Massoud M. Engineering Thermofluids: Thermodynamics, Fluid Mechanics, and Heat Transfer

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882 VIe. Applications: Nuclear Heat Generation

Hence, T

≅ 600 F

– the profile of the water temperature in the average channel from Equa-

tion VIe.3.8

– steam generator inlet temperature from T

h,in

= T

= 600 F

– the profile of water in the steam generator tubes from Equation VIa.6.8. Note,

h,o

= T

= 550 F

– steam generator secondary-side inlet temperature, T

c,i

= T

= 430 F.

– turbine exit temperature from the condenser pressure, T

turbine,o

= 100 F.

Condenser or

Cooling Tower

Ultimate Heat Sink

Turbine

Steam Generator

Reactor

Vessel

Feedwater Heater

To Cold

Leg

Cold Leg

Hot Leg

550 F

600 F

Primary

side

Secondary

side

Balance of

Plant

and

430 F

550 F

530 F

Feedwater

100 F

Feedwater

100 F

70 F

80 F

Steam Generator

Primary

Side

Secondary

Side

5. Shutdown Power Production

Unlike other power producing systems, nuclear reactors continue to produce

power, albeit at a much smaller rate, even after being shutdown. Power generation

in nuclear reactors following shutdown is due to two sources: the power produced

by fission caused by the delayed neutrons and the power due to

and

decays of

radioisotopes. Power produced by delayed neutrons is short lived. It can be

calculated by solving the neutron kinetic equation with the insertion of a large

negative reactivity (–0.09). Such solution would show that the reactor power due

to delayed neutrons would decrease exponentially over a period of about 80

seconds (the half-life of the longest lived delayed neutron precursor). Hence, the

most dominant source of power following a reactor shutdown is the decay of

radioisotopes.

The rate of decay heat, as shown in Figure VIe.5.1 is generally obtained from

the models developed by the American Nuclear Society (ANS). In this figure,

ANS 1971_1 refers to the nominal value for the decay of fission products. ANS

1971_2 refers to the nominal value plus the decay of the heavy elements (U-239

and NP-239). ANS 1971_3 is the same as ANS 1971_2 but it accounts for 20%

uncertainty in the nominal and 10% uncertainty in the decay of the heavy ele-

ments. ANS 1971_4 applies 20% instead of 10% uncertainty to the decay of the

heavy elements.

ANS 1979_1 refers to the nominal value for the decay of fission products plus

the decay of the heavy elements. The ANS 1979_2 model also accounts for 2

5. Shutdown Power Production 883

uncertainty. The Branch Technical Position (BTP) in this figure is similar to ANS

1971_3. To highlight the differences between these model, the bottom figure fo-

cuses on the first 1000 seconds after shutdown.

Figure VIe.5.1. Various models for the estimation of decay power

884 VIe. Applications: Nuclear Heat Generation

As shown in Figure VIe.5.1, reactor decay power following shutdown drops

rapidly in the short term (about 1000 s) and in the long term approaches zero as-

ymptotically. Obtaining a general formula for decay power is difficult due to such

factors as dependency on the fuel cycle and duration of operation (resulting in dif-

ferences in heavy nuclide concentration and their decay characteristics). See

Problem 55 for a best estimate prediction of decay heat as recommended by El-

Wakil. This correlation is applicable for time greater than 200 s after shutdown.

QUESTIONS

– What is the diameter of the chlorine atom?

– What are subatomic particles?

– What is an isotope? What are the isotopes of uranium?

– Define atomic mass unit. How much energy in MeV is associated with 1 amu?

– Explain the difference between a chemical and a nuclear reaction

– What is the abundance of the U-235 isotope in naturally occurring uranium?

– What is the process by which we increase the mass of certain isotopes in natu-

rally occurring substance?

– What is mass defect? Why is the mass of a nucleus less than the total mass of

its constituents?

– Why are heavy elements such as uranium and plutonium more amenable to fis-

sion?

– In how many ways may a neutron interact with a nucleus?

– What are the differences between elastic and inelastic scattering?

– Are microscopic and macroscopic cross sections properties of the neutron or of

the nucleus?

– What does the macroscopic cross section physically represent?

– Why do we refer to slow neutrons as thermal neutrons?

– What major assumption constitutes the basis of the diffusion equation?

– Mathematically speaking, what do temperature distribution in a rectangular

plate (Figure VIIb.2.1) and neutron flux distribution in a cylindrical core have

in common?

– Why, in an elastic scattering between a fast neutron and a nucleus, is most en-

ergy lost in collision with light nuclei than with heavier nuclei?

PROBLEMS

1. The atomic nucleus contains protons and neutrons while the electrons are orbit-

ing the nucleus on specific shells or orbits. Each shell is filled with a certain

number of electrons. The shells are identified with quantum numbers 1, 2, 3, …,

etc. The shell with the quantum number 1 is the closest orbit to the nucleus.

These are also referred to as orbits K, L, M, N, etc. Usually the shells closest to

the nucleus are filled first. The number of electrons each shell is filled is given by

. Thus, shell K is filled with 2, shell L with 8, shell M with 18, and shell N

with 32 electrons. Electrons that orbit in the outermost shell of an atom are called

Questions and Problems 885

the valance electrons. Shown in the figure are the structure and the valance elec-

trons for hydrogen, helium, lithium, and neon. Draw similar atomic structures for

sodium, phosphorous, and xenon.

10p

10n

2. How much energy corresponds to 1 lbm?

3. If the energy released by the Hoover dam in 2.5 days is 2.7E14 J, find the

equivalent mass associated with this amount of energy. [Ans. 3 grams].

4. Treating neutrons as a gas, we may describe the total number of neutrons per

unit volume by the Maxwellian distribution. If n(E) is the number of neutrons per

unit volume having energy E per unit energy, then n(E)dE is the number of neu-

trons per unit volume having energies in the range of E and E + dE so that:

πκ

/2/1

2/3

)(

−

where N is the total number of neutrons and T is the absolute temperature of the

medium. In this relation, ț is Boltzmann’s constant ț = 1.3806E–23 kJ/K =

8.617E –5 eV/K. Use the above information and find:

a) similar distribution for neutron velocity. [Hint: Substitute for E from the K.E.]

b) the most probable energy, the most probable velocity, and the energy corre-

sponding to the most probable velocity.

c) the average energy

[Hint: use the averaging method given by

()

(

)

nEdEEnE

= ].

[Ans.: a) E

T/2, b) V

= (2

T/m)

1/2

, and c)

= 3

T/2].

5. Calculate the most probable neutron velocity and the neutron energy corre-

sponding to the most probable velocity. Use room temperature of 20 C. [Ans.:

= 2200 m/s, and E = 0.0253 eV].

6. Steady state neutron flux in a bare spherical reactor of radius R is approxi-

mately expressed as:

()

EEr

/sin

exp

),,(

κπ

−=Ω

where

is the maximum flux at the center of the reactor. Use Equation VIe.1.4

and the relation between energy and velocity to find the number of neutrons in the

reactor. Gamma function properties are given in Chapter VIIb. [Ans.

1/2

(

3/2

886 VIe. Applications: Nuclear Heat Generation

7. Show that the atom density of an element is given by N =

/M where N

Avogadro’s number (6.023E23) and M is the molecular weight. Find the atom

density of C-12. Use the data for scattering and absorption cross sections and find

the total macroscopic cross section of C-12. Since the mean free path is

= 1/Σ,

show that C-12 is an excellent moderator.

8. Collision between neutrons and nucleus of the moderator results in slowing

down the newly born fast neutrons. Such a collision is depicted in the figure. The

striking fast neutron has an initial energy E

and an initial momentum p

. The tar-

get nucleus is initially at rest. Considering an elastic scattering, following the col-

lision, the scattered neutron has an energy of

E and momentum of

p while the

recoiling nucleus has an energy of

E and momentum of

p . Use the conserva-

tion of momentum and energy to drive a relation for energy of the scattered neu-

tron in terms of the initial neutron energy and mass number of the target nucleus.

[Hint: Find the momentum of the recoiling nucleus in terms of the momentum of

the initial and the scattered neutron. Substitute for momentum terms (p

= 2mE)

and for the recoiling energy from the energy balance].

Striking

Neutron

Target

Nucleus

Recoiling

Nucleus

Scattered

Neutron

′

9. The energy of the scattered neutron following an elastic scattering between the

neutron and the target atom is given as (note that the molecular mass of the nu-

cleus, M divided by the mass of neutron, m is M/m = A):

sincos

)1(

−+

′

ϕϕ

Find the minimum energy of the scattered neutron following a collision with the

atom of C-12. The striking neutron has an initial energy of 5 MeV.

10. An isotropic neutron source emitting S

neutrons/s⋅cm

is located on the sur-

face of a sphere of radius R. Find a) neutron flux at the center of the sphere and b)

neutron current at the center of the sphere through a mid plane. [Ans.: S

/2 and

/4 downward].

Questions and Problems 887

Problem 10 Problem 11

11. A plane (P) is located between a disk (D) and a point source. The disk emits

particles isotropically and is located at a distance

from plane P. The point

source emits S particles isotropically and is located at a distance

P from point O

on plane P. Find neutron flux and current at point O.

12. The collision in Problems 8 and 9 is described from the point of view of a sta-

tionary observer, referred to as the laboratory (LAB) system. Now, consider a

case were the observer is instead located at the center of momentum of the neutron

and nucleus, referred to as the center of momentum (COM) system. In this case

the total momentum before and after the collision is zero. Show that the velocity

of the center of momentum (which for non-relativistic events is the same as the

center of mass) for the stationary nucleus is given by V

COM

= V

n,LAB

/(A + 1) where

n,LAB

is the neutron velocity in the LAB system before collision. Also show that

n,COM

= A V

n,LAB

/(A + 1) and V

N,COM

= – V

n,LAB

/(A + 1) where V

N,COM

is the veloc-

ity of the nucleus before the collision in the COM system.

COMN

COMn

COMN

′

COMn

′

COM

LABn

′

COMn

′

LAB

COM

Problems 12, 13, and 15

13. Use the diagram showing neutron velocity before and after a collision to con-

clude that:

888 VIe. Applications: Nuclear Heat Generation

)1(

1)cos(2

′

COM

αα

cos

1 −

where a = [(A – 1)/(A + 1)]

is known as the collision parameter. Use this relation

to:

a) find the angle corresponding to the minimum energy of the emerging neutron

(

min

′

)

b) find

min

′

, the minimum energy of the emerging neutron

14. Neutron lethargy is defied as

)/ln(lnln EEEE

′

−=

′

−=

. Use the result

of Problem 13 and show that neutron lethargy in terms of the nucleus mass num-

ber may be expressed as

§ 2/(A + 2/3). [Hint: Finde an energy-averaged leth-

argy. The probability distribution function for elastic scattering and isotropic in

the center of mass is 1/(1 –

α)E].

15. Use the diagram to conclude that the cosine of the scattering angle in the LAB

system in terms of the cosine of the scattering angle in the COM system is given

as:

1)cos(2

)cos(

COM

LAB

16. Use the results of Problems 13 and 15 to plot E’/E as a function of both

LAB

and

COM

17. Consider the case of linearly anisotropic elastic scattering in the COM system

(

COM

) =

COM

where

and

are known constants. Find and plot the

distribution of the nuclear recoil energies

18. Find the probability of isotropic scattering into a differential solid angle is

Ω/4

where the differential solid angle dΩ is given as 2

rsin(

COM

)(rd

COM

[Hint: Integrate over

to get scattering in the segment]

Ω

Ω'

Ω

Ω'

Questions and Problems 889

19. Show that scattering in the COM system is isotropic. For this purpose, find

)cos(

COM

and interpret the result. [Hint: Multiply the cosine of the scattering

angle in the COM by the probability of scattering and integrate from 0 to

20. Use the result of Problem 15 and the method of Problem 18 to find the cosine

of the scattering angle in the LAB system,

)cos(

LAB

. Does the result show that

scattering in the LAB system is backward, isotropic, or forward scattering? [Ans.:

2/(3A)].

21. In Problem 13 it is shown that there is a one-to-one relation between the

change in neutron energy and the change in the scattering angle. Thus, it can be

concluded that p(E ĺ E’)dE’ = –p(

Ω ĺ Ω’)dΩ’ where p is probability and the

minus sign reflects the fact that the larger the scattering angle, the lower the en-

ergy of the scattered neutron. We represent p(

Ω ĺ Ω’) = 4π

(Ω ĺ Ω’)/

where

(Ω ĺ Ω’) is the differential scattering cross section and ³

Ω’

(Ω ĺ Ω’)dΩ’ =

Use this information and find p(E ĺ E’) for an elastic scattering and isotropic in

the COM where

(Ω ĺ Ω’)=

. [Ans.: p(E ĺ E’) = 1/(1 –

)E].

22. Use the result of Problem 21to find the average fractional energy loss in an

elastic scattering collision. The average fractional energy loss is defined as

∆ /E. [Ans. (1 –

)/2].

23. Regarding neutron-nucleus interaction, so far we dealt with elastic collision

for isotropic and anisotropic scatterings. In this problem, we want to find E’/E for

an inelastic scattering in which the target nucleus absorbs an amount of energy Q.

Use the energy equation, which now accounts for Q and the velocity diagram of

Problem 12 to show that:

)1(

1cos2

COM

′

ϕξξ

;

2/1

−=

24. Consider two groups of isotopes. Group A consisting of U-233, U-235, Pu-

239, Pu-241 and group B of Th-232, U-238, Pu-240, and Pu-242. Identify the

group that represents fissile and the group that represents fissionable nuclides.

25. Find velocity (m/s) and kinetic energy (eV) of a thermal neutron at a tempera-

ture of 500 F. [Ans.: 2964].

26. Find the temperature of a thermal neutron having energy of 0.11 eV. [Ans.:

1000 C].

27. Start with the one-group neutron diffusion equation and derive the relation for

neutron flux in a bare critical slab reactor. Find the maximum to average flux for

this reactor.

28. An isotropic surface source of S

neutrons/s⋅cm

is located on the surface of a

sphere of radius R. The sphere consists of a non-absorbing material. Find a) the

890 VIe. Applications: Nuclear Heat Generation

flux density at the center of the sphere and b) the net current of neutrons at the

center of the sphere through a mid plane. [Ans.: a) S

and b) 0].

29. An isotropic surface source emits S

neutrons/s⋅cm

and is distributed on the

surface of a hemisphere of radius R. The hemisphere consists of a non- absorbing

material. a) Find the flux density at the center of the hemisphere. b) Find the net

current of neutrons at the center of the hemisphere through a mid plane. [Ans.:

/2 and S

/4].

30. The left side of the slab shown in the figure is exposed to a monoenergetic

neutron beam of an intensity I

neutrons/s cm

. The slab material is homoge-

nously distributed and has an atom density of N atoms/cm

and a cross section of

t for interactions with incident neutrons. Show that the neutron distribution in-

side the slab is given by I(x) = I

exp(-Σ

x) where Σ

= N

. Find a) the probability

that a neutron does not have an interaction when moves a distance dx, b) the frac-

tion of neutrons without any interaction at x = L, and c) the average distance a

neutron travels before interacting with a nucleus located in dx.

x = 0

x = L

I(x)

31. Show that the following integral is zero:

()

[]

∂

∞

Σ−

βαβαβ

0coscossin

drdder

32. Consider a plane neutron source emitting S

neutrons/s⋅cm

in a non-

multiplying, infinite, homogenous medium. The material of the medium has high

scattering and low absorption cross section for neutrons. Find an expression for

neutron flux in terms of D and L of the medium. [Hint: Solve Equation VIe.2.12

in the x-direction with s = 0 subject to the boundary conditions given by VIe.2.15].

[Ans.:

(x) = (S

L/2D)e

–

(x)

Questions and Problems 891

33. An isotropic point source emits S

neutrons per second in an infinite non-

multiplying weakly absorbing medium. Find neutron flux as a function of r, D,

and L of the medium. [Ans.:

(x) = (S

Dr)e

–r/L

34. A plane neutron source emitting S

neutrons/s⋅cm

is located at the center of a

non-multiplying homogenous bare slab. The material of the medium has high

scattering and low absorption cross section for neutrons. Find an expression for

the neutron flux in the slab. [Ans.:

(x) = (S

L/2D) sinh(c

)/cosh(c

) where c

(b – 2x)/2L and c

= b/2L with b =

+ a/2].

(x)

a/2 a/2

35. The slab of Problem 32 is now placed between two slabs of weakly absorbing

materials. Find the neutron flux profile in each slab. Slab 2, which is blanketing

slab 1 is referred to as a blanket or reflector.

Reflector

(x)

a/2a/2