Parents of two-phase flow and theory of " gas-lift "

This paper gives a brief overview of types of two-phase flow. Subsequently, it deals with their mutual division and problems with accuracy boundaries among particular types. It also shows the case of water flow through a pipe with external heating and the gradual origination of all kinds of flow. We have met it in solution of safety condition of various stages in pressurized and boiling water reactors. In the MSR there is a problem in the solution of gas-lift using helium as a gas and its secondary usage for clearing of the fuel mixture from gaseous fission products. Theory of gas-lift is described. 1 Boiling Heat Transfer Suppose we first consider a heating element submerged in a pool of stationary fluid. A sketch of the heat flux transferred from the heating surface to the fluid versus the temperature difference between the surface and the bulk fluid temperature is shown in figure 1. From this sketch it is evident that one can distinguish several different regions of heat transfer. Figure 1 Heat flux versus temperature difference for poolboiling heat transfer 0 a: There is little liquid superheat; heat transfer is by natural convection. a b: A few bubbles are formed but collapse after leaving surface (although agitation will increase heat transfer). a c: The number of bubbles formed increases rapidly this is nucleate boiling regime. c: "Burnout" point or DNB is reached. c d: Bubbles become so numerous that they begin to coalesce and clump near the heating surface (the vapor covering the surface acts as a heat insulator) this is known as partial nucleate boiling. d e: A continuous blanket of vaper forms over the heating surface film boiling regime. e f: Thermal radiation from the surface comes into play film and radiation regime. It should particularly be noted that if the heat flux q" is the independent variable, then increasing q" beyond the DNB point will result in a large increase in the surface temperature for a given coolant temperature (see figure 1). This phenomenon is known as burnout and must be avoided in reactor operation. These concepts also apply in a flowing channel. Suppose we consider a channel with a uniform wall heat flux . Then one will find temperature profiles up the channel as shown in figure 2. Once again we can distinguish several regions of heat transfer: 1. Region A: Single-phase convection. In this region we can use the analysis we developed in the previous section to determine the convective heattransfer coefficient arising in Newton ́s law of cooling (1) 2. Region B: Subcooled boiling: but . Once the clad surface temperature reaches the saturation temperature of the coolant, the temperature drop is no longer linear in the heat flux. In this region, the heat transfer process is extremely efficient, and the temperature difference is quite small compared to the other temperature differences in the core. In LWRs it is customary to use an empirical correlation developed [19] in order to calculate the temperature difference between the clad surface and the coolant: !"# $ %&'() (2) where p is the pressure in units of bar and q" is the heat flux in units in MW/m. This correlation is valid for DOI: 10.1051/ C © Owned by the authors, published by EDP Sciences, 2014 ,


Boiling Heat Transfer
Suppose we first consider a heating element submerged in a pool of stationary fluid.A sketch of the heat flux transferred from the heating surface to the fluid versus the temperature difference between the surface and the bulk fluid temperature is shown in figure 1.From this sketch it is evident that one can distinguish several different regions of heat transfer.a -b: A few bubbles are formed but collapse after leaving surface (although agitation will increase heat transfer).
a -c: The number of bubbles formed increases rapidly -this is nucleate boiling regime.c: "Burnout" point or DNB is reached.c -d: Bubbles become so numerous that they begin to coalesce and clump near the heating surface (the vapor covering the surface acts as a heat insulator) -this is known as partial nucleate boiling.
d -e: A continuous blanket of vaper forms over the heating surface -film boiling regime.
e -f: Thermal radiation from the surface comes into play -film and radiation regime.
It should particularly be noted that if the heat flux q" is the independent variable, then increasing q" beyond the DNB point will result in a large increase in the surface temperature for a given coolant temperature (see figure 1).This phenomenon is known as burnout and must be avoided in reactor operation.
These concepts also apply in a flowing channel.Suppose we consider a channel with a uniform wall heat flux ‫̶ݍ‬ ௪ .Then one will find temperature profiles up the channel as shown in figure 2. Once again we can distinguish several regions of heat transfer: 1. Region A: Single-phase convection.
In this region we can use the analysis we developed in the previous section to determine the convective heattransfer coefficient ݄ ௦ arising in Newton´s law of cooling Once the clad surface temperature reaches the saturation temperature of the coolant, the temperature drop is no longer linear in the heat flux.In this region, the heat transfer process is extremely efficient, and the temperature difference is quite small compared to the other temperature differences in the core.In LWRs it is customary to use an empirical correlation developed [19] in order to calculate the temperature difference between the clad surface and the coolant: where p is the pressure in units of bar and q" is the heat flux in units in MW/m 2  pressures, temperatures, and coolant flow rates characteristic of both PWRs and BWRs.For example, in a PWR at 155 bar with a heat flux of 1.5Â10 6 W/m 2 , one would find a temperature difference of only 3°C.There will be very little change in the temperature difference with heat flux in this boiling regime.Once the clad surface temperature of the coolant, it tends to remain near that level unless the heat flux reaches the DNB value.One can use these relationships to determine the position z SCB at which subcooled boiling begins.We equate T fl = T sat in (2): Then noting also that We can find and then solve this for the position z SCB .
3. Region C: Saturated or bulk boiling: Eventually sufficient heat is transferred to the coolant that it reaches its saturation temperature and begins bulk boiling.Subsequent heat addition will cause further boiling, and hence the coolant temperature will remain essentially constant and equal to T sat up the remainder of the channel.

Two-Phase Flow
When the coolant is a two-phase mixture of liquid and vapor, the coolant flow pattern can become quite complicated.One can distinguish several types of two-phase flow, as show in figure 3. It is useful to introduce some definitions characterizing the vapor content of a coolant at rest: We can compute these quantities in terms of the cross-sectional areas in the channels occupied by liquid (A l ) and vapor (A g ).The void fraction is given by ߙ ൌ ‫ܣ‬ ‫ܣ‬ ‫ܣ‬ (7) Figure 3 Flow patterns in a heated channel while the static quality is For example, the void fraction is given for water at various qualities and pressures in figure 4. 1 When the two-phase mixture is flowing, we must be more careful in our definitions since the vapor has a tendency to slip past the liquid because of its buoyancy.We then must define the quality in terms of a parameter S known as the slip ratio which relates the different flow velocities:

Figure 4 Void fraction versus quality for bulk boiling
Then we can define a flow quality One can invert this to write the void fraction Į as This latter equation is sometimes known as the ̶ܵ െ ߙ െ ‫̶ݔ‬ relationship.

Description of Flow Patterns
When two phases flow co-currently in a channel, they can arrange themselves in a number of different configurations, called flow patterns or flow regimes.Each flow pattern is characterized by a relatively similar distribution of the two phases and of their interfaces.Transition from one flow pattern to another takes place whenever a major change occurs in geometry of the gas-liquid interface.

Hewitt and Roberts
Axis x: Axis y: Dispersed bubbles flow regime will be in: Axis x: Axis y: where indicated in Hewitt and Roberts are: where: ܸ ሶ Drift Flux model ܿ ് ܿ ǡ ‫ݏ‬ ൌǫ Thermodynamic equilibrium.The model is derived in Zuber, Findlay -Journal of Heat Transfer, where was introduced shifting velocity ܿ (drift velocity) that local velocity vapor (gas): ܿ ൌ ݆ ܿ ՜ ܿ ݆ see ( 16) is derived: ሼఈሽሼሽ is called the distribution parameter which expresses the distribution of the steam (or gas) content on the surface (for uniform distribution ‫ܥ‬ ൌ ͳ).Is necessary to know ‫ܥ‬ ǡ ܿ .The latest set of parameters ‫ܥ‬ ǡ ܿ can be found in [14,18].

Dispersion Bubble Regime
Dispersion bubble regime occurs when turbulence shear forces are equal to or larger than the buoyancy force in the fluid.
Buoyancy force: Strength of turbulence Critical fluid velocity ܿ ௧ ௗ௦ ௨ for transition to dispersed bubble flow can be determined from a comparison of forces. where: drag coefficient of liquid For ‫ܥ‬ ௩ we use the empirical relationship ܿ ሾ݉Ȁ‫ݏ‬ሿ velocity of liquid ܹ ሾ݇݃ ‫ݏ‬ Τ ሿ mass flow rate These relationships apply for ߙ ͲǤͷ and a vertical pipe.

Conditions for the Use of "Gas-Lift"
Specification of the parameters of the core must be based on the requirements for the performance of MSR.The performance MSR will be used for production of electricity for technological purposes, or in combination.The volume of the core using the center determines specific power q.
It is necessary to structurally arrange the primary circuit of the reactor and our "Two-phase flow demonstrator" (TFD), so that the total pressure losses were minimal.Effectiveness of natural flow and "gas-lift" is strongly dependent.

Procedure to Calculate the Flow
We obtain this relation for natural heat flowing in the primary circuit -on condition that the temperature changes only in the active zone and the heat exchangerfrom the solution of the continuity equation and the Bernoulli's equation: where ߯ are total hydraulic losses in the primary circuit ‫ܣ‬ ҧ is the medium flow cross-section of the circuit It applies : where : ߩҧ is the medium density of the coolant in the circuit where P [W] is the total heat output of the zone ܹ ൌ ߩ ሺܶ ௫ ሻ ή ‫ܣ‬ ή ܿ ௩ and ܹ ൌ ߩ ൫ܶ ൯ ή ‫ܣ‬ ή ܿ ௩ where the cross-sectional area of the tractive "chimney" is ‫ܣ‬ ൌ ߨ ή ‫ܦ‬ ଶ Ͷ Τ , ȡ is density, index l is the fluid, g is the gas, ܹ ஶ ሾ݇݃ ‫ݏ‬ Τ ሿ is the weight flow rate.
Using ܹ ஶ ௦ି௧ it is necessary to use in general ܿ and ȡ for a two-phase mixture and simultaneously determine the new losses, because ܹ ஶ ௦ି௧ increases the flow velocity in the whole primary circuit.The amount of the gas supplied by the m vents into the tractive cylinder of the gas-lift ܹ ሾ݇݃ ‫ݏ‬ Τ ሿ is determined by the following relation from [3].
where ݉ ሾͳሿ is the number of holes with a diameter d, ‫ܥ‬ ௗ ሾͳሿ is the loss factor, which must be determined experimentally, ‫‬ ଵ ሾܲܽሿ is the pressure of gas before the throttle hole, ‫‬ ଶ ሾܲܽሿ is the pressure of gas after the throttle hole, ߢ ൌ ͳǡ for monatomic gases (He, Ar, Kr), ߢ ൌ ͳǡͶ for O 2 , N 2 , CO, air, ቃ the mass of one mole of gas Changes and flow properties at various temperatures and pressures of the gas and the liquid in tension and chimney losses in two-phase flow in the entire circuit are described in [20].
Homogeneous mixture of liquid and gas density ߩ varies speed ܿ .Modeling of two-phase flow forecast includes a two-stage friction factor and the properties of the mixture.
Bertil and Whalley (1982) [21] recommended use Colebrook equation for friction properties using a twophase Reynolds numbers listed for the homogeneous model, which describes the current phase of medium properties of individual phases.
The Colebrook equation is implicit in the two-phase friction factor and its determination requires an iterative approach.In 1983 S.E.Heeland determine approximate relationship for the two-phase friction factor, which differs from the equation Colebrook to 2% and does not require iteration.
where ݂ ሶ ் two-phase friction factor ߝ ோ wall roughness where ߩ ் is density two-phase mixture in the pipe the gas lift.
Two-phase Reynolds number Two-phase dynamic viscosity ߟ is determined by the experimental relationship The overall average speed homogeneous mixture can be easily determined

Effect of Temperature and Pressure on the Flow
We consider that we have a bubble of radius R filled with gas at pressure ‫‬ ଶ and temperature ܶ ଵ .It is located in the local fluid pressure ‫‬ ଵ ሺͲሻ and temperature ܶ ଵ .Bubble moves towards the liquid relative velocity ܿ .Usually is ܶ ଵ ܶ ଵ and ‫‬ ଵ ‫‬ ଶ .
We determine the amount of heat that goes from liquid to bubble in time dt.
For outer wrap ball says Whitaker (1972) [22] universal correlation valid for forced flow and use that to measure the heat transfer to the bubbles.(34) Member of Re 1/2 corresponds to the heat transfer in laminar boundary layer; Re 2/3 corresponds to the heat transfer in wake.Correction for variable uses dynamic viscosity coefficient 0.25, which is used by more than 0.14 coefficient of relationship Sieder-Tate.For Re ĺ 0, Nu approaching asymptotic value of 2 (corresponding to heat transfer in the upright environment) ie molecular heat transfer.
For true balance bubble ‫‬ ‫‬ ଵ , by which we determine the initial bubble size where ߪ is surface tension of the liquid.From (33) we determine the amount of heat that goes into the bubble from the surrounding fluid.At an initial temperature ܶ ଵ of the gas bubble energy where ܿ ௩ is specific heat of gas at constant volume ܸ is the volume of gas in the bubble Divide the height of the chimney tensile K intervals ȟ݄ , where ݇ ‫א‬ ‫ͳۃ‬ǡ ‫.ۄܭ‬In time ȟ‫ݐ‬ ଵ ൌ ȟ݄ ܿ Τ goes into heat ȟܳ ଵ bubbles that increase the gas temperature ȟܶ ൌ ܶ ଶ െ ܶ ଵ .
For a general interval k applies: and thence: For the calculation must be finely dividing the flow path in tension channel and iteratively determine the increase in flow velocity and the growth rate of losses.
Besides the heat transfer to the bubble we have to consider the influence of changes in pressure and flow velocity of the liquid in the pressure with a height of chimney.
Next, we determine the size of the bubbles, depending on the pressure and temperature of the liquid velocity in tension chimney.
We expect: Above the surface of liquid gas pressure ‫‬ ௦௬௦ .The point ‫ݖ‬ ൌ Ͳ is the pressure in the fluid flow velocity with consideration equal to: The radius of the bubble The bubble will grow with the increasing influence of pressure reduction in tension in the liquid chimney and heat transfer from the liquid to the gas in the bubble.This further increases the lifting force and the mass flow quantity ܹ ஶ ௦ି௧ , ie speed mixture flow, but also the losses in the circuit.
If we consider the distribution of tensile chimney on volumes ܸ ൌ ‫ܣ‬ ή ȟ݄ ሺ‫ݖ‬ሻ and the assumption of medium-sized bubbles (where we identify ܴ according to (45) we can determine the number of bubbles ܰ in the volume ܸ .

Note:
The molecular surface energy is then characterized by variable ߱ the product of the surface tension ߪ and values ܸ ଶ ଷ Τ where ܸ ൌ ‫ܯ‬ ߩ Τ is the mol volume.
Expressing molar surface energy is not calculated with a geometric factor ሺ͵ ή ߨሻ ଵ ଷ Τ , which is the same for all liquids.
The dependence of ߱ሺ‫ݐ‬ሻ on the temperature ‫ݐ‬ሾι ሿ determine Eötvös (1886) empirical relationship [22]: where: ݇ constant of proportionality ‫ݐ‬ critical temperature of the liquid Knowing the value of the surface tension ߪሺ‫ݐ‬ ሻ for temperature ‫ݐ‬ , then using equation (50) we determine the constant of proportionality k and then when knowledge of temperature ‫ݐ‬ and density ߩ to easily determine ߪሺ‫ݐ‬ሻ.

Description of the experimental device
The research of two-phase flow was the KKE assembled experimental device on which to plan further treatment.The experimental device consists of the series of construction parts that have been named after the parts of the primary circuit that they represent.The model of the whole device including the description of its main parts is shown in the figure 11.
There were used two types of the construction material -metal from the carbonaceous steel and plexiglass.The most parts are welded from the 2 mm thick metal plates.The remaining parts that have to be transparent in order to perform the PIV measurements are made from the 5 mm or 10 mm thick plexiglass depending on the strain of the part.The maximum dimensions of the device are 2160 x 1100 x 360 mm (height x length x width).
The active zone is represented here by the cylindrical metal container of 360 mm diameter and height of 300 mm.There is a conic crossing in its upper part and they are of air intake for the gas-lift follows.Gas phase must be separated from liquid.The centrifugal separator is for this purpose.Its aerodynamic design is provided according [24].The heat source in the model core secures a 2000 W heating unit that is bolted and correctly sealed to the wall of the container in its bottom part in order not to obstruct the flowing water and therefore not to increase the pressure losses.The bottom of the core, that has to be removable in order of assembling and potential exchange of the heating unit, is connected with the cylindrical part of the core using the flange and twelve screws.There is bolted a ball valve for filling and drainage in the bottom most part of the core.

Figure 1
Figure 1 Heat flux versus temperature difference for poolboiling heat transfer 0 -a: There is little liquid superheat; heat transfer is by natural convection.a -b: A few bubbles are formed but collapse after leaving surface (although agitation will increase heat transfer).a-c: The number of bubbles formed increases rapidly -this is nucleate boiling regime.c: "Burnout" point or DNB is reached.c -d: Bubbles become so numerous that they begin to coalesce and clump near the heating surface (the vapor covering the surface acts as a heat insulator) -this is known as partial nucleate boiling.

Figure 2
Figure 2 Boilling channel temperature profiles Figure 5 shows the flow patterns observed in a vertical circular duct, and figure 6 illustrates typical flow regimes reported for horizontal flow in a pipe.These include: Bublle flow … The liquid is the continuous phase and the gas is dispersed in the liquid in the form of bubbles of variable shape and size.Slug flow … is characterized by large gas bubbles almost filling the channel and separated by slugs of liquid.Churn flow … transition state between Slug and Annular flow.Annular flow … consists of an annular liquid film and of a gas core with or without drops of liquid in it.Spray, mist, dispersed or fog flow … The gas occupies most of the cross-sectional area and the liquid is in the form of small droplets dispersed in the gas.Stratified flow … Because of gravity forces the liquid flows along the bottom of a horizontal or inclined channel, and the gas flows above it.

Figure 5 Figure 6
Figure 5 Flow patterns in the vertical direction

Figure 7
Figure 7 Distribution of the types of a two-phase flow (Hewitt & Roberts)

Figure 11
Figure11 The model of the two-phase flow demonstrator First, we set the corresponding diagram of experimental apparatus MSR primary circuit high in integral structure.The experimental device consists of the series of construction parts that have been named after the parts of the primary circuit that they represent.The model of the whole device including the description of its main parts is shown in the figure11.There were used two types of the construction material -metal from the carbonaceous steel and plexiglass.The most parts are welded from the 2 mm thick metal plates.The remaining parts that have to be transparent in order to perform the PIV measurements are made from the 5 mm or 10 mm thick plexiglass depending on the strain of the part.The maximum dimensions of the device are 2160 x 1100 x 360 mm (height x length x width).The active zone is represented here by the cylindrical metal container of 360 mm diameter and height of 300 mm.There is a conic crossing in its upper part and they are of air intake for the gas-lift follows.Gas phase must be separated from liquid.The centrifugal separator is for this purpose.Its aerodynamic design is provided according[24].The heat source in the model core secures a 2000 W heating unit that is bolted and correctly sealed to the wall of the container in its bottom part in order not to obstruct the flowing water and therefore not to increase the pressure losses.The bottom of the core, that has to be removable in order of assembling and potential exchange of the heating unit, is connected with the cylindrical part of the core using the flange and twelve screws.There is bolted a ball valve for filling and drainage in the bottom most part of the core.

Figure 12
Figure 12 Two-phase flow demonstrator

Figure 13 Figure 14
Figure 13 . This correlation is valid for DOI: 10.1051/ C Owned by the authors, published by EDP Sciences, 2014