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Konvektiv värmeöverföringConvective heat transfer
värmeledande kropptf
tw(y)Heat conducting body
y
x
w(y)
t(x,y)fluidbody
x = 0 ⇒ u, v, w = 0 ⇒ värmeledning i fluiden,
AttQ fw )( −=α&
heat conduction in the fluid
0
fx
i fluidenin the fluid
tQ Ay
λ=
⎛ ⎞∂= − ⎜ ⎟∂⎝ ⎠
&
14243
Q
t⎛ ⎞∂
0/ (6 3)f
x
w f w f
tyQ A
t t t t
λα =
⎛ ⎞∂⎜ ⎟∂⎝ ⎠= = − −
− −
&Q
C h fConvective heat transfer
Målsättning: Bestämma α och de parametrar som bestämmer den vid givet t (x) eller q (x) = Q/Avid givet tw(x) eller qw(x) = Q/A
Objective: Determine α and theObjective: Determine α and the parameters influencing it for prescribed tw(x) or qw(x) = Q/A
Storleksordning för värme-övergångskoefficienten α, Order of magnitude for αmagnitude for αMedium α W/m²KLuft, air (1bar); naturlig konvektion 2-20
natural convectionLuft, air (1bar); forcerad konvektion 10-200
forced convectionLuft, air (250 bar); forcerad konvektion 200-1000
forced convectionVatten, Water; forcerad konvektion 500-5000
forced convectionOrganiska vätskor; forcerad konvektion 100-1000Organic liquids; forced convectionOrganic liquids; forced convection
Kondensation (vatten)Condensation (water) 2000-50000Kondensation (organiska ångor) 500-10000Condensation (organic vapors)
Förångning (vatten) 2000-100000Förångning (vatten) 2000 100000Evaporation, boiling, (water)Förångning (organiska vätskor) 500-50000Evaporation, boiling (organic liquids)
Konvektiv VärmeöverföringConvective heat transfer
Verktyg - Tools
Massbalans ⇒ KontinuitetsekvationenMass balance ⇒ Continuity eq., mass
conservation eq.
Rörelseekvationer ⇒ Navier-Stokes`ekvationerMomentum balance (eqs. of motion) ⇒ Navier-Stokes eqs.
Energi- ⇒ Temperaturfälts-ekvationen ekvationene vat o e e vat o eEnergy eq. (First law of thermodynamics) ⇒ tempera-ture field eq.
Kontinuitetsekv (K E ) ContinuityKontinuitetsekv. (K.E.), Continuity eq. y
dy
x: dzdyum 1x ρ=&
∂ &
xz dx
dy
dz
Netto ut i x-led net mass flow out in x-direction
2 1
1
( )
xx x
mm m dxx
u dy dz u dx dy dzx
ρ ρ
∂= + =
∂∂
= +∂
& &
Netto ut i x-led, net mass flow out in x-direction
Analogt i y- och z-led, analogous in y- and z-directions
dzdydx)u(x
mx ρ∂∂
=Δ &
dydxdzwz
mdzdxdyvy
m zy )( )( ρρ∂∂
=Δ∂∂
=Δ &&
zyx mmmutströmmatNetto &&& Δ+Δ+Δ:out flownet ,
Netto utströmmat, net mass flow out ⇒minskning av massa inom volymelementet, reduction in mass within volume element
y
Forts kontinuitetsekv, cont. i icontinuity eq.
Minskningen per tidsenhet, reduction per time unit:∂ dzdydxτ∂ρ∂
Q
)w()v()u( ρ∂
+ρ∂
+ρ∂
=ρ∂
− )w(z
)v(y
)u(x
ρ∂
+ρ∂
+ρ∂τ∂
)46(0)w(z
)v(y
)u(x
−=ρ∂∂
+ρ∂∂
+ρ∂∂
+τ∂ρ∂
Spec. stationärt, inkompressibelt, tvådim., especially forsteady state, incompressible flow, two-dimensional case
y
⇒
)56(0yv
xu
−=∂∂
+∂∂
Navier – Stokes’ ekvationer (eqs.)(eqs.)
d d d
Famrr
=⋅
m = ρ dx dy dz
⎟⎞
⎜⎛ dwdvdur
men, but
⎟⎠⎞
⎜⎝⎛
τττ=
ddw,
ddv,
ddua
r
u = u(x, y, z, τ), v = v(x, y, z, τ),
w = w(x, y, z, τ)
A lAcceleration, Inertia
du u u dx u dy u dzd x d y d z dτ τ τ τ τ
∂ ∂ ∂ ∂= + ⋅ + ⋅ + ⋅ =∂ ∂ ∂ ∂
u u u uu v wx y zτ
∂ ∂ ∂ ∂= + + +∂ ∂ ∂ ∂
Forces
Fr
l k ft (F F F ) äka. volymkrafter (Fx , Fy , Fz) räknas per massenhet, volume forces per unit mass, N/kg
b. ytkrafter – spänningar, stresses σij N/m2
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
σσσ
σσσ
σσσ
=σ
zzzyzx
yzyyyx
xzxyxx
ij
⎦⎣ y
{ { {"z""y""x"
Teckenkonvention för ytspänningarna, Signs for the stresses
.
dyy
σxxσxx
σyyσyx
σxy
R lt d ä i lti t
σyy
xx
x
Resulterande spänningar, resulting stressesdy)(
y yyyy σ∂∂
+σ
∂
dy)(y yxyx σ∂∂
+σ
σyy
σxx
σyxσxy
dx)( xxx
xx σ∂∂
+σ
dx)(x xyxy σ∂∂
+σ
Nettokraft i x- led, net force in x-direction:
dzdxdy)(dydxdz)(dxdydz)( zxyxxx σ∂∂
+σ∂∂
+σ∂∂
zyx y ∂∂∂
dxdydz)(x ji
jσ
∂∂
Have a look at stress-Have a look at stressstrain in solids
S fl dStress-strain fluids
All ä ll.Allmännare, generally
⎥⎥⎤
⎢⎢⎡
+⎥⎥⎤
⎢⎢⎡
−−
=
=+δ−=σ
xzxyxx
ijijij
ddd
ddd
0p000p
dp
⎥⎥
⎦⎢⎢
⎣
+⎥⎥⎦⎢
⎢⎣ −
=
zzzyzx
yzyyyx
ddd
dddp00
0p0
)146()1e(2d ijijij −δΔ−μ= )146()3
e(2d ijijij δΔμ=
⎥⎥⎤
⎢⎢⎡
μ=⎥⎥⎤
⎢⎢⎡ xzxyxxxzxyxx
eee
eee
2ddd
ddd
⎥⎥
⎦⎢⎢
⎣
μ=⎥⎥
⎦⎢⎢
⎣ zzzyzx
yzyyyx
zzzyzx
yzyyyx
eee
eee2
ddd
ddd
⎥⎤
⎢⎡Δ 003/
⎥⎥⎥
⎦⎢⎢⎢
⎣ ΔΔμ−
3/0003/02
uu1 ii ⎟⎞
⎜⎛ ∂∂ )156(
xu
xu
21e
i
i
j
iij −⎟
⎟⎠
⎜⎜⎝ ∂
∂+
∂∂
=
)166(zw
yv
xueii −
∂∂
+∂∂
+∂∂
==Δ
l fExamples of stresses
.
upep ∂+=+= μμσ 22
xpep xxxx
∂+−=+−= μμσ 22
⎟⎟⎞
⎜⎜⎛ ∂
+∂
===vue μμσσ 2 ⎟⎟⎠
⎜⎜⎝ ∂
+∂ xy
exyyxxy μμσσ 2
vpep ∂+−=+−= μμσ 22
ypep yyyy
∂+−=+−= μμσ 22
Resulting momentumResulting momentum equations
.
⎟⎟⎞
⎜⎜⎛ ∂
+∂
+∂
−=⎟⎟⎞
⎜⎜⎛ ∂
+∂
+∂ 22
:ˆ uupFuvuuux μρρ ⎟⎠
⎜⎝ ∂
+∂
+∂⎟
⎠⎜⎝ ∂
+∂
+∂ 22
:yxx
Fy
vx
ux x μρτ
ρ
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
+∂∂
−=⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
+∂∂
2
2
2
2
:ˆyv
xv
ypF
yvv
xvuvy y μρ
τρ
Energiekv., Energy eq. (First law of thermodynamics of an open system), ⇒Temperaturfälts-ekv., temperature field eqeq.
y
xz dx
dy
dz
dHQd =&
.Värmeledning i fluiden, heat conduction in the fluid
Qd &
tt ∂∂&
tt
dxx
QQQ
xtdydz
xtAQ
xxdxx
x
∂∂∂
=∂∂
+=
∂∂
λ−=∂∂
λ−=
+&
&&
dxdydz)xt(
xdydz
xt
∂∂
λ∂∂
−∂∂
λ−=
dxdydz)t(QQQ xdxxx ∂∂
λ∂∂
−=−=Δ +&&& y)
x(
xQQQ xdxxx ∂∂+
Analogt i y- och z-led, analogous in y- and z-directions
.in y- and z-directions
dzdxdy)yt(
yQy ∂
∂λ
∂∂
−=Δ &
dydxdz)zt(
zQz ∂
∂λ
∂∂
−=Δ &
{ }QQQQd &&&& ΔΔΔ{ }zyx QQQQd Δ+Δ+Δ−=↑
heatfor conventionsign ,Qdför
konventiontecken&
ttt ⎫⎧ ∂∂∂∂∂∂
Q
(6-27)
dzdydx)zt(
z)
yt(
y)
xt(
xQd
⎭⎬⎫
⎩⎨⎧
∂∂
λ∂∂
+∂∂
λ∂∂
+∂∂
λ∂∂
=&
(6 27)
Enthalpy flows andEnthalpy flows and changes
x-direction
hdzdyuhmH xx ρ== && yxx ρ
h∂∂ dzdydxxhudzdydx
xuhHd x
∂∂
+∂∂
=⇒ ρρ&
h l hEnthalpy changes
y- and z-directions
dzdydxyhvdzdydx
yvhHd y
∂∂
+∂∂
= ρρ&
dzdydxzhwdzdydx
zwhHd z
∂∂
+∂∂
= ρρ&
l h h lTotal change in enthalpy
.
dddhhhdddwvuh
HdHdHdHd zyx
⎟⎞
⎜⎛ ∂∂∂⎟
⎞⎜⎛ ∂∂∂
=++= &&&&
dzdydxz
wy
vx
udzdydxzyx
h ⎟⎟⎠
⎜⎜⎝ ∂
+∂
+∂
+⎟⎟⎠
⎜⎜⎝ ∂
+∂
+∂
= ρρ
Energy equation,Energy equation, intermediate step
.
t t t⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂t t tx x y y z z
h h h
λ λ λ⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂
+ + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞∂ ∂ ∂⎜ ⎟u v w
x y zρ + +⎜ ⎟∂ ∂ ∂⎝ ⎠
h lEnthalpy vs temperature
.
( )tphh = ( )
dtthdp
phdh
tphh
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
=⇒
=
,
tp pt ⎠⎝ ∂⎠⎝ ∂
h lEnthalpy vs temperature
.p
p thc ⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
=
For ideal gases the enthalpy is independent of pressure, i.e.,
0)/( ≡∂∂ tph
. For liquids, one commonly assumes that the derivative
ph )/( ∂∂ tph )/( ∂∂is small and/or that the pressure variation dp is small compared to the change in temperature.
Then generally one statesThen generally one states
dtcdh p=
Temperature Equation
.
⎟⎟⎞
⎜⎜⎛ ∂
+∂
+∂
=∂
+∂
+∂ 222 ttttwtvtu λ
⎟⎟⎠
⎜⎜⎝ ∂
+∂
+∂
=∂
+∂
+∂ 222 zyxcz
wy
vx
upρ
Boundary layer approximations
)U∞
. u(x,y)
δ(x)
U∞
y x
δ T
twt(x,y)t∞
y x
Boundary layer approximations –Boundary layer approximations Prandtl’s theory
. vu >>
vvuu ∂∂∂>>
∂yxxy ∂∂∂
>>∂
,,
tt ∂∂xt
yt
∂∂
>>∂∂
Boundary layer approximations –Boundary layer approximations Prandtl’s theory
.)(xpp =
2
2
yu
dxdpF
yuv
xuu x
∂∂
+−=⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂ μρρ
2
2ttvtu∂∂
=∂∂
+∂∂
ρλ
2ycyx p ∂∂∂ ρ
Boundary layer approximations –Boundary layer approximations Prandtl’s theory
.21 konstant
2p Uρ+ =
dxdUU
dxdp ρ−=
λ
μ
λ
ρν pp cc==Pr
λλ
Boundary layer equations
. 0=∂∂
+∂∂
yv
xu
2
2
yu
dxdUU
yuv
xuu
∂∂
+=∂∂
+∂∂
ρμ
2ttvtu ∂=
∂+
∂ μ Pr 2yy
vx
u∂∂
+∂ ρ
Boundary layers
.
U∞
U∞
yx
fully turbulentlayer
buffer layerviscous sublayer
xc
x
laminar boundarylayer
transition turbulent boundary layer
viscous sublayer
/R U ν/Re cc xU∞= 5105 ⋅
Pr)(Re,Nu 7f=