# Transfer of heat through a heat transfer surface - cvut. Transfer of heat through a heat transfer

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x18 1128, Tepelne procesy/Heat Processes, 2011/2012 2Martin Dostal, martin.dostal@fs.cvut.cz

Transfer of heat through a heat transfer surface

is generally described by Fouriers equation

% cPT

t= 2T + Q(g) , where2T

2T

x2+2T

y2+2T

z2,

1r

r

(rT

r

)+

1r22T

2+2T

z2.

Basic heat transfer mechanism

conduction ~q = Tconvection q = (Tf T )radiation q = (S) T 4

For plain wall

d2Tdx2

= 0 . . . T = C1x+ C2 =TW2 TW1

x+ Tw1 . . . Q = qxS =

dTdx

S =

S (TW1 TW2)

For cylindrical wall1r

ddr

(r

dTdr

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

(dTdrS

)r

=2Lln R2R1

(TW1 TW2)

For 2D temperature field (approximation)

Q = ST (TW1 TW2) , where ST is so called shape factor. The shape factors for geometries mentioned above canbe expressed as ST = S/, ST = 2L/ 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).

Square channel of length L

Ww < 1.4 ST =

2L0.785 ln(W/w)

Ww > 1.4 ST =

2L0.930 ln(W/w)0.05

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

ST =2L

ln 4HD 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 =2L

ln[

2WD sinh

(2 HW

)]

Overall heat transfer coefficient Q = k ST = k S (T1 T2)Q = 1S (T1 TW1)

Q =

S (TW1 TW2)

Q = 2S (TW2 T2)

Q =1

11

+ +12

k

S (T1 T2) = kS (T1 T2)

Concept of the heat resistance U = RI T = RT Q . . .RT, = 1S , RT, = S , RT, =ln

R2R1

2L ,RT = 1ST , RT =

1kS

For plane wall

RT = RT,1 +RT, +RT,2 . . .1kS

=11S

+

S+

12S

. . . k =1

11

+ +12

and similarly for cylindrical wall (e.g. overall heat transfer coefficient, denoted here as k1, related to the inner heattransfer surface S1 = 2R1L)

RT = RT,1 +RT, +RT,2 . . .1

k1S1=

11S1

+ln R2R12L

+1

2S2. . . k1 =

111

+ S12L lnR2R1

+ S12S2When we take into account heat resistance of dirt layer on both sides heat transfer surface for case of heat transferthrough cylindrical wall (where S1 = 2R1L and S2 = 2R2L) we can rewrite equation above, as

k1 =1

11

+Rf1 +R1

lnR2R1

+Rf2 +R1R2

12

.

Transfer of heat by convection

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

fer coefficient represented by Nusselt number Nu = D , where D is the characteristic length (e.g. inner pipediameter in case of forced convection in pipes) and is the thermal conductivity of fluid (in W m1 K1) and otherdimensionless quantities.

Convection

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

Re =uD

=uD

=uD%

, Pr =

a=

% %cP

=cP

, Gr =gTD3

2, Gz =

RePrL/D

, Pe = Re Pr, Ra = Gr Pr

(g is the gravitational acceleration, in m s2, is the coefficient of volumetric thermal expansion, in K1, 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.

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(

W

)0.14

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 = jcPGPr2/3 (/W )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 , asp = fG2N/

{% (/W )

0.14}

, where N is the number of tranverse rows.

March 19, 2012

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