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Lecture 31

Reactor Heat Transfer

Objectives

In this lecture you will learn the following

We shall discuss the various coolants used in reactors.

Then we shall look into the definitions associated with thermal analysis of reactors.

Then we shall present the analysis to map the coolant, clad and fuel temperature distributions.

Finally we shall understand a procedure to estimate the decay power.

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Lecture 31

Reactor Heat Transfer

Reactor Coolants

Most commercial nuclear power plants use turbines for producing electricity.

Thermodynamically, the working fluid has to be at a higher temperature to get goodefficiency.

Thus, the coolant used should operate at these temperatures.

The desirable characteristics are:High specific heat. This would lead to less amount of fluid to be handled for a giventemperature difference.Should be easily available and economical.Should have less neutron absorption and should not produce highly active products(due to neutron activation).It will be ideal, if it can also be used as a moderator.Should be compatible with structural material and hence, should not be corrosive.

Seldom all properties will be satisfied. Usually, it is a compromise. Some of the competitivecoolants are:

Light WaterHeavy WaterCarbon-dioxideHeliumLiquid MetalsOrganic CoolantsMolten Salts

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Lecture 31

Reactor Heat Transfer

Light Water

Used as coolant and moderator.

It is cheap.

Has low boiling point and therefore, requires high pressure.

Decomposes into H2 and O2 by the action of radiation.

Corrosive.

Strong absorber of neutrons.

Heavy Water

Expensive.

Lower thermal neutron absorption cross section than light water.

Otherwise the same advantages and disadvantages as light water.

Tritium production can occur.

Liquid Metals

High boiling point, hence lower operating pressures.

High melting point requires heating of pipes when not in operation.

Coolant becomes radioactive.

Corrosive.

Excellent heat transfer properties.

Water must be kept out of reactors to avoid violent reactions.

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Lecture 31

Reactor Heat Transfer

Figure of Merit

A figure of merit has been evolved to compare their utility as a coolant.

Using basic principles, it can be shown that parameter represents the inverse of pumpingpower for the same amount of heat extracted with same temperature rise in the core, whenflow is turbulent.

Thus, lower this figure, smaller is pumping power.

The value of this parameter relative to sodium is summarised in the following table.

It may be observed that water has the highest value, indicating that it is the best coolant.

Gas coolants are very poor as they need a lot of pumping power.

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Lecture 31

Reactor Heat Transfer

Heat Generation in Nuclear Reactor

Heat generation in nuclear reactor is not spatially uniform.

Heat generated per unit volume depends on the number of fissions occurring per unit volume.

This can be shown to be proportional to the variation of neutron density in the reactor.

This information is provided to a thermal hydraulic engineer by the reactor physicist.

While the exact evaluation is fairly complex, some simplifying approximations shall be madein our further discussion.

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Lecture 31

Reactor Heat Transfer

Heat Distribution

Heat generation in nuclear reactor is distributed amongstKinetic energy of fission fragments (KE-FF).Kinetic Energy of neutrons (KE-N).Betas, Gammas and Neutrinos.

Further, not all energy is liberated instantly.

We can typically approximate them as follows

Receiver Instantaneous fraction(~90%)

Delayed fraction (~10%)

Fuel Element KE- FF - 83% Betas – 4%Dispersed in Fuel,Mod., structure

KE-N 2.5%, PromptGamma 4%

Capture Gamma 3% Fission Product Gamma 3.5%

Approximately 90% is deposited in fuel and 10% at other places.

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Lecture 31

Reactor Heat Transfer

Thermal Constraints

The objective of a thermal hydraulic analysis is to satisfy the following Constraints:The maximum fuel temperature has to be within the allowable limit set by the materialsengineer.Typically this is the metal softening temperature.Similarly, the clad temperature shall be within its applicable safety limit.This is dictated by the metal water reaction temperature.In addition, the heat flux should be below the Critical Heat Flux (CHF limit).We shall discuss this during boiling heat transfer.

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Lecture 31

Reactor Heat Transfer

Heat Distribution

The spatial variation for bare cylindrical reactors has been shown to be Cosine axially and

Bessel function of zeroth order in radial direction as shown.

For complex systems, these arecomplicated.

Shape supplied by physicist.

We shall keep the discussion generalso as to grasp the fundamentals.

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Lecture 31

Reactor Heat Transfer

Definitions

The reactor core by definition is the assembly of the fuel material along with moderator andcoolant material.

It can be assumed to be an array of rods distributed in a uniform manner.

As discussed, each rod has an outer clad and inner fuel pellets.

The discussion is presently illustrated using a square pitch (p). It should be fairly straightforward to extend it to any type of unit cell.

For a unit cell, the volume of the core can be viewed as the total volume of the unit cell.

Power density (Q''') is defined as the thermal power generated per unit core volume.

Linear heat rate (q') is defined as the thermal power generatedper unit length of the fuel rod.

Heat flux (q'')is defined as the thermal power generated per unitexternal surface area of the fuel rod.

Volumetric heat generation (q''') rate is defined as the thermal power generated in the coreper unit fuel pellet volume.

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Lecture 31

Reactor Heat Transfer

Consider an infinitesimal length (dl) of unit cell and let the thermal power generated be (dQ).The power density can be written as

Similarly, the linear heat rate can be written for the controlvolume of length dl as

Similarly, the heat flux for the control volume can be written as

Finally, the volumetric heat generation rate can be written as

Since dQ is same in all the above equations, we note that

Thus, the parameters defined differ only by a geometric factor. Hence if any one of them isknown, the others may be computed from the geometry.

Further, since dQ varies as the neutron density, all the parameters would vary accordingly.

Lecture 31

Reactor Heat Transfer

Thermal Analysis

To keep the analysis general for any shape distribution, we shall introduce the followinganalysis.

For the purpose of analysis, the case of the cylindrical reactor is analysed. This is so as mostpower reactors are cylindrical. The argument can be extended for any other case similarly.

For a cylindrical bare core, the variation of neutron density varies as

This would imply that

where

and

P(r) and P(z) are called radial and axial power factors.

Thus the maximum value can be found from the average value if P(r) and P(z) are known.

The values of maximum heat flux, linear heat rate and volumetric heat generation rates can becomputed using the geometric factors.

Single channel Hot Channel Analysis is generally carried out to assess the overall safety of anuclear reactor.

The steps are:

Identify the hot channel, which is usually the central fuel channel.

Compute the maximum linear heat rate in central channel by dividing the global averagelinear heat rate by the product of radial and axial power factor.

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Compute the mass flow rate associated with the channel by dividing the average mass flowrate per channel by any radial factor (introduced by orifices).

The analysis illustrated for a single phase constant property flow to begin with.

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Lecture 31

Reactor Heat Transfer

Fluid Temperature

Energy Balance for an infinitesimal slice at adistance z can be written as

On integration from the entrance to any generalpoint z, we get

Having found the temperature distribution for the fluid, we turn our attention to the outerclad temperature.

From the definition of convective heat transfer coefficient, we can write

To find the clad temperature distribution we need to perform conduction analysis in clad.

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Lecture 31

Reactor Heat Transfer

Clad Temperature

The governing equation for conduction is

For steady and non-heat generating system, we can write

In the above expression Rco stands for clad outer radius.

By definition

Integration with kc assumed to be constant gives

Taking the other limit to the clad inner radius, Rci,

Inner clad temperature would be the highest clad temperature and if this is safe, then thedesign is OK.

Now we shall perform conduction analysis in fuel pellet.

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Lecture 31

Reactor Heat Transfer

Fuel Temperature

The governing equation for conduction in fuel is

In steady state

Transposing and performing the integration we get,

Using the boundary condition that on dTf/dr = 0 at centre

Transposing and performing integration once again

If kf is assumed to be constant, then

Taking the other limit of r = 0, we get

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Lecture 31

Reactor Heat Transfer

Summary

The axial variation of temperatures are summarised below and are shown in the adjoiningfigure.

TBulk increases monotonously.

Tclad has a maxima after the centreline.

TF(0) also has a maxima after thecentreline.

Lecture 31

Reactor Heat Transfer

Decay Heat

We have understood the behaviour of temperature in steadily operating reactors.

Unlike thermal power plants where the reaction ceases when fuel supply is cut off, nuclearreactor continues to produce heat even after shut down due to radioactive decay of fissionproducts.

Estimation of the power produced after shut down is very complex. However, the simplemodel of Way and Wigner is useful in understanding the trends.

Referring to the figure shown above. The coordinate t* starts when the reactor startedproducing power, whereas the coordinate Ts starts from the time the reactor is shut down.

During operation, it is assumed that the power of the reactor is constant.

Since there are several radioactive species present, there is no specific half time that can beassigned.

Way and Wigner fitted a curve for the decrease in energy release with time as a power law.

The energy release rate in the form of β and γ at time t after occurrence of a fission wasexpressed as

In the above expressions, t is time expressed in seconds.

Our aim is to express the decay power as a fraction of the original full power at any giventime Ts after the reactor is shut down. It is assumed that the reactor was operating for a timeTo at a uniform power.

Let us consider the fissions that occurred between t* and t*+ dt*. Also, let q"' be the volumetricheat generation rate. The number of fissions during this time shall be,

Since is same as ,

Similarly

Hence total decay power can be written as,

American Nuclear Society (ANS-1971) had suggested a curve that was fitted to the followingexpression

where, a and b are given in the following table

TS(s) a b0.1-1 12.05 0.0639

10-150 15.31 0.18078

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150-8x10 27.43 0.2962