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Fundamentals of Nuclear Engineering

Module 11: Single Phase Heat Transfer and Fluid Flow

Joseph S. Miller, P.E. and Dr. John Bickel

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Objectives:Previous Lectures described core heat transfer and

reactivity feedback. This lecture will:1. Describe Systems2. Describe Basic Fluid Flow Equations3. Describe fluid flow, and pressure drops in single

phase system 4. Describe heat transfer rates from fuel to coolant

in PWR5. Describe steady state core temperature profiles 6. Describe critical heat flux limitations7. Describe analytical processes for establishing

limiting power density and heat flux limitations

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Some Systems of Interest• All fluid flow systems associated with Power Plant

• BWR System

• PWR System

Some Important Water Systems in a Nuclear Power Station

• About 100,000 feet of water piping• 40 to 50 Large Heat Exchangers• About 10,000 valves• Reactor Vessel and Core• ECCS pumps, valves and piping • Service Water• Turbine and reheat• Condenser

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BWR System

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PWR System

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Basic Systems Reactor Systems – 4 Loop Westinghouse PWR Reactor

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PWR Steam Generator

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Two Other Types of PWRs:

Major Fluid Flow and Heat Transfer Situations in a PWR

• Steady State Operation• Transients• Accidents

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2. Describe Basic Fluid Flow Equations

Basic Single Fluid Flow Equations

• Navier-Stokes Equation (Ref. 6)

• Euler Equation where viscous Effects is Relatively Unimportant

• Conservation of mass• Conservation of momentum• Conservation of energy.

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Homogeneous flow theory

One-dimensional steady homogeneous equilibrium flow in a duct (Wallis page 18, Ref. 3)

Continuity

Momentum

Energy

.constvAM m == ρ

θρτ cosgAPdzdpA

dzdvM mw −−−=

z

A: area

P: perimeterstressshear wall:wτ

θ

g˅

mρ

++=− ge zgvh

dzdM

dzdw

dzdq

2

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Heat transfer and work rate

dqe /dz is the heat transfer per unit area

dw /dz is the work per unit area

dp /dz is the pressure per unit area

dv /dz is the velocity per unit area

h: enthalpy

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3. Fluid Flow and Core Pressure Drop

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Key Fluid Flow and Pressure Drop IssuesSteady State:• Flow distribution in core needs to be uniform• New reactor: fueled with one fuel assembly design, pretty

much guarantees uniform flow distribution in core• Operating reactors contain different fuel bundle designs• Flow distribution will seek path of least resistanceTransients:• Flow resistance impacts safety analysis assumptions• Loss of Flow following pump coastdown depends on flow

resistance• Efficiency of Natural Circulation cooldown following loss of

offsite power• LOCA blowdown, reflood impacted by flow resistance and

two phase behavior

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Normal Steady State Balance

• Under normal plant operation:

• Operation of reactor coolant pump provides ΔPRCP which matches net flow losses: ΔPfriction for a specific flow velocity

• Largest ΔPfriction losses are steam generator tube sheet transition, steam generator tubes, reactor core and flow channels around fuel rods and grid spacers

• Large diameter pipe runs are relatively minor sources of losses.

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Pressure Drop is Function of Flow Speed• Single phase flow pressure drop in psi can be expressed:

• -where:• f is wall friction factor (see: Moody friction factor chart)• L is length of fuel channel in ft.• Dh is equivalent Hydraulic Diameter (corrects original data

from circular geometry (inside of tubes) – to actual geometry (outside of tubes)

• Dh = 4 x flow area / wetted perimeter• ρ is fluid density in lbm/ft3.• ν, νi are local coolant velocities in ft/hr.• Ki is form friction factor due to “i”th change in cross-section

or restriction in flow channel (sometimes from: tests)

∑

+

=∆

i

ii

hfriction K

DfLP

22

22 ρυρυ

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Friction Factor for Turbulent Flow: Simple fit for fuel rods: f ~ 0.184/Re0.2

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Recommended K-factors

2

2i

iK ρυ

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Hydraulic Diameter Dh Corrects for Actual Flow Channel Geometry

• Examples:• W 17 x 17 standard fuel assembly: d=0.94 cm., p=1.25 cm.• Dh=(0.94 cm)[(4/π)(1.25 cm/0.94 cm)2-1] = 1.176 cm.

= 3.858 x 10-2 ft.

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Complex Flow Restriction Examples:Fuel Assembly & Grid Spacer

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Fuel Rod Channel Pressure Drop(Pressure Drop Due to Grid Spacer Would Need to be Added !)

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Example Actual Pressure Drops:

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Establishing Actual Hydraulic Flows:

Involves combination of:

• Calculations based upon basic principles of hydraulics and fluid flow

• Use of test data to address complex flow geometries (grid spacers, fuel bundle support plates, etc.)

• Experience gained from scaling up similar vessels and piping geometries

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Transient Flow Coastdown Behavior

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Flow Coastdown Behavior

• RCS hydraulic performance during 4-pump loss of flow is important characteristic in plant safety

• Delay in flow coastdown allows sensing event, generating reactor trip, assuring transient core heat flux is removed

• Coastdown sets up long term Natural Circulation Cooling

• Understanding coastdown characteristics is important to understand loss of offsite power performance

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Transient Flow Coastdown Model

Flow coastdown from sudden loss of pump power is governed by following equation:

Where: L is effective length of flow loop in ft.ρ is density in lbm/ft3

Cf is a net form friction factor ν is local core coolant velocity in ft/hr.

2

2ρννρ fCdtdL −=

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Transient Flow Coastdown Model Solution• Assuming ν(0) is at 100%

rated flow• Integrating differential

equation yields:

• Loss of flow analysis for ANO-1 indicates:

• Cf /2L = 0.1116 ft-1

LtCt

tL

Ct

tL

Cdt

LCd

f

f

ft t

f

2)0(

1

)0()(

2)0(1

)(1

22

)(

)0( 02

ννν

νν

ννν

ν

+=

−=+−

−=−=∫ ∫

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Natural Circulation Cooling

• Hydraulic coastdown model is highly accurate until natural circulation cooling develops sufficient ΔP to maintain circulation

Natural circulation relies on:• Heat sink elevated significantly above reactor core heat

source• Single phase fluid capable of supporting siphon action• Hot water (less dense) rises, cooled water (more dense)

flows downward• Larger temperature gradients across height of core

produce larger flow velocities

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Natural Circulation Cooling

2),( RxSGRCS

COREHH

TPTgTP −

∂∂

∆=∆ρ

• Pressure differential of water column is: ΔP = ρg ΔH• Net driving pressure difference is function of elevation

differences of thermal centers:

• Figures below show location of relative thermal centers

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Natural Circulation Cooling Tests• Startup test have measured natural circulation differently• Maine Yankee (1972) tripped RCPs from 37% Rx Power

generating a transient ΔTCORE that stabilized• Zion 1 (1974) operated at various low powers measuring

steady state natural circulation flows at different powers

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4. Heat transfer rates from fuel to coolant in PWR and 5. steady state

core temperature profiles

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Local PWR Fuel Rod Heat Transfer

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Tf(r), Tc(r) Expressions Derived Previously• Overall solution for fuel pin temperature:

• Overall solution for clad temperature:

• Value for: hfilm ~ 4.5 Watts/cm2 K• We now describe how to calculate hfilm

+

++

−

+=filmcc

o

c

gapof

occoolantf hRk

RR

hRkRr

qRTrT 12

ln1

2

1

2)()(

2

2

π

+

+=filmcc

c

ccoolantc hRkrR

qRTrT 12

ln

2)()(

π

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Heat Transfer from Clad to Coolant• q, heat flux (Watts/cm2), from clad to coolant across “film” is

described via Newton’s law as: q = hfilm ΔT = hfilm (Tclad – Tcool)

• Film conductance: hfilm is actually fairly complex function of heat flux, flow, geometry, temperature, pressure……

• Historical approach uses experimental correlations for heat transfer of fluids within pipes

• Assuming forced convection turbulent flow, hfilm corrected for actual flow channel geometry is expressed:

• hfilm = (k / Dh) Nu -where:• k is fluid thermal conductivity for water in Watts/cm.°K• Dh is hydraulic diameter in cm.• Nu is dimensionless Nusselt Number = hfilm (Dh / k)

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Nusselt Number Can Be Derived from Dittus-Boelter Correlation

• Nu = 0.023 Pr0.4 Re0.8 -where:• Pr is dimensionless Prandtl Number• Re is the dimensionless Reynolds Number• Pr = f(p,Tfilm) can be expressed: Pr = ν/α -where:• ν = ν(p,Tfilm) is kinematic viscosity in ft2/sec defined: ν = μ / ρ• μ = μ(p,Tfilm) is dynamic fluid viscosity in lbm/ft sec • ρ = ρ(p,Tfilm) is fluid density in lbm/ft3

• α = α(p,Tfilm) is thermal diffusivity in ft2/sec, defined: α = k/ρCp• Cp = Cp(p,Tfilm) is heat capacity at constant pressure in

BTU/lbm°F

• Tabulated values of Pr = f(p,Tfilm) are found in Steam Tables with: Tfilm = (Tclad +Tcool ) / 2

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Nusselt Number Can Be Derived from Dittus-Boelter Correlation

• Re is dimensionless Reynolds Number• Reynolds Number is measure of fluid turbulence • Flow is laminar when: Re < 2100• Flow is turbulent when: 2100 < Re < 100,000• Re = f(p,T,V) can be expressed: Re = Dh V ρ / μ -where:• Dh is Hydraulic Diameter (as previously defined)• V is local fluid velocity in ft./hr.• ρ = ρ(p, Tfilm) is fluid density in lb./ft3

• μ = μ(p, Tfilm) is viscosity in lb./hr. ft.

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hfilm Derived from Dittus-Boelter Correlation

• Combining expressions for hfilm and Nu:

• hfilm = (k / Dh)Nu

• Nu = 0.023 Pr0.4 Re0.8

• Thus: hfilm = (k / Dh) 0.023 Pr0.4 Re0.8

• We now look at an example application using data from Callaway NPP

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Example: hfilm via Dittus-Boelter Correlation

• Calloway NPP: total core heat output: 3565 MW• Average LPD: 5.69 kW/ft (186.7 W/cm )• Peak LPD: 14.23 kW/ft (465.9 W/cm )• Reactor Coolant Tin: 556.8 °F (564.7 °K)• Reactor Coolant Tavg: 593.1 °F (584.8°K)• Reactor Coolant Tout: 620.0 °F (599.8 °K)• RCS pressure: 2274 psia (15.68 MPa)• RCS flow velocity: 14.9 ft/sec (4.54 m/sec)• Fuel Rod Outer Diam.: 0.36in (0.9144cm)• Fuel Rod Pitch: 0.496in (1.259 cm)

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Example: hfilm via Dittus-Boelter Correlation

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PWR Axial Heat Transfer

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Coolant, Fuel Temperatures Derivedfrom Simple Energy Balance

• Axial and radial power distribution derived earlier

• Highly simplified picture has uniform k throughout core:Φ(r,z) = Φo Jo(2.405r/R) Cos(πz/H)

• Linear power density from individual fuel rod:q(z) = qo Cos(πz/H) -where:

qo = (πRc2) Ef ∑f Φo Jo(2.405r/R)

• Energy balance along single fuel rod is expressed:W dh(z) = W Cp dT(z) = q(z) dz

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Coolant, Fuel Temperatures Derivedfrom Simple Energy Balance

• Integrating along length of fuel rod yields:

• Upon inserting expression for q(z), and rearranging this yields:

∫∫−

=z

H

zT

Tinp dzzqzdTWC

2/

)(

)()(

+=− )

2sin()sin(

2)(

effeffp

effoin H

HH

zWC

HqTzT ππ

π

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Example 3411MWt PWR Center Peak:

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Some Axial Power Shapes From ANO-2:

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Example 3411MWt Trapezoidal Shape:

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6. Critical Heat Flux Limitations to Coolant

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Exceeding Critical Heat Flux → Large ΔT

DNB

Unstable

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Heat Transfer Limitations to Coolant• Nucleate Boiling (q < qCHF ) is most efficient for heat transfer

• Exceeding qCHF or: Departure from Nucleate Boiling “DNB”

• Results in:

• Reduced heat transfer

• Surface burnout (rapid clad temperature rise)

• Fuel rod failure

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Departure from Nucleate Boiling - DNB• Specific flow, ΔT, pressure, core heat flux (Watts/cm2) lead

to DNB and depend on fuel bundle design.• Fuel bundle design: rod diameter, pitch, grid spacer effects• Predicting DNB from first principles: not reliable• Experimentally determine DNB onset using electrically

heated fuel bundle in pressurized flow test loop• Develop correlation of form: qDNB=f(G,P,Hin,χ…)• Quality or: χ = vapor mass/ total mass (χ = 1 is pure steam,

χ =0 is pure liquid)• Reactor fuel vendors each developed DNB correlations

based upon tests with electrically heated fuel bundles• Example: W3, CE2

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Columbia University Flow Test Loop

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Example Critical Heat Flux Correlation• In uniformly heated bundle, W-3 correlation is:

qDNB(P, χ,G,Dh,Hsat,Hin) ={ (2.022-0.0004302P) +(0.1722 – 0.0000984P)exp[(18.177 – 0.004129P)χ]}x [(0.1484 – 1.596 χ + 0.1729 χ│χ│)G/(106) + 1.037]x [1.157 – 0.869χ] x [0.2664 + 0.8357exp(-3.151Dh)]x [0.8258 + 0.000794(Hsat –Hin)] - in units of: 106 BTU/hr ft2

• For: 1000 < P < 2300 psia1.0 x 106 < G < 1.0 x 106 lb/hr ft2.

0.2 < D < 0.7 in.χ ≤ 0.15

Hin ≥ 400 BTU/lb.10 < L < 144 in.

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CHF For Non-Uniform Axial Power Distribution

• Previous DNB correlation assumed uniform axial power distribution.

• For case of non-uniform axial power distribution: q(z), Tong & Weissman, Thermal Analysis of Pressurized Water Reactors, shows:

• DNB would occur at location: ZDNB given by:

inchesG

C

dzzZzqCZZq

CF

FqZq

DNB

DNB

Z

DNBDNB

DNBDNBDNB

DNB

72.16

9.70

)10/()1(44.0

)exp()()]exp(1)[(

)(

χ−=

−−−

=

=

∫

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Consideration of DNB in Design/Safety:Avoiding DNB limits is accomplished by:

• Understanding range of possible power distributions q(z),operating conditions (flow, temperature, pressure) and margin to DNB typically expressed as: DNBR = qCHF(z)/q(z)

• Typical Minimum DNBR ≥ 1.3 for W-3

• Evaluating operating transients which cause sudden changes to DNBR margin: rapid pressure decrease, rapid increase in Tin, rapid reduction in RCS flow, rapid change in local power distribution q(z)

• Assuring via analysis and operational limits that rapid reduction in q(z) (e.g. reactor trip) occurs before reaching CHF condition

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Margin Against DNB is Statistical

• Critical Heat Flux (DNB) correlations are based upon fitting scattered experimental data from test facilities

• Scatter in experimental data results in quantifiable uncertainties

• Margin against exceeding DNB criteria is based on 95% confidence of less than 5% probability that anticipated operational occurrences would exceed DNB limits based upon specific DNBR correlation

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7. Describe analytical processes for establishing limiting power density

and heat flux limitations

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Design & Operating Limits

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Hottest Channel Limits Operating Power• Recall simple diffusion model for power distribution:

Φ(r,x) = Φo Jo(2.405r/R) Cos(πz/H)

• Hottest location is at midpoint plane in central fuel bundles

• Peak centerline temperature for core is as this point

• Limiting DNBR would likely be in channels above this point

• Reactor is operated using information on average heat output

• Important to know ratio of Peak Values to Average Values

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Peaking Factors: Relate Peak to Average

• Average Rod Power is total thermal power divided by total number of fuel rods

• Peaking Factors typically noted in FSARs:• Heat flux hot channel factor: FQ

T = max q(z)/Avg q(z)• Nuclear hot channel factor: FQ

N = max Φ(r,z)/Avg Φ(r,z)• Defining FQ

N = FRN x FZ

N

• Radial, axial peaking factors: FRN, FZ

N

• Engineering heat flux hot channel factor: FQE, is an

allowance on heat flux which accounts for manufacturing tolerances, local variations in enrichment, pellet density, diameter)

• Peaking factors are related as follows: FQT = FQ

N x FQE

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Example Calculation of: FRN, FZ

N

• Carrying out: FQN = FR

N x FZN

• FQN = (2.32)(1.571) = 3.64

• This value is very largebecause of assumed high central core region peaking

• High peak is from Jo Bessel function which originated from assumption of uniform k∞ in neutron diffusion model

• Actual core loading pattern objective is to flatten FR

N

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Peaking Factor Example: Diablo Canyon

• Adjacent figure shows total Peaking Factor:FQ

T < 2.45 in lower coreFQ

T < 2.45 – 2.3 upper core• Engineering hot channel

factor is: FQE = 1.03

• Quick “guestimate” is that core loading gets

FRN ~ 1.5 to 1.6

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PWR Heat Transfer and Fluid Flow:

• Hydraulic analysis assures sufficient flow near fuel rods and appropriate flow coast-down characteristics

• Local flow and thermal conditions impact heat transfer rates from fuel rods – hfilm can be calculated

• Critical heat flux limitations exist which limit how much heat can be moved from fuel pellets to coolant

• Methods (traceable back to test data correlations) exist to provide margin against fuel centerline melt, and DNB

• Manufacturing, materials, and operating uncertainties are incorporated into Peaking Factors to assure safety margins

Interesting Links

• http://cmmt.gatech.edu/Bruno/Publications/Frazier_01_Review_Lam_Sing_Phas_Flow_Mchannels.pdf

• http://www.cfd.com.au/cfd_conf97/papers/hov025.pdf

• http://www.nea.fr/html/dbprog/cpsabs_h.html• http://www.iasmirt.org/iasmirt-3/SMiRT7/M5-5

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References1. L.S. Tong & Joel Wiesman, ”Thermal Analysis of

Pressurized Water Reactors”, third edition, American Nuclear Society, La Grange Park, Il, (1996)

2. M.M. El-Wakil, “Nuclear Energy Conversion”, American Nuclear Society, La Grange Park, Il, Third Printing, (January 1982)

3. Wallis G.B., One-dimensional two-phase flow, McGraw-Hill Book Company, New York (1969)

4. N. Todreas and M Kazimi, ”Nuclear Systems I –Thermal Hydraulic Fundamentals”,Taylor & Francis”, (1993).

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References (cont)5. N. Todreas and M Kazimi, ”Nuclear Systems II –

Elements of Thermal Hydraulic Design”,Hemishere Publishing Corporation, (1990).

6. Bird, R.B., Steward, W.E., and Lightfoot, E.N., ”Transport Pheomena”, New York: Wiley, (1960)

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