REACTION RATES It is possible to determine the rate at which a nuclear reaction will take place based on the neutron lux, cross section or the interaction, and atom density o the target. This relationship illustrates how a change in one o these items aects the reaction rate. EO 2.10 EO 2.11 Given the neutron lux and macroscopic cross section, CALCULATE the reaction rate. DESCRIBE the relationship between neutron lux and reactor power. Reaction Rates I the total path length o all the neutrons in a cubic centimeter in a second is known, (neutron lux (1)), and i the probability o having an interaction per centimeter path length is also known (macroscopic cross section (*)), multiply them together to get the number o interactions taking place in that cubic centimeter in one second. This value is known as the reaction rate and is denoted by the symbol R. The reaction rate can be calculated by the equation shown below. R = 1 * (2-6) R = reaction rate (reactions/sec) 1 = 2 neutron lux (neutrons/cm -sec) * = -1 macroscopic cross section (cm ) Substituting the act that * = N ) into Equation (2-6) yields the equation below. 1 ) ) = microscopic cross section (cm 2 ) N = atom density (atoms/cm 3 ) NP-02 Page 18 Rev. 0
Reactor Theory (Neutron Characteristics) DOE-HDBK-1019/1-93 REACTION RATES The reaction rate calculated will depend on which macroscopic cross section is used in the calculation. Normally, the reaction rate o greatest interest is the ission reaction rate. Example: Solution: I a one cubic centimeter section o a reactor has a macroscopic ission cross section o -1 13 2 0.1 cm, and i the thermal neutron lux is 10 neutrons/cm -sec, what is the ission rate in that cubic centimeter? R 1 x 10 13 neutrons cm 2 sec 1 x 10 12 issions 0.1 cm 1 In addition to using Equation (2-6) to determine the reaction rate based on the physical properties o material, it is also possible to algebraically manipulate the equation to determine physical properties i the reaction rate is known. Example: Solution: 13 2 20 A reactor operating at a lux level o 3 x 10 neutrons/cm -sec contains 10 atoms o 3 12 3 uranium-235 per cm. The reaction rate is 1.29 x 10 ission/cm. Calculate and. Step 1: The macroscopic cross section can be determined by solving Equation (2-6) or and substituting the appropriate values. R R 1.29 x 10 12 issions 3 x 10 13 neutrons cm 2 sec 0.043 cm 1 Rev. 0 Page 19 NP-02
Step 2: To ind the microscopic cross section, replace with (N x ) and solve or. R N N R 1.29 x 10 12 issions 1 x 10 20 atoms 3 x 10 13 neutrons cm 3 cm 2 sec 4.3 x 10 22 cm 2 1 barn 1 x 10 24 cm 2 430 barns Reactor Power Calculation Multiplying the reaction rate per unit volume by the total volume o the core results in the total number o reactions occurring in the core per unit time. I the amount o energy involved in each reaction were known, it would be possible to determine the rate o energy release (power) due to a certain reaction. In a reactor where the average energy per ission is 200 MeV, it is possible to determine the number o issions per second that are necessary to produce one watt o power using the ollowing conversion actors. 1 ission = 200 MeV 1 MeV = -6 1.602 x 10 ergs 1 erg = -7 1 x 10 watt-sec 1 watt 1 erg 1 x 10 7 watt sec 1 MeV 1.602 x 10 6 erg 1 ission 200 MeV 3.12 x 10 10 issions second 10 This is equivalent to stating that 3.12 x 10 issions release 1 watt-second o energy. NP-02 Page 20 Rev. 0
Reactor Theory (Neutron Characteristics) DOE-HDBK-1019/1-93 REACTION RATES The power released in a reactor can be calculated based on Equation (2-6). Multiplying the reaction rate by the volume o the reactor results in the total ission rate or the entire reactor. Dividing by the number o issions per watt-sec results in the power released by ission in the reactor in units o watts. This relationship is shown mathematically in Equation (2-7) below. P th V 3.12 x 10 10 issions watt sec (2-7) P = power (watts) 2 th = thermal neutron lux (neutrons/cm -sec) -1 = macroscopic cross section or ission (cm ) V = volume o core (cm 3 ) Relationship Between Neutron Flux and Reactor Power In an operating reactor the volume o the reactor is constant. Over a relatively short period o time (days or weeks), the number density o the uel atoms is also relatively constant. Since the atom density and microscopic cross section are constant, the macroscopic cross section must also be constant. Examining Equation (2-7), it is apparent that i the reactor volume and macroscopic cross section are constant, then the reactor power and the neutron lux are directly proportional. This is true or day-to-day operation. The neutron lux or a given power level will increase very slowly over a period o months due to the burnup o the uel and resulting decrease in atom density and macroscopic cross section. Rev. 0 Page 21 NP-02
Summary The important inormation in this chapter is summarized below. Reaction Rates Summary The reaction rate is the number o interactions o a particular type occurring in a cubic centimeter o material in a second. The reaction rate can be calculated by the equation below. R = 1 * Over a period o several days, while the atom density o the uel can be considered constant, the neutron lux is directly proportional to reactor power. NP-02 Page 22 Rev. 0