THERMAL CONDUCTIVITY

The semi-infinite medium with the initial temperature

e,

is again considered to directly measure thermal conductivity. At T = O, the surface is suddenly exposed to a constant heat flux q0 • The solution for this case is (see Holman [1981])

q⁰

{2v;;T (-x

^{2) ( x )}}

0(x,r) -

ei = -

- - e x p - - - x 1-e r f - - .

k

Ji

^4ar 2ya:i

Once the thermal diffusivity a is known, therefore, the thermal conductivity k can be obtained by measuring q0 , r, x, and E>(x,r).

A four by four inch heater is placed in the granular medium, and a ther- mocouple is located at the distance x from the center of the heater. After the heater is turned on, the heat flux, the elapsed time, and the temperature at x are recorded at regular intervals. The power output of the heater was usually about 1500 W / m², and the distance x was about 2 cm. Various values for r were used in calculating k using the previous measurements of a.

The specific heat capacity c was then obtained using the values of a and k as follows:

c = - - -k PpVoa

Since the density of the interstitial gas is far lower than that of the solid particles, the above approximation is valid. All the properties were measured at the critical solid fraction v0 •

beads beads beads

diameter (mm) 3.04 1.26 0.50

standard deviation (%) 7.2 2.9 14.9

critical solid fraction 0.59 0.61 0.60

solid density (kg/m³⁾ 2500 2500 2500

thermal conductivity (W /m · °C) 0.20 0.20 0.20 thermal diffusivity (m² /s) 1.5

x

10-¹ 1.5

x

10-¹ 1.5

x

10-¹

Table 1. Properties of granular materials

beads 3.15 19.7 0.58 1010 0.073 0.93

x

10-¹

seeds 2.22 12.6 0.64 1100 0.092 0.87

x

10-¹

... (0

VARIABLE, HIGH

SPEED ELEVATOR

FLOW CONTROL GATE

UPPER FEED

HOPPER

CHUTE INTAKE

HOPPER

FLOW CONTROL GATE

Figure 3.1. Schematic of the experimental facility.

CANTILEVER BEAM

COMPUTER A ID

CONVERTER

DIGITAL VOLTMETER

Figure 3.2. Schematic of the shear gauge.

ALUMINUM PLATE

AMPLIFIER

co c..:>

I

r-0.25cml

OJ 6 cm

~

RECEIVING FIBRES

Figure 3.3. Geometry of the faces of the two displacement probes used for velocity measurements with the 1.26 mm diameter glass beads.

~ co I

VOLTAGE OUTPUT

6.t = 6.360 ^X 10-S

2400

2200

2000 I I I I ! I ! I I I I I I ! ! I ! I I I I I I 1 1 1

100 200 300

t

flt

400 500 600

Figure 3.4. Typical signals from two neighboring fibre optic probes. Voltage output (1 unit = 2.44 millivolts) against the time, t, normalized by the time in- terval between each sample, !:t.t

=

6.360 x 10-⁵ sec. Threshold voltages shown in dotted lines.

r.o Cl-.

I

CROSS CORRELATION

0 10 20 30 40 50

T

ll.t

Figure 3.5. Output of cross-correlation of two signals shown in Figure 3.4.

co 0)

I

1.1

1.0 r

u

0.9 [

u., -

0.8

r

0.7 -0.5

1.1

a

1

^1.0

r

^D

^e

^v

• ..

a

^x ~

^{-u. u 0.9}

^.

a

1

^0.8

^~

0.7

0 0.5 -0.5 0

z z

b b

Figure 3.6. The transverse velocity profiles (a) at the chute base and (b) at the free surface; mean velocity normalized by mean velocity at the center against the lateral location, z, normalized by the chute width, b = 76.2 mm. D, 8

=

17.8°, Vm

=

0.54, Uw

=

0.898 m/sec, and u11 = 1.118 m/sec; 6, 8

=

22.7°, ^Vm

=

0.50, Uw

=

1.386m/sec, and u11

=

1.639m/sec; \], 8

=

32.2°, ^Vrn.

=

^{0.49, Uw}

=

2.055 m/sec, and u11

=

2.263 m/sec. ^Uw, velocity at the center on the chute base; u11 , velocity at the center on the free surface.

v

!Iii

~

_CD ^I

-:(

I 0.5

e

6

l

• t-

3

r

;; II

2 t-

0 t ^{I I I}

0

I I

~ f

^{I I}

.. ..

^{-I I- a}

, , '

,

D' , ^{~ I-} a ^-t I- ⁰

, , ,

a,' ' 1 t a 1 t

.

, , ,

a :

,

r ^a

..,

_r

' ' , a,'

l I

a

'

l I

, a

a. ' D• ,

.

a

a . .

^{' a}

a

.

^a

I

.

I

...

O.li 1.0 1.li 0 0.05 0.10 0.15 0.20 0 0.2 0 .•

u(m/Bec) u' (m/Bec) J.110

Figure 3. 7. The vertical profile at the sidewall, 0. Vertical location, y, normalized by the particle diameter d against (a) mean velocity, (b) velocity fluctuation, and (c) linear concentration. x, data at the center of the chute. Dotted line, the assumed velocity profile at the center of the chute. Data were taken at 6

=

^{17 .8° on} the rubberized surface; ^Vm

=

^{0.30, h}0

=

25.4 mm, and d = 3.04 mm.

)(

a

a

ll

a

a

ll

al

^{co 00 I}

0.6 0.8

6

5

l

^I

4 t-

II 3

d '

2 t"

0

0 0.5

I

D

l ^r

D

l ^r

D

>ll ^{D )( D )(}

D D D

D a ^D

D D D

D D D

D a ^D

D D D

D a a

D D

a ^D

1.0 1.5 2.0 0 0.05 0.10 0.15 0.20 0.26 0.30 0 0.2 0.4 0.6

u (m/aec) u'(m/aec) ¹¹¹⁰

Figure 3.8. The vertical profile at the sidewall, ^0. Vertical location, y, normalized by the particle diameter d against (a) mean velocity, (b) velocity fluctuation, and (c) linear concentration. x, data at the center of the chute. Data were taken at 0

=

22.7° on the rubberized surface; ^I.Im

=

0.10, h0

=

^15.9mm,

and d

=

3.04mm.

a co

D co

I 0.8

6

5

4

JI 3

;;

2

t

0 0

I I I

~ ~ .

a a l r a

a

~ ~

^{a l r a}

a

~ ~

^{a l r}

a a

a a

a a

a a

# g

^a

a a

a a

a a

0.5 1.0 1.5 2.0 0 0.05 0.10 0.15 0 0.2 0.4

u(m/aec) u' (m/aec)

Figure 3.9. The vertical profile at the sidewall, ^D. Vertical location, y, normalized by the particle diameter d against (a) mean velocity, (b) velocity fluctuation, and ( c) linear concentration. Data were taken at () ⁼ 22. 7° on the smooth surface; h0 = 15.9mm and d = 1.26mm.

a

a a a a

f}

t--0 0 0

I

0.6 0.8

lltD

1.0

++c:10+0 + ^QJo 0 0

.:t

_{+ + + +}

b. + +

0.8

I-

b.

It

_{b. +}

Uw

0.6

r- ⁱ

^{b. b. b. b. b..}

- ^4\

fr

_It ^b.

u, _{b. b. b.} ¹¹

IS. 11

0.4

~

^I

r ~

0.2

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Vm

Figure 3.10. The ratio of velocity at the chute base wall to velocity at the free surface,

uw/u,,

against mean solid fraction, ^Vm. D, the smooth surface;

+,

the moderately smooth surface; 6, the rubberized surface.

...

0 ...

I

40f1111I1111I1111I111'I'111I111•11r11 .fO

f'

^{I I I I' I I I I} I I I I I I I I I I I I I I I I I I I I' I I·~ 80

t A

A A

30 t- I 30 t- ⁶⁰

A A A A A

A A A A

A A A

0

1 ^wr

A A

Au

r

^A

T ^{20 AA A} A 40 ^A

D D AA A

(sec~¹) D D DD

0

D D A

D D D D

D

10 I- Cb

1

¹⁰

^~

^{20 0}

D a ^{D D}

D 1

0 111111111111111111111111111111111111 0 I I I I I 111111111111111111111111111111 I 0 111111111111111111111111111111IJ1111

0 0.1 0.2 0.3 0.4 O.li 0.6 0.7 0 0.1 0.2 0.3 0.4 O.li 0.6 0.7 0 0.1 0.2 0.3 0.4

Vm Vm v.,.

Figure 3.11. The shear rate, llu/h, against mean solid fraction, v,.,..: (a) the smooth surface, (b) the moderately smooth surface, and (c) the rubberized surface.

D, d

=

3.04 mm; b., d

=

1.26 mm.

O.li 0.6 0.7

>--"' 0 ~

I

u' (m/aec)

0.4

A 0.3

r

1 ^{A A}

j t ^:

^A

^j t

^{v v A}

.. r

^{A ..:..}

_C\

_A ^A _D ^v _{A a} ^v _AD

A+ b v v

+A A A v a ^v AD

++t +A+ C A a a

0.1 I- ^{+ A +D+ +} v v

A +A a+ A A

A A+ A A

a + ~

Ill

0 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 111111111111111111111111111111 111111111111111111111111 11111 111 I 11 J

0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4

Vw v,,. Vw

Figure 3.12. The longitudinal velocity fluctuation at the wall, u~, against solid fraction with the rubberized surface. (a) Data ford= 1.26mm and d = 3.04mm.

D, the smooth surface;

+,

the moderately smooth surface; 6, the rub- berized surface. (b) Data ford= 1.26mm. D, h0 = 38.1 ,..., 50.Smm; 6, h₀ = 25.4mm; \}, h0 = 12.7,..., 15.9mm. (c} Data as in (b).

0.5 0.6 0.7

...

c...:i 0 I

0

0 0.1 0.2 0.3 0.4 0.5

llw

Figure 3.13. The longitudinal velocity fluctuation at the chute base wall normalized by mean velocity,

u'w/uw,

against wall solid fraction, Vw· D, the smooth surface; +,the moderately smooth surface; ~,the rubberized surface.

0.5 ~ A A

AA ~ /iA £ _AA 0.4 t-

A

J&

A AA

A A

0.3

r _!~

₊

~

rs

++ + +

+ -

+

TN

++

Q. r!J _D

~

+ +

D

Cb

^D

0.2 t-

D

0.1

~

0

0 0.1 0.2 0.3 0.4 0.5

llw

Figure 3.14. Friction coefficient at the wall, f = rs/rN, against wall solid fraction,

Vw. D, the smooth surface;

+,

the moderately smooth surface; 6, the rubberized surface.

~

0 Ol I

0.5 L

0.4 t-

0.3

r

-TS TN

0.2 t-

0.1 ~

0 0

• +t

+ ~

+++

.t t

C!IP a +

fl

a

0.05 0.10

u' _w

Uw

A A AA A &A fr,A. A AA AA

~

A·

A

j -i

0.15 0.20

Figure 3.15. Friction coefficient at the wall,

f

= rs/TN, against longitudinal veloc- ity fluctuation at the wall normalized by mean velocity,

u'w/uw.

^{D, the}

smooth surface;

+,

the moderately smooth surface; 6., the rubberized surface.

f-' 0

""

^I

TN

p _p u'2 _w

100

[

^{I I I A + I I}

J' ~ ~

^{I I I I A}

+ a

A

I 1 ^r

^A

10

l-

_{A A} A _-

A r:J- n // ^+A

N:.. + +

f

_+~ ^{A 6 A +cr-+ a} + ^{A A 121+}

0 A+

i~ ⁺

~~

1 I:-

A

t

^t!J

i

+ A A ,,./Ji

A ~ 1

0.1

I .

^I

~Ii,

^{I I I}

^I

I I I I

I

I I I I

I

I I I I

I

I I I I

I

I I I I

I I ... ^li.1, .1 •••• 1,., .1.,,, 1, ••• 1,,,, I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6

llw llm

Figure ^3.16. The normalized normal stress,

TN/

Pvu',}, against (a) wall solid fraction,

Vw, and (b) mean solid fraction, Vm. D, the smooth surface;

+,

the moderately smooth surface; !:;:,., the rubberized surface. The solid lines, the results of Lun et al. [1984].

0.7

I-' 0 ~

I

100 ~

10 I:-

Ts

~

(

^{~U) I}

p,,

dh

^uw

1 I:-

0.1 0

I I

' ^I

e"

1

"-' 0.951

~ f

^{' ' ' I e,, I 0.95}

+

/J,.

a

II I I

^D

+ +

+ _{0. /J,.}

+a _{D 4} ⁺

""

^{+ +}

t.. _~ /J,. /J,. !j! ^/J,.

'*Cl\.

^/J,.

+ +1'\. +EB

+f~ At-+

^{,6.+ 04}

+1t

a a

"'

^{440 a}

t.. ~ _a ^/J,.

t

_{4 D}

/J,. 6.

/J,. t.. &

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6

Vw V,n

Figure 3.17. The normalized shear stress, rs/Pvd(Ll..u/h)u'w, against (a) wall solid frac- tion, ^Vw, and (b) mean solid fraction, Vm· D, the smooth surface;

+,

^the

moderately smooth surface; 6., the rubberized surface. The solid lines, the results of Lun et al. [1984J.

~

00 0 I

0.7

TN

( 6u)2

Pr•

dh

1000

~

^{I I I I}

a+ a

+ +

+ +

100 ~ +a a +

a ^A a

~+ ^{+ a}

+ +

+ ++ :f: _A ++ + ⁺

10

~A

^if..

!

^+A

^~A

^{11 +++ a 'A +}

~~ A !1iA A

A A~ri ~a ~a

1

l\

^{~ /} _{~ ~}

'

/ e , . = 0 . 8

0.1

e,. = 0.8

t

^{I I I I I I}

1 t

^{I I I I I I}

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6

Vw Vrn

Figure 3.18. The normalized normal stress, ^{TN/ Pv(d~u/h)2,} against (a) wall solid frac- tion, ^Vw, and (b) mean solid fraction, vYn. ^D, the smooth surface; +,the moderately smooth surface; ^!:::,., the rubberized surface. The solid lines, the results of Lun et al. (1984).

.

^...

co 0 I

1

0.7

1000

100

"\.

+ a

rs

a +

t.

er>= 0.95 _er>= 0.95

a p,,

(d~u )2

¹⁰ _{++ +:\:} l!t. +

^&

+

+ t.+ + t.

t. t6. +

t. +:t+ +

+ +

:t

t.

t &ti.

^t.

t. t.* t. a a

1

t.

t.

~.

a

0.1 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1

Vw

t.

t.~ !

% a a

a

0.2 0.3 0.4 0.5 0.6

v,n

Figure 3.19. The normalized shear stress, rsf Pv(d!:!,..u/h)^{2 ,} against (a) wall solid frac- tion, Vw, and (b) mean solid fraction, v,,.,. D, the smooth surface; +,the moderately smooth surface; ^'6., the rubberized surface. The solid lines, the results of Lun et al. ^[1984).

0.7

lo-' lo-'

0 I

100

+ +

+ a c1

a

10

I:

⁺

+ ++ +fl.+

a +

+

fl.

+ +

a aD

++ + +

.p + +

+

+ +

TN tan2 (}

r ~ ...

.6.u)

^{2 + +}

p,,(d-,;

^{+ +'1 l1 l1}

fl. a c:Jlll. A+ll. _fl. aa a '1+ ^l1

l:i. l1 l1

!fl. fl.A f l . \

1 I:- ^fl.

~ fl. ....

l1 A l1

.. fl. b.l1 q

l1

l1 l1 l1 l1

0.1 0

fl. A l1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6

llw Vrn

Figure3.20. The normalized normal stress, ^TNtan28/pv(dfl.u/h)^{2 ,} against (a) wall solid fraction, ^Vw, and (b) mean solid fraction, v"'. ^D, the smooth surface;

+,

the moderately smooth surface; 6, the rubberized surface. The solid lines, the results of Lun et al. (1984).

0.7

~

~

~

100 .--~--.~~--.-~~-.--~~.--~~,-~~n-~-:i

TS

( Llu)-

^tan(}

p,. d~

h

10

1

0.1 0

+

+ D

+ +

D

+ + A

+ ++ + AA+

A -A + A

~ IP- R+

A A A -n

A

Jl

^A

..

A DA a

D

0.1 0.2 0.3

D A

+

A

0.4 Vw

0.5

6.

\

0.6 0.7 0 0.1

+

t+

+ + + +6.

~+

A A

• A ~DA

6. 6. a

D

0.2 0.3

D + + +

D CD

+ + + A

A AA

If A

0.4 Vm

0.5 +

0.6

Figure 3.21. The normalized shear stress, r₈ tan9/pv(df),.ujh)², against (a) wall solid fraction, Vw, and (b) mean solid fraction, v,,... D, the smooth surface;

+,

the moderately smooth surface; 6-, the rubberized surface. The solid lines, the results of Lun et al. (1984].

0.7

...

...

t...:>

I

b.

1.5

b.

s

^1.0

^~ /

b. b.

ev = 0.95

- 1

D

~

0.5 I- /~ b. ~A

•+ + +

OD D

0.- +

~y

"1"

I I

0

0 0.1 0.2 0.3 0.4 0.5

Vw

Figure 3.22. The parameter, S

=

d(!l.u/h)/u'w, against wall solid fraction, l.lw· 0, the smooth surface;

+,

the moderately smooth surface; ~' the rubberized surface. The solid lines, the results of Lun et al. [1984].

I ...

...

w

I

s

tanO

I::..

5

eP = 0.6

4 _I::..

e,, = 0.95 3

d

2 ^I::..

~ 1 a

I::.. I::..

I::.. I::.. +

I::..

,t

,al::.. A+ + +

A 1Jt:.. 0" + +

t:.. + + ++ + I + OT _._ a ~

+

0 ^I I I I I I I I I I I I I I I I I I I I I I I I I I

0 0.1 0.2 0.3 0.4 0.5

Vw

Figure 3.23. The parameter, S/tanO = d(Ll.u/h)/u'w tanO, against wall solid fraction,

Vw. D, the smooth surface;

+,

the moderately smooth surface; 6, the rubberized surface. The solid lines, the results of Lun et al. [1984].

...

...

H:>..

I

0

4

r-

0

+

3 a

tan ^(J +

-

f

^{+ 0 + a}

+ 2

A ,P. ⁺

0

&A + a+

+A+

~ ^{A AA}

+++ ⁰

l:i. AA A+A +

]

1 I- A ~

'1

IA M l:i.

0

0 0.1 0.2 0.3 0.4 0.5

Vw

Figure 3.24. The ratio of tan 0 to

f

against wall solid fraction, llw. D, the smooth surface;

+,

the moderately smooth surface; !::,., the rubberized surface.

I-' I-' CTI I

100

Ts

( flu ² tanO

Pv d-,;)

tan ti

- , - < 1.25

II i I

- , - < ^{tan ti} 1.25

JO

~

A

r

^{A A} _A

1 I:- ^A A~

II

I I

.

^A If'

0.1

A 6 A

A A t.

0 0.1 0.2 ^0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6

Vw Vm.

Figure 3.25. The normalized shear stress, Ts tan 0 / Pv(d!lu/ h)², against (a) wall solid fraction, Vw, and (b) mean solid fraction, Vm. Data only with tan 0 /

f

<

1.25. The solid lines, the results of Lun et al. [1984).

0.7

"""'

"""'

0)

I

J

_{-,- <} ^tan8 _1.25 ^h.

ev = 0.6

4

r:- ___---;:-

--

s

^{I- / ~ e1, = 0.95}

- 3

tanO

2

1

0

0 0.1 0.2 0.3 0.4 0.5

Vw

Figure 3.26. The parameter, S/tanfJ = d(.6.u/h)/u'w tanfJ, against wall solid fraction,

Vw. Data only with tan fJ /

f <

1.25. The solid lines, the results of Lun et al. [1984].

I-' I-' - l I

•o

lO

y• 20

10

0.

0. 0.5 1 1.5 2 0. 0.05 0.1 0.15 0.2 0. 1

(~:)I

..

^{II -u• u•}

..

Figure 4.1. (a) Nondimensionalized square root of granular temperature,

(T* /T:;J ! ,

(b) solid fraction,

v,

and (c) nondimensionalized mean velocity,

u* /u:V,

against nondimensionalized distance from the chute base, y*. Parameters;

eP

=

^0.95 and tan 0

=

^0.4. Boundary values at the wall; T~

=

^{2, llw}

=

^0.2,

and

u:V

= 10.

2 l

...

...

00 I

10

y• 5

0.

0. 0.5

(r·)

_r·

..

^t

l 0. 0.1

II

0.2 0.3 0. 0.5

u•

u:.

Figure 4.2. (a) Nondimensionalized square root of granular temperature, (T* /T:i) ! ,

(b) solid fraction, v, and (c) nondimensionalized mean velocity, u* /u:,, against nondimensionalized distance from the chute base, y*. Parameters;

ep

=

0.6 and tan (J

=

0.4. Boundary values at the wall;

r:i =

2, Vw

=

0.2, and u:, = 10.

1.5 2

...

...

ID I

tO

JO

y• 20

10

0.

0. 0.5 1 1.5 2 0. 0.1 0.2 O.J O.t o. 1 2

(~:)

..

^{i II -}u• ^u•

^..

Figure 4.3. (a) Nondimensionalized square root of granular temperature, (T* /T~) ! ,

(b) solid fraction,

v,

and (c) nondimensionalized mean velocity,

u* /u:U,

against nondimensionalized distance from the chute base, y*. Parameters;

eP

=

^0.9 and tan 8

=

^0.4. Boundary values at the wall; T~

=

^{2, Vw = 0.3,}

and

u!i

⁼ 10.

J

...

1:-.j

0

' I

to

30

y• 20

10

0.

0.

tan I= 0.3

lanl = 0.4

lanl ~ 0 4 land;;:; 0.3

.._ ... ~~-'~-'---'C....-'---'~~ ... ~~~~-~ ... ~~ ...

0.5 l

(r· !

1··)

"'

1.5 2 o. 0.05 0.1 0.15

,,,

0.2 0. 1

u•

u•

..

Figure 4.4. The effect of the variation of the chute inclination angle. (a) Nondimen- sionalized square root of granular temperature,

(T* /T:,J 1,

(b) solid frac- tion, v, and ( c) nondimensionalized mean velocity, u * / u:,, against nondi- mensionalized distance from the chute base, y*. Parameters; ev

=

^0.95,

tan (J = 0.3 and 0.4. Boundary values at the wall;

r:i

= 2, Vw = 0.2, and

u:,

= 10.

2 3

...

t-.j

...

•O

JO

!/ ²⁰

lU

u.

u.

r:. = 1 7'.: =I

T,: =2

U.5 1 l.5 2 0. o.05 O.l 0.15 0.2 o. 1

(")t

_{1 ..}

..

^{JI -}u• ^u•

..

Figure 4.5. The effect of the variation of granular temperature at the wall. (a) Nondi- mensionalized square root of granular temperature, (T*

/T:;,)

! , (b) solid fraction,

v,

and (c) nondimensionalized mean velocity,

u* /u:V,

against nondimensionalized distance from the chute base, y*. Parameters; ep

=

0.95 and tan fJ

=

0.4. Boundary values at the wall;

T:;, =

1 and 2, Vw = 0.2, and

u:i

^{= 10.}

-T,: = 2

2

...

I:,,;) I:,,;)

l I

40

30

y• 20

10

0.

0. 0.5

... w =0.2

Vw = 0.15

... w = 0.15

1 1.5 2 o. 0.05 0.1 0.15 0.2 0. 1

(~:) t

.,.

_-^u•

..

u•

..

Figure 4.6. The effect of the variation of solid fraction at the wall. (a) Nondimension- alized square root of granular temperature, (T* /T:,)

1,

(b) solid fraction, v, and (c) nondimensionalized mean velocity, u* /u:,, against nondimen- sionalized distance from the chute base, y*. Parameters; eP

=

^{0.95 and}

tan 0 = 0.4. Boundary values at the wall;

r:,

^{= 2, Vw =} 0.15 and 0.2, and

u:,

⁼ 10.

Vw = 0.2

2 3

t--1 1:-.j

~

I

10

y• 5

A A A A

"

A

0. I ^{I I I I}

0. 1

11., = 0 304 _Vw = 0.304

11., = 0.205

2 3 4 0. 0.1

(~:)

..

^j

0.2 0.3

II

o. 4 0.

u•

u•

..

Figure 4.7. Comparison with the experimental data of Ahn et al. [1989a]. The smooth aluminum surface used for the chute base, and d = 1.26 mm. 6, data from the experiment. Solid line, results of the present analysis. (a) Nondimen- sionalized square root of granular temperature, (T* /T:i)~, (b) solid frac- tion, v, and (c) nondimensionalized mean velocity, u* /u:i, against nondi- mensionalized distance from the chute base, y*. Parameters; ev = 0.95 and tan 8

=

0.418. Boundary values at the wall; T:i

=

0.935, Vw

=

^0.205

and 0.304, and u:i = 11.3.

2 3

...

~

~

I

10 _I

y• 5 I -

0. I I I I I I

0. 0.5

~

"· =0.244

l'w = 0.1711

I I I I 't: ^I I I I I

1 1.5

(")•

_1··

...

... -

A

2 0. 0.1 0.2

"

...

0. 3 0. 0 5

l'w =0.1711

1

u•

u• ...

Figure 4.8. Comparison with the experimental data of Ahn et al. [1989a]. The rubber- coated surface used for the chute base, and d = 3.04 mm. ~,data from the experiment. Solid line, results of the present analysis. (a) Nondimension- alized square root of granular temperature, (T* /T:,)

1,

(b) solid fraction,

v,

and ( c) nondimensionalized mean velocity,

u

* /

u:i,

against nondimen- sionalized distance from the chute base, y*. Parameters; ep = 0.95 and tan 0

=

0.418. Boundary values at the wall; T:, = 0.440, ^Vw

=

0.179 and 0.244, and u:i = 4.39.

l'w = 0.244

...

1.5 2

...

t...:i CJl I

20

15

y• 10

5

0.

0.

lanl = 0.S77

Ian 0 = 0.577 land= 0.45

lanS = 0.45

0.5 1.5 2 0. 0.1 0.2

(~:)

..

^{i v}

A

0.3 0.4 0. 5 0. 2

u•

u• .,

Figure 4.9. Comparison with the computer simulations by Campbell and Brennen [1985b). 6, data from the computer simulations. Type A simulation with

(J = 30°, ep = 0.6, and ew = 0.8. Solid line, results of the present analysis.

Parameters; eP = 0.6, and tan 0 = 0.577 and 0.45. Boundary values at the wall;

r;

^{= 7.00, vw =} 0.140, and u:, = 14.0. (a) Nondimensionalized square root of granular temperature, (T* /T,:)

! ,

(b) solid fraction, v, and (c) nondimensionalized mean velocity,

u* /u:U,

against nondimensionalized distance from the chute base, ^y*.

l '

...

tv Cj)

I

fp=0.6 ~

tan 8 0.5

e,,

⁼ 0.8

e,. = 0.95

0.

0. 0.1 0.2 0.3 0.4 0.5 0.6

v

Figure 4.10. Zeros of the granular conduction term for various ev.

'"""'

t.,j

-.;s I

'0

30

y• 20

10

0.

-2 -1.5

10

'°

30

5 ²⁰

10

0. o.

-1 -.5 o. ^{0. 5 -.1 0. 0.1 0.2 O.l 0., 0.5 0.6 -1.5 -1 -.5}

Q·, q• Q·, q• Q·, q•

Figure 4.11. Nondimensionalized granular conduction, Q*, and fluctuation energy flux, q*, against nondimensionalized distance from the chute base, ^y*. Parame- ters and boundary values at the wall; (a) as in Figure 4.1, (b) as in Figure 4.2, and (c) as in Figure 4.3.

0. 0.5

...

t.,j 00 I

20

IS

·~ = 0.95

y• 10

•• = 0.9

5 f- ' • ; 0.9

0.

0. 0.5 1 0. 0.1 0.2 0.J 0.,

u=)·

^T•

_..

^II

Figure 4.12. The study of the case in which no shear motion and mean velocity exist.

(a) Nondimensionalized square root of granular temperature,

(T* /T:;J ! ,

and (b) solid fraction, v, against nondimensionalized distance from the chute base, y*. Parameters; e,, = 0.9 and 0.95, and tan 8 = 0. Boundary values at the wall;

r:, =

5 and ^11w = 0.2.

,_.

tv co I

•O

30

y• ²⁰

10

0.

_,

-3

10

•o

30

5 20

10

0. 0.

-2 -1 0. 0. 0.5 1 ^-1 -.5 0.

Q· q· Q·

..,. ..,. ., .

Figure 4.13. The ratio of granular conduction to dissipation, Q* /1*, against nondi- mensionalized distance from the chute base, ^y*. Parameters and bound- ary values at the wall; (a) as in Figure 4.1, (b) as in Figure 4.2, and (c) as in Figure 4.3.

0.5 1

,,...

I'.."

0 I

2 ^ep = 0

gf g2g5

( gr )

g2Y5 v=O

e₁, = 1 1

0.

0. 0.1 0.2 0.3 0.4 0.5 0.6

v

Figure 4.14. The function _{grf g2g5} for various ep, normalized by g~/g2g5 at v = 0.

~

""

^~

FLOW.

ALUMINUM CHUTE BASE

LOCATIONS OF lHERMOCOUPLES UNDER THE COPPER PLAlE

x

25

TOP VIEW

TOP HEATER

SIDE VIEW

x

II II

E5

x

""- SETSCREW

Figure 5.1. Schematic of the heating plate.

Vm

~ v v v

0.6t-

o

i1<

txW

+Ii+

Fl

Xxlfj

0 +

0.5~

x

~ ^ll.

ll. + ^ll.

11 ll.

0.4r v

At!.A

~ v

0.31-

x

ti ..

jV

vj I

0.2 t-

x

o•

XX +

0.1 I-

x

0 I I I I !Ill!! I I ! Ill!!! I I I !11111 I I I 1111!1 ! ! 1111111

10-³ 10-² 10-¹ 1 10¹ 102

u

Fr= Jghcos()

Figure 5.2. The mean solid fraction, Vm., against the Froude number, Fr. D, 3.04 mm glass beads;

+,

1.26mm glass beads; 6., 0.50mm glass beads; x, 3.15mm polystyrene beads; V, 2.22 mm mustard seeds.

,_.

w w I

A A

300

~ _A

+ _A

+ + A

A +

+ ^A&

h ₂₀₀ A

A

(W

/m

^{2 ·} °C) + _"11 1-

100

v _Qb

A +

0 v

ix x

Xx

VI

+

0 ~

I

Vv++

a, j(

ll

^{v r}

~

O' 10-3 I 11111111 I ,.1

-l...1l..1.i.1Jii~!1 ~~~I ...L.lllli~_JLJ..J..Wul_LU.illH

^{- 11111111 I • I} I I I I I l l I I • I I I 1111

10-2 10-1 1

io•

¹⁰²

u

Fr =

.jg

h cos 0

Figure 5.3. The heat transfer coefficient, h, against the Froude number, Fr. D, 3.04 mm glass beads; +, 1.26 mm glass beads; b.., 0.50 mm glass beads;

x, 3.15 mm polystyrene beads; \), 2.22 mm mustard seeds.

...

w ~

I

1

Nu*=

.Ji f l

0.025

+ 2V Pe*

x

'2t

a 15

* hd

lOr Id

^v

N u = - ^VV

kg

*v El

V/

xv

AA + v

~ ~

"'v

I

5 t- ^{,,A A}

~ ^a

'

⁺

0 I I I I II II!! I I I II !Ill I I I Ill Ill I I I I 11111 1 J 1 I ' 11u

1 10¹ 10² 103 10⁴

lOS

Pe*=

UL(~)2(kc)2

Ge L kg

Figure 5.4. The modified Nusselt number, Nu*, against the modified Peclet number, Pe*. The curve is from equation (5.3). D, 3.04 mm glass beads;

+,

1.26 mm glass beads; !::.., 0.50 mm glass beads; x, 3.15 mm polystyrene beads; ^\J, 2.22mm mustard seeds.

...

~ Cl1

I

1 10 10 10 10

Pe;ff = UL ( d ) ^{2 (} ke ) 2

ae L kg

Figure 5.5. The effective Nusselt number, Nu:ff, against the effective Peclet number, Pe!ff· The curve is from equation (5.7). D, 3.04mm glass beads;

+,

1.26 mm glass beads; 6, 0.50 mm glass beads;

hd kc

* - - - Nueff - kg ke

25 Nu;H =

20

15

10

5

A

+ _ip; ~

~~

AX A

.a.;~

v

<!" ~ ^{x x}

⁾⁽

§

0^{1 ,} 1 1111111 1 1 1111111 • 1 I " ' " ' , . . .

10⁴

101 102 103

1

* _ UL

(!!.)2

^(ke

)2

Peefr - ae l ^J k ^g

Figure 5.6. The effective Nusselt number, Nu:ff, against the effective Peclet number, Pe;ff. The curve is from equation (5.7). O, 3.04mm glass beads;

+,

l.26mm glass beads; 6., 0.50mm glass beads; x, 3.15mm polystyrene beads; \J, 2.22 mm mustard seeds.

I-' w

- I

I

y*

0.5 ~ I I I I

I

I I I I

I

^{I I •}

I l ^f

k. ; k.

- =6.7 o - = 35

oA J- 1 1 r kg ¹ kg

0.3

~

0.21

0.1 t-

I , , '

/ l

^[" _, ^,

, , , ,

~

-,

_I

,:/

0.1 0.2 0.3 0.4 0.5 0 1 2 3

v k*

Figure 6.1. The profiles of (a) the solid fraction and (b) the nondimensionalized ther- mal conductivity against the nondimensionalized distance from the flat plate for the case of Vm. = 0.4.

i

^...

""

⁰⁰

I

4

15

Nu* - hd

--;;- 10

g

5

or::, , '""" , , """' , , "'"" , , , ...

10⁴

1 ¹⁰¹ 102 103

Pe**= UL

(!!_)2(kc)~

ac L kg Ve

Figure 6.2. The results from the numerical analysis: The modified Nusselt number, Nu*, against the Peclet number, Pe**, with the variation of the mean solid fraction, Vmi ka/kg = 35.

...

c.AJ

co I

Nu*

hd

kg

15

10

5

0 ^{I I I I I I I I l l I I I I} " " ' I I I I I I I l l I I I I I I " '

1 10• 102 103 104

Pe**=

UL(~)2(kc)_!_

O!c L kg Ve

Figure 6.3. Comparison with the constant wall heat flux boundary condition for the case of k./k₀

=

35: Solid line, the constant wall temperature boundary condition; dotted line, the constant wall heat flux boundary condition.

f--4

""'"

0 I

Nu* kg

5

10• 102 103 104

Pe**

= UL (!!_)2 (kc)_!_

ac L kg Ve

Figure 6.4. The effect of k₁₁ ^/ ku, the ratio of the thermal conductivity of solid particle to that of gas phase: Solid line, k₁₁/ku

=

35; dotted line, k11/ku

=

6.7.

...

,f:>..

...

4

3

y*

2 I

"

^.J

t

Pe**= IO

~

~

I

0.2 0.4 0.6 0.8 1

E>*

Figure 6.5. The profile of the nondimensionalized temperature, 0*, at the trailing edge against the nondhnensionalized distance from the flat plate, y*, with the variation of the Peclet number, Pe^{0 ;} vm. = 0.4.

,...

,i::...

t..:)

I

0.8

0.6

y*

0.4 1- ""-.. -f

r ""

^~

¹

0.2

00 0.2 0.4 0.6 0.8 1

0*

Figure 6.6. The profile of the nondimensionalized temperature, 0*, at the trailing edge against the nondimensionalized distance from the flat plate, y*, with the variation of the mean solid fraction, vm; Pe**

=

1000.

I

...

~ w

I

kg

5

10¹ 102 ₁₀³ 10⁴

Pe**= UL

(d)2(kc)_!_

ll:'.c L kg ^lie

Figure 6.7. Comparison of the numerical results for ka/ku = 35 with the experimental data for glass beads. x, Vin < 0.15; \J, 0.15 < Vm. < 0.25; D, 0.25 < ^Vm. <

0.35; 6, 0.35 < Vm < 0.45; T, 0.45 < Vm < 0.55;

+,

0.55 < ^Vm·

...

.i:...

.i:...

I

Nu*= hd

kg

15

10

5

I I

I I 1111111

0 c: 10₁

1

*

+ ⁺⁺

... + + +

102

"

+

11111111

10³

Pe**

= UL (!}_)2 (kc)_!__

ac L kg Ve

I I I ¹ I I Ill

10⁴

Figure 6.8. Comparison of the numerical results for

k./ k.,

= 35 with the experimental data for glass beads, polystyrene beads, and mustard seeds. x, Vm < 0.15;

v,

0.15 < Vni. < 0.25; D, 0.25 < Vni. < 0.35; l::., 0.35 < Vni. < 0.45; T' 0.45

<

^Vm

<

0.55;

+,

0.55

<

^Vm·

...

"""

Cll

I