Thermodynamics, Heat Transfer, And Fluid Flow. V.2. Heat Transfer (776131), страница 9
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The density of the fuel,the neutron flux, and the type of fuel all affect the fission rate and, therefore, the heat generationrate. The following equation is presented here to show how the heat generation rate ( Q̇ ) isrelated to these factors. The terms will be discussed in more detail in the Nuclear Sciencemodules.Q̇G N σf φVf(2-14)where:Q̇ =heat generation rate (Btu/sec)G=energy produced per fission (Btu/fission)N=number of fissionable fuel nuclei/unit volume (atoms/cm3)σf =microscopic fission cross-section of the fuel (cm2)φ=neutron flux (n/cm2-sec)Vf =volume of the fuel (cm3)The thermal power produced by a reactor is directly related to the mass flow rate of the reactorcoolant and the temperature difference across the core. The relationship between power, massflow rate, and temperature is given in Equation 2-14.Q̇ṁ cp ∆T(2-15)where:Q̇ṁcp∆T====heat generation rate (Btu/hr)mass flow rate (lbm/hr)specific heat capacity of reactor coolant system (Btu/lbm-°F)temperature difference across core (°F)For most types of reactors (boiling water reactor excluded), the temperature of the coolant isdependent upon reactor power and coolant flow rate.
If flow rate is constant, temperature willvary directly with power. If power is constant, temperature will vary inversely with flow rate.Rev. 0Page 45HT-02HEAT GENERATIONHeat TransferFlux ProfilesOnce the type and amount of fuelis determined, the shape of theneutron flux distribution along thecore is established. Both radialand axial flux distributions mustbe determined. A radialdistribution looks at flux from thecenter of the core out to the edges.An axial distribution looks at fluxfrom the bottom to the top of thecore. As seen in Equation 2-14,the fission rate directly affects theheat generation rate within areactor core. In the core regionsof highest flux, the highest heatgeneration rate will be present.Many factors affect the axial andFigure 14 Axial Flux Profileradial flux distributions, includingthe number and type of controlrods, the geometry and size of core, the concentration of fission product poisons, and reflectorproperties.
The peak power production regions within each distribution normally occurs near thecenter of the core, as indicated in Figures 14 and 15, but can vary during transients or as the coreages.The above figures represent theneutron flux profiles withoutconsidering the effects of controlrods.Once control rods andreflectors are taken into account,the flux profiles become muchflatter although the peak stilloccurs near the center.The shape of the profiles can bedetermined by measuring the ratioof the peak flux to the averageflux in the distribution.Thispeaking factor is referred to as thehot channel factor.
A hot channelfactor of 1.0 would imply a flatflux profile.HT-02Figure 15Page 46Radial Flux ProfileRev. 0Heat TransferHEAT GENERATIONThermal LimitsHot channel factors are calculated values used to take into account various uncertainties intolerances used in core manufacturing. For example, consider a coolant channel of the minimumacceptable width and length, that happens to be adjacent to a fuel plate with the maximumacceptable fuel loading.
In this channel, we would now have less water than in the averagechannel, receiving more heat than the normal coolant channel. For any given values of corepower and flow, this hypothetical channel would be closest to a thermal limit. Therefore, alldesign considerations are based upon the hot channel factor for each core. The nuclear heat fluxhot channel factor (HFHCF) is the ratio of the maximum heat flux expected at any area to theaverage heat flux for the core. The nuclear enthalpy rise hot channel factor is the ratio of thetotal kW heat generation along the fuel rod with the highest total kW to the total kW of theaverage fuel rod.Thus the limitation of the peak flux value in a core is directly related to the hot channel factor.However, in discussing flux profiles, "average" values of flux in the core are usually referred torather than peaks.Average Linear Power DensityIn nuclear reactors, the fuel is usually distributed in individual components which sometimesresemble rods, tubes, or plates.
It is possible to determine the average power produced per unitlength of fuel component by dividing the total thermal output of the core by the total length ofall the fuel components in the core. This quantity is called the average linear power density.Common units for measuring average linear power density are kW/ft.Example:Calculate the average linear power density for an entire core if a 3400 MW reactor isoperating at full power.Core data is:each fuel rod is 12 ft long264 rods/fuel assembly193 fuel assemblies in the coreSolution:Rev. 0Average linear power density=total thermal powertotal fuel rod lengthAverage linear power density=3.4 x 106 kW12 (264) (193)=5.56 kW/ftPage 47HT-02HEAT GENERATIONHeat TransferMaximum Local Linear Power DensityThe maximum local linear power density when compared to the average linear power densityresults in the definition of the nuclear heat flux hot channel factor.
The nuclear heat flux hotchannel factor can be looked at as having axial and radial components that are dependent uponthe power densities and, thus, the flux in the radial and axial planes of the core. Once the hotchannel factor is known, the maximum local linear power density anywhere in the core can bedetermined, as demonstrated in the following example.Example:If the nuclear heat flux hot channel factor is 1.83, calculate the maximum local linearpower density in the core for the previous example (the average linear power densityproblem).Solution:Maximum linear power density= HFHCF (Av linear power density)= 1.83 (5.56) kW/ft= 10.18 kW/ftNormally, nuclear facility operatorsare provided with the above corepower and heat generationdistributions, rather than having tocalculate them.
In addition, variousmonitoring systems are alwaysemployed to provide the operator witha means of monitoring coreperformance and the proximity of theexisting operating conditions to coreoperational limitations.Temperature ProfilesAdditional areas of interest are thetemperature profiles found within thecore. A typical axial temperatureprofile along a coolant channel for apressurized water reactor (PWR) isFigure 16 Axial Temperature Profileshown in Figure 16. As would beexpected, the temperature of thecoolant will increase throughout the entire length of the channel.HT-02Page 48Rev. 0Heat TransferHEAT GENERATIONHowever, the rate of increase will vary along with the linear heat flux of the channel.
The powerdensity and linear heat rate will follow the neutron flux shape. However, the temperaturedistributions are skewed by the changing capacity of the coolant to remove the heat energy.Since the coolant increases in temperature as it flows up the channel, the fuel cladding and, thus,the fuel temperatures are higher in the upper axial region of the core.A radial temperature profile across a reactor core (assuming all channel coolant flows are equal)will basically follow the radial power distribution.
The areas with the highest heat generationrate (power) will produce the most heat and have the highest temperatures. A radial temperatureprofile for an individual fuel rod and coolant channel is shown in Figure 17. The basic shapeof the profile will be dependent upon the heat transfer coefficient of the various materialsinvolved. The temperature differential across each material will have to be sufficient to transferthe heat produced. Therefore, if we know the heat transfer coefficient for each material and theheat flux, we can calculate peak fuel temperatures for a given coolant temperature.Figure 17 Radial Temperature Profile Across aFuel Rod and Coolant ChannelRev. 0Page 49HT-02HEAT GENERATIONHeat TransferVolumetric Thermal Source StrengthThe total heat output of a reactor core is called the heat generation rate.
The heat generationrate divided by the volume of fuel will give the average volumetric thermal source strength. Thevolumetric thermal source strength may be used to calculate the heat output of any section of fuelrod, provided the volume of the section is known.Volumetric Thermal Source StrengthQ̇coreVfuelFuel Changes During Reactor OperationDuring the operation of a nuclear reactor, physical changes occur to the fuel that affect its abilityto transfer heat to the coolant.
The exact changes that occur are dependant on the type and formof fuel. Some reactors use fuel assemblies that consist of zircalloy tubes containing cylindricalceramic pellets of uranium dioxide. During manufacture, a small space or gap is left betweenthe fuel pellets and the zircalloy tube (clad). This gap is filled with pressurized helium.
As thereactor is operated at power, several physical changes occur in the fuel that affect the gapbetween the pellets and clad. One change occurs due to high pressure in the coolant outside theclad and the relatively high temperature of the clad during reactor operation.
The hightemperature and high pressure causes the clad to be pushed in on the pellets by a process referredto as creep. Another physical change is caused by the fission process. Each fission event createstwo fission product atoms from a fuel atom. Even though each fission product atom is roughlyhalf the mass of the fuel atom, the fission products take up more volume than the original fuelatom. Fission products that are gases can collect together and form small gas bubbles within thefuel pellet. These factors cause the fuel pellets to swell, expanding them out against the clad.So the two processes of pellet swell and clad creep both work to reduce the gap between the fueland clad.This change in the gap between the pellet and clad has significant impact on heat transfer fromthe fuel and operating fuel temperatures.
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