Internally Ridged Heat Transfer Tube

Abstract

Metal heat transfer tube has a single start helical ridge on its inner surface which conforms to a range of values of a disclosed equation relating the height of the ridge to its pitch and to the inner diameter of the tube. A method of designing a tube for maximum performance is also disclosed. The improved tube provides especially good results in systems, such as steam condensation systems, wherein a single phase fluid is carried by the tube.

Description

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.

As explained at some length in British Pat. No. 1,230,196, U.S. Pat. No. 3,612,175, and especially in U.S. Pat. No. 3,217,799, substantial improvements in heat transfer over plain tubing can be achieved by providing special configurations on the inner and/or outer surfaces of tubes. Where the tubing is to be used in a steam condensation apparatus where a single phase fluid such as water is on the inside of the tube, it has been found that the major modification which can be made to a plain tube to increase its overall heat transfer efficiency is one wherein the interior surface is modified. The objective of the surface modification is to increase heat transfer by corrugating the inner surface to promote fluid turbulence without, at the same time, providing such an increase in the resistance of flow through the tube as to nullify the overall efficiency thereof.

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:

h.sub.i Di/k =Ci (D.sub.i G/.mu.).sup.0.8 (C.sub.p .mu./k ).sup.1/3 (.mu./.mu..omega.).sup. 0.14

Where

h.sub.i = inside coefficient of heat transfer, Btu/hr-sq ft-.degree. F

d.sub.i = tube inside diameter, ft

k = water thermal conductivity at bulk water temperature, Btu/hr-sq ft-.degree. F/ft

C.sub.i = inside heat transfer coefficient constant, dimensionless

G = mass flow rate, lb/hr-sq ft

C.sub.p = specific heat, Btu/lb-.degree.F

.mu. = water viscosity at average bulk water temperature, lb/ft-hr

.mu..omega.= water viscosity at average wall temperature, lb/ft-hr

The dimensionless inside heat transfer coefficient constant C.sub.i for the particular tube can be determined by means of a modified Wilson plot technique as described at pages 19-30 of Industrial Engineering Chemistry Process Design & Development, Vol. 10, No. 1, 1971, in an article entitled "Steam Condensing On Vertical Rows Of Horizontal Corrugated And Plain Tubes" by J.G. Withers and E.H. Young. Although it is generally desirable to design a tube so that C.sub.i is a maximum, there are many instances where one might desire that C.sub.i be of a lower but predetermined value. This latter situation could prevail in the case where allowable pressure drop is severely restricted. Another desirable design feature is to have the corrugated section of the tube have a diameter equal to the diameter of the tube ends since a tube will exhibit less friction loss and pressure drop if its corrugated portion has a diameter as large as the tube ends rather than a smaller one.

In view of the many variables that affect the heat transfer and pressure drop properties of a tube it would be highly desirable to be able to predict the performance of a particular tube configuration and to be able to predict the configuration which will provide the maximum performance.

SUMMARY

It is an object of this invention to provide a single helix metal heat transfer tube having an internal configuration which will provide maximum heat transfer performance.

It is another object of this invention to provide a means for enabling one to predict the heat transfer performance of the inside surface of a tube.

These and other objects are attained by the metal heat transfer tube of the present invention which includes a single start helical ridge on its inner surface. The function of the ridge is to perturb the liquid flowing in the tube so that the liquid can not build up boundary layers along the tube wall which would inhibit the transfer of heat from the fluid to 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 heat transfer coefficient C.sub.i to variations in geometrical considerations will be predictable. Rodgers U.S. 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 tube performance could be predicted or maximized.

After thoroughly studying data from many tubes we have found that there is a geometrical parameter that correlates well with C.sub.i. This parameter is a dimensionless severity parameter, .phi., which involves ridge height (e), pitch (p) and inside diameter (d.sub.i), in such a way that:

.phi.= e.sup.2 /p d.sub.i

Data from many different single start helical ridged tubes have established a rather remarkable correlation between C.sub.i and .phi. and a plotting of the data indicates that there is a maximum possible C.sub.i and that this maximum value occurs at a specific value of .phi. rather than over a range of values of .phi.. Since the maximum value of C.sub.i has been found to occur when .phi. = 0.365 .times. 10.sup.-.sup.2, it is possible to tailor the tube configuration so that it can provide any desired value of C.sub.i up to the maximum and down to that for plain tube. Although the C.sub.i vs .phi. correlation has been found to hold true for the vast majority of tubes studied, it has been noted that in a few of the tubes the ridge cap dimensions of the helical ridge have been found to be critical in that the measured value of C.sub.i for these few tubes did not correspond to the value predicted by the C.sub.i vs .phi. correlation curve. Fortunately, this situation can be resolved by means of a reinforcing criterion involving a parameter .chi., which is defined as:

.chi.= ety/d.sub.i where t and y are the width and height of the ridge cap, e is the ridge height and d.sub.i is the inside diameter of the tube. A plot of C.sub.i vs .chi. has been made which indicates that the maximum value of C.sub.i corresponds to the extreme maximum value of .chi.. Although the .chi. correlation is not as uniform as the .phi. correlation, it does seem to predict C.sub.i within 10 percent of its measured value. If both the .chi. and .phi. correlation curves are used whenever the .phi. of a particular single start helically ridged tube exceeds 0.25 .times. 10.sup.-.sup.2 and the lower value of C.sub.i predicted by the two correlation curves is selected, one can predict with a high degree of accuracy the intube heat transfer performance for turbulent flow of single phase fluid inside the particular tube. For values of .phi. below 0.25 .times. 10.sup.-.sup.2 there is no need to use the C.sub.i vs .chi. correlator. An alternative procedure to avoid the necessity of using the C.sub.i vs .chi. correlation for values of .phi. above 0.25 .times. 10.sup.-.sup.2 would be to simply avoid values of t below 0.085 inches since the defect in the C.sub.i vs .chi. correlation was found to occur only at low values of ridge cap thickness.

An upper limit of 0.365 .times. 10.sup.-.sup.2 for the severity factor .phi. is very desirable since beyond this value the value of C.sub.i drops off while the friction factor, a direct indicator of pressure drop, increases. Values of .phi. greater than 0.365 .times. 10.sup.-.sup.2 should only be considered for single-phase turbulent intube flow when the controlling thermal resistance is associated with the external surface, and a severely contoured external surface is justified by its improvement contribution, and the internal configuration is incidental to that of the external surface of the tube. Although the correlation of C.sub.i vs .phi. seems to hold true down to a value of .phi.=0 where the tube inner surface would be plain, an arbitrary lower limit of .phi.=0.1 .times. 10.sup.-.sup.2 has been set since the improvement in the value of C.sub.i over that of a plain tube for lower values of .phi. is relatively minor.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a partially sectioned side plan view of a plain ended corrugated tube;

FIG. 2 is an enlarged sectional view of a portion of the corrugated tube section in FIG. 1;

FIG. 3 is a fragmentary sectional view similar to FIG. 2 but showing a modified corrugation shape;

FIG. 4 is a graph illustrating the heat transfer performance of a plurality of single-helix internal-ridged tubes which plots the Sieder-Tate-Equation Constant, C.sub.i as a function of the Severity Parameter .phi.;

FIG. 5 is a graph illustrating the heat transfer performance of a plurality of single-helix internal-ridged tubes which plots the Sieder-Tate-Equation Constant, C.sub.i, in relation to a function, .chi., which includes the ridge cap dimensions of the tube;

FIG. 6 is a graph illustrating the heat transfer performance of single-helix, internal-ridged tubes, expressed as an improvement ratio over a plain tube;

FIG. 7 is a graph illustrating the Pressure Drop characteristics of single-helix internal-ridged tubes taken at an arbitrary reference Reynolds Number equal to 35,000 as a function of Severity Parameter, .phi.;

FIG. 8 is a graph illustrating the effect of helix pitch on outside tube diameter when internal single-start helical ridges are formed by an external corrugating operation; and

FIG. 9 is a graph illustrating a correlation of helix pitch reguired for a uniform diameter corrugated tube with the product of the outside diameter and the wall thickness.

DESCRIPTION OF THE PREFERRED EMBODIMENT

FIG. 1 shows a corrugated tube indicated generally at 10 having a plain end 12 and a corrugated section 14. The outer diameter AB of the plain end 12 is preferably equal to or very slightly greater than the outer diameter CD of the corrugated section 14 while the plain end wall thickness BE is equal to the corrugated section wall thickness CF. The distance GH between identical points on adjacent internal ridges produced by the corrugations is defined as the pitch p.

In the enlargement of the corrugated section 14 shown in FIG. 2, one can see that the internal corrugations comprise ridge portions indicated generally at 20 and connecting portions indicated generally at 22. The ridge portion 20 is generally convex toward the inside of the tube while the connecting portion 22 is generally concave. The portions 20 and 22 join each other smoothly at points of inflection 26 where the ridge arc 20' and the connecting arc 22' have a common tangent. The convex curved portion 27 of the ridge 20 between the points 26 is termed the ridge cap. The ridge cap has a width between points 26, 26 defined as t and a height y between its crest 28 and points 26. The ridge height e is the radial distance between ridge crest 28 and the outermost point 30 on the inner surface of connecting portion 22. The internal diameter d.sub.i is the diametral distance between points 30 on opposite sides of the tube. The pitch, p, is the distance between any pair of identical points on adjacent ridges 20, such as the points 28.

FIG. 3 illustrates a modification of the tube shown in FIGS. 1 and 2 in that connecting portions 122 are altered in shape as compared to the concave connecting portions 22 of FIG. 2. The connecting portion 122 is flat over a portion 34 of its length. The outer surface of the tube is broken away in FIG. to illustrate the fact that the tube could have a number of different outer surface configurations other than the shape shown in FIG. 2. Since our invention is concerned with improving the tube side heat transfer properties, there is no need to discuss particular external shapes since these will depend on the external heat transfer conditions.

FIG. 4 is a plot of the data derived from testing a plain tube and many single-helix internally ridged tubes using the modified Wilson plot technique previously referenced to determine the value of the Sieder-Tate Equation constant C.sub.i. The abcissa of the plot is the severity parameter .phi. which is equal to e.sup.2 /p d.sub.i where e is the height of the corrugations (FIG. 2), p is the pitch and d.sub.i is the internal diameter. The parameter .phi. is defined as a severity parameter since it is strongly dependent on the ridge height or severity of the corrugations. From the curve 36 it can be seen that C.sub.i reaches a peak value when .phi. = 0.365 .times. 10.sup.-.sup.2 and then drops off as .phi. increases. The right hand portion of the curve 36 represents several prior art tubes. Point 38 represents the 1 in. tube and point 40 represents the 5/8 in. tube discussed in the aforementioned Withers and Young article.

The single-helix ridged tubes tested represent variations in ridge height depth e from 0.014-0.046 in., pitch p from 0.240-0.625 and internal diameter d.sub.i from 0.530-1.288 in. These values are not meant to be limiting, however, since it is felt that e could be at least as large as 0.09 in., the pitch p at least as great as 1.2 in. and the internal diameter d.sub.i any value up to about 3 inches.

Although the severity parameter .phi. shows an excellent correlation between tube geometry and the Sieder-Tate Constant C.sub.i which makes it most easy to design a tube by choosing e, p and d.sub.i to provide the necessary value of .phi. for the value of C.sub.i desired, the correlation (curve 36) was shown not to hold for a few tubes as evidenced by points 38, 44 and 46 in FIG. 4. These non-conforming tubes proved to provide lower values of C.sub.i for their particular values of .phi. than would be predicted by the curve 36 of FIG. 4. Fortunately, it was found that the tubes which failed to fall on the curve had rather critical ridge cap dimensions. By avoiding tubes with a ridge cap width t of less than 0.85 in., the designer can insure that the correlation curve 36 plotted in FIG. 4 will hold. Alternately, another parameter which is, in part, a function of the ridge cap dimensions can be used to predict the value of C.sub.i. This parameter is termed .chi. and is equal to e t y/d.sub.i where e is the ridge height, t is the width of the ridge cap and d.sub.i is the internal diameter. As can be seen in FIG. 5, there is a fairly good correlation between C.sub.i and .chi. in that C.sub.i increases as .chi. increases. The C.sub.i vs .chi. curve 48 of FIG. 5 need not be considered for tubes having values of .phi. which are less than 0.25 .times. 10.sup.-.sup.2. When .phi. is greater than 0.25 .times. 10.sup.-.sup.2, both the .phi. correlation curve 36 of FIG. 4 and the .chi. correlation curve 48 of FIG. 5 should be considered with the lower value of C.sub.i being considered to be the more accurate.

FIG. 6 is a plot similar to FIG. 4 except that it relates, by curve 50, the improvement ratio over plain tube [C.sub.i /(C.sub.i)p] to the .phi. parameter. This alternative method of displaying the C.sub.i vs .phi. correlation is useful in comparing results from different laboratories since the base value, (C.sub.i)p, for plain tube may vary somewhat among different test setups.

FIG. 7 illustrates a correlation of pressure drop characteristics of single-helix, internal-ridge tubes as a function of the severity parameter .phi. where the pressure drop is expressed as Friction Factor, f, at a reference Reynolds number of 35,000. It is commonly understood that the friction factor, f, is a direct index of pressure drop per unit length of tube, as long as one compares tubes of a given diameter at the same Reynolds number. Since it is evident from the curve 56 of FIG. 7 that pressure drop increases significantly with increases in the severity parameter .phi., it is desirable that tubes be configured so that .phi. not be permitted to increase beyond the optimum value of 0.365 .times. 10.sup.-.sup.2. Such an increase in .phi. would not only result in a lower value of C.sub.i, but would also cause a presumably undesirable increase in pressure drop. In certain instances, design limitations on length, pressure drop, diameter, etc. could render appropriate the selection of .phi. below 0.365 .times. 10.sup.-.sup.2 even though entailing a lower value of C.sub.i.

FIG. 8 illustrates the effect of the helix pitch, p, on the outside diameter of a corrugated tube when internal single-start helical ridges are formed by an external corrugating operation of the type shown in Anderson U.S. Pat. No. 3,128,821. The curve 58 shows that by varying the pitch, p, the outside diameter CD (FIG. 2) of the corrugated section 14 can be varied so as to either decrease or increase relative to the outside diameter AB of the uncorrugated section 12 of the tube 10. The curve 58 is obtained for any particular alloy, diameter and wall thickness by arbitrarily selecting a given corrugation depth, corrugating the tube at various helix angles, and measuring the resulting outer diameter and corresponding pitch for each of the helix angles. By connecting the test points with a curve as shown in FIG. 8, the pitch required to provide a uniform diameter can be readily determined.

FIG. 9 is a graph illustrating the helical pitch required to obtain a uniform diameter corrugated tube for any particular product of the tube outside diameter times its wall thickness. The particular correlation curve 60 shown was determined from data derived from a given tube material (90-10 cupronickel) and given groove depth (0.032 in.) where the tube was corrugated in a single helix style by apparatus such as shown in Anderson U.S. Pat. No. 3,128,821. A family of such curves could be determined for other tube materials and groove depths. The correlation is possible since experiments have shown that there exists a certain helix pitch, (p).sub.u.d., which will yield a uniform diameter product in the sense that the maximum projected outer diameter of the corrugated section is essentially equal to the outside diameter of the plain starting tube.

In order to apply the teachings of the invention to the design of a single start, internally grooved tube where it is desired to achieve maximum heat transfer between a single phase liquid in the tube and the tube surface, the following procedures should be followed:

1. Select a material, outside diameter, and wall thickness which will provide the necessary corrosion resistance, strength and cost, for example, for the intended use.

2. Assuming that a uniform diameter product is desired, multiply the outside diameter times the wall thickness and read the corresponding pitch from a curve such as curve 60 in FIG. 9. If the curve 60 has not been determined for the particular material and corrugation depth, the proper helix pitch for various groove depths may be determined by trial and error by selecting various helix angles and groove depths until the diameter remains constant. This should be done until several combinations are known which will provide a constant outside diameter.

3. Using the equation, .phi. = e.sup.2 /Pd.sub.i = 0.365 .times. 10.sup.-.sup.2, various values of p should be tried until a resulting value of e is found which is identical to the groove depth which must be used with the particular value of p to achieve a constant diameter.

If it is desired to design a tube so that C.sub.i is a particular value less than its maximum, the value of .phi. corresponding to the desired value of C.sub.i can be found on curve 36 in FIG. 4. The values of p, and e which should be used can then be determined as set forth in the preceding example. When designing for either a maximum or a particular C.sub.i, the designer should also check curve 48 in FIG. 5 when .phi. is between 0.25-0.365 .times. 10.sup.-.sup.2 and t is less than 0.085 in. to be certain that a C.sub.i as high as predicted by curve 36 will be obtained.

The teachings of the present invention relative to designing tubes for maximum internal heat transfer are applicable to any of the common tube materials such as cuprous alloys, titanium, stainless steel, carbon steel and aluminum and are independent of outside diameter and the outer configuration of the tube.

Of all the tubes used to establish the various correlations previously set forth herein, one seemed to exactly correspond to the predicted criteria for a single-helix, internally grooved tube which would have a maximum value of C.sub.i. This tube was made of 90-10 Cupronickel and had the following dimensions: Outer Diameter (plain end) = 1.250 inches; Outer Diameter (corrugated end) = 1.249 inches; Wall = 0.050 inch; d.sub.i = 1.149 inch; e = 0.046 inch; p = 0.505 inch; t = 0.120 inch; y = 0.010 inch; .phi. = 0.365 .times. 10.sup.-.sup.2 ; X = 0.48 .times. 10.sup.-.sup.4 in..sup.2 ; C.sub.i = 0.0693; C.sub.i /(C.sub.i).sub.p = 2.62.

Claims

We claim as our invention:

1. A metal heat transfer tube having single start internal helical ridging such that the inner boundary of the tube wall, when viewed in longitudinal sectional profile, is comprised of alternating convex and concave portions which join at common tangents, and wherein the internal tube surface can be described by the equation: .phi. = e.sup.2 /pd.sub.i where .phi. is a dimensionless severity parameter, e is the height of the helical ridge, p is the helical pitch and d.sub.i is the inside diameter, and where .phi. is greater than 0.1 .times. 10.sup.-.sup.2 and equal to or less than about 0.365 .times. 10.sup.-.sup.2.

2. The metal heat transfer tube of claim 1 wherein .phi. is equal to about 0.365 .times. 10.sup.-.sup.2.

3. The metal heat transfer tube of claim 1 wherein the helical ridge has a convex ridge cap, the axial thickness t of said cap being equal to or greater than 0.085 inches.

4. The metal heat transfer tube of claim 2 wherein the helical ridge has a convex ridge cap, the axial thickness t of said cap being equal to or greater than 0.085 inches.

5. The metal heat transfer tube of claim 1 wherein said tube is corrugated and has a uniform wall thickness.

6. The metal heat transfer tube of claim 2 wherein said tube is corrugated and has a uniform wall thickness.

7. The metal heat transfer tube of claim 3 wherein said tube is corrugated and has a uniform wall thickness.

8. The metal heat transfer tube of claim 4 wherein said tube is corrugated and has a uniform wall thickness.