momentum heat mass transfer

Momentum Heat Mass TransferMHMT11

Thermal boundary layer. Forced convection (pipe, plate, sphere). Natural convection.

Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

Heat transfer-convection

sourceDt

D

Rohsenow W.M., Hartnett J.P., Cho Y.I.: Handbook of Heat Transfer. McGraw Hill, 3rd Edition, 1998

Heat transfer - convectionMHMT11

Heat flux from a surface to fluid is evaluated using heat transfer coefficient , assuming linear relationship between heat flux and driving force, which is the temperature difference between solid surface and temperature of fluid far from surface (outside a thermal boundary layer)

)( fw TTq Heat transfer coefficient is related to thickness of thermal boundary layer

T

T

y

xTw

T

Thermal boundary layerMHMT11

Integral boundary layer theorem can be derived in the same way like for the momentum boundary layer

2

20 0

( )H H

x y

T T Tu u dy a dy

x y y

0

0

0 0

0

0

0 0

( ) [ ]

[ ] |

|

H

H

H

H

H H

y yx x xy

yxy y y

x xy

u Tu T u u TT T dy a

x x y x y

u T Tdy u T a

x y

u T u Tdy T dy a

x x y

0

0

0

0this is zero at boundary

( ( )) |

( ) ( ) | |

H

H

H

x y

Hx x y

Tu T T dy a

x y

dd Tu T T dy u T T a

dx dx y

Upper bound H

MHMT11

Final form of integral boundary equation

0

heat flux thermal energy thickness

( )[( ( ) ) ( ) ]

( )( ( ) )

wT

x ww

w p

q

u T T qdT x T U x dy

dx U x T x T c

This equation can be applied as soon as the momentum boundary layer was identified (velocity profile inside the thermal boundary layer must be known).

This equation is used for prediction of heat transfer at flows along plates, pipes, cylinder, spheres…

Thermal boundary layer

However, in the following slides we shall analyze heat transfer in a pipe and along a plate in a simpler way, using the penetration theory (instead of integral theorem). The penetration theory predicts not so accurate but still qualitatively correct results.

Heat transfer – pipeMHMT11

Duchamp

Forced convection: Reynolds number is given, Nusselt number is to be calculated.

3 Re Pr /Nu D L

y

x

Tw

T0

D

parabolic velocity profile

slope of the velocity profile

behind the boundary layer is the inlet temperature T0

Heat transfer – pipe (laminar)MHMT11

Convective heat transfer in a pipe for the laminar flow and fully developed velocity profile (therefore it is not necessary to solve the problem of momentum boundary layer).

Velocity profile D

yuyu

8)(

3

2

2

4

( ) 2axD

u

ax axD

u u

2

32D D uD

Nuax

penetration theory

2

Re Pr

RePr

Graetz number

Pe

Gz

uD uD D

ax a x

Leveque formula

2

33x x

D uDNu c c Gz

ax

Remark: This correlation predicts local value of (x), which is stressed by lower index in Nusselt and Graetz number


Anyway, qualitatively the same result can be obtained using integral theorem

D

yuyu

8)(

00

0

( )T

w wx

p T

q T Tdu T T dy a

dx c

assuming linear temperature and velocity profile in the thermal boundary layer

0( )w wT

yT T T T

00

0

2

3

8( )(1 )

8( )

6

9

8

T

ww

T T

T

T

T

T Td y yu T T dy a

dx D

d aD

dx u

aDx

u

…this results differs only by a constant (9/8) from the previous result (1/2) but this constant is anyway usually modified using experiments or more accurate assumptions.


Local value (x) increases to infinity with decreasing distance from the tube inlet. From practical point of view a more important is the average value of at pipe of the length L .

333ln

0 0

1 3Re Pr Re Pr 1.6

2

L Lc D c Ddx dx Gz

L D L x D L D

ln2/3

0.06683.66

1 0.04

D GzNu

Gz

3ln 1.67D

Nu Gz

index ln is used because this ln is related to the mean logarithmic

temperature difference used in the heat exchanger design

Correlations used in practice for heat transfer prediction in laminar flows in pipes

Léveque formula for Gz>50 (short pipes)

Hausen formula for arbitrary long pipes

Note the fact that at very long tubes the Nu (and ) is constant 3.66, and that the Hausen correlation reduces to Leveque for Gz

Heat transfer – pipe (turbulent)MHMT11

Turbulent flow is characterised by the energy transport by turbulent eddies which is more intensive than the molecular transport in laminar flows. Heat transfer coefficient and the Nusselt number is therefore greater in turbulent flows. Basic differences between laminar and turbulent flows are:

Nu is proportional to in laminar flow, and in turbulent flow.

Nu doesn’t depend upon the length of pipe in turbulent flows significantly (unlike the case of laminar flows characterized by rapid decrease of Nu with the length L)

Nu doesn’t depend upon the shape of cross section in the turbulent flow regime (it is possible to use the same correlations for elliptical, rectangular…cross sections using the concept of equivalent diameter – this cannot be done in laminar flows)

3 u 0.8u

The simplest correlation for hydraulically smooth pipe designed by Dittus Boelter is frequently used (and should be memorized)

0.80.023Re PrmNu m=0.4 for heating

m=0.3 for cooling

Heat transfer - plateMHMT11

Duchamp

3Re Prx xNu

Analysis of heat transfer at external flows (around a plate, cylinder, sphere…) differs from internal flows (for example heat transfer in pipe) by the fact that velocity profile at surface is not known in advance and therefore not only the thermal boundary layer but also the momentum boundary must be solved


Heat transfer in parallel flow along a plate is characterised by simultaneous development of thermal and momentum boundary layers. It will be assumed, that the thickness of momentum boundary layer H is greater than thermal boundary layer T

T

y

x

Tw

T

H

)(

42

TT u

ax

4.6H

x

U

Linear velocity profileH

yUyu

)(

Thermal boundary layer thickness4 4 4

4.6( )T HT T T

ax ax ax x

u U U U

4.6H

x

U

( ) TT

H

u U

1/3 1/6(18.4 ) ( )T

xa

U

Note the fact, that the ratio of thermal and momentum boundary layer thickness is independent of x and velocity U

1/21/6 1/2 1/3

1/3 1/3

loca value of this constant should PrReheat transfer be replaced by 0.332in accurate solution

( ) 1( ) ( ) ( )

(18.4 ) 18.4T

xU xUx xNu

a a

(this is previously obtained result for thickness of the momentum boundary layer)


Mean value of heat transfer is obtained from the previous formula by integration along the length of plate L

1/30.664 Re PrLNu

y

x

1/3/ PrT H

H

x

U

L

Turbulent flow regime (Re> 500000)

Laminar flow regime

0.8 1/30.0365Re PrLNu

Compare these results with the heat transfer in pipe

Pipe: Laminar flow NuRe1/3 turbulent flow NuRe0.8

Plate: Laminar flow NuRe1/2 turbulent flow NuRe0.8

Heat transfer – sphere…MHMT11

1/2 2/3 0,42 (0.4Re 0.06Re ) PrNu

0,71Pr380 3,5Re7,6.104 .

1/2 2/3 0.370.25 (0.4Re 0.06Re ) PrNu 0,67Pr300 1Re105 .

Flow around a sphere (Whitaker)

Cross flow around a cylinder (Sparrow 2004)

Cross flow around a plate (Sparrow 2004)

0.6 RefrontNu 2/30.17 RerearNu

See next slide

Front sideRear side

(wake)Important for heat transfer from droplets…

Important for shell&tube and fin-tube heat exchangers

Heat transfer – sphereMHMT11

Conduction outside a sphere, see 1D solution of temperature profile

Tw

TD

r

22

1( ) 0

d dTr

r dr dr

( ) ( )2 w

DT r T T T

r

/2

2( ) | ( )w r D w

Tq T T T T

r D

2D

Nu

Natural convectionMHMT11

Duchamp

Velocity and Reynolds number is not known in advance. Flow is induced only by buoyancy. Instead of Reynolds it is necessary to use the Rayleigh number.


um

x

y

Vertical wall. Wall temperature Tw>Tf. Heated fluid flows upward along the plate due to buoyant forces.

Forces equilibrium (wall shear stress=buoyant force)

gwmu g T

Penetration depth at distance x2

2

44 m

m

x axa uu

43

4 4

ax axg T

g T

Substituting um into force balance

3 344 4

4

x x g Tx g TxNu c c Ra

a a

where Ra is Rayleigh number

(x represents a characteristic dimension in the direction of gravity, e.g. height of wall)


Turbulent flow regime occurs at very high Ra>109 and instead the 4th root of Ra the 3rd root is used in correlations

3turbNu c Ra

Note the fact, that at turbulent flow regime the heat transfer coefficient is almost independent of x (size of wall)

3

3 3turb turb

x g Tx g Tc c

a a

EXAMMHMT11

Forced convection

Natural convection

What is important (at least for exam)MHMT11

)( fw TTq T

What is it heat transfer coefficient and how is related to thermal boundary layer thickness

3 3ln 1.6 1.6 Re Pr /D

Nu Gz D L

Heat transfer in a pipe (laminar)

Heat transfer a plate (laminar) 1/30.664 Re PrLNu

Heat transfer turbulent 0.8 1/3Re PrNu

Forced convection Nu(Re,Pr)

What is important (at least for exam)MHMT11

Free convection Nu(Ra)

344

x g TxNu c c Ra

a

Laminar flow (Re<1010)

333

t t

x g TxNu c c Ra

a

Turbulent flow

momentum heat mass transfer

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