the dominant thermal resistance approach for heat transfer

1. Nuclear Engineering Division, U. Idaho, Idaho Falls, Idaho 83402 U.S.A.

2. Professor Emeritus, U. Arizona, Tucson, Ariz. 85721 U.S.A.

3. Institut für Kernergetik und Energiesysteme (IKE), Uni. Stuttgart, D-70569 Stuttgart, Germany

4. Mechanical Engineering Dept., U. Sheffield, Sheffield S1 3JD, England

5. Now at Daresbury Laboratory, Science and Technology Facilities Council, Warrington, WA4 4AD England

The Dominant Thermal Resistance Approach forHeat Transfer to Supercritical-Pressure Fluids

Donald M. McEligot1,2, Eckart Laurien3,

Shuisheng He4 and Wei Wang4,5

sCO2 Recuperative Recompression Brayton Cycle'Advanced Design' of the MIT-Study

criticalpoint

200 bar77 bar

region of interest

Outline

Introduction

Test Case : Direct Numerical Simulation (Wang, He)

Dominant Thermal Resistance Approach

Results

Conclusions and Outlook

Physical Properties of Supercritical Waterpressure : 23.5 MPa

2case 1

Approach to provide approximate predictions and improved analyses with varying fluid properties

Possibly a useful basis for extending constant property correlations to variable properties

A reasonable, sensible, simple analysis will (may) provide better predictions than empirical correlations for fluids with significant property variation

Improved treatment for wall functions in CFD

Why Develop an Approximate Method ?

Outline

Introduction



Results


DNS of Supercritical Pipe or Channel Flows (Water or CO2)

DNS : Direct Numerical SimulationComputational Fluid Dynamics (CFD) without turbulence model, all scales resolved

Rynolds Number Re must below (typically: Re < 6000)

authors case HT mode published year

Bae, Yoo, Choi(Korea) upward vertical pipe wall heating Phys Fluids 2006

Nemati et al. (Delft) vertical pipewhith/no buoyancy wall heating IJHMT 2015

Chu and Laurien(Stuttgart) horizontal pipe wall heating J. Supercritical

Fluids 2016

Wang and He(Sheffield, UK)

plane channelwhith/no buoyancy

constant T at the walls NURETH-16 2015

Pandey and Laurien(Stuttgart) vertical pipe wall heating

or cooling this conference 2018

usedin thepresentwork

Description of the ‘Wang-DNS’ (no accelation, no buoyancy)p = 23.5 MPa, G = 108.6 kg/m2s, Tc = 367 °C

Geometry case: 1, Th = 367 °C case 2, Th = 380 °C

Turbulence structures visualizedby the second-largest eigenvalueof the stress tensor

W. Wang & S. He, NURETH-16, 2015

2Re 5730mb

b

u

Re 5670b

Forced-Convection Correlations for NuDittus Boelter (DB), Gnielinski (VG), Mokry and Pioro (Mok)

605040

30

20

10 650 660 670 680

pseudo-criticaltemperature 652.5 Kat the wall

case 1

Nu

2( )

w

w b b

qNuT T k

Forced-Convection Correlations for NuDittus Boelter (DB), Gnielinski (VG), Mokry and Pioro (Mok)

30

10

4

2650 660 670 680

pseudo-criticaltemperature 652.5 Kat the wall

case 2

Nu

2( )

w

w b b

qNuT T k

Importance of the Turbulent CoreDNS data from Bae, Yoo and Choi, Phys. Fluids 2005

/r R

laminarsub layer

y

5 %

Outline

Introduction



Results


Steady state,

Quasi-established velocity and temperature profiles

Constant shear layer and heat flux layer approximations

Negligible buoyancy, negligible acceleration

Turbulent core - high turbulence,

high , high cp ---->

Dominant Thermal Resistance Approachassumptions

centerline laminar conductingsublayer sublayer

bT T T T

Wang-DNS Mean Temperature ProfileCase 1

0.99

0.995

1

1.005

1.01

-1 -0.5 0 0.5 1

WangF653-Profs-T.qpc

2

h

c pc

ref

TT

y2

region of interest

cooledside

wall distance

heatedside

left wall

right wall

conductingsub-layer

csy

0

0.5

1

1.5

-1 -0.5 0 0.5 1

WangF653-Profs.qpc

2

h

c pc

Wang-DNS Mean Velocity ProfileCase 1

U

y2

region of interest

wall distance

ucooledside

heatedside

viscoussub-layer

vsy

Integration of the Thermal Energy Equationin the laminar, conducting sub-layer: the region of dominant thermal resistance

0 ( ) .p wT qc U q y const qx y

Near the wall we have Fourier‘s law:

( ) ( ) Tq y k Ty

Define

( )

0

( ) ( )w

y T y

T

q y d y k T d T Integrate:

( ) ( )ref

T

T

T k T d T a property

then

( ) ( )b ww

cs

T Tqy

At y = ycs

( ) ( )w wq y T T

May be a good approximationIf one has a good estimateof ycs

Prandtl [1910] approach ycs+ ≈ yvs

+

Two-layer approximation yvs+ ≈ 11.6

where y+ = y (w/)1/2/ or

y+ = (y/Dh) ReDh (Cf/2)1/2

DO WE HAVE A GOOD ESTIMATE OF Cf ?

How to get good Estimates for ycsusing the universal wall units

Forced-Convection Correlations for wall friction used in Gnielinski (VG), Drew, Koo, McAdams (DKM), M.F. Taylor (MFT), Blasius (Blas)

660 680 700 720 740

wT K

fccase 1

fc

20.5w

fw m

cu

Forced-Convection Correlations for Nuused in Gnielinski (VG), Drew, Koo, McAdams (DKM), M.F. Taylor (MFT), Blasius (Blas)

Case 2

660 680 700 720 740 wT K

fc

20.5w

fw m

cu

Integration of the Momentum Equationin the laminar, conducting sub-layer: the region of dominant flow resistance

( ) ( ) . wUy y consty

Near the wall we have Newton‘s law and Fourier‘s law:

( )

( )

w

w

dy T dUdydUT

Define

0

( )( )

b b

w

U Tw

w T

k Td U d Tq T

and

( )( )( )

ref

T

T

k TT d TT

a property

( ) ( )wb b w

w

U T Tq

Integrate

( )( )

wq dy k T dTk T dTdy

q

( )( )

w

w

k T d TdUT q

Solve for

( ) ( )b w

wb w

U qT T

Integration of the Momentum Equation (contd.)in the laminar, conducting sub-layer: the region of dominant flow resistance

( ) ( )w cs cs wq y T T Substitute into

cscs

w

yy

with

( ) ( )csw cs w

w

yq T T

( ) ( )

( ) ( )

csw cs w

b w

w b

yq T TU q

T T

i.e. into

to give

and solve for

2

2

( ) ( )( ) ( ) ( )

b wbw

cs w b

T TUqy T T

Outline

Introduction



Results


ResultHeat Transfer

case 1present

650 660 670 680

wT K

Nu

11.6vs cs

w

w w

y y

yy

upper index +in „wall units“

2( )

w

w b b

qNuT T k

ResultHeat Transfer

case 2

present

650 660 670 680

wT K

Nu

2( )

w

w b b

qNuT T k

ResultFriction

case 3

present

650 670 700 740 wT K

fc

20.5w

fw m

cu

ResultHeat Transfer

case 3

Th = 655 K

present

650 660 670 680

wT K

Nu

2( )

w

w b b

qNuT T k

Demonstrated a closed-form, approximate, coupled analysis for Nu for ScPF (with negligible buoyancy and acceleration)

Some reasonable agreement with DNS of Wang+He

Nu is sensitive to choice of yvs+

Useful approach to provide approximate predictions and improved analyses

Improved treatment for wall functions in CFD

Can provide a first estimate for interative processes in "more sophisticated" analyses

Concluding Remarks

Extend to significant buoyancy and acceleration

Revise analysis to treat differing ycs+ and yvs

+

Add thermal resistance for turbulent core?

Outlook

DNS data: 80 bar, CO2, D = 2 mm

upward downward

250260270280290300310320330340350

300 350 400 450

T [K]

h [kJ/kg]

250260270280290300310320330340350

300 350 400 450

T [K]

h [kJ/kg]1 18 8

qW = 61 kW/m2K30

bulk

qW = 61 kW/m2K

30

3030

bulk

Backup Slides

Forced convection Mixed convection (w/ gravity influence)

almost the same result forupward and downward flow

State-of-the-Art Correlations for Narrow Channels (2 mm)G = 60 kg/m2s, qw = -30 kW/m2 (cooled wall)

index 0 : constant propertiesw : wallb : bulk

Nu : Nusselt numberGr : Grashof numberRe : Reynolds number+ upward, - downward flow

DNS-Results of a Heated PipeFlow at Re = 5400

upwards:thermallystable

buoyancyfirst dampslaterinducesturbulence(recovery)

downwards:thermallyunstable

buoyancyinducesturbulence

Visualization of turbulence structures using the λ2 criterium

g g

0 50 100 150 200T [°C]

100

10

p[bar]

73.1

30.9

coolingcriticalpoint

pseudo-critical line

bulk temperature (Tb) of aheated/cooled pipe/channel

pseudo-criticalpoint (ppc, Tpc)

ρ = 100 kg/m3

50

150

200250300

saturation line

heating

Terminologyof Single-Pipe or Channel Experiments at Super-Critical Pressure

sCO2 Recuperative Recompression Brayton Cycle'Advanced Design' of the MIT-Study: Net Efficiency ~47 %

RejectHX

InputHX

MainCompressor

Alternator

Re-Com-pressor

HighTempRecuperator

Low TempRecuperator

Turbine

60%

40%

6 12

34

5

7 8

200 bar

77 bar p TMPa °C

1 7,7 32,02 20 61,13 20 157,14 20 488,85 20 650,06 7,7 534,37 7,7 165,88 7,7 68,9

critical point

200 bar77 bar

region of interest

Example: Two-Layer Model for Turbulent Boundary LayersPrandtl 1910, Kays and Crawford 1980, constant properties

T

y

lT

bTT R

2R D

pw p

w

c uT y T T y c u R

q

The temperature in wall units

Can be considered a non-dimensional thermal resistance. Expand to

p lam turbT y c u R R

And compare to the Kays and Crawford relation

PrPr ln lntlam lam bT R y R y T

The total resistance is the sum of two individualresistances fore the two layers

wT turbulentcore layer

laminarsub-layer

the dominant thermal resistance approach for heat transfer

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