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Influence of aspect ratio and hydraulic diameter on flat oval elbow loss coefficients.

INTRODUCTION

ASHRAE Research Project RP-1319 was undertaken to test flat oval elbows to determine their total pressure loss coefficients, and update the ASHRAE Duct Fitting Database (ASHRAE 2006). A complete description of the test program and its results are provided in Idem et al. (2008) and Kulkarni et al. (2008). Testing in the project complied with ANSI/ ASHRAE Standard 120-1999, Method of Testing to Determine Flow Resistance of HVAC Ducts and Fittings (ASHRAE 1999). Loss coefficients were measured for (1) the easy bend 90[degrees] five-gore flat oval elbow, (2) the hard bend 90[degrees] five-gore flat oval elbow, (3) the mitered easy bend flat oval elbow with and without vanes, and (4) the mitered hard bend flat oval elbow with and without vanes.

Refer to Figure 1 for a sketch of flat oval elbows in the database. There are limited flat oval elbow loss coefficient data available in the literature, such as Smith and Jones (1976) and Townsend et al. (1996). However, the existing flat oval elbow loss coefficient data are rather sparse, as relatively few elbow geometries have been studied in detail. Furthermore, the data are rarely in perfect agreement, and often exhibit considerable scatter about a central tendency. This is likely due to a combination of random experimental errors and differences in test methodology, data reduction procedures, and fitting characteristics such as surface effects and joints. In an attempt to reconcile data from these disparate sources, this paper presents a method of correlating flat oval elbow loss coefficient data as a function of aspect ratio and hydraulic diameter.

DATA ANALYSIS

In a hard bend flat oval elbow the fluid turns about an axis that is parallel to the minor axis of the fitting, whereas for an easy bend elbow the fluid turns about an axis that is parallel to the major axis of the fitting. For hard bend elbows, the aspect ratio is defined as the minor duct dimension divided by the major duct dimension, i.e., AR = a/A. Similarly, for easy bend elbows, the aspect ratio is defined as the major duct dimension divided by the minor duct dimension, i.e., AR = A/a. The hydraulic diameter is customarily defined in terms of duct cross section A and duct perimeter P, such that

[D.sub.h] = 4[A.sub.cs]/P.(1)

To correlate the data, the hydraulic diameter was normalized by the largest hydraulic diameter considered in the study, i.e., round elbows where [D.sub.h] = 762 mm (30 in.). The dimensionless hydraulic diameter is therefore defined as follows:

[D'.sub.h] = [D.sub.h]/762 (2 SI)

[D'.sub.h] = [D.sub.h]/30 (2 I-P)

It is noted that in the limit of a round cross section elbow, the aspect ratio has a value of unity. In this study the flat oval elbow loss coefficient data are correlated by means of the following expression:

C = [alpha][([D.sub.h]').sup.[beta]][(AR).sup.[lambda]] (3)

Assuming constant physical properties of air, the specific values of the coefficient a and the exponents [beta] and [gama] vary with fitting geometry. The geometry, in turn, influences the characteristics of flow. The constant parameters were evaluated by performing a least-squares curve-fit to the log-linearized experimental data. In that case it was necessary to solve the following matrix equation:

[MATHEMATICAL EXPRESSION NOT REPRODICIBLE IN ASCII] (4)

Measured data may contain both bias and precision (random) errors. Bias errors will either tend to shift the entire data set above or below the true line curve or change the slope. Precision errors will cause the data to scatter about the apparent line. The objective of curve-fitting is to average out the precision errors by calculating a curve that follows the apparent central tendency of the scattered data. The independent variable or the dependent variable may include both precision and bias errors. The least-squares curve-fitting method implicitly assumes the precision error in the dependent variable is much greater than that in the independent variable. Furthermore, least-squares curve-fitting cannot reduce bias error.

The linear correlation coefficient is a measure of how the variance in the dependent variable is accounted for by a linear curve-fit. It is interpreted as the ratio of the variation assumed by the fit to the actual measured variation in the data. Hence,

[r.sup.2] = explained variation/total variation. (5)

As outlined in Bethea et al. (1995), for a data set comprised of n variables, the correlation coefficient can be calculated by

r = [n.summation over (i - 1)] ([x.sub.i] - [bar.x]) ([y.sub.i] - [bar.y])/[[n.summation over i = 1] [([x.sub.i] - [bar.x]).sup.2] [n.summation over (i = 1)] [([y.sub.i] - [bar.y]).sup.2].sup.1/2] (6)

The variables [X.sub.i] and [y.sub.i] - are defined by

[x.sub.i] = log[AR.sub.i] (7)

and

[y.sub.i] = log[[C/[D'.sub.h.sup.[beta]]].sub.i] (8)

Similarly, in Equation 6, x and y are the mean values of the x and y values, respectively, defined as

[bar.x] = 1/n [n summation over (i = 1)] [x.sub.i] (9)

and

[bar.y] = 1/n [n summation over (i = 1)] [y.sub.i]. (10)

The linear correlation coefficient, as defined by Equation 6 is a dimensionless quantity. In general, -1[less then or equal to] r [less than or equal to] 1, wherein a "+" sign indicates positive linear correlation, and a "-" sign implies negative linear correlation. Values of r [right arrow][+ or -1] imply a greater likelihood that a linear relationship exists between the independent and dependent variables. In order to evaluate the adequacy of the regression analyses, linear correlation coefficients were calculated for each set of loss coefficient data.

RESULTS

Individual hard bend elbow loss coefficient data are summarized in Table 1 as a function of the nominal fitting geometry. Similarly, Table 2 presents easy bend elbow loss coefficient data. Figures 2 through 4 represent a summary of the flat oval elbow loss coefficient data obtained in this project, and data from previous ASHRAE-sponsored research projects, including Townsend et al. (1996), Smith and Jones (1976), and Mahank and Mumma (1997). Data from these projects are in Appendix A. The Mahank and Mumma study employed computational fluid dynamics (CFD) to predict the loss coefficient of several flat oval five-gore elbows without turning vanes. Measured fitting dimensions were unavailable from Smith and Jones. Presumably the loss coefficient calculations in Mahank and Mumma employed nominal duct dimensions. In Figures 2 through 4, the aspect ratio and dimensionless hydraulic diameter values are based on nominal dimensions; refer to Tables 1 and 2.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]
Table 1. Hard Bend Elbow Loss Coefficient Data

A x a mm x mm  a/A   [D.sub.h]   [D'.sub.h]            C
 (in. x in.)          mm (in.)

                                              Mitered--  Mitered--
                                             With Vanes  No Vanes

  838 x 152    0.18  264 (10.4)     0.478       1.68       2.40
  (33 x 6)

  584 x 152    0.26  250 (9.86)     0.292       1.42       1.51
  (23 x 6)

  965 x 254    0.26  417 (16.4)     0.326       1.10       2.61
  (38 x 10)

  381 x 102    0.27  166 (6.55)     0.218       1.79       2.44
  (15 x 4)

  940 x 254    0.27  414 (16.3)     0.547       1.23       1.91
  (37 x 10)

  559 x 152    0.27  248 (9.78)     0.329       1.45       2.11
  (22 x 6)

  356 x 152    0.43  222 (8.75)     0.348       1.07       1.68
  (14 x 6)

  787 x 356    0.45  511 (20.1)     0.544       0.90       1.45
  (31 x 14)

  559 x 254    0.45  363 (14.3)     0.670       0.91       1.66
  (22 x 10)

  762 x 762    1.00  762 (30.0)     1.000       0.72       1.49
  (30 x 30)

Table 2. Easy Bend Elbow Loss Coefficient Data

A x a, mm  A/a   [D.sub.h],  [D'.sub.h]                C
x mm(in.         mm (in.)
x in.)

                                         Five-Gore  Mitered  Mitered--
                                         R/a = 1.5  --With      No
                                                     Vanes     Vanes


762 x 762  1.00  762 (30.0)    1.000        0.44      0.72      1.49
(30 x 30)

559 x 254  2.20  363 (14.3)    0.478        0.51      0.89      1.65
(22 x 10)

787 x 356  2.21  511 (20.1)    0.670        0.45      0.69      1.50
(31 x 14)

356 x 152  2.33  222 (8.75)    0.292        0.50      1.03      1.45
(14 x 6)

559 x 152  3.67  248 (9.78)    0.326        0.50      1.03      1.40
(22 x 6)

940 x 254  3.70  414 (16.3)    0.544        0.87      1.17      1.73
(37 x 10)

381 x 102  3.75  166 (6.55)    0.218        0.82      1.93      2.17
(15 x 4)

965 x 254  3.80  417 (16.4)    0.547        0.32      0.51      1.26
(38 x 10)

584 x 152  3.83  250 (9.86)    0.329        0.54      1.22      1.81
(23 x 6)

838 x 152  5.50  264 (10.4)    0.348        1.06      1.53      2.00
(33 x 6)


It was noted that for a family of dimensionless hydraulic diameter values, the loss coefficients obtained experimentally tended to follow straight-line curves when plotted as a function of aspect ratio on log-log axes, implying the existence of a power-law relationship. This was not the case for the loss coefficient values predicted using CFD, reported in Mahank and Mumma (1997), which were apparently independent of either aspect ratio or hydraulic diameter. Therefore, the 1997 CFD report values were not considered further in the present study.

Separate curve-fits were performed for the flat oval elbow geometries by solving Equation 4. In every instance nominal dimensions were employed in the calculations. Linear correlation coefficients were calculated for each case by Equation 6. This yielded the following empirical loss coefficient correlations and their corresponding linear correlation coefficients.

Hard bend five-gore elbows:

C = 0.301 [([D'.sub.h]).sup.0.072] [(AR).sup.0.427]; r = 0.520 (11)

Easy bend five-gore elbows:

C = 0.439[([D'.sub.h]).sup.-0.065) [(AR).sup.0.075]; r = 0.089 (12)

Hard bend mitered elbows-with vanes:

C = 0.760 [([D'.sub.h]).sup.-0.178] [(AR).sup.-0.141]; r = -0.178 (13)

Easy bend mitered elbows-with vanes:

C = 0.581[([D'.sub.h]).sup.-0.647] [(AR).sup.-0.0004]; r = -0.001 (14)

Hard bend mitered elbows-no vanes:

C = 1.341 [([D'.sub.h]).sup.-0.011] [(AR).sup.-0.325]; r = -0.693 (15)

Easy bend mitered elbows-no vanes:

C = 1.366[([D'.sub.h]).sup.-0.156] [(AR).sup.-0.040]; r = -0.129 (16)

In calculating Equations 11 through 16 by means of Equation 4, the loss coefficient data obtained for the 762 mm (30 in.) round five-gore or mitered elbows were incorporated into the curve-fits both for the hard bend and easy bend configurations.

[FIGURE 5 OMITTED]

Figures 5 through 7 display the resulting least-squares curve-fit expressions as solid/dashed lines, which are, in turn, compared to the experimental data. Several trends in these results are apparent. The hard bend data for five-gore and mitered elbows with no vanes were correlated satisfactorily by the proposed curve-fit expressions, Equations 11 and 15, respectively, in that |r| [greater than or equal to]However, the data for hard bend mitered elbows with vanes were not as well correlated by Equation 13, as |r| [less than or equal to] 0.178 . This was due in part to possible outliers associated with data from Townsend et al. (1996); refer to Figure 6. The easy bend data exhibited very little correlation with the aspect ratio, as indicated by the fact that the coefficient [gamma] [write arrow] 0 in each instance. This is further substantiated by the small linear correlation coefficients associated with Equations 12, 14, and 16, collectively, where, in general, |r| [less than or equal to] 0.129 .

[FIGURE 6 OMITTED]

The curve-fit expressions provided in Equations 11 through 16 correlate flat oval elbow loss coefficient data as a function of aspect ratio and dimensionless hydraulic diameter. Since these equations were shown to be reliable they were used to generate tabular loss coefficient values suitable for inclusion in the ASHRAE Fitting Database (ASHRAE 2006). These results are presented in Tables 3 through 5. For each elbow geometry, the least-squares expression yielded different values for the loss coefficient in the limit as AR [right arrow] 1, i.e., for round elbows. This often resulted in distinct discontinuities in calculated loss coefficient values in the vicinity of AR = 1, particularly for the largest and smallest dimensionless hydraulic diameters considered in the present analysis. Hence, in Tables 3 through 5, the entries for AR = 1 were calculated by averaging the predictions generated by the respective hard bend and easy bend cases.
Table 3. Calculated 5-Gore Flat Oval Elbow Loss Coefficients, hard
Bend: R/A = 1.5, Easy Bend: R/a = 1.5

D'h                         Aspect Ratio (AR)

     0.15  0.3   0.45  0.6   0.75   0.9    1     2     3     4     5

0.2  0.12  0.16  0.19  0.22  0.24  0.26  0.38  0.51  0.53  0.54  0.55
0.4  0.13  0.17  0.20  0.23  0.25  0.27  0.37  0.49  0.51  0.52  0.53
0.6  0.13  0.17  0.21  0.23  0.26  0.28  0.37  0.48  0.49  0.50  0.51
0.8  0.13  0.18  0.21  0.24  0.26  0.28  0.37  0.47  0.48  0.49  0.50
 1   0.13  0.18  0.21  0.24  0.27  0.29  0.37  0.46  0.48  0.49  0.50
1.2  0.14  0.18  0.22  0.25  0.27  0.29  0.37  0.46  0.47  0.48  0.49
1.4  0.14  0.18  0.22  0.25  0.27  0.29  0.37  0.45  0.47  0.48  0.48
1.6  0.14  0.19  0.22  0.25  0.28  0.30  0.37  0.45  0.46  0.47  0.48
1.8  0.14  0.19  0.22  0.25  0.28  0.30  0.37  0.45  0.46  0.47  0.48
 2   0.14  0.19  0.22  0.25  0.28  0.30  0.37  0.44  0.46  0.47  0.47

Table 4. Calculated Mitered With Vanes Flat Oval Elbow Loss
Coefficients

D'h                      Aspect Ratio (AR)

     0.15  0.3   0.45  0.6   0.75  0.9    1     2     3     4     5

0.2  1.32  1.20  1.13  1.09  1.05  1.03  1.33  1.65  1.65  1.65  1.65
0.4  1.17  1.06  1.00  0.96  0.93  0.91  0.97  1.05  1.05  1.05  1.05
0.6  1.09  0.99  0.93  0.89  0.87  0.84  0.82  0.81  0.81  0.81  0.81
0.8  1.03  0.94  0.89  0.85  0.82  0.80  0.73  0.67  0.67  0.67  0.67
 1   0.99  0.90  0.85  0.82  0.79  0.77  0.67  0.58  0.58  0.58  0.58
1.2  0.96  0.87  0.82  0.79  0.77  0.75  0.63  0.52  0.52  0.52  0.52
1.4  0.94  0.85  0.80  0.77  0.75  0.73  0.59  0.47  0.47  0.47  0.47
1.6  0.91  0.83  0.78  0.75  0.73  0.71  0.56  0.43  0.43  0.43  0.43
1.8  0.89  0.81  0.77  0.74  0.71  0.69  0.54  0.40  0.40  0.40  0.40
 2   0.88  0.80  0.75  0.72  0.70  0.68  0.52  0.37  0.37  0.37  0.37

Table 5. Calculated Mitered No Vanes Flat Oval Elbow Loss Coefficients

D'h                    Aspect Ratio (AR)

       0.15    0.3     0.45    0.6     0.75    0.9

0.2    2.44    1.95    1.71    1.56    1.45    1.36
0.4    2.46    1.96    1.72    1.57    1.46    1.37
0.6    2.47    1.97    1.73    1.57    1.46    1.38
0.8    2.48    1.98    1.73    1.58    1.47    1.38
 1     2.48    1.98    1.74    1.58    1.47    1.39
1.2    2.49    1.99    1.74    1.59    1.48    1.39
1.4    2.49    1.99    1.74    1.59    1.48    1.39
1.6    2.50    1.99    1.75    1.59    1.48    1.39
1.8    2.50    2.00    1.75    1.59    1.48    1.40
 2     2.50    2.00    1.75    1.60    1.48    1.40

D'h        Aspect Ratio (AR)

         1      2        3      4        5

0.2    1.58    1.81    1.83    1.86    1.87
0.4    1.50    1.62    1.65    1.67    1.68
0.6    1.45    1.52    1.55    1.56    1.58
0.8    1.42    1.45    1.48    1.50    1.51
 1     1.40    1.40    1.43    1.44    1.46
1.2    1.38    1.37    1.39    1.40    1.42
1.4    1.36    1.33    1.35    1.37    1.38
1.6    1.35    1.31    1.33    1.34    1.35
1.8    1.34    1.28    1.30    1.32    1.33
 2     1.33    1.26    1.28    1.30    1.31


CONCLUSIONS

A power law expression is proposed to correlate the flat oval elbow loss coefficient data for each type of flat oval elbow tested as a function of aspect ratio and hydraulic diameter. Limited loss coefficient data from previous projects sponsored by ASHRAE were also included in the correlations. The family of curves corresponding to different area ratios and dimensionless hydraulic diameters collapsed to a single straight line by plotting the results in terms of the ratio C/[([D.sub.h]').sup.[beta]] as a function of AR with varying degrees of success. However, to a reasonable approximation, the analysis indicated the loss coefficients of easy bend flat oval elbows were primarily a function of dimensionless hydraulic diameter, since in every instance there was very little correlation with the aspect ratio. By contrast, the loss coefficients of hard bend flat oval elbows were correlated with sufficient accuracy as a function of both aspect ratio and dimensionless hydraulic diameter. The curve-fit analysis was used to generate tables of flat oval elbow loss coefficient values suitable for inclusion in the ASHRAE Duct Fitting Database (ASHRAE 2006). In many instances, the flat oval data exhibited trends that are similar to those for rectangular elbows, i.e., loss coefficients generally increased with decreasing major-to-minor ratio. Likewise, for easy bend elbows, there was a tendency for the loss coefficient to decrease with increasing hydraulic diameter.

REFERENCES

ASHRAE. 1999. ANSI/ASHRAE Standard 120-1999. Method of Testing to Determine Flow Resistance of HVAC Ducts and Fittings. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

ASHRAE. 2006. ASHRAE Duct Fitting Database, Version 4.0.3. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

Bethea, R.M., B.S. Duran, and T.L. Boullion. 1995. Statistical Methods for Engineers and Scientists, 3d ed. New York: Marcel Dekker, Inc.

Idem, S., D. Kulkarni, and S. Khaire. 2008. Laboratory testing of duct fittings to determine loss coefficients. Final Report, ASHRAE RP-1319. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

Kulkarni, D., S. Khaire, and S. Idem. 2008. Measurements of flat oval elbow loss coefficients. ASHRAE Transactions. In press.

Mahank, T.A., and S.A. Mumma. 1997. Flow modeling of flat oval ductwork elbows using computational fluid dynamics. ASHRAE Transactions 103(1):172-77.

Smith, J.R., and J.W. Jones. 1976. Pressure loss in high velocity flat oval duct fittings. ASHRAE Transactions 82(1):244-55.

Townsend, B., F. Khodabakhsh, and S. Idem. 1996. Loss coefficient measurements for flat oval elbows and transitions. ASHRAE Transactions 102(2):159-69.

NOMENCLATURE

[A.sub.cs] = cross-sectional duct area, [mm.sup.2] ([in.sup.2])

A = major duct dimension, mm (in.)

a = minor duct dimension, mm (in.)

AR = aspect ratio

C = loss coefficient

[D.sub.h] = hydraulic diameter, mm (in.)

[D'.sub.h] = dimensionless hydraulic diameter

n = number of data points

P = perimeter, mm (in.)

R = elbow turning radius, mm (in.)

r = linear correlation coefficient

x, y = variables

[bar.x], [bar.y] = mean values

[alpha], [beta], [gamma] = coefficients

Table A-1 Hard Bend Flat Oval Elbows (a)

(a) Townsend et al. (1996)
Table A-1 Hard Bend Flat Oval Elbows (a)

A x a mm x mm (in. x in.)   a/A                      C

                                  Five-Gore R/A = 1.5  Mitered w/vanes

    279 x 76 (11 x 3)      0.268        0.205               1.296
    483 x 127 (19 x 5)     0.258        0.091               0.492
    838 x 152 (33 x 6)     0.179        0.232               0.660

(a) Townsend et al. (1996)

Table A-2  Easy Bend Flat Oval Elbows (a)

A x a mm x mm (in. x in.)   A/a          C

                                  Five-Gore R/a = 1.5

    279 x 76 (11 x 3)      3.73           0.475
    483 x 127 (19 x 5)     3.88           0.372
    838 x 152 (33 x 6)     5.60           0.266

(a) Townsend et al. (1996)

Table A-3 (a)

          Hard Bend (a/A), R/A=1.5        Easy Bend (A/a), R/a=1.5

AR  0.25  0.33  0.5   0.67  0.75  1.0   1.33  1.5   2.0   3.0   4.0
C   0.22  0.22  0.22  0.24  0.24  0.23  0.24  0.24  0.24  0.23  0.23

(a) Mahank and Mumma (1997)

Table A-4 (a) Hard Bend Flat Oval Elbows

A x a mm x mm (in. x in.)  a/A            C

                                 Five-Gore, R/A=1.5

   305 x 152 (12 x 6)      0.5           0.19
   483 x 152 (19 x 6)      0.32          0.18
   635 x 152 (25 x 6)      0.24          0.17
   508 x 254 (20 x 10)     0.5           0.15
   737 x 254 (29 x 10)     0.34          0.14
  1041 x 254 (41 x 10)     0.24          0.13

(a) Smith and Jones (1976)


D. Kulkarni

S. Khaire

S. Idem, PhD

Member ASHRAE

This paper is based on findings resulting from ASHRAE Research Project RP-1319.

D. Kulkarni and S. Khaire are research assistants and S. Idem is a professor in the Department of Mechanical Engineering, Tennessee Tech University, Cookeville, TN.
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Author:Kulkarni, D.; Khaire, S.; Idem, S.
Publication:ASHRAE Transactions
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2009
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