Prof. Nico Hotz ME 150 – Heat and Mass Transfer 1 Principles of Convection Convective heat transfer = Heat transfer between a fluid and a surface in contact with the fluid flow General case: arbitrary body, 3-dimensional Simple case: flat plate, 1-dimensional Chap. 12: Introduction to Convection
ME 150 intro convection
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Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
1
Principles of Convection
Convective heat transfer = Heat transfer between a fluid and a surface in contact with the fluid flow
General case: arbitrary body, 3-dimensional
Simple case: flat plate,
1-dimensional
Chap. 12: Introduction to Convection
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
2
)( 0 ∞−⋅=ʹ′ʹ′ TThq
∫∫ ⋅⋅−=⋅ʹ′ʹ′= ∞
00
000 )(AA
dAhTTdAqq
)( 00 ∞−⋅⋅= TTAhq
∫ ⋅⋅=L
dxxhL
h0
)(1
General description of convection:
Velocity and temperature of the fluid depend on the position:
∫ ⋅=0
00
1
A
dAhA
h
Goal: Calculation of convective heat transfer
coefficient h [W/m2.K]
Practical assumption for some problems: local can be replaced by average :
General body:
Flat plate (1D):
Chap. 12: Introduction to Convection
hh
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
3
Conservation Equations for flowing fluid:
Fluid motion is descriped by 5 variables:
5 equations necessary: Continuity (mass conservation) = 1 equation Momentum conservation = 3 equations Energy conservation = 1 equation
We will consider 2-dimensional problems, therefore 4 equations necessary.
Chap. 12.1: Conservation Equations
T
wvuUUUU zyx
ρ
),,(),,( ==
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
4
Continuity Equation – Steady-state
Considering an infinitesimal volume element
Chap. 12.1.1: Mass Conservation
AreaVelocityDensitym ⋅⋅=
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
5
dzdxvdzdyumin ⋅⋅⋅+⋅⋅⋅= ρρEntering mass flows: (in the following: dz = 1)
dxdyvy
vdydxux
umout ⎥⎦
⎤⎢⎣
⎡⋅⋅
∂
∂+⋅+⎥⎦
⎤⎢⎣
⎡ ⋅⋅∂
∂+⋅= )()( ρρρρLeaving mass
flows:
( ) ( ) 0)()(0 =∂
⋅∂+
∂
⋅∂→=⋅⋅⋅
∂
∂+⋅⋅⋅
∂
∂
yv
xudydxv
ydydxu
xρρ
ρρ
Mass conservation: Difference between in and out = 0:
0=∂
∂+
∂
∂
yv
xu0)( =⋅udiv
ρ
Simplification for incom-pressible liquid (ρ = const.)
General form in vector notation:
Chap. 12.1.1: Mass Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
6
Momentum Conservation (steady-state)
Consider momentum fluxes through stationary control volume
Example: x-direction, 2-D (dz = 1)
x-momentum flux in x-direction
x-momentum flux in y-direction
VelocityflowMass
udzdyu ⋅⋅⋅⋅ ρ
udzdxv ⋅⋅⋅⋅ )(ρ
Chap. 12.1.2: Momentum Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
7
( ) ( ) dxvudyuu ⋅⋅⋅+⋅⋅⋅ ρρ
( ) ( )( ) ( ) ( )( ) dydxvuy
dxvudxdyuux
dyuu ⋅⋅⋅⋅∂
∂+⋅⋅⋅+⋅⋅⋅⋅
∂
∂+⋅⋅⋅ ρρρρ
Entering momentum fluxes:
Leaving momentum fluxes:
( )[ ] ( )[ ]
dydxyv
xuu
yuv
xuu
dydxyuv
yvu
xuu
xuu
dydxvuy
uux
dydxvuy
dxdyuux
equationContinuity
⋅⋅
⎪⎪
⎭
⎪⎪
⎬
⎫
⎪⎪
⎩
⎪⎪
⎨
⎧
⎥⎦
⎤⎢⎣
⎡
∂
⋅∂+
∂
⋅∂⋅+
∂
∂⋅+
∂
∂⋅=
⋅⋅⎥⎦
⎤⎢⎣
⎡
∂
∂⋅+
∂
⋅∂+
∂
∂⋅+
∂
⋅∂=
⋅⋅⎥⎦
⎤⎢⎣
⎡⋅⋅
∂
∂+⋅⋅
∂
∂=⋅⋅⋅⋅
∂
∂+⋅⋅⋅⋅
∂
∂
=
0
)()(
)()(
)()(
ρρρρ
ρρ
ρρ
ρρρρ
Difference between in and out = Change in momentum
Chap. 12.1.2: Momentum Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
8
dydxyuv
xuu ⋅⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅ρ
Rate of change in momentum in the control volume:
Newton‘s Second Law: Rate of change in momentum = Force
Surface forces: • Viscous normal stress • Normal stress from pressure • Shear stress
Possible forces:
Volume forces (e.g. gravitation)
Chap. 12.1.2: Momentum Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
9
Net force = Difference between opposing areas
Terminology: First index: Area (perpendicular to direction) Second index: Direction of force
pxx
pxy pyx
dy
dx
y
x
Oberflächenkräfte
dyxxp
dyxyp
dxyxp dxyyp
1dxdyxgρ
dy]dx)xxp(xxxp[ ∂
∂+
dy]dx)xyp(xxyp[ ∂
∂+
dx]dy)yxp(yyxp[ ∂
∂+dx]dy)yyp(yyyp[ ∂
∂+
Chap. 12.1.2: Momentum Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
10
dydxyp
dydxxp
dxpdypdxdypy
pdydxpx
p
yxxx
yxxxyxyxxxxx
⋅⋅∂
∂+⋅⋅
∂
∂=
⋅−⋅−⋅⎥⎦
⎤⎢⎣
⎡
∂
∂++⋅⎥
⎦
⎤⎢⎣
⎡
∂
∂+ )()(
xu
yv
xuPPp xxx ∂
∂⋅⋅+⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅⋅−−=+−= µµσ 2
32
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂===
yu
xvp xyyxyx µττ
Net force in x - direction:
From Fluid Dynamics (without derivation):
normal: external pressure + viscous
tangential: viscous
Chap. 12.1.2: Momentum Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
11
dydxyu
xv
yxu
yv
xu
xxP
⋅⋅⎪⎭
⎪⎬⎫⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅
∂
∂+
⎪⎩
⎪⎨⎧
⎥⎦
⎤⎢⎣
⎡
∂
∂⋅+⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅−
∂
∂+
∂
∂− µµµ 2
32
dydxgx ⋅⋅⋅ρ
Sum of surface forces (x - direction):
Volume force: gravity
xgyu
xv
yxu
yv
xu
xxP
yuv
xuu ⋅+
⎥⎥⎦
⎤
⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅
∂
∂+
⎢⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
∂
∂⋅+⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅−
∂
∂+
∂
∂−=⎥
⎦
⎤⎢⎣
⎡
∂
∂+
∂
∂⋅ ρµµµρ 2
32
Momentum Conservation (x-direction): Navier Stokes
Momentum change Normal viscous stress
Pressure
Shear stress
Gravity
Chap. 12.1.2: Momentum Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
12
0=∂
∂+
∂
∂
yv
xu
equationContinuity
yv
xu
0
32
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅− µfor incompres-
sible fluids:
x
equationContinuity
x
x
gyu
yv
xu
xxu
xP
gyu
yxv
xu
xu
xP
gyu
xv
yxu
xP
⋅+∂
∂⋅+⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂
∂
∂⋅+
∂
∂⋅+
∂
∂−=
⋅+∂
∂⋅+
∂⋅∂
∂⋅+
∂
∂⋅+
∂
∂⋅+
∂
∂−=
⋅+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂
∂
∂⋅+
∂
∂⋅⋅+
∂
∂−
=
ρµµµ
ρµµµµ
ρµµ
2
2
0
2
2
2
22
2
2
2
2
2
2
2
Rearranging the right-hand side of Navier Stokes:
Chap. 12.1.2: Momentum Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
13
xgyu
xu
xP
yuv
xuu ⋅+⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅+
∂
∂−=⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅ ρµρ 2
2
2
2
ygyv
xv
yP
yvv
xvu ⋅+⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅+
∂
∂−=⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅ ρµρ 2
2
2
2
( ) gUPUU ⋅+∇⋅+∇−=∇•⋅ ρµρ 2
2
2
2
22
yxyj
xivjuiU
∂
∂+
∂
∂=∇
∂
∂⋅+
∂
∂⋅=∇⋅+⋅=
Navier Stokes in x - direction (incompressible fluid):
and in y - direction, respectively:
In vector notation:
2-D vector operators:
Chap. 12.1.2: Momentum Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
14
Energy Conservation (steady-state)
We have to condsider:
• Convective heat transfer
• Thermal conduction
• Internal heat sources
• Work due to friction and volume forces
We can neglect:
• Kinetic energy
• Potential energy
Applicable for problems with: low Mach numbers, Δh < 1000 m
Chap. 12.1.3: Energy Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
15
Components of Heat Transfer:
D = Conduction V = Convection
Chap. 12.1.3: Energy Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
16
Balance for convection:
e = specific internal energy (J/kg)
Chap. 12.1.3: Energy Conservation
( ) ( )
( ) ( )
( ) ( )
⎥⎦
⎤⎢⎣
⎡⋅⋅⋅⋅
∂
∂+⋅⋅⋅⋅
∂
∂−=
=⎥⎦
⎤⎢⎣
⎡⋅⋅⋅⋅
∂
∂+⋅⋅⋅−⋅⋅⋅+
+⎥⎦
⎤⎢⎣
⎡⋅⋅⋅⋅
∂
∂+⋅⋅⋅−⋅⋅⋅=
=−+−= ++
dydxvey
dydxuex
dydxvey
dxvedxve
dxdyuex
dyuedyue
EEEEE dyyVyVdxxVxVtotV
)()(
)(
)(
,,,,,
ρρ
ρρρ
ρρρ
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
17
dydxyTk
yxTk
x
dyyTdxk
ydx
xTdyk
xE totD
⋅⋅⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂⋅
∂
∂+⎟⎠
⎞⎜⎝
⎛∂
∂⋅
∂
∂=
=⋅⎭⎬⎫
⎩⎨⎧
∂
∂⋅⋅−
∂
∂−⋅
⎭⎬⎫
⎩⎨⎧
∂
∂⋅⋅−
∂
∂−=,
Balance for conduction:
Using Fourier‘s Law:
( ) ( )
⎥⎦
⎤⎢⎣
⎡⋅
∂
∂+⋅
∂
∂−=
=⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
∂
∂+−+⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅
∂
∂+−=
=−+−= ++
dyEy
dxEx
dyEy
EEdxEx
EE
EEEEE
yDxD
yDyDyDxDxDxD
dyyDyDdxxDxDtotD
,,
,,,,,,
,,,,,
Chap. 12.1.3: Energy Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
18
Rate of work due to friction (surface forces) and volume forces:
Rate of work = Power = Force ● Velocity
total of 10 force components for a 2D-volume element
Chap. 12.1.3: Energy Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
19
dydxgvdydxgu
fasdxdypvx
dydxpuy
dydxpvy
dxdypux
W
yx
xyyx
yyxxtot
⋅⋅⋅⋅+⋅⋅⋅⋅+
+⋅⋅⋅∂
∂+⋅⋅⋅
∂
∂+
+⋅⋅⋅∂
∂+⋅⋅⋅
∂
∂=
ρρ
)()(
)()(
Rate of work of these 10 force components:
yxxyyxxy
yyy
xxx
ppPpPp
ττ
σ
σ
===
+−=
+−=Substitute for the surface forces pressure and viscous components:
Chap. 12.1.3: Energy Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
20
dydxgvvx
Pvy
vy
dydxguuy
Pux
ux
W
yxyy
xxyxtot
⋅⋅⎭⎬⎫
⎩⎨⎧
⋅⋅+⋅∂
∂+⋅
∂
∂−⋅
∂
∂+
+⋅⋅⎭⎬⎫
⎩⎨⎧
⋅⋅+⋅∂
∂+⋅
∂
∂−⋅
∂
∂=
ρτσ
ρτσ
)()()(
)()()(
Using this substitution:
dVqWEE stottotVtotD ⋅+++= ,,0
Condition for steady-state energy balance:
Chap. 12.1.3: Energy Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
21
Using the individual components:
Convection
Conduction
Work due to forces
Sources
Chap. 12.1.3: Energy Conservation
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
s
yxyy
xxyx
q
gvvx
Pvy
vy
guux
Pux
ux
yTk
yxTk
x
vey
uex
+
+⎥⎦
⎤⎢⎣
⎡⋅⋅+⋅
∂
∂+⋅
∂
∂−⋅
∂
∂+
+⎥⎦
⎤⎢⎣
⎡ ⋅⋅+⋅∂
∂+⋅
∂
∂−⋅
∂
∂+
+⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂⋅
∂
∂+⎟⎠
⎞⎜⎝
⎛∂
∂⋅
∂
∂+
+⎥⎦
⎤⎢⎣
⎡⋅⋅
∂
∂+⋅⋅
∂
∂−=
ρτσ
ρτσ
ρρ0
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
22
Next steps:
- Use explicit expressions for σ und τ
- Neglect volume force
- Combine friction term to Φ
sqyv
xuP
yTk
yxTk
xyev
xeu +Φ⋅+⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅−⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂⋅
∂
∂+⎟⎠
⎞⎜⎝
⎛∂
∂⋅
∂
∂=
∂
∂⋅⋅+
∂
∂⋅⋅ µρρ
2222
322 ⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+⎟
⎠
⎞⎜⎝
⎛∂
∂⋅+⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂=Φ
yv
xu
yv
xu
xv
yu
Φ = Effect of viscous friction
Convection Conduction Pressure Sources
Chap. 12.1.3: Energy Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
23
0=∂
∂+
∂
∂
yv
xu
dTcdTcdTcde vp ⋅=⋅=⋅=
sqyTk
yxTk
xyTv
xTuc +Φ⋅+⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂
∂
∂+⎟⎠
⎞⎜⎝
⎛∂
∂
∂
∂=⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅⋅ µρ
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂⋅
∂
∂⋅−⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂=Φ
yv
xu
xv
yu 4
2
More simplifications: Fluid is incompressible
constant=ρ Continuity Pressure term = 0
Energy purely thermal: KE = PE = 0
Energy conservation in terms of temperature:
with simplified Φ
Chap. 12.1.3: Energy Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
24
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂⋅=⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂⋅+
∂
∂⋅ 2
2
2
2
yT
xT
yTv
xTu α
TTU 2∇⋅=∇•
α
yTj
xTiTjviuU
∂
∂+
∂
∂=∇⋅+⋅=
Neglecting friction (Φ = 0) and taking k = constant:
pck⋅
=ρ
α
Heat Transfer Equation for laminar, incompressible flows without friction and with constant thermal conductivity:
Using vector operators:
Chap. 12.1.3: Energy Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
25
dtTc p ∂⋅⋅ρ
Remark on transient problems:
For transient problems, an additional term is needed
Transient change of energy content of a control volume:
Chap. 12.1.3: Energy Conservation
Prof. Nico Hotz
ME 150 – Heat and Mass Transfer
26
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