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. € ηSterling=ηCarnot= 1−TrTa= 1−120 + 273( )650 + 273( )= 57.42%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 Boiling Water Reactor 1. Control rods 1. Control rods 2. Containment vessel with steam suppression spray 2. Containment vessel with steam suppression spray 3. Accumulator tanks with coolant under nitrogen pressure 3. Pressure suppression pool to quench steam 4. Residual heat removal system 4. Residual heat removal system 5. High Pressure Coolant Injection system 5. High Pressure Coolant Injection system 6. Low Pressure Coolant Injection system 6. Low Pressure Coolant Injection system 7. Boron injection tank 7. Boron injection tank 8. Extra coolant in refueling storage 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=Peηth=1000MWe.3333MWeMWth= 3000.3MWth a. Annual burnup rate: € BR = 1.1121MWth( )Pgday= 1.1121MWth( )3000.3MWth( )365dayyr( )BR = 1.22 ×106gyrBR = 1.22tonyr b. Annual consumption rate: € CR = 1.2991MWth( )Pgday= 1.1121MWth( )3000.3MWth( )365dayyr( )CR = 1.42 ×106gyrCR = 1.42tonyr 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=CRxp=1.42tonyr0.72%= 198.58tonyrHomework 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=Peηth=1000MWe.3333MWeMWth= 3000.3MWth 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. € BR = 1.1121MWth( )Pgday= 1.1121MWth( )3000.3MWth( )365dayyr( )BR = 1.22 ×106gyrBR = 1.22tonyr € CR = 1.2991MWth( )Pgday= 1.2991MWth( )3000.3MWth( )365dayyr( )CR = 1.42 ×106gyrCR = 1.42tonyr 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=xp− xt( )xf− xt( )Mp=xp− xt( )xf− xt( )CRxp=5% − 0.2%( )0.72% − 0.2%( )1.42tonyr5%Mf= 262.15tonyr While the mass of uranium-235 (Mp) required annually by a PWR is considerably less when compared to a CANDU reactor (€ 28.4tonyr versus € 198.58tonyr), 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 € M = 0.0055%( )234( )+ 0.72%( )235( )+ 99.27%( )238( )= 237.967gmol≈ 238gmol € N =ρAvM=18.9gcm3( )6.02 ×1023nucleimol( )238gmol= 4.78 ×1022nucleicm3 b. Total Macroscopic cross section € Σ = Nσ= N234σ234+ N235σ235+ N238σ238Σ = 4.78 ×1022nuccm3( )0.0055%( )119.2bnuc( )+ 0.72%( )698.2bnuc( )+ 99.27%( )12.09bnuc( )[ ]10−24cm2b( )Σ = 4.78 ×1022nuccm3( )17.04bnuc( )10−24cm2b( )Σ = 0.8141cm c. Total mean free path € λ=1Σ=10.8141cm= 1.23cm 2. Beryllium a. Number Density € N =ρAvM=1.848gcm3( )6.02 ×1023nucmol( )9gmol= 1.24 ×1023nuccm3 b.Total Macroscopic cross section € Σ = Nσ= 1.24 ×1023nuccm3( )6.159bnuc( )10−24cm2b( )= 0.7611cm c. Total mean free path € λ=1Σ=10.7611cm= 1.31cm 3. Graphite a. Number Density € M = 98.89%( )12( )+ 1.11%( )13( )= 12.01gmol≈ 12gmol € N =ρAvM=2.03gcm3( )6.02 ×1023nucleimol( )12gmol= 1.02 ×1023nucleicm3 b. Total Macroscopic cross section (C13 cross sections are negligible at thermal E) € Σ = Nσ= 1.02 ×1023nuccm3( )4.75bnuc( )10−24cm2b( )= 0.4841cm c. Total mean free path € λ=1Σ=10.4841cm= 2.07cmHomework 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∇φ( )= −D∇2φ Since it is stated that the diffusion coefficient is constant, it can be pulled outside of the divergence that leaves: € ∇ ⋅ −D∇φ( )= −D∇ ⋅ ∇φ= −D∇2φ Therefore, what needs to be proved is € ∇ ⋅ ∇ = ∇2. Cartesian coordinates are used to simplify the proof, but the proof may be applied in any coordinate system. €