Homework 15 Solutions Due Monday 3 1 A Stirling cycle engine using a radioactive isotope for space power applications operates at a hot end temperature of 650 C and rejects heat through a radiator to the vacuum of space with a cold end temperature at 120 C Calculate its ideal Stirling cycle efficiency Ideal Sterling cycle efficiencies can be estimated by Carnot cycle efficiencies Sterling Carnot 1 Tr 120 273 57 42 1 Ta 650 273 Homework 16 Solutions Due Wednesday 3 3 Construct a table comparing the PWR and the BWR designs in terms of 1 Their Engineered Safety features ESFs 2 Their technical specification and operational characteristics Engineered Safety Features Pressurized Water Reactor 1 Control rods 2 Containment vessel with steam suppression spray 3 Accumulator tanks with coolant under nitrogen pressure 4 Residual heat removal system 5 High Pressure Coolant Injection system 6 Low Pressure Coolant Injection system 7 Boron injection tank 8 Extra coolant in refueling storage tank Boiling Water Reactor 1 Control rods 2 Containment vessel with steam suppression spray 3 Pressure suppression pool to quench steam 4 Residual heat removal system 5 High Pressure Coolant Injection system 6 Low Pressure Coolant Injection system 7 Boron injection tank 8 Extra coolant in refueling storage tank 9 An internal core spray system The technical specifications for PWRs are given in Table 1 on page 6 of Chapter IV 2 Pressurized Water Reactors The same characteristics for a BWR are given in Table 1 on page 6 of Chapter IV 3 Boiling Water Reactors Homework 17 Solutions Due Wednesday 3 10 An executive at an electrical utility company needs to order natural uranium fuel from a mine The utility operates a single 1 000 MWe power plant of the CANDU type using natural uranium and operating at an overall thermal efficiency of 33 33 percent What is the yearly amount in metric tons of a U235 burned up by the reactor b U235 consumed by the reactor c Natural uranium metal that the executive has to contract with the mine per year as feed to his nuclear unit Since the burnup and consumption rate equations deal with thermal power we need to convert the power output of the reactor using the given thermal efficiency Pth Pe 1000MWe 3000 3MWth MWe th 3333 MWth a Annual burnup rate 1 1 BR 1 112 P dayg 1 112 MWth 3000 3MW th 365 dayyr MWth BR 1 22 10 6 yrg BR 1 22 ton yr b Annual consumption rate 1 1 CR 1 299 MWth P dayg 1 112 MWth 3000 3MW th 365 dayyr CR 1 42 10 6 yrg CR 1 42 ton yr c Amount of natural uranium required annually Since a CANDU plant uses natural uranium we do not need to factor mass balances for an enrichment plant As a result the amount of uranium needed from a mine is simply the amount of uranium needed at the power plant Mp CR 1 42 ton yr 198 58 ton yr x p 0 72 Homework 18 Solutions Due Friday 3 12 An executive at another electrical utility company needs to order uranium fuel from a mine This utility operates a single 1 000 MWe PWR power plant operating at an overall thermal efficiency of 33 33 percent The fuel needs to be enriched to the 5 w o in U235 Consider that the enrichment plant generates tailings at the 0 2 w o in U235 level Calculate the yearly amount of natural uranium metal that the executive has to contract with the mine as feed to his nuclear unit Compare the natural uranium fuel needs in the case of the PWR design to the CANDU design First we need to convert the power output of the plant to thermal power Pth Pe 1000MWe 3000 3MWth MWe th 3333 MWth Since the power output of this plant is the same as the CANDU plant we looked at previously the burnup and consumption rates should be the same 1 1 BR 1 112 MWth P dayg 1 112 MWth 3000 3MW th 365 dayyr BR 1 22 10 6 yrg BR 1 22 ton yr 1 1 CR 1 299 MWth P dayg 1 299 MWth 3000 3MW th 365 dayyr CR 1 42 10 6 yrg CR 1 42 ton yr The difference comes when we look at the mass of uranium needed for the power plant since the enrichment for this PWR plant is different However since this plant requires enriched uranium we need to take that mass balance into account as well Mf x x p xt f xt Mp x x p x t CR f xt x p 5 0 2 1 42 ton yr 0 72 0 2 5 M f 262 15 ton yr While the mass of uranium 235 Mp required annually by a PWR is considerably less when ton compared to a CANDU reactor 28 4 ton yr versus 198 58 yr the required mass of uranium contracted from the mine Mf is larger for a PWR as opposed to a CANDU Homework 19 Solutions Due Wednesday 3 17 For 2 200 m sec or thermal neutrons calculate the following quantities 1 Uranium a Number Density g g M 0 0055 234 0 72 235 99 27 238 237 967 mol 238 mol g 23 nuclei Av 18 9 cm 3 6 02 10 mol N 4 78 10 22 nuclei g cm 3 M 238 mol b Total Macroscopic cross section N N 234 234 N 235 235 N 238 238 4 78 10 nuc 4 78 10 22 cm 3 22 nuc cm 3 0 0055 119 2 0 72 698 2 99 27 12 09 10 17 04 10 b nuc b nuc b nuc 24 cm 2 b 0 814 cm1 c Total mean free path 1 1 1 23cm 0 814 cm1 2 Beryllium a Number Density g 23 nuc Av 1 848 cm 3 6 02 10 mol nuc N 1 24 10 23 cm 3 g M 9 mol b Total Macroscopic cross section 24 nuc b N 1 24 10 23 cm 3 6 159 nuc 10 cm 2 b 0 761 1 cm c Total mean free path 1 1 1 31cm 0 761 cm1 3 Graphite a Number Density g g M 98 89 12 1 11 13 12 01 mol 12 mol 2 03 cmg 3 6 02 10 23 nuclei Av mol N 1 02 10 23 nuclei g cm 3 M 12 mol b Total Macroscopic cross section C13 cross sections are negligible at thermal E 24 cm 2 nuc b N 1 02 10 23 cm 0 484 cm1 3 4 75 nuc 10 b c Total mean free path 1 1 2 07cm 0 484 cm1 b nuc 24 cm 2 b Homework 20 Solutions Due Friday 4 2 Prove that the divergence of the gradient leads to the Laplacian operator in the leakage term of the neutron diffusion equation for a constant diffusion coefficient D D D 2 Since it is stated that the diffusion coefficient is constant it can be pulled …
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