ACAD BASIC CURRICULUM NUCLEAR SCIENCE CHAPTER 5 NEUTRON LIFE CYCLE 346 RESONANCE LOSSES p 038 THERMAL NEUTRON 2 THERMAL NEUTRON LEAKAGE 52 THERMAL ABSORBED BY NON-FUEL ATOMS L th 07 THERMAL f 965 THERMAL NEUTRON MODERATOR 384 FAST U-235 FUEL 400 FAST BORN η 435 FROM THERMAL FISSION 58 FAST NEUTRON LEAKAGE L f 442 FAST U- 235 238 Pu- 239 400 FAST START CYCLE HERE ε 42 FROM FAST FISSION STUDENT TEXT TM 2003 General Physics Corporation, Elridge, Maryland All rights reserved. No part of this boo may be reproduced in any form or by any means, without permission in writing from General Physics Corporation.
TABLE OF CONTENTS FIGURES AND TABLES... ii OBJECTIVES... iii STEADY STATE NEUTRON BALANCE... Six Factor Formula... 2 Fast Fission Factor - ε... 2 Fast Non-Leaage Probability - L f... 2 Resonance Escape Probability - p... 2 Thermal Non-Leaage Probability - L th... 3 Thermal Utilization Factor - f... 3 Reproduction Factor - η... 4 The Six Factors... 4 REACTIVITY... 7 SUMMARY... 0 PRACTICE EXERCISES... GLOSSARY... 2 EXAMPLE EXERCISE ANSWERS... 4 PRACTICE EXERCISE ANSWERS... 5 2003 GENERAL PHYSICS CORPORATION i NUCLEAR SCIENCE - CHAPTER 5 -
FIGURES AND TABLES Figure 5- Neutron Multiplication Factor... Figure 5-2 Characteristic Resonance Absorption Cross Section... 3 Figure 5-3 Neutron Reproduction Factor, η... 4 Figure 5-4 Neutron Cycle... 4 Table 5- Reactor Reactivity Values... 9 2003 GENERAL PHYSICS CORPORATION ii NUCLEAR SCIENCE - CHAPTER 5 -
OBJECTIVES Upon completion of this chapter, the student will be able to perform the following objectives at a minimum proficiency level of 80%, unless otherwise stated, on an oral or written exam.. DEFINE ective multiplication factor and discuss its relationship to the state of the reactor. 2. DEFINE the following terms with respect to the reactor: a. Neutron generation time b. Critical c. Subcritical d. Supercritical 3. DESCRIBE the neutron life cycle using the following terms: a. Fast fission factor b. Fast non-leaage probability factor c. Resonance escape probability factor d. Thermal non-leaage probability factor e. Thermal utilization factor f. Reproduction factor 4. DEFINE reactivity. 5. STATE the relationship between reactivity and ective multiplication factor. 6. Given specific values of, CALCULATE values of reactivity. 7. Given specific values of reactivity, CALCULATE values of. 8. CONVERT given values of reactivity to other expressions of reactivity. 2003 GENERAL PHYSICS CORPORATION iii NUCLEAR SCIENCE - CHAPTER 5 -
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STEADY STATE NEUTRON BALANCE The neutron, from its birth as a fission neutron to its absorption in the core, undergoes several processes. The neutron life cycle is used to represent these various processes and the ect each has on sustaining a steady state condition. The neutron population in any given volume depends on the processes that add or remove neutrons from the volume. The time dependent behavior of the neutron population in any reactor at power is given by the mathematical expression: Rate of Neutron Production - Rate of Neutron Removal Equation 5- = Rate of Change of Neutron Population When the reactor is in a steady state condition, the rate of neutron production is equal to the rate of neutron removal. Under these conditions, the rate of change of the neutron population is zero, and reactor power will remain constant. Neutrons are primarily produced by fission and are removed by either absorption or leaage from the reactor. Several processes determine a neutron s fate. The neutron life cycle is used to represent these various processes and the ect each has on sustaining a steady state condition. They are discussed in greater detail later in this chapter. For purposes of simplification, the following assumptions can be made regarding the neutron life cycle: All neutrons are born as fast neutrons. Some fast neutrons can be absorbed by fuel and cause fast fission. Some fast neutrons can lea out of the reactor core. Some fast neutrons can be resonantly captured while slowing down. All remaining neutrons become thermalized. Some thermal neutrons can lea out of the core. Some thermal neutrons can be absorbed by non-fuel material. Some thermal neutrons can be absorbed by fuel and not cause fission. All remaining thermal neutrons are absorbed by fuel and cause thermal fission. Neutron generation time is defined as the time from the birth of one generation of neutrons to the time of the birth of the next generation of neutrons. By comparing the number of neutrons produced from fission in one generation to the number of neutrons produced from fission in the next generation, an indication of the rate of change in neutron population is obtained. The ective neutron multiplication factor ( ) is defined as the factor by which the number of neutrons produced from fission in one generation is multiplied to determine the number of neutrons produced from fission in the next generation, as shown in Figure 5-. Neutrons In (Generation #) Effective Neutron Multiplication Factor In the Reactor Neutrons Out (Generation #2) Figure 5- Neutron Multiplication Factor The ective neutron multiplication factor can be mathematically expressed as: # = # of of neutrons in one neutrons produced generation produced by by fission fission in the previous generation Equation 5-2 The ective multiplication factor is the product of several factors that address everything that can happen to a neutron during its lifetime. The values of determine whether the neutron population in the core is increasing, decreasing, or remaining the same. 2003 GENERAL PHYSICS CORPORATION of 6 NUCLEAR SCIENCE - CHAPTER 5 -
If the number of neutrons produced by fission in one generation equals the number of neutrons in the previous generation, =. This indicates a steady state condition and defines an exactly critical reactor. SIX FACTOR FORMULA The six factor formula is used to describe the processes that occur during the neutron life cycle. The starting point in the neutron generation process is taen to be the birth of all the fast neutrons from thermal fission events and represents the numerator in the formula. FAST FISSION FACTOR - ε In light water reactors most fissions are caused by thermal neutrons, however there are an appreciable number of fast neutrons that cause fission in U-235, U-238, and Pu-239. These fissions, nown as fast fissions, produce fast neutrons in addition to the fast neutrons starting the cycle that were produced from thermal fissions. The fast fission factor (ε) accounts for the neutrons produced by fast fission and is given by the equation: fast neutrons produced by ALL fission events ε = fast neutrons produced by THERMAL fission events Equation 5-3 Because the fast fission factor represents a net gain in neutron population, the fast fission factor is slightly greater than one, typically between.03 and.0. FAST NON-LEAKAGE PROBABILITY - L f As the fast neutrons produced by fission begin their process of slowing down, there exists a possibility that a given neutron will be lost from the core due to leaage. The fast non-leaage probability (L f ) represents that fraction of fast neutrons that do not lea out of the core and is given by the equation: L f fast neutrons that start to slow down = fast neutrons produced from ALL fission events Equation 5-4 The fast non-leaage probability represents a net loss in neutron population and has a typical value of 0.96. This means that 96 percent of fast neutrons remain in the core. The ective core size and moderator density impacts the value of the fast non-leaage probability. RESONANCE ESCAPE PROBABILITY - p All nuclei have some probability of absorbing a neutron, as indicated by the microscopic cross section for absorption (σ a ). The microscopic cross section for absorption is not a constant value but is dependent on the energy level of the neutron. In general, the cross section for absorption increases as the neutron energy level decreases. However, certain nuclei (U-238 and Pu-240, in particular) show an extremely high absorption cross section for neutrons at specific energy levels. At certain neutron energy levels the cross section can be as much as,000 times the cross section for a neutron of a slightly higher or lower energy level (Figure 5-2). 2003 GENERAL PHYSICS CORPORATION 2 of 6 NUCLEAR SCIENCE - CHAPTER 5 -
SLOW(THERMAL )(E PITHERMA L) FAST RESON PEA KANCE σa 0. 0 00. 70-8- 50-6 0203 30-40- 20-.0 0405 0607 0.0 NEUT RONENER GY KFN05Sr02_Neutron Life Cycle 08Jun25.doc σa SLOW (THERMAL) 0 2 0 - - 0 8 0-7 0-0 -6 0-5 INTERMEDIATE (EPITHERMAL) 2 0 RESONANCE PEAK 3 0 4 0 0-4 0-3 2 NEUTRON ENERGY 5 0 6 0 FAST 7 0 ev 0-0.. 0 0 MeV Figure 5-2 Characteristic Resonance Absorption Cross Section The resonance escape probability (p) is the fraction of neutrons that are not absorbed while slowing to thermal energy. fast neutrons that become thermal p = fast neutrons that start to slow down Equation 5-5 The resonance escape probability represents a net loss in neutron population and has typical value of approximately 0.75. There are several factors that affect the value of the resonance escape probability, such as the moderator-to-fuel ratio, fuel temperature, core age, and fuel enrichment. THERMAL NON-LEAKAGE PROBABILITY - L th As thermal neutrons begin the diffusion process, there is a possibility that some of the neutrons will be lost to core leaage. The thermal non-leaage probability (L th ) represents the probability that a thermal neutron will not lea out of the core and is given by the following equation: L th = thermal neutrons absorbed in the core fast neutrons that become thermal Equation 5-6 The ective core size and moderator density impacts the thermal non-leaage probability. The ect of these parameters is small because the distance that a neutron travels in the thermal energy range is much less than that of a fast neutron. The thermal non-leaage probability represents a net loss in the neutron population and has a typical value of 0.98. As with the fast non-leaage probability, this leaage term is often neglected due to the relative infinite size of the reactor. A factor so close to.0 does not change the value of very much. THERMAL UTILIZATION FACTOR - f All materials in the reactor absorb neutrons to some extent. By carefully selecting the materials that go into the reactor, control of neutron absorption is accomplished and non-fuel absorption is minimized. The thermal utilization factor (f) is the ratio of the number of thermal neutrons absorbed in the fuel to the number of thermal neutrons absorbed in the core. The term core includes the fuel, moderator, fuel cladding, structural members, control rods, etc. thermal neutrons absorbed f = thermal neutrons absorbed Equation 5-7 in in fuel core The thermal utilization factor represents a net loss in neutron population and has a typical value of 0.95. 2003 GENERAL PHYSICS CORPORATION 3 of 6 NUCLEAR SCIENCE - CHAPTER 5 -
REPRODUCTION FACTOR - η The reproduction factor (η) represents the number of fast neutrons produced from fission compared to the number of thermal neutrons absorbed in the fuel, as shown in Figure 5-3. Thermal Neutrons In Neutron Reproduction Factor η Fast Neutrons Out Figure 5-3 Neutron Reproduction Factor, η In equation form, η becomes: η = fast neutrons produced by thermal fission events thermal neutrons absorbed in the fuel Equation 5-8 The reproduction factor represents a net gain in neutron population and has a typical value of approximately.45. The value varies with fuel enrichment and core age. THE SIX FACTORS 346 RESONANCE LOSSES 58 FAST NEUTRON LEAKAGE p MODERATOR 384 FAST L f 038 THERMAL NEUTRON 442 FAST 2 THERMAL NEUTRON LEAKAGE 52 THERMAL ABSORBED BY NON-FUEL ATOMS L th U- 235 238 Pu- 239 ε FROM FAST FISSION 07 THERMAL 400 FAST 42 f 965 THERMAL NEUTRON U-235 FUEL η 400 FAST BORN Figure 5-4 Neutron Cycle 435 FROM THERMAL FISSION START CYCLE HERE Using Figure 5-4, assume that the neutron life cycle begins with,400 fast neutrons. Recall that these fast neutrons are born from thermal fission of U-235 fuel. Of these neutrons, some will cause fast fission in U-235, U-238, and Pu-239, producing additional fast neutrons. The fast fission factor (ε) is represented by the equation: fast neutrons produced by ALL fission events ε = fast neutrons produced by THERMAL fission events Equation 5-9 The number of fast neutrons has increased from,400 to,442. Calculate the fast fission factor. ε = Example 5- Thus, the fast fission factor (ε) in this example is.03. The,442 fast neutrons exist to continue through the neutron life cycle. Some of the fast neutrons will be lost due to fast leaage. The fast non-leaage probability (L f ) represents the fraction of fast neutrons that do not lea out of the core and is given by: L f = fast neutrons that start to slow down fast neutrons produced from ALL fission events Equation 5-0 The number of fast neutrons decreased from,442 when 58 fast neutrons lea out of the core. Calculate the fast non-leaage factor. L f = Example 5-2 2003 GENERAL PHYSICS CORPORATION 4 of 6 NUCLEAR SCIENCE - CHAPTER 5 -
In this example,,384 fast neutrons remain in the core and begin to slow down. Therefore, the L f is determined to be 0.96. As the remaining neutrons begin to slow down, they will pass though the resonance region and are subject to resonance capture. The resonance escape probability defines the probability that a given neutron will escape capture and is given by: Of the,038 neutrons that are thermalized it is determined that 2 thermal neutrons lea out of the core, that leaves,07 to be absorbed in the core (fuel and non-fuel materials). Calculate the thermal non-leaage probability. L th = p = fast neutrons that become thermal fast neutrons that start to slow down Equation 5- Example 5-4 Therefore, in this example, L th is equal to 0.98. Of the,384 neutrons that begin to thermalize, it is determined that 346 neutrons are absorbed in the resonance pea regions. Calculate the resonance escape probability. p = Example 5-3 Therefore,,038 neutrons reach thermal energy and p = 0.75. A fraction of the thermal neutrons will be lost to thermal leaage. The fraction of neutrons that are not lost is given by the thermal non-leaage probability (L th ) and is given by: thermal neutrons absorbed in the core L th = fast neutrons that become thermal Equation 5-2 The next factor to be determined in this neutron life cycle is the thermal utilization factor (f). It is written mathematically by: f = thermal neutrons absorbed in fuel thermal neutrons absorbed in the core Equation 5-3 Of the,07 thermal neutrons that remain in the core it is determined that 52 thermal neutrons are absorbed into non-fuel atoms in the core, that would leave 965 neutrons to be absorbed into the fuel. Calculate the thermal utilization factor. f = Example 5-5 The thermal utilization factor (f = 0.95) denotes the ratio of the thermal neutron absorbed in the fuel to those absorbed in the core. The last factor to be considered is the reproduction factor (η). The reproduction factor is given by the equation: η = fast neutrons produced by thermal fission events thermal neutrons absorbed in the fuel Equation 5-4 2003 GENERAL PHYSICS CORPORATION 5 of 6 NUCLEAR SCIENCE - CHAPTER 5 -
There are 965 thermal neutrons available for absorption into the U-235 fuel. As a result of these absorptions, fast neutrons are born from fission. The fission process produces,400 fast neutrons. Calculate the reproduction factor. η = Using the six factors as determined from Figure 5-4, becomes:,442,384,038,07 965,400 =,400,442,384,038,07 965 =.03 0.96 0.75 0.98 0.95.45 =.0 (The reactor is critical.) Example 5-6 In this example, the neutron reproduction factor (η) is equal to.45. Note that the number of neutrons produced by fission in this generation equals the number of neutrons produced in the previous generation. By definition, is equal to one, and the reactor is exactly critical. The ective multiplication factor,, is equal to the product of the six factors and is independent of neutron sources other than fission. Where: = εl f p L th f η Example 5-7 The ective multiplication factor ( ) is essentially a measure of the probability that one fission event will cause another fission. As illustrated by the six factor formula, core size and materials affect this probability. However, it is not affected by the introduction of non-fission neutrons. = ε L f p L th = f η ective multiplication factor = fast fission factor = fast non-leaage probability = resonance escape probability thermal non-leaage probability = thermal utilization factor = reproduction factor Equation 5-5 2003 GENERAL PHYSICS CORPORATION 6 of 6 NUCLEAR SCIENCE - CHAPTER 5 -
REACTIVITY Reactivity is the measure of the departure of a reactor from criticality. Reactivity is defined as the fractional change in fission neutron population per generation and is indicated by the Gree letter rho (ρ). The fractional change in neutron population per generation (reactivity) can be shown by the equation given below. Where: = ρ ρ ective multiplication factor = reactivity ( /) Equation 5-6 Calculate the reactivity level of a core with a of 0.985. ρ = 0.985 ρ = 0.985 ρ = 0.052 / Example 5-8 The following notational changes are used to simplify the discussion of reactivity: Where: ρ = = ρ = = = ective multiplication factor = reactivity ( /) Equation 5-7 Besides the / unit for reactivity, the fractional change in neutron population may also be expressed in terms of % /. The % / unit may be obtained as follows: ρ 00% = ρ Equation 5-8 (% / ) Reactivity is also expressed by the term pcm. Where: pcm pcm = 0 5 / % / =,000 pcm = percent milli rho (ρ). Equation 5-9 Core reactivity is -0.025 /. Calculate the value of reactivity in pcm. ( 0.025 / ) 00% = 2.50% / 000 pcm 2.50% / = 2500 pcm % Example 5-9 2003 GENERAL PHYSICS CORPORATION 7 of 6 NUCLEAR SCIENCE - CHAPTER 5 -
Given that the reactivity level of a core is 0.052 /. Calculate the core reactivity value in % /. ρ 00% = ρ (% / ) ( 0.052 / ) 00% =.52 % / Example 5-0 PCM is the acronym for. A control rod withdrawal results in the of a reactor changing from 0.97 to 0.975. Calculate how much reactivity was added to the core by the control rod withdrawal. ρ 0.975 ρ 2 = = 0.0256 / 0.975 0.97 ρ = = 0.0309 / 0.97 ρ = ρ2 ρ Example 5- Since is a dimensionless quantity, reactivity ( /) is also dimensionless. It is convenient, however, to tal about reactivity in units of / or % /. or ρ = 0.0256 / ( 0.0309 / ) ρ = 0.0053 / ρ = 0.53% / ρ =530pcm Example 5-2 For values of very close to, ρ. If the reactivity of the reactor is nown, then can be determined by: Where: = ρ = ρ ective multiplication factor = reactivity ( /) Equation 5-20 2003 GENERAL PHYSICS CORPORATION 8 of 6 NUCLEAR SCIENCE - CHAPTER 5 -
A shutdown reactor has a core reactivity of 0.0028 /. Calculate the core. = ρ = ( 0.0028) = 0.9972 Example 5-3 A shutdown reactor has a core reactivity of 0.0028 /. A control rod movement inserts a negative 940 pcm. Calculate the final core reactivity. Example 5-4 Reactivity is a convenient term to use when discussing deviations from criticality. For any power, if the reactor is critical ( =, ρ=0), the reactivity associated with the reactor is zero. For a supercritical reactor ( > ), reactivity is a positive value, and for a subcritical reactor ( < ), reactivity is a negative value. The and reactivity equation can be rearranged to solve for reactivity: = ρ = ρ Equation 5-2 Table 5- Reactor Reactivity Values Reactor Status Critical 0 Supercritical > Positive Subcritical < Negative If is equal to, substituting into the equation we find that reactivity is equal to zero. ρ = = = 0 ρ = 0 If is greater than, substituting into the equation we find that reactivity is a positive value. ρ = ρ = = 0.000999.00 ρ is positive If is less than, substituting into the equation we find that reactivity is a negative value. ρ = = = 0.00 0.999 ρ is negative Example 5-5 2003 GENERAL PHYSICS CORPORATION 9 of 6 NUCLEAR SCIENCE - CHAPTER 5 -
SUMMARY The ective neutron multiplication factor ( ) is defined as the factor by which the number of neutrons produced from fission in one generation is multiplied to determine the number of neutrons produced from fission in the next generation. The ective neutron multiplication factor can be mathematically expressed as: # = # of neutrons produced by fission in one generation of neutrons produced by fission in the previous generation Critical is the condition where the neutron chain reaction is self-sustaining and the neutron population is neither increasing nor decreasing. Subcritical is the condition in which the neutron population is decreasing each generation. Supercritical is the condition in which the neutron population is increasing each generation. The number of neutrons present at any point in the neutron life cycle can be calculated as the product of the number of neutrons present at the start of the generation and all the factors preceding that point in the life cycle. The enrichment of uranium-235, the temperature of the fuel, and the temperature of the moderator affect the resonance escape probability. The thermal utilization factor is affected by the enrichment of uranium-235, the amount of neutron poisons, and the moderator-to-fuel ratio. The reproduction factor is affected by the enrichment of uranium-235. Reactivity is a measure of the departure from critical. If: is equal to, ρ = 0 is >, ρ = positive is <, ρ = negative 2003 GENERAL PHYSICS CORPORATION 0 of 6 NUCLEAR SCIENCE - CHAPTER 5 -
PRACTICE EXERCISES. Define. 2. The reactor is critical if neutrons produced by fission in one generation are (equal to/greater than) neutrons produced by fission the previous generation. 3. A control rod withdrawal results in the of a reactor changing from 0.975 to 0.980. Calculate how much reactivity was added to the core by the control rod withdrawal. 3. The fast fission factor (ε) will always be less than/greater than one. 4. (TRUE or FALSE) The Thermal Utilization factor can vary from 0.9 to. in a commercial nuclear reactor. 5. Define Reactivity. 6. The number of fast neutrons has increased from,500 to,560 due to fast fission. Calculate the fast fission factor. 7. The number of fast neutrons decreased from,560 when 47 fast neutrons lea out of the core. Calculate the fast non-leaage factor. 8. Of the,53 neutrons that begin to thermalize, it is determined that 332 neutrons are absorbed in the resonance pea regions. Calculate the resonance escape probability. 9. Of the,8 neutrons that are thermalized it is determined that 30 thermal neutrons lea out of the core. Calculate the thermal non-leaage probability. 0. Of the,5 thermal neutrons that remain in the core it is determined that 5 thermal neutrons are absorbed into non-fuel atoms in the core. Calculate the thermal utilization factor.. The absorption of,00 thermal neutrons in U-235 results in the production of 500 fast neutrons. Calculate the reproduction factor. 2. Calculate the reactivity level of a core with a of 0.987. 2003 GENERAL PHYSICS CORPORATION of 6 NUCLEAR SCIENCE - CHAPTER 5 -
GLOSSARY Critical Effective Multiplication Factor ( ) Fast Fission Factor (ε) Fast Non-Leaage Probability (L f ) Neutron Generation Time Reactivity (ρ) Reproduction Factor (η) Resonance Escape Probability (p) Six Factor Formula Subcritical Supercritical Thermal Non-Leaage Factor (L th ) The condition of the reactor where the number of neutrons produced by fission in one generation equals the number of neutrons produced by fission in the previous generation ( = ) (ρ = 0). The factor by which the number of neutrons produced by fission in one generation must be multiplied to determine the number of neutrons produced by fission in the next generation. The ratio of fast neutrons produced from all fission events divided by fast neutrons produced by thermal fission events. The ratio of the number of fast neutrons that start to slow down divided by the number of fast neutrons produced from all fissions. The time from the birth of one generation of neutrons to the time of the birth of the next generation of neutrons. The fractional change in fission neutron population per generation, or the measure of the departure of a reactor from criticality. Reactivity is zero when the reactor is exactly critical. If positive reactivity is added, reactor power will increase. If negative reactivity is added, reactor power will decrease. The ratio of fast neutrons produced by thermal fission events divided by the number of thermal neutrons absorbed in the fuel. The ratio of fast neutrons that become thermal divided by the number of fast neutrons that start to slow down. Used to describe the processes that occur during the neutron life cycle. The condition in which the number of neutrons produced by fission in one generation is less than the number of neutrons produced by fission in the previous generation ( < ) (negative ρ). The condition in which the number of neutrons produced by fission in one generation is greater than the number of neutrons produced by fission in the previous generation ( > ) (positive ρ). The ratio of the number of thermal neutrons absorbed in the core divided by the number of fast neutrons that become thermal. 2003 GENERAL PHYSICS CORPORATION 2 of 6 NUCLEAR SCIENCE - CHAPTER 5 -
GLOSSARY Thermal Utilization Factor (f) The ratio of the number of thermal neutrons absorbed in fuel divided by the number of thermal neutrons absorbed in the core. 2003 GENERAL PHYSICS CORPORATION 3 of 6 NUCLEAR SCIENCE - CHAPTER 5 -
r EXAMPLE EXERCISE ANSWERS The number of fast neutrons has increased from,400 to,442. Calculate the fast fission factor. ε =,442,400 =.03 Example 5- The number of fast neutrons decreased from 442 when 58 fast neutrons lea out of the core. Calculate the fast non-leaage factor.,384 L f = = 0. 96,442 Example 5-2 Of the 384 neutrons that begin to thermalize, it is determined that 346 neutrons are absorbed in the resonance pea regions. Calculate the resonance escape probability.,038 p = =,384 0.75 Example 5-3 Of the 038 neutrons that are thermalized it is determined that 2 thermal neutrons lea out of the core, that leaves 07 to be absorbed in the core (fuel and non-fuel materials). Calculate the thermal non-leaage probability.,07 L th = = 0. 98,038 Example 5-4 Of the,07 thermal neutrons that remain in the core it is determined that 52 thermal neutrons are absorbed into non-fuel atoms in the core, that would leave 965 neutrons to be absorbed into the fuel. Calculate the thermal utilization factor. 965 f = =,07 0.95 Example 5-5 There are 965 thermal neutrons available for absorption into the U-235 fuel. As a result of these absorptions, fast neutrons are born from fission. The fission process produces,400 fast neutrons. Calculate the reproduction factor. η =,400 965 =.45 Example 5-6 PCM is the acronym for. (percent milli rho (P)) Example 5-2003 GENERAL PHYSICS CORPORATION 4 of 6 NUCLEAR SCIENCE - CHAPTER 5 -
r A shutdown reactor has a core reactivity of -0.0028 /. A control rod movement inserts a negative -940 pcm. Calculate the final core reactivity. -0.0028 / + (-0.00940 /) = -0.0220 / Example 5-4 If is equal to, substituting into the equation we find that reactivity is equal to zero. ρ = = = 0 ρ = 0 If greater than, substituting into the equation we find that reactivity is a positive value. ρ = = = 0.000999.00 ρ is positive If less than, substituting into the equation we find that reactivity is a negative value. ρ = = = 0.00 0.999 ρ is negative Example 5-5 PRACTICE EXERCISE ANSWERS. Define. If the number of neutrons produced by fission in one generation equals the number of neutrons in the previous generation, =. This indicates a steady state condition and defines an exactly critical reactor. 2. The reactor is critical if neutrons produced by fission in one generation are (equal to/greater than) neutrons produced by fission the previous generation. 3. The fast fission factor (ε) will always be less than/greater than one. 4. (TRUE or FALSE) The Thermal Utilization factor can vary from 0.9 to. in a commercial nuclear reactor. 5. Define Reactivity. The measure of the departure of a reactor from criticality or the fractional change in neutron population from one generation to another. 6. The number of fast neutrons has increased from,500 to,560 due to fast fission. Calculate the fast fission factor. ε =,560,500 =.04 7. The number of fast neutrons decreased from,560 when 47 fast neutrons lea out of the core. Calculate the fast non-leaage factor.,53 L f = = 0. 97,560 8. Of the,53 neutrons that begin to thermalize, it is determined that 332 neutrons are absorbed in the resonance pea regions. Calculate the resonance escape probability. 2003 GENERAL PHYSICS CORPORATION 5 of 6 NUCLEAR SCIENCE - CHAPTER 5 -
r,8 p = =,53 0.78 9. Of the,8 neutrons that are thermalized it is determined that 30 thermal neutrons lea out of the core. Calculate the thermal non-leaage probability.,5 L th = = 0. 975,8 0. Of the,5 thermal neutrons that remain in the core it is determined that 5 thermal neutrons are absorbed into non-fuel atoms in the core. Calculate the thermal utilization factor.,00 f = =,5 0.956. The absorption of,00 thermal neutrons in U-235 results in the production of,500 fast neutrons. Calculate the reproduction factor. η =,500,00 =.36 2. Calculate the reactivity level of a core with a of 0.987. ρ = 0.987 ρ = 0.987 ρ = 0.032 / 3. A control rod withdrawal results in the of a reactor changing from 0.975 to 0.980. Calculate how much reactivity was added to the core by the control rod withdrawal. ρ 0.98 ρ 2 = = 0.0204 / 0.98 0.975 ρ = = 0.0256 / 0.975 or ρ = ρ2 ρ ρ = 0.0204 / ( 0.0256 / ) ρ = 0.0052 / ρ = 0.52% / ρ =520pcm 2003 GENERAL PHYSICS CORPORATION 6 of 6 NUCLEAR SCIENCE - CHAPTER 5 -