Peaceful Use of Nuclear Energy

7/31/2019 Peaceful Use of Nuclear Energy

1/23

15.4 Thermal Utilization Factor

Homogeneous Reactor

By definition, the thermal utilization factor (f) is the ratio of the number of thermal neutrons

absorbed in the fuel to the total number absorbed by all processes. Denoting the fuel by the

subscribe 1, we then write for a reactor in which the different components having volumes V1, V2,

V3 ect. Face the neutron fluxes 1, 2, 3, ect.., we have

absorbedneutronsthermjalofno.Total

fuelinabsorbedneutronsThermal=f

+++=

....333222111

111

VVV

V

aaa

a (15.4-1)

For a homogeneous reactor, 1= 2 = 3 = and V1=V2=V3= So we get

.....321

1

+++=

aaa

af

.....332211

11

+++=

aaa

a

NNN

N

(15.4-2)

Here N1 = No, of nuclei of the fuel per unit volume;

N2 = No, of nuclei per unit volume of the i th component;

as are the microscopic absorption cross sections

Assuming only duel (u) and moderate (m) nuclei to be present, we then get

+=

+=

amau

au

ammauu

auu

NN

Nf

(15.4-3)

I should be noted that 1a for the fuel includes both fission and radioactive capture. For natural

uranium, = aua1 includes absorption in both 235 U and 238 U.

Heterogeneous Reactor

As we shall see below, in graphite moderated natural uranium reactor, the moderator and fuel

assembly has a heterogeneous arrangement. The thermal utilization factorfis smaller in this case

than in a homogeneous assembly, because the average thermal neutron flux in the fuel is less than

that in the moderator. Since the rate of neutron capture in a given material is equal to the product

of the flux and the absorption cross section in that material, we get

.....333312222111

1111

+++= VNVNVNVN

f aaa

a

(15.4-4)

Assuming only fuel (u) and moderator (m) to be present, we can write

FFNN

Nf

amau

au

ammauu

auu

+

=+

=

(15.4-5)


2/23

Where )/( uumm VVF = . The ratio may be called the thermal disadvantage factor.

The value ofFdepends on the size of the fuel element, its absorption cross section, spacing

within the fuel elements and neutron diffusion properties of the moderator. Larger the size of the

fuel element, greater is the value ofFand smaller is the thermal factorf.

15.5 Calculation of

The value of, the number of fast neutrons produced per thermal neutron absorbed in the

fuel, can be estimated as follows. For natural uranium as fuel, if we denote the two of nuclei by the

subscripts 1 and 2 for235 U and 238U respectively, we get for thermal neutrons.

rrf

f

NN

Nvvg

22111

11

)(

++==

rrf

f

NNv

21211

1

)/(

++= (15.5-1)

For natural uranium, N2/N1 = 139. For thermal neutrons, 1f = 280 b, 1r = 112 b, 2r = 2.8 b.

Assuming v = 2.5 for the fission of235U by thermal neutrons, we then get

34.18.2139)112580(

5805.2=

++

=

It may be noted that the value of is determined by the type of fuel. From the calculation given

above, we see that for every 100 thermal neutrons absorbed in the natural uranium fuel, 134 fast

neutrons are produced. Since k=pfwith slightly greater than 1, it is possible in principle in

principle to make an assembly of natural uranium with a moderator go critical (k= 1>1) by

suitably adjusting the values ofp andf.

If the fuel used is enriched in 235U content, then value of is different from that given above.

If for instance pure 235U is used as the fuel, we get

rif

if

rf

f v

N

vN

1111

11

)(

+=

+=

07.2112580

58047.2=

+

= (15.5-2)

Thus is much larger in this case if the fission is induced by thermal neutrons. For fission

induced by fast neutrons in 235U which is of importance in the atomic bomb, the value of of

different. Table 15.1 below given the values ofv and for the principal isotopes useful for nuclear

power production.

Table 15.1

NucleusThermal Fast

v v 233U

235U

239U

2.52

2.47

2.91

2.28

2.07

2.09

2.70

2.65

3.00

2.45

2.30

2.70


3/23

15.6 Multiplication Factor

The multiplication factor k depends on the four factors, p and f, apart from the non-

leakage factors. Considering for simplicity, an infinite reactor for which leakage can be neglected,

we note that depends on the nature of the fuel, over which we have no control. The fast fission

factor can be slightly changed by changing the radius of the fuel rods. However, is close to

unity. So its effect on kis small. Thus we are left with p andfto adjust. Unfortunately, the factors

which increase p tend to decrease fand vice-versa. So in practice, we have to find an optimum

condition for which kbecomes maximum. It turns out that with an optimum design of the blocks,

this can be realized more easily in a heterogeneous reactor, rather than in a homogeneous one,

since the gain a heterogonous p over-compensates for the loss due to decrease in fin the former.

With natural uranium as fuel, criticality condition (k>1) can be achieved in a homogenous

assembly only if heavy water is used as moderator which is the best moderator available. However,

in a heterogeneous assembly, criticality can be achieved even with a moderator of inferior quality,

e.g., graphite, using natural uranium as fuel. Thus, for a uranium-graphite reactor in which the

molar ratio of carbon to uranium is 215, the product pf= 0.823 for a heterogeneous assembly

giving k=v pf> 1.

On the other hand, in a homogeneous assembly for the same ratio of graphite to U nuclei, pf

= 0.595 giving k< 1.

In fig. 15.2 is shown the variation of k with the molar ratio of graphite to uranium in a

heterogeneous reactor for different diameters of the fuel rod. It can be seen that the multiplication

factor is greater than unity over a considerable range of values of the molar ratio.

In table 15.2 are listed the value ofp, f, and k for heterogeneous reactors using natural

uranium as fuel and graphite as moderator for different values of the cell radius in the uranium

graphite lattice. The fatter is defined as the radius of a circle having the same area as the unitsquare upon which the lattice is based. For example if the unit square is of 8 sides with a

cylindrical uranium fuel rod at each corner, then the cell radius is

=

81r inches. The radius of

the uranium rod is 0.55.

Table 15.2

Cell radius (cm) p f k10

11

12

13

0.866

0.905

0.9085

0.923

0.907

0.888

0.8765

0.846

1.056

1.0805

1.071

1.0455

The values of and used in these calculations are 1.308 and 1.028 respectively


4/23

15.7 Reactor Theory

For a given arrangement of the fuel and the moderator, in which khas a value greater than 1,

there is a definite and finite size of the reactor at which the neutron distribution, once established,

will maintain itself, neither increasing nor decreasing. This is known as the critical size of the

reactor. The critical size is determined by the condition that the rate of neutron leakage is equal to

the rate of neutron production (source strength) minus the rate of neutron absorption.

We assume that the neutrons are created and absorbed at the thermal energy only. This is the

basis of the one-group theory. We also consider a homogeneous arrangement of the fuel and the

moderator. These consideration will apply to a heterogeneous arrangement if the value of k

corresponding to the latter is used, provided the critical size is large compare to the size of the unit

cell in the lattice.

From the diffusion equation (13.20-12) we have, under steady state condition

=+ 02 QD a

where3

=D is the diffusion coefficient is the neutron flux, a is the macroscopic

absorption and Q is the source term. Ifkdenotes the multiplication factor, then = akQ . Wethen have

=+ 02 aakD (15.7-1)

Or,( )

01

2 =

+ D

k a

Writing

( )2

2 11

L

k

D

kB

a =

= (15.7-2)

Where ;3//2

== auaDL L is the diffusion length defined in 13.20. We then have022 =+ B (15.7-3)

B2 is known as thegeometrical bucklingand determines the critical size. Eq. (15.7-3) can be solved

for different geometrical shapes of the reactor by applying appropriate boundary condition.

15.8 Critical size of a reactor when slowing down takes place

The neutrons produced in fission are fast and must be slowed down to thermal energies to

produce further fission. Thus the calculation of the critical size of reactor involves consideration of

the slowing down process of the fast neutrons and the absorption of the thermal neutron.

As before, we start with the diffusion equation. However, the source term Q is now different.


5/23

We have seen in 13.22, that the slowing down density q(E) in a weakly absorbing medium

for the energyEis related to q0 (E), the slowing down density for the resonance escape probability.

Hence for thermal neutrons, we can write the source term as

),(0 thth rpqQ = (15.8-1)

Where ( )thrq ,0 is obtained by solving the Fermi age equation (13.23-9). If we write q forq0 we

can write the diffusion equation as

0),()()(2 =+ tha rpqrrD (15.8-2)

Now for each thermal neutron absorbed in the medium, the number absorbed in the fuel isf,

wherefis the thermal utilization factor. Hence the number of fast neutrons produced per thermal

neutron absorbed is p

kf =

. Since the number of thermal neutrons absorbed per unit volume

per second is a , the number of fast neutrons produced per unit volume per second at a point rwithin the medium is

== pkfrq aa /)0,( (15.8-3)Here ( )0,rq is the slowing down density for the source neutrons for which the Fermi age 0=

To solve the diffusion equation (13.20-11), it is necessary to know ( )thrq , for the thermal

neutron by solving the age equation.

0/2 = qq

Where is the Fermi age defined by Eq. (13.23-7). To solve the above equation, we apply the

method of separation of variables

)()(),( TrRrq = (15.8-4)

Then ddTrRqRTq /)(/,)( 22 ==

Substituting in the age equation we get

ddTrRrRT )()()()(

2 =

Or,22 )(

)(

1)(

)(

1B

d

dT

TrR

rR=

(15.8-5)

WhenB2 is a constant.

The -equation then becomes

2Bd

dT= (15.8-6)

Or, )exp()(2 BAT =

)exp()(),( 2 BrARrq = (15.8-7)

Hence for the fission neutrons for which =0, we get

)()0,( rARrq =


6/23

Using Eq. (15.8-3), we then get

p

krAR

a = )( (15.8-8)

)exp()(),( 2 Brpk

rq a

= (15.8-9)

So the diffusion equation becomes

0)exp()()()( 22 =+ BrkrrD aa

Or,{ }

0)(1)exp(

)(

2

2 =

+ rD

Bkr

a

But the square of the diffusion length is given by (see Eq. 13.20-16a)

== 3//2 axaDL

Hence we get finally the critical thermal diffusion equation as

{ }0)(

1)exp()(

2

22 =

+ r

L

Bkr

(15.8-10)

Eq. (15.8-7) shows that (r) is proportion to R(r). If the extrapolation distance is the same for

(r)=0 and q(r) = 0 will satisfy the same differential equation asR(r):

0)()( 22 =+ rBr (15.8-11)

Comparing the Eq. (15.8-10) and (15.8-11), we get

222 1)exp( BLBk th = (15.8-12)

Or, 11

)exp(22

2

=+

=

BL

Bkk theff

(15.8-13)

This is the critical equation for a bare homogeneous reactor. B2, as before is the geometrical

buckling. Knowledge of the physical characteristics of the reactors (k, andL2) givesB2 which is

a geometrical quantity depending on the shape and size of the reactor. With the help of Eq. (15.8-

13), it is thus possible to determine the critical size of a reactor of a given shape from the known

physical properties of the material.

We shall see below how to obtain the critical size of a reactor of different shapes in terms of

the buckling by solving Eq. (15.8-11) for different shapes.

SinceB2 is inversely related to the reactor size, for a large reactorB2 must be small. It is then

possible to write.

Exp ( )

2

22

1

11

BBB

+==

Hence the critically equation (15.8-13) becomes

( )( )( )22222

2

111

exp

LBB

k

BL

Bkk

th

theff ++

=+

=

Neglecting the term inB4, we than get


7/23

( )theff

LB

kk

++=

221

We define the migration area

>> showing intense absorption. In D2O, thL >> .

This is the reason for D2O being the best moderator.

15.9 Critical size of reactors of different shapes


8/23

We now consider the solution of Eq. (15.8-11) for three commonly used reactor shapes, viz,

spherical, rectangular parallelepiped and cylindrical. We shall consider the reactor to be bare

(without any reflector surrounding it) and homogeneous with pure 235U as the fuel.

(1) Spherical

Using spherical polar coordinates, we get from Eq. (15.8-11)

01 22

2=+

Bdr

dr

dr

d

r (15.9-1)

Here the flux is a function ofronly, the centre of the sphere being taken as the origin. does

not depend on the angular coordinates. Writingr

ryr

)()( = , we get from Eq. (15.9-1)

02

2

2

=+ yBdryd

The solution is Brr

ABrAry cossin)( 21 += , which gives

Brr

ABr

r

Ar cossin)( 21 +=

We now apply the boundary conditions. The first of these states that the flux must be finite at

every point including r= 0.

This makesA2 = 0 so that

Brr

Ar sin)( 1= (15.9-2)

The second boundary condition states that the flux vanishes at the extrapolated boundary (see

13.21) which is at RRrr ,0 71.0 +== being the actual radius of sphere. Hence

0sin

0

01 =r

BrA

Or sinBr0 = 0 which gives

..)..........,.........3,2,1(0 == nnBr

0r

nB

= (15.9-3)

So finally we have, writingA1 =A

0/sin)( rrnr

Ar = (15.9-4)

The differential equation (15.8-11) is satisfied for an infinite number of values of the bucking B2

corresponding to n = 1, 2, 3 Only the lowest eigenvalue with n = 1 is of significance, since it is

the only value ifB2 for which the flux is positive everywhere. So takingB = /r0 orr0 = /B we

get the critical size of the spherical reactor as

33

43

0

130

3

4

3

4

BBrVsph ===

(15.9-5)


9/23

(2) Rectangular parallelepiped of sides a, b, c including extrapolated distances:

Eq. 915.8-11), in Cartesian coordinates with the origin at the centre of the rectangular

parallelepiped, gives

02

2

2

2

2

2

2

=+

+

+

Bzyx

We write )()()(),,( zZyYxXzyx =

Separating the variable we get

0111 2

2

2

2

2

2

2

=+++ Bdz

Zd

Zdy

Yd

Ydx

Xd

X (15.9-6)

The first three terms of the l.h.s of the above equation can be separated and put equal to a

constant each:

2

2

22

2

22

2

2 1,

1,

1 ===

dz

Zd

Zdy

Yd

Ydx

Xd

X

Than2222 B=++ (15.9-7)

The solution of thex-equation is of the form

xAxAX sincos 21 +=

This must be symmetric for +x and x. HenceA2 = 0. So we have

xAX cos1= (15.9-8)

The flux vanishes at the extrapolated boundary. We assume that these are at 2/ax += ,

2/by += and 2/cz += . Hence

02/cos1 =aA

Or,

+=

2

12/ 1na

With n1 = 0, 1, 2, 3, . We then get

an /)12( 1 += (15.9-8a)

and( )

+=

a

xnAX

12cos 11 (15.9-8b)

The solution ofy andzequations can be similary found:

( )b

ynBY

12cos 21

+= (15.9-9)

( )c

znCZ

12cos 31

+= (15.9-10)

With bn /)12(2

+=

(15.9-9a)

cn /)12( 3 += (15.9-10a)

Where n2, n3= 0,1,2,3. As in the case of the spherical reactor, the critical size is determined by

the values n1 = 0, n2=0, n3=0. So we get finally


10/23

c

z

b

y

a

xA

coscoscos= (15.9-11)

From Eq. (15.9-7) we then have

++=++= 2222222 111

cbaB (15.9-12)

For a cube of side a, we get the minimum critical volume. In this case

222 /3 aB = (15.9-12a)

The critical volume of the reactor in this case is

( )3

33 161/33B

BnaVcr === (15.9-13)

(3) Cylinder of radius a and height h:

We use cylindrical; polar coordinates ),,( zr

and write the Laplacian as

2

2

2

2

22

22 11

dzdrdrrr

+

+

+

=

Because of symmetry, there is no -dependence So we can write Eq. (15.8-11) as

011 2

2

2

2

2

22

2

==

+

+

+

B

dzdrdrrr (15.9-14)

We write )()(),( zZrRzr =

Then 0111 22

2

2

2

=++

+ B

dzZd

ZdrdR

rdrRd

R

We can then split the above equation into the following two:

2

2

2

2

2

2

1

11

=

=

+

dz

Zd

Z

dr

dR

rdr

Rd

R

and are constants satisfying the relation222 B=+ (15.9-15)

Thezequation has the solution

zAZ cos1 =

We ignore the term in the solution due to symmetry considerations ( is the same for z)

The flux vanishes at the extrapolated boundaries. Hence = 0 at z = h/2 where h/2

includes the extrapolated on each side.

cos (h/2) = 0

Or, h/2 = (n + )

With n = 0, 1, 2, 3, ..

= (2n + 1) /h

and( )

h

znAZ

12cos1

+=


11/23

As before, it is the fundamental with n = 0 which determines the critical size. So we have

h/= and )/cos( hzAZ = (15.9-16)

The radial equation is

Rdr

dR

rdr

Rd2

2

2 1=+

Where222 =B . The above equation can be rewritten as

0)( 22

22 =++ Rr

dr

dRr

dr

Rdr

Putting =r , we get

d

d

dr

d

d

d

dr

d== . and 2

22

2

2

d

d

dr

d=

Substitution gives

022

22 =++ R

d

dR

d

Rd

This is nothing but the Bessel equation of order zero. The solution is the Bessel function of order

zero:

)()( 0202 rJAJAR == (15.9-17)

From Eqs (15.9-16) and (15.9-17) we then get

hzrAJzr cos)(),( 0

= (15.9-18)

From the table of Bessel functions, it is found that the first zero ofJ0 (r) is reached at r= a where

a = 2.405

So that = 2.405/a

Here a is radius of the cylinder (including the extrapolated distance).

Eq. (15.9-15) then gives the buckling as

22222

)/()/405.2( haB +=+=Sing Eqs. (15.9-19) we can express the radius of the cylinder in terms its height:

222

22 )405.2(

=

hB

ha (15.9-20)

The volume of the cylinder is

222

322 )405.2(

==

hB

hhaV

The smallest critical volume is obtained by differentiating Vw.r.t. h ad equating it to zero

0)(

)405.2(2)405.2(3).(2222

42222222

=

=

hB

hBhhB

dh

dV

Or, 222 /3 Bh =

Hence the minimum critical volume is


12/23

33

22

min

3.148)405.2(

2

33

BBV ==

(15.9-21)

The flux distribution in the reactor of the different shapes considered above follows different

mathematical laws. However, in all cases, the flux decreases from the centre outwards. There is not

much difference in the nature of the flux distribution in the reactors of different shapes. So for

ready and rough calculation, a cosine function is commonly used.

5.10 Reactor Materials

A very large number of nuclear reactors are in operation at the present time in different parts

of the world. Though these differ widely in design and construction, their main components have

certain common features which guide the choice of the materials comprising them. These

components include fuel, moderators, reflectors, coolants, and control systems, claddings for the

fuels, structures and radiation shields. Special materials have for to be used for their fabrication,

because of the special needs associated with the fission chain reaction. We shall briefly discuss

these below.

1) Fuel

As bready stated, natural uranium in which the isotope 235U is present to the extent of

0.715%, is a commonly used reactor fuel. In many reactors, uranium enriched in

235

U has been asfuel.

Theoretically, any material fissionable with thermal neutrons can be used as the reactor fuel.

We have seen that there are only three isotope 233U, 235U, 239Pu which satisfy this criterion. Of

these, only 235U occurs naturally. The other two must be produced from the fertile materials 232Th

and 238U respectively (see Ch. XIV)

2) Moderators

The commonly used moderators are water, heavy water, graphite, beryllium and its oxide

and some organic compounds.

As seen before, a good moderator must have good slowing down property and must have

very low neutron absorption cross section (see 13.19). the performance of a moderator is

determined by the moderating ratio given by

a

s, .Table 13.7 shows that deuteron and heavy

water have the highest moderating ratios. Hence D2O is considered the best moderator. However, itis expensive. Even so, many power generating reactors use D2O as the moderator.

Carbon, in the form of graphite, is also a fairly good moderator. When a solid moderator is

required, graphite is commonly used. Chain reaction can be achieved in reactors using natural

uranium as fuel, moderated either by D2O or graphite.


13/23

Ordinary water (H2O), though best from the point of view of slowing down power is not as

good a moderator as the other two considered above, because it has high neutron absorption cross

section. Chain reaction cannot be achieved in a natural uranium ordinary water reactor. The fuel

must be enriched in the isotope 235U in an ordinary water moderated reactor.

Beryllium and its oxide are also good moderators, though rather expensive. Further, they are

toxic and have poor mechanical properties.

While choosing a moderator, the radiation damage in the intense radiation field within the

reactor core must be kept in mind.

3) Reflectors

The reactor theory discussed above applies to a bare reactor in which there is no reflector

surrounding the reactor. Actually most reactors use reflectors around them to reflect back the

neutrons leaking out of the reactor to the latter. This helps in achieving neutron economy, which in

turn reduces the amount (and cost) of fuel to be used. The neutrons, leaking out of the reflector

material and a large fraction of them are reflector for thermal neutron should have the same

characteristics as that for the moderator, so that it should have small absorption and large scattering

cross sections.

The efficiency of a reflector is measured by its reflection-coefficient oralbedo which is the

ratio of the number of neutrons reflected back to the number entering the reflector. The albedo

depends on the size and shape of the reflector. Generally, for smaller diffusion coefficient (D) and

large diffusion (L), the albedo increases, its limiting value being unity. For a thicker reflector, the

albedo increases. In practice a reflector with thickness equal to 2L is almost equivalent to one with

infinite thickness.

For fast neutron, heavy materials (uranium) are better as reflectors.

As stated above, the use of a reflector help in neutron economy. For a bare reactor, the

neutron flux goes to zero at the extrapolated boundary. With a reflector, the variation of

outwards from the reactor core becomes more flat, so that the flux is quite considerable near the

outer regions of the reactor.

4) Coolants

Intense heat is generated within the reactor core due to nuclear fission chain reaction. This

heat must be removed for the safe operation of a reactor by using suitable coolants. Apart from the

conventional coolant like air and water, the other coolants which have been used are heavy water

liquid metals.

A reactor coolant should have the following properties;

(a) It should have good thermal properties, e.g., high specific teat and high thermal

conductivity to be a good heat transferring agent; (b) It should have low power requirement for

pumping; (c) It should have high boiling point and low melting point, so that its vapor pressure is


14/23

not too high in the high temperature environment within the reactor nor should it solidify when the

reactor is shut down; (d) It should be stable against heat and radioactive radiations, both of which

are intense within the reactor; (e) Its neutron capture cross section should be small; (f) it should not

be toxic or otherwise hazardous; (g) It should not acquire long-lived radioactivity due to intense

neutron bombardment inside the reactor which can pose health hazard when the coolant is released

to the environment; (h) It should be readily available at low cost.

All the above characteristic cannot of course be expected a single material. So an optimum

choice has to be made, depending on the requirement in an individual case.

For low power reactor, air has been used as coolant in some cases, though it suffers from the

disadvantage that nitrogen has a relatively large neutron absorption cross section. It is not good at

high temperature, since in chemically reacts with many of the materials within the reactor at high

temperature. Hydrogen would be the best gaseous coolant, but constitutes serious explosion.

Carbon-dioxide has been used in some power reactors. It suffers from the disadvantage that it

reacts with the graphite, used as moderators, at high temperature.

Liquid coolants are preferred over gases from the point of view of heat transfer property.

Pure ordinary water (H2O) has been used as a coolant in many reactors because of its easy

availability and low cost. However, it suffers from some disadvantage because of the high neutron

absorption cross section in hydrogen.

Heavy water has been used as coolant in some cases, because of its low neutron absorption

cross section. It is also quite expensive. For reactors operating at high temperature with high

thermal flux, liquid metals serve as good coolant. They should have low melting points and small

thermal neutron absorption cross section. Some of the suitable metals are bismuth (M.P.271 oC),

lead (327oC), sodium (98oC), tin (232oC) and potassium (62oC). of these, bismuth has the smallest

thermal neutron absorption cross section (a = 0.032b). Sodium has a = 0.5b). An alloy of sodium

and potassium has been found to be the best liquid metal coolant (66 oC:0.96b). The drawbacks of

sodium or the Na-K alloy are that these metals are highly reactive with many substances including

water. Further, neutron capture in sodium produces radioactive 24Na ( = 5h) emitting both -

particles and penetrating -rays, which liquid shielding of the coolant tanks, piping ect. The

pumping of the liquid metal coolant like sodium is carried out with the help of electromagnetic

pumps in which advantage is taken of its electrical conductivity to force it to flow in a pipe under

the influence of a magnetic field. This type of pump is leak-free because it has no moving part or

packings. So it can circulate the coolant, even if it is contaminated with induced radioactivity,

without any environmental pollution.

5) Structural and cladding materials

All reactors use some structural materials which are required as mechanical frame work for

the various components within the reactor. Besides, containers are required for fuel; coolants

control rods and measuring instruments. Uranium as fuel readily reacts with air, water another


15/23

fluids used as coolants. Hence the fuel element must be suitably packed within a claddingwhich

also prevents escape of the fission fragments.

The choice of the structural materials is guided by their mechanical properties, thermal

conductivity (which should be high) and coefficient of thermal expansion (which should be low).

In addition, they must be able to withstand severe thermal stress. They must be corrosion resistant

and should have low neutron absorption cross section.

Aluminium (a ~ 0.26 b) in relatively pure 2S form has been extensively used as a reactor

structural material, for cladding of fuel element and for other purpose not involving exposure to

high temperatures.

Zirconium (a ~ 0.18 b) has been found to be an excellent structural and cladding material,

especially where water under high pressure is used as a coolant. Zirconium must be freed from

hafnium (which is always present with it) since it has high neutron absorption cross section. An

alloy of zirconium known as zircaloy-2 has better corrosion resistance than pure Zr. The cost of

zirconium is rather high.

Another metal which has been found to be a good structural material is titanium. It is

however quite costly. Some ceramics have been found to be fairly good structural materials at

higher temperature. They can also be used in the fabrication of fuel elements, moderators, ect.

6) Control rodsLike any other energy generating device, a number reactor requires a proper control device

to ensure steady and smooth operation and to provide safeguards against accidents.

Since the neutrons are the agents responsible for the progress of the fission chain reaction in

the reactor, any control mechanism of the reactor involves the use of suitable neutron absorbers.

Two of the most frequently used thermal neutron absorbers for reactor control are cadmium

and boron. These materials have large neutron absorption cross sections. The control procedure

involves the insertion or withdrawal of these materials, usually in the form of rods or strips, into or

from the reactor core. Cadmium is used when the temperature is not too high, since it has a

relatively low melting point (3210C). It can also be used, alloyed with other metals, e.g. silver and

indium, since the alloy has a higher melting point. Boron, which the most common control material

is used in the form of boron steel.

Because of the loss of neutron economy in the case of the control rods within the reactor

core, these are also used within the reflectors in some reactors. An alternative method would be to

combine the motion of the core material (fuel and/or moderator) with that of the neutron absorber

on such a way that when the neutron absorber is inserted, some of the core is removed at the

sometime.

In some reactors, instead of the use of wasteful neutron absorbers like Cd or B, a more useful

absorbere.g. 238U or232Th has been used. Absorption of the neutrons in these materials produces

the useful fissile materials 239Pu or233u.


16/23

In fast neutron reactors, control by the use of neutron absorbers is in general not satisfactory.

The control can be achieved by movement of the fuel material either into or from the core or by

moving the reflector.

In a heavy water moderated reactor some degree of control can be achieved by adjusting the

level of the heavy water.

As stated in 14.5, reactor control is made possible because of the emission of delayed

neutrons in the fission process. We shall revert to this subject in 15.11.

7) Reactor Shielding

A reactor must be provided with adequate shielding to minimize the effects of the

biologically harmful radiations (mainly -rays and neutrons). The shield used for this purpose is

known as the biological shield.

The most commonly used material for the biological shield is ordinary concrete. Layers of

concrete, about 2 m (6 to 8 feet) thick surrounding the reactor is usually adequate for this purpose.

There are two special requirements. For reducing the intensity of the neutrons, these must be

slowed down to thermal energies and then absorbed as thermal neutrons. Hydrogenous material is

good for both these purposes. Special water bearing concrete has been used in some ordinary

concrete for the above purpose seems to be doubtful. Concrete with a special neutron absorber, e.g.

boron added to it has also been used.For reduction of the -rays intensity, some heavy elements (of high Z) should be added to the

concrete. Thus barites concrete, in which cheap barium sulphate is added to ordinary concrete has

been found to be a good -ray shield, barium having Z=56. The resulting heavy concrete has a

density about 1.5 times that of ordinary concrete. The cost in only marginally increased. Other

types of heavy concrete e.g. iron aggregate concrete, Ferro-phosphorous concrete or poured-lead

concrete are usually much more expensive.

Apart from the biological shield, a thermal shield is also needed. This is an inner wall,

usually of steel, which is placed between the reactor and the biological shield to protect the latter

from damage, due to excessive heating. Concrete shielding is also sometimes provided with

cooling arrangement on the inner side. Thermal shield uses materials which are effective for

absorbing -rays and for inelastic scattering of neutrons, so that a large portion of the energy

leaking out of the reactor is converted into heat in the thermal shield and only a small fraction of

these radiations enter the main biological shield.

15.11 Reactor Control: Effect of delayed neutrons

In 14.5 we saw that a small fraction of the neutrons emitted in fission are delayed neutrons.

If all the neutrons were emitted as prompt neutrons with a mean life 10 -14 s, the reactors control

would have been impossible.


17/23

A neutron life time is dependent on three factors: (a) mean time of fission neutron emission

tf; (b) slowing down time of fast neutrons to thermal energies ( ts); (c) diffusion time of the thermal

neutrons before capture (td).

Of these tf~ 10-14 s for prompt neutrons. The slowing down time is the time required by a fast

neutron produced in fission till it is therma-lised and is given by the relation.

E

dE

vt

thE

Es

s =

0

1

>1), the rate of energy release increases with time, because of the

increasing rate of fission from one generation to the next.

We introduce a quantity called reactivity

eff

eff

k

kk

1= (15.11-3)

If 0 is the neutron flux in the reactor, then aftern generations of fission, the flux increases

to

nk)1(0 += (15.11-4)

or,

+=+= ....3

)(

2

)()1ln()/ln(

32

0

kkknkn

Since kis usually small, we can neglect the higher powers and write

kn )/ln( 0 = (15.11-5)

This gives )exp(0 kn= (15.11-6)


18/23

If the time that elapse after n generations of fission have occurred in the reactor is t, then

0l

tn = So that

)/exp( 00 lkt= (15.11-7)

If, for instance, the reactivity k=0.005 (the effective multiplication factor increases to keff =

1.005), then after 1 second, the flux rises to (taking l0 = 10-3 s)

00

3

0 150)5exp()10/005.0exp( ===

So even if a very slight increase occurs in the value ofkeff suddenly, the neutron flux level in

the reactor and hence the reactor power level would rise by a factor of 150 within one second, if

the entire neutrons are emitted as prompt neutrons. Obviously it will be impossible to control the

reactor under this condition.

Fortunately such an exigency can be avoided due to the emission of a small fraction of

delayed neutrons in the fission process, which provides a built-in safety device in the reactor.

Thought the percentage of the delayed neutrons is quite small compared to that the mean-life of

fission neutron emission is actually of the order of tf ~ 0.1 s instead of 10-14 s for the prompt

neutrons alone. Thus the neutron life time from production in fission till its absorption is

determined by tf = 0.1 s and get td. In the example considered above, we can now write l0 = 0.1 s

and get

00 05.1)1.0/005.0exp( ==

Thus the power level rises by only 5% per second. The reactor period Tis defined as the time

in which the neutron flux increases by a factore: / 0 = e. Comparing with Eq. (15.11-7) we get

T= l0/k, so that

)/exp(0 Tt= (15.11-8)

For the example cited above, the reactor period is T = 0.1/0.005=20 second, which is

sufficiently long for accurate control of the reactor.

The fraction of delayed neutron emission is = 0.755% for235U fission. The reactor period T

can be made reactivity k


19/23

Research and development reactors are built to rest new method of reactors design, to supply

neutron beams for physical, chemical and biological research or for radioisotope production. The

requirements for a research reactor are safety, simplicity of operation, relatively high neutron flux

at low power level, easy accessibility for experiments and moderate cost.

The neutron flux and the power levelPof a reactor are related to each other. For a reactor

of volume V the rate of fission is fV where = ff N is the macroscopic fission crosssection Nbeing the number of fuel nuclei per unit volume. f is the macroscopic fission cross

section. The power level of the reactor is

P = Rate of fission x Energy released per fission

sJV f /106.120013 =

watts101.3101.3 1010

=

= NVV ff (15.12-1)

The mass of fissionable nuclei present in the reactor is

0N

VNAm =

WhereA = atomic weight of the fuel andN0 is Avogadro number.

For235U, A = 235 ; so that we get

kgVNVNA

m2426 1056.210025.6

=

= (15.12-2)

The power of the reactor then becomes

10

24

101.3

1056.2

=m

Pf

watts1026.813

mf= (15.12-3)

Thus for a given neutron flux the reactor power can be reduced if the mass m of the fuel used is

small. This can be achieved by using enriched uranium as fuel (which would reduce the resonance

absorption in238

U) and by using a good moderator, e.g., heavy water.

Research reactors fall into five main groups: (i) Natural uranium-graphite; (ii) Natural

uranium-heavy water; (iii) Enriched uranium-heavy water; (iv) Homogeneous enriched uranium-

ordinary water (water boiler); (v) Heterogeneous enriched uranium-ordinary water (swimming pool

and MTR).

(i) Natural uranium-graphite reactor:

Historically, self-sustaining nuclear chain reaction was first achieved on December 2, 1942

in a natural uranium-graphite assembly, called a pile designed and constructed by the physicists at

the University of Chicago under the guidance of Enrico Fermi. This was known as CP-1 and was

operated at a very low power level, because to shielding was provided.


20/23

Originally it had been planned to have an approximately spherical assembly. But since chain

reaction was achieved before the completion of the sphere, the actual shape of the pile was an

oblate spheroid with a flat top. The whole assembly was a matrix formed of horizontal layers of

graphite bricks within which lump of uranium were placed at the corners of square in alternate

layers.

Initially the pile was operated at a power level of 0.5 watt, which was raised to 200 watts.

Lack of shielding had posed serious radiation hazard for the operating personnel when the power

level was raised. So the pile was dismantled in 1943 and rebuilt at the Oak Ridge National

Laboratory with a radiation shield and was redesignated as CP-2.

The chief value of the CP-1 pile was to show that controlled fission chain reaction was

possible in a natural uranium-graphite assembly.

The rebuilt CP-2 was used for many early measurements on the basis of which the Hanford

production reactors were designed.

I had a vertical thermal column to transport the thermal neutrons out of the active core and

make them available for research purpose.

In addition to CP-2, the X-10 reactor at Oak Ridge and the Brookhaven National Laboratory

(BNL) reactors in the U.S.A. use natural uranium as fuel and graphite as moderator. The graphite

Low Energy Experimental Pile (GLEEP) and the British Experimental Pile (BEPO), both in

England, also use the exception of CP-2, all these reactors were cooled by forced convection of air,

CP-2 operated at a very low power (2 kW) had no provision for heat removal except by conduction

through graphite moderator and concrete shield.

The main advantage of the air-cooled natural uranium-graphite reactors lies in its large size

and adaptability. The X-10 and B.N.L. reactors radioisotope. However, the large size of these

reactors is also a draback. The minimum amounts required for criticality are about 30 tonne twice

as large. The BNI reactor with the shield has size 11.6 m x 16.8 x 9.1 m. Its total weight is about

20,000 tonne.

The schematic diagram of the graphite-uranium lattice in a graphite moderated natural

uranium reactors is shown in Fig 15.3.

(ii) Natural uranium-heavy water reactors (Heterogeneous)

Heavy water is a much better moderator than graphite. Hence reactors with D2O as

moderator have much smaller size than the graphite moderated reactors. Because of the very low

absorption cross sections of heavy water for neutrons, higher multiplication factor can be achieved

in these reactors and greater neutron flux is available.

The first heavy water moderated reactors CP-3 was built at Oak Ridge in the U.S.A. It used

about 3 tonne of natural uranium metal and about 6.5 tonne of heavy water in an aluminium tank

1.83 m in diameter and 2.75 m in height. In spite of the smaller mass of the fissionable material

used, the mean thermal neutron flux in the CP-3, operating at 300 kW was about the same as in the


21/23

X-10 pile operating at 3800 kW (5 x 1015 neutrons per m2 per s). the uranium metal rods about 2.8

cm in diameter and 1.83 m in length were arranged to form a square lattice with centre to center

distance of 13.65 cm. There were 120 such rods suspended in the heavy water which was used

both as moderator and coolant.

Amongst the other reactors of this type, mention may be made of the Canadian ZEEP (Zero

Energy Experimental Pile) and the NRX (National Research Experimental) reactors. The latter

uses ordinary water as coolant flowing through the annular spaces around the fuel rods in the

reactor. A large Canadian reactor of this type, known as NRU, is several times as powerful as the

NRX. It has a very high neutron flux (3 x 1014 neutrons per m2 per s) and is used for testing and

plutonium production. In most heavy water reactors, the fuel elements are cooled by convection of

the moderator (D2O) itself, which is circulated and cooled in an external heat-exchanged. In the

French P-2 reactor at Saclay, nitrogen gas is used as the coolant. The reflector and the shield are

cooled by air in this reactor.

Fig. 15.4 shows the schematic diagram of the fuel rod and coolant channel of the D 2O

moderated NRX reactor.

(iii) Enriched uranium-heavy water reactor (Heterogeneous)

The original CP-3 reactor was modified in 1950 by replacing the natural uranium fuel with

uranium in 235U. This revised CP-3 reactor was replaced by CP-5. The fuel elements are made of

enriched uranium-aluminium in the shape of curved sandwich plates with 0.05 cm thick central

layers clad on each side with 0.05 cm aluminium. The coolant is heavy water itself. A graphite

reflector surround the reactors tank. The reactor contains only 1.2 kg of 235U and 6.6 tonne of

heavy water-within an aluminium tank 1.8 m in diameter and 2.3 m in height. The design power of

the reactor is 4000 kW at a neutron flux of 6.2 x 1017 per m2 per s.

(iv) Homogeneous enriched uranium-ordinary water reactor (Water-Boiler)

The first reactor of the type, nicknamed Water Boiler, was put into operation in 1994. This

low power (LOPO) Water Boiler contained about 13 liters of a solution of 6 kg of uranyl sulphate

in ordinary water enriched in 235U to the extent of 14.6% (0.57 kg of 235U). The container was a

stainless steel sphere about 1 ft. in diameter.

The name Water Boiler is a misnomer, because the temperature of water is maintained below

the boiling point in these reactors.

The LOPO Water Boiler with a power level of 0.05 watt was reconstructed as HYPO later in

1944 having a power rating of 6 kW. It contained more than 10 kg of uranyl nitrate (0.87 kg of U-

235) in about 13 liters of the solution. Cooling coils were provided through which water could be

circulated. In a further improvement, known as SUPO, the power level was raised to 45 kW with a

neutron flux of 1.7 x 1016 neuts/m2 s. It was completed in 1951. It used uranium with an enrichment

factor of 88.1%. About 15 kg uranyl nitrate containing 0.87 kg of U-235 was used in it.


22/23

Because of the simplicity of its design and intrinsic safety, this type or reactor is considered

especially suitable as an experimental tool for the universities and industrial laboratories.

The main advantage of a homogeneous reactor is that it permits continuous processing of the

fuel to remove the fission products and if necessary of plutonium. They have a simpler mechanical

device, operates with liquids which are easily transported by pumping and do not require costly

metallic fuel elements. One of the chief disadvantages of this type of reactor is the problem of

corrosion due to the use uranyl nitrate (or sulphate). Also decomposition of the moderator is

serious problems.

If natural uranium is used as fuel, expensive heavy water is to be used in place of ordinary

water as the moderator.

(v)Heterogeneous enriched uranium-ordinary water reactor

Use of enriched uranium has the advantage that the multiplication factor can be increased

considerably, the limiting value of being 2.1 compared to 1.34 for natural uranium. This makes is

possible to use stainless steel and other moderate absorbers pf neutrons as the structural materials

within the reactor. Further, the critical mass of the fuel can be reduced to a few kilogramme only.

Reactors using enriched uranium yield higher neutron flux for a given power level.

The first heterogeneous ordinary water moderated enriched uranium reactor, known as the

bulk-shielding reactor was built at the Oak Ridge National Laboratory in the U.S.A. in 1951. Thereactors of this type are more popularity known as the swimming pool type reactors. The low

intensity test reactor (LITR) of this type has a power level of 3000 kW. The core a about 22

11

ft in size containing 3.2 kg of uranium-235 with an enrichment of ~90% alloyed with aluminium.

The core is suspended in a large tank (20 x 40 x 20 deep) containing ordinary water (hence the

name swimming pool reactor). The water serves as moderator, coolant and shield.

LITR was actually used as a mock up of the much more powerful Material Testing Reactor

(MTR) which has power level of 30 MW. Its core is similar to that of the LITR and is contained in

an aluminium tank2

14 ft in diameter. There is a beryllium reflector inside the tank and a graphite

reflector outside. Cooling is increased by placing the reactor in a pressure vessal and providing

high velocity cooling water. The maximum thermal neutron flux is 4 x 1018 per m2 per s. The fast

neutron flux is 1 x 1018 per m2 per s. The reactor used almost exclusively for the study of radiation

damage of proposed reactor components.

Many swimming pool type reactors have been built in different parts of the world, including

one in India (see later). The heterogeneous enriched uranium reactor is the cheapest and most

versatile reactor for neutron fluxes in the region of 1016 to 017 per m2 per s.


23/23

A number of MTR type reactors have been constructed to provide high fast neutron fluxes

over large volumes. A very large reactor of this type, known as the Engineering Test Reactor

(ETR) having a power level of 175 MW has been built at Arco, Idaho in the U.S.A. The main

problem of these reactors is the rapid burn up of the fuel. Refueling is necessary about once every

month.

(vii) Pulsed reactors

Intense neutron fluxed of very short duration can be obtained in a pulsed reactor. Peak power

output of 10 MW (neutron flux ~1022 per m2 per s) of 0.1 s duration has been obtained in a graphite

moderate reactor in Russia. The Russians have also built a pulsed fast reactor which uses two

plutonium cylinders separates by a gap. A steel disc with two uses two uranium blocks rotate at

5000 rpm between the cylinder butts. When the U blocks pass between the cylinders, chain

reaction takes place, producing an intense fast neutron beam of very short duration with a peak

power output of 150 MW.

The neutron flux enters a kilometer long tube of 1 m diameter, within which velocity

dispersion of the neutrons takes place by the time of flight method making possible resolution of

very narrow closely spaced neutron resonances in different materials.

Date post:	05-Apr-2018
Category:	Documents
Upload:	basry-cengil-dlibrazz
View:	213 times
Download:	0 times

Peaceful Use of Nuclear Energy

Documents