Report-Choking flow application in nuclear reactor core

7/31/2019 Report-Choking flow application in nuclear reactor core

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Choking flowas a application in nuclear reactor core

(Report)


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Choking is term in fluid mechanics represent that the flow is choked. When

flow is choked, the flow rate attained its maximum value for other parameter

like stagnation pressure, length, diameter of pipe etc. Choking generally can

be seen in Nozzles, Orifice. At minimum cross section, the flow attain M=1

i,e. sonic flow.

m.= mass flow rate, kg/s | C= discharge coefficient, dimensionless

A= discharge hole cross-sectional area, m\ | k= cp/cv of the gas

= real gas density at P and T, kg/m

P0= absolute upstream pressure of the gas, Pa

At the choked area we can measure the mass flow rate and amazingly

further reduction in downstream pressure doesnt affect mass flow rate.

Hence we called conditions to be choked at critical location.

Fig(1.1)

Likewise choking also occur in liquid flow through pipe, but equations

representing choking water flow or fluid are different than gas choking flow

as gas is compressible but water is not and there are many other reasons.


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Choking flow has many applications in our life. Indian nuclear power station

production is about 3.2% of overall electricity production. Nuclear power

plant has very basic setup. Nuclear core where fuel rods are mounted

undergoes nuclear fission and produce extent of heat that heats water and

heated water allowed to pass through heat exchanger in which it evaporate

water. The evaporated water allowed to rotated turbine blades whichproduce electric power.

Inside core Uranium atoms undergoes fission reaction and produce heat.

To control extent of heat control rods are introduced. Main purpose of

control rod is to absorb neutrons produced in fission process and break

/control chain reaction. To cool / rapid cool the fuel rods coolant is use.

Generally water serves the purpose of coolant in most of nuclear power

plants.

But in case if something went wrong and coolant stats escape from the

core, fuel rods will heat up and results can be devastating. To avoid such

consequences we need to prepare for backup plants like ECCS

(Emergency core cooling system) and RELAP. This fill up core will coolant

through different methods so that fuel rod remain at low temp.


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Fig 1.3

Water inside core has tendency to for two phase mixture or superheated

vapour because of heat of fuel rods but the pressurizer apply pressure on

water and make it subcooled (Compressed) as shown in above (see fig 1.3)

Temperature can reach upto 250oc 300 oc. Hence pipe which used to flow

high pressure high temp. fluid may fail due to metallurgical problem. This

cause cracks to form or even it can break. At this condition our ECCS canbe used for emergency backup for coolant. But how to design ECCS?

ECCS need mass flux (mass flow rate per unit area) at break location and

other parameters like leak rate and depressurization rate. Hence we need

dedicated calculations for the mass flux and depressurization rate at break

or leak.

In this reports we are going to discuss how to form the basic models that

can predict the mass flow rate (critical) and depressurization rate. Many

models are proposed for choked flow, we are going to discuss few of them

and based upon that we are going to form our ECCS and other cooling

arrangement plans.


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It is experimentally proved that flow choked at break or leak and hence it is

useful tool for our calculations of mass flow rate. We have given that our

inlet conditions are subcooled with known temperature and pressure. Then

it would be so easy to from direct relation between mass flow rate and inlet

and exit parameters. But is it?

No. As we can see from fig 1.3 water is subcooled at inlet but as pressure

decreases water is losing the temperature. At certain point water will starts

to convert into steams. Hence our pipe flow is no longer single phase flow.

Both steam and water at high temp will start to escape from break of leak.

All calculations that we are going to make must be applicable to both

vapour/steam and water and we know water and steam have different

properties. So we need to understand different property of steam and water.

Little observation can be made like water and steam velocity is same or

different at critical location or water and steam are in thermodynamic

equilibrium or not. If it became possible to form relations between water and

steam then it will be lot easier to formulate relations of mass flow rate.

Based upon this conditions few models are proposed are given below.

1. Homogeneous Equilibrium Model(HEM)2. Non-Homogeneous Equilibrium Model (NHEM)

3. Homogeneous Non-Equilibrium Model (HNEM)

4. Non-Homogeneous Non-Equilibrium Model(NHNEM)

Homogeneous means that our two phases has same velocities.

Non-Homogeneous means two phase velocities are different.

Equilibrium means that two phases are in thermodynamic equilibrium.

Non-Equilibrium applied to two phased with thermodynamic instability.

Based on those models it is very easy to understand choking flow criterion.

This models may not give exact value but sufficient to design and operate

ECCS.

Many researchers worked on this topic and Moody and Henry-Fauske

Models are most followed. Based on their studies and other paperworks we

are going to form our models.


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1. Homogeneous Equilibrium Model(HEM)

At exit we have two phases; vapour and water. In this model we are going

to treat the fluid as pseudo mixture that can be described by same

equations as a single equivalent flow. Hence, our velocities u (l) and u (g)

are same and two phases are in thermodynamic equilibrium.

The main criteria to achieve thermodynamic equilibrium are long length. In

long sections, there is enough time to achieve thermodynamic equilibrium

and hence HEM is good way to analyse critical flow rate in long pipe flow.

The basic conditions are (key)

1. Same Velocity

2. Phase Equilibrium

3. Pseudo fluid (considering two phase mixture as a fluid)

The Bernoulli equation in its general differential form of energy balance per

unit mass can be written as

Where;

P= pressure

v= specific volume

V= velocity

g= gravity

D= diameter

F= friction factor

L= length W

W= work done

c = critical & e=external


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Assuming no work interaction and no elevation changes.

Our formula become

As we already mentioned that we are going to assume the two phase

mixture as pseudo fluid, hence we must not hesitate to take specific volume

of pseudo fluid as specific volume of mixture and hence velocity of pseudo

mixture can be written as

Where;

m= total mass (mtotal=mwater+ mvapour)

A= orifice area (break area)

V= velocity

v= specific volume (total)

Replacing Velocity from (3) in equation (2).

The final equation become

Substituting = 1/ v and integrate we get


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Friction factor can be calculated from colebrooks formula or blasius

correlation for Re< 105

Based on experimental data and calculations we find value of f can be

taken approx. 0.003. Later we can compare our calculated data and

measured data based on different value off.

To find maximum flow rate, following things are calculated,

1. Inlet pressure Pb is measured

2. Exit pressure Pe is measured

3. Quality at exit is calculated

Considering isentropic and adiabatic flow, hence no heat interaction to

surrounding. By enthalpy conservation we can write

Hb = X Hg + (1-X) Hf

Where Hb is enthalpy at entrance of pipe

Hf is enthalpy of water at exit

Hg is enthalpy of vapour at exit

X is quality at exit

Hence X = (Hb Hf ) / Hfg

To calculate density and specific volume of mixture, we need to consider

specific volume of each phase and hence

vm = X vg + (1-X)vf and = 1/ vm

As we studies in choking flow reduction in back pressure increases mass

flow rate. Hence for each pressure drop (exit) interval density is obtained

from (4) and arithmetic mean value is used in (5) to calculate mass flux.


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From this graph we can see that as exit pressure decreases our flow tends

towards constant mass flux line. But after some pressure decrease in exit

pressure will not lead to increase in mass flux (like in graph after 4.0 atmany decrease in pressure does not cause mass flux to change). Each graph

is drawn for different inlet pressure. As we can see that as pressure

decreases mass flow flux for same exit pressure decreases. In nuclear

reactor core, we have depressurization rate hence we can plot series of

graphs for different inlet pressure.

But in case of real life problem, we have same exit pressure. Based on

experiments done so far, it is prove that the pressure inside nuclear reactor

core is more than sufficient to cause choking in flow for exit pressure equal

to atmospheric pressure. Hence we need not to worry about exit pressure

and hence mass flow rate or mass flux is independent of exit conditions and

it can easily predict.

Restrictions:

As we are talking about HEM (Homogeneous Equilibrium Model), for

phases to be in thermal equilibrium, sufficient time must be allowed for

thermodynamic stabilization. Hence for HEM, flow must be taken in long

pipe section where there is sufficient time for phase equilibrium. HEM can

give best predicted value in long sections and deviate from observed value

for short pipes and also pipes with short bore.


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Above diagram is plot between mass flux observed at exit and pipe length

for testing HEM.

Looking at Exp. Data graph (with surface tension) and Model (smooth

entrance) we see huge deviation for pipe length less than 600 mm. Though

600 mm is very short length but we can see appreciable convergence of

Experimental data and data tabulated by HEM.

This table shows that mass flux decreases as time passes due to

depressurization in core. Total water/coolant blow down can be calculated

by integrating mass flow arte Vs time equation or simply area under cure

(shown above) multiplied by orifice area. Using this model we can designECCS and amount of coolant/water needed.


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2. Homogeneous Non-Equilibrium Model(HNEM)

The major phenomenon that are ignored in HEM two phase flow are Bubble

nucleation, interphase heat, mass and momentum transfer. All this

parameter if taken into account, our model will be more complicated to

understand and to solve.

Hence in HNEM we will solve one dimension mass and momentum

conservation ad transfer equations with bubble nucleation phenomenon and

phase change. As HNEM is complicated model to solve and different

approached as been developed which can be categorized as follows:

1. Empirical Model (based on observation and experiments)

2. Physically Based model for thermal non-equilibrium

3. Two.-Fluid models

In our report we are going to discuss Henry-Fauske Empirical Model, which

is rather simple to explain and understand.

They introduce coefficient N which is fraction of equilibrium vapourgenerated. From this value we can calculate non-equilibrium steam

generation. Also assuming are VL = VG we are going to form our HNEM

Empirical Model.

Henry-Fauske assume that non-equilibrium steam X quality is proportional

to equilibrium steam quality X

As discussed N is dimensionless constant determined by empirical

calculations of Henry-Fauske and observed that it depends on geometry of

break or leak or test section. N can vary from 7-100. Flashing is process of

rapid conversion of liquid into steam. As we assumed the fluid at inlet is

subcooled means it is compressed and when it brought isentropically tolower pressure (fig.1.3) it start to flashing into vapour. This phenomenon

occurs below some pressure called depressurization pressure.


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First we will discuss the flow under subcooled inlet and next based upon

that we will determine flashing process and calculate mass flow rate.

Critical Flow with Subcooled Inlet:

Fig 2.1

This fig. is graph between pressure (local pressure) and distance along testsection (Reocreux Test) with subcooled inlet.

For this test we are taking inlet pressure about 1.8 bar which is very low

compared to actual nuclear core pressure. As we move from inlet of test

section toward downstream local pressure P starts decreasing and

saturation pressure PTi corresponding to inlet water temp. Ti reached. After

pressure PTi water became superheated and at pressure Pb there is enough

superheat first bubble to form. See fig2.2 [A]. After that point local pressure

continue to decreasing and reach Pd where significant amount of steams

begins to form, See fig 2.2 [D]


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[A] [B] [C] [D]

Fig 2.2 we can see bubbles are forming before vaporization.

From Reocreux test fig 2.1 we can see that Pd is reached at end of pipe

(constant diameter) or at critical location. Hence Pd=Pc. Where Pc is

pressure at critical location.

From fig.2.1 we can see that steam volume fraction is zero (0) upto the

critical location and starts increasing rapidly beyond that point. At that

location (here at critical location) one can define critical pressure difference

Pd which can be defined as pressure below saturated pressure

corresponding to water temperature required to form significant amount of

steam.

In diffuser section local pressure continues to decrease and steam volume

fraction and steam content increases. From calculated steam volume

fraction and local pressure P we can calculate non-equilibrium steam

quality X. For equal water and steam velocity we can write

Where g is vapour density and l is liquid density. As we are considering all

process in this model are isentropic, hence from isentropic relations we can

calculate liquid entropy

Where So is stagnation entropy and Sg is saturated steam entropy at local

pressure P. We can find P + Pd from below


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Fig 2.3

From this we can calculate Pd along diffuser section and can be seen infig. 2.1. We can see that Pd relatively remain constant along diffuser

section. This determine strong non-equilibrium between two phases.

To measure degree of non-equilibrium in pipe and diffuse r section we need

to calculate equilibrium steam quality from isentropic calculations.

So is stagnation entropy and Sg and Sf are vapour entropy and liquid

entropy at saturated pressure Ps taken equal to the local pressure.

Corresponding calculated from equation 3 called as eq called equivalent

volume fraction. This is shown in fig 2.1 by dashed line.

Conclusions: (examining fig 2.1)

1. Nucleated bubble nucleated at pressure Pb do not get opportunity to

grow due to limited time for heat transfer between them and superheated

liquid. As heat transfer rate between steam and water is low due to equal

velocities.

After studying other Reocreux test (with less subcooled inlet) we found that

flashing need not to occur at critical location and bubble can get more time

to grow but at last there will be non-equilibrium in critical location. We foundthat Pd in both case are same and hence we can find Pd where significant

amount of steam begins to form. Corresponding non-equilibrium steam

volume fraction can be plotted and shown in fig 2.4


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Fig 2.4

From Alamgir and Lienhard subcooled depressurization model we arrived at

Where is surface tension, k is Boltzmanns constant, Tc is critical

temperature, is rate of depressurization, Tr is reduced temp (Ti/Tc)

For pipe depressurization Alamgir have recommended

Where KL is liquid thermal conductivity, is liquid viscosity, Hfg is heat of

vaporization, Pr is prandlt number, T* is difference between stagnation

temp and temp at critical location.


17/20

Above equation represent relation between different term including nozzle

diameter or Critical location diameter D. V* is critical velocity.

Flashing and Critical Flow Model:

Proposed flashing model was formulated from result obtained from fig 2.1and fig 2.4. We assumed

1. The first significant steam form at Pd below saturation temp

corresponding to liquid temp. Hence liquid temp has to be superheated by

amount Tdcorresponding to Pd.

2. Any temp formed will be at saturation conditions corresponding to

pressure P.

Assuming isentropic flashing process we can calculate non-equilibrium

steam quality

Where so is stagnation entropy, sg is saturated vapour entropy at pressure

p and sf is superheated liquid entropy at temp T+Td where T is sat. tempcorresponding to P. As explained earlier we can write

From enthalpy balance over control volume give by equation

For frictionless isentropic fluid, the local pressure gradient with distance z is

given by

After lengthy iteration and approximation in model we arrived at


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Where is depressurization rate, and Gc is flow rate at critical location and

Gt is maximum flow rate.

Using this relation we can find decompression pressure change

and neglecting any density change at

critical location we from

Comarison:


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Comparison Between observed mass flux with calculated mass flux ofHEM and HNEM


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Documents

Report-Choking flow application in nuclear reactor core