OF SUMMARY.TEX
SUMMARY CONVECTION CORRELATION EQUATIONS ME 353 Heat Transfer 1 Department of Mechanical Engineering University of Waterloo M.M. Yovanovich November 10, 1997
The attached material is a summary of some of the important results for forced and natural convection heat transfer from isothermal or iso ux surfaces. Correlation equations for local and area-average heat transfer for external and internal ow are given. Many empirical and analytic correlation equations have been developed for the local and area-average values of the Nusselt number for limited ranges of the forced and buoyancyinduced ow parameters: Reynolds, Peclet, Grashof and Rayleigh numbers; for laminar or turbulent ows; and for various uids which are characterized by the Prandtl number. One should consult the course text for the de nitions of the various dependent and independent parameters and the basis for the uid properties evaluation for the particular correlation equation. This summary does not cover the numerous forced and buoyancy-induced internal ows through and within complex con gurations. One should consult the course text or the several handbooks which deal with these topics.
1. Laminar and Turbulent Forced External Flow
De nitions of Local and Area-average Values ,k @T x; y = 0 Z 1 L hx = T ,@y , h = L hx dx = 2hx = L T
w
1
0
x Nux = hkx ,
Rex = U1 x ReL = U1 L
1
h NuL = kL ,
Flat Plate, Laminar Boundary Layer Flow
Correlation
= 5xRe,1=2 x Cf;x = 0:664Re,1=2 x C f;x = 1:328Re,1=2 x = Pr,1=3 Nux = 0:3387Pr1=3 1=2
Limits
100 100 100 100 100
Conditions
500; 000 500; 000 500; 000 500; 000 500; 000 Local Local Area-Average Local Local, UWT, Pr ! 1
Rex Nux 1=2 1 Rex=2 = 0:564Pr Nu=x2 = h 0:3387Pr1=3 i1=4 1 Rex 1 + 0:0468=Pr2=3 Nux = 0:4637Pr1=3 1 Rex=2 Nu=x2 = 0:886Pr1=2 Re1
x
Rex Rex Rex Rex Rex
100 Rex 500; 000 Local, UWT, Pr ! 0 100 Rex 500; 000 Local, UWT, 0 Pr 1 100 Rex 500; 000 Local, UWF, Pr ! 1 100 Rex 500; 000 Local, UWF, Pr ! 0
Nu=x2 = h 0:4637Pr1=3 i1=4 100 Rex 500; 000 Local, UWF, 0 Pr 1 1 Rex 1 + 0:0205=Pr2=3 Nu=2 = h 0:6774Pr1=3 i 100 ReL 500; 000 Average, UWT, 0 Pr 1 L 1 1=4 ReL 1 + 0:0468=Pr2=3 NuL = h 0:9274Pr1=3 i 100 ReL 500; 000 Average, UWF, 0 Pr 1 1 ReL=2 1 + 0:0205=Pr2=3 1=4
Flat Plate, Turbulent Boundary Layer Flow
Correlation
= 0:37xRe,1=5 x Cf;x = 0:0592Re,1=5 x ,1=5 , 1742Re,1 C f;L = 0:074ReL L = Pr,1=3 100 Rex 500; 000 Nux = 0:0296Re4=5 Pr1=3 100 Rex 500; 000 x
1=3 NuL = 0:037Re4=5 , 871 Pr 100 Rex 500; 000 L 2
Limits 5 105 Rex 108 5 105 Rex 108 Rex;c = 5 105
Conditions
Local Local Mixed-Average Local Local, UWT, 0:6 Pr 60 Average, UWT, 0:6 Pr 60
Cross Flow Over Circular Cylinders
" 4=5 1=2 1=3 ReD 5=8 ? + " 0:6ReD Pr NuD = SD 2=3 1=4 1+ 282; 000
? SD
Correlation
Limits
100 ReD 107 0 L=D 8
Conditions
Average; UWT; 0 Pr 1
0 1 4 ? A 1 SD = p @ q 1 1 + 0:5D=L ln2L=D
Flow Over Spheres
L=D 0:76 4 = 1 + 0::869L=D 05+
1+ 0:4 Pr
!
ReD ! 0 ReD ! 0;
Asymptote
L=D 8
Correlation
24 CD = 0:4 + ReD +
Limits
Conditions
6 q 0 ReD 2 105 Total Drag, 10 1 + ReD 4=5 = " Average; UWT; 0:6Re1=2Pr13 1+ ReD 5=8 NuD =2+ " D 2=3 1=4 100 ReD 107 282; 000 0 Pr 1 1 + 0:4 Pr
2. External Flow Over Isothermal Oblate and Prolate Spheroids
The following universal correlation equation:
0 p
2 !1=2 3 P 1=2 p NupA = Nu A + 40:15 p RepA + 0:35Re0:5665 Pr1=3 A A
was developed by Yovanovich 1988 from two accurate correlation equations proposed by Yuge 1960 for air cooling of isothermal spheres, and the correlation equations developed by several researchers for convection heat and mass transfer from isothermal oblate and prolate spheroids. In the above correlation equation the Nusselt and Reynolds numbers are both based p p on the length scale L = A. The di usive limit Nu0 A corresponding to Re ! 0 is the ? dimensionless shape factor SpA . Yovanovich 1988 blended the two Yuge equations and introduced the parameter P= A which accounts for the blockage of the body as the uid ows around it. Also the Yuge 3
p
correlation equations which were developed for air were extended to account for large Prandtl number uids, i.e. Pr 0:7. The correlation equation is valid for the wide Reynolds number range: 0 RepA 2 105 . The general correlation equation is in very good agreement with numerous analytical and experimental correlation equations over various ranges of the Reynolds number for Pr = 0:7. It is also in good agreement with the empirical correlation equation of Pasternak and Gauvin 1960 which was developed from 20 di erent convex body shapes to account for both body shape and orientation. The body length scale which they proposed was based on the ratio of the total surface area of the body divided by the maximum projected area of the body perpendicular to the air ow. They achieved good correlation of their heat and mass transfer data with a single power-law equation which was converted to the body scale p length L = A p NupA = 0:914Re0:514Pr1=3 886 RepA 8860 A This equation correlated data for spheres, nite circular cylinders with axes parallel and perpendicular to the ow, prisms, cubes in various orientations, and hemispheres positioned with the at section at the rear. The turbulence intensity was reported to be in the range: 9 to 10 in all their experiments. The single equation correlated all data with a deviation of only 15 in the speci ed range. The general correlation equation agrees with the Pasternak-Gauvin correlation equation within the given Reynolds number range to within 15 . Therefore, the general equation of Yovanovich can be used for arbitrary convex isothermal bodies over a much wider range of the Reynolds number.
3. Laminar Forced Internal Flow
De nitions and Notation Reynolds number: Laminar Flow: Hydraulic Diameter: Dimensionless axial distance: Local, Isothermal Wall, Nusselt number: Mean-value, Isothermal Wall, Nusselt number: Local, Iso ux Wall, Nusselt number: Mean-value, Iso ux Wall, Nusselt number: 4
ReD = UDh ReD 2300 A Dh = 4 P = Cross Section Area Wetted Perimeter x 1 ?= x = x Dh Pe Dh RePr = Gz 4 Nux;UWT = Tw qw xx Dh , Tm k w Num;UWT = Tw ,qTmx Dh k Nux;UWF = Tw x qw Tmx Dh k , Num;UWF = Twx qw Tmx Dh k ,
Local Nusselt Number for Thermally Developing Flow Churchill and Ozoe 1973 propose the following correlation equations for the local Nusselt number for the developing thermal eld for the UWT and UWF cases:
" Nux;UWT + 1:7 = 1 + 388 x?,8=9 3=8 5:357 " Nux;UWF + 1 = 1 + 220 x?,10=9 3=10 5:364
5 5
They developed these expressions based on asymptotic solutions valid for small and large values of x?. Area-Average Nusselt Number for Fully Developed Hydraulic, Thermally Developing Flow following The approximations of Shah 1975 for fully developed hydraulic ow and thermally developing in an isothermal UWT or an iso ux UWF circular pipe are based on the analytic solutions of the Graetz-type problems. Expressions for area-mean Num;UWT ; Num;UWF versus the local dimensionless position x? = x=Dh =RePr are given below. The approximations are quite accurate over the entire range: 0:005 x? 1: The maximum di erence with respect to accurate analytic results is less than 4:4. For very small values x? 0:005 the approximate expressions approach the Leveque asymptotes which were obtained by the method of similarity transformation. For large values x? 0:25, the approximations go to the fully-developed hydraulic and thermal solutions: NuUWT = 3:656 and NuUWF = 4:354 which were obtained by the method of separation of variables which leads to a di erential equation of the Sturm-Liouville type. The solution is presented as an in nite series expansion of eigenfunctions and corresponding eigenvalues.
8 1:615 , 0:2; 0:005 x? 0:03 x?1=3 Num;UWT = : 3:656 + 0:0499 ; x? 0:03 x? 8 1:953 ; x? 0:03 x?1=3 Num;UWF = : 4:354 + 0:0722 ; x? 0:03 x?
The circular cylinder results may be used to nd approximate values for isothermal and iso ux tubes having other cross-sections e.g. square or triangular pipes by the use of the hydraulic diameter in the Nusselt and Reynolds numbers.
5
4. Laminar and Turbulent Natural External Flow
De nitions of Local and Area-average Values ,k @T x; y = 0 @y y hx = 3 h = 4 hx = L T , T ,
w
1
x Nux = hkx ,
h NuL = kL
3 Rax = g Tw , T1 x , 3 RaL = g Tw , T1 L ,
y Grx =
g Tw , T1 x3 , 2 3 y GrL = g Tw , T1 L , 2
Rax = GrxPr RaL = GrLPr
y For UWF cases, use the midpoint
temperature di erence: Twx = L=2 , T1
Flat Plate, Buoyancy-Induced Laminar Boundary Layer Flow
Nux Ra1=4 = 0:5027 x Nu=4 = 0:6004Pr1=4 x Ra1 x Nux 0:5027 Ra1=4 = h1 + 0:492=Pr9=16 i4=9 x Nux Ra1=4 = 0:5627 x Nu=4 = 0:6922Pr1=4 x Ra1
x
104 Grx 109 Local, UWT, Pr ! 1 104 Grx 109 Local, UWT, Pr ! 0 104 Grx 109 Local, UWT, 0 Pr 1
? 104 Grx 109 Local, UWF, Pr ! 1 ? 104 Grx 109 Local, UWF, Pr ! 0
Nux 0:5627 4 ? 9 Ra1=4 = h1 + 0:437=Pr9=16 i4=9 10 Grx 10 Local, UWF, 0 Pr 1 x NuL = h 0:6703 i 104 GrL 109 Average, UWT, 0 Pr 1 1=4 9=16 4=9 RaL 1 + 0:492=Pr NuL = 0:7503 ? 104 GrL 109 Average, UWF, 0 Pr 1 Ra1=4 h1 + 0:437=Pr9=16 i4=9 L
6
Flat Plate, Buoyancy-Induced Turbulent Boundary Layer Flow
NuL = h
92 0:387RaL i = 10,1 RaL 1012 NuL = 0:825 + h 8=27 : 1 + 0:492=Pr9=16 ;
1=6
8
0:150Ra1=3 i L 16=27 1 + 0:492=Pr9=16
109
GrL 1012
Average; UWT; 0 Pr 1 Average; UWT; 0 Pr 1
Long Horizontal Isothermal Circular Cylinders, Laminar and Turbulent Flow
2 NuD = 40:60 +
0:387Ra1=6 D 5 1 + 0:559=Pr9=16 8=27
32
0 Pr 1;
10,5
RaD 1012
Finite Horizontal Isothermal Circular Cylinders, Laminar Flow
? NuD = SD +
1 0:518RaD=4 1 + 0:559=Pr9=16 4=9
0 Pr 1;
0 RaD 109
Isothermal Spheres, Laminar Flow
NuD = 2 +
0:589Ra1=4 D ; 1 + 0:469=Pr9=16 4=9
0 Pr 1;
0 RaD
1011
5. General Correlation Equation for Arbitrary Isothermal Convex Bodies
NuL = Nu0 + F PrGL Ra1=4; L L
0 Pr 1 0 RaL 1011
where the characteristic body length is L = A and A is the total active or wetted surface area. The universal Prandtl number function valid for all isothermal convex bodies is given by: 0:670 F Pr = 1 + 0:5=Pr9=16 4=9 which for air Pr = 0:71 has the value F Pr = 0:71 = 0:513. The di usive limit Nu0 or L p ? shape factor SL with L = A is a weak function of body shape and its aspect ratio. For 7
p
? example, its range is 3:20 SpA 7:55 for a solid circular cylinder whose length-to-diameter ratio varies from 0 a circular disk to 100 very long cylinder. For long axisymmetric bodies e.g. circular cylinder and long square cuboid the shape factor can be accurately q approximated by SpA = 4 L=D= ln2L=D where D is the diameter of the circular cylinder and it is equal to the geometric-mean of the diameters of the inscribed and circumscribed circular cylinders respectively, and L is the le...