P= pressure Q= heat transfer rate   =heat transfer from the vapour to the droplets qL= heat flux from the wall to the droplets   = total wall heat flux T = ambient temperature Ts = surface temperature Tsat = saturation temperature Tv= superheated vapour temperature U= velocity   = actual velocity of the liquid   = actual quality   = equilibrium quality = density σ = surface tension coefficient