Download - Transfer of heat through a heat transfer surface - cvut.czusers.fs.cvut.cz/martin.dostal/tp/FILES/2012/hp-tutorial-2012-2.pdf · Transfer of heat through a heat transfer surface ...

Transcript
Page 1: Transfer of heat through a heat transfer surface - cvut.czusers.fs.cvut.cz/martin.dostal/tp/FILES/2012/hp-tutorial-2012-2.pdf · Transfer of heat through a heat transfer surface ...

x18 1128, Tepelne procesy/Heat Processes, 2011/2012 2Martin Dostal, [email protected]

Transfer of heat through a heat transfer surface

is generally described by Fourier’s equation

% cP∂T

∂t= λ∇2T + Q(g) , where∇2T

∂2T

∂x2+∂2T

∂y2+∂2T

∂z2,

1r

∂r

(r∂T

∂r

)+

1r2∂2T

∂ϕ2+∂2T

∂z2.

Basic heat transfer mechanism

conduction ~q = −λ∇Tconvection q = α (Tf − T )radiation q = εσ(S) T 4

♠ For plain wall

d2T

dx2= 0 . . . T = C1x+ C2 =

TW2 − TW1

δx+ Tw1 . . . Q = qxS = −λdT

dxS =

λ

δS (TW1 − TW2)

♠ For cylindrical wall

1r

ddr

(r

dTdr

)= 0 . . . T = C1 ln r + C2 = . . . . . . Q = (qrS)|r = −λ

(dTdrS

)∣∣∣∣r

=2πLλln R2

R1

(TW1 − TW2)

♠ For 2D temperature field (approximation)

Q = λST (TW1 − TW2) , where ST is so called shape factor. The shape factors for geometries mentioned above canbe expressed as ST = S/δ, ST = 2πL/ ln(R2/R1) and some examples are in following table (other shape factors canbe found in literature, e.g. Sestak, J., Rieger, F.: Momentum, heat and mass transfer, Czech Technical University inPrague (1998) (in czech), Serth, R. W.: Process Heat Transfer: principles and applications, Elsevier Academic Press(2007) and in other literature).

Page 2: Transfer of heat through a heat transfer surface - cvut.czusers.fs.cvut.cz/martin.dostal/tp/FILES/2012/hp-tutorial-2012-2.pdf · Transfer of heat through a heat transfer surface ...

Square channel of length L

Ww < 1.4 ST = 2πL

0.785 ln(W/w)Ww > 1.4 ST = 2πL

0.930 ln(W/w)−0.05

Circular cylinder of diame-terD centered in a rectanu-lar solid of length L

ST =2πL

ln 4HπD − f(W/H)

,

where [H/W, f ] are [1, 0.1658], [1.25, 0.07926],

[1.5, 0.03562], [1.75, 0.00816], [2, 0.00746],

[2.25, 0.00340], [2.5, 0.00156], [3, 0.00032]

Row of cylinders of di-ameter D and same tem-perature buried in a semi-infinite medium (for onecylinder)

ST =2πL

ln[

2WπD sinh

(2π HW

)]

Overall heat transfer coefficient Q = k S∆T = k S (T1 − T2)Q = α1S (T1 − TW1)

Q =λ

δS (TW1 − TW2)

Q = α2S (TW2 − T2)

Q =1

1α1

+ δλ + 1

α2︸ ︷︷ ︸k

S (T1 − T2) = kS (T1 − T2)

Concept of the heat resistance ∆U = RI ∼ ∆T = RT Q . . .RT,α = 1αS , RT,λ = δ

λS , RT,λ =ln

R2R1

2πLλ ,RT = 1

λST, RT = 1

kS

For plane wall

RT = RT,α1 +RT,λ +RT,α2 . . .1kS

=1α1S

λS+

1α2S

. . . k =1

1α1

+ δλ + 1

α2

and similarly for cylindrical wall (e.g. overall heat transfer coefficient, denoted here as k1, related to the inner heattransfer surface S1 = 2πR1L)

RT = RT,α1 +RT,λ +RT,α2 . . .1

k1S1=

1α1S1

+ln R2

R1

2πLλ+

1α2S2

. . . k1 =1

1α1

+ S12πLλ ln R2

R1+ S1

α2S2

When we take into account heat resistance of dirt layer on both sides heat transfer surface for case of heat transferthrough cylindrical wall (where S1 = 2πR1L and S2 = 2πR2L) we can rewrite equation above, as

k1 =1

1α1

+Rf1 +R1

λlnR2

R1+Rf2 +

R1

R2

1α2

.

Transfer of heat by convection

is in engineering practice described with relations between dimensionless quantities, such as dimensionless heat trans-

fer coefficient α represented by Nusselt number Nu = αDλ , where D is the characteristic length (e.g. inner pipe

diameter in case of forced convection in pipes) and λ is the thermal conductivity of fluid (in W m−1 K−1) and otherdimensionless quantities.

Convection

{forced Nu = f(Re,Pr, geometry)natural Nu = f(Gr,Pr, geometry)

Page 3: Transfer of heat through a heat transfer surface - cvut.czusers.fs.cvut.cz/martin.dostal/tp/FILES/2012/hp-tutorial-2012-2.pdf · Transfer of heat through a heat transfer surface ...

Re =u∞D

ν=uD

ν=uD%

µ, Pr =

ν

a=µ

%· %cPλ

=µcPλ

, Gr =gβ∆TD3

ν2, Gz =

RePrL/D

, Pe = Re ·Pr, Ra = Gr ·Pr

(g is the gravitational acceleration, in m s−2, β is the coefficient of volumetric thermal expansion, in K−1, for idealgases = 1/T and ∆T is the characteristic temperature difference between wall and bulk)

For non-circular cross-sectional flow areas we usually use concept of the equivalent (hydraulic) diameter defined

De = 4AP , where A is cross-section area and P is wetted perimeter.

In the following table we can find examples of the Nusselt number correlations for various geometries.

Page 4: Transfer of heat through a heat transfer surface - cvut.czusers.fs.cvut.cz/martin.dostal/tp/FILES/2012/hp-tutorial-2012-2.pdf · Transfer of heat through a heat transfer surface ...

Sieder-Tate correction

takes into account dependency of thermophysical properties of fluid on temperature (at vicinity of the heat transfersurface, thermophysical properties of flowing fluid are affected by different temperature of the heat transfer surface –the higher temperature of flowing fluid, in case of heating, the lower dynamic viscosity of fluid and the higher intensityof heat transfer than heat transfer calculated with using of bulk temperatures).

Nucorr. = Nu(µ

µW

)0.14

Page 5: Transfer of heat through a heat transfer surface - cvut.czusers.fs.cvut.cz/martin.dostal/tp/FILES/2012/hp-tutorial-2012-2.pdf · Transfer of heat through a heat transfer surface ...

Shell side heat transfer coefficient

Recommendation of shell side heat trans-fer coefficient choosing from Foust, A.S., Wenzel, L. A., Clump, C. W., Maus,L., Andersen, L. B.: Principles of UnitOperations, John Wiley & Sons., NewYork (1960) (all pictures in this part isfrom . . . ). For more exact estimation theBell-Delaware method can be used.

The shell side heat transfer coefficient can be calculated from the Colburn j-factor as α = jcPGPr−2/3 (µ/µW )0.14,where G is the mass flux of shell-side fluid at the shell centerline cross-flow area A (area between tubes), i.e. G =m/A. The Reynolds number Re is based on the outer diameter of tube (DO in chart).

The pressure drop during fluid flow through ideal tube bank can be calculated from the Fanning friction factor f , as∆p = fG2N/

{% (µ/µW )0.14

}, where N is the number of tranverse rows.

March 19, 2012