US 3847212 A
Improved externally finned metal heat transfer tube has multiple-start helical ridging on its inner surface which conforms to a range of values of a disclosed equation relating the size and shape of the ridging to its pitch and to the inner diameter of the tube. The inside tube wall profile, in axial section, is defined by a straight line interrupted by the internal ridges. The ridge tips are connected to the straight line segments of the profile by convex and concave portions which meet at a point of common tangency. The improved tube provides especially good results in systems, such as submerged-tube, refrigerated coolers, wherein the film resistance of the intube fluid would represent a substantial portion of the total thermal resistance if the improved internal surface were not provided. A method of designing a tube for a particular coefficient of internal heat transfer is also disclosed.
Description (OCR text may contain errors)
United States Patent [1 1 Withers, Jr. et a1.
1 1 Nov. 12, 1974 HEAT TRANSFER TUBE HAVING MULTIPLE INTERNAL RIDGES 175] Inventors: James G. Withers, Jr., pearborn,
Mich.; Klaus K. Rieger, Decatur,
1731 Assignec: Universal Oil Products Company, Des Plaines, Ill.
 Filed: July 5, I973  Appl. No.: 376,507
FOREIGN PATENTS OR APPLICATIONS Italy 165/179 Primary Examiner-Charles J. Myhre Attorney, Agent, or Firm-James R. Hoatson. Jr.; Barry L. Clark; William H. Page. II
57 ABSTRACT Improved externally finned metal heat transfer tube has multiple-start helical ridging on its inner surface which conforms to a range of values of a disclosed equation relating the size and shape of thc ridging to its pitch and to the inner diameter of the tube. The inside tube wall profile, in axial section, is defined by a straight line interrupted by the internal ridges. The ridge tips are connected to the straight line segments of the profile by convex and concave portions which meet at a point of common tangency. The improved tube provides especially good results in systems. such as submerged-tube, refrigerated coolers, wherein the film resistance of the intube fluid would represent a substantial portion of the total thermal resistance if the improved internal surface were not provided. A method of designing a tube for a particular coefficient of internal heat transfer is also disclosed 5 Claims, 12 Drawing Figures PAIENTED NOV 2 I974 Figure l/ saw a or 2 Figure /0 Figure l2 30! Prior An) Friction Fae/or, f, ((D NR5 35, 000
BACKGROUND OF THE INVENTION This invention relates to metal tubing for heat transfer purposes and particularly to such tubing wherein a special configuration is given to the inner surface to improve its performance.
A explained at some length in U.S. Pats. Nos. 3,217,799, 3,463,997, 3,481,394, 3,559,437 and in copending applications Ser. No. 674,611 filed Oct. 11, 1967, now abandoned, Ser. No. 224,095 filed Feb. 2, 1972, now U.S. Pat. No. 3,768,291 granted Oct. 30, 1973, and Ser. No. 232,571 filed Mar. 7, 1972, now U.S. Pat. No. 3,779,312 granted Dec. 18, 1973, all assigned to a common assignee, and incorporated by reference herein, substantialimprovements in heat transfer over plain tubing can be achieved by providing special configurations on the inner and/or outer surfaces of tubes. Regarding the outer surface, the state of the finning art is such that external finned tubes can be produced having greatly extended surface and other beneficial properties which enable major improvements-in the outside film coefficient of heat transfer. Thus it follows that for certain heat transfer systems, an additional improvement is logically to be sought by modification of the internal surface of external finned tube. One approach is to provide helical or annular ridging of the inner surface to promote'fluid turbulence, as
shown by the aforementioned patents and applications and by U.S. Pats. Nos. 2,432,308, 2,913,009, 3,088,494, and 3,612,175.
In order to enable comparisons of the tubeside heat transfer performance of different tubes having different internal configurations, the following specialized form of the Sieder-Tate equation may be used:
where h; inside coefficientof heat transfer, Btu/hr-sq ftd,- tube inside diameter, ft.
k thermal conductivity of intube fluid at bulk fluid temperature, Btu/hr-sq ft-F/ft.
C, inside heat transfer coefficient constant, dimensionless.
G mass velocity, lb/hr-sq ft.
u viscosity of intube fluid at average bulk fluid temperature, lb/ft-hr.
u viscosity ofintube'fluid at average wall temperature, lb/ft-hr.
This equation is applicable to singlephase fluid flowing turbulently inside of plain or internal ridged tubes, providedthe correct value of C is utilized. The dimensionless insideheat transfer coefficient constant, 'C,-", for a particular tube can be determined experimentally by means of a modified Wilson-plot technique as described at pages 19-30 of Industrial Engineering Chemistry Process Design and Development, Vol. 10, No. 1, 1971, in an article entitled Steam Condensing On Vertical Rows of Horizontal Corrugated and Plain Tubes by .l. G. Withers and E."-H.Young. Although it is generally desirable to design a tube so that C, is a maximum, there are many instances where one might desire that C be of a lower but predetermined value. This latter situation could prevail in the case where allowable pressure drop is severely restricted. In case where the designer is constrained in selection of internal configuration elements by limited metal working capability or the necessity to conserve materials, it is important to obtain, not the ultimate-maximum C,, but the maximum possible C within the constraints existing. Thus the ability to predict heat transfer performance as a function of geometrical elements is highly desirable.
SUMMARY It is an object of this invention to provide a metal heat transfer tube having an internal configurtion which will provide improved heat transfer performance.
' It is another object of this invention to provide a means for enabling one to predict the heat transfer performance ofthe inside surface of a tube.
These and other objects are attained by the metal heat transfer tube of the present invention which includes multiple start helical ridging integrally formed on its inner cylindrical surface. The function of the ridges is to perturb the fluid flowing in the tube so that the fluid cannot build up boundary layers along the tube wall which would inhibit the transfer of heatbetween the'fluid and the tube'wall. Although the prior art has intimated some of the significant geometrical considerations which affect heat transfer performance, it has failed to relate the geometrical characteristics in a way that the response of the intube heat-transfeb coefficient to variations in geometrical configuration will be predictable. Rodgers Pat. No. 3,217,799 singles out the ratio of the axial spacing dimension between adjacent ridges to the ridge height dimension as the significant parameter. Although this relationship is an important consideration, it is not sufficiently specific to narrow down the most favorable tube design in such a manner that the intube heat transfer performance could be predicted or maximized.
A co-pending patent application, Ser. No. 232,571, filed Mar. 7, 1972 and assigned to a common assignee, discloses a correlation between C, and a geometrial-parameter called the severity factor. This parameter,-, is a dimensionless grouping which involves ridge height (e), pitch (p) and inside diameter (d,-), in such a way that:
lt wasbrought out in application Ser. No. 232,571 that there is a maximum possible C,- for tubes having singletube.
Although the correlation of C, vs "41 was developed for tubes having single-helix internal ridging, the correlation is also of interest in the design of multiple start internally ridged tubes wherein it has been found that it is possible to design tubes having even higher heat transfer coefficients for a given degree of either severity or of pressure drop than is possible with single helix internal ridging. The present invention incorporates the severity factor, 4), as a framework element. It also advances the art by clarifying the role of ridge shape, and ridge and tube dimensions, in the improvement of intube heat transfer performance in a multiple start internally ridged tube. The instant invention relates specifically to internal tube walls which are so shaped as to define. in a longitudinal-section profile, intermediateflat portions between the internal ridges and convex and concave connecting portions between the ridge tips and the intermediate-flat portions.
The aforementioned Pat. No. 3,481,394 discloses several tube embodiments wherein the tube has a plurality of external fins and a single internal rib or ridge. Although the pitch of the single internal rib of Pat. No. 3,481,394 is of course greater than the pitch of the plural external fins, the rib has the same lead as the fins since it is formed incident to the deepening of the groove defining two adjacent external fins. The reduction in the outside root diameter of the tube adjacent the internal ridge results in the prior art tube being less stiff (and therefore more sensitive to vibration) than the tubing of the present invention wherein the lead of the plurality of helical ridges is greater than the lead of the fins. The improved tube also provides additional latitude in design since the size, shape, number of starts and lead angle of the internal ridges can be chosen for their effect on heat transfer and pressure drop characteristics of the tube rather than be fixed relative to the external fins. The tubing of the current application presents a uniform wall thickness under the fins except for thickened portions at the ridges; whereas the tube of Pat. No. 3,481,394, when manufactured by at least one method (U.S. Pat. No. 3,559,437), can result in thinning of the tube wall in the vicinity of the internal ridges. Thus, for a given strength, the improved tube should require less metal.
Afte designing and testing a number of tubes having a variety of multiple start ridge profiles and dimensional aspects it has been possible to develop a mathematical model. or equation which permits the internal heat transfer coefficient constant, C,-, to be predicted quite accurately. Conversely, where a specific value of the constant, C is desired, it is possible to predict certainparameters of the tube, such as the base width of the ridge, which will provide the desired constant. Within the range of applicability of the equation, the heat transfer performance appears to be enhanced as ridge height is increased and also as ridge width is decreased. However, there are many factors which affect the ridge geometry. For instance, metal working characteristics may limit the extent to which tube metal may be moved radially inwardly, and therefore limit the maximum ridge height. When narrow ridges are desired, there may be a problem in fabricating the appropriate metal working tools to shape the ridges. On the other hand, ridges of greater width are easier to form to a greater depth than narrow ridges and also provide greater resistance to wear if contacted with an erosive fluid; however these benefits may obtain at the cost of excess tube material or the loss of some external heat transfer surface.
The aforementioned equation which has been developed for predicting C, can be written as follows:
c, =0.0264 22.1 (l-b/p) (er/y) [Eqn. 3]
where d) the severity factor (Equation 2) b ridge base width (measured in an axial direction) p pitch measured between corresponding points on adjacent ridges in an axial direction e ridge height y ridge cap height measured radially from ridge peak to the point of inflection of the ridge profile boundary. The equation is applicable to internally helically ridged tubes having cylindrical internal wall portions between adjacent internal ridges. Constraints on application of the equation to provide superior tubes are as follows: b/p should be between 0.l0 and 0.20, 4) should be less than 0.25 X 10*; and e/y should be between 1.50 and 500.
BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a fragmentary axial sectional view of a tube made in accordance with the invention;
FIG. 2 is a graph showing the inside heat transfer factor C,- as a function of the severity parameter q5 for certain multiple-helix internally ridged tubes having an intermediate fiat inner-wall profile;
FIG. 3 is a graph showing the correlation of experimental versus predicted values of the inside heat transfer factor C for several multiple-helix internally ridged tubes having an intermediate flat inner wall profile;
FIGS. 4-7 show several alternative ridge profiles, which may be used in a tube of a given severity, with the profiles shown in cross-section in a plan normal to the ridge;
FIG. 8 is a view identical to FIG. 5 except that the ridge is widened and the cross-section is taken in an axial plane;
FIGS. 9-11 are graphs illustrating the effect on the inside heat transfer factor, C caused by changes in the severity factor, (I), and in the ridge dimension ratios blp and e/y, as calculated from the equation 0, 0.0264 22.1 a Hi/in [e/yl;
DESCRIPTION OF THE PREFERRED EMBODIMENT FIG. 1 is an axial section through a tube indicated generally at 10 made in accordance with the invention. The tube 10 includes a plurality of external fins I2, 14 and plurality of multiple start internal ridges 16, 18. The external fins 12, 14 and the internal ridges 16, 18 are preferably integrally and simultaneously formed from the wall portion 20 of the tube on a grooved mandrel (not shown). The inner wall 22 of the tube is of cylindrical cross-section except where it is interrupted by the internal ridges l6, 18. The width of the ridges is b, the ridge pitch is p, and the ridge helix angle is 0, with 6 being measured from a perpendicular to the axis. Specific dimensions for an actual tube having a crosssection as shown in FIG. I are: e 0.0178 inch; p 0.333 inch;d,-=0.820 inch; 4 =0.l66 X 10 b=0.064
inch; y 0.0089 inch; b/p 0.2; e/y 2.00; C, (predicted) 0.052; C, (actual) 0.052; 39; fin starts 3; ridge starts 6; and material is copper.
FIG. 2 is a graph plotting the severity factor (1) against the inside heat transfer factor C, for a number of tubes. The lower curve 26 represents the performance line for single-helix corrugated tubes having a curvilinear inner wall profile as disclosed in copending application Ser. No. 232.571 filed Mar. 7, 1972. The upper curve 28 represents a performance line for multiple-helix internal-ridged tubes having an intermediate flat inner-wall profile in accordance with the present invention. The curves intersect at a value of C, 0.0264, the value for plain tube, when 0. The lines 26, 28 generally illustrate the heat transfer characteristics of the tubes and their degree of improvement as compared to plain tube. The relationship between heat transfer and pressure drop is given in FIG. 12 in terms of C,- and friction factor, f. Pressure drop is directly proportional to friction factor when one compares tubes of a given diameter at the same Reynolds number. The improvement of C, at a given degree of pressure drop for the tubes of this invention (shown by line 29) as compared with prior art tubes (shown by line 30) is evident in FIG. 12. The prior art tubes represented by line 30 are of the type disclosed in copending application Ser. No. 232,571, filed Mar. 7, 1972, and comprise single start internally ridged tubes having a curvilinear inner wall profile.
FIG. 3 is a graph plotting the experimental inside heat transfer factor C obtained experimentally for a number of multiple start internal ridgedtubes having an intermediate-flat inner wall profile and varying ridge shapes, against the predicted heat transfer factor C,- predicted by the aforementioned Equation 3. The graph shows a good correlation between the predicted and experimental values as evidenced by the closeness of the various test points on the graph to the 45 line 32 which represents a perfect correlation.
As can be seen from Equations 2 and 3, the height, width, shape and pitch of the internal ridges 16, 18 each play an important part in determining the heat transfer and pressure drop characteristics of a particular tube design. FIGS. 47 illustrate four different ridge profiles, as viewed perpendicular to the ridge path, which have a common ridge width, b(cos 0), and a common ridge height, e, which in this illustration is taken equal to b(cos 0)]2. Each of the ridge profiles shown in FIGS. 4-7 has its side boundary determined by a concave curved line 36 and a convex curved line 38. The lines 36, 38 meet at a point of inflection 40. The ridges 44 defined in sectional edge profile by the combination of lines 36, 38 include a ridge cap portion 46 which has a height y equal to the radial distance between ridge peak 48 and the point of inflection 40. The ridges 44 also include a ridge base portion 50 which has a width of b(cos 0) and a height of (e-y). The various ridge profiles shown in FIGS. 4-7 differ in that the y dimension varies in each figure so that the four figures have values of e/y equal to 1.50, 2.00, 3.00, and 4.00, respectively. The FIG. 8 ridge profile is identical to the FIG. profile except that the ridge cap and base are wider by the dimension, f (co's 0), which corresponds to the width of the flat ridge cap peak 48. Since FIG. 8 is a view of the ridge cross section in an axial plane, the circular arcs of FIG. 5 become elliptical arcs, being elongated along withf(c0s 6) by the factor, llcos 0. Within a tube having given values of severity factor and ridge pitch, the wider base of the FIG. 8 ridge profile provides a lower coefficient of heat transfer than theprofile of FIG. 5, for example, but has advantages in fabrication. It is easier to prepare a mandrel than a narrow grooved mandrel, and it is easier to move the metal of the tube during forming so as to form a wide ridge than a narrow one. In the case of an erovise or corrosive internal fluid, the wider ridge would afford longer wear. It is quite difficult to grind grooves in a mandrel to provide the curved profiles of FIGS. 4-8. It has been found that satisfactory results can be obtained when the curved profiles 36, 38 of FIG. 4, for example, are approximated by straight lines such as the dotted lines 36',36" and 38',38. The advantage of using a straight line approximation to a curve is that the very thin grinding wheels used to form the grooves in the mandrel can be provided with a straight edge profile which is easier to achieve and maintain than a curved profile.
FIG. 9 is a graph which illustrates the relationship between the severity factor (I) and theaforementioned heat transfer coefficient constant, C,, when Equation 3 is plotted for a value of b/p of 0.15 with e/y as parameter. The graph shows that for a given value-of the severity factor (b, the value of C, increases as the value of e/y is increased from 1.5, to 5.0. Lines 51, 52, 53, and 55 illustrate the increase for values of e/y of 1.5, 2, 3, and 5 respectively. By means of a graph such as the one shown in FIG. 9, one can readilydetermine the ridge shape to use to achieve a given degree of heat transfer for a particular degree of severity and for a particular value of b/p. For example, referring to FIG. 9, for a severity factor of 0.15 X 10 a value of C, 0.067 should be obtained with a ridge shape having an e/y of 3.0 as shown in FIG. 7. The wider-capped, but easier to fabricate, ridge shape of FIG. 4 which hasan e/y of 1.5 would provide a value of C,- 0.059 for a severity factor of 0.15 X 10 FIG. 10 is a graph similar to FIG. 9 but different in that various values of b/p are plotted against C, for Equation 3. The graph, which is plotted with a constant value of severity factor, 4), of 0.1 X 10 indicates that the value of C, decreases for any given value of e/y, such as values of e/y 2 and 5 represented by lines 62 and 65, as the value of b/p increases. The graph thus indicates that heat transfer efficiency is improved by decreasing the width b of the ridge relative to the pitch,
FIG. 11 is a graph similar to FIGS. 9 and 10 but differing in that various values of e/y are plotted against C,- for Equation 3. The graph, which is plotted for a fixed value of the severity factor, d), of 0.1 X 10 illustrates that for a given value of b/p, the value of the heat transfer coefficient C, increases as e/y increases. Lines 71 and 72 represent values of b/p of 0.1 and 0.2 respectively.
As can be seen from Equation 3 and the graphs of FIGS. 9, 10 and 11, it is possible to design an externally finned,multiple internal ridged tube with intermediate flat walls between its ridges which is superior to prior art tubes and which can have a particular value of the heat transfer coefficient C,. For example, assuming one has a tube having an inside diameter of 0.8 inch and that a particular heat transfer situation requires that the value of C, be equal to 0.056, the following steps might be utilized to determine the width, b, of the ridge. (a)
Assume, based on known metalworking limitations, that the maximum producable ridge height e is 0.0175 inch. (b) Assume that the least pitch, ,0, producable on a 6-start ridge configuration is 0.3 inch. The ridge helix angle will be implicit when the diameter, ridge height, number of starts, and pitch are known. (c) Compute the value of d) from Equation 2. e lpd 0.128 X 10 2. (d) Select a ridge shape, such as the e/y 2 shape shown in FIG. 5, which has good formability and wear resistance. (e) Solve Equation 3.
but p 0.3 therefore, b 0.167 (0.3) 0.050 inch We claim as our invention:
1. A metal tube having an improved heat transfer rate to or from an internally flowing fluid, characterized in that said tube has at least one integral external helical tin of a predetermined pitch distance and lead angle and a plurality of integral multiple start internal helical ridges extending radially inwardly from the inner wall of the tube, said plurality of internal helical ridges having a lead angle of less than 60 (as measured from a perpendicular to the tube axis} and a pitch distance which is greater than the pitch distance of said at least one external fin but wherein the respective lead angles of said at least one external fin and said plurality of internal ridges are different from each other in magnitude and/or sense, said inner wall of said tube being so shaped as to define, in a longitudinal sectioned profile, an intermediate flat portion between adjacent internal ridges, with said ridges having a ridge cross-sectional profile which includes a pair of side boundaries connecting the flat inner wall portions of the tube intermediate adjacent pairs of ridges and the tips of the ridges, said side boundaries being comprised of concave and convex portions whichconnect at a point of inflection radially outward from the tip and at a distance from the tip which is less than the height of the ridge.
2. A metal tube in accordance with claim 1 wherein the ratio of the width, b, of a ridge, as measured in an axial direction, to it pitch, p, is between 0. i0 and 0.20; the ratio of the height, e, of a ridge to the height from its tip to the point of inflection, y, is between 1.50 and 5.0; and the value of a severity factor, ii, is less than 0.0025, where d: e /pd and where d; is the maximum projected internal diameter of the tube.
3. A metal tube in accordance with claim 2 characterized in that the inside heat transfer coefficient con- 8 stant, C is approximately equal to 0.0264 (22.1)()(lb/p)(e/y)" and wherein: (ii is within the range 0.00057 0.0025; 2 is within the range 0.0125 0.075; p is within the range 0.25 0.70; d. is within the range 0.20 3.00; b is within the range 0.02 0. l 5; and, y is within the range 0.0065 0.05.
4. A metal tube having an improved heat transfer rate to or from an internally flowing fluid, characterized in that said tube has annular integral external fins of a predetermined pitch distance and a plurality of integral multiple start internal helical ridges extending radially inwardly from the inner wall of the tube, said plurality of internal helical ridges having a pitch distance which is greater than the pitch distance of said external fins but wherein the lead angle of said plurality of internal ridges is less than 60 (as measured from a perpendicular to the tube axis), said inner wall of said tube being so shaped as to define, in a longitudinal sectioned profile, an intermediate flat portion between adjacent internal ridges, with said ridges having a ridge crosssectional profile which includes a pair of side boundaries connecting the flat inner wall portions of the tube intermediate adjacent pairs of ridges and the tips of the ridges, said side boundaries being comprised of concave and convex portions which connect at a point of inflection radially outward from the tip and at a distance from the tip which is less than the height of the ridge.
5. A metal tube having an improved heat transfer rate to or from an internally flowing fluid, characterized in that said tube has a plurality of integral radial external fins and a plurality of integral multiple start internal helical ridges extending radially inwardly from the inner wall of the tube, and wherein the lead angle of said plurality of internal ridges is less than 60, as measured from a perpendicular to the tube axis, but wherein the respective lead angle of said plurality of radial external fins and said plurality of internal ridges are different from each other in magnitude and/or sense, said inner wall of said tube being so shaped as to define, in a longitudinal sectioned profile, an intermediate flat portion between adjacent internal ridges, with said ridges having a ridge cross-sectional profile which includes a pair of said boundaries connecting the flat inner wall portions of the tube intermediate adjacent pairs of ridges and the tips of the ridges, said side boundaries being comprised of concave and convex portions which con nect at a point of inflection radially outward from the tip and at a distance from the tip which is less than the height of the ridge.