Lecture 2 - Thermal Design Principles

MASTER NUCLEAR ENERGY

NUCLEAR THERMOHYDRAULICSLecture 2 - Thermal Design Principles

Nicola Pedroni

17/09/2012

Outline

� Overall plant characteristics influenced by thermal-hydraulic considerations

� Coolant selection

� Plant thermal performance

� Energy production and transfer parameters

� Volumetric energy (or heat) generation rate

� Surface heat flux

� Linear heat generation rate or power rating

� Rate of energy generation

22


� Core power

� Thermal design limits*

� Fuel Pins With Metallic Cladding

� Graphite-Coated Fuel Pellets

� Figures of merit of core thermal performance

� Core power density

� Specific power

� Examples

2

Overall Plant Characteristics

333

Overall Plant Characteristics Influenced by Thermal-Hydraulic Considerations

Overall Plant Characteristics Influenced by Thermal-Hydraulic Considerations

Primary system temperature Tp and pressure Pp

Coolant selection

Plant thermal performance

Plant thermal performance depends on:� maximum allowable primary coolant outlet temperature, Tp = Thot (heat source)� minimum achievable condenser coolant inlet temperature, Tcond (heat sink)

= Tp

Hot leg to the SG

q = UA∆Tm

444

Tcond fixed (atmosphere)

= Tp

Turbine

Condensate

Feedwater train

piping

Boiling

Pumping+pipingTcond

~ Power prod.

Heat tranfered ~

↑ thermal performance = ↑ Thot

Ts

Maximum allowable primary coolant outlet temperature, Tp = Thot

IT DEPENDS ON THE COOLANT TYPE

LIQUID COOLANTS

Key aspect: relation between saturation temperature and pressure (Ts-Ps)(which does not exist for gas)

“Ideal” condition: primary coolant liquid at the higher T possible!

GAS COOLANTS

Pressure and temperature “decoupled”

555

Pressure and temperature “decoupled”


Liquid coolants

The Tsat-Psat relation

666

Water:� (Ideal) plant efficiency (Carnot) limited by critical point (Tmax ≈ 374 °C)

� Tmax achieved at very high Pp (≈22 Mpa)� For Light Water Reactors (LWRs), high Thot requires high Pp (≈7-15 MPa)

� high stored energy in primary coolant (dangerous in accidents)� increased structural piping and component wall thickness

Metals:� Psat less than atmospheric pressure at Tp of interest (≈500-550 °C)

� For Liquid Metal Fast Breeder Reactors (LMFBRs) Tp not limited by boilingpoint of coolant, but by creep lifetime characteristics of the stainless steelprimary system material


Gas coolants

� Single-phase coolants offer the potential for high outlet temperatures Tp = Thot without such inherently coupled high primary pressure Pp

� System pressure Pp is dictated by the desired core heat transfer capabilities which strongly depend on pressure

� Tp = Thot = 635-750 °C and Pp = 4-5 MPa

777

No saturation line

General Considerations: PWR vs BWR Thermal Efficiencies

� Plant thermal efficiency depends on the maximum temperature in the secondary or power generation system (saturation temperature Ts at which vapor is produced)

� PWR: Ts ≠ Tp = Thot, due to the temperature diffeence needed to transfer heat between the primary and secondary systems in the steam generator

� BWR: Ts ≈ Tp = Thot (neglecting losses), limited to saturation condition (vapor directly in turbine: average quality x ≈ 15%)

Turbine steam conditions in PWR and BWR are similar(although T and P are very different)

888

PWR and BWR have similar thermal efficiencies

(although Tp and Pp are very different)

Power Spent per Power Produced

Tcold

Thot The fluid through the core suffers pressure drops and require pumping power:

vA∆p

W

flow ⋅⋅== timeunitperdistancethroughforce&

∆pwhere ∆p = pressure drop through the circuit

Aflow = cross-sectional area of the coolant passagev = average coolant velocity

W 2.0µ& LIQUID BETTER THAN GAS:

2

2v

D

Lfp

ρ=∆

999

It can be seen that:

pcQ

W

⋅∝

2.0

2.0

ρµ

&

& LIQUID BETTER THAN GAS:cpl >> cpg

ρl >> ρg

Coolant hi [W/m2] W/Q (normalized towater)

Water 2000-10000 1

Organic liquid 1000-4000 4-10

Liquid metals 3000-20000 3-7

Gas 50-500 100

Remember:

µρvD=Re2.0Re−⋅= Cf (smooth tubes)

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Energy Production and Transfer Parameters

101010



� Energy production in a nuclear reactor is expressed by a variety of terms reflecting the multidisciplinary nature of the design process:



( )rqr

'''

( )Sq ''r

[W/m3]

[W/m2]

1111



� Rate of energy generation (in the n-th fuel pin � )

� Core power

11

( )Sq ''

( )zq'

q& nq&

Q&

[W/m ]

[W/m]

[W]

[W]

Energy Production and Transfer ParametersVolumetric Energy (or heat) generation rate


� energy-generation rate per unit volume at position of the fuel material

� computed by the reactor physicists using fission reaction rates and flux

� Assumption: energy released by a fission reaction is recovered at the position of

( )rqr

''' [W/m3]

rr

1212

� Assumption: energy released by a fission reaction is recovered at the position of the fission event (except for the fraction carried away by neutrinos and the fraction deposited in nonfuel materials)

12

( ) ( )rRRrq f

rr ∝'''

( ) ( )rRRrdepositionEnergy f

rr ∝(Fission fragments, γ’s, β’s) Total fission reaction rate

rrV


� The total fission reaction rate is obtained by summing the fission reaction rates of all the isotopes j considered (e.g., U-233, U-235, Pu-239, …)

� Let

( ) ( )∑=j

jff rRRrRRrr

( )Ejaσ

= microscopic absorption cross section for isotope j [barn] = [10-24 cm2]= equivalent projected area of an atom for an absorbtion reaction= proportional to the probability that one atom of type j absorb an incident neutron

of energy E

131313

� Macroscopic fission cross section = sum of all microscopic cross section for a fission reaction due to all atoms of type j within a unit volume interacting with an incident neutron per unit time

of energy E

( ) ( )EE jc

jf σσ += (fissions + captures)

( ) ( ) ( )ErNEr jf

jjf σrr ≡Σ ,

jM

AN

j

jvj isotopeofdensityatomic==ρ

jj isotopeofdensitymass=ρ

( )molmoleculesAv /106.022numberAvogadro 23⋅=

jM j isotopeofmassmolecular=


� The fission rate of isotope j at position due to the netron flux within the interval of neutron energy of E to E + dE is obtained from

� The fission reaction rate for isotope j due to neutrons of all energies is

rr ( )Er ,

rφ

( ) ( ) ( )dEErErdEErRR jf

jf ,,,

rrr φΣ=

( ) ( )∫∞

=0

, dEErRRrRR jf

jf

rr

1414

� Defining the energy per fission reaction of isotope j as , the heat-generation rate per unit volume at due to isotope j is

� Summing over all the isotopes yields the total volumetric heat-generation rate:

14

0

rr

jfχ

( ) ( )∫∞

=0

,''' dEErRRrq jf

jfj

rr χ

( ) ( ) ( ) ( )∑∫∑∫∞∞

Σ==j

jf

jf

j

jf

jf dEErErdEErRRrq

00

,,,'''rrrr φχχ


� In practice, the energy is subdivided into a few intervals or groups; a multi-energy group model is then used to calculate neutron fluxes. Thus,

( ) ( ) ( )∑∑=

Σ=j

K

kk

jfk

jf rrrq

1

'''rrr φχ

where K = number of energy groups

= equivalent macroscopic fission cross section and neutron fluxrespectively for the energy group k

( ) ( )rr kjfk

rr φ,Σ

1515

� If we use a one energy group approximation (OK for homogeneous thermal reactors at locations far from the reactor core boundaries), we have:

� If we also assume uniform fuel material composition, then:

15

( ) ( ) ( )∑ Σ=j

jf

jf rrrq

rrr11''' φχ

( ) ( )∑ Σ=j

jf

jf rrq

rr11''' φχ


� Assuming that for all fissile material, the volumetric core heat-generation rate is given by:

fjf χχ =

( ) ( )rrq ff

rr11''' φχ Σ= ∑=

j

jff ΣΣ 11where

161616

j

NOTICE: the thermal neutron flux in a typical LWR is only 15% of total neutron flux!BUT

Fission cross sections of U-235 and Pu-239 are so large at thermal energiesthat thermal fissions are 85-90% of total fissions

Energy Production and Transfer ParametersSurface Heat Flux

� Fuel element surface heat flux

� Important to the thermal designer

� Related to the volumetric energy generation rate as

( )Sq ''r

[W/m2]

( )rqr

'''

( ) ( )∫∫∫∫∫ =⋅VS

dVrqdSnSqrrr

'''''

171717

Where: S is the surface area that bounds the volume V where the heat generation occurs

VS

is the outward unity vector normal to the surface S

rrV

nr

Snr

nr

Energy Production and Transfer ParametersLinear heat generation rate or power rating


� Important to both the thermal and metallurgic designers

� Related to the volumetric energy generation rate as

( )zq' [W/m]

( )rqr

'''

( ) ( )∫∫∫∫ =VL

dVrqdzzqr

''''

181818

VL

Where: L is the length of the volume V bounded by the surface S where the heat generation occurs

rrV

Sz

L

Energy Production and Transfer ParametersRate of energy generation


( )∫∫∫=V

dVrqqr

& ''' Heat-generation rate in the heat-generating volume V

q& [W]

� If the volume V is taken as the entire heat-generating volume of the n-th fuel pin Vfn, the quantity is the heat-generation rate in the n-th pin

q&nq&

191919

( ) ( ) ( )∫∫∫∫∫∫ =⋅===LSV

n dzzqdSnSqdVrqqqnfn

''''''rrr

&&

Where: Vfn is the volume of the energy-generating region of the n-th fuel pin

is the outward unity vector normal to the surface Sn surrounding Vfnnr

L is the length of the active fuel element (pin)

These general expressions take different forms depending on the size and shapeof the region generating the heat (e.g., cylindrical pin, …)

Energy Production and Transfer ParametersThermal Parameters Averaged Over A Pin

� Define the following (mean) quantities:

( )fn

n

Vfnn V

qdVrq

Vq

fn

&r == ∫∫∫ '''1

'''

( )n

n

Snn S

qdSnSq

Sq

n

&rr =⋅= ∫∫ ''1

''

( )L

qdzzq

Lq n

Ln

&== ∫ '

1'

Mean volumetric generation rate in the n-th pin

Mean heat flux through surface Sn of the n-th pin

Mean linear power rating of the n-th pin

202020

For any cylindrical fuel rod

nfoncoconn qRLqRLqLq '''''2' 2ππ ==⋅=&

Energy Production and Transfer ParametersCore Power

� Core power

� It is obtained by summing the heat generated per pin over all the N pins in the core and the heat generated in the nonfueled regions (moderator and structural materials)

Q& [W]

N

∑

2121

� Defining γ as the fraction of power generated in the fuel:

21

nonfuel

N

nn QqQ &&& +=∑

=1

∑=

=N

nnqQ

1

1&&

γ ( )typereactortheondependsbut96.093.0 −≅γ

Energy Production and Transfer ParametersCore-Wide Thermal Parameters For An “Average” Pin

� Rimembering that nfoncoconn qRLqRLqLq '''''2' 2ππ ==⋅=&

∑∑∑∑====

==⋅==N

nnfo

N

nncoco

N

nn

N

nn qRLqRLqLqQ

1

2

111

'''1

''21

'11 π

γπ

γγγ&&

� Define the core-wide thermal parameters for an “average” pin

qqN

N

n && ≡∑1

(heat generation rate in the “average” pin)

222222

'''''2'1 2 qR

LqR

Lq

Lq

N

Qfococo π

γπ

γγγ==== &

&

N nn∑

=1

� NOTICE:

fuelfn V

Q

NV

Qq

&& γγ =='''- Core-average volumetric heat generation rate in the fuel:

- Core power density:

coreV

QQ

&='''

≠

(Vcore = Vfuel + Vmoderator + Vstructures)

Thermal Design Limits

232323



� All reactors (except High Temperature Gas Reactor-HTGR)

� Fuel pins with metallic cladding

2424

� HTGR

� Graphite-coated fuel pellets

24

Thermal Design Limits Fuel Pins With Metallic Cladding

� Thermal design limits � integrity of the clad

� In principle: limits expressed in terms of structural design parameters (e.g., strain, fatigue limits)

� In practice: complex behavior of materials in radiation and thermal environment � limits expressed on temperatures and heat fluxes

Cylindrical, metallic clad, oxide fuel pellets

252525

Thermal Design Limits Fuel Pins With Metallic Cladding - LWRs

� Clad average temperature, Tclad

� Steady-state: inherent characteristics of LWRs limit Tclad to a narrow band above the coolant saturation temperature � not necessary to set a

steady-state limit on Tclad

� Transient (in particular, Loss of Coolant Accident-LOCA): most important criterion is T < 2200°F to prevent extensive metal (Zirconium)-water

2626

criterion is Tclad < 2200°F to prevent extensive metal (Zirconium)-water reaction from occurring

26


� Fuel centerline temperature

� It is maintained well within its design limit due to restrictions imposed by the critical heat flux limit

272727


� Critical Heat Flux (CHF)

� It results from a sudden reduction of the heat transfer capabilities of the two-phase coolant (i.e., reduction of heat transfer coefficient h)

� For fuel rods:

fobco hD

Rq

h

qTT

2''''' ==− TbTb

282828

cobco hDh

TT ==−

Tco

TbTb

q’’ = “independent” parameter + Tb = nominally fixed (saturation)

↓ h � ↑ Tco

Thermal design limit on q’’



� PWR: low void fractions � heated surface cooled by nucleate boiling

Departure from Nucleate Boiling (DNB)

(vapor blanket)

292929

DNB depends on local conditions� correlations and limits in terms of heat flux

ratios between limiting and operating



� PWR: low void fractions � heated surface cooled by nucleate boiling

= Departure from Nucleate Boiling Ratio (DNBR)

Minimum Departure from Nucleate Boiling Ratio (MDNBR)

303030

Design limit: MDNBR ≥ 1.3 at 112% power (to account for overpower margin)



� BWR: high void fractions � heated surface cooled by a liquid film

Dryout depends on the channel thermal hydraulicconditions upstream of the dryout location, ratherthan on the local condition of the dryout:

� correlations and limits in terms of power

313131

� correlations and limits in terms of power

ratios between limiting and operating

Design limit: MCPR ≥ 1.2 at 100% power(Minimum Critical Power Ratio)

(see second part of the course)

Thermal Design Limits Fuel Pins With Metallic Cladding - LMFBRs

� Present practice is to require a level of soobcoling such that the sodium temperature does not exceed its boiling temperature for transient conditions

� Additionally, considerable effort is applied to ensure that coolant voiding in accident situations can be satisfactoritly accommodated

LMFBR limits placed on fuel and clad temperatures

323232

Thermal Design LimitsGraphite-Coated Fuel Pellets

� Coated particles deposited in holes symmetrically drilled in graphite matrix blocks to provide passages for helium coolant

� Inner layer and fuel kernel designated to accommodate expansion of the particle and trap gaseous fission products

� Fission gas release due to:

333333

� Diffusion

� Coating rupture

� Limits imposed to control gas release rates from coatings

� Limits on fuel particle center temperature:

� 100% power � 1300°C (minimize steady-state diffusion)

� Peak transient � 1600°C (minimize cracking of particle coatings)

Figures of Merit for Core Thermal Performance

343434



�Core power density

3535

�Specific power

35

Figures of Merit for Core Thermal PerformanceCore Power Density

� Core power density

� It is a measure of the energy generated per unit volume of the core

� Since the size of the reactor vessel and hence the capital costs are related to the core size, the power density is a measure of the capital cost of a design concept

coreV

QQ

&=''' (Vcore = Vfuel + Vmoderator + Vstructures)

core

fuel

V

Vq

γ'''

=

363636

( ) ( )22

2 '1'''4141'''

P

q

dzP

dzqRQ fo

square γπ

γ=

⋅=−∞

( ) ( )2

2

23

'1

23

2

'''6131'''

P

q

dzPP

dzqRQ fo

triang γπ

γ=

⋅=−∞

( ) ( ) squaretriang QQ −∞−∞ ⋅= '''155.1'''(LMFBRs) (LWRs: loosely-packed

� better moderation)

Figures of Merit for Core Thermal PerformanceSpecific Power

� Specific power

� It is a measure of the energy generated per unit mass of fuel material

� It is usually expressed as watts per grams of heavy fuel atoms (U, Pu, Th, but not O, C, etc.)

� While power density is a measure of the capital cost, specific power has direct implications on the fuel cycle cost and core inventory requirements

atomsheavyofMass

QSP

&=

373737

fLR

Lq

QSP

pelletfo ρπγ 2

'1

atomsheavyofMass

=

=&

fuelofgrams

atomsheavyfuelofgrams

fueltheinatomsheavyoffractionmass

=

=f*

* See later

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� Introduce the “smeared density” ρsmeared:

�

� ρsmeared includes the void that is present as the gap between the fuel pellet and the clad inside diameter

( ) fLR

LqSP

smearedgfo ρδπγ 2

'1

+=

3838

pellet and the clad inside diameter

38

( )22

gfo

pelletfosmeared

R

R

δπ

ρπρ

+=

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�

� Heavy atoms include all the U, Pu, Th isotopes and are therefore composed of fissile atoms (Mff) and nonfissile atoms (Mnf), where M is the molecular weight

� Fuel is the entire fuel-bearing material, i.e., UO2 but not the clad

� For oxid fuel:

fuelofgrams

atomsheavyfuelofgramsfueltheinatomsheavyoffractionmass ==f

+

3939

� The enrichment (r) is the mass ratio of fissile atoms to total heavy atoms

39

22 OOnfnfffff

nfnfffff

MNMNMN

MNMNf

+++

= (N = atomic density [atoms/cm3])

nfnfffff

ffff

MNMN

MNr

+=

nfnfffff

nfnf

MNMN

MNr

+=−1and


� It can be observed that for UO2:

� Using the expressions

� Divide each term of by

nfffO NNN +=2

nfnfffff

ffff

MNMN

MNr

+=

nfnfffff

nfnf

MNMN

MNr

+=−1and

r

N

M

MrN nf

ff

nfff −

=

1we obtain ( )

r

N

M

MrN ff

nf

ffnf

−= 1

nfnfffff MNMNf

+= NNN +=

4040

� Divide each term of by

� Substitute the expressions for Nff, Nnf and (Nff + Nnf) into the expression for f to obtain

40

22 OOnfnfffff

nfnfffff

MNMNMN

MNMNf

+++

=

( )( ) ( ) ( )

( )( ) ( ) ( ) 2

2

1/1

/1

1/1

/1

Onfffnf

ffnfff

nfffnf

ffnfff

UO

MMrMMr

rM

MMrr

r

MrMMr

rM

MMrr

r

f+

−+−+

−+

−+−+

−+=

nfffO NNN +=2


� For the case where Mff ≈ Mnf

( )( ) ( ) ( )

( )( ) ( ) ( ) 2

2

1/1

/1

1/1

/1

Onfff

nfffnf

ffnfff

UO

MMrMMr

rM

MMrr

r

MrMMr

rM

MMrr

r

f+

−+−+

−+

−+−+

−+=

414141

( )( ) ( ) ( ) 21//1 Onfffnf

ffnfff

MMrMMr

MMMrr

+−+

+−+

( )( )

2

2 1

1

Onfff

nfffUO MMrrM

MrrMf

+−+−+

=

Examples

424242

Examples

Determination of the neutron flux at a given power in a thermal reactor – Example 3.1 Nuclear Systems I

� A large PWR designed to produce heat at a rate of 3083 MW has 193 fuel assemblies each loaded with 517.4 kg of UO2. If the average isotopic content of the fuel is 2.78 weight percent 235U, what is the average thermal neutron flux in the reactor?

4343

reactor?

Assume uniform fuel composition, and Xf = 190 MeV/fission (3.04 x 10-11 J/fission). Also assume that γ = 95% of the reactor energy, i.e., of the recoverable fission energy is from the heat generated in the fuel. The effective thermal fission cross section of 235U (σf

235) for this reactor is 350 barns.

43

Determination of the neutron flux at a given power in a thermal reactor – Solution

� Use to find

� Since in this reactor, then

� The core-average value of the neutron flux in a reactor with uniform density of 235U is obtained from the average heat-generation rate by

( ) ( )rrq ff

rr1''' φχ Σ= ( ) ( )

ff

rqr

Σ=

χφ

rr '''

1

235

ff Σ=Σ 235235235

ff N σ=Σ

'''q

4444

� Multiplying both the numerator and denominator by the UO2 volume (VUO2) we get:

44

2352351

'''

ff N

q

σχφ =

22

2

2352352352351

'''

UOff

fuel

UOff

UO

VN

Q

VN

Vq

σχσχφ

&

==


� To use the above equation only N235 needs to be calculated. All other values are given. The value of N235 can be obtained from the uranium atomic density if the 235U atomic fraction (a) is known:

� The uranium atomic density is equal to the molecular density of UO2, as each molecule contains one uranium atom (NU = NUO2).

UaNN =235

4545

UO2, as each molecule contains one uranium atom (NU = NUO2). Thus:

� Multiplying each side by VUO2 we get

45

2

22235

UO

UOvUOU

M

AaaNaNN

ρ===

2

2

2

2

2

235

UO

UOv

UO

UO

UO M

mAaVaNVN ==


� Now

� The molecular mass of UO2 is calculated from

g109858.9kg

g1000

assembly

kg4.517]assemblies[193 72

2⋅=

⋅

⋅= UO

mUO

]g/mole[0439.235235235 =MU

]g/mole[0508.238238

238 =MU]g/mole[9994.15Oxygen =OM

MMM 2+=

4646

� To obtain the value of a we use the known 235U weigth fraction, or enrichment (r):

46

OUUO MMM 22

+=( ) 238235 1 MaaMM U −+=( ) OUO MMaaMM 21 2382352

+−+=

( ) 238235

235235

1 MaaM

aM

M

Mar

U −+==


� Expression can be solved for a:

� Then,

( ) 238235

235235

1 MaaM

aM

M

Mar

U −+==

( ) ( )028146.0

0278.010508.2380439.235

0278.0

0278.0

1238

235

=−+

=−+

=r

M

Mr

ra

( ) ]g/mole[9645.26921 2382352=+−+= OUO MMaaMM

g109858.9moleatoms

100225.6 723 ⋅⋅⋅mA

4747

� Now,

� Finally, the value for the average flux is obtained as:

47

Uatoms1027.6g/mole269.9645

g109858.9mole

100225.6028146.0 23527235

2

2

2⋅=

⋅⋅⋅==

UO

UOv

UO M

mAaVN

( )smneutrons/c1038.4

atoms1027.6neutronatom

cm10350

fissionJ

103.04

MWW

10MW308395.0

213

272

2411-

6

2352352352351

22

⋅⋅=

⋅

⋅⋅

⋅

⋅===

−UOff

reactor

UOff

fuel

VN

Q

VN

Q

σχγ

σχφ

&&

Heat transfer parameters in various power reactors – Example 3.2 Nuclear Systems I

� For the set of reactor parameters given below, calculate for each reactor type:

� Equivalent core diameter and core length

� Average core power density Q'" (MW/m3)

� Core-wide average linear heat-generation rate of a fuel rod , <q’> (kW/m)

� Core-wide average heat flux at the interface between the rod and the coolant, <q’’co> (MW/m2)

484848

Heat transfer parameters in various power reactors – Solution

� Only the PWR case is considered in detail here; the results for the other reactors are summarized

� Equivalent core diameter and length calculation

( ) 222 043.0207.0)spacinglateralassembly(areaassemblyFuel m===22 36.10assemblies241043.0areaassemblyFuelareaCore mmN assemblies =⋅=⋅=

mDmD

64.336.10areacore4

:diametercircularEquivalent 22

=⇒==π

mL 81.3lengthrodFuellengthCore ===

4949

� Average core power density

49

mL 81.3lengthrodFuellengthCore ===32 65.3981.336.10lengthcoreareacorevolumecoreTotal mmm =⋅=⋅=

3

3/85.95

65.39

3800

volumeCore

levelpowerCore''' mMW

m

MW

V

QQ

core

====&

Heat transfer parameters in various power reactors – Solution

� Average linear heat generation rate in a fuel rod:

� Average heat flux at the interface between rod and coolant

( ) mkWMW

NL

Qq /8.16

m/rod81.3assemblies241blyrods/assem236

380096.0' =

⋅⋅⋅==

&γ

5050

� Average heat flux at the interface between rod and coolant

50

( ) 23

/552.00097.0

/10/8.16'

2'' mMW

m

kWMWmkW

D

q

DLN

Q

RLN

Qq

cococo

co =⋅

====−

πππγ

πγ &&

Power density and specific power for a PWR – Example 2.1 Nuclear Systems I

� Confirm the core power density listed in Table 1 (Table 2.3 Nuclear Systems I) for the PWR case and comment the results. Calculate the specific power of the PWR with characteristics given in Tables 1 and then Table 2 (Tables 2.3 and 1.3, respectively, in Nuclear Systems

5151

then Table 2 (Tables 2.3 and 1.3, respectively, in Nuclear Systems I). Assume a pellet density of 95% of the UO2 theoretical density.

51

(Notice: consider γ = 1 in this case)

Power density and specific power for a PWR

525252

Table 1 (Table 2.3 Todreas & Kazimi)

Power density and specific power for a PWR

535353

Table 2 (Table 1.3 Todreas & Kazimi)

Power density and specific power for a PWR – Solution

� The core power density is given as

� From Table 1 (Table 2.3),

� From Table 2 (Table 1.3), P = 12.6 mm

� Then,

( )2

'1'''

P

qQ square γ

=−∞

mkWq /8.17' =γ

( ) ( )kW

orMWkWq

Q 6 11210112.08.17'1

''' =⋅===

5454

� Notice that Table 1 (Table 2.3) lists the average power density as Q’’’PWR = 105 kW/L. The difference arises because our calculation is based on the core as an infinite square array, whereas in practice a finite number of pins form each array so that edge effects within assemblies must be considered.

54

( ) ( ) Lor

mmPQ square 33

6

23211210112.0

106.12''' =⋅=

⋅==

−−∞ γ


� The specific power is given by

� Evaluating f using for the enrichment r of 2.6% listed in Table 2 (Table 1.3)

fLR

LqQSP

pelletfo ρπγ 2

'1

atomsheavyofMass==

&

( )( )

21

1

Onfff

nfff

MMrrM

MrrMf

+−+−+

=

( )( ) 8815.0

0508.238)026.01(0439.235026.01=−+⋅=

−+= nfff MrrM

f

5555

� For a pellet density of 95% of the UO2 theoretical density of 10.97 g/cm3 and a fuel pellet diameter of 8.2 mm (Table 2), the specific power is

55

( )( ) 8815.0

9994.1520508.238)026.01(0439.235026.0

0508.238)026.01(0439.235026.0

12

=⋅+−+⋅

−+⋅=+−+

=Onfff

nfff

MMrrMf

atomsheavyfuelg

W7.36

8815.010

97.1095.02102.8

/1078.1'1

atomsheavyofMass3

26

23

4

2=

⋅

⋅

⋅===

−

−

m

gm

mW

fR

qQSP

pelletfo πρπγ

&


� Alternatively, Table 1 (Table 2.3) lists the total core loading of fuel material as Mfm = 101∙103 kg of UO2. In this case,

� If Q is evaluated from the PWR dimensions as in Tables 1 and 2 (Tables 2.3 and 1.3, respectively)

fmMf

QQSP

⋅===

&&

atomsheavyfuelofloading

powercore

atomsheavyofMass

( ) tMWmmMWLNqQ 331926419366.3/0178.0'1 =⋅⋅⋅==γ

&

5656

� Then,

� Unlike the case of power density, specific power is closely estimated, as the fuel mass, not the core volume, is utilized.

56

γ

atomsheavyfuelofg

W28.37

101018815.0

3319

atomsheavyofMass 3=

⋅⋅=

⋅==

kg

MW

Mf

QQSP t

fm

&&

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Lecture 2 - Thermal Design Principles