SUMMARY AND

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CHAPTER 7: SUMMARY AND CONCLUSION

SUMMARY AND CONCLUSION

7.1 SUMMARY

This study addressed the modelling of the effective thermal conductivity in a packed

t..:

rl.

of

mono-sized spheres

.

Although several accepted correlations for the effective thermal

conductivity exist, these are based on porosity alone which does not fully account for the

packing structure, especially in the near-wall region

.

Therefore, this study aimed to develop a

new method of characterising the effective thermal conductivity, whilst accounting for the

porous structure in a more fundamental manner (coordination number

,

contact angle).

In Chapter 1

,

the need to have a fundamental understanding of the impact of any wall region

on not only porous structure, but also the effective thermal conductivity was identi

fi

ed.

Previous studies suggested multiple constant correction factors in the wall region and the

failure of existing models in the near-wall and wall regions.

Chapter 2 introduced various methods to represent the porous structure; with porosity the

most widely used parameter. In addition to the literature reviewed, several parameters were

developed for a randomly packed bed to help quantify the porous structure in such a way to

be used for effective thermal conductivity calculations. This was achieved by using the

coordinates of the positions of the spheres as generated by a

OEM

code

.

In addition

,

it was

shown that the porous structure cannot be defined using poros

i

ty alone and that various

characterisation methods (coordination number, contact angle, coordination flux number)

should be used in conjunction with each other in order to successfully characterise the po

r

ous

structure.

The limitations on using porosity as the main parameter to quantify porous

structure were also indicated.

Chapter 3 presented a summary of the effective thermal conductivity correlations found in

literature. These correlations were grouped into three main components: heat transfer by

point conduction

,

heat transfer by conduction through contact area and heat transfer by

thermal radiation. It has been shown that relatively good accuracy can be obtained with

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CHAPTER 7: SUMMARY AND CONCLUSION

several unit cell approaches in the bulk region of a packed bed, but that significant uncertainty

arises in the near-wall and wall regions. A summary of all the correlations will be presented in

Appendix A and Appendix B.

Chapter 4 presented a broad overview of the newly built HTTU. For the purpose of this study,

experimental results conducted with the vacuum configuration to a temperature of 1200°C

were used. The effective thermal conductivity results were extracted using a polynomial

method and an uncertainty analysis was conducted in order to determine the confidence

interval of the experimental results.

Chapter 5 presented the derivation of the Multi-sphere Unit Cell Model. This model was

developed in such a way that it can be seen as a summary of all separate effects found for

heat transfer in a porous structure of mono-sized spheres surrounded by a stagnant gas.

This was achieved by using a thermal resistance network methodology for the conduction

component and a radiative conductivity for the thermal radiation component.

A clear

distinction was made between the bulk and wall regions, for which separate heat transfer

parameters were developed. In addition, a distinction was made between conduction through

spheres with rough surfaces (rough contact network) and perfectly smooth surfaces (Hertzian

contact network). Thermal radiation was grouped into two components: thermal radiation for

spheres in contact (short-range radiation) and thermal radiation for spheres further apart

(long-range radiation). Some uncertainty arose for the long-range radiation in the bulk,

near-wall and near-wall regions for temperatures above 1200°C and materials with low conductivity.

Chapter 6 illustrated the validation and verification of the Multi-sphere Unit Cell Model and

widely accepted existing correlations. It was shown that the Multi-sphere Unit Cell Model

conduction component compares quite well with experimental data.

For the SANA-I

experimental test facility, the effective thermal conductivity measurements were extracted in a

different manner to that used in past studies with effective thermal conductivity a function of

radial position and not temperature. This illustrated the near-wall and wall effects more

clearly.

In addition, this highlighted the contribution of natural convection more clearly,

showing two peaks near the inner and outer wall. Comparison between the Multi-sphere Unit

Cell Model and the IAEA ZS Total correlation demonstrated the breakdown of the IAEA ZS

model in the near-wall and wall region. For the HTTU experimental test facility, the

Multi-sphere Unit Cell Model (total) and the IAEA ZS Total correlation demonstrated good

comparison in the bulk region with experimental effective thermal conductivity values in both

the 20 kW and 82.7 kW steady-states.

However, the IAEA ZS Total correlation again

demonstrated breakdown in the near-wall and wall regions where the Multi-sphere Unit Cell

Model slightly under-predicted the experimental effective thermal conductivity results. A

flowchart and summary of the Multi-sphere Unit Cell Model is presented in Figure 7.1 and

Table 7.1.

MODELLING THE EFFECTIVE THERMAL CONDUCTIVITY IN THE NEAR-WALL REGION

160

OF A PACKED PEBBLE BED

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ll00tm'NES-1J!I!Vt:RSI'Tn'1' CHAPTER 7: SUMMARY AND CONCLUSION

INSERT PARAMETERS

c

Yes-RCN

r··

NUMBER10 Calculated Joint ... ~ Thertnal Resistance,

MULTI-SPHERE UNIT CELL MODEL

( Yes- Wall Region

INSERT PARAMETERS

Yes- Near-wall, Bulk Region

NUMBER 1 Calculate

s(z) Eq. (2.7) +-·-s' (z) Eq. (2.30)

f,k9,ks-,F,Es,J.lp•J.lm,Hs,<rRMs,

mRMs, Pr,y,P0,Pg,A..z, rp,s,,u RJ,W Eq. (5.51)

NUMBER12 Calculate Effective

Thermal Conductivity

(Conduction},

R},W Eq. (5.52)

.... f,k₉,k0, F, E., i'p• I'm·

Pr~r.P0,Pg.A.,z,rp,sr, Rough Contact or HertZlan Contact Network ( Yes-HCN ) ~.s.w Eq. (5.74) F2~1 Eq. (5.75)

A.z Eq. (5.83) NUMBER15 Short-Range Thermal

·•

Radiation In Wall Region Calculated L-···-···-·· .. --. ~.s.w ' kg.c.W Eq. (5.68) k~·s,w Eq. (5.74) F.,ltf, Eq. (5.72) A.z Eq. (5.83) No Long-R.anfje Thermal R-adiation is

Calculated from the Wall into the Pebble

Bed

______ !

~ L_ ______ _

NUMBER 1~

Calculate Thermal Radiation in

...

"

Wall Region + ···---t{,·w =1{,-s.w +~·L.WEq. (5.69)

*

NUMBER20 NUMBER18 Long-Range Thennal Radiation in Wall Reglon CalCUlated JC.oL.W

.

···-·--···.J INSERT PARAMETERS f,kg;ks,F,Es,Jlp,J.lm,Ha,uRMS, mRMs,Pr,y,P0,P9 ,A...z,rp,s,,a-~,t,W Eq. (5.81) F.{;:f_vg Eq. (5.79) A.z Eq. (5.83)

... J

···--···-~

"

+

NUMBERS NUMBER4 Calculated Joint Calculated Joint ·---® Thermal Resistance, Thermal Resistl.nce1

-RJ Eq. (5.3) Rj Eq. (5.4)

L

NUMBERS

L

...

... Then'l)a! Conductivity Calculate Effective

i

(Conduction),

~·• Eq. (5.34)

1 I

Short and Long j

!

1

Range Th~rmal

1

!

Radiation

NUMBERS NUMB£R7

Calculate Short-Range Calculate Long-Range

~,s Eq. (5.43) ~·t Eq. (5.50)

I_.

NUMBERS

. ... 1 ····---~

Calculate Thermal Radiation in

Near..wall~ Bulk Region

_14·

il.=t{,·s+~·L Eq. (5.35)

.),

NUMBER9

calculate Effective 'Thermal Conductivity in Wall Calculate Effective Thermal Conductivity in

Near-Region wall, Bulk R:eglon

~ Eq. (5.2) kew

Figure 7.1: Flowchart of Multi-sphere Unit Cell Model

Eq.(5.1)

INSERT PARAMETERS

. f,~<g,k.,F,E •. Ilp.i'm· Pr, y, _!'l,,P9,,t, z,rp,s,,

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Table 7.1: Summary of Multi-sphere Unit Cell Model

' BlOCK:''·

NUMBER,,

2 3 e(z)

=

2.14z2 -2.53z +1 e(z)

=

Eb +0.29exp(-0.6z) z ::;;0.637 x [ cos(2.3n(z -0.16))] + 0.15exp(-0.9z) e* = -0.0127z2 + 0.0967z- 0.2011 z >0.637

N

0

=

25.952& 3 - 62.364&2 + 39. 724&- 2.0233

¢a=

-6.1248iV: + 73.419N₀-186.68

(

_J

-1 1 1 . 1 + + +

-Rin,1,2 Rmid,1,2 Rout,1,2

1 RL12=--" 2k _sar. (2.7) (2.30) (2.22) (2.27) (5.3) (3.105) (5.5) (3.119) (5.6) Use when z > 3.8 from wall. Use when 0.5 < z::;; 3.8 in near-wall region. (0.2398::;; E::;; 0.54) (0.2398::;; E::;; 0.54) 1.3 ::;;H₈::;;7.6 GPa De Klerk (2003:2022)

Derived in current study

Bahrami eta/. (2006:3691)

CHAPTER 7: SUMMARY AND CONCLUSION

Calculation of radial porosity.

Calculation of porosity correction factor in near-wall region.

Calculation of average coordination number in a randomly packed bed.

Calculation of average contact angle in a randomly packed bed.

Calculation of joint thermal resistance with the Rough Contact Network (RCN) configuration.

Calculation of microcontact thermal resistance.

ra

is calculated with Eq. (3.113), contact area radius is renamed in the Multi-sphere Unit Cell Model as

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.. "+· ., ]

g a a1+V/2J2aRMs) R· _ (dp-Wo) m,1,2- ₂ k₅trr₈ 2 RJt= trkg r)i, In I A )I,-2

B)i, I+ 2B)i,-2C)i,) AJt-2CJt (dp-Wo) Rmid,1,2 = _{( 2 2)} k5tr fJt -ra 2 RG= trkg(

~lnl ~

_~-2BGI-2BG

J

In I Aout +Bout I A out-Bout R t _au_,1,2-- _{k B} str out (3.123) (5.7) (5.12) (5.22) (5.23) (5.27) (5.32)

CHAPTER 7: SUMMARY AND CONCLUSION

Calculation of the thermal resistance of the conduction Bah rami eta/. (2006:3691)

through interstitial gas in contact region.

Calculation of the thermal resistance of the bulk solid

Derived in current study material in the inner region,

me

= rl f2rp,eq ' 'p,eq =rp.

Calculation of the thermal resistance of the interstitial gas in the middle region. AJt = 2rp + j-aJo,

~

JR

BJt= , CJt= rp -r8 and

rJt =

~rff

-(rp -0.50Jo -52)2

Calculation of the thermal resistance for the middle

solid region.

Calculation of thermal resistance of the interstitial gas

Derived in current study in the micro-gap.

~=2rraJo,and BG=~rt-rf

Assume isothermal Calculation of the thermal resistance of the bulk solid temperature Derived in current study material in the outer region.

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CHAPTER 7: SUMMARY AND CONCLUSION

6_{r;~;B()UN[)~RY} &

i'

~1i:~.~~~u~r1ioN~l~:·i

( ) -1 1 1 1 + + +

-Rin,1,2 Rmid,1,2 Rout,1,2

(5.4) Derived in current study Calculation of joint thennal resistance with the Hertzian

contact network configuration.

4

5

R _ 0.64 HER7Z,1,2 - k r.

sc (5.8) Chen & Tien (1973:302)

(5.12) Derived in current study

(5.27), (5.32) Derived in current study

Hertzian contact radius

rc

calculated by Eq. (3.101). Calculation of the bulk solid material in the inner region, w₀

=

r~

j2rp .

Calculation of the thennal resistance of the interstitial gas in the middle region.

A

A. =

2rp

+ j -

wa,

Calculation of the thennal resistance for the middle solid region.

Same as given in block number 3. Calculation of effective thennal conductivity (conduction component),

(¢a

is in degrees).

wa

=

r: j2rp,eq

or w

₀

=

r~ j2rp

depending on contact network used.

MODELLING THE EFFECTIVE THERMAL CONDUCTIVITY IN THE NEAR-WALL REGION_

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«

1 for up to 2.25dp 0.01:::;1/As :::;10 fk ,

f

stays the same for thermal radiation between surfaces further

apart than d p .

CHAPTER 7: SUMMARY AND CONCLUSION

and near-wall region.

fi_

2 = 0.0756 ,

2 2

As= 4Jrrp , A,= dp

Calculation of long-range thermal radiation in the bulk and near-wall region.

!il:.2,avg = 0.0199 , As = 4Jrr; , A, = dt and

'iilong =4.7

Summation of short and long-range thermal radiation. Calculation of effective thermal conductivity due to therm>JI conduction and radiation in bulk and near-wall region. The limiting parameters for the Multi-sphere Unit Cell Model not reaching temperatures above

1200°C is: 1/As and the long-range radiation assumptions. The inverse of the dimensionless solid conductivity 1/ As must be extended for lower conductivity solid materials. The long-range thermal radiation over estimates

Jfe•L

with

f

> 1200 °C .

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1 1 )-1

RA.,W %,w RL12+ + -" Rg Rs ( ) -1 1 1 1 + + + -R;n,1,W Rmid,1,W Raut,1,W R;.,w

( I

B;.w-A;.wl

J

2trkg A;t,wln • · +B;.w-C;.w C;,w-A;,w • ' (5.51) (5.53) (5.54) (3.123) (5.7) (5.56) (5.61)

1.35Ha 57.6GPa Bahrami eta/. (2006:3691)

Bahrami eta/. (2006:3691)

Bahrami eta/. (2006.:3691)

CHAPTER 7: SUMMARY AND CONCLUSION

Calculation of joint thennal resistance with the Rough Contact Network (RCN) configuration in the wall region.

Calculation of microcontact thennal resistance.

fa is calculated with Eq. (3.113), contact area radius is renamed in the Multi-sphere Unit Cell Model as

Calculation of conduction through interstitial gas in contact region.

of the gas in the middle region. AA.,W

=

rp + j -0Jo ,

BA.,w=~rff-r1w, CA.,w=~rff-rl

and

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:NO,.,BER

·.~F

'>+'"·.,,,,,,;,,.,.,

" "' ,<' '·-~;~<·:<. ~',' 10 11 lniAout,W + Bout,W

I

R _ Aout,w-Bout,W out,1,W - _2k~a S" out,W

(

)

~ 1 1 1 Riw= + + -, RHERTZ,1,2 R,t,W J%,w + ( - 1 - + 1 + ____ 1_)-1

Rin,1,W Rmid,1,W Rout,1,W

R _ 0.64 HERTZ,1,2-k* src

R;.,w=

_2trkg

(

\B;.w-A;.w\

J

A;.wln

c ·

A· +B;.w-C;.,w

' ;.,w- ;.,w ' (5.62) (5.65) (5.66) (5.62) (5.54) (5.56) (5.61) Assume isothermal temperature boundary.

Chen & Tien (1973:302)

CHAPTER 7: SUMMARY AND CONCLUSION

Calculation of the thermal resistance for the middle solid region in the wall region.

Calculation of thermal resistance of the interstitial gas in the micro-gap of the wall region.

~.w

=

rp- too and l%,w

=

~rff-

rl,w

Calculation of the thermal resistance of the bulk solid material in the outer region of the wall region.

Aout,w=rp-2(too+10A.) and

B

0 ut,w=~rff-rl,w

Calculation of joint thermal resistance with the Hertzian contact network configuration in the wall region.

Hertzian microcontact modified in this equation to accommodate k~ . Hertzian contact radius rc

calculated by Eq. (3.1 01 ).

Calculation ofthe bulk solid material in the inner region, m₀=

r~

j2rp .

Calculation of the thermal resistance of the interstitial gas in the middle region. A,t,W = rp + j-too ,

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AT/f «1

0.01~1/As ~10 0.2:;;

e,

~ 1 fiT/f

«1

0.01 :'> 1/As =" 10 fiT/f

«

1 for up to 2.25dp 0.01~1/As :'>10 0.2:'>e, :'>1 fk ,

f

stays the same for thermal radiation between surfaces further

CHAPTER 7: SUMMARY AND CONCLUSION

Calculation of the thermal resistance for the middle solid region in the wall region.

Same as given in block number 10.

Calculation of effective thermal conductivity in the wall region lliQ = rt j2rp .

Calculation of short-range thermal radiation in the wall region (Wall to Sphere).

w

2 2

f=:2_

1

=

0.01976,

A,=

dp, .Aj

=

41E'rp and

Calculation of short-range thermal radiation in the wall region (Sphere to Wall).

w

2 2 d

F_{1_2}= 0.315, A,= dp, .Aj = 41E'rp an

A2 = 63.687E'rff

Calculation of long-range thermal radiation in the wall region.

F.ffavg = 0.02356 , A,

=

d~

, .Aj

=

47E'rff ,

A2

=

63.687E'rff and

n:ng

= 1

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CHAPTER 7: SUMMARY AND CONCLUSION

Summation of short and long-range thermal radiation in wall region.

Calculation of effective thermal conductivity due to thermal conduction and radiation in wall region. The limiting parameters for the Multi-sphere Unit Cell Model not reaching temperatures above 1200°C is: 1/As and the long-range radiation assumptions. The inverse of the dimensionless solid conductivity

1J

As must be extended for lower conductivity solid materials. The long-range thermal radiation over estimates k~·L with

T

> 1200°C.

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7.2 CONCLUSION

CHAPTER 7: SUMMARY AND CONCLUSION

It can thus be concluded that all the outcomes defined in Section 1.4 were met.

Comprehensive research was conducted in order to understand and implement the various

porous structure characterisation methods, and to develop a new empirical correlation relating

the various methodologies. It was also demonstrated that the relation between porosity and

coordination number is different in a randomly packed bed to what has been found in other

studies. A new definition and method to calculate the effective thermal conductivity was

presented that demonstrates relatively good agreement with the various experimental data

sets. However, some additional research should be done expanding the Multi-sphere Unit

Cell Model to calculate effective thermal conductivity beyond 1200°C and materials with low

conductivity.

7.3 RECOMMENDATIONS FOR FURTHER RESEARCH

Recommendations can be made regarding further research with the objective of improving

the Multi-sphere Unit Cell Model:

• Several shortcomings in experimental data sets can be identified in the relation to the

development of a PBR:

o Thermal conductivity tests should be conducted at low temperatures with

graphite spheres. These tests need to be conducted using various applied

forces and different surface roughness for different packings in the wall,

near-wall and bulk regions. This is important because the surface roughness in a

PBR changes with the constant bombarding of neutron flux. This research

should also be conducted at vacuum and elevated pressures with helium as

the gas medium.

o Thermal radiation tests conducted at temperatures above 12oo·c should also

be conducted in the bulk, near-wall and wall regions. This is important

because of the uncertainty that arose in effective thermal conductivity

experiments above 12oo·c in the HTO experimental test facility. This test

should also be conducted at vacuum conditions and elevated pressure

conditions with helium eliminating natural convection to an extent.

o The impact on the effective thermal conductivity with the presence of a

convex or concave curve should also be investigated at low and high

temperatures. This is important because the HTTU could only achieve higher

temperatures at the inner wall, where in general the outer wall and bulk

regions should achieve equally high temperatures. CFD can possibly be

used.

170

OF A PACKED PEBBLE BED

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CHAPTER 7: SUMMARY AND CONCLUSION

• Further research should also be conducted on the long-range radiation conductivity,

for which the following areas are identified:

o The long-range average diffuse view factor and average geometrical length

needs to be derived as a function of radial position. Currently, the long-range

average view factor is assumed to be the same in the bulk and near-wall

regions, which is not the case in general.

o

The long-range average view factor must also be weighted according to the

number of spheres that contribute to long-range thermal radiation.

o The long-range average diffuse view factor in the wall region for thermal

radiation heat transfer from a sphere to the wall should be investigated for a

convex curved and a concave curved wall. Subsequently, the simulation

answers need to be compared with relevant experimental data.

o A new long-range non-isothermal correction factor needs to be derived using

the same methodology applied for the derivation of the short-range

non-isothermal correction factor.

o The impact of the assumption that the average temperature between spheres

not in contact is the same as that of the adjacent spheres should be further

investigated.

• The non-isothermal correction factor Figure 5.11 should be extended

1f

A

5