boundary layes
DESCRIPTION
boundary layersTRANSCRIPT
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
1
Laminar Boundary Layers
To calculate the convective heat transfer coefficient, we have to analyze the boundary layer close to a rigid boundary:
Conservation Equations
Specific boundary conditions are necessary to simplify these equations.
Within boundary layer: low velocities normal to wall
Outside of boundary layer: Potential flow with u∞ und T∞, no effect of viscosity
Chap. 12.2: Laminar Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
2
2D Boundary Layer: Forced convection over plate
Flow in x-direction,
y-direction ┴ to surface,
z-direction is ∞
Boundary layers: Velocity: δ, Temperature δT
( ) ( ) ( )WWT TTTyTuyu −⋅=−=⋅== ∞∞ 99.099.0 δδ
Chap. 12.2: Laminar Boundary Layers
Potential flow
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
3
Goal of this analysis:
Friction force on the surface (Newtonian fluid)
Heat transfer on the surface
∫ ⋅⋅=L
w dxWF0
τ
dxyuWF
yu
y
L
yw ⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂⋅=→⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂⋅=
==∫
000
µµτ
∫ ⋅⋅ʹ′ʹ′=L
w dxWqq0
dxyTWkq
yTkq
y
L
y
⋅⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂⋅−=⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂⋅−=ʹ′ʹ′
==∫
000
Chap. 12.2: Laminar Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
4
Laminar Boundary Layers: Solving the governing equations with three different methods: - Order of Magnitude Method (Approximation) - Approximate Integral Method - Exact Similarity Method
Chap. 12.2: Laminar Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
5
Governing equations:
pca
yT
xTa
yTv
xTu
yv
xv
yP
yvv
xvu
yu
xu
xP
yuv
xuu
yv
xu
⋅=⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂=
∂
∂+
∂
∂
=
⎪⎪
⎩
⎪⎪
⎨
⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅+
∂
∂−=
∂
∂+
∂
∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅+
∂
∂−=
∂
∂+
∂
∂
=∂
∂+
∂
∂
ρλ
ρµ
ρ
ρ
2
2
2
2
2
2
2
2
2
2
2
2
1
1
0
E
I
K
νν
ν
Continuity
Momentum
Energy α= αk
Chap. 12.2: Laminar Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
6
∞∞∞ =∞=∞=∞∞→
====
PPTTuuy
TTvuy W
)(,)(,)(
)0(,0)0(,0)0(0
Boundary conditions for laminar boundary layers:
In the following, an estimation of phenomena in the boundary layer by analyzing orders of magnitude
Order of Magnitude Solution by Prandtl (1904)
Chap. 12.2: Laminar Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
7
Velocity Boundary Layer
vvv
UuUu
yy
LxLx
Δ→
≈Δ→
≈Δ→
≈Δ→
∞∞
?.........0:
........0:
........0:
........0:
δδ
Variables and orders of magnitudes:
LUvvv
LU
yv
xu δ
δ ∞∞ ≈=Δ→≈
Δ+→=
∂
∂+
∂
∂ 00
Approximation of order of magnitude: v
Signs are irrelevant !
Chap. 12.2.1: Velocity Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
8
( ) ( ) yyxx PdPPdP Δ≈Δ≈
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
+⋅+⋅≈⋅⋅+⋅ ∞
≈
∞∞∞
∞∞ 2
0
2
1δρδ
δ ULU
LPU
LU
LUU x ν
Δ
Rearranging of pressure terms (still unknown)
x - momentum equation:
L>>δ
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅+
∂
∂−=
∂
∂+
∂
∂2
2
2
21yu
xu
xP
yuv
xuu ν
ρ
Chap. 12.2.1: Velocity Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
9
FrictionPressure
y
Inertia
y
LUP
LU
LUP
LLU
⋅⋅+
Δ⋅≈
⋅→⋅⋅+
Δ⋅≈⋅ ∞∞∞∞
δδρδδ
δδρδ νν 11
2
2
2
2
Similarly for y – momentum equation:
Significance of terms:
ibungDruck
x
Trägheit
ULP
LU
Re
2
2
1
12δρ∞∞
≈
⋅+⋅≈⋅ νΔ
Inertia Pressure Friction
yxyx PPP
LLP
PdyyPdx
xPdP Δ+Δ=⋅
Δ+⋅
Δ=Δ→⋅
∂
∂+⋅
∂
∂= δ
δ
Pressure terms:
Chap. 12.2.1: Velocity Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
10
Pressure drop is caused by friction forces →
Pressure and friction terms have the same oder of magnitude
222
1δ
µδ
νρδ
νρ
∞∞∞ ⋅⋅=⋅⋅⋅≈Δ→⋅≈Δ⋅
ULULPULP
xx
LU
LUP
LUP
yy ∞∞∞ ⋅=⋅⋅≈→
⋅⋅≈⋅ µρδδρ
νν ΔΔ1
Similarly for y – component:
LUL
LUP ∞∞ ⋅+⎟
⎠
⎞⎜⎝
⎛⋅≈Δ µδ
µ2
Total pressure difference:
L>>δ xy PP ΔΔ <<
Chap. 12.2.1: Velocity Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
11
dxxPdPandxPP ⋅∂
∂== !)(
Conclusion:
)()( xPxP ∞=
2
21yu
dxdP
yuv
xuu
∂
∂⋅+⋅−=
∂
∂⋅+
∂
∂⋅ ∞ ν
ρ
Therefore:
x – momentum equation simplified:
very often: P1(∞) = P2(∞)
since: P1(0) = P1(∞) and P2(0) = P2(∞)
→ P1(0) = P2(0)
Chap. 12.2.1: Velocity Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
12
equationmomentumxyu
yuv
xuu
Continuityyv
xu
−∂
∂⋅=
∂
∂⋅+
∂
∂⋅
=∂
∂+
∂
∂
2
2
0
ν
Summary: Results of “Order of magnitude analysis “ so far:
• Pressure term in x – momentum equation = 0
• y – momentum equation is negligible
Relevant equations for velocity boudary layer:
∞=∞→===
Uuyvuy 00
with boundary conditions:
Chap. 12.2.1: Velocity Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
13
Temperature Boundary Layer
Thickness of velocity and temperature boundary layers are not identical !
δδ >>T
(1) Analysis with assumption:
Chap. 12.2.2: Temperature Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
14
∞
∞∞
∞∞
−≈Δ→
⋅≈Δ→⋅
≈Δ→
≈Δ→
≈Δ→
∞TTTTTTL
UvL
Uv
UuUu
yy
LxLx
WW
TT
TT
.....:
........0:
........0:
........0:
........0:
δδ
δδ
Order of magnitudes for energy equation:
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂=
∂
∂+
∂
∂2
2
2
2
yT
xT
yTv
xTu α
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−+
−⋅≈
−⎟⎠
⎞⎜⎝
⎛ ⋅+−
⋅
−
∞
−
∞∞∞
∞∞
directionyinConduction
T
W
directionxinConduction
W
T
WTW TTLTTTT
LU
LTTU 22 δ
αδ
δ
Chap. 12.2.2: Temperature Boundary Layers
L >> δT
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
15
2
2
yT
yTv
xTu
∂
∂⋅=
∂
∂⋅+
∂
∂⋅ α
∞=∞→
==
TTyTTy W
::0
Simplified energy equation with boundary conditions:
δδ >>T ∞=Uu T )(δ
δδ <<T ∞< Uu T )(δ
with: meaning:
(2) Analysis with assumption: Temperature boundary layer is smaller than velocity boundary layer
Orders of magnitude are different !
Chap. 12.2.2: Temperature Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
16
For order of magnitude, we can assume a linear velocity profile
T
uUδδΔ
≈∞
δδTUu ⋅≈Δ ∞
⎟⎠
⎞⎜⎝
⎛⋅⋅≈Δ→Δ
≈Δ
→=∂
∂+
∂
∂∞ L
UvvLu
yv
xu TT
T
δδδ
δ0
Velocity component v from continuity:
Chap. 12.2.2: Temperature Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
17
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−+
−⋅≈
−⋅⎟⎠
⎞⎜⎝
⎛⋅⎟⎠
⎞⎜⎝
⎛ ⋅+−
⋅⎟⎠
⎞⎜⎝
⎛⋅
−
∞
−
∞∞∞
∞∞
directionyinConduction
T
W
directionxinConduction
W
T
WTTWT TTLTTTT
LU
LTTU 22 δ
αδδ
δδδδ
Energy equation for δT < δ (with orders of magnitude)
additional term
An additional term appears in both inertial terms.
Summary of all findings:
- no effect of pressure - no y-momentum equation - no thermal diffusion in x-direction
Chap. 12.2.2: Temperature Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
18
EnergyyT
yTv
xTu
Momentumxyu
yuv
xuu
Continuityyv
xu
2
2
2
2
0
∂
∂⋅=
∂
∂⋅+
∂
∂⋅
−∂
∂⋅=
∂
∂⋅+
∂
∂⋅
=∂
∂+
∂
∂
α
ν
Complete simplified equation system for boundary layer:
With boundary conditions:
yallforTTUuxTTUuyTTvuy W
∞∞
∞∞
===
=∞=∞∞→
====
0)()(:)0(0)0()0(:0
Chap. 12.2.2: Temperature Boundary Layers
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
19
2
2
2
2
δδδ ∞∞
∞∞ ⋅≈⋅⎟
⎠
⎞⎜⎝
⎛ ⋅+→∂
∂⋅=
∂
∂⋅+
∂
∂⋅
UUL
ULU
yu
yuv
xuu νν
22
2 12δδ⋅≈→⋅≈⋅ ∞∞∞ νν
LUU
LU
Surface friction from order of magnitude solution
Momentum equation:
Factor 2 negligible:
( ) 21
Re21
−
≈→⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅≈
∞LLU
L δδ
νµ
ρ LULUL
⋅⋅=
⋅= ∞∞
νRe
Solving for δ:
with
Chap. 12.2.3: Surface Friction
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
20
( )
( ) 2122
1
21
0
Re
Re
−∞
∞
∞∞∞
∞∞
=
⋅⋅≈⋅
⋅⋅⎟
⎠
⎞⎜⎝
⎛ ⋅⋅⋅≈
⋅⋅≈⋅≈→⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂⋅=
L
LWy
W
UUULU
LU
LUU
yu
ρρρ
νµ
µδ
µτµτ
Wall shear stress from velocity gradient:
( ) 212 Re −
∞ ⋅⋅≈ LW Uρτ
( ) 212 Re2
2−
∞ ⋅=⋅⋅≈ LffW cwithUc ρτ
Friction coefficient:
Chap. 12.2.3: Surface Friction
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
21
( ) LWUFdxWF L
L
W ⋅⋅⋅⋅→∫ ⋅⋅= −∞ 2
12
0Re~ ρτ
Friction force = Wall shear stress integrated over the entire area
Chap. 12.2.3: Surface Friction
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
22
Heat Transfer from order of magnitude solution (δ << δT)
( ) ( )22
2
T
W
T
WTW TTTTL
ULTTU
yT
yTv
xTu
δα
δδ
α ∞∞∞
∞∞
−⋅≈
−⋅⋅+
−→
∂
∂⋅=
∂
∂⋅+
∂
∂⋅
21
212
1
2 Re ⎟⎠
⎞⎜⎝
⎛⋅≈⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅≈→≈
−
∞
∞
νααδ
δα
LULLU T
T
Energy equation:
Simplified and solved for δT/L:
Prandtl Number Pr αν
=Pr Peclet Number Pe PrRePe ⋅=
Chap. 12.2.4: Convective Heat Transfer
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
23
21
21
21
PeRePr −−−≈⋅≈ L
T
Lδ
Temperature boundary layer thickness for δ << δT
1PrRe 21
21
>>≈→≈−−
δδδ T
LL
Material property
Flow property
Precondition for δ << δT
1Pr <<True for low viscosity and/or high thermal conductivity, e.g.liquid metals, Hg: Pr < 0.03 at room temperature
Chap. 12.2.4: Convective Heat Transfer
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
24
Summary of dimensionless parameters:
Reynolds Number
Prandtl Number
Peclet Number
ForcesViscousForcesInertialLU
=⋅
= ∞
νRe
yDiffusivitThermalyDiffusivitMomentum
==ανPr
TransferHeatTransferMassLU
=⋅
=⋅= ∞
αPrRePe
Chap. 12.2.4: Convective Heat Transfer
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
25
Calculation of convective heat transfer coefficient from Fourier‘s Law:
Using the result for the temperature boundary layer:
21
21RePr LL
kh ⋅⋅≈
21
21RePrNu LL k
Lh⋅≈
⋅≡
Nusselt Number Nu as basis for calculating h:
TW
T
W
W
y kTT
TTkh
TTyTk
Tqh
δδ
≈−
−⋅
≈→−
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
⋅−
=Δ
ʹ′ʹ′=
∞
∞
∞
=0
Chap. 12.2.4: Convective Heat Transfer
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
26
( ) ( ) 21
21RePr LWW kTTWTThWLq ⋅⋅⋅−⋅≈−⋅⋅⋅= ∞∞
Heat flux through a rectangular area (W = width, L = length)
Note: Nusselt Number NuL leads to h averaged over 0 …. L: !
Second case: δ >> δT:
( ) ( )22
2
T
W
T
WTTWT TTTTL
ULTTU
yT
yTv
xTu
δα
δδ
δδ
δδ
α ∞∞∞
∞∞
−⋅≈
−⋅⋅⋅+
−⋅⋅→
∂
∂⋅=
∂
∂⋅+
∂
∂⋅
Energy equation (with ‘additional terms‘):
h
Chap. 12.2.4: Convective Heat Transfer
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
27
21
T
T
LU
δα
δδ
⋅≈⋅
⋅∞ ( ) 21
Re −⋅≈ LLδ
232
1
3
3
ReRe −
∞
−
⋅≈⋅
⋅≈ L
LT
ULL νααδ
21
31RePr
−−⋅≈ L
T
Lδ
1Pr 31
<<≈−
δδT
Energy equation simplified:
where
Leading to:
and results in:
Range of validity: 1Pr >
Chap. 12.2.4: Convective Heat Transfer
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
28
( ) 21
31
21
31
21
31
213
1
RePr
RePrNu
RePr
RePr
LW
L
L
LT
kTTWq
kLh
Lkh
L
⋅⋅⋅−⋅≈
⋅≈⋅
≡
⋅⋅≈
⋅≈
∞
−−δ
Summary for case 2: δ > δT
Derivation analogously to case 1:
Difference between both cases:
δ < δT δ > δT 21
Pr∝ 31
Pr∝
Chap. 12.2.4: Convective Heat Transfer
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
29
Thickness of velocity boundary layer
21
Re−≈ LLδ
Wall shear stress: 212 Re
−∞ ⋅⋅≈ LW Uρτ
Friction coefficient: 21
Re−
≈ Lfc
Friction force: 212 Re
−∞ ⋅⋅⋅⋅≈ LLWUF ρ
Summary for velocity boundary layer
(independent of δT)
Chap. 12.2.4: Convective Heat Transfer
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
30
Chap. 12.2: Laminar Boundary Layers
Laminar Boundary Layers: Solving the governing equations with three different methods: - Order of Magnitude Method (Approximation) - Approximate Integral Method - Exact Similarity Method
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
31
Approximate Integral Method
Principle Idea: - Consider boundary layer equations in integral form - Integration over height H > δ and δT - Approximation of profiles within the boundary layer
Results:
- determine surface gradients more accurately,
- determine h and τ more accurately
Chap. 12.2.5: Approximate Integral Method
Velocity
Temperature
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
32
2
22
2
2
)()(yuvu
yu
xyu
yuv
xuu
∂
∂⋅=⋅
∂
∂+
∂
∂→
∂
∂⋅=
∂
∂⋅+
∂
∂⋅ νν
continuitytodue
yv
xuu
yuv
xuu
yvu
yuv
xuuvu
yu
x0
2 2)()(
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅+
∂
∂⋅+
∂
∂⋅=
∂
∂⋅+
∂
∂⋅+
∂
∂⋅=⋅
∂
∂+
∂
∂
2
2
2
2
)()(yTTv
yTu
xyT
yTv
xTu
∂
∂⋅=⋅
∂
∂+⋅
∂
∂→
∂
∂⋅=
∂
∂⋅+
∂
∂⋅ αα
Derivation of Approximate Integral Method
Momentum equation:
Equations are identical for incompressible fluids:
Similarly for energy equation:
Chap. 12.2.5: Approximate Integral Method
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
33
( )Hy
y
Hy
y
H
yuvudy
xdd
=
=
=
= ∂
∂⋅=⋅+⋅∫
00
0
2u ν ( )Hy
y
Hy
y
H
yTTvdyTu
xdd
=
=
=
= ∂
∂⋅=⋅+⋅⋅∫
00
0
α
( )
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂−⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂⋅=⋅−⋅+⋅
=
=
==
==∞∫0
00
00
2
yHyyHy
H
yu
yuvuvUdyu
xdd
ν
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂−⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂⋅=⋅−⋅+⋅⋅
=
=
==
=∞=∫0
00
00 yHy
WyHy
H
yT
yTTvTvdyTu
xdd
α
Integration over the boundary layer (H >> δ, δT)
Momentum equation Energy equation
Using known values at the boundary:
Momentum equation
Energy equation
Chap. 12.2.5: Approximate Integral Method
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
34
00
2
==∞ ⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂⋅−=⋅+⋅∫
yHy
H
yuvUdyu
dxd ν
00 ==∞ ⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂⋅−=⋅+⋅⋅∫
yHy
H
yTvTdyTu
dxd
α
xu
yv
yv
xu
∂
∂−=
∂
∂→=
∂
∂+
∂
∂ 0
0
0000
=
==−=⋅=⋅−=⋅− ∫∫∫ yHy
HHH
vvdydydvdy
dxdudyu
dxd
Leading to simplified equations:
next step: determining Hyv =
using continuity:
and integrating over boundary layer:
Momentum equation Energy equation
Chap. 12.2.5: Approximate Integral Method
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
35
0000
2 )(=
∞∞ ⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂⋅−=⋅−⋅=⋅⋅−⋅ ∫∫∫
y
HHH
yudyUuu
dxddyu
dxdUdyu
dxd ν
0000
)(=
∞∞ ⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂⋅−=⋅−⋅=⋅⋅−⋅⋅ ∫∫∫
y
HHH
yTdyTTu
dxddyu
dxdTdyTu
dxd
α
Combining integrals:
E
M
∞==
=
∞∞∞ ∫∫∫ ⋅−⋅+⋅−⋅=⋅−⋅
UHuu
HH
dyUuudyUuudyUuu
)()(since,0
00
)()()(
δ
δ
δ
∞==
=
∞∞∞ ∫∫∫ ⋅−⋅+⋅−⋅=⋅−⋅
THTT
HH
T
T
T
dyTTudyTTudyTTu
)()(since,0
00
)()()(
δ
δ
δ
Dividing into two parts: inside + outside of boundary layer
E
M
Chap. 12.2.5: Approximate Integral Method
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
36
Boundary Layer Equations in Integral Form
Variables of interest
Integral equations are exact, valid as the differential equations
Still unknown: u(y), T(y)
00
00
)(
)(
=
∞
=
∞
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂⋅−=⋅−⋅
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂⋅−=⋅−⋅
∫
∫
y
y
yTadyTTu
dxd
yudyUuu
dxd
Tδ
δ
ν
α
Chap. 12.2.5: Approximate Integral Method
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
37
δδ
<+⎟⎠
⎞⎜⎝
⎛⋅=
∞
yAyAUyu
21)(
TTw
w yByBTTTyT
δδ
<+⎟⎟⎠
⎞⎜⎜⎝
⎛⋅=
−
−
∞21
)(
∞
∞
==
==
===
TTyUuy
TTuy
TT
W
)(:)(:
)0(0)0(:0
δδ
δδ
1010
12
12
==
==
BBAA
Assumption of linear profiles inside the boundary layer
Boundary conditions to determine constants Ai, Bi
Leading to:
T
Chap. 12.2.5: Approximate Integral Method
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
38
δδδ
δ∞
∞∞∞ ⋅−=⋅⎟⎠
⎞⎜⎝
⎛−⋅⋅∫
UdyUyUyUdxd
ν0
( )δ
ζζζδ∞
−=
−=
⎪⎪
⎭
⎪⎪
⎬
⎫
⎪⎪
⎩
⎪⎪
⎨
⎧
⋅−⋅∫ Ud
dxd ν
61
1
0
1
( ) xU
xdxU
dUdx
d⋅
⋅=→⋅
⋅=⋅→
⋅=⋅
∞∞∞
ννν 62
66 2δδδ
δδ
Momentum equation: Using assumed linear profile
Dividing by U∞2, substituting with new variable:
δζ
y=
Integration over x:
Chap. 12.2.5: Approximate Integral Method
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
39
Result: δ = f( ) x
21
Re464.3Re12:Re12 −∞
∞
⋅==⋅
=⋅⋅
= xx
x xUxwith
Ux δ
νδ
ν
( ) 212
0 Re12 xyw
UUdydu
⋅
⋅=⋅=⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅= ∞∞
=
ρδ
µµτ
21
2Re577.0
21
−
∞
⋅=⋅⋅
= xw
fU
cρ
τ
Wall shear stress:
Wall shear stress coefficient:
Chap. 12.2.5: Approximate Integral Method
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
40
Energy equation: Using assumed linear profiles
[ ] [ ]T
ww
Tw
TTdyTTyTTyUdxd T
δα
δδ
δ−
⋅−=⋅⎟⎟⎠
⎞⎜⎜⎝
⎛−−⋅−⋅⋅ ∞
∞∞∞∫0
( )T
TTTT
Ud
dxd
δα
ζζζδδ
⋅−=
⎪⎪
⎭
⎪⎪
⎬
⎫
⎪⎪
⎩
⎪⎪
⎨
⎧
⋅−⋅∞
−=
∫
61
1
0
2
1
Dividing by U∞.(T∞ -TW), using new variable:
TT
yδ
ζ =
Chap. 12.2.5: Approximate Integral Method
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
41
31
Pr1
23
3
Pr112 −
∞
=⎟⎠
⎞⎜⎝
⎛→=⋅
⋅⋅=
δδ
δδδ TT a
Uxa
ν
Integration over x: Assumption: δT/δ = constant
( ) ( )
∞
∞
⋅⋅=⋅→
⋅=⋅⋅⎟
⎠
⎞⎜⎝
⎛=⋅⋅⎟⎠
⎞⎜⎝
⎛=⎟⎟⎠
⎞⎜⎜⎝
⎛
Ux
Udxd
dxd
dxd
T
T
T
TTT
TTT
αδ
δδ
αδ
δδ
δδδδ
δδ
δ
12
621
2
22
Using earlier result for δ:
31
213
1PrRe464.3Pr −−−⋅⋅=⋅= x
T
xxδδ
Result for δT :
Chap. 12.2.5: Approximate Integral Method
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
42
T
W
yW
TTkdydTkq
δ∞
=
−⋅−=⋅−=ʹ′ʹ′
0
( )∞−⋅⋅⋅⋅=ʹ′ʹ′ TTxkq wxw
31
21PrRe289.0
( ) 31
21PrRe289.0 ⋅⋅⋅=
−
ʹ′ʹ′=→−⋅=ʹ′ʹ′
∞∞ x
w
www x
kTT
qhTThq
31
21PrRe289.0Nu ⋅⋅=
⋅= xk
xα
Heat flux from Fourier‘s Law:
Using solution of δT :
Calculating convective heat transfer coefficient:
Dimensionless local Nusselt number Nu(x):
Chap. 12.2.5: Approximate Integral Method
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
43
Chap. 12.2: Laminar Boundary Layers
Laminar Boundary Layers: Solving the governing equations with three different methods: - Order of Magnitude Method (Approximation) - Approximate Integral Method - Exact Similarity Method
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
44
xv
yu
∂∂ψ
∂∂ψ
−==
EnergyyT
yTv
xTu
Momentumxyu
yuv
xuu
Continuityyv
xu
2
2
2
2
0
∂
∂⋅=
∂
∂⋅+
∂
∂⋅
−∂
∂⋅=
∂
∂⋅+
∂
∂⋅
=∂
∂+
∂
∂
α
ν
Exakt Solution using Similarity Solution (according to Blasius)
Governing equations:
Analytical solution exists for potential flows, i.e. a potential function Ψ exists with:
Chap. 12.2.6: Exact Solution
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
45
νν∞∞ ⋅
=⋅=⋅
⋅=Uxwith
xy
xUy xx ReRe 2
1η
ηddfUu ∞=
Ψ⋅⋅⋅
=
⋅=Ψ⋅∂
∂
∂
∂=
∂
∂⋅
∂
Ψ∂=
∂
Ψ∂=
∞
∞
∞
xUf
fUy
fUyy
u
ν1)(
)(
η
ηη
ηη
η
Similarity variable to transform governing equations to ODEs:
Instead of potential function, we use a function f(η):
Correlation to potential function Ψ:
Chap. 12.2.6: Exact Solution
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
46
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
⋅+
∂
∂⋅⋅⋅−=⋅⋅⋅
∂
∂−=
∂
Ψ∂−= ∞
∞∞ )(21)()( η
ηη f
xU
xfxUfxU
xxv ννν
xxU
xy
xxf
xf
⋅−=
⋅⋅−=
∂
∂
∂
∂⋅
∂
∂=
∂
∂ ∞
22)()( ηηη
ηηη
ν
( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛−
∂
∂⋅⋅
⋅= ∞ η
ηη
ην ffxUv
21
Calculation of v from the definition of potential flow:
Derivations:
v as a function of f(η) and η:
Chap. 12.2.6: Exact Solution
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
47
3
32
2
2
2
2
2
2
2
η
η
ηη
∂
∂⋅
⋅=
∂
∂
∂
∂⋅
⋅⋅=
∂
∂
∂
∂⋅⋅
⋅−=
∂
∂
∞
∞∞
∞
fx
Uyu
fx
UUyu
fx
Uxu
ν
ν
02 2
2
3
3=
∂
∂⋅+
∂
∂⋅
ηη
fff
ηddfUu ∞=
Derivations of u necessary for momentum equation:
Momentum equation with similarity variable: ODE
Chap. 12.2.6: Exact Solution
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
48
∞→∞→
==
η
η
yy 0:0
∞=∞== Uxuxvxu ),(0)0,()0,(
1000
====∞=∞=∞==∞ ηη ηη d
dfUu
ddf
Uu
yy
y∝η
Boundary conditions for similarity variable:
Boundary conditions for u:
Boundary conditions for potential function f(η) :
Chap. 12.2.6: Exact Solution
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
49
η = y u∞νx f
dfdη
=uu∞
d2 fdη2
0 0 0 0.3320.4 0.027 0.133 0.3310.8 0.106 0.265 0.3271.2 0.238 0.394 0.3171.6 0.420 0.517 0.2972.0 0.650 0.630 0.2672.4 0.922 0.729 0.2282.8 1.231 0.812 0.1843.2 1.569 0.876 0.1393.6 1.930 0.923 0.0984.0 2.306 0.956 0.0644.4 2.692 0.976 0.0394.8 3.085 0.988 0.0225.2 3.482 0.994 0.0115.6 3.880 0.997 0.0056.0 4.280 0.999 0.0026.4 4.679 1.000 0.0016.8 5.079 1.000 0.000
Numerical solution (using series expansion or numerical integration)
See later!
4.92 0.99
Chap. 12.2.6: Exact Solution
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
50
0 0.5
21
Re xxy⋅=η
1.00
2
4
6
99.092.4 ==∞U
uη
ηddf
Uu =
∞
Determining boundary layer thickness δ:
99.0==∞ ηd
dfUu
21
99 Re92.4 xx⋅==→
δη
21
Re92.4 −⋅⋅= xxδ
Back in x – coordinates:
Compared with ‘Order of Magnitude‘ and ‘Integral‘ Solutions:
21Re−≈ LLδ
21
Re464.3 −⋅= xx
δ
Chap. 12.2.6: Exact Solution
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
51
21
332.0
02
2
2
0 Re664.02
21
−
=
=
∞
∞∞
= ⋅=⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
⋅⋅
⋅=
⋅
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅⋅
=xx
tablefrom
yf d
fdx
UUU
yu
c η
ηρ
∂∂
ρ
νν
ν
Wall shear stress coefficient: local value
21
0 0
Re328.11664.01 −
∞
⋅=⋅⋅
⋅⋅=⋅⋅= ∫ ∫ L
L L
fxfL dxxUL
dxcL
c ν
Average over entire length L:
LWLWUcF LfLW ⋅⋅=⋅⋅⋅⋅= ∞ τρ 2
2
Total force on surface:
Chap. 12.2.6: Exact Solution
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
52
Temperature boundary layer
aTTTT
W
W ν=
−
−=
∞
Prθ
02Pr
2
2=⋅⋅+
ηθ
η
θddf
dd
1:0:0
=∞→==
θηθη
0
21
0
21
0
Re==
∞
=∞⎟⎟⎠
⎞⎜⎜⎝
⎛⋅⋅=⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅⎟
⎠
⎞⎜⎝
⎛⋅
⋅=⎟⎟⎠
⎞⎜⎜⎝
⎛⋅=
−
ʹ′ʹ′=
ηη ηθ
ηθθ
dd
xk
dd
xUk
dydk
TTqh x
yW
W
ν
Definition excess temperature:
Equation to be solved (numerically):
Boundary conditions:
Results (without derivation):
Chap. 12.2.6: Exact Solution
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
53
31
0
Pr332.0:6.0Pr ⋅=⎟⎟⎠
⎞⎜⎜⎝
⎛≥
=ηηθddFor
6.0PrPr332.0Re 3121
≥⋅⋅⋅= xxkh
6.0PrPrRe332.0Nu 31
21
x ≥⋅⋅=⋅
= xkxh
Results from nume-rical solution:
Local h:
Nusselt number:
6.0Pr)(2Pr332.01)(1
0
3121
0
≥⋅=⋅⋅⋅⎟⎠
⎞⎜⎝
⎛⋅
⋅⋅=⋅⋅= ∫∫ ∞ Lhdxx
UkL
dxxhL
hLL
ν
Averaged over the range x = 0 ….. L
Chap. 12.2.6: Exact Solution
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
54
6.0PrPrRe664.02 3121 ≥⋅⋅=⋅=⋅
= LLNukLhNu
3121 PrReNu ⋅≈ L31
21
x PrRe289.0Nu ⋅⋅= x
Nusselt number averaged over range x = 0 …. L
L
xx
NuNu
Nu
⋅=≤
⋅⋅=
26.0Pr
Pr)(Re565.0 21
Numerical solution for small Prandtl numbers:
Compared with ‘Order of Magnitude‘ and ‘Integral‘ Solutions:
Chap. 12.2.6: Exact Solution
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
55
( ) 41
32
31
21
xx
Pr0468.01
PrRe3387.0Nu :100PrRePe
⎥⎦
⎤⎢⎣
⎡ +
⋅⋅=→>⋅= x
x
Semi-empirical solution (curve fitting) for any Prandtl number (Churchill and Ozoe 1973):
( ) LLhhxx
xh x Nu2Nu)(2Re
212
1
⋅=⋅=→≈≈−
Averaged values:
Chap. 12.2.6: Exact Solution
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
56
124.92 Rexx
δ −= ⋅
21
Re664.0 −⋅= xfxc
31
Pr=Tδδ
21
Re328.1 −⋅= xfxc
Heat transfer
6.0Pr >
100PrRePex >⋅= x ( ) 41
32
31
21
Pr0468.01
PrRe3387.0Nu
⎥⎦
⎤⎢⎣
⎡+
⋅⋅= x
Summary Results: Exact solution
Boundary layer thickness:
Friction coefficient:
6.0Pr < 21
Pr)(Re565.0Nu ⋅⋅=⋅
= xx kxh
31
21PrRe332.0Nu ⋅⋅=
⋅= xx k
xh
Chap. 12.2.6: Exact Solution