Reactivity Coefficients

Transcription

1 Revision 1 December 2014 Reactivity Coefficients Student Guide GENERAL DISTRIBUTION

2 GENERAL DISTRIBUTION: Copyright 2014 by the National Academy for Nuclear Training. Not for sale or for commercial use. This document may be used or reproduced by Academy members and participants. Not for public distribution, delivery to, or reproduction by any third party without the prior agreement of the Academy. All other rights reserved. NOTICE: This information was prepared in connection with work sponsored by the Institute of Nuclear Power Operations (INPO). Neither INPO, INPO members, INPO participants, nor any person acting on behalf of them (a) makes any warranty or representation, expressed or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this document, or that the use of any information, apparatus, method, or process disclosed in this document may not infringe on privately owned rights, or (b) assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus, method, or process disclosed in this document. ii

3 Table of Contents INTRODUCTION... 1 TLO 1 REACTIVITY, K EFF AND SHUTDOWN MARGIN... 2 Overview... 2 ELO 1.1 Reactivity... 2 ELO 1.2 Reactivity Conversions... 5 ELO 1.3 Excessive Reactivity... 6 ELO 1.4 Shutdown Margin... 9 ELO 1.5 Shutdown Purpose ELO 1.6 Sufficient Reactivity Conversions to Calculate Reactor Shutdown Margin TLO 1 Summary TLO 2 MODERATOR, VOID AND PRESSURE REACTIVITY Overview ELO 2.1 Reactivity Coefficients ELO 2.2 Moderator Temperature Coefficient (MTC) ELO 2.3 Moderator to Fuel Ratio Effects on MTC ELO 2.4 Void and Pressure Reactivity Coefficients TLO 2 Summary TLO 3 FUEL TEMPERATURE AND POWER COEFFICIENTS Overview ELO 3.1 Fuel Temperature Reactivity Coefficient ELO 3.2 Doppler and Self-shielding ELO 3.3 Moderator Temperature Effects on the Fuel Temperature Coefficient ELO 3.4 Power Reactivity Coefficient ELO 3.5 Power Defect on Reactor Power Operations Definition TLO 3 Summary TLO 4 REACTIVITY BALANCES AND BORON REACTIVITY Overview ELO 4.1 Reactivity Balance ELO 4.2 Purpose of Boron Reactivity Control ELO 4.3 Changes in Boron Worth with Changes in Boron Concentration ELO 4.4 Changes in Boron Concentration over Core Life TLO 4 Summary REACTIVITY COEFFICIENTS SUMMARY iii

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5 Reactivity Coefficients Revision History Revision Date Version Number Purpose for Revision Performed By 11/5/ New Module OGF Team 12/11/ Added signature of OGF Working Group Chair OGF Team Introduction This module includes key concepts that will help the operator understand power operations of reactivity coefficients. Rev 1 1

6 Objectives At the completion of this training session, the trainee will demonstrate mastery of this topic by passing a written exam with a grade of 80 percent or higher on the following Terminal Learning Objectives (TLOs): 1. Describe reactivity, k eff and shutdown margin and their effect on the reactor operational status. 2. Describe moderator, void and pressure reactivity coefficients and how they are affected by changing reactor conditions. 3. Describe the fuel temperature and power reactivity coefficients and describe how they are affected by changing reactor conditions. 4. Describe how a reactivity balance is performed and methods used to compensate for excess reactivity. TLO 1 Reactivity, k eff and Shutdown Margin Overview Previous sections explained k eff, the effective multiplication factor, the ratio of the neutrons produced by fission in one generation to the number of neutrons lost through absorption and leakage in the preceding generation. This section introduces reactivity and its relationship to k eff. Objectives Upon completion of this lesson, you will be able to do the following: 1. Describe the term reactivity and its relationship to k eff and criticality. 2. Convert between alternate units of reactivity. 3. Define excess multiplication factor (k excess ) and excess reactivity (ρ excess ). 4. Define shutdown margin. 5. Evaluate plant parameters or design features that affect shutdown margin. 6. Given sufficient reactivity information, calculate the reactor shutdown margin. ELO 1.1 Reactivity Introduction Reactivity is a measure of the fractional change in neutron population per generation. Reactivity is a function of k eff, defined as the ratio of the neutrons produced by fission in one generation to the number of neutrons lost through absorption and leakage in the preceding generation. Reactivity, like k eff, describes the reactor's deviation from criticality. Reactivity units measure the reactor s deviation from criticality. Reactivity versus k eff It is possible to determine the number of neutrons after a certain number of generations if you know the original number of neutrons (N o ) at the start of the first generation and the value of k eff, with k eff at a constant value from generation to generation. We use the formula below for this purpose: 2 Rev 1

7 Example: ( ) Where: n = number of generations N n = number of neutrons in the n th generation N o = number of neutrons at the start of the first generation The number of neutrons in the core at time zero is 1,000 and k eff = Calculate the number of neutrons after 50 generations. Solution: Using: ( ) ( ) If there are N o neutrons in the preceding generation, then there are N o (k eff ) neutrons in the present generation. The numerical change in neutron population is (N o k eff - N o ). We express reactivity (ρ) as a fraction, therefore the count rate expressed as a fraction is: Cancelling out the term N o from the numerator and denominator, reactivity relates to k eff as: Reactivity, as shown in the above formula, is the fractional change in neutron population per generation. Reactivity versus Criticality Reactivity is the term used when discussing a nuclear reactor's deviation from criticality. If the reactor is critical (k eff = 1) then reactivity = 0 for any reactor power level. Reactivity is a positive value >0 for a supercritical reactor (k eff > 1). Reactivity is negative value <0 for a subcritical reactor (k eff < 1). From the reactivity equation below, ρ may be positive, zero, or negative, depending upon the value of k eff. Rev 1 3

8 The larger the absolute value of reactivity in the reactor core, the further the reactor is from criticality. Example: Calculate the reactivity in the reactor core when k eff is equal to and For each value of k eff, state whether the reactor is critical, supercritical, or subcritical. Solution: The reactivity for each case is determined by substituting the value of k eff into the formula for reactivity: Reactivity is positive, therefore the reactor is supercritical Reactivity is negative, therefore the reactor is subcritical You can determine k eff by transforming the equation to solve for k eff in terms of the reactivity if you do not know k eff and you know reactivity. The result is: Example: Given a reactivity of x 10-4 k/k, calculate k eff. Solution: ( ) 4 Rev 1

9 Knowledge Check Reactivity is defined mathematically as the fractional change in. A. reactor power per second B. neutron population per second C. reactor period from criticality D. the effective multiplication factor from criticality Knowledge Check Given a reactivity of x 10-4 k/k, calculate k eff to the nearest thousandth. A B C x 10-4 k/k D x 10-4 k/k ELO 1.2 Reactivity Conversions Introduction Reactivity is a dimensionless number. It is simply a ratio of two quantities, expressed either as a ratio, or in percent (such as ρ). Reactivity conversions Step-by-Step Table The value of reactivity is often a small decimal value, often expressed in special units to make this value easier to express. The value for reactivity that results directly from the calculation of k eff is in units of k/k by definition. Alternative units for reactivity are percent k/k and pcm (percent millirho). The table below shows conversions between these units of reactivity: Step Action 1. Determine the unit of reactivity to be used. 2. Convert using appropriate step below or ( ) Rev 1 5

10 Reactivity Conversion Demonstration Example: Convert the values of reactivity listed below to the indicated units. a k/k = pcm b k/k = percent k/k c. 16 x 10-4 k/k = k/k Solution: a pcm b percent k/k c k/k Knowledge Check Convert the values of reactivity listed below to the indicated units. A x 10-4 k/k = pcm B. 350 x 10-4 k/k = % k/k C. 2,500 pcm = k/k ELO 1.3 Excessive Reactivity Introduction Excess reactivity (k excess ) is the reactivity from excess fuel loaded into a nuclear reactor core beyond the minimum amount necessary to achieve criticality at the beginning of core life. This is necessary to provide for longer operational periods between refueling. Excess positive reactivity must be available to compensate for the following: Fuel burnup Fission product poisons (xenon and samarium) Increases in resonance capture from plutonium-240 buildup Raising temperature and power to their normal full power values Excess Reactivity A critical reactor has a k eff = 1. In order to maintain a value of 1 throughout core life, we must add excess reactivity (extra fuel) to the core at the beginning of a fuel cycle. This means that we must also add negative reactivity to the core to counter the positive reactivity from the "excess" fuel loading. Operators cancel (offset) the excess reactivity from the fuel using control rods, soluble boron, and fixed burnable poison rods that provide negative reactivity. Excess Multiplication Factor The excess multiplication factor (k excess ) is the amount of excess fuel loading that causes k eff to exceed 1.0. The equation below shows the mathematical expression: 6 Rev 1

11 We express excess reactivity (ρ excess ) in terms of k excess by the following formula: Example: Consider the refueling of a reactor. The refueling of the total core increases k eff to a value of 1.5. What is the value of the excess multiplication factor (k excess ) and excess reactivity (ρ excess ) after refueling? Solution: Solve for k excess using the following equation: Then solve for ρ excess : Or, we express ρ as: We generally define excess multiplication factor (k excess ) and excess reactivity (ρ excess ) for specific reactor conditions. Commonly used conditions are: Cold, xenon-free, no control rods Hot, xenon-free, no control rods Hot, rated power, equilibrium fission product poisons (xenon and samarium) Changes in Excess Multiplication Factor over Core Life The value of k excess varies over core life due to changing neutron poison concentrations in the reactor core and fuel burnout. The following figure is a generic example k excess over core life. Rev 1 7

12 Figure: Core Age versus k eff 1. At the beginning of core life, k excess decreases due to the buildup of xenon and samarium (fission product poisons) in the reactor (A to B in the figure above). For core fuel loads that include burnable poison rods, this reduction would be less significant (removal of negative reactivity). 2. Toward the middle of core life, k excess increases to a maximum value because of the depletion of burnable poisons (B to C in the figure above). Depending on fuel load /burnable poisons, this peak may be lower. 3. From middle of core life to end of core life, k excess decreases due to fuel burnout, until k excess is eventually exhausted (C to D in the figure above). Core coastdown begins at point D to maintain a reduced power level. Knowledge Check A. 0.8 B. 1.6 C. 0.6 After a core reloading p excess has been calculated to equal 37,500 pcm. What is k excess equal to? D Rev 1

13 ELO 1.4 Shutdown Margin Introduction Shutdown margin (SDM) is the instantaneous amount of reactivity by which the reactor is subcritical or would be subcritical from its present condition, assuming complete insertion of all full-length rod cluster assemblies (shutdown and control) and the most reactive control rod fully withdrawn from the core at any time during the core cycle. The shutdown value (SDV) is the reactivity amount by which nuclear reactor core is subcritical; or SDV is the additional amount of reactivity that would make a reactor subcritical from its present condition. These two terms are closely related. However, most importantly each commercial nuclear plant has specific SDM requirements required by their operating license. Shutdown Value Determination The SDV is the reactivity amount by which nuclear reactor core is subcritical or the additional amount of reactivity that would make a reactor subcritical from its present condition. We calculate the SDV by using the following equation: The SDV is simply the actual reactivity value by which the reactor is subcritical or the amount needed to make it subcritical. For example if k eff is 0.99, the SDV is equal to Δk/k. Control rod position, moderator temperature, poisons, boron concentration, etc. affect SDV. Shutdown Margin Determination The plant's technical specifications specify SDM requirements. The SDM is the instantaneous amount of reactivity by which a nuclear reactor core is subcritical, or would be subcritical from its present condition with the most reactive control rod fully withdrawn from the core. Notice from the definition that SDM exists if the reactor is operating at 100 percent power, or is shut down. Nuclear reactor technical specifications require reactors to maintain a specific minimum SDM, assuming the most reactive rod fully withdrawn from core. A typical value ranges from 1.0 to 1.7 percent Δk/k. The required value for SDM will change with core life. Rev 1 9

14 The SDM is calculated using same equation as used for SDV: Example: Calculate SDM of shutdown reactor with a core reactivity value of Δk/k. Solution: First, find k eff : ( ) Then, use the SDM equation: Since SDM and SDV have units of reactivity (Δk/k or percent Δk/k) the value can be determined directly from the Δk/k given in the problem - just change it to a positive value. Question: If the plant requires an SMD of 1.0 percent Δk/k, is the above SDM sufficient? Answer: No. Knowledge Check Calculate shutdown margin of shutdown reactor in Δk/k with a k eff of 0.9. A. 100 Δk/k B. 10 Δk/k C. 1 Δk/k D. 0.1 Δk/k ELO 1.5 Shutdown Purpose Introduction The time in core life, control rod position, reactivity poison, boron concentration, and other reactivity related core conditions determine the amount of reactivity that actually shuts a reactor down, and therefore determines the SDM. 10 Rev 1

15 Shutdown Margin Definition The SDM is the instantaneous amount of reactivity by which a nuclear reactor core is subcritical, or would be subcritical from its present condition with the most reactive control rod fully withdrawn from the core. Understanding this definition is key to understanding how reactivity conditions in the core can affect its actual value. The following parameters or design features will affect SDM: Moderator temperature Reactor coolant system boron concentration Fuel temperature (Doppler) Control rod position Xenon/samarium and other reactivity poisons concentration Number of fuel assemblies loaded in core Time in core life Reactor power level Reactivity effects to Shutdown Margin Example Each of the following reactivity parameters affects the SDM during reactor shutdown conditions: Moderator temperature - an increase in moderator temperature adds negative reactivity. This increases SDM. During a plant cooldown, the decreased moderator temperature adds considerable positive reactivity. It is necessary to increase the RCS boron concentration to compensate to maintain the required SDM. Boron concentration in the reactor coolant system - increasing boron concentration causes a decrease in the thermal utilization factor, which adds negative reactivity; resulting in an increase in SDM. Fuel temperature (Doppler) - when in a shutdown condition and the reactor core is cooling, fuel temperature is maintained constant, and SDM is unaffected. As the RCS cools, fuel temperature will also decrease, causing the resonance escape probability to increase. This adds positive reactivity, with a resulting decrease in SDM. Control rod position normally during shutdown conditions, the control and shutdown rods are in the fully inserted position. If they are withdrawn, this will add positive reactivity, causing the SDM to decrease. Xenon/samarium and other reactivity poisons concentration during shutdown conditions, fission product poisons such as xenon and samarium will either peak or decay off, depending on the power history and length of shutdown. If poisons increase, this adds negative reactivity causing SDM to increase. SDM will decrease from the addition of positive reactivity if poisons are decaying off. Number of fuel assemblies loaded in core It is possible to maintain SDM by a minimum boron concentration during refueling and shutdown verification via the performance of 1/m plots during fuel loading and unloading. As fuel is loaded to the core, add positive reactivity since the concentration of the fuel is increasing. Therefore, as fuel is loaded, SDM will decrease. Rev 1 11

16 Time in core life - when the reactor is shut down, there is no change in core life and therefore no effect on SDM. However, the time in core life does affect k excess, requiring a lower boron concentration (with increasing core life) to meet minimum SDM requirements. Reactivity Effects to Shutdown Margin During Operating Conditions During reactor operations, the second half of the SDM definition applies - the instantaneous amount of reactivity by which a nuclear reactor would be subcritical from its present condition with the most reactive control rod fully withdrawn from the core. Therefore, for each of the following reactivity parameters, operators must consider the reactivity change immediately following the reactor shutdown or trip. Moderator temperature - RCS temperature following the shutdown will level off at the no-load value (less than full load), adding positive reactivity, and causing SDM to decrease. Reactor coolant system boron concentration - immediately following the trip or shutdown, boron concentration does not change and there is no effect on SDM. Fuel temperature (Doppler) - the cooler fuel temperature from the trip adds positive reactivity. This is a large effect, causing a large decrease in SDM. Control rod position - on a reactor trip, personnel insert all control and shutdown rods into the core and add a very large amount of negative reactivity. This results in a large increase in SDM. A reactor shutdown produces a similar effect; however, since the shutdown rods may not be inserted, the increase in SDM may be less. During power operation (power dependent), the control rods must be above a certain minimum height to ensure adequate SDM on a trip. Xenon/samarium and other reactivity poisons concentration - immediately following a trip or shutdown xenon will peak (first 8 hours). This adds negative reactivity causing an increase to the SDM. The immediate effect from samarium is much less. Following the xenon peak, xenon decay is greater than production and positive reactivity will be added, reducing SDM. Time in core life - the time in core affects control rod worth, fuel temperature and moderator temperature reactivity, boron worth, and other factors. Therefore, time in core life will have an effect on SDM following a trip or shut down. Reactor power level - as the power level increases; moderator and fuel temperatures also increase. Therefore, more positive reactivity would be added on a trip because of the greater fuel and moderator temperature decrease. The SDM would be lower on a trip or shutdown from higher power levels. 12 Rev 1

17 Knowledge Check A nuclear power plant is operating at 70 percent power with manual rod control. Which one of the following conditions will increase shutdown margin? Assume that no unspecified operator actions occur and the reactor does not trip. A. The reactor coolant system is diluted by 10 ppm. B. A control rod in a shutdown bank (safety group) drops. C. Power is decreased to 50 percent using boration. D. The plant experiences a 3 percent load rejection. ELO 1.6 Calculate Reactor Shutdown Margin Introduction With a known value of k eff, SDM can be determined using the formula: However, during reactor operation we probably do not know the exact value of k eff. We express SDM in units of reactivity so it is possible to determine the SDM by accounting for all of the positive and negative reactivities existing in the reactor core at a given time. Reactivity Conversions Each plant has a specific procedure for determining the SDM or a method for establishing adequate SDM. This lesson will use a generic method for demonstration purposes. At most plants when the reactor is at power SDM is maintained (and known to exist) by ensuring that the control rods are above a certain minimum height. This minimum height is the insertion limit. The insertion limit ramps higher as reactor power level is increased. Recall that the SDM is less on a trip or shutdown as power increases, requiring higher insertion limits to ensure adequate SDM. When we shut the reactor down, the rods are on the bottom, so how do we calculate SDM? Step Action 1. Obtain last critical data as a starting point, where reactivity = 0 2. Determine all reactivity changes from last critical data 3. Sum the reactivities to determine reactivity and change the value to a positive number for the SDM. Rev 1 13

18 Calculating Shutdown Margin Demonstration Example: The following critical conditions exist just prior to a reactor trip: Power level = 100 percent Boron concentration = 660 ppm Power defect = 1,500 pcm (power defect includes reactivity from the fuel and moderator temperature coefficients to be discussed in detail later) Control rod fully withdrawn 5,000 pcm Xenon at equilibrium Samarium at equilibrium RCS temperature at full load Middle of core life Given these reactivity parameters, what is the SDM immediately following a reactor trip? Solution: Just prior to the reactor trip, the core reactivity is 0 pcm: Reactor critical at 100 percent power Reactivity from power decrease Reactivity from control rod insertion Boron, Xe, Sm no change Core life no change RCS temperature at no load SDM = 3.5 Note Knowledge Check In this example, the SDM is determined immediately following the trip. If the SDM were to be calculated a day or more later, xenon and samarium would change in reactivity value, boron concentration may change, and the plant may be cooled down or in the process of cooling down. With a nuclear power plant operating at 85 percent power and rod control in manual, the operator borates the reactor coolant system an additional 10 ppm. Assuming reactor power does not change during the boration, shutdown margin will A. decrease and stabilize at a lower value. B. decrease, then increase to the original value as coolant temperature changes. C. increase and stabilize at a slightly higher value. D. increase, then decrease to the original value as coolant temperature changes. 14 Rev 1

19 TLO 1 Summary 1. Reactivity and its relationship to k eff and criticality If the reactor is critical (k eff = 1), then reactivity = 0 for any reactor power level. Reactivity is a positive value >0 for a supercritical reactor (k eff > 1). Reactivity is negative value <0 for a subcritical reactor (k eff < 1). 2. Convert between alternate units of reactivity. The value for reactivity that results directly from the calculation of k eff is in units of k/k. Alternative units for reactivity are percent k/k and pcm (percent millirho). 3. Excess multiplication and (k excess ) and excess reactivity (ρ excess ). Excess multiplication factor (k excess ) is the amount of excess fuel loading that causes k eff to exceed 1.0. Excess reactivity (ρ excess ) is = k excess / k eff. 4. Shutdown Margin The plant's technical specifications specify SDM requirements. SDM is the instantaneous amount of reactivity by which a nuclear reactor core is subcritical, or would be subcritical from its present condition with the most reactive control rod fully withdrawn from the core. SDM is expressed in units of reactivity. Determine the SDM by accounting for all of the positive and negative reactivities existing in the reactor core at a given time. 5. Plant parameters or design features that affect SDM. The following parameters of design features will affect the SDM: Moderator temperature Reactor coolant system boron concentration Fuel temperature (Doppler) Control rod position Xenon/samarium and other reactivity poisons concentration Number of fuel assemblies loaded in core Time in core life Reactor power level Summary Now that you have completed this lesson, you should be able to: 1. Describe the term reactivity and its relationship to k eff and criticality. 2. Convert between alternate units of reactivity. 3. Define excess multiplication factor (k excess ) and excess reactivity (ρ excess ). 4. Define shutdown margin. 5. Evaluate plant parameters or design features that affect shutdown margin. 6. Given sufficient reactivity information, calculate the reactor shutdown margin. Rev 1 15

20 TLO 2 Moderator, Void and Pressure Reactivity Overview This session introduces reactivity coefficients. This section will explore how moderator temperature, pressure, and voids affect reactivity in the core. The concepts covered in this lesson are some of the most important to your reactor operation responsibilities. In particular, the moderator temperature coefficient provides an inherent safety feature, along with the fuel temperature coefficient of a PWR. This lesson includes an explanation of the moderator temperature coefficient, as well as how boron concentration affects the coefficient. We define SDM in terms of reactivity coefficients. Objectives Upon completion of this lesson, you will be able to do the following: 1. Explain differences between reactivity coefficients and reactivity defects and explain their use to balance reactivity parameters. 2. Describe the moderator temperature coefficient of reactivity. 3. Describe how the magnitude of the moderator temperature coefficient varies with changes in the following parameters: a. Overmoderation and undermoderation of the moderator-to-fuel ratio b. Moderator temperature c. Core age d. Boron concentration 4. Describe the void and pressure coefficients of reactivity. ELO 2.1 Reactivity Coefficients Introduction The amount of reactivity (ρ) in a reactor determines the neutron population and/or reactor power state. Many factors affect reactivity, such as fuel depletion, temperature, pressure, or fission product poisons. This section discusses the factors affecting reactivity and tells how they control or predict reactor behavior. Reactivity Coefficients Reactivity coefficients quantify the effect that a variation in a reactor parameter (i.e. a change in temperature, control rod position, boron changes, etc.) has on the overall reactivity of the core. Reactivity coefficients define the amount of reactivity change for a given change in the parameter (per F, per ppm boron, etc.). As an example, a moderator temperature increase causes a decrease in the reactivity of the core. The amount of reactivity change per unit increase of moderator temperature is the moderator temperature coefficient. Units for the moderator temperature coefficient are pcm/ F. 16 Rev 1

21 Generally, α x symbolizes reactivity coefficients, where x represents the reactor parameter affecting reactivity. The equation below shows reactivity coefficients expressed as a formula: Where: x = reactivity coefficient for plant parameter x Δρ = change in reactivity (Δk/k) Δx = a unit increase in plant parameter x If the parameter x increases resulting in an addition of positive reactivity, then x is positive. If the parameter x increases resulting in an addition of negative reactivity, then x is negative. Reactivity Defects Reactivity defects are the total reactivity change caused by variation in a parameter. The term "reactivity defect" (ρ x ) describes the total amount of reactivity added, positive, or negative, due to changing a certain nuclear reactor parameter by a given amount. Reactivity defects are determined by multiplying the total change in the parameter by its average coefficient value. The equations below relate reactivity coefficients to reactivity defects. ( )( ) ( ) ( ) Where: Example: ρ x = reactivity defect (Δk/k) x = specific parameter (fuel temperature, moderator temperature, etc.) Δ x = change in parameter x α x = parameter x reactivity coefficient (fuel temperature, moderator temperature, etc.) The moderator temperature coefficient for a reactor is -8.2 pcm/ F. Calculate the reactivity defect that results from a temperature decrease of 5 F. Solution: Rev 1 17

22 ( ) ( ) The reactivity addition due to the temperature decrease was a positive 41 pcm because of the negative temperature coefficient. Knowledge Check Moderator temperature coefficient is the change in core reactivity per degree change in. A. fuel temperature B. fuel clad temperature C. reactor vessel temperature D. reactor coolant temperature ELO 2.2 Moderator Temperature Coefficient (MTC) Introduction The moderator temperature coefficient (MTC) of reactivity is the change in reactivity per degree change in moderator temperature. We discussed the moderator temperature effect on k eff with the six-factor formula, and we will further review it later in this lesson. Moderator Temperature Coefficient The reactivity change per degree change in moderator temperature is the moderator temperature coefficient (MTC) of reactivity. Its magnitude and sign (+ or -) is primarily a function of the moderator-to-fuel ratio, density of the moderator, and boron concentration. Commercial PWRs are designed with an undermoderated moderator-to-fuel ratio that normally provides a negative moderator temperature coefficient. Early in core life, the MTC may be positive with the initial high boron concentration. If a reactor is overmoderated, it will have a positive MTC as the change in thermal utilization factor overrides the resonance escape probability. However, a negative MTC is more desirable because of its power level regulating effect in the power range. Assuming reactor power is in the power range, a power increase will cause moderator temperature to increase. If the core is undermoderated (negative MTC), the temperature increase will insert negative reactivity into the core and will slow the power rise. Since power is in the power range and steam demand has not changed, reactor power will level off at the initial value and moderator temperature will stabilize at a new value depending on the amount of reactivity that initially caused the power increase. The MTC responds to a power decrease in the power range in the opposite manner (adding positive reactivity to slow the power decrease). The MTC in equation form is: 18 Rev 1

23 ( ) Where: m = moderator temperature coefficient (MTC) (Δk/k/ F) Δρ = change in reactivity associated with change in moderator temperature (Δk/k) ΔT mod = change in moderator temperature ( F) The symbol m as well as the symbol T represent moderator temperature coefficient. This text uses the symbol m. Example: A reactor is operating at 480 F with an effective multiplication factor of (k eff = 1.0). The moderator temperature increases to 490 F and k eff decreases to What is the value of the moderator temperature coefficient? Solution: First, convert k eff values to reactivity. Then, calculate the value of MTC. ( ) ( ) ( ) Value of Moderator Temperature Coefficient Boron concentration and time in core life affect the value of MTC. A good approximation of the MTC is -1 x 10-4 Δk/k/ F for the normal operating range of moderator temperatures in a commercial nuclear reactor. Rev 1 19

24 Knowledge Check A reactor is operating at 560 F with k eff = 1.0. The reactor operator borates the reactor an equivalent of 200 pcm (negative reactivity). RCS temperature responds by dropping 10 degrees. Assuming no other reactivity effects what is the MTC? A. 5 x 10-4 Δk/k/ F B. 20 x 10-4 Δk/k/ F C. 10 x 10-4 Δk/k/ F D. 2 x 10-4 Δk/k/ F ELO 2.3 Moderator to Fuel Ratio Effects on MTC Introduction Moderator temperature coefficient (MTC) values are not constant throughout core life. As we have learned, the moderator-to-fuel ratio has an effect on whether or not k eff increases or decreases with a moderator temperature change. In terms of a reactivity coefficient, this translates to either a positive or negative moderator temperature coefficient. This section discusses how the following parameters affect MTC: Overmoderation and undermoderation of the moderator-to-fuel ratio Moderator temperature Core age Boron concentration Moderator to Fuel Ratio Effects on MTC The moderator-to-fuel ratio (N m /N u ) is very important in the discussion of moderators. The reactor designer adjusts the amount of moderator and fuel in the core (N m /N u ratio) to an optimum value that establishes a negative MTC throughout core life based on this ratio; however, it is possible during specific core age and core parameters for a positive MTC to exist. Moderator temperature affects moderator density and causes the moderatorto-fuel ratio to change. Changes in the moderator-to-fuel ratio affect the thermal utilization factor (f) and the resonance escape probability (p) which in turn affect k eff and reactivity or more precisely the MTC. It is possible to design the moderator-to-fuel ratio to be either undermoderated (too little moderator) or overmoderated (too much moderator). An overmoderated condition leads to a positive MTC (undesirable) while an undermoderated condition leads to a negative MTC. The amount of over or under moderation determines the magnitude of the MTC. Commercial PWRs are designed to operate in an undermoderated condition because of the design requirement to have a negative MTC. The following graphic illustrates this: 20 Rev 1

25 Figure: Moderator to Fuel Ratio Curves Undermoderation The area to the left of the dotted vertical line is the undermoderated region. Notice also that at the dotted line, the k eff curve peaks. In the undermoderated region, a decrease in the moderator-to-fuel ratio results in a decrease in k eff, equivalent to negative reactivity. Relating this to temperature, as temperature is increased, the concentration of the moderator (N m ) decreases, causing N m /N u to decrease (move to the left). This is a negative MTC. As you previously learned, the changes in thermal utilization factor (f) and the resonance escape probability (p) are the main causes for the change in k eff. Using the same illustration of a temperature increase with a decreasing N m /N u, we see that the thermal utilization factor increases while the resonance escape probability decreases. In this case, the effect from the resonance escape probability overrides the effect from the thermal utilization factor leading to a negative MTC. It is the balance of these two factors (the curves have different slopes) that determines the magnitude of the MTC (while undermoderated) because one of these is a positive effect and the other is negative. Recall that the nonleakage factors have a small influence on MTC, which also cause it to be negative. Operating in the undermoderated region is very important to reactor control. The moderator temperature will rise, inserting negative reactivity, thereby limiting the magnitude of the power excursion if reactor power suddenly increases. Commercial nuclear reactors are designed with a moderator-tofuel ratio such that MTC is negative in the normal operating temperature range. Overmoderation The area to the right of the dotted vertical line is the overmoderated region. In the overmoderated region, a reduction in moderator density (temperature increase) has a greater effect on thermal utilization factor than the resonance Rev 1 21

26 escape probability. With f greater than p, k eff increases, equivalent to positive reactivity. If the reactor operates in the overmoderated region, any increase in reactor power would result in an increase in moderator temperature. This effect feeds itself; the increase in moderator temperature adds more positive reactivity, resulting in an additional increase in reactor power, and even higher temperatures and higher power. Safe control of the reactor and maintaining operation within the core operating limits is much more difficult with a positive MTC (also referred to as PTC). Moderator Temperature Effects on MTC As illustrated with the explanation of under- and overmoderation, the moderator density change affects the moderator-to-fuel ratio, not the moderator temperature. An increase in moderator temperature results in a decrease in moderator density. Conversely, a decrease in moderator temperature results in an increase in moderator density. As we know, commercial reactors (in USA) use light water as both a coolant and a moderator. Another feature of water is that at higher temperatures, the density change per degree F of water is greater. The figure below shows this relationship: Figure: Water Density Change versus Moderator Temperature A greater density change at higher moderator temperatures means a larger change in the moderator-to-fuel ratio leading to a larger value MTC. The result is larger absolute value of MTC at high temperatures (500 F to 550 F) than at lower temperatures (100 F to 150 F range). 22 Rev 1

27 Note Two points of clarification about the lower absolute value of MTC at lower moderator temperature: 1. The reactor is only made critical at normal operating temperatures (around 550 F), so lower MTC values at lower temperatures are of no concern when critical. 2. A lower MTC at lower temperatures is a good thing in regards to a steam break accident, where a rapid cooldown would cause a large insertion of positive reactivity for a possible reactor restart accident. However, accident analysis considers worst case, which would be a higher MTC. Boron Concentration Effects on MTC The discussion so far has considered the moderator to be pure water. This makes the moderator-to-fuel ratio effect on MTC easier to explain. However, the moderator is not pure water. Commercial PWRs use soluble boron, referred to as boric acid, added to the moderator to provide a variable reactivity poison for control of k excess, maintaining T avg in the program band during power changes, compensating for fission product poisons, and reactivity adjustment to "trim" the control rods fully withdrawn at 100 percent power. Boron has a high thermal neutron absorption cross section, adding negative reactivity to the core much as control rods do - the higher the concentration of boron the more negative reactivity. Boron concentration is decreased (diluted) adding positive reactivity to compensate for the negative reactivity from fuel depletion over the life of a reactor core, as fuel depletes. The presence of boron in the moderator affects the value of the MTC. Higher boron concentrations have a greater the effect on the MTC. The presence of boron in the coolant results in a reduction in the value of the thermal utilization factor (f) since boron is a neutron absorber. Remember the ratio for f: From the formula, as boron absorbs more neutrons, the number of thermal neutrons absorbed in all reactor material increases, causing f to decrease. Therefore, increasing the soluble boron concentration causes f to decrease, which, in turn causes k eff to decrease, adding negative reactivity. When soluble boron is added to the moderator, it becomes an integral part of the moderator and therefore affects the moderator-to-fuel ratio. Consider the figure below that illustrates the response of the thermal utilization factor (f) on moderator/coolant boron concentration. Rev 1 23

28 Figure: Boron Effect on the Thermal Utilization Factor On this family of curves, the area of interest is toward the left side. Notice that as boron concentration is increased, the slope of the curve (change in f) becomes steeper. This means that at high boron concentrations, for a given change in N m /N u (or N mod /N fuel ), the thermal utilization factor will have a greater change in value. When the density of the moderator changes, N m changes, and so does N B (boron concentration). Higher boron concentrations (atoms/cm 3 ) yield a greater change in N B for the same temperature (density) change. The thermal utilization factor (f) and resonance escape probability (ρ) are two factors affected by moderator temperature. They determine both the magnitude of MTC, and whether the reactivity coefficient is positive or negative. Recall that f is the positive factor while ρ is the negative factor to MTC. The figure above shows that with high boron concentrations, thermal utilization becomes a bigger factor. In fact, at very high boron concentrations (possible after refueling), the thermal utilization factor can override the negative effect from resonance escape probability resulting in a positive MTC. Boron has minimal effect on the resonance escape probability since it is predominantly a thermal neutron absorber. Remember the ratio for ρ: The MTC becoming less negative as the boron concentration of the moderator/coolant increases means that the boron concentration must be limited to prevent the MTC from becoming positive during power operations. Some plants are allowed to operate with a positive MTC up to 24 Rev 1

29 some designated power level for a short period, but beyond that, a negative MTC is required for safety considerations. By the time the unit is at full power (or before), sufficient buildup of fission product poisons has occurred, requiring the operators to reduce boron concentration to compensate, and thereby establishing a negative MTC. For example, at the beginning of core life (BOL), when the boron concentration is high, the MTC may be +0.1 x 10-4 Δk/k/ F. At the end of core life (EOL), after significant boron dilution, the MTC is approximately x 10-4 Δk/k/ F. Alternate Explanation Another way to look at this concept is to consider a moderator temperature increase of one degree Fahrenheit (1 F). This temperature increase causes three effects: The boron concentration (atoms/cm 3 ) decreases, resulting in a positive reactivity insertion. Thermal utilization factor increases. Decreased moderator density, fewer water atoms (N m ) causes the thermal utilization factor (f) to increase slightly, causing a positive reactivity insertion. This insertion is smaller than the insertion due to the boron effects (depending on boron concentration). This positive reactivity insertion is a result of fewer water molecules and boron atoms per cubic centimeter (cm 3 ) available for absorption reactions within the reactor core. The resonance escape probability (ρ) decreases due to fewer moderator molecules per cm 3 being present in reactor core. Therefore, neutrons travel further and resonance capture is more likely, resulting in an insertion of negative reactivity. The processes listed above are three competing effects that take place with a moderator temperature increase. For higher boron concentrations, MTC tends to be less negative (or even positive). Conversely, as boron concentration approaches zero, MTC tends to be more negative. Therefore, as previously explained, MTC at the beginning of core life (BOL) can be slightly positive, whereas the MTC at the end of core life (EOL) will be at its most negative value. Core Age Effects on MTC The MTC becomes more negative over core life. The primary reason for this effect is the decrease in RCS boron concentration as discussed previously. Rev 1 25

30 Note Commercial PWRs are also limited on how negative the MTC can become. This restriction is required because of the Main Steam Line Break Accident. During a steam line break accident, the reactor coolant system (RCS) will undergo a rapid cooldown because the steam system begins to act like an infinite heat sink. This rapid cooldown will result in large positive reactivity insertion to the reactor core from the MTC. Some plant accident analyses demonstrate that the reactor could actually be rendered supercritical with all control rods fully inserted. An example of such a limit on the MTC is a value such as -44 pcm/ F (-4.4 x 10-4 Δk/k/ F). Knowledge Check As the reactor coolant boron concentration increases, the moderator temperature coefficient becomes less negative. This is because a 1 F increase in reactor coolant temperature at higher boron concentrations results in a larger increase in the. A. fast fission factor B. thermal utilization factor C. total nonleakage probability D. resonance escape probability ELO 2.4 Void and Pressure Reactivity Coefficients Introduction Void (steam bubbles) and pressure coefficients play a very small role in the reactivity balances for a commercial PWR compared to MTC. Rules of thumb for pressure are 100 psi is equal to 1 F temperature change and at full power voids may occupy about 0.5 percent of the total moderator volume. Any changes in pressure and voiding large enough to make significant reactivity changes in normal operating bands do not occur. The pressure coefficient of reactivity is the result of the effect of pressure on the density of the moderator. The pressure coefficient of reactivity is the change in reactivity per unit change in pressure (Δk/k/psi). This implies that for a given pressure change, a certain amount of water density change occurs, which, causes a change in reactivity (like the moderator temperature effect on density). As pressure increases, density increases, increasing the moderator-to-fuel ratio. In the undermoderated core, the increase in the moderator-to-fuel ratio results in positive reactivity addition. Therefore, the pressure coefficient is a positive reactivity coefficient. 26 Rev 1

31 A 100-psi increase in pressure causes approximately the same reactivity as a one-degree decrease in temperature relating the pressure coefficient to MTC. A typical value for the pressure coefficient of reactivity in a commercial PWR is 1 x 10-6 Δk/k/psi. For PWRs, the overall reactivity effect of the pressure coefficient is a minor factor in normal operation because it is much smaller than the MTC. Void Coefficient The void coefficient quantifies the effect that the formation of steam voids in the moderator has on the MTC. The void coefficient is the change in reactivity per percent change in void volume (Δk/k/percent void). In commercial PWRs, the amount of voids is very small; however, in boiling water reactors (BWR) it is very significant. This discussion is limited to PWRs. Voiding may occur in a PWR when power increases to higher levels. These voids displace moderator from the coolant channels within the core. This reduces the moderator-to-fuel ratio, and in an undermoderated core, results in a negative reactivity addition limiting further power increase. The void coefficient is a negative coefficient. Moderator Density Effects on Void Coefficient Bulk boiling of the moderator/coolant does not occur in a PWR; however, steam bubbles will form in the moderator/coolant around the fuel elements as reactor power increases. The moderator/coolant sweeps these bubbles into the bulk coolant where they collapse. Voids have the effect of reducing the moderator density in the area of the void. The result is similar to an increase in moderator/coolant temperature that lowers moderator density. A decreased density causes a decrease in the resonance escape probability (ρ), an increase in the thermal utilization factor (f), and an overall decrease in k eff. As with MTC, the dominant effect is the decrease in resonance escape probability making the void coefficient negative. An approximate value in a commercial PWR reactor is -1 x 10-3 Δk/k/percent void. Voids occupy about 0.5 percent of the total moderator/coolant volume at full power, so like the pressure coefficient, total reactivity inserted by the void fraction is very small compared to MTC. Example: Compute the approximate negative reactivity due to voids in a pressurized water reactor (PWR) at 100 percent reactor power. Given: Void fraction at 100 percent power = 0.6 percent Rev 1 27

32 Solution: Note Many plants combine and include the pressure and void coefficients into the power coefficient because their reactivity effect is relatively small. Knowledge Check Concerning the reactivity affects from the void and pressure coefficients, which one of the following statements is true? A. The pressure and void coefficient are both negative. B. The pressure and void coefficients are both positive. C. The void coefficient is negative and the pressure coefficient positive. D. The voice coefficient is positive and the pressure coefficient negative. TLO 2 Summary 1. Reactivity coefficients and reactivity Reactivity coefficients are the amount that the reactivity will change for a given change in the parameter (per F, per ppm boron, etc.). Generally, α x symbolizes reactivity coefficients, where x represents some variable reactor parameter that affects reactivity. Reactivity defects ( ρ) are the total reactivity change caused by a variation in a parameter. Reactivity defects are determined by multiplying the total change in the parameter by its average coefficient value. The equation below relates reactivity coefficients to reactivity defects. 2. Moderator temperature coefficient of reactivity. The reactivity change per degree change in moderator temperature is the moderator temperature coefficient (MTC) of reactivity. MTC is primarily a function of the moderator-to-fuel ratio, density of the moderator, and boron concentration. PWRs are designed with an undermoderated moderator-to-fuel ratio that provides a negative moderator temperature coefficient except sometimes early in core life. Negative MTC is more desirable because of its power level regulating effect. 28 Rev 1

33 MTC works by turning power down when a power increase causes moderator temperature to increase. The increase in moderator temperature, adds negative reactivity (MTC) causing reactor power to stop its increase. The MTC in equation form is: ( ) Approximation of the MTC is -1 x 10-4 Δk/k/ F. The reactor designer adjusts the amount of moderator with the fuel in the core (N m /N u ratio) to an optimum value to ensure a negative MTC throughout core life. Changes in the moderator-to-fuel ratio affect the thermal utilization factor (f) and the resonance escape probability (ρ), in turn affecting k eff and reactivity or more precisely the MTC. It is possible to design the moderator-to-fuel ratio to be either undermoderated (too little moderator) or overmoderated (too much moderator). 3. Moderator temperature coefficient variations. An overmoderated condition leads to a positive MTC (undesirable) while an undermoderated condition leads to a negative MTC. Commercial PWRs are designed to operate in an undermoderated condition because of the design requirement to have a negative MTC. In the undermoderated region, a decrease in the moderator-to-fuel ratio results in a decrease in k eff, equivalent to negative reactivity. Relating this to temperature, as temperature is increased, concentration of the moderator (N m ) decreases, causing N m /N u to decrease (move to the left). This is a negative MTC. Thermal utilization factor increases while the resonance escape probability decreases. The balance of these two factors (curves have different slopes) determines the magnitude of the MTC (while undermoderated). In the overmoderated region, a reduction in moderator density (temperature increase) has a greater effect on thermal utilization factor than the resonance escape probability. With f greater than ρ, k eff increases, equivalent to positive reactivity. The density change per degree F of water is greater at higher temperatures. A temperature increase causes three effects on boron in the moderator: The boron concentration (atoms/cm 3 ) decreases, resulting in a positive reactivity insertion. Thermal utilization factor increases. Decreased moderator density, fewer water atoms (N m ) causes the thermal utilization factor (f) to increase slightly, causing a positive reactivity insertion. This insertion is smaller than the insertion due to the boron effects (depending on boron concentration). Rev 1 29

34 This positive reactivity insertion is a result of fewer water molecules and boron atoms per cubic centimeter (cm 3 ) available for absorption reactions within the reactor core. 4. The resonance escape probability (ρ) decreases due to fewer moderator molecules per cm 3 being present in reactor core, neutrons travel further, resonance capture is more likely, resulting in an insertion of negative reactivity. 5. With the MTC becoming less negative as the boron concentration of the moderator increases, the boron concentration must be limited to prevent the MTC from becoming positive during power operations. 6. Some plants may operate with a positive MTC up to some designated power level for a short period, but beyond that, a negative MTC is required for safety considerations. By the time the unit is at full power (or before) sufficient buildup of fission product poisons has occurred, requiring the operators to reduce boron concentration to compensate, and thereby reestablishing a negative MTC MTC becomes more negative as a nuclear reactor core life increases - the primary reason is the decrease in RCS boron concentration. 7. Void and pressure coefficients of reactivity The pressure coefficient of reactivity is the result of the effect of pressure on the density of the moderator. The pressure coefficient of reactivity is the change in reactivity per unit change in pressure (Δk/k/psi). This implies that for a given pressure change, a certain amount of water density change occurs, which like the moderator temperature effects to density, causes a change in reactivity. As pressure increases, density increases, increasing the moderator-tofuel ratio. In the undermoderated core, this results in positive reactivity addition. Therefore, the pressure coefficient is a positive reactivity coefficient. A 100-psi increase in pressure causes approximately the same reactivity as a one-degree decrease in temperature. The pressure coefficient of reactivity has a typical value of 1 x 10 6 Δk/k/psi. The pressure coefficient effect is much smaller than the MTC effect. The void coefficient quantifies the effect that the formation of steam voids in the moderator has on the MTC. The void coefficient is the change in reactivity per percent change in void volume (Δk/k/percent void). In commercial PWRs, the amount of voids is very small. Voiding (steam bubbles) may occur when power increases to higher levels. These voids displace moderator from the coolant channels within the core, reducing the moderator-to-fuel ratio, and in an undermoderated core, results in a negative reactivity addition. An approximate value in a commercial PWR reactor is -1 x 10-3 Δk/k/percent void. At full power, voids occupy about 0.5 percent of the total moderator/coolant volume Void and pressure coefficients total reactivity is very small compared to MTC. 30 Rev 1

35 Now that you have completed this lesson, you should be able to: 1. Explain differences between reactivity coefficients and reactivity defects, and how they are used to balance reactivity parameters. 2. Describe the moderator temperature coefficient of reactivity. 3. Describe how the magnitude of the moderator temperature coefficient varies with changes in the following parameters: a. Overmoderation and undermoderation of the moderator-to-fuel ratio b. Moderator temperature c. Core age d. Boron concentration 4. Describe the void and pressure coefficients of reactivity. TLO 3 Fuel Temperature and Power Coefficients Overview This session discusses the fuel temperature coefficient, otherwise known as Doppler broadening or Doppler and power coefficient/defect. It is important to understand all reactivity coefficients and defects for safe reactor operations. The MTC provides an inherent safety feature for PWRs; the fuel temperature coefficient (FTC) is just as much an inherent safety feature in that it adds negative reactivity on a power/fuel temperature increase and, as an added benefit, it is fast acting. This lesson explains Doppler functions and the Doppler effects on reactor operation. Objectives Upon completion of this lesson, you will be able to do the following: 1. Describe the fuel temperature coefficient of reactivity. 2. Explain resonance absorption, Doppler broadening, and selfshielding. 3. Describe how the magnitude of the fuel temperature coefficient varies with changes in the following parameters: a. Moderator temperature b. Fuel temperature c. Core age 4. Describe the components of the power coefficient of reactivity and the magnitude of their overall effect over core life. 5. Explain how the power defect affects the reactivity balance on reactor power operations. ELO 3.1 Fuel Temperature Reactivity Coefficient Introduction Another temperature coefficient of reactivity, the FTC, has a large effect on reactivity. The FTC is the change in reactivity per degree change in fuel temperature (Δk/k/ F). Usually, the two dominant temperature coefficients in a reactor are the moderator temperature coefficient and the FTC. Rev 1 31

36 This FTC also responds quicker to an increasing power transient than MTC, because reactor power causes an immediate increase in fuel temperature. The moderator lags due to the time for the transfer of heat from the fuel to the moderator. This is also true for decreasing power (fuel temperature decrease). The exception to this is when a change in steam demand initiates the power transient by changing moderator temperature and causing reactivity to be inserted into the core. A negative FTC is an important safety feature inherent to PWRs, similar to the MTC. In the event of a large positive reactivity insertion, because of the delay in the moderator temperature change, MTC cannot slow the reactor power rise for several seconds, whereas the FTC starts adding negative reactivity immediately. Fuel Temperature Reactivity Coefficient Another name applied to the FTC is the Doppler reactivity coefficient, often shortened to Doppler. This coefficient was named after the Doppler Effect or Doppler broadening of the resonance peaks of U-238 and Pu-240. The phenomenon of Doppler broadening occurs when the fuel temperature increases and causes the target nucleus to have more energy. As a result, the relative energy between the target nucleus and the incident neutron changes and the acceptable neutron energy band that the nucleus will absorb will widen. The actual peak for the microscopic cross-section will lower. However, the dominant effect is that the nucleus will absorb a broader band of neutrons (off-peak neutrons). This effect is plays a dominant role in low enriched cores since there is much more U-238 in the core. Figure: Doppler Broadening The broadening of the peaks occurs as fuel temperature increases, making resonance capture more likely. Therefore, the resonance escape probability decreases, causing k eff to decrease due to the addition of negative reactivity. Uranium-238 and plutonium-240 are the two significant nuclides with large resonant peaks. 32 Rev 1

37 Fuel Temperature Coefficient or Doppler Coefficient The FTC is the change in reactivity per unit change in fuel temperature. Where: ( ) D = Doppler coefficient (FTC) (Δk/k/ F) Δρ = change in reactivity associated with change in fuel temperature (Δk/k) ΔT fuel = change in fuel temperature ( F) In low enrichment reactor fuel (commercial reactors), most of the uranium found in the fuel rods is uranium-238 (plutonium-240 builds in over core life). The magnitude of the Doppler coefficient in PWRs is about -1 x 10-5 Δk/k/ F, or -1 pcm/ F. Doppler Defect Although the coefficient is small, the defect can be a very high value because of reactor power level changes from 0 to 100 percent during power operations. The average fuel temperature at 100 percent reactor power is about 2,200 F; however, peak fuel temperature in some fuel rods could be greater than 3,000 F. Because of this, the magnitude of the change in reactivity due to fuel temperature changes is large. The figure below shows an example plot of Doppler defect and rated power: Example Figure: Doppler Defect vs. Rated Reactor Core Power A reactor with an effective multiplication factor of (k eff = 1.009) has a fuel temperature of 100 F. When fuel temperature is raised to 600 F, k eff = What is value of Doppler coefficient? Rev 1 33

38 Solution: First, solve for ρ initial and ρ final. ( ) Then, use the above equation to solve for D using ρ initial and ρ final. ( ) ( ) ( ( ) Doppler Coefficient Mechanism A fuel temperature increase causes higher vibrational frequency of the fuel atoms. This increases neutron absorption by uranium-238 and plutonium- 240 (Doppler). As shown in the previous Doppler Broadening figure, the movement of uranium-238 atoms relative to incident high velocity neutrons results in a broadening and flattening of the resonance absorption peaks; however, the total area under the resonance peak curve will remain essentially the same. The overall effect is that the incident neutrons encounter a higher absorption cross section over a wider range of neutron energies, resulting in more resonance absorptions and a decrease in k eff. Later sections will provide more detail on this. Importance of the Doppler Coefficient The importance of the Doppler coefficient is that fuel temperature immediately increases following an increase in reactor power. Uranium oxide (UO 2 ) (the fuel pellets) is a relatively poor conductor of heat and the cylindrical fuel rods have a small heat transfer surface per unit volume. It requires a relatively long time for transfer of the heat generated at any instant to the moderator/coolant. This time may be 7 to 9 seconds. In the event of a large positive reactivity addition to the reactor, the MTC will be subject to this time delay, and therefore have a delayed effect in countering the insertion of positive reactivity. On the other hand, the Doppler coefficient, because of its direct association with the fuel itself, responds immediately. This is why some refer to Doppler coefficient as the "prompt" coefficient, and MTC as the "delayed" coefficient. With the Doppler coefficient responding first to an accidental, 34 Rev 1

39 large positive reactivity addition, Doppler is of paramount importance in the event a rod ejection accident or other rapid positive reactivity insertion. Knowledge Check If fuel temperature decreases by 50 F, the area under the resonance peak curve will and positive reactivity will be added to the core because. A. decrease; fewer neutrons will be absorbed by uranium- 238 overall B. decrease; fewer 6.7 ev neutrons will be absorbed by uranium-238 at the resonance energy C. remain the same; fewer neutrons will be absorbed by uranium-238 overall D. remain the same; fewer 6.7 ev neutrons will be absorbed by uranium-238 at the resonance energy ELO 3.2 Doppler and Self-shielding Introduction Doppler is generally associated with the physics of sound and light, but it also apples to nuclear physics. The Doppler Effect (or Doppler shift) is the change in frequency of a sound wave for a listener as the source moves. It is heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an observer. Compared to the emitted frequency, the received frequency is higher during the approach, identical at the instant of passing and lower during the recession. We use a source of sound waves moving toward the listener to explain this phenomenon. As the source moves toward the listener, the source emits each successive sound wave peak from a position closer to the listener than the previous sound wave. Therefore, each sound wave takes slightly less time to reach the listener than the previous one, and the time between successive sound wave peaks deceases. This is the increase in sound frequency. The opposite is true when the source of sound is moving away. In nuclear reactor fuel, Doppler Effect explains the probability of resonant absorption as a function of the fuel's temperature. Assume a stationary nucleus will absorb only neutrons of a specific energy E o. If the nucleus is moving away from the neutron, the velocity (and energy) of the neutron must be greater than E o to undergo resonance absorption. If the nucleus is moving toward the neutron, the neutron needs less energy than E o to be absorbed. Raising the nuclei temperature causes more rapid vibration within their lattice structures, in effect broadening the energy range of neutrons for resonance capture, known as Doppler broadening. Rev 1 35

40 Doppler Broadening and Resonance Capture Neutrons give up energy incrementally via collisions with the nuclei of materials present in the reactor; this is the purpose of the moderator. The microscopic cross section for absorption (σ a ) for uranium-238 is 5,500 barns for neutrons at 21 ev. However, the microscopic cross-section for absorption is only 15 to 20 barns for a neutron with an energy level of 20 or 22 ev; either side of 21 ev. These "resonance" peaks, where absorption is most likely to occur, are where the neutron losses occur from resonance capture or resonance absorption. The resonance escape probability is the probability that a neutron will pass through these energy levels without capture. The figure below shows the U-238 resonance capture cross sections as a function of neutron energy for two different fuel temperature conditions, room temperature vs. reactor operating conditions. Figure: Uranium-238 Cross-Section for Absorption Curve The relative motion between the incident neutron and the target nucleus (Doppler Effect) influences the resonance capture cross section for uranium-238. The average kinetic energy of the uranium-238 nucleus increases as the temperature increases. The cross section peak decreases, but the energy spectrum broadens with increasing temperature. Overall, the likelihood of a neutron capture increases. This is the Doppler Effect. The motion (KE) or vibration of the nucleus has a direct impact on its magnitude of capture cross section. 36 Rev 1

41 To demonstrate this Doppler Effect with different neutron and nucleus energies, consider the three neutron reactions depicted in the following figure. Figure: Doppler Effect in Uranium-238 Resonance Capture Suppose an incident neutron having 21 ev of kinetic energy impinges on a target nucleus at room temperature (roughly ev), as shown in a. in the previous figure. The microscopic cross section for absorption for uranium- 238 at 21 ev is 5,500 barns and the neutron is likely to be absorbed. Next, consider a 20 ev neutron interacting on a nucleus that is vibrating toward it with kinetic energy of 1eV, shown in b. in the previous figure. The relative energy between the incident neutron and target uranium-238 nucleus is, once again 21 ev. The effective absorption cross section is about 5,500 barns and the neutron is likely to be absorbed as with the previous example. In the last example, c. above, the incident neutron possesses KE of about 22 ev, and the target uranium-238 nucleus is vibrating away from the neutron with KE of 1 ev. The relative energy between the incident neutron and the target uranium-238 nucleus is, once again 21 ev. The effective absorption Rev 1 37

42 cross section is about 5,500 barns and the neutron is likely to be absorbed as with the previous examples. These examples depict the Doppler Effect. The KE of the fuel atoms increases, resulting in neutrons of both higher and lower KE (than required at room temperature) having an equal probability of resonance absorption by the fuel atoms as fuel temperature increases. The figure below provides another illustration of Doppler Effect. Figure: Resonance Capture in Nucleus Vibrating at 5 ev The figure above illustrates the effect of heat energy applied to a nucleus. Upon adding 5 ev of heat energy to the nucleus, the nucleus vibrates rapidly in all directions. The nucleus still prefers a 21 ev neutron, and has a high cross section only for neutrons of 21 ev. The nucleus now absorbs any neutron within the KE range of 16 ev to 26 ev (+ or - 5 ev), depending upon the neutrons' angle of approach to the nucleus because of the relative motion between the nucleus and the surrounding neutrons. The motion between the neutron and the nucleus must be sufficient for a neutron to "appear" to the nucleus as a 21 ev neutron. Its speed and area of motion due to vibration increases; however, because it is vibrating faster, it now spends less time at any given energy within its KE range if more heat energy is added to the nucleus. The nucleus now has the capability of capturing "off-resonance" neutrons of 16 ev and 26 ev respectively. The probability for capturing a 21 ev "resonance" neutron has decreased, but the probability of capturing neutrons in the 16 ev to 26 ev range has increased. 38 Rev 1

43 The net result of heating nuclear fuel is to "broaden" and flatten the uranium-238 resonance capture cross-section curve. This shift in resonant capture cross section peaks for uranium-238 is Doppler broadening. The effects of Doppler broadening result in a modified capture cross section curve, as shown in a previous figure of the uranium-238 cross-section for absorption curve. The area under both the original and the broadened curve is theoretically the same. Therefore, you might assume that the overall capture of neutrons by uranium-238 would not change significantly. However, research proves that broadening of the uranium-238 capture cross section curve increases the resonant neutron capture in uranium oxide (UO 2 ) fuel pellets. We consider the effects of self-shielding within the fuel pellet to explain this. Self-Shielding The fuel in a commercial nuclear reactor is constructed of ceramic pellets that are housed in a helium gas-filled, Zircaloy tm -clad, cylindrical fuel pin. The surrounding moderator slows down neutrons (thermalizes). Highenergy neutrons pass through the fuel pellets and the surrounding cladding into the moderator. The moderator slows the neutrons down into the epithermal (intermediate) and thermal energy range. A neutron entering the fuel pellet with the exact resonant energy has a very high probability of absorption at low fuel temperatures, most likely in the outer edge of fuel pellet. Epithermal neutrons of other than resonant energies are more likely to pass directly through the pellet without being absorbed. The outer fuel atoms tend to shield the inner fuel atoms from resonant energy neutrons. This is termed self-shielding. Consider two uranium oxide fuel pellets, one at room temperature and another at operating reactor fuel temperature, to further explain selfshielding. Refer to the figure below: Figure: UO 2 Fuel Pellet at Room and Operating Reactor Temperature At room temperature (part a), only resonance neutrons would be captured, as shown by the 21 ev resonance neutron with the UO 2 fuel pellet. Off- Rev 1 39

44 resonance neutrons would pass right through and not be "seen" by the UO 2 fuel pellet. The inner region of the pellet is termed "self-shielded" by the outer periphery because the resonance neutron is captured immediately as it enters the fuel pellet and off-resonance neutrons are not captured. Part b of the previous figure illustrates what happens when the fuel pellet is at an elevated temperature. The uranium-238 nuclei tend to capture both resonance and off-resonance neutrons because of increased vibration due to increased heat energy (Doppler Effect). The central portion of the fuel pellet now tends to capture both off-resonance and resonance neutrons because there is a reduction in fuel pellet self-shielding with the higher temperatures. We must consider two issues to determine the amount of self-shielding that occurs: Physical size of the fuel pellet Design characteristics of the fuel pellet The combination of these two effects determines the overall effect of fuel temperature on resonance capture within a nuclear reactor core. Physical Size of Fuel Pellets The physical size of the fuel pellets and the average distance that a neutron can travel into a pellet prior to resonance absorption determines if a neutron will pass through the pellet without absorption. Recall that the mean free path (Σ) is the average distance that a neutron travels before being absorbed. The equation below gives the mean free path for absorption: Where: Σ a = mean free path (cm) N = atomic density (atoms/cm 3 ) σ a = microscopic cross section for absorption (barns) The atomic density (N) is approximately 2 x atoms/cm 3 for the uranium-238 contained in a fuel pellet. For this discussion, assume that every neutron is absorbed in three (3) mean free paths. If 100 neutrons, all at 21 ev, enter the fuel pellet, then all neutrons are absorbed if the fuel pellet is three mean free paths wide. (At 21 ev, uranium-238 has a resonance peak of 5,500 barns). Recall that 1 barn = cm 2. Therefore: ( )( )( ) 40 Rev 1

45 Since the average fuel pellet is 1.0 cm in diameter, all 100 neutrons at 21 ev entering the fuel pellet will be absorbed (0.009 cm x 3 = cm < 1 cm). For neutrons that are not at an energy level of a resonance peak for uranium-238, the microscopic cross section for absorption is about 15 barns. This makes the mean free path for these neutrons 3.33 cm. ( )( )( ) The fuel pellet would have to be about 10 cm (3 x 3.33 cm) in order for all of these neutrons (not at 21 ev) to be absorbed in the uranium-238, or approximately 4.0 inches in diameter. The uranium-238 in the fuel pellet will absorb very few of the off-resonance neutrons. Assume that 100 neutrons enter the fuel pellet at 22 ev and two of these are absorbed in the pellet. The uranium-238 fuel pellet absorbs 102 of the 200 neutrons (we add the two absorptions to the ev neutrons). Now consider an increase in the fuel temperature. The microscopic crosssection for absorption of neutrons at energy levels equal to uranium-238 resonance peaks decreases, but the absorption cross section for neutrons with energy levels near the resonance peaks increases due to Doppler broadening. This means that for the 1.0 cm fuel pellet there are still 102 neutrons absorbed within the pellet. However, now not all of the neutrons at an energy level corresponding to the resonance peak (21 ev) are absorbed and more of the neutrons not at resonance peak energy are absorbed. For this example, assume that at 600 F fuel temperature, 99 of the resonant energy (21 ev) neutrons are absorbed and 3 off-resonance energy neutrons are absorbed. The total number of neutrons absorbed is the same (102) but the number of resonant and non-resonant energy neutrons absorbed has changed. The microscopic cross section for absorption has decreased for the 21 ev neutrons and increased for the 22 ev neutrons at this higher temperature. Therefore, there is now a slight possibility that some of 21 ev-neutrons will escape the fuel pellet without capture. This decreasing of the microscopic cross section for absorption has the effect of decreasing the self-shielding occurring within the fuel pellet. A 21 ev-neutron is likely to travel farther into the fuel pellet prior to capture, and some may pass completely through the pellet without capture. The off-resonance neutrons that normally would have passed completely through the pellet now have an increased probability of capture by uranium- 238 within the fuel pellet at this higher temperature. At lower temperatures, the average fuel pellet has a diameter smaller than the three mean free paths needed for total neutron absorption and the internal portion of the fuel pin does not see neutron flux from neutrons at resonance peak(s) energy. Rev 1 41

46 If the fuel temperature is increased, the mean free path increases due to decreased microscopic cross section (Doppler broadening) and more of the fuel pellet now experiences resonance neutron flux energy levels. In other words, as fuel temperature increases, self-shielding decreases. If the diameter of the fuel pellet is sufficiently large compared to the mean free path, the effect of self-shielding can be quite pronounced. Not all paths that a neutron can take will lead through the center of the fuel pellet even though the diameter of the fuel pellet may be 1 cm. Not all neutrons entering a fuel pellet have the opportunity to travel 1 cm through the pellet. In fact, the average straight-line distance a neutron travels through a fuel pellet is about cm. Using this information, three mean free paths at cm would equal a distance of cm divided by three or 0.21 cm of travel for one mean free path. Using the mean free path equation, this yields a value of approximately 238 barns as the microscopic cross section for absorption with a 0.21 cm mean free path, as shown below. ( )( ) For a real fuel pellet, any neutron at an energy level equal to a microscopic cross section of greater than 238 barns will appear as a resonant energy neutron and be absorbed in the fuel pellet. Looking at the figure: Uranium-238 Cross Section for Absorption Curve, for the energy levels with cross sections for absorption above 238 barns, if the temperature of the fuel were to increase to 600 F, as in our example, the energy levels for resonant neutron absorption in uranium-238 with cross sections above 238 barns are greatly expanded. Therefore, the Doppler Effect, when combined with the decrease in self-shielding, results in an increased resonance absorption by uranium-238 at higher fuel temperatures. The above examples discuss uranium-238; however, all resonant absorbers found in a nuclear reactor exhibit similar behavior as uranium-238. Fuel Pellet Design Characteristics The characteristics of fuel pellet design are a second issue that affects selfshielding. To understand this effect, we must investigate the temperaturedependent characteristics of the fuel pellets. Manufacturers produce nuclear reactor fuel pellets as ceramic pellets (uranium oxide). Like other ceramic materials, fuel pellets are poor conductors of heat. This results in large temperature gradients from the center to the outer surface of the pellet. This is a major contributor to the reduction in self-shielding as the fuel temperature is increased. 42 Rev 1

47 The figure below shows temperature gradients encountered for fuel pellets located in low and high power regions of the core. Figure: Fuel Pellet Temperature Profile Consider the two gradient curves for high and low temperature conditions as shown in the figure above. The change in temperature across the fuel pellet increases as well as the center temperatures. For fuel pellets in high power regions of the core, the fuel centerline temperatures may be above 3,000 F, while temperatures near the fuel pellet surface are closer to 1,000 F. The centerline temperature may be 1,500 F, whereas the temperature at the surface of the pellet is closer to 700 F for fuel pellets in lower power regions of the core. The next figure shows the effect of the increasing temperature gradient on self-shielding. Figure: Fuel Pellet Shielded Areas An epithermal neutron that is not at resonance energy, as it penetrates deeper into a pellet may appear as a resonance energy neutron in a low power region of the core. The off-resonance energy neutron may pass completely through the pellet and not be captured because the temperature gradient is not as large as that found in a pellet located in a higher power region of the core. However, the same neutron entering a fuel pellet in a high power region of the core would have a higher probability of appearing as a resonance energy neutron upon entering the pellet and as it penetrates deeper into the pellet. The result is that as the fuel temperature increases, the effective resonance capture area for epithermal neutrons also increases. Only a very small Rev 1 43