Reactor Theory (Nuclear Parameters)DOE-HDBK-1019/2-93NEUTRON LIFE CYCLEAn equation can be developed for the thermal utilization factor in terms of reaction rates asfollows.f rate of absorption of thermal neutrons by the fuelrate of absorption of thermal neutrons by all reactor materialsf SUaf^{U}V^{U}SUaf^{U}V^{U} Smaf^{m}V^{m} Spaf^{p}V^{p}The superscripts U, m, and p refer to uranium, moderator, and poison, respectively. In aheterogeneous reactor, the flux will be different in the fuel region than in the moderator regiondue to the high absorption rate by the fuel. Also, the volumes of fuel, moderator, and poisonswill be different. Although not shown in the above equation, other non-fuel materials, such ascore construction materials, may absorb neutrons in a heterogeneous reactor. These othermaterials are often lumped together with the superscript designation OS, for "other stuff." Tobe completely accurate, the above equation for the thermal utilization factor should include allneutron-absorbing reactor materials when dealing with heterogeneous reactors. However, for thepurposes of this text, the above equation is satisfactory.In a homogeneous reactor the neutron flux seen by the fuel, moderator, and poisons will be thesame. Also, since they are spread throughout the reactor, they all occupy the same volume. Thisallows the previous equation to be rewritten as shown below.(3-1)f SUaSUaSmaSpaEquation (3-1) gives an approximation for a heterogeneous reactor if the fuel and moderator arecomposed of small elements distributed uniformly throughout the reactor.Since absorption cross sections vary with temperature, it would appear that the thermalutilization factor would vary with a temperature change. But, substitution of the temperaturecorrection formulas (see Module 2) in the above equation will reveal that all terms change bythe same amount, and the ratio remains the same. In heterogeneous water-moderated reactors,there is another important factor. When the temperature rises, the water moderator expands, anda significant amount of it will be forced out of the reactor core. This means that N^{m}, the numberof moderator atoms per cm^{3}, will be reduced, making it less likely for a neutron to be absorbedby a moderator atom. This reduction in N^{m} results in an increase in thermal utilization asmoderator temperature increases because a neutron now has a better chance of hitting a fuel atom.Because of this effect, the temperature coefficient for the thermal utilization factor is positive.The amount of enrichment of uranium-235 and the poison concentration will affect the thermalutilization factor in a similar manner as can be seen from the equation above.Rev. 0NP-03Page 5