Download Integral Method - Thermal

Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
4.7 Integral Method
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As noted before, the similarity solution provides an
analytical exact solution for laminar boundary layer
conservation equations. However, there are
limitations in terms of geometry and boundary
conditions as well as laminar flow restrictions.
Integral methods are approximate closed form
solutions which have much less limitations in terms
of geometry and boundary condition. It can also
apply to both laminar and turbulent flow situations.
The integral method easily provides accurate
answers (not exact) for a complex problem.
4.7 Integral Method
Chapter 4: External Convective Heat
and Mass Transfer
1
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
In integral methods, one usually integrates
the conservative differential boundary
layer equation over the boundary layer
thickness by assuming a profile for
velocity, temperature and concentration,
as needed.
 The better approximate shape for the
profile, such as velocity and temperature,
is the better prediction for drag force and
heat transfer (friction coefficient or heat
transfer coefficient).

4.7 Integral Method
Chapter 4: External Convective Heat
and Mass Transfer
2
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


Integral methodology has applied to a variety of
configurations to solve transport phenomenon
problems (Schlichting and Gersteu, 2000).
To illustrate the integral methodology, it is
applied for flow and heat transfer over a wedge
with non-uniform temperature and blowing at the
wall.
4.7 Integral Method
Chapter 4: External Convective Heat
and Mass Transfer
3
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Consider two dimensional laminar steady flow with constant
properties over a wedge, as shown in figure 4.16, with an unheated
starting length, x0.
U∞
T∞
_
β
Figure 4.16 Momentum and heat transfer over a wedge with an unheated starting length.
4.7 Integral Method
Chapter 4: External Convective Heat
and Mass Transfer
4
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

The governing boundary layer equation for mass, momentum and energy
for constant property, steady state and laminar flow including boundary
conditions for convective heat transfer over a wedge are presented below:




Continuity equation
Momentum equation
Energy equation
Boundary conditions
∂u ∂v
+
=
0
∂x ∂y
(4.151)
∂u
∂u
∂ 2u
dU
u
+v
= ν 2 +U
∂x
∂y
∂y
dx
(4.152)
∂T
∂T
∂ 2T
u
+v
=
α 2
∂x
∂y
∂y
(4.153)
u ( 0, x ) = 0
v ( 0, x ) = vω
u (δ , x ) = U ( x )
(4.154)
T ( ∞, x ) =
T∞
4.7 Integral Method
=
T ( 0, x ) T∞
for x < X 0
=
T ( 0, x ) Tw
for x > X 0
Chapter 4: External Convective Heat
and Mass Transfer
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Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

It should be noted that U is known from
1 ∂p
dU
U
−
=
potential flow theory ρ ∂x dx . If U is
constant, then
dU
(4.155)
dx

=0
As for the case of flow over a flat plate, If
x0 = 0, U = constant and vw = 0, the
problem will be similar to case presented
by similarity solution before.
4.7 Integral Method
Chapter 4: External Convective Heat
and Mass Transfer
6
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


In integral methods, it is customary to assume a
profile for u and obtain v from continuity, equation
(4.151).
Let us integrate equation (4.151) with respect to y
from y = 0 to y = δ . The velocity and temperature
field outside δ is uniform.
δ ∂u
δ ∂v
0
∫0 ∂x dy + ∫0 ∂y dy =
(4.156)
The second term can be easily integrated
∫
δ
0
4.7 Integral Method
∂v
dy=
v y δ=
=
−v y
∂y
0
vδ − vw
=
(4.157)
Chapter 4: External Convective Heat
and Mass Transfer
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Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


Combining (4.156) and (4.157)
δ ∂u
∫0 ∂x dy= vw − vδ
(4.158)
Applying Leibnitz’s formula to the left hand side of
(4.158) yields
∂ δ
dδ
udy
−
u
x
δ
=
vw − vδ
,
(
)
∫
∂x 0
dx

Or rearrange and using
u ( x, δ ) = U
dδ ∂
vδ =
vw + U
−
dx ∂x
4.7 Integral Method
( ∫ udy )
(4.159)
δ
0
(4.160)
Chapter 4: External Convective Heat
and Mass Transfer
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Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Let us rearrange the momentum equation,
equation (4.152):
 ∂u ∂v 
dU
∂u 2 ∂vu
∂2u
+
− u
+ = U
+ν 2
dx
∂x
∂y
∂y
 ∂x ∂y 

(4.161)
The last term is in parenthesis of the right hand
side of the above equation is zero because of
continuity equation. Upon integration of equation
(4.161) from y = 0 to y = δ.
∫
δ
0
δ ∂vu
∂u 2
dy + ∫
dy =
0
∂x
∂y
4.7 Integral Method
2
δ ∂ u
dU
∫0 U dx dy +ν ∫0 ∂y2 dy
δ
(4.162)
Chapter 4: External Convective Heat
and Mass Transfer
9
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Upon further integration and simplification the above equation
reduces to
∫
δ
0

∂u 2
dy + vu δ −=
vu 0
∂x
∫
δ
0
U
dU
∂u
∂u
dy + ν
−ν
dx
∂y δ
∂y 0
(4.163)
Using equation (4.162) for vδ , no slip boundary condition at wall
u ( 0, x ) = 0 , and assuming no velocity gradient at the outer edge of
boundary layer at y = δ based on physical ground
∫
δ
0
∂u 2
∂
dδ
−U
dy + Uvw + U 2
∂x
∂x
dx
(∫
τw = − µ
∂u
∂y
where
is the shear stress at the wall.
4.7 Integral Method
δ
0
)
τ
ρ
δ
− w +∫ U
udy =
0
dU
dy
dx
(4.164)
(4.165)
0
Chapter 4: External Convective Heat
and Mass Transfer
10
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Apply Leibnitz’s rule and rearrangement will provide the final
form.
∂  δ
+
u
u
U
dy
−
(
)

∂x  ∫0


(∫
δ
0
)
δ
τ + ρUvw
dU
dU
udy
dy =
−∫ U
− w
0
dx
dx
ρ
(4.166)
The only dependent unknown variable in the above equation
ν using continuity. U, τ w and vw
is u since is eliminated
should be known quantities.
Equation (4.166) can be further rearranged
∂  2 δ u
u
U
−
1

∂x  ∫0 U  U
4.7 Integral Method
u
   δ
dy
+
−
1
   ∫0 
    U
  dU τ w + ρUvw
=
 dy  U
ρ
  dx
(4.167)
Chapter 4: External Convective Heat
and Mass Transfer
11
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


It is customary that assume a third order
polynomial equation is for the velocity profile to
obtain a reasonable result.
2
3
u =c1 + c2 y + c3 y + c4 y
(4.168)
c1, c2, c3 and c4 are constant and can be obtained
from boundary conditions for velocity and shear
stress at the wall and outer edge. Once the
constants are obtained, they are substituted in the
momentum integral equation (4.167) and solved
for momentum boundary layer thickness,δ .
4.7 Integral Method
Chapter 4: External Convective Heat
and Mass Transfer
12
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Similarly, the energy equation (4.153) can be rearranged
into the following form to make the integration process
easier.
 ∂u ∂u 
∂uT ∂vT
∂ 2T
(4.169)
+
−T 
+ =
α 2
∂x

∂y
 ∂x
∂y 
∂y
Let’s integrate the above equation from y = 0 to y = δ T ,
knowing the last term in parenthesis on the right hand
side is zero because of continuity equation.
∫
δT
0
4.7 Integral Method
2
δ T ∂vT
δT ∂ T
∂uT
dy + ∫
dy =
dy
α∫
2
0
0 ∂y
∂x
∂y
(4.170)
Chapter 4: External Convective Heat
and Mass Transfer
13
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Upon integration and using continuity equation to obtain
ν δ equation (4.160) and assuming no temperature
gradient at the outer edge of the thermal boundary layer, we
get the following equation
δ ∂
dδ T
∂ δ
−k ∂T
(4.171)
−
+
−
+
=
uT
dy
T
udy
T
v
v
T
T
u
( )
∞
∞ w
∞ δ
w w
∫
0

T
∂x
∂x ∫0
T
T
dx
ρ c p ∂y
y =0
Upon using Leibnitz rule and rearrangement
∂  δT
u ( T − T∞ ) dy  +
∫

∂x  0
4.7 Integral Method
(∫
δT
0
)
qw′′
dT∞
= + vw ( Tw − T∞ )
udy
dx
ρcp
(4.172)
Chapter 4: External Convective Heat
and Mass Transfer
14
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


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
Once again the integral form of energy equation is in
terms of the unknowns temperature assuming the velocity
profile is known.
Similar to momentum integral equation, a temperature
profile should be assumed and substituted in integral
energy equation (4.172) to obtain δ T.
To illustrate the procedure, we use the above
approximation to solve the classical problem of flow and
heat transfer over a flat plate when U = U∞ = constant, no
blowing or suction at the wall and constant wall and flow
stream temperature.
Momentum and energy integral equations (4.167) and
(4.172) will reduces to the following form using the above
assumptions.
4.7 Integral Method
Chapter 4: External Convective Heat
and Mass Transfer
15
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
  τw
(4.173)
 dy  =
ρ
 
qw
∂  δT

(4.174)
−
=
u
T
T
dy
(
)
∞
 ρ c
∂x  ∫0
p
Let’s assume the polynomial velocity profile is third degree with the
following boundary conditions.
u ( 0) = 0
(4.175)
∂  2 δ u
U ∫

0 U
∂x 

 u
1 −
 U
u (δ ) = U ∞
(4.176)
∂u
∂y
(4.177)
=0
y =δ
∂ 2u
∂y 2
4.7 Integral Method
=0
(4.178)
y =δ
Chapter 4: External Convective Heat
and Mass Transfer
16
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


It is assumed that shear stress at boundary layer
edge is zero, which is a good approximation for
this configuration. Equation (4.178) is obtained
by using equation (4.177) and applying the xdirection momentum equation at the boundary
layer edge.
Upon applying equations (4.175) through (4.178)
in equation (4.168), one obtains four equations
and four unknowns (c1, c2, c3 and c4). The final
velocity profile is
3
u 3 y 1 y
=
 −   ,
U 2δ  2δ 
4.7 Integral Method
y≤δ
(4.179)
Chapter 4: External Convective Heat
and Mass Transfer
17
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Shear stress at the wall τ ω is calculated using equation
(4.179)
3µU ∞
∂u
(4.180)
−µ
=
τω =
∂y

y =0
2δ
Substituting equations (4.179) and (4.180) in (4.173),
and performing the integration we get
2

d 39U ∞ δ

dx  280
4.7 Integral Method
 3ν U ∞
=
2δ

(4.181)
Chapter 4: External Convective Heat
and Mass Transfer
18
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


Integrating the above equation and assuming
δ = 0 at x = 0, we get
1/ 2
 280ν x 
δ =
(4.182)

13
U

∞ 
or
δ 4.64
= 1/ 2
(4.183)
x Re
The friction coefficient is found as before
cf
=
2
τ
w
=
U∞2
ρ
2
µ
∂u
∂y
y =0
2
∞
(4.184)
U
ρ
2
or using equation (4.180) and (4.182)
cf
4.7 Integral Method
0.323
= 1/ 2
2 Re
(4.185)
Chapter 4: External Convective Heat
and Mass Transfer
19
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


The prediction of the momentum boundary layer thickness and
friction coefficient cf by integral method are 7% and 3% lower than
the exact solution obtained using the similarity method respectively.
Use the same general third order polynomial equation for a
temperature profile with the following boundary conditions:
T ( 0=
constant
) T=
w
(4.186)
T (δ T=
constant
) T=
∞
(4.187)
∂T
∂y
(4.188)
∂ 2T
∂y 2
4.7 Integral Method
=0
y =δT
=0
y =δT
(4.189)
Chapter 4: External Convective Heat
and Mass Transfer
20
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Upon using equations (4.186) through (4.189), to find
constants for the temperature profile, we get
T − T∞
3 y 1 y 
=
+  
1−
2 δT 2  δT 
Tw − T∞
3
3 k
∂T
qw =
−k
=
(Tw − T∞ )
∂y y =0 2 δ T
4.7 Integral Method
(4.190)
(4.191)
Chapter 4: External Convective Heat
and Mass Transfer
21
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Substitution of equation (4.188), (4.189) and (4.190) into
equation (4.173) and approximate integration for δT < 1
δ
yields:
1 u  T −T
∂ δT
∂ 
∞
−
=
−
δ
u
T
T
dy
U
T
T
(
)
(
)


w
T
∞
∞
∞
∫0 U ∞  Tw − T∞
∂x ∫0
∂x 
  y
d 
  δT

δT 2 
∂  3 δT 2 
= 
U T − T∞ ) 
1 −
2  ∞( w
∂x  20 δ  14δ 

=
4.7 Integral Method



(4.192)
qw
3α
=
(Tw − T∞ )
ρ c p 2 δT
Chapter 4: External Convective Heat
and Mass Transfer
22
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
Upon further simplification
δ T2   10v 1
d  δ T2 
=
 1 −
2 
dx  δ  14δ   Pr U ∞ δ T
(4.193)
The only unknown in the above equation is δ T since δ is known.
δ
Assume ς = T .
δ
 ς 2   10 v 1
d 
2
ς
 δ 1 −
dx 

 =
14   Pr U ∞ ςδ
(4.194)
The solution of the above equation for ζ=0 at x=x0 yields
Pr   x0 
1 −  
=
ς
1.026   x 

−1
4.7 Integral Method
3
3
4



1
3
(4.195)
Chapter 4: External Convective Heat
and Mass Transfer
23
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
The local heat transfer coefficient can now be calculated
∂T
since δT is known.
−k
∂y y =0 3 k (Tw − T∞ )
qw
(4.196)
h =
=
=
2
δT
Tw − T∞
Tw − T∞
or
3 k
(4.197)
h=
2 ςδ T
Using ζ for equation (4.195) and
δ from equation (4.69)
hx
the local Nusselt number Nu = k is
x
1
Nu x =
1
0.332 Pr 2 Re x 2
 x 
1 −  0 
  x 
3
4



1
3
(4.198)
The above equation in this case with no unheated
starting length (ζ=0) reduces to the exact solution
obtained by the similarity solution.
1
1
2
(4.199)
Nu x = 0.332 Pr Re x 2
4.7 Integral Method
Chapter 4: External Convective Heat
and Mass Transfer
24