NQE Qualifying Exam Problem Bank 200

Categories			Problems	Rate
Major	Introduction to Nuclear Engineering(1) - Ch.2 원자핵구조 (20) - Ch.3 방사선상호작용 (20) -Ch.5 중성자 수송 (15) -Ch.6 중성자 분포 (15) -Ch.8 원자로 열출력 (20) -Ch.9 방사선량 (10)	100Problems	140 Problems	70%
	Radiation Detection(2)	10Problems
	Materials(3)	10Problems
	Fuel Cycles(4)	20Problems
Mathematics	Advanced Engineering Mathematics(5)		60 Problems	30%
Total	2 Subjects (Major + Mathematics)		200 Problems	100%

→ References : (1) J. R. Lamarsh and A. J. Baratta의 Introduction to Nuclear Engineering

3rd edition, Chapter 2, 3, 5, 6, 8 & 9.

(2) G. F. Knoll의 Radiation Detection and Measurement, 3rd edition,

Chapter 3-IV, 4-IV, VI & VII, 6-I, 7-IV, 14-II, 10-II & III, 17-VI.

(3) Donald R. Askeland, Pradeep P. Phule, The Science and Engineering of

Materials, 5th edition, 2006. Chapter 6,7,8,11,12.

William F. Smith, Foundations of Materials Science and Engineering,

3rd edition, 2004. Chapter 12.

B. M. Ma, Nuclear Reactor Materials and Applications, 1983, Chapter 4,5

(4) W. Loveland, D. J. Morrissey, and G. T. Seaborg, Modern Nuclear

Chemistry, Wiley Interscience, Chapter 15 & 16.

G. R. Choppin, J. O. Liljenzin, and J. Rydberg, Radiochemistry and

Nuclear Chemistry, Butterworth-Heinemann, 3rd edition의 Chapter 15 & 21.

(5) D. G. Zill and M. R. Cullen, Advanced Engineering Mathematics

3rd edition, Chapter 2, 3, 8, 12, 13 & 15.

I. Introduction to Nuclear Engineering

1. Chapter 2 (원자핵 구조 : 20문제)

1. Problem 6  p.46

Natural uranium is composed of three isotopes: 234U, 235U, and 238U. Their abundances

and atomic weights are given in the following table. Compute the atomic weight of              natural uranium.

Isotope	Abundance, a/o	Atomic weight
234U	0.0057	234.0409
235U	0.72	235.0439
238U	99.27	238.0508

2. Problem 7  p.46

A beaker contains 50g of ordinary (i.e., naturally occurring) water.

(a) How many moles of water are present?

(b) How many hydrogen atoms?

(c) How many deuterium atoms?

3. Problem 13  p.46

Using Eq. (2.3), estimate the density of nuclear matter in g/㎤; in Kg/㎥. Take the              mass of each nucleon to be approximately 1.5×10-24g.

4. Problem 14  p.46

The planet earth has a mass of approximately 6×1024kg. If the density of the earth            were equal to that of nuclei, how big would the earth be?

5. Problem 15  p.46

The complete combustion of 1 kg of bituminous coal releases about 3×107 J in heat            energy. The conversion of 1 g of mass into energy is equivalent to the burning of how          much coal?

6. Problem 16  p.46

The fission of the nucleus of 235U releases approximately 200 MeV. How much energy          (in kilowatt-hours and megawatt-days) is released when 1 g of 235U undergoes fission?

7. Problem 25  p.47

An electron moves with a kinetic energy equal to its rest-mass energy. Calculate the           electron's

(a) total energy in units of [eqn: m _{e} c ^{2}];

(b) mass in units of [eqn: m _{e}];

(c) speed in units of [eqn: c];

(d) wavelength in units of the compton wavelength.

8. Problem 28  p.47

Using the chart of the nuclides, complete the following reactions. If a daughter nucleus          is also radioactive, indicate the complete decay chain.

(a) 18N →

(b) 83Y →

(c) 135Sb →

(d) 219Rn →

9. Problem 29  p.48

Tritium (3H) decays by negative beta decay with a half-life of 12.26 years. The                atomic weight of 3H is 3.016.

(a) To what nucleus does 3H decay?

(b) What is the mass in grams of 1 mCi of tritium?

10. Problem 33  p.48

After the initial cleanup effort at Three Mile Island, approximately 400,000 gallons of           radioactive water remained in the basement of the containment building of the Three           Mile Island Unit 2 nuclear plant. The principal sources  of this radioactivity were               137Cs at 156 [eqn: mu]Ci/㎤ and 134Cs at 26 [eqn: mu]Ci/㎤. How many atoms per ㎤ of these                   radionuclides were in the water at that time?

11. Problem 37  p.48

Since the half-life of 235U (7.13×108 years) is less than that of 238U(4.51×109 years),           the isotopic abundance of 235U has been steadily decreasing since the earth was                formed about 4.5 billion years ago. How long ago was the isotopic abundance of 235U           equal to 3.0 a/0, the enrichment of the uranium used in many nuclear power plants?

12. Problem 39  p.49

Consider the chain decay

A→ B→C→,

with no atoms of B present at t=0.

(a) Show that the activity of B rises to a maximum value at the time [eqn: t _{m}] given by

[eqn: t _{m} = {1} over {lambda _{B} - lambda _{A}} 1n( {lambda _{B}} over {lambda _{A}} ),]

at which time the activities of A and B are equal.

(b) Show that for [eqn: t<t _{m}], the activity of B is less than that of A, whereas the reverse              is the case for [eqn: t>t _{m}].

13. Problem 40  p.49

Show that if the half-life of B is much shorter than the half-life of A, then the                 activities of A and B in Problem 2.39 eventually approach the same value. In this case,          A and B are said to be in secular equilibrium.

14. Problem 43  p.49

According to U.S. Nuclear Regulatory Commission regulations, the maximum permissible          concentration of radon-222 in air in equilibrium with its short-lived daughters is 3              pCi/liter for nonoccupational exposure. This corresponds to how many atoms of                 radon-222 per ㎤?

15. Problem 45  p.49

Complete the following reactions and calculate their [eqn: Q] values.

[Note: The atomic weight of 14C is 14.003242]

(a) 4He(p, d)

(b) 9Be([eqn: alpha], n)

(c) 14N(n, p)

(d) 115In(d, p)

(e) 207Pb(g, n)

16. Problem 50  p.50

Tritium (3H) can be produced through the absorption of low-energy neutrons by                deuterium. The reaction is

[eqn: {} ^{2} H`+`n` -> ` ^{3} H`+ gamma]

where the [eqn: gamma]-ray has an energy of 6.256 MeV.

(a) Show that the recoil energy of the [eqn: {} ^{3} H] nucleus is approximately 7 keV.

(b) What is the [eqn: Q] value of the reaction?

(c) Calculate the separation energy of the last neutron in [eqn: {} ^{3} H].

(d) Using the binding energy for [eqn: {} ^{2} H] of 2.23 MeV and the result from part (c),

compute the total binding energy of [eqn: {} ^{3} H].

Where, M(1H), Mn, M(2H) and M(3H) are 1.00797, 1.00867, 2.01410, and 3.01605, respectively.

17. Problem 52  p.50

Using atomic mass data, compute the average binding energy per nucleon of the                following nuclei:

(a) 2H

(b) 4He

(c) 12C

(d) 51V

(e) 138Ba

(f) 235U

18. Problem 57  p.51

Calculate the atom density of graphite having density of 1.60 g/㎤.

19. Problem 60  p.51

Plutonium-239 undergoes [eqn: alpha]-decay with a half-life of 24,000 years. Compute the

activity of 1 gram of plutonium dioxide, 239PuO2. Express the activity in terms of Ci             and Bq. Assume the density of plutonium dioxide, 239PuO2, is 11.0 g/cm3 and the atomic          weight of Pu239 and natural oxygen is 239.05 amu and 15.999 amu, respectively.

20. Problem 62  p.51

Compute the atom densities of 235U and 238U in UO2 of physical density 10.8 g/㎤ if             the uranium is enriched to 3.5 [eqn: w/o`] in 235U.

2. Chapter 3 (방사선 상호작용 20 문제)

21. Example 1  p.55

A beam of 1-Me V neutrons of intensity 5×108 neutrons/㎠-sec strikes a thin 12C              target. The area of the target is 0.5㎠ and is 0.05㎝ thick. The beam has a                     cross-sectional area of 0.1㎠. At 1 Me V, the total cross-section of 12C is 2.6 b. (a)           At what rate do interactions take place in the target? (b) What is the probability that a          neutron in the beam will have a collision in the target?

22. Example 7  p.61

A certain research reactor has a flux of 1×1013 neutrons/㎠-sec and a volume of               64,000 ㎤. If the fission cross-section, [eqn: SIGMA _{f}], in the reactor is 0.1cm-1, what is the               power of the reactor?

23. Example 9  p.67

The value of [eqn: sigma _{gamma }] for 1H at 0.0253 eV is 0.332 b. What is [eqn: sigma _{gamma }] at 1 eV?

24. Example 15. p.99

Here, m/r = 0.0757 cm2/g for uranium and 0.0636cm2/g for oxygen.

Calculate the mass attenuation coefficient of UO2 for 1 Me V [eqn: gamma]-rays. What is their             mean free path? The density of UO2 is about 10 g/㎤.

25. Example 16. p.100

Here, ma/r = 0.0274 cm2/g.

It is proposed to store liquid radioactive waste in a steel container. If the intensity of          g-rays incident on the interior surface of the tank is estimated to be 3×1011 g-rays/            ㎠-sec and the average g-ray energy is 0.8 Me V, at what rate is energy deposited at          the surface of the container?

26. Problem 3  p.110

A monoenergetic beam of neutrons, [eqn: phi]=4×1010 neutrons/cm2-sec, impinges on a target          1 cm2 in area and 0.1㎝ thick. There are 0.048×1024 atoms per cm3 in the target, and          the total cross-section at the energy of the beam is 4.5b.

(a) What is the macroscopic total cross-section?

(b) How many neutron interactions per second occur in the target?

(c) What is the collision density?

27. Problem 9  p.111

What is the probability that a neutron can move one mean free path without interacting          in a medium?

28. Problem 26  p.112

A 1-Me V neutron strikes a 12C nucleus initially at rest. If the neutron is elastic                scattered through an angle of 90°

(a) What is the energy of the scattered neutron?

(b) What is the energy of the recoiling nucleus?

(c) At what angle does the recoiling nucleus appear?

29. Problem 39  p.113

Cross-sections of 235U at 1 Me V are as follows: [eqn: sigma _{e}]=4.0b, [eqn: sigma _{i}]=1.4b, [eqn: sigma _{f}]=1.2b, and [eqn: sigma _{a}]            =1.3b. Here, subscripts e, i, f, and a denote elastic scattering, inelastic scattering,               fission, and absorption, respectively. The cross-sections for neutron-producing and             charged-particle reactions are all negligible. Compute at this energy

(a) the total cross-section,

(b) the capture-to-fission ratio [eqn: alpha].

30. Problem 52  p.115

A 2-Me V photon is Compton scattered through an angle of 30°.

(a) What is its energy after scattering?

(b) What is the recoil energy of the struck electron?

(c) At what angle does the electron appear?

31. Problem 54  p.115

Show that the minimum energy of the scattered photon in Compton scattering is given          by

[eqn: (E') _{min} = {EE _{e}} over {2E+E _{e}}]

and that for [eqn: E] >> [eqn: E _{e}],

[eqn: (E prime ) _{min`} =E _{e} /2=0.255] Me V

32. Problem 61  p.115

A beam of 0.1-Me V g-rays with an intensity of 5×106 g-rays/㎠-sec is incident on            thin foils of (i) aluminum, (ii) water. At this energy, the Compton cross-section per             electron is 0.4929 b, and the Compton energy absorption cross-section per electron is          0.0685 b. Calculate the energy extracted from the beam per unit volume of the foils            due to

(a) Compton scattering,

(b) the photoelectic effect

33. What is “ionizing radiation”?

34. What are the three main energy loss mechanisms of g -ray when it enters a material ?

35. Explain Bragg peak. Explain the reason why Bragg peak exists.

36. What is the difference between LET and stopping power ?

37. If the energies of two particles, an electron and a heavy charged particle, are the              same, which of the two has a higher range ? What is the reason ?

38. Make a plot of cross section versus energy of g-ray for photoelectric effect, Compton          scattering, and pair production.

39. Explain charged-particle reactions induced by a neutron and given example of the

reaction.

40. Derive the collisional parameter in elastic scattering collision between a neutron

and a nucleus with a mass number A.

3. Chapter 5-6 (중성자 수송 및 원자로내 분포 : 30문제)

(Chapter 5) (중성자 수송 : 15문제)

41. Problem 1  p.260

A point source emits [eqn: S] neutrons/sec isotropically in an infinite vacuum.

(a) Show that the neutron flux at the distance [eqn: r] from the source is given by

[eqn: phi = {S} over {4 pi r ^{2}}].

(b) What is the neutron current density vector at the same point?

[Note: Neutrons do not diffuse in a vacuum]

42. Problem 2  p.260

Three isotropic neutron sources, each emittings [eqn: S] neutrons/sec, are located in an              infinite vacuum at the three corners of an equilateral  triangle of side [eqn: a].

43. Problem 4  p.260

The neutron flux in a bare spherical reactor of radius 50cm is given by

[eqn: phi =5 TIMES 10 ^{13} {sin0.0628r} over {r} ``] neutrons/cm2-sec,

where r is measured from the center of the reactor. Suppose that the diffusion                 coefficient for the system is 0.80cm.

(a) What is the maximum value of the flux in the reactor?

(b) Calculate the neutron current density as a function of position in the reactor.

(c) How many neutrons escape from the reactor per second?

44. Problem 5  p.261

Isotropic point sources each emitting [eqn: S] neutrons/sec are placed in an infinite                   moderator at the four corners of a square of side [eqn: a]. Compute the flux and current at          the midpoint of any side of the square and at its center.

45. Problem 7  p.261

An isotropic point source emits [eqn: S] neutrons/sec in an infinite moderator. Using the            diffusion theory ;

(a) Compute the net number of neutrons passing per second through a spherical                    surface of radius [eqn: r] centered on the source.

(b) Compute the number of neutrons absorbed per second within the sphere.

(c) Verify the equation of continuity for the volume within the sphere.

46. Problem 8  p.261

Two infinite planar sources each emitting [eqn: S] neutrons/㎠sec are placed parallel to one           another in an infinite moderator at the distance [eqn: a] apart. The moderator is                      characterized by [eqn: SIGMA _{a}] and D. Calculate the flux and current by diffusion theory as a              function of distance from a plane midway between the two.

47. Problem 9  p.261

Suppose the two planar sources in the preceding problem are placed at right angles to          one another. Derive expressions for the flux and current as a function of distance from          the line of intersection of the sources in a plane bisecting the angle between the               sources.

48. Problem 10  p.261

An infinite moderator contains uniformly distributed isotropic sources emitting [eqn: S]                neutrons/㎤-sec. Determine the steady-state flux and current at any point in the                medium.

49. Problem 11  p.261

An infinite bare slab of moderator of thickness [eqn: 2a] contains uniformly distributed                sources emitting [eqn: S] neutrons/㎤-sec.

(a) Show that the flux in the slab is given by diffusion theory

[eqn: phi = {S} over {SIGMA _{a}} LEFT ( 1- {cosh``x/L} over {cosh` LEFT ( {a+d} over {L} RIGHT )} RIGHT ) ``,]

where [eqn: x] is measured from the center of the slab.

(b) Verify the equation of continuity by computing per unit area of the slab the total    number of neutrons (i) produced per sec within the slab; (ii) absorbed per second  within the slab; and (iii) escaping per second from the slab.

[Hint: The solution to an inhomogeneous differential equation is the sum of         solutions to the homogeneous equation plus a particular solution. Try a constant    for the particular solution.]

50. Problem 12  p.261

A point source emitting [eqn: S] neutrons/sec is placed at the center of a spherical                   medium of radius [eqn: R]. Assume that the diffusion coefficient and absorption cross section	of the medium are D and [eqn: SIGMA _{a}], respectively.

(a) Show that the one-group neutron flux in the sphere is given by

[eqn: phi = {S} over {4 pi D`sinh LEFT ( {R+d} over {L} RIGHT )} ` {sinh {1} over {L} (R+d-r)} over {r} ``,]

where [eqn: r] is the distance from the source and d is the extrapolated distance from

the boundary, and L is the diffusion length defined as [eqn: L ^{2} =D/ SIGMA _{a}].

(b) Show that the number of neutrons leaking per second from the surface of the                   sphere is given by

No. leaking/sec = [eqn: {(R+d)S} over {L`sinh LEFT ( {R+d} over {L} RIGHT )} `.]

(c) What is the probability that a neutron emitted by the source escapes from the                  surface?

51. Problem 13  p.262

An infinite planar source limitting [eqn: S] neutrons/㎠-sec is placed between infinite slabs of beryllium and graphite of thickness [eqn: a] and [eqn: b], respectively, as shown in Fig. 5.9.  Derive an expression for the neutron flux in the system.

[Note: Since the media are different on opposite sides of the source, this problem is   not symmetric and the conventional source condition, [eqn: lim _{x -> 0} J(x)=S/2], is not valid. The appropriate  boundary conditions for this problem are

(i) lim[eqn: {} _{x -> 0} [ phi (x>0)- phi (x<0)]=0]     (ii) lim[eqn: {} _{x -> 0} [J(x>0)-J(x<0)]=S] .

Condition (i) states, in effect, that [eqn: phi] is continuous at the source, whereas (ii) accounts for the neutrons emitted from the source.]

Figure 5.9 Infinite planar source between two finite slabs of material.

52. Problem 14  p.262

A sphere of moderator of radius [eqn: R] contains uniformly distributed sources emitting [eqn: S]           neutrons/㎤-sec.

(a) Show that the flux in the sphere is given by

[eqn: phi = {S} over {SMALLSUM _{a}} LEFT [ 1- {R+d} over {r} ` {sinh`r/L} over {sinh LEFT ( {R+d} over {L} RIGHT )} RIGHT ] ``.]

(b) Derive an expression for the current density at any point in the sphere.

(c) How many neutrons leak from the sphere per second?

(d) What is the average probability that a source neutron will escape from the sphere?

53. Problem 15  p.263

The three-group fluxes for a bare spherical fast reactor of radius [eqn: R]=50cm are given           by the following expressions:

[eqn: phi _{1} (r)= {3 TIMES 10 ^{15}} over {r} sin LEFT ( {pi r} over {R} RIGHT )]

[eqn: phi _{2} (r)= {2 TIMES 10 ^{16}} over {r} sin LEFT ( {pi r} over {R} RIGHT )]

[eqn: phi _{3} (r)= {1 TIMES 10 ^{16}} over {r} sin LEFT ( {pi r} over {R} RIGHT )]

The group-diffusion coefficients are [eqn: D _{1}]=2.2cm, [eqn: D _{2}]=1.7cm, and [eqn: D _{3}]=1.05cm. Calculate           the total leakage of neutrons from the reactor in all three groups.

[Note: Ignore the extrapolation distance.]

54. Problem 18  p.263

The thermal flux in a bare cubical reactor is given approximately by the function

[eqn: phi _{T} (x,y,z)`=`A`cos LEFT ( {pi x} over {tilde{a}} RIGHT ) `cos LEFT ( {pi y} over {pile{tilde{a}}} RIGHT ) `cos LEFT ( {pi z} over {tilde{a}} RIGHT ) ``,]

55. Problem 34  p.265

An infinite slab of moderator of extrapolated thickness [eqn: tilde{a}] contains sources of fast               neutrons distributed according to the function

[eqn: s(x)`=`S```cos LEFT ( {pi x} over {tilde{a}} RIGHT ) ``.]

Using the two-group method, derive an expression for the thermal flux in the slab.

(Chapter 6) (원자로 중성자 플럭스 : 15문제)

56. Problem 3  p.322

A bare-cylinder reactor of height 100cm and diameter 100cm is operating at a                 steady state power of 20 MW. If the origin is taken at the center of the reactor, what          is the power density at the point [eqn: r]=7cm, [eqn: z]=-22.7cm?

57. Problem 4  p.322

In a  homogeneous spherical reactor of radius 45cm, the fission rate density is                 measured as 2.5×1011 fissions/㎤-sec at a point 35cm from the center of the reactor.           Let’s assume that the flux is zero on the surface of the core and the recoverable              energy per fission is 200 MeV (3.2E-11 J). You can use the one-group diffusion               theory.

(a) At what steady state power is the reactor operating?

(b) What is the fission rate density at the center of the reactor?

58. Problem 6  p.322

The core of a certain reflected reactor consists of a cylinder 10-fit high and 10 ft in          diameter. The measured maximum-to-average flux is 1.5. When the reactors is                 operated at a power level of 825 MW, what is the maximum power density in the              reactor in kW/liter?

59. Problem 10  p.322

An infinite slab of moderator of thickness [eqn: 2a] contains at its center a thin sheet of             235U of thickness [eqn: t] . Show that in one-group theory the condition for criticality of this          system can be written approximately as

[eqn: {2D} over {( eta -1)L SMALLSUM _{aF} t} coth {a} over {L} =1``,]

where [eqn: D`] and [eqn: L] are the diffusion parameters of the moderator and [eqn: SMALLSUM _{a}]F is the                 macroscopic absorption cross-section of the 235U.

60. Problem 21  p.324

The flux in a bare-finite cylindrical reactor of radius [eqn: r] and height [eqn: H] is given by

[eqn: phi =AJ _{0} LEFT ( {2.405r} over {tilde{R}} RIGHT ) `cos` LEFT ( {pi z} over {tilde{H}} RIGHT ) ``.]

Find [eqn: A] if the reactor is operated at a power of [eqn: P] watts.

61. Problem 23  p.324

Find the flux distribution in a bone cubical  reactor of side a with extrapolation                 distance d.

(Use one group diffusion theory)

[eqn: phi =A LEFT ( cos {pi } over {tilde{a}} x RIGHT ) ^{3} ``.]

62. Problem 24  p.324

If the reactor in Problem 6.23 is operating at P watts, show that the constant [eqn: A] is

[eqn: A= {P} over {E _{R} SMALLSUM _{f} {8} over {pi ^{3}} tilde{a} ^{3} LEFT ( sin {pi } over {2} {a} over {tilde{a}} RIGHT ) ^{3}} ``.]

63. Problem 30  p.325

Show that the solution to the one-speed diffusion equation in an infinite cylinder made          of a nonmultiplying media is

[eqn: phi =AK _{0} +CI _{0``,}]

where [eqn: K _{0}] and [eqn: I _{0}] are modified bessel functions.

64. Problem 31  p.325

An infinite-cylindrical reactor core of radius [eqn: R] is surrounded by an infinitely thick             reflector. Using one-group theory, (a) find expressions for the fluxes in the core and           reflector; (b) show that the condition for criticality is

[eqn: D _{c} B {J _{1} (BR)} over {J _{0} (BR)} = {D _{r}} over {L _{r}} {K _{1} (R/L _{r} )} over {K _{0} (R/L _{r} )} ``,]

where [eqn: J _{0}] and [eqn: J _{1}] are ordinary Bessel functions and [eqn: K _{0}] and [eqn: K _{1}] are modified Bessel             functions.

65. Problem 34  p.325

A spherical-breeder reactor consists of a core of radius [eqn: R] surrounded by a breeding           blanket of thickness [eqn: b]. The infinite multiplication factor for the core [eqn: k _{INF }] is greater              than unity, whereas that for the blanket [eqn: k _{INF }] is less than unity. Using one-group theory,          derive the critical conditions and expressions for the flux throughout the reactor.

66. Problem 39  p.326

In a hexagonal lattice (also called a triangular lattice), each fuel rod is surrounded by          six nearest neighbor rods, equally spaced at a distance equal to the pitch [eqn: s] of the             lattice. Show that the radius of the equivalent cell in this lattice is given by

[eqn: b=0.525`]S .

67. Problem 40  p.326

Calculate [eqn: k _{INF }] for a hexagonal lattice of 1.4-cm-radius natural uranium rods and                 graphite if the lattice pitch is 20cm.

[Note: The fast fission factor for this lattice is 1.03.]

68. Example 2  p.282

A bare spherical reactor of radius 50cm operates at a power level of 100                      megawatts =108 joules/sec. [eqn: SMALLSUM _{f}]=0.0047cm-1. What are the maximum and average values          of the flux in the reactor? You can use the one-group diffusion theory and zero-flux

boundary condition.

69. Example 3  p.283

A fast reactor assembly consisting of a homogeneous mixture of 239Pu and sodium is          to be made in the form of a bare sphere. The atom densities of these constituents are        [eqn: N _{F}]= 0.00395×1024 for the 239Pu and [eqn: N _{S}]=0.0234×1024 for the sodium.

Estimate the critical radius [eqn: R _{c}] of the assembly. For estimation of the infinite                  multiplication    factor, you can assume that both resonance escape probability and fast        fission factor are      unity (1.0). Also, the diffusion coefficient can be determined by          1/(3Σtr) and the extrapolation distance d=2.13D.

70. Example 4  p.285

Consider a bare spherical fast reactor assembly consisted of homogeneous mixture of          Pu and sodium (Na). Let’s assume that radius of the assembly is 48.5 cm and diffusion         coefficient (D) and absorption cross section (Σa) are 3.21 cm and 0.00835 1/cm,                respectively. What is the average probability that a fission neutron in the assembly is          absorbed within the system? You can use the one-group diffusion theory for the                calculation of the non-leakage probability in the spherical reactor.

4. Chapter 8 (원자로 열출력 : 20문제)

71. Example 3  p.413

The extrapolated dimensions of a certain pressurized water reactor are [eqn: tilde{R}] = 67 in and          [eqn: tilde{H}] = 144 in. The reactor operates at the thermal power of 1,893 MW. It contains 193 fuel assemblies, each consisting of 204 UO2 fuel rods 0.42 in, in diameter. Assuming that the assemblies are uniformly distributed throughout the reactor, calculate the total energy production rate and maximum energy production rate per ft3 in rods located

(a) at the axis of the reactor, (b) 20 in from the axis.

Use: J0(0.70)=0.8812 J0(0.75)=0.8642

72. Example 4  p.416

A certain 235U-fueled reactors operates at a thermal power of 825 MW for 1.5 years           and is then shut down.

(a) Using Fig. 8.3, compute the decay energy at the following times; at shutdown, 1                hour after shutdown, 1 year after shutdown.

(b) If the conversion factor for the reactor is C = 0.88, what are the contributions to               the decay energy at the prior times due to 239U and 239Np ?

73. Example 5  p.426

The fuel rods for the reactor described in Example 8.3 consist of a fueled portion 0.42          in. in diameter that is clad with Zircaloy-4, 0.024 in thick. Each rod is 12 ft long.              Given that the center temperature of the fuel is 3,970° F at the midpoint of the central          rod, calculate at this point the (a) heat flux out of the rod; (b) outer temperature of            the cladding.

cf. Example 8.3 :

Thermal conductivity of fuel : 1.1 Btu/hr-ft-°F

Thermal conductivity of clad : 10 Btu/hr-ft-°F

74. Example 7  p.435

Pressurized water enters the core of the reactor described in Example 8.3 and  8.5 at          a temperature of 543°F and passes along the fuel rods at the rate of 3,148 lb/hr               per channel.

(a) What is the exit temperature of the coolant from the hottest channel?

(b) What is the maximum temperature of the cladding and the fuel in this channel?

75. Example 10  p.448

For the PWR lattice described in Example 8.3 through 8.8, calculate for the onset              of the boiling crisis the : (a) cladding temperature, (b) heat transfer coefficient, and (c)          critical heat flux.

76. Example 12  p.457

The core of a PWR consists of a lattice of fuel rods 12 ft long and 0.5 in. in diameter.         The net electric power excluding in-house power requirements is 1000Mwe with 33%          of thermal efficiency of this reactor. When the reactor has a maximum calculated heat          flux of 1.7Mw/m2 and an overall hot channel factor of 3.2, answer the followings:

(a) (5) the total heat transfer area

(b) (5) the total number of fuel rods in the core

77. Problem 2  p.459

An experimental LMFBR operates at 750 MW. Sodium enters the core at 400°C and            leave at 560°C. At what rate must the sodium be pumped through the core?

78. Problem 3&4  p.459

(Problem 3)

Show that if water passes through the core of a BWR at the rate of [eqn: w] lb/hr, then [eqn: w _{g}]          lb/hr of steam will be produced, where

[eqn: w _{g} = {q-w(h _{f} -h _{i`n} )} over {h _{fg}} `;]

[eqn: q] is the thermal power of the reactor; [eqn: h _{f}] and [eqn: h _{}]in are, respectively, the specific                enthalpies of the saturated water leaving the core and the water entering the core; and          [eqn: h _{fg}] is the heat of vaporization per pound.

(Problem 4)

A BWR operates at a thermal power of 1,593 MW. Water enters the bottom of the             core at 526°F and passes through the core at a rate of 48×106 lb/hr. The reactor             pressure is 1,035 psia. Using the result of the previous problem, compute the rate in           lb/hr at which steam is produced in this reactor.

79. Problem 6  p.459

The decommissioned Fort St. Vrain HTGR was designed to produce 330 MW of         electricity at an efficiency of 39.23%. Helium at a pressure of 710 psia enters the core  of the reactor at 760°F and exits at 1,430°F. What is the helium flow rate(lbm/hr)      through the core?

Data:

Helium enthalpy at inlet and outlet of the core: 1525.38/2355.87 Btu/lbm

1kw=3412.142Btu/hr

80. Problem 7  p.459

A small PWR plant operates at a power of 485 MWt. The core, which is approximately         75.4 in. in diameter and 91.9 in high, consists of a square lattice of 23,142 fuel tubes          of thickness 0.021 in and inner diameter of 0.298 in on a 0.422-in pitch. The tubes            are filled with 3.40 [eqn: w/o]-enriched UO2. The core is cooled by water, which enters at           the bottom at 496°F and passes through the core at a rate of 34×106 lb/hr                     at 2,015 psia. The constant pressure specific heat of water at this condition is 4.866           kJ/kg·K. Compute:

(a) the average temperature of the water leaving the core;

(b) the average power density in kW/liter;

(c) the maximum heat production rate, assuming the reactor core is bare.

81. Problem 8  p.459

The core of a BWR consists of 764 fuel assemblies, each containing a square array of          49 fuel rods on a 0.738-in pitch. The fuel rods are 175 in long, but contain fuel over          only 144 in of their length. The outside diameter of the fuel rods is 0.563 in, the              cladding(fuel tube) is 0.032 in thick, and the UO2 fuel pellets are 0.487 in. in diameter.          This leaves a gap of (0.563-2×0.032-0.487)/2=0.006 in between the pellets and                the cladding. The UO2 has an average density of approximately 10.3g/㎤. The radius of          the core is 93.6 in, and the reactor is designed to operate at 3,293 MW. The                   peak-to-average power density is 2.62. Calculate for this reactor: (a) the total weight          of UO2 and uranium in the core; (b) the specific power in kW/kg U; (c) the average            power density in kW/liter; (d) the average linear rod power [eqn: q '' _{av}] in kW/ft; (e) the               maximum heat production rate.

82. Problem 13  p.460

A BWR power  plant operating at an efficiency of 34% has an electrical output of              1,011 MW.

(a) What is the maximum fission product decay energy in the reactor at shutdown?

(b) what is the decay energy 6 months after shutdown?

83. Problem 24  p.462

The fuel rod in the reactor described in Problem 8.8 with the highest heat flux                 contains uranium enriched to 2.28 [eqn: w/o].

(a) Calculate the maximum heat-generation rate in this rod when the reactor is                     operating at design power.

(b) What is the maximum thermal neutron flux in this rod?

[Note: In part (b), use the average fuel temperature from Problem 8.23]

84. Problem 29  p.462

Compute and plot the temperatures of the water and the outer surface of the cladding          as a function of distance along the hottest channel of the reactor under the following           conditions. Write down the value and the location of the maximum coolant temperature          and the maximum cladding outer surface temperature, respectively.

The extrapolated dimensions of the reference PWR R = 67 in. and H = 144 in

Thermal Power = 1,893 MW

Number of Fuel Assemblies = 193

Fuel Rods per Assembly = 204 UO2 Fuel Rods

Energy per fission = 200 MeV

Energy deposited in the fuel = 180 MeV

Assume that the assemblies are uniformly distributed throughout the cylindrical reactor      core, and neutron flux shape follows chopped cosine shape.

Fuel Rod Outer diameter = 0.42 in.

Fuel Rod Height = 12 ft

Core inlet Coolant Temperature = 543 0F

Mass flow rate per a channel

Coolant Specific heat = 1.3 Btu/lb-0F

Heat Transfer Coefficient = 7,500 Btu/hr-ft2-0F

* Note: 1Btu = 1,055J, 1lb = 0.454kg, 1ft= 0.305m

84 수정본 for 2018-2회

Pressurized water enters the core of the reactor at a temperature of 543°F and passes

along the fuel rods at the rate of 3,148 lb/hr per channel. Answer using the following         data:

Data:

Extrapolated dimensions of a certain pressurized water reactor: R = 67 in and H = 144 in.

Fuel: 0.42 in. in diameter that is clad with Zircaloy-4, 0.024 in thick, and 12 ft long

Reactor thermal power: 1,893 MW

Reactor data: 193 fuel assemblies, each consisting of 204 U02 fuel rods 0.42 in. in             diameter.

Maximum volumetrically heat transfer rate in hottest channel:

Water property: cp=1.3Btu/lb.F

Heat transfer coefficient on cladding: 7,500Btu/hr-ft2-F

(a) What is the exit temperature of the coolant from the hottest channel?

(b) What is the maximum temperature of the cladding in this channel

85. Problem 31  p.463

For the hottest channel in the reactor described in Problem 8.7, calculate, assuming            the reactor is bare, the (a) exit temperature of the water; (b) maximum temperatures           of the fuel and the cladding surface. [Note: Take [eqn: h] = 7,600 Btu/hr-ft2-°F.]

86. Problem 33  p.463

Water at about 2,000 psia and 613°F leaves a PWR vessel via a 29-in I.D. primary             coolant pipe with an average flow velocity of 49.5 ft/sec.

(a) What is the mass flow rate out this pipe in lb/hr?

(b) What is the Reynolds number for the water in the pipe?

(c) If the water returns to the reactor vessel at the same pressure and velocity but at             555°F, how large should the return pipe be?

87. Problem 37  p.463

For the LMFBR core described in Problem 8.9, calculate the: (a) equivalent diameter of          a coolant channel; (b) total coolant flow area; (c) sodium flow rate in lb/hr; (d) average          sodium exit temperature; (e) Reynolds number for the sodium entering and leaving (at          the temperature in part [d]) the core; (f) average heat transfer coefficient for the              sodium entering and leaving the core.

88. Problem 38  p.464

Water enters the core of the BWR described in Problem 8.8 at 525°F and flows                through the core at a rate of 106.5×106 lb/hr. The heat production rate in the fuel             along the hottest channel can be approximated as

[eqn: q ^{''' } (z)`=`q _{max} ^{''' } `cos( pi z/165)`,]

where [eqn: z] is in inches. The reactor pressure is 1,020 psia. Compute the (a)                      maximum-to-average power density in the [eqn: z] direction; (b) total flow area in the core;          (c) average water velocity near the bottom of the core; (d) water/fuel volume ratio; (e)          equivalent diameter of a coolant channel; (f) Reynolds number near the entrance to the          hottest channel; (g) heat transfer coefficient for the convective part of the hottest              channel; (h) location of the onset of local boiling.

89. Problem 43  p.464

The maximum heat flux in the HTGR described in Problem 8.6 is 120,000 Btu/hr-ft2            and the hot channel factor is 2.67.

(a) What is the average heat flux?

(b) What is the total heat transfer area?

90. Problem 45  p.465

The power distribution in the axial direction of the 12-ft-long fuel rods of a PWR is           well represented by

[eqn: q ''' `=`constant` TIMES cos( pi z/144)`,]

where [eqn: z] is in inches. The minimum DNBR is 2.1 and occurs 90 in up the hottest               channel, where the critical heat flux is computed to be 1.08×106 Btu/hr-ft2. The hot            channel factor for the reactor is 2.78, and the total heat transfer area is 48,000 ft2.

Calculate (a)[eqn: q '' _{max}] ; (b)[eqn: q '' _{av}]  ; (c) the operating power of the reactor.

5. Chapter 9 (방사선량 : 10문제)

91. Example 1  p.470

In a 1-mR/hr X-ray field, at what rate are ions being produced per ㎤/sec?

Data : air density : 0.001293 g/㎤ of air

1R = 2.58e-7 Coul/gram

92. Example 2  p.475

It can be shown that a beam of 1 MeV g-rays having an intensity of 105 g-rays/㎠ sec          deposits in tissue approximately 5×10-3 ergs/g-sec. Calculate the absorbed dose rate           and the dose-equivalent rate.

93. Example 3 ("from cancer"→"from bone cancer")  p.495

In a radiation accident, a 30-year-old male worker receives an acute whole-body dose          of 25 rems. what is the probability that the worker will die from cancer as a result of          this exposure? [Note: In the United States, a 30-year-old male, on the average, lives          to an age of 77 years.

94. Example 5  p.498

A radiation worker takes his first job at the age of 18. He continues to be so                  employed for 50 years and retires at the age of 68. If he receives an average annual          occupational dose of 5 rems (which is the maximum acceptable dose according to               current regulations; sec Section 9.8) throughout his working career, what is the                 probability that he will incur and die of cancer as the result of this exposure?            Use the linear hypothesis.

95. Example 6  p.513

What beam intensity of 2-MeV g-rays is required to give an exposure rate of 1                mR/hr?

96. Example 7  p.515

The flux near a 50-kV x-ray machine is 2.4×105 x-rays/㎠-sec. What is the                   dose-equivalent rate in bone, muscle, and fat to an operator standing nearby?

97. Example 8  p.518

At a point near a neutron source, the fast (greater than 1 MeV) neutron flux is 20             neutrons/㎠-sec, and the thermal flux is 300 neutrons/㎠-sec. There are no g-rays.             How long may a radiation worker remain at this point if he is not to exceed his                normal MPD dose?

98. Example 10  p.525

(a) Compute the inhalation dose-commitment factor for 131I for the thyroid. In an adult           individual, this organ has a mass of 20g. Use the following equation and the table	below.

(b) Using the results from (a), compute the dose commitment to the thyroid when a             person, breathing at a normal rate of 2.32×10-4㎥/sec, stands for 2 hours in a                  radiation plume from a nuclear power plant containing 131I at a concentration of                 2×10-9Ci/㎥.

99. Example 12  p.534

In certain parts of the United States, the radon concentration has been measured at 1          pCi/l in older uninsulated homes and about 6 pCi/l in more modern, well-insulated              homes. If a person spends an average of 3,500 hours in his home per year, roughly            how many additional cases of lung cancer would result over a period of 50 years from          insulating (and not ventilating) one million homes, each with four residents.

100. Example 13  p.535

During one year, a radiation worker anticipates receiving a uniform external dose of 3          rems and simultaneously inhaling some airborne 131I. How much of the radionuclide may          the worker inhale if he is not to exceed recommended limits?

II. Radiation Detection

101. If a disk type GM detector made of pressurized air (30 atm) with a radius of 2 cm,            thickness of 0.5 cm is placed 1 meter away from a point-like gamma source of energy          1 MeV as seen in the following figure.  [G. Knoll Chap 4. VI & VII]

(a)  What is the geometrical efficiency ?

(b)  If the attenuation coefficient of the gas at 1 atm in the gamma-ray energy of 1                 MeV in the problem is 0.02  [1/cm/atm] , then what is the interaction efficiency ?

If the measured count for 10 min is 54,580 and the GM detector dead time is 1                msec, what is the incident rate [1/sec] of gamma ray in the detector ?  Hint, GM              detector follows the non-paralyzable dead-time system. n=m/(1-mt )

102. If the spectrum of radiation sample in a simple gas counter set-up is given as below,          sketch the counting curve (a count rate as a function of bias voltage)  [G. Knoll Chap          4. IV]

103. The following figure is an idealized pulse height spectrum of a high mono-energetic            gamma-ray source by a HP detector. Explain the origins of the five features (a to e)           of the spectrum. [G. Knoll Chap 10. II & III]

104. A student measured a gamma-ray check source by a radiation detector with the                 interaction efficiency of 10% in a laboratory. Answer the following questions when the

data obtained was given in the following table. [G. Knoll Chap 3. IV] (상)

(a) Calculate the activity of the source in Bq. Consider the total counting efficiency           and the background.

(b) Calculate the uncertainty of the activity in Bq for the answer of the above                question.

Condition	Counts	Meas. Time (sec)
With Source	360,505	60
Background	38,192	60

105. A radioactive source was measured with a GM counter having the absolute efficiency

of 2 %. The measured count with the source was 72,000 counts for 2 min and the             background count without the source was 14,400 counts for 1 min. What is the

source activity and its error in Bq units ? [G Knoll Chap 3. IV]

106. Explain the differences of operation principles and applications for three types of gas           detectors; an ionization chamber, a proportional counter, a GM counter. [G. Knoll               Chap 6. I]

107. Explain the dead time, resolving time and recovery time of a Geiger Muller counter            using a figure of pulses in time. [G. Knoll Chap 7. IV]

108. Typical counting system can be assembled with several signal processing components          and a sensor. Draw a black box diagram for a typical counting system with the                 following components; A sensor, A timer, A counter, A Discriminator, A preamplifier, A          linear amplifier, A high voltage power supply. [G. Knoll Chap 17. VI]

109. Compare the advantages and disadvantages between a NaI(Tl) scintillation detector and          a HPGe detector in terms of the detection efficiency and the energy resolution.

110. Isotope of Boron-10 is used for neutron detector as (1) BF3 gas or (2) Boron-lined            P10 gas proportional counters.   We can assume the following reaction as a neutron            conversion process for thermal neutron detection. [G. Knoll Chap 14. II]

[eqn: n`+` ^{10} B`` -> ` ^{7} Li`+` alpha ```````(94%)`````````Q=2.310`[MeV]]

(a) Estimate the kinetic energy of Li-7 and alpha particles in MeV.

(b) Draw the spectrum of BF3 gas proportional counter and explain the 3 components          of spectrum

(c) Draw the spectrum of B-lined gas proportional counter and explain the 2                  components of spectrum

III. Materials

111. There are several ways to strengthen the metals and alloys, such as grain size             control, cold working, solid solution hardening, and dispersion hardening.

(a) Explain how the metals and alloys are strengthened by cold working.

(b) Precipitation hardening is one of the dispersion hardening mechanisms. What are      the required characteristics of the precipitates to achieve the significant hardening?

112. Explain the following aging mechanisms of the structural materials used in the             commercial water reactors. Also explain, if any, the mitigative or remedial actions to       prevent or slow down the progress of each aging mechanism.

(a) Irradiation embrittlement of the reactor pressure vessel steels.

(b) Stress corrosion cracking of the steam generator tube materials.

113. Explain the following aging mechanisms of the structural materials used in the              commercial water reactors. Also explain, if any, the mitigative or remedial actions to       prevent or slow down the progress of each aging mechanism.

(a) Flow accelerated corrosion of the secondary piping materials.

(b) Dimensional change of a CANDU pressure tube materials.

114. Answer the followings;

(a) List and briefly explain strengthening mechanisms of metals and alloys.

(b) Shown below is the Fe-rich portion of the Fe-C phase diagram. Mark the               eutectic and eutectoid points on the plot and write down the reactions.

115. Stainless steels are widely used as structural materials in nuclear power plants. One        of such stainless steel is a type SS304 whose typical composition is Fe-18Cr-8Ni         with about 0.06% of carbon.

(a) Explain the role of each element (Fe, Cr, Ni, C) in terms of phase stability and         corrosion resistance.

(b) Stainless steels can be sensitized during heat treatment or welding process.             Explain how the sensitization happens.

116. Austenitic stainless steels, such as SS304, SS316 are widely used as structural             materials in nuclear power plants. They are known to be susceptible to sensitization       during welding process.

(a) Explain how the sensitization happens during welding process

(b) How would you do to prevent sensitization of 304 SS during welding? Please            focus on the role of carbon in your explanation.

117. If a zinc surface is corroding at a current density of 3.45*10-7 A/cm2, what thickness       of metal will be corroded in 210 days? (Atomic weight of Zn : 65.38 g/mol.  Density        of Zn : 7.133 g/cm3)

118. Consider the ductile fracture processes of the round bar tensile specimen under             tensile loading. Assume the brittle inclusions are dispersed within the specimens.

(a) Explain and schematically draw the formation of initial crack.

(b) Explain and schematically draw the propagation of crack.

(c) Explain and schematically draw the final fracture.

119. Shown bellow are the mechanical properties changes of a precipitation hardening

nickel-base superalloy and Ti3Al intermetallics. Discuss the followings:

(a) What are the advantages and disadvantages of nickel-base supperalloys in high          temperature application.

(b) Repeat a) for Ti3Al intermetallics.

(c) Why the nickel-base superalloy loses strength above ~600C.

(d) What would be the proper choice of strengthening mechanism if the nickel-base         alloys are to be used over 600C?

120.It is known that the material properties are gradually affected by the fast neutron

irradiation during the reactor operation. The degree of fast neutron irradiation is          measured as dpa (displacement per atom).

(a) From the elastic scattering of 2 MeV fast neutron with atom nucleus, calculate         the maximum and average energy transfer to Fe (M = 56 amu, aniostropy             correction factor f = 0.57)

(b) Calculate the dpa/days when 2 MeV fast neutrons (flux = 1*1015/cm2-sec) are         irradiated on Fe atoms. Assume displacement cross-section (σd) is equal to elastic      scattering cross-section (σel = 3 barns)

IV. Fuel Cycles

121.  Define or describe the following terms or concepts:

(a) crud, (b) radiolysis of water, (c) accelerator transmutation of waste, (d) PUREX             process, (e) pyroprocessing, (f) LLW, (g) TRU, (h) HLW, (i) SNF, and (j) back-end              fuel cycle

122. Define or describe the following terms or concepts:

(a) yellowcake, (b) orange oxide, and (c) green salt

(d) Write balanced chemical equations for three different methods to produce metallic               uranium.

123. Describe the following chemical reactions:

(a) UCl6 is also volatile similar to UF6. Why is UCl6 inappropriate for the enrichment                of uranium?

(b) Even though UO2 can be converted directly to UF6, why is it first converted to                 UF4 and then to UF6? And write the chemical reactions from UO2 through UF4 as               intermediate product to UF6.

(c) Nitric acid readily dissolves UO2. Why doesn't hydrochloric acid?

124. Natural uranium is 0.72 atom % [eqn: {} _{92} ^{235} U``], and the fuel used in light water reactors is  typically 3 - 5 atom % [eqn: {} _{92} ^{235} U``]. This means the refined uranium must be enriched in the lighter 235 isotope prior to fuel fabrication. Describe the following enrichment          techniques: (a) gaseous diffusion, (b) gas centrifuges, (c) electromagnetic separation,   and (d) laser enrichment. (e) What is the theoretical maximum separation factor for    separation by gaseous diffusion of  [eqn: {} _{2} ^{3} H``e] and [eqn: {} _{2} ^{4} He``]? (f) If a gaseous diffusion plant     produces uranium (UF6) with a 235/238 ratio of 5, what is the expected 234/235 ratio  in the resulting material?

125. Discuss the disposal of the following examples of radioactive wastes:

(a) Water solutions containing 1 mCi of [eqn: {} _{1} ^{3} H``] from a research lab (4 points)

(b) Gas escaping from the dissolution of 1 kg of irradiated reactor fuel (3 points)

(c) Ion exchange resin used to purify the cooling water of a 1-MW research reactor                (4 points)

126. Describe the front-end fuel cycle and the back-end fuel cycle and explain various physical and chemical processes in each fuel cycle step with pictorial flow sheet.

127. Of all the radiation-chemical reactions that have been studied in aqueous solution, the most important is the decomposition of pure water itself. This is called the radiolysis of water which results in a large variety of chemical changes. Therefore it draws interests in various fields of reactor chemistry, nuclear fuel cycle, nuclear waste disposal, and biology.

(a) Write the governing reactions of the water radiolysis and describe its intermediate              products and final products. (5 points)

(b) Given a G-value of 1.59 for the production of hydrogen gas by the irradiation of   TBP (tributyl phosphate) with 1-MeV electrons, calculate the rate of hydrogen gas  evolution in a liter of TBP irradiated for 1 h at a dose rate of 200 W/liter. Assume the standard conditions for temperature and pressure. (5 points)of water which results in a large variety of chemical changes. Therefore it draws interests in various fields of reactor chemistry, nuclear fuel cycle, nuclear waste disposal, and biology.

(a) Write the governing reactions of the water radiolysis and describe its intermediate              products and final products. (5 points)

(b) Given a G-value of 1.59 for the production of hydrogen gas by the irradiation of               TBP (tributyl phosphate) with 1-MeV electrons, calculate the rate of hydrogen gas              evolution in a liter of TBP irradiated for 1 h at a dose rate of 200 W/liter. Assume              the standard conditions for temperature and pressure. (5 points)

128. In an UO2 fueled reactor, the transuranium elements are produced in the course of  nuclear reactions. Describe the generating nuclear reaction mechanism of the         transuranic elements (Np, Pu, Am, Cm), and how much of each transuranic element is approximately produced in spent nuclear fuel, although it depends on the burn-up of nuclear fuel ?

129. By 1955, scientists at Berkeley had prepared an equilibrium amount of about 109   atoms of [eqn: {} _{```99} ^{253} Es``]by neutron irradiation of plutonium in The Materials Testing Reactor in Idaho. They thought it might be possible to prepare element 101 using the reaction

[eqn: {} _{```99} ^{253} Es```+` _{``2} ^{``4} He`` -> ` _{``101} ^{`256} Md```+` _{``0} ^{``1} n````]

How many atoms of the element 101 are produced under the following conditions               below?

(Use: s is the reaction cross section (10-27cm2), f is the helium ion flux                         (1014particles/s), l is the decay constant of [eqn: {} _{```99} ^{253} Es``] (10-4s-1), and t is the length of               each bombardment (104 s)

130. The favored method for permanent storage of radioactive waste is deep geologic   repositories. This option is the only option for unprocessed spent fuel assemblies      and for most high level waste. The prevailing design strategy for waste repositories is that of multiple barriers. What is the multi-barrier system in a deep geologic           repository? Define each barrier and describe its technical roles.

131. Discuss advantages and disadvantages of the Th-U fuel cycle and the U-Pu fuel   cycle from the perspective of the fuel reprocess and the yield of long-lived isotopes  and explain why [eqn: {} _{90} ^{228} Th``] and [eqn: {} _{92} ^{232} U``]are a nuisance in the thorium-uranium fuel cycle.

132. Fuel reprocessing has three objectives: (a) to recover U or Pu from the spent fuel          for reuse as a nuclear reactor fuel or to render the waste less hazardous, (b) to               remove fission products from the actinides to lessen short-term radioactivity                    problems, and (c) to convert the radioactive waste into a safe form for storage. The            hydro-metallurgical process, PUREX process, is used for almost all fuel reprocessing           today. An alternative to the aqueous separation process, "dry" process, called                   pyroprocess, has also been studied, but none has been used on an industrial scale.             Describe both processes and discuss advantages and disadvantages.

133. Consider transmutation of plutonium 239 by a light water reactor. We assume that          fresh fuel initially contains N28 atoms of 238U and N29 atoms of 239Pu. To simplify,             we assume that the fresh fuel contains no 235U. We also neglect the production of              239Pu in the core by absorption of resonance neutrons in 238U. Plutonium 239,                   therefore, is generated only by thermal neutron capture by 238U and consumed by its           fission. Assume that the thermal flux ΦM is constant with time and uniform in the core.           The number, N28, of 238U atoms is assumed to be constant with time. The absorption           cross section of 239Pu is σ29. The absorption cross section of 238U is σ28. t is time.

(a) Write the governing equation and the initial condition for the number, N29(t), of                 239Pu atoms in the reactor core. Solve the equation. (6 points)

(b) Define and formulate the burn-up for this system in terms of the parameters in                 this model. (4 points)

134. Consider extraction of UO22+ in the aqueous phase by the organic solvent TBP by the following equilibrium:

[eqn: UO _{2} ^{2+} `(aq)``+`2NO _{3} ^{-} (aq)+2TBP(org)`` LRARROW ``UO _{2} (NO _{3} ) _{2} BULLET 2TBP(org)]

(a) Write the equilibrium constant KUO2 in terms of the concentrations of the species                appearing in the above reaction. (3 points)

(b) Write the definition of the distribution coefficient DUO2 for UO22+. (3 points)

(c) Assume that the concentration of total TBP, both combined and uncombined, in the              organic phase is C. Obtain the equation for the distribution coefficient DUO2 for                  UO22+ in terms of C, DUO2 and aqueous concentrations. Assume DUO2 = 12, C = 1.5              mol/liter, and [NO3-] = 3 mol/liter. Would DUO2 increase or decrease if NO3- of the              aqueous phase increase? (4 points)

135. Answer the following questions

(a) How and where is 14C generated in a nuclear fuel cycle? Why is this nuclide of                 concern in the waste management? (2.5 points)

(b) Select a radionuclide (other than 14C) that is important in radioactive waste                      management for nuclear fuel cycle. Discuss how and where mainly that nuclide is               generated in the fuel cycle, and explain why that nuclide is so important. (2.5                  points)

(c) Answer the following true/false questions. Explain briefly your answer. (each 1                 point)

(c-1) Radioactivity of any radionuclide in spent fuel from a PWR	decreases monotonically with time after it is discharged from a reactor.

(c-2) Because wastes from reactor decommissioning are mostly concrete	and

structural materials consisting of reactor buildings, decommissioning

wastes can be regarded as non-radioactive.

(c-3) With careful choice of solvents and operating conditions, perfect

separation of one material from another is possible by a multi-stage

solvent-solvent extraction process.

(c-4) All fission-product isotopes in a spent fuel at discharge from a	reactor

core are radioactive.

(c-5) Radionuclides included in activated materials are so short-lived that	these

can be regarded as non-radioactive after a storage period of 10 y.

136. In a radioactive decay chain

[eqn: A```` REL -> {lambda _{1}} {} `````B````` REL -> {lambda _{2}} {} `````C````(stable)]

(a) A sample of pure A nucleus is isolated at time zero. Derive a general expression               for the time at which the daughter's (B) activity is a maximum and the raio of                  daughter activity at that time to initial parent activity

(b) Explain the transient and secular radioactive equilibrium and give the conditions

137. Calculate the time required for preprocessing cooling of irradiated fuel such that the inhalation toxicity (C) of the contained 131I will have decayed to that of 129I. Assume 235U fission at constant power and an irradiation time of 3 years. Use the following data.

	T1/2	Yield	C (in air, mCi/m3)
129I	1.7×107 yr	0.008	2×10-11
131I	8.05 d	0.0293	1×10-10

138. Consider a 1000 MWe PWR with

In-core inventory of coolant water = 8000 kg

Average thermal neutron flux = 3.5×1013 n/cm2·s

Capacity factor of the power plant = 0.8

(a) Calculate the tritium activity (Ci/yr) produced by D(n,γ)T (T1/2 of tritium = 12.3 yr)               during the one calendar year. The corresponding effective cross section =                      3.35×10-4 b

(b) Assume water contains 0.015 a/o (atomic percent) D.

Assuming an average dissolved boron concentration of 1000 ppm in the coolant,                calculate the tritium activity (Ci/yr) produced yearly from the reaction in an                     average fast neutron flux 7×1013 n/cm2·s.

[eqn: {} ^{10} B```+`n`` -> ``2` ^{4} He``+``T`````( sigma =42`mb)]

Natural boron contains 19.61% of 10B.

139.

(a) Describe the radioactive materials existing inside the reactor coolant of PWR in                 terms of more than 4 different classes and list the important radionuclides in each              classes

(b) Describe various gaseous, liquid, and solid radioactive waste treatment systems                 with respect to treatment methods, system flow sheets, characteristics of the                   system and the important radionuclides in each system.

140. Pure 239Pu is placed in a reactor. How much 240Pu and 241Pu will be produced, when 50% of 239Pu is consumed? The thermal neutron flux is 1012 n/cm2·s.

The decay and neutron reaction data are:

	T1/2	l (s-1)	sc (barn)	sa (barn)
239Pu	24,400 yr		268.8	1011.3
240Pu	6,580 yr	3.34×10-12	289.5	289.5
241Pu	13.2 yr	1.66×10-9	368	1377

V. Engineering Mathematics

1. Chapter 2,3,8

141. 2-1 Exercise 21  p.44

Find the critical points and phase portrait of the given autonomous                              first-order differential equation. Classify each critical point as asymptotically stable,            unstable, or semi-stable. By hand sketch typical solution curves, in the regions in the         [eqn: xy]-plane determined by the graphs of the equilibrium solutions.

[eqn: {dy} over {dx} =y ^{2} -3y]

142. 2-1 Exercise 25  p.44

In problem, fine the critical points and phase portrait of the given autonomous                  first-order differential equation. Classify each critical point as asymptotically stable,             unstable, or semi-stable. By hand, sketch typical solution curves, in the regions in the           [eqn: xy]-plane determined by the graphs of the equilibrium solutions.

[eqn: {dy} over {dx} =y ^{2} (4-y ^{2} )]

143. 2-3 Example 5   p.56

A function [eqn: y(x)] is defined in [eqn: [0, INF ]], and the function itself is continuous within the           defined domain (but, its derivatives are not necessarily continuous). Find the function        [eqn: y(x)] satisfying the following first order inhomogeneous ordinary differential equation:

[eqn: {dy} over {dx} +`2y`=`f(x),````y(0)=0] where [eqn: f(x)= {cases{2 phi ,&`0 LEQ x LEQ 1#0,&`x>1`.}} `]

Note that [eqn: phi] is a constant.

144. 2-3 Exercise 17  p.58

In problem, find the general solution of the given differential equation. Give the                 largest interval over which the general solution is defined. Determine whether there            are any transient terms in the general solution.

[eqn: cos`x {dy} over {dx} +(sin`x)y`=`1]

145. 2-4 Example 1  p.62

(Solving an Exact DE)

Solve [eqn: 2xydx+(x ^{2} -1)`dy=0.]

146. 2-4 Exercise 27  p.65

In Problem, find the value of [eqn: k] so that the given differential equation is exact.

[eqn: (y ^{3} +kxy ^{4} -2x)dx`+`(3xy ^{2} +20x ^{2} y ^{3} )dy=0]

147. 2-5 Example 2  p.68

Solving a Bernoulli DE

Solve [eqn: x {dy} over {dx} +y=x ^{2} y ^{2}].

148. 2-5 Exercise 19  p.70

Solve the given differential equation by using an appropriate substitution.

[eqn: t ^{2} {dy} over {dt} +y ^{2} =ty]

149. 2-7 Example 2  p.76

A breeder reactor converts relatively stable uranium-238 into the isotope                       plutonium-239. After 15 years it is determined that 0.043% of the initial amount [eqn: A _{0}]            of the plutonium has disintegrated. Find the half-life of this isotope if the rate of               disintegration is proportional to the amount remaining.

150. 2-9 Exercise 1  p.98

(Radioactive Series)

We have not discussed methods by which systems of first-order differential equations           can be solved. Nevertheless, systems such as (2) can be solved with no knowledge             other equation. Find a solution of (2) subject to the initial conditions                            [eqn: x(0)=x _{0} ,``y(0)=0,``z(0)=0`.]

151. 3-1 Exercise 15  p.115

Determine whether the given set of functions is linearly dependent or linearly                  independent on the interval [eqn: (- INF ,` INF )].

[eqn: f _{1} (x)=3], [eqn: f _{2} (x)=3sin ^{2} x], [eqn: f _{3} (x)=4cos ^{2} x]

<< 변형 for 2017-1회

[eqn: f _{1} (x)=x], [eqn: f _{2} (x)=x ^{2}], [eqn: f _{3} (x)=4x-3x ^{2}]  >>

152. 3-1 Exercise 23  p.115

Verify that the given functions from a fundamental set of solutions of the differential           equation on the indicated interval. Form the general solution.

[eqn: y '' -y'-12y=0;`e ^{-3x} ,e ^{4x} ,(- INF , INF )]

153. 3-2 Example 1  p.117

(Finding a Second Solution)

Given that [eqn: y _{1} =e ^{x}] is a solution of [eqn: y '' -y=0] on the interval [eqn: (- INF , INF )], use                     reduction of order to find a second solution [eqn: y _{2``.}]

154. 3-2 Exercise 1  p.118

In Problem, the indicated function [eqn: y _{1} (x)] is a solution of the given equation. Use                reduction of order or formula (5), as instructed, to find a second solution [eqn: y _{2} (x).]

155. 3-3 Example 4  p.122

(Fourth-Order DE)

Solve [eqn: {d ^{4} y} over {dx ^{4}} +2 {d ^{2} y} over {dx ^{2}} +y=0`.]

156. 3-3 Exercise 17  p.124

Find the general solution of the given higher-order differential equation.

[eqn: y '' prime -5y '' +3y prime +9y=0]

157. 3-4 Example 2  p.128

(Particular Solution Using Undetermined Coefficients)

Find a particular solution of [eqn: y '' -y'+y=2`sin`3x.]

158. 3-4 Example 10  p.133

(Third-Order DE-Case I)

Solve [eqn: y ''' +y '' =e ^{x} `cos``x.]

159. 3-11 Example 2  p.186

(Solution by Elimination)

Solve      [eqn: x'-4x+y '' =t ^{2}]

[eqn: x'+x+y'=0`.]

160. 3-11 Exercise 3  p.188

Solve the given system of differential equations by systematic elimination.

[eqn: {dx} over {dt} =-y+t]

161. 8-6 Example 7  p.391

Use the inverse of the coefficient matrix to solve the system

[eqn: ```````2x _{1} +````````````````````````x _{3} =2# -`5x _{1} +5x _{2} +6x _{3} =-1# -2x _{1} +3x _{2} +4x _{3} =4`.]

162. 8-6 Exercise 57  p.395

The system of equations for the currents [eqn: i _{1`,``} i _{2} ,`and``i _{3}] in the network shown in Figure           8.6 is

[eqn: ````````````````i _{1} `+`````````i _{2} +````i _{3} ```````=0# -R _{1} i _{1} `+`R _{2} i _{2} ``````````````````````````=`E _{2} -E _{1}# ```````````````````````-`R _{2} i _{2} +R _{3} i _{3} =E _{3} -E _{2}]

where [eqn: R _{k}] and [eqn: E _{k`} `,`k=1,`2,`3,`] are constants.

(a) Express the system as a matrix equation AX = B.

(b) Show that the coefficient matrix A is nonsingular.

(c) Use X = A-1 B to solve for the currents.

Figure 8.6  Network in Problem 57

163. 8-7 Example 1  p.397

(Using Cramer's Rule to Solve a System)

Use Cramer's rule to solve the system

[eqn: 3x _{1} +2x _{2} +````x _{3} =7# ````x _{1} -````x _{2`} `+`3x _{3} =3# 5x _{1} +4x _{2} -2x _{3} =1``.]

164. 8-8 Example 4  p.401

(Finding Eigenvalues and Eigenvectors)

Find the eigenvalues and eigenvectors of  [eqn: A= {pmatrix{9&1&1#1&9&1#1&1&9}} ```.]

165. 8-8 Exercise 15  p.403

Find the eigenvalues and eigenvectors of the given matrix.

[eqn: {pmatrix{5&-1`````&0#0&-5`````&9#5&-1`````&0}}]

166. 8-9 Example 1  p.406

([eqn: A ^{m}] for a 3 × 3 Matrix)

Compute [eqn: A ^{m}] for [eqn: A=` {pmatrix{`````2&`-1&```````0#``````0``&````1&````````0#``````-3&``-2&```-1}} ```.]

Hint: Use the characteristic equation and eigenvalues of the given matrix A.

<< 변형 for 2017-1

Compute [eqn: A ^{m}] for [eqn: A= {pmatrix{`````1&`1&-2#-1``&2&``````1#``````0&``1&-1}} ```.] >>

167. 8-10 Example 3  p.412

(Constructing an Orthogonal Matrix)

In Example 1, we verified that the eigenvectors

[eqn: K _{1} = LEFT ( pile{pile{```1#```0}#``1} ``` RIGHT ) `,```K _{2} = LEFT ( pile{pile{-1#``````1}#``````0} RIGHT ) ``,`K _{3} = LEFT ( pile{pile{``````1#```````2}#-1} RIGHT )]

of the given symmetric matrix A are orthoonal. Now the norms of the eigenvectors             are [eqn: LEFT ∥ K _{1} RIGHT ∥ = sqrt {K _{1} ^{T} K _{1}} = sqrt {2} ``,] [eqn: LEFT ∥ K _{2} RIGHT ∥ = sqrt {K _{2} ^{T} K _{2}} = sqrt {3} ``,]  [eqn: LEFT ∥ K _{3} RIGHT ∥ = sqrt {K _{3} ^{T} K _{3}} = sqrt {6} `]

Thus, an orthonormal set of vector is

[eqn: LEFT ( pile{`` pile{` {1} over {sqrt {2}} ````#eqalign{# 0# }}#{1} over {sqrt {2}}} RIGHT ) ``,`` LEFT ( pile{pile{- {1} over {sqrt {3}}#{1} over {sqrt {3}}}#{1} over {sqrt {3}}} RIGHT ) ``,`` LEFT ( pile{pile{{1} over {sqrt {6}}#{2} over {sqrt {6}}}#- {1} over {sqrt {6}}} RIGHT ) ``.]

Using these vectors as columns, we obtain the orthogonal matrix

[eqn: P= {pmatrix{{1} over {sqrt {2}}&- {1} over {sqrt {3}}&{1} over {sqrt {6}}#0&{1} over {sqrt {3}}&{2} over {sqrt {6}}#{1} over {sqrt {2}}&{1} over {sqrt {3}}&- {1} over {sqrt {6}}}} `````.]

You should verify that PT = P-1 .

We will use the technique of constructing an orthogonal matrix from the eigenvectors           of a symmetric matrix in the next section.

Do not misinterpret Theorem 8.28. We can always find [eqn: n] linearly independent                  eigenvectors for an [eqn: n TIMES n] real symmetric matrix A. However, the theorem does not            state that all the eigenvectors are mutually orthogonal. The set of eigenvectors                 corresponding to distinct eigenvalues are orthogonal; but the different eigenvectors             corresponding to a repeated eigenvalue may not be orthogonal. Consider the                    symmetric matrix in the next example.

168. 8-12 Example 2  p.425

Diagonalize the following matrix if possible.

[eqn: A= {bmatrix{-5&```9#-6&```10}}]

If it is not diagonalizable, state the reason.

If it is diagonalizable, then create a matrix [eqn: P] with eigenvectors that supports                diagonalizing the original matrix [eqn: A], and show how the matrix [eqn: P] works in this                diagonalization.

169. 8-15 Example 2  p.441

(Least Squares Line)

Find the least squares line for the data in Example 1. Calculate the sum of the square           errors [eqn: E] for this line and the line [eqn: y=x+1``.]

170. 8-15 Exercise 1  p.442

Find the least squares line for the given data.

(2, 1), (3, 2), (4, 3), (5, 2)

2. Chapter 12,13,15

171. 12-1 Example 1  p.654

(Orthogonal Set of Functions)

Show that the set [eqn: LEFT { 1,`cos``x,``cos``2x,`..... RIGHT }] is orthogonal on the interval [eqn: LEFT [ - pi ,` pi RIGHT ]]

172. 12-1 Exercise 5  p.657

Show that the given functions are orthogonal on the indicated interval.

[eqn: f _{1} (x)`=`x,`f _{2} (x)`=`cos``2x;` LEFT [ - pi /2,` pi /2 RIGHT ]]

173. 12-2 Example 1  p.660

(Expansion in a Fourier Series)

Expand         [eqn: f(x)= LEFT { pile{0`,``````````````- pi <x<0#pi -x,`````````0 LEQ x< pi } RIGHT .]

in a Fourier series

174. 12-2 Exercise 1  p.662

Find the Fourier series of [eqn: f] on the given interval.

[eqn: f(x)= LEFT { pile{0`,``````````````- pi <x<0#sin``x,`````````0 LEQ x< pi } RIGHT .]

175. 12-2 Exercise 9  p.662

ind the Fourier series of [eqn: f] on the given interval.

[eqn: f(x)= LEFT { pile{0`,``````````````- pi <x<0#sin``x,`````````0 LEQ x< pi } RIGHT .]

176. 12-3 Example 1  p.664

(Expansion in a Sine Series)

Expand [eqn: f(x)`=`x,``-2<x<2,`] in a Fourier series.

177. 12-3 Example 2  p.665

Expand the given function in an appropriate cosine or sine series

[eqn: f(x)= LEFT { pile{-1`,``````- pi <x<0#```1,```````````````````0 LEQ x< pi } RIGHT .]

<<

(Expansion in a Sine Series)

The function [eqn: f(x)= LEFT { pile{-1`,``````- pi <x<0#```1,```````````````````0 LEQ x< pi } RIGHT .]  shown in Figure 12.8 is odd on the

interval           [eqn: (- pi , pi )`.] With [eqn: p= pi] we have from (5)

[eqn: b _{n} = {2} over {pi } int _{0} ^{pi } {(1)} ``sin``nx``dx`= {2} over {pi } ` {1-(-1) ^{n}} over {n} ````,] and so

[eqn: f(x)= {2} over {pi } sum _{n=1} ^{INF } {1-(-1) ^{n}} over {n} `sin``nx.]  >>

178. 12-3 Exercise 11  p.668

Expand the given function in an appropriate cosine or sine series..

[eqn: f(x)= LEFT { pile{-1`,`````````- pi <x<0#````1,```````````````````0 LEQ x< pi } RIGHT .]

179. 12-4 Example 3  p.672

(Frequency Spectrum)

Find the frequency spectrum of the periodic square wave with a period of 1. The

wave is the periodic extension of the function [eqn: f] :

[eqn: f(x)= LEFT { pile{````0`,`````````- {1} over {2} <x`<`- {1} over {4}#eqalign{1,`````````- {1} over {4} <x< {1} over {4}# 0,```````````````` {1} over {4} <x< {1} over {2}}} RIGHT .]  .

180. 12-4 Exercise 7  p.673

Find the frequency spectrum of the periodic wave that is the periodic extension of             the function [eqn: f] in Problem 1.

181. 13-1 Example 1  p.690

(Using Separation of Variables)

Find product solutions of [eqn: {Partial ^{2} u} over {Partial x ^{2}} =4 {Partial u} over {Partial y} ````.]

182. 13-1 Exercise 3  p.693

Use separation of variables to find, if possible, product solutions for the given partial           differential equation.

[eqn: u _{x} +u _{y} =u]

183. 13-2 Exercise 1  p.699

A rod of length [eqn: L] coincides with the interval [0, L] on the [eqn: x]-axis. Set up the                 boundary-value problem for the temperature  [eqn: u(x,t)]

The left end is held at temperature zero, and the right end is insulted. The initial              temperature is [eqn: f(x)] throughout.

184. 13-2 Exercise 3  p.699

A rod of length [eqn: L] coincides with the interval [0, L] on the [eqn: x]-axis. Set up the                 boundary-value problem for the temperature  [eqn: u(x,t)]

The left end is held at temperature 100°, and there is heat transfer from the right             end into the surrounding medium at temperature zero. The initial temperature is [eqn: f(x)]           throughout.

185. 13-3 Exercise 1  p.701

Solve the heat equation (1) subject to the given conditions. Assume a rod of length             [eqn: L].

[eqn: u(0,t)`=`0,``u(L,t)`=`0# u(x,0)=` LEFT { pile{1,#0,} ```` pile{0<x<L/2#L/2<x<L} RIGHT }]

186. 13-3 Exercise 3  p.701

Find the temperature [eqn: u(x,t)] in a rod of length [eqn: L`] if the initial temperature is [eqn: f(x)]             throughout and if the ends [eqn: x=0] and [eqn: x=L] are insulated.

187. 13-4 Exercise 1  p.705

Solve the wave equation (1) subject to the given conditions.

[eqn: u(0,t)=0,`````````u(L,t)=0# u(x,0)= {1} over {4} x(L-x),`` {Partial u} over {Partial t} LEFT | pile{#{} _{t=0}} `=0]

188. 13-4 Exercise 3  p.705

A function [eqn: u(x,`t)] satisfies the below wave equation with the following boundary and            initial conditions:

Given wave equation: [eqn: {Partial ^{2} u} over {Partial x ^{2}} - {Partial ^{2} u} over {Partial t ^{2}} =0`]   for [eqn: 0<x<L] and [eqn: t>0]

Boundary conditions: [eqn: u(0,`t)=0] and [eqn: u(L,`t)=0]

Initial conditions: [eqn: {Partial u} over {Partial t} LEFT | pile{#{} _{t=0}} `=0 RIGHT .] and [eqn: u(x,`0)`=`f(x)] where [eqn: f(x)] is shown below.

Note that the function [eqn: u(x,`t)] is not a damping system. Find a non-trivial solution [eqn: u(x,`t)].

189. 13-5 Exercise 1  p.711

Solve Laplace's equation (1) for a rectangular plate subject to the given boundary               conditions.

[eqn: u(0,y)=0,u(a,y)=0# u(x,0)=0,`u(x,b)`=f(x)]

190. 13-8 Example 1  p.724

(Temperatures in a Plate)

Find the temperature [eqn: u(x,y,t)] in the plate shown in Figure 13.21(a) if the initial               temperature is [eqn: f(x,y)] throughout and if the boundaries are held at temperature zero           for time [eqn: t>0``.]

191. 15-2 Example 1  p.748

(Laplace Transform of a PDE)

Find the Laplace transform of the wave equation [eqn: a ^{2} {Partial ^{2} u} over {Partial x ^{2}} = {Partial ^{2} u} over {Partial t ^{2}} ``,``t>0```.]

192. 15-2 Example 2  p.749

Using the Laplace Transform to Solve a BVP

Solve [eqn: {Partial ^{2} u} over {Partial x ^{2}} = {Partial ^{2} u} over {Partial t ^{2}} ``,``0<x<1,`t>0`.]

subject to  [eqn: u(0,t)`=0,`u(1,t)`=0,`t>0# u(x,0)=0,` {Partial u} over {Partial t} LEFT | pile{#{} _{t=0}} =sin` pi x,```0<x<1``. RIGHT .]

193. 15-2 Exercise 1  p.752

In the following problems use tables as necessary.

A string is stretched along the [eqn: x]-axis between (0,0) and [eqn: (L,0)]. Find the                       displacement [eqn: u(x,t)] if the string starts from rest in the initial position [eqn: A] sin[eqn: (pix/L)].

194. 15-2 Exercise 3  p.752

The displacement of a semi-infinite elastic string is determined from

[eqn: a ^{2} {Partial ^{2} u} over {Partial x ^{2}} = {Partial ^{2} u} over {Partial t ^{2}} ``,``x>0,``t>0# u(0,t)=f(t),`` lim _{x -> INF } {u(x,t)=0,`t>0}# u(x,0)=`0,``` {Partial u} over {Partial t} LEFT | pile{#{} _{t=0}} `=`0,``x>0``.]

Solve for [eqn: u(x,t) .]

195. 15-3 Example 1  p.756

(Fourier Integral Representation)

Find the Fourier integral representation of the function

[eqn: f(x)= LEFT { pile{pile{````0,#``1,}#0,} ``````` pile{pile{x<0#`````````````0<x<2}#x>2.} RIGHT .]

196. 15-3 Exercise 3  p.759

Find the Fourier integral representation of the given function.

[eqn: f(x)= LEFT { pile{pile{0,#x,}#0,} ``` pile{pile{x<0#`````````````0<x<3}#x>3}]

197. 15-4 Example 1  p.763

Use the Fourier integral transforms of the function [eqn: u(x,`t)] to solve the following given partial differential equation.

[eqn: k {Partial ^{2} u} over {Partial x ^{2}} = {Partial u} over {Partial t} ,```- INF <x< INF ,``t>0# u(x,0)=`e ^{- LEFT | x RIGHT |} ``,`- INF <x< INF]

Hint: You may find that the following Euler’s identity, [eqn: e ^{ik} =cos(k)+isin(k)], useful with the properties of odd and even functions.

198. 15-4 Example 2  p.764

(Using the Cosine Transform)

The steady-state temperature in a semi-infinite plate is determined form

[eqn: {Partial ^{2} u} over {Partial x ^{2}} + {Partial ^{2} u} over {Partial y ^{2}} =0,```0<x< pi ,``y>0# u(0,y)=0,```u( pi ,u)=e ^{-y} ,``y>0# {Partial u} over {Partial y} LEFT | pile{#{} _{y=0}} =0,``0<x< pi .]

Solve for [eqn: u(x,y).]

199. 15-4 Exercise 1  p.764

Use the Fourier integral transforms of this section to solve the given boundary-value           problem. Make assumptions about boundedness where necessary.

[eqn: k {Partial ^{2} u} over {Partial x ^{2}} = {Partial u} over {Partial t} ,```- INF <x< INF ,``t>0# u(x,0)=`e ^{- LEFT | x RIGHT |} ``,`- INF <x< INF]

200. 15-4 Exercise 13  p.765

Use the Fourier integral transforms of this section to solve the given boundary-value           problem. Make assumptions about boundedness where necessary.

Find the steady-state temperature [eqn: u(x,y)] in a plate defined by [eqn: x GEQ 0,`y GEQ 0] if the            boundary [eqn: x=0] is insulated and, at [eqn: y=0,]

[eqn: u(x,0)= {cases{50,&0<x<1#0,&``````````````x>1.}}]