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=