Introduction to Reactor Physics

Transcription

1 Introduction to Reactor Physics Vasily Arzhanov Reactor Physics, KTH

2 Course Objectives Having finished the course, you will be able to: Derive and Solve Equations Describing Multiplying Media in Several Approximations. Evaluate Important Reactor Parameters Including Performance and Safety. Describe and Compare Various Reactor Designs. Characterize Various Fuel Cycles. Represent Various Waste Management Strategies. Use Industry Adopted Soft- and Hardware for Evaluating Basic Reactor Parameters. HT 2008 Introduction 2

3 Nuclear Engineering Nuclear Engineering is an endeavor that makes use of radiation and radioactive material for the benefit of mankind. Like their counterparts in chemical engineering, nuclear engineers endeavor to improve the quality of life by manipulating basic building blocks of matter. Unlike chemical engineers, nuclear engineers works with reactions that produce millions of times more energy per reaction than any other known material. HT 2008 Introduction 3

4 Nuclear Energy It is free from the problems of fossil fuels: greenhouse gas emissions. A typical 1000 MW coal-burning plant emits yearly: tons of SO tons of NO x tons of fly ash USA generates 20% of the electricity at NPP; it avoided in 1999 the emission of 150 million tonnes of CO 2 On contrary, there is still the association of nuclear power with the tremendous destructive force. HT 2008 Introduction 4

5 Use of Nuclear Energy Energy generation (electricity, heating) Propulsion of naval vessels Nuclear-powered spacecraft Production of radioisotopes Activation analysis HT 2008 Introduction 5

6 The International System of Units Le Système International d Unités. Abbreviated as SI. World s most widely used in everyday life, commerce and science, notable exceptions: US and UK. SI was developed in 1960 from metrekilogram-second, MKS, rather than centimetre-gram-second, CGS. HT 2008 Introduction 6

7 Base Units Base quantity Name Symbol Length meter m Mass kilogram kg Time second s Electric current ampere A Thermodynamic temperature kelvin K Amount of substance mole mol Luminous intensity candela cd HT 2008 Introduction 7

8 Some Derived Units Derived quantity Name Symbol Other SI units Base units Plane angle radian rad m m -1 = 1 Force newton N m kg s -2 Energy, work, heat joule J N m m2 kg s-2 Power watt W J/s m2 kg s-3 Electric charge coulomb C s A Electric potential volt V W/A m2 kg s-3 A-1 Celsius temperature degree ºC K Activity becquerel Bq s -1 Absorbed dose gray Gy J/kg m2 s-2 Dose equivalent sievert Sv J/kg m2 s-2 HT 2008 Introduction 8

9 SI Prefixes Factor Name Symbol Factor Name Symbol yotta Y 10-1 deci d zetta Z 10-2 centi c exa E 10-3 milli m peta P 10-6 micro μ tera T 10-9 nano n 10 9 giga G pico p 10 6 mega M femto f 10 3 kilo k atto a 10 2 hecto h zepto z 10 1 deka da yocto y HT 2008 Introduction 9

10 Accepted Non-SI Units Name Symbol Value in SI minute (time) min 1 min = 60 s hour h 1 h = 60 min = 3600 s day d 24 h = s degree (angle) º 1º = (π/180) rad minute (angle) 1 = (1/60)º = (π/10800) rad second (angle) 1 = (1/60) = (π/ ) rad liter L 1 L = 1 dm 3 = 10-3 m 3 tonne t 1 t = 1000 kg electronvolt ev 1 ev = J unified atomic mass unit amu, u, 1 u = m u = kg m u atomic unit of mass m e m e = kg Astronomical unit ua m HT 2008 Introduction 10

11 Currently Accepted Non-SI Units Name Symbol Value in SI nautical mile 1852 m bar bar 1 bar = 0.1 MPa = 100 kpa = 10 5 Pa ångström (angstrom) Å 1 Å = 0.1 nm = m barn b 1 b = 100 fm 2 = m 2 = cm 2 curie Ci 1 Ci = Bq roentgen R 1R = C/kg rad rad 1 rad = 1 cgy = 10-2 Gy rem rem 1 rem = 1 csv = 10-2 Sv 1 kwh = (1000 W) (3600 s) = J 1 mmhg = 1 Torr = (1/760) atm = Pa HT 2008 Introduction 11

12 Some Units in Reactor Physics Very often, centimeter will be used rather than meter. Mass density in g/cm 3 : ρ w 1 g/cm 3 Number density: #/cm 3 : n = n/cm 3 ; N = atom/cm 3 Velocity in m/s: v th = 2200 m/s Energy in ev: ε f 200 MeV per 1 fission of 235 U HT 2008 Introduction 12

13 Fundamental Particles Particles of Interest to Nuclear Engineering Electron, e or e - Positron, e + Neutrino, ν, ν e Classified as leptons (λεπτος small, thin) do not experience the strong interaction. Neutron, n Proton, p Classified as hadrons, any strongly interacting composite subatomic particles (composed of quarks): HT 2008 Introduction 13

14 Fundamental Constants Universal Name Speed of light Planck constant Symbol = Value c = m/s h = Js Electromagnetic Atomic and nuclear Physico-chemical Elementary charge e = C Electron mass m e = kg Neutron mass m n = kg Proton mass m p = kg Atomic mass const. m u = kg Avogadro constant N A = mol -1 Boltzmann constant k = J K -1 Molar gas constant R = J mol -1 K -1 HT 2008 Introduction 14

15 Atoms and Nuclei Atoms (ατομος - indivisible) are building blocks of gross matter. α-particle Simplified helium model Atomic number, Z, is the total number of protons. Neutron number, N, is the total number of neutrons. Atomic mass number, A, is the total number of nucleons. He He A = Z+N Q = Ze He α HT 2008 Introduction 15

16 Nuclides and Isotopes A particular value of (Z) defines a chemical element. A particular value of (Z,N) defines a nuclide. Each nuclide (Z,N) is considered an isotope of the corresponding chemical element (Z). By extension, two nuclides with the same Z value are called isotopes of each other. Oxygen has three stable isotopes, 16 O, 17 O, 18 O and five known unstable, 13 O, 14 O, 15 O, 19 O and 20 O. In nature: Isotope Abundance 16 O 99.8 % 17 O % 18 O % a w 17 ( ) 17 ( ) 17 N ( O) 16 ( O) + 17 ( O) + 18 ( O) O 100% N N N 17 m( O) 16 ( O) + 17 ( O) + 18 ( O) O 100% m m m HT 2008 Introduction 16

17 Some Important Nuclides Z Nuclide Abundance a/o Half-life 0 n 12 m 1 H H H y 5 10 B 11 B C C C 5736 y 234 U y U y 238 U y HT 2008 Introduction 17

18 Example Problem. A glass of water is known to contain atoms of hydrogen. How many atoms of deuterium ( 2 H) are present? Solution. Isotopic abundance of 2 H is a/o. The fraction of 2 H is therefore The total number of 2 H is then = Z Nuclide Abundance a/o Half-life 0 n 12 m H 2 H 3 H 10 B 11 B 12 C 13 C 14 C 234 U 235 U 238 U y y y y y HT 2008 Introduction 18

19 Unified Atomic Mass Unit "The AME2003 atomic mass evaluation (I). Evaluation of input data, adjustment procedures". A.H. Wapstra, G. Audi, and C. Thibault. Nuclear Physics A729, 129 (2003). "The AME2003 atomic mass evaluation (II). Tables, graphs, and references". G. Audi, A.H. Wapstra, and C. Thibault. Nuclear Physics A729, 337 (2003). Let m( 12 C) be the mass of neutral 12 C. Arbitrarily, we set m( 12 C) = 12 u. m 1u 1 m 12 u 6 ( 12 C) HT 2008 Introduction 19

20 Atomic Weight The atomic weight of an atom is the mass of the neutral atom expressed in atomic mass units. M ( A ) Z ( A ) ( A ) ZX m ZX ( 12 m ) u m 6C m X = 12 (unitless number!!) The mass of any atom in amu is numerically equal to the atomic weight of atom in question. ( A ) ( A X X) Z = Z u m M m In practice, it is acceptable: M ( A ) Z X A HT 2008 Introduction 20

21 Atomic Weight of Mixtures The atomic weight of an element is the average atomic weight of the mixture. M γ M i i i γ i is isotopic abundance in a/o. M i is atomic weight of the ith isotope. Isotope Abundance [a/o] Weight 16 O 99.8 % O % O % M(O) = M(O nat ) = HT 2008 Introduction 21

22 Molecular Weight The total mass of a molecule relative to the mass of neutral 12 C is called the molecular weight. To a very good precision, the molecular weight is merely the sum of atomic weights of the constituent atoms. M(O 2 ) = = HT 2008 Introduction 22

23 Gram Atomic Weight Atomic and molecular weights are unitless numbers. By contrast, gram atomic (molecular) weight is defined as the amount of a substance having a mass, in grams, equal to the atomic (molecular) weight of the substance. This amount (number of entities) of material is also called a mole. Thus 1 g.a.w. or 1 mole of 12 C is exactly 12 grams of this isotope. 1 mole of natural O g. HT 2008 Introduction 23

24 Avogadro s Number The number of structural elements in one mole The mass of 1 mole of A Z A ( Z ) A ( X) 10 3 A A Z kg ( ZX) ( ZX) X M X g = = M = m N = M m N u kg 23 # N = N A m mol u One mole of any substance contains the same number of entities, namely N A. HT 2008 Introduction 24

25 Mole as Unit The mole (symbol: mol) is the SI base unit that measures an amount of substance. The mole is a counting unit. A mole is much like "a dozen." A mole is the amount of substance of a system, which contains as many elementary entities as there are atoms in kilogram (or 12 grams) of 12 C, where the carbon-12 atoms are unbound, at rest and in their ground state. According to the SI, the mole is not dimensionless, but has its very own dimension, namely "amount of substance", comparable to other dimensions. The SI additionally defines the Avogadro constant as having the unit reciprocal mole. Subatomic (Neutrons, protons, electrons, photons) Mole Atomic (Neutral atoms, ions) Molecular (Neutral molecules, ions) HT 2008 Introduction 25

26 Atomic Radii Atomic radius is not a precisely defined physical quantity, nor is it constant in all circumstances. The value assigned to the radius of a particular atom will always depend on the definition chosen for "atomic radius", and different definitions are more appropriate for different situations. A reasonable definition is an average distance Except for a few of the lightest elements, these average radii are approximately the same for all atoms, about m. HT 2008 Introduction 26

27 Radii and Periodic Table HT 2008 Introduction 27

28 Nuclear Radii Various types of scattering experiments suggest that nuclei are roughly spherical and appear to have essentially the same density. The data are summarized in the expression called the Fermi model: r = r A 0 13 r = 1.25 fm = m The constant density, V ~ A, suggests that nuclei are similar to liquid drops. HT 2008 Introduction 28

29 Mass and Energy Mass and energy are equivalent and convertible, one to the other. Complete annihilation of m 0 releases Erest = m c g E = J = kwh. 2 mc e = MeV e - e + 2 mc u = MeV HT 2008 Introduction 29

30 Particles in Motion Apparent mass: m = m 0 1 v c 2 2 p = mv Total energy: E mc E E 2 tot = = rest + k Kinetic energy: Ek = mc m0c = m0c v c E k m 0 v when v c HT 2008 Introduction 30

31 Relativistic Effects E = m v is accurate enough when v 0.2c or E 0.02E k k rest Electrons: E k 10 kev (relativistic formula should be used). Neutrons: E k 20 MeV (classical formula may be used). [ ev] = [ s] E= m 2 4 n v v E 2 m HT 2008 Introduction 31

32 Particle Wavelengths Planck: hc h E E = hν = λ λ = c Einstein: 2 ( ) 2 ( 2) 2 m E 0 = E= mc = pc + m 0 0c p= c Photon: λ = h p De Broglie: h h λ = = p mv E k E rest HT 2008 Introduction 32

33 Neutron Wavelength Non relativistic: Relativistic: λ = h me 2 n λ = E tot hc E rest λ = E 9 cm ev HT 2008 Introduction 33

34 Ionization The process of removing an electron from an atom is called ionization ev λ = hc E λ = E 6 88 kev m ev λ = = m Pb (Z = 82) HT 2008 Introduction 34

35 Atomic Excited States In a neutral atom, the electrons can be in a variety of different orbits or states. The state of lowest energy is the ground state. When the atom possesses more energy, it is said to be in an excited state or energy level. The highest energy state corresponds to the situation in which the electron has been completely removed from the atom and the atom is ionized Energy, ev HT 2008 Introduction 35

36 Nuclear Excited States Nucleons in nuclei are also moving in various orbits. The orbits are not as well defined and understood. In any case, there is a state of lowest energy, ground state; except for very lightest nuclei, all nuclei have excited states. 0 HT 2008 Introduction 36 Energy, MeV 12 C Energy levels

37 Number (Atom) Density Mass # Mole Mass = Volume Volume # Mole ρ ( A X) Z ( A ) ( A X X) ( A X) Z Z Z N M N A = N N A A N ( A X) Z ( A X) ( A X) Z ρ N = M ρ ( A X) Z A Z A A N HT 2008 Introduction 37

38 Example 1 For water of normal (unit) density, compute: the number of H 2 O molecules per cm 3 ; the atom densities of hydrogen and oxygen; the atom density of 2 H. Isotope Abundance 1 H a/o 2 H a/o HT 2008 Introduction 38

39 Solution 1 The atom weights: M H = M O = The molecular weight of water is M = 2 M H + M O = (natural mix.) ρ N N A 2 M N(H) = 2 N H O = ( H O) = = = N( O) ( ) 2 N( H) = N(H) = HT 2008 Introduction 39

40 Chemical Composition Let a substance be given by a chemical formula, X m Y n, for example Fe 3 O 4. Then the atom density of X or Y is N = m N and N = n N X X Y Y X Y m n m n The weight fraction of X is easily evaluated as w X = mmx mm + nm X Y HT 2008 Introduction 40

41 Weight Percent Usually, the components of mixtures are given in percent by weight. Let ρ be the physical density of the mixture. Then the density of ith component is ρ i ρ N w ρn i A i A = wiρ Ni = = Mi Mi to be compared with the case of isotopic abundance. N i = γ ρn i M A Gram atomic weight HT 2008 Introduction 41

42 Enrichment in Weight Percent Natural uranium, U nat must be enriched in 235 U. Often, we disregard Isotope Abundance, a/o 234 U U U It is the practice to specify enrichment in weight percent. The atomic weight of the enriched uranium may be evaluated as follows Total number of U atoms in cm 3 ρn M A = N = Ni = i i wiρn M i A Atomic weigh of mixture 1 wi = M M i i Atomic weigh of ith isotope HT 2008 Introduction 42

43 Example 2 A reactor is fueled with 1500 kg of uranium rods enriched to 20 w/o in 235 U. The remainder is 238 U. The density of uranium is 19.1 g/cm 3. 1) How much 235 U is in the reactor? 2) What are the atom densities of 235 U and 238 U in the rods? The atomic weights of 235 U and 238 U are and HT 2008 Introduction 43

44 Solution 2 1) 20 w/o means the that 20% of the total uranium mass is 235 U. The amount of 235 U is therefore kg = 300 kg. 2) The atomic weights of 235 U and 238 U are and N 235 w A = = = M ρn N 238 w ρn A = = = M HT 2008 Introduction 44

45 Example 3 The fuel for a reactor consists of pellets of uranium dioxide (UO 2 ) which has a density of 10.5 g/cm 3. The uranium is enriched to 30 w/o in 235 U. What is the atom density of uranium 235 U in the fuel? The atomic weights of 235 U and 238 U are and cm g ρ ( UO ) = 10.5 cm 2 3 Fuel pellet: U = 238 U+ 235 U UO 2 2 cm m 235 m( U) ( U) + m( U) = 30% HT 2008 Introduction 45

46 N ρ 235 Solution U ρ w ( UO ) U U 2 U = = = ρ w w M M ρ N ρ U = = 2 A U M M + 2M 1 w w = + M M M UO U O U M HT 2008 Introduction 46

47 Maxwellian Distribution In a gas being at thermal equilibrium, the energies of atoms or molecules are distributed according to the Maxwellian distribution function. Let N(E) be the density of particles per energy. NEdE= ( ) number of particles per unit volume having energies in de about E. 2 N NE ( ) = ( EkT) 12 e π kt EkT Boltzmann s constant: k = = J K ev K N = N( E) de 0 T is the absolute temperature in o K For solids and liquids, the energy distributions are more complicated. However, to a first approximation this formula is also valid for solids and liquids. But the parameter T differs somewhat from the actual temperature. The difference is small for temperatures above 300 o K. HT 2008 Introduction 47

48 Distribution Function 1 Ep = kt E= EN( E) de kt N = 2 0 N(E) T 0 kt = K 0 1 = ev ev E p kt E kt EkT HT 2008 Introduction 48

49 Gas Law Ideal gas: PV = n RT M P n N R T M A = V NA P = NkT HT 2008 Introduction 49

50 The END HT 2008 Introduction 50