convection 3
TRANSCRIPT
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Convection 3
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Energy equation
First law of thermodynamics
dWdQdE dE increment in the (kinetic plus thermal energy) of the system
dQ heat transfer to the system
dW work done on the system
Internal energy per unit mass of the fluid consists of the sum of thekinetic energy ( ) and thermal internal energy e = cv T
Writing the above equation in the substantial derivative form, so that
it applies to transport ofEby a moving system
2
22vu
Dt
DW
Dt
DQ
Dt
DE x1
y1
z1
dx
dy
dz
x
y
z
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Rate of increaseof energy E offluid element
Rate of heat transferto fluid element Rate of work doneon the fluid elementby surface and bodyforces
Rate of increaseof E in CV
Rate at which Eenters throughsurface of CV
Rate at which Eleaves throughsurface of CV
Rate of heat transferinto CV by conduction Rate of surface and body forces dowork on CV
Rate of increaseof E in CV
dxdydzt
E
Rate at which Eenters throughsurface of CV
Rate at which Eleaves throughsurface of CV
dxdydzy
Ev
x
uE
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dxdydzx
uEuEdydz
xy
z
ji
KdydzEu
dxdzEv
dydxdz
y
EvvEdxdz
Rate of increaseof E in CV
Rate at which Eenters throughsurface of CV
Rate at which Eleaves throughsurface of CV
dxdydzy
Ev
x
uE
t
E
CV showing rate of transport of
energy through the facesnormal to the x and y axes
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Rate of increaseof E in CV
Rate at which Eenters throughsurface of CV
Rate at which Eleaves throughsurface of CV
dxdydzy
Ev
x
uE
t
E
dxdydzy
Ev
y
vE
x
Eu
x
uE
tE
t
E
dxdydzyEv
xEu
tEdxdydz
yvE
xuE
tE
dxdydzy
Ev
x
Eu
t
Edxdydz
y
v
x
u
tE
dxdydzvu
e
tD
D
2
22
dxdydztD
DEdxdydz
y
Ev
x
Eu
t
E
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Rate of heat transferinto CV by conduction
x
y
z
ji
Kdydzqxdxdydz
x
qdydzq xx
dxdzqy
dxdydzy
qdxdzq
yy
dxdydzy
q
x
q yx
dxdydzy
T
kyx
T
kx
Negative sign arises because heat
transfer is counted as positive in the
positive coordinate direction
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x
y
z
ji
KudydzP dxdydzxuPdydzPu
vPdxdz
dxdydzy
vPdxdzPv
dxdydz
yvP
xuP
dxdydzy
vPdxdzPvPvdxdzdxdydz
x
uPdydzPudydzPu
Outward normal stresses are positive. Positive normal
stresses are tensile stresses; that is, they tend to stretchthe material. Compressive normal stress wil give positive
valuefor p
RATE OF WORK DONE BY PRESSURE FORCES
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xxxyxz
yy
yzyx
xyxzxx
First subscript denotes the direction of the normal to the
plane on which the stress actsSecond subscript denotes the direction of the stress
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x
y
z
ji
Kudydzxx
dxdydz
y
vdxdzv
yyyy
dxdydzy
v
x
u
dxdydzy
vdxdzvvdxdzdxdydz
x
udydzudydzu
yyxx
yyyyyy
xxxxxx
vdxdzyy
dxdyd
x
udydzu xxxx
RATE OF WORK DONE BY NORMAL STRESSES
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RATE OF WORK DONE BY SHEAR STRESSES
x
y
z
ji
Kvdydzyx
dxdydzy
u
x
v
dxdydzy
udxdzuudxdzdxdydz
x
vdydzvdydzv
xyyx
xyxyxy
xxxxyx
dxdzuxy
dydzdxx
vvdx
x
yxyx
dxdzdyy
uudy
y
xyxy
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Rate of increaseof E in CV
Rate at which Eenters throughsurface of CV
Rate at which Eleaves throughsurface of CV
Rate of heat transferinto CV by conduction Rate of surface and body forces dowork on CV
2
22vu
e
tD
D
y
Tk
yx
Tk
x
y
vP
x
uP
y
v
x
u yyxx yx
xyyxvfuf
y
u
x
v
xzxyxxx fzyxx
P
Dt
Du
X- momentum equation
xyxxx uf
y
u
x
u
x
Pu
Dt
Duu
A
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xyxxx ufyx
u
x
Pu
Dt
uD
Dt
Duu
2
2
xyxxx vfyx
v
x
Pv
Dt
vD
Dt
Dvv
2
2
yxyyyxyxxx vfufyx
v
yx
u
y
Pv
x
Pu
Dt
vuD
122
22
BA -B
y
v
x
v
y
u
x
u
y
v
x
uP
y
T
x
Tk
Dt
Deyyyxxyxx
1
2
2
2
2
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y
v
x
v
y
u
x
u
y
v
x
uP
y
T
x
Tk
Dt
Deyyyxxyxx
1
2
2
2
2
y
v
x
u
x
uxx
3
22
y
v
x
u
y
vyy
3
22
x
v
y
uyxxy
2222
3
222 xvyuyvxuyvxu
D
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P
eh Dt
PD
Dt
De
Dt
Dh
y
v
x
uP
Dt
DP1
y
v
x
uP
Dt
DP1
Dt
DP
Dt
DP1
Dt
P
D
22
y
v
x
v
y
u
x
u
y
v
x
uP
y
T
x
Tk
Dt
De
yyyxxyxx
1
2
2
2
2
y
v
x
uP
Dt
DP1
y
v
x
uP
y
T
x
Tk
Dt
Dh2
2
2
2
Dt
DP1
y
T
x
Tk
Dt
Dh2
2
2
2
0V.tD
D
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Dt
DP1
y
T
x
Tk
Dt
Dh2
2
2
2
Assuming a fluid of constant specific heat TCh P
Dt
DP
y
T
x
Tk
Dt
DTCP 2
2
2
2
ndissipatioviscousworkpressure
conductionconvection
P Dt
DP
y
T
x
T
ky
T
vx
T
ut
T
C
2
2
2
2
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Dt
DP
y
T
x
Tk
Dt
DTCP 2
2
2
2
2
* * * * * * s
s
T T x y u v P x , y , u v , P , and T
L L V V T T V
PRINCIPLE OF SIMILARITY
y
Pv
x
Pu
y
T
x
Tk
y
Tv
x
TuCP
2
2
2
2
For steady flows
2
23
2
2
2
2
2 L
V
y
Pv
x
Pu
L
V
y
T
x
T
L
TTk
y
Tv
x
Tu
L
TTVC
*
*
**
*
**
*
*
*
*s
*
**s
P
2222
2
22222
3
222
3
222
*
*
*
*
*
*
*
*
*
*
*
**
x
v
y
u
y
v
x
u
y
v
x
u
L
V
x
v
y
u
y
v
x
u
y
v
x
u
223
2
2
2
2
2L
V
TTVC
L
y
Pv
x
Pu
TT
L
VCL
V
y
T
x
T
TT
L
VCL
TTk
y
Tv
x
Tu
sP
*
*
**
*
**
sP*
*
*
*
sP
s*
**
sP*
*
**
*
**
sP*
*
*
*
P
*
**
TTLC
V
y
Pv
x
Pu
TTC
V
y
T
x
T
VCL
k
y
Tv
x
Tu
2
2
2
2
2
***** 222
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Re
Ec
y
Pv
x
PuEc
y
T
x
T
PrRey
Tv
x
Tu
*
*
**
*
**
*
*
*
*
*
**
2
2
2
21
PrReC
k
VLVLC
k
PP
1
sP*
*
**
*
**
sP*
*
*
*
P*
**
TTLC
V
y
Pv
x
Pu
TTC
V
y
T
x
T
VCL
k
y
Tv
x
Tu
2
2
2
2
2
sP TTCV
Ec
2
Eckert number is measure of the dissipation effects in the flow. Since this grows in
proportion to the square of the velocity, it can be neglected for small velocities.
In an air flow, V = 10 m/s, Cp = 1050 J/kg.K and a reference temperature difference of
10K,Ec 0.01.
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NONDIMENSIONALIZED CONVECTION AND SIMILARITYWhen viscous dissipation is negligible, the continuity, momentum,
and energy equations for steady incompressible, laminar flow of a
fluid with constant properties
0u v
x y
2
2
u u u Pu v
x y xy
2 2
2 2p
T T T T C u v k
x y x y
At 0 0 0
At 0 0 0 0 0 0
As
s
x u , y u , T , y T
y u x, , v x, ,T x, T
y u x, u , T x, T
With the boundary conditions
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2
* * * * * * s
s
T T x y u v P x , y , u v , P , and T
L L V V T T V
0
* *
* *
u v
x y
2
2
1* * * *
* *
* * * *L
u u u dPu v
Re x y y dx
2
2
1* * *
* *
* * *L
T T Tu v
Re Pr x y y
0 1 0 0 1 0 0
0 1 0 0 1
* * * * * * * *
* * * * * *
u , y , u x , , u x , , v x , ,
T , y , T x , , T x ,
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212f1f NuNuCC 2121 PrPrReRe
The number of parameters is reduced greatly by non-
dimensionalizing the convection equations
L1
Re1
L2
Re2
V1Water
V2Air
L,V ,T , ,
Re,Pr
Parameters before nondimensionalizing
Parameters after nondimensionalizing
F i t th l ti f * b d
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For a given geometry, the solution for u* can be expressed as
1* * * Lu f x , y ,Re
20 0
*
*
*s L*
y y
u V u V f x ,Re y L Ly
2 2 32 22
2 2
* * *s f ,x L l L
L
V LC f x ,Re f x ,Re f x ,Re
ReV V
Friction coefficient for a given geometry can be expressed in terms
of the Reynolds number Re and the dimensionless space variable x*alone (instead of being expressed in terms ofx, L, V, and ).This is a very significant finding, and shows the value of
nondimensionalized equations.
L
*
x,f Re,xC
*
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Dimensionless temperature T* for a given geometry
Note that the Nusselt number is equivalent to the dimensionlesstemperature gradient at the surface, and thus it is properly
referred to as the dimensionless heat transfer coefficient
* * * LT g x , y ,Re ,Pr
20
*
*
*x L*
y
hL T Nu g x ,Re ,Pr k y
0
0 0* *
* *y s
* *s s y y
k T yk T T T k Th
T T L T T Ly y
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Nusselt number is equivalent
to the dimensionless
temperature gradient at thesurface
m nL Nu C Re Pr
*y
0*
*
*
y
TNu
y
*
x
*T
Laminar
*
x L
x L
Local Nusselt number :
Nu f ( x , Re , Pr)
Average Nusselt number :
Nu f (Re , Pr)
A common form of Nusselt
number:
ANALOGIES BETWEEN MOMENTUM AND HEAT TRANSFER
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ANALOGIES BETWEEN MOMENTUM AND HEAT TRANSFER
Reynolds analogyand Chilton-Colbourn analogy
2
2
1* * *
* *
* * *L
u u u
u v Re x y y
2
2
1* * *
* *
* * *L
T T Tu v
Re x y y
1Pr
20 0
*
**
s L*y y
u V u V f x ,Re
y L Ly
2 2 32 22
2 2
* * *s f ,x L l L
L
V LC f x ,Re f x ,Re f x ,Re
ReV V
20
*
**
x L*
y
hL T Nu g x ,Re ,Pr
k y
2L
f ,x x
ReC Nu
REYNOLDS ANALOGY
h Nu
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p L
h NuSt
C V Re Pr
2
L f ,x x
Re
C Nu2
f ,xxC St
* *
* *
* *
* *
y y
L f , x x
Pr ofile : u T
u TGradients :
y y
Re Ana log y : C Nu
0 0
2
Laminar flow over a flat plate
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1 20 664 f ,x xC . Re
1 3 1 20 332 x Nu . Pr Re
2
f ,x
x
C
St p L
h Nu
St C V Re Pr
Laminar flow over a flat plate
1 3
2
L f ,x x
ReC Nu Pr
2 3
2
f ,x xH
p
C hPr j
C V
For . Here is called the Colburn j-factor. Although this
relation is developed using relations for laminar flow over a flat plate(for which = 0 ), experimental studies show that it is also
applicable approximately for turbulent flow over a surface, even in
the presence of pressure gradients. For laminar flow, however, the
analogy is not applicable unless = 0.
0 6 60. Pr Hj
**P x
**P x
Th f i d l l i fl i i