rajai1 y b. 2 applications v heat and mass transfer rates are enhanced by the oscillation of the...
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![Page 1: Rajai1 y b. 2 APPLICATIONS v Heat and mass transfer rates are enhanced by the oscillation of the surrounding fluid. Useful in combustion, drying and the](https://reader031.vdocuments.site/reader031/viewer/2022013004/56649d785503460f94a5a1c3/html5/thumbnails/1.jpg)
Rajai 1
y
b
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Rajai 2
APPLICATIONAPPLICATIONSS
Heat and mass transfer rates are Heat and mass transfer rates are enhanced by the oscillation of the enhanced by the oscillation of the surrounding fluid. Useful in combustion, surrounding fluid. Useful in combustion, drying and the passage of sound waves drying and the passage of sound waves through particulate systems.through particulate systems.
Particle-laden flows, Brownian motion, Particle-laden flows, Brownian motion, suspension rheometry, colloidal suspension rheometry, colloidal suspension, and particle motion in suspension, and particle motion in filters.filters.
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Rajai 3
MODES OF MODES OF HEAT HEAT TRANSFERTRANSFER
RADIATIONRADIATION
CONDUCTIONCONDUCTION CONVECTION (forced & CONVECTION (forced &
free)free)
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Rajai 4
GOVERNING EQUATIONS
Conservation of momentumConservation of momentum
wFpw)w(t
w 2
)w(w)2/w(w)w( 2
)w()w(w2
where
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Rajai 5
First law of thermodynamicsFirst law of thermodynamics
Conservation of massConservation of mass
0w
T)Tw(tT 2
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Rajai 6
VECTOR RELATIONS
2
1 2 3 1
2 3
1 1 2
1 3
2 2 3
1 2
3 3
1
h h h q
h hh q q
h hh q q
h hh q
( ) ( ) ( )
ds h dq dqij i jji
2 2
1
3
1
3
h
xq
yq
zqi
i i i
2 2 2 2 ( ) ( ) ( )
333
222
111
uqh
1u
qh
1u
qh
1
)hhF(q
)hhF(q
)hhF(qhhh
1F 213
3312
2321
1321
332211
321
332211
321
hFhFhFqqq
uhuhuh
hhh
1F
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Rajai 7
w
1231
2132 q
whh,q
whh
Define:
332211
321
332211
321
hwhwhwqqq
uhuhuh
hhh1
w
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Rajai 8
q
h
h h q q
h
h h qh h
1
2
1 3 1 2
1
2 3 21 2( ) ( )
h ht q q h q q h
qh
h h qh
qh
h h qh
hq
h Fq
h F
1 22 1 3 1 2 3
1
2
1 3 13
2
1
2 3 23
31
2 22
1 1
( ) ( )
then,
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Rajai 9
and,
h h hT
t q
T
q q
T
q
q
h h
h
T
q q
h h
h
T
q
1 2 32 1 1 2
1
2 3
1 1 2
1 3
2 2
( ) ( )
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Rajai 10
a
g
x r sin cos y r sin sin z r cos
SPHERICAL COORDINATES
SCALE FACTORSh h r h rr 1 , , sin
r
T
Ts
U U to cos( )
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Rajai 11
ln( / )r a
/ U ao2
a Uo/
( ) / ( )T T T Ts
t U t ao /
U U U S to / cos( )
VARIABLES
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Rajai 12
Re /2aUo
Gr g T T as ( )( ) /2 3 2
Pe = Re Pr
S a Uo /
Pr /
T
Ts
DIMENSIONLESS NUMBERS
kinematic viscosity
thermal diffusivity
volumetric expansion coef.
frequency of oscillation
surface temperature
far-field temperature
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Rajai 13
e32
2
2
2 0
sin cot
ee
e
2
2
2
2
2 2
2
t
Gr2Re
2
sin( ) ( cot )
Recot
sin
sin cos
ee
Pe2
2
2
2
2
2
t
sincot
USING THE SPHERICAL COORDINATE SYSTEM, THE EQUATIONS REDUCE TO:
(1)
(2)
(3)
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Rajai 14
0 1 0, and at
e S t and e S t
ore
S t
as
2 2 2
22
2
sin cos( ) , sin cos cos( )
, sin cos( )
0 0, and as
BOUNDARY CONDITIONS
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Rajai 15
• Equations are two dimensional, time-dependent, nonlinear, coupled, and of infinite domain.
• No explicit boundary conditions for vorticity on the surface.
• Difficulties with finite- differences such as, indeterminate forms, etc..
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Rajai 16
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Rajai 17
THE METHOD OF SOLUTION
e f t P dn n
zn
/ ( , ) ( )21
1
g t P zn nn
( , ) ( )1
1
z cos
h t P zn nn
( , ) ( )0
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Rajai 18
HOW ?!
t
h
tPn
nn
0
ee
Pe2
2
2
2
2
2
t
sincot
eh
tP P dzi
i ji
2
1
1
0
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Rajai 19
INTEGRALS NEEDED !!
P z P z dznk
mk( ) ( )( ) ( )
1
1
z P z P z dzn m2 1 1
1
1( ) ( )( ) ( )
P z P z P z dzn i j( ) ( )( ) ( ) ( )1 1
1
1
P z P z P z dzn i j( ) ( ) ( )( ) ( ) ( )1 1 2
1
1
and in general,
P z P z P z dzjm
jm
jm
1
1
2
2
3
3
1
1
( ) ( ) ( )
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Rajai 20
P z P z P z dzj m j mj m j m
nj j n j j n
m m m m
n m mn m m
P z P z dz
jm
jm
jm
m m
n
jm
nm m
1
1
2
2
3
3
1 2
3
3 2 1
1
12 2 1 1
2 2 1 1
1 2 1 2
1 2 1 2
1 2
1 2 1
1
1 2 10 0 0
( ) ( ) ( )( )!( )!( )!( )!
( ) ( )
( )!( )!
( ) ( )
j j n j j1 2 1 2
where
After Mavromatis & Alassar:
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Rajai 21
P z P z dzm m m m
j m j m
j m j m
Gk m m
k m m
Fm m k m m k m m
m mm
jm
jm
m
m m
k
k m m
1
1
2
2
2
2 1
2 1
1
1
2 1 2 1 2 1
1 1 2 2
1 1 2 2
2 1
2 1
3 22 1 2 1 2 1
2 1
1
21
2 2
3
2 2
1 1
1
2 2 21 1
3
( ) ( )( )
( ) ( )
( )!( )!
( )!( )!
( ( ) )( )!
( )!
, , ; ,
( )
2 1
21
m;
G kj j k j j k
m m m mm m
( ) ( )1 2 1
0 0 01 2 1 2 1 2
1 2 1 2j j k j j1 2 1 2
where
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Rajai 22
Saalschutz’s Theorem
If c d a b n then
F a n b c dc a n c b n c c a b
c a c b c n c a b n
In thiscase
am m k
nk m m
bm m
c m m
dm m
1
1
1
2
2
21
1
3
2
3 2
1 2
1 2
1 2
1 2
1 2
,
, , ; , ;
:
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Rajai 23
3-j SYMBOLS
Represent the probability amplitude that three angular momentum j1, j2, and j3 with projections m1, m2, and m3 are coupled to yield zero angular momentum. They are related to Clebsch-Gordan coefficients (C) by:
j j j
m m mCj m j
j j m j mj m1 2 3
1 2 3
2 12 1
1 3 3 1
3 1 1 2 2
3 3
( )
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Rajai 24
Cabc b c
a b c a b
a c ca b b c
F a b c a c a b b c
a bc b
,
/
( )( )( )!
( )!( )!
( )!( )!( )( )!( )!( )!( )!
, , ; , ;
1
2 1
1 1 1
1 2
3 2
( )( )!( )!( )!
( )!
/
abca b c a b c a b c
a b c
1
1 2where
Clebsch-Gordan Coefficients
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Rajai 25
2
22 5 21 2 1
fn f n n e gn
n n ( / ) ( ) /
eg
t
g gn n g
en
n hh
nn h
hS
n n nn
nn
nn
n
22
2
11
11
21
1
2 11
1
2 32
Re( )
( )( )
( )( )
Gr
2Re2
eh
t Pe
h hn n h Hn n n
n n2
2
2
21
( )
(1)
(2)
(3)
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Rajai 26
S e fg
g gf
fn ijn
ij
j ijn
ji
ijiji
/ ( ) ( )2
1111
1
2
H en
ih
ff f
hn nj
ij
ii ij
ni
j
jiji
/ ( )2
0101
2 1
2 1
1
2
ijn n
j j
n n
n i j n i j
( )
( )
( )2 1
1
1 1 0 1 0 0 0
ijn n
j j j
n n i i
n i j n i j
( )
( )( )
( ) ( )2 1
1 2
1 1 1 1 2 0 0 0
2
ijn n
n i j
( )2 1
0 0 0
2
where,
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Rajai 27
MODES BOUNDARY CONDITIONS
f tf
t and h tnn
n n( , ) ( , ) , ( , )0 0 0 0 0
f t e S t
f te S t as
n n
nn
( , ) cos( ) ,
( , )cos( )
/
/
3 21
3 21
3
2
g t and h t asn n( , ) , ( , ) 0 0
e g d S tnn n
( ) cos( )2
0
1
3
2
2
22 5 21 2 1
fn f n n e gn
n n ( / ) ( ) /
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Rajai 28
NUMERICAL METHOD
eg
t
g gn n g
en
n hh
nn h
hS
n n nn
nn
nn
n
22
2
11
11
21
1
2 11
1
2 32
Re( )
( )( )
( )( )
Gr
2Re2
eh
t Pe
h hn n h Hn n n
n n2
2
2
21
( )
CRANK-NICOLSON F.D. SCHEME
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Rajai 29
g
tq tn
n ( , )
q eg g
n n g Snn n
n n
22
2
21
Re
( ) *
1
12
tg t g t t
q t q t t
n n
n n
( , ) ( , )
( , ) ( , )
A t g t B t g t
C t g t D t t E tn n
n n n
( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
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Rajai 30
SPECIALIZED STEP-BY-STEP METHOD
xf
n fnn
n
( / )1 2
Qf
n fnn
n
( / )1 2
fQ x
nnn n2 1 2( / )
2
22 5 21 2 1
fn f n n e g rn
n n n ( / ) ( ) ( )/
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Rajai 31
x
n x rnn n ( / )1 2
Q
n Q rnn n ( / )1 2
at 0x Qx Q
n nn n
0
x h t x t e e r z t dzn nn n x
n
h
( , ) ( , ) ( , )( / ) ( / )
1 2 1 2
e n h( / )1 2 1
x h t x t r t nn n n( , ) ( , ) ( ) ( , ) / ( / ) 1 1 2
Q t e tn n( , ) cos( )/ 3 3 21
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Rajai 32
x h t x th n h n n
r t
h n h n nr h t
h n h nr h t
n n n
n
n
( , ) ( , )( / ) ( / ) /
( , )
( )
( / ) ( / ) /( , )
( / ) ( / )( , )
01
1 2
1 3
2 1 2 1 20
2 1
1 2
2
1 2
1
1 2
1
1 2
1
2 1 22
2 3 2
2 3 2
2 3 2
x h t x th n h n n
r t
h n h nr h t
h n h n nr h t
n n n
n
n
( , ) ( , )( / ) ( / ) /
( , )
( )
( / )
( )
( / )( , )
( / ) ( / ) /( , )
21
1 2
3 1
2 1 2 1 2
2 1
1 2
2 1
1 2
1
1 2
3
2 1 2
1
1 22
22
2 3
2
2
2
2
2 3
2
2
2
2 3
2
2
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Rajai 33
PHYSICAL PARAMETERS
Nh
Pun
nn
( , ) (cos )
2 20 0
N N du u( ) ( , )
1
0
N N du u( ) ( , ) sin
1
2 0
Nusselt Number
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Rajai 34
Nh
u ( )
2 0
N N x dxu u( ) ( )/
/
1 2
1 2
N n F n nh
n
n
h
un
n
n
n
( ) ( ) ( / , , / ; , ; )
( / )
!
2 1 3 2 3 2 2 2 1
2 1 2
23 2
2
0
2
22
0
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Rajai 35
3.00 3.25 3.50 3.75 4.00
-0.10
0.00
0.10
0.20
R e = 50
205
St
2
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Rajai 36
CD
U aDo
2 2
C C CD DF DP
C dDF 4
0 2
0Re( , ) sin
CU
p dDPo
1
0 22
0
( , ) sin
Drag Coefficient
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Rajai 37
( )Re Re
sin
p Gr
0
02
4
C gDF 16
301Re
( , )
C gg Gr
DP
8
30 0
2
311
2Re( , ) ( , )
Re
p p p dGr
d*
Re( )
Resin
4
2
pGr
Pg
gnn
n
Nn
n*( , )
Re( cos )
Re(cos ) ( ) ( , ) ( , )
2
1
14
1 0 0
Pressure Coefficient
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Rajai 38
Re Present
Wong
et al.
Sayegh&
Gauvin
Dennis&
Walker Whitaker
Dennis&
Walker
1 2.263 - 2.232 2.260 2.402 2.22
10 3.326 3.282 3.323 3.358 3.349 3.18
20 4.046 3.971 4.022 4.065 3.950 3.86
30 4.584 - 4.560 - 4.421 4.36
60 5.855 5.889 - - 5.511 5.65
100 7.153 7.351 - - 6.625 7.11
Re Gr Present Wong et al.
Percentage
Difference
60 720 5.943 6.021 1.312
60 14400 7.331 7.183 2.019
100 2000 7.190 7.509 4.437
VALIDATION
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SOME RESULTSR e G r G r / R e 2
N uN N f o r c e du u/ ( )
5 0 . 0 0 . 0 2 . 2 3 9 4 1 . 0 0 0 05 1 2 . 5 0 . 5 2 . 5 5 8 6 1 . 1 4 2 65 2 5 1 . 0 2 . 6 9 5 7 1 . 2 0 3 85 1 2 5 5 . 0 3 . 1 1 8 6 1 . 3 9 2 65 2 5 0 1 0 . 0 3 . 3 7 4 1 1 . 5 0 6 7
2 0 0 . 0 0 . 0 2 . 5 2 9 6 1 . 0 0 0 02 0 1 0 0 0 . 2 5 3 . 1 6 0 2 1 . 2 4 9 32 0 2 0 0 0 . 5 3 . 3 8 6 2 1 . 3 3 8 62 0 4 0 0 1 . 0 3 . 6 5 8 7 1 . 4 4 6 42 0 2 0 0 0 5 . 0 4 . 5 3 8 0 1 . 7 9 4 02 0 4 0 0 0 1 0 . 0 5 . 0 8 5 8 2 . 0 1 0 5
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0 30 60 90 120 150 180
1
2
3
4
5
=0.125
0.0 , 1 .0
0 .25
0.375
0.5
0.625
0.75
0.875
fo r th e c a se o f R e = 5 0 , S = , G r = 0 d u rin g o n e co m p le te c y c le .
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19.0 19.5 20.0
2
3
4
5
6
R e = 200
100
50
2010
fo r th e case o f S = , G r = 0 a t d iffe ren t R ey n o ld s n u m b ers d u rin g cy c le n o . 2 0 .
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0 30 60 90 120 150 180
2
3
4
5
0.125
= 0.0 , 1 .0
0.25
0.375
0.5
0.625
0.75
0.875
fo r th e ca se o f R e = 5 , S = , G r = 2 5 0 d u rin g o n e co m p le te cy c le .
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9.00 9.25 9.50 9.75 10.00
2
3
4
5
6
G r = 4000
fo r th e case o f S = , R e = 2 0 a t d iffe ren t G rash o f n u m b ers d u rin g o n e co m p le te o sc illa tio n .
2000
400
200
100
0
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-10 -8 -6 -4 -2
r T em p e ra tu re d is trib u tio n a lo n g a t = 0 .0 fo r th e ca se o f R e = 2 0 .
G r = 0.0 100 400
0.0
0.2
0.4
0.6
0.8
1.0
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10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
r
T em p era tu re d is trib u tio n a lo n g a t = 0 .0 fo r th e ca se o f R e = 2 0 .
G r = 0.0 100
400
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0.23000.0000 0.2500
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0.2625 0.2725
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0.2750 0.3125
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0.5000 0.5550 0.5650
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0.6000 0.7250 0.7500
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Rajai 51
FUTURE RESEARCH
•Heat transfer from a sphere in a spinning infinite fluid.
•Heat transfer from other geometries such as oblate and prolate spheroids.
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Rajai 52
SPHEROIDS
x c cosh sin cos y c cosh sin sin z c sinh cos
h h c sinh cos2 2
h c cosh sin
c'
A x is o f sy m m e try
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(sinh cos ) (cosh sin
) (cosh sin
)
Re cosh(cosh )
sin(sin )
2 2
2 1 1
t
cosh sin (sinh cos ) cosh (cosh
)
sin (sin
)
2 2 1
10
Stream Function
Vorticity
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Rajai 54
(sinh cos )cosh sin
tanh cot
2 2
2
2
2
2
1
2
t
Pe
1
22 2cosh sin ( )F t as
0 0, and as
0 1, at o
Boundary Conditions
Energy
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A NOTE ON THE PROBLEM OF SPINNING STREAM
ee2
2
2
2
2
2
t
sin Recot
e as2 2 sin
y t P zn nk
n
( , ) ( )1
332211
321
332211
321
hwhwhwqqq
uhuhuh
hhh1
w
wFpw)w(tw 2
w
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Rajai 56