flat oval elbow loss coefficients

©2009 ASHRAE 35

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

ABSTRACT

Pressure loss coefficients were determined for the follow-ing types of flat oval elbows having various aspect ratios: 5-gore 90° easy bend, mitered 90° easy bend with and withoutvanes, and mitered 90° hard bend with and without vanes. Thetests were performed in accordance with ANSI/ASHRAE Stan-dard 120-1999. Data are presented graphically, and the losscoefficient for each fitting is tabulated. A linear correlationcoefficient is calculated for each case.

INTRODUCTION

This paper presents results of an experimental program todetermine loss coefficients for several flat oval elbows, i.e.,90° easy bend elbows, mitered 90° easy bend elbows with andwithout vanes, and mitered 90° hard bend elbows with andwithout vanes. A complete description of the test program isprovided in Idem et al. (2008). In a hard bend flat oval elbowthe fluid turns about an axis which is parallel to the minor axisof the fitting while in an easy bend elbow the fluid turns aboutan axis which is parallel to the major axis of the fitting. Thisresearch was motivated by the need to increase the populationof flat oval fittings in the ASHRAE Duct Fitting Database(DFDB) (2006). Presently only straight flat oval duct and the5-gore hard bend elbow are in the DFDB.

EXPERIMENTAL PROGRAM

Refer to Figure 1 for a sketch of the various elbowsconsidered in this project. The dimensions of the flat oval andround elbows are listed in Tables 1 through 3. The test speci-mens were constructed using 20 gage galvanized steel. The

ratio of major-to-minor dimensions ranged from 1.0 to 5.5. Forthe 5-gore easy bend elbow the dimensionless turning radiusratio R/a equaled 1.5.

The elbow test setups shown in Figure 2 included a 30 hpcentrifugal fan. A cylindrical nozzle chamber was used forflow measurement. A variable frequency drive (VFD) wasused to control the fan speed, and hence the air flow throughthe system. Flow control screens were mounted upstream anddownstream of the nozzle board to settle the flow. The systemwas blow through in nature. The nozzle board contained fourlong-radius spun aluminum flow nozzles having throat diam-eters of 51 mm (2 in.), 102 mm (4 in.), 152 mm (6 in.) and 203mm (8 in.). The nozzles were mounted on a 25 mm (1 in.) thickplywood board. Various combinations of flow nozzles wereselected to obtain a desired flow rate; unused nozzles wereblocked by means of smooth vinyl balls. The nozzle pressuredrop was measured by two piezometer rings located 38 mm(1.5 in.) on each side of the nozzle board, with both sidesconnected to a manometer. The nozzle chamber wasconstructed in accordance with ANSI/ASHRAE Standard120-1999.

Pressure taps constructed from 6.4 mm (1/4 in.) diametercopper tubing were soldered onto the ductwork upstream anddownstream of the test section. Flexible tubing was used toconstruct piezometer rings at both measurement locations.The piezometer rings were connected to a single micro-manometer using flexible tubing so as to measure the pressuredrop across the test section. Static gage pressure was measuredat each location by inserting tees into the pressure tubing.

Pressure drop measurements over the test section andacross the nozzle board were performed using liquid-filled

Measurements of Flat Oval Elbow Loss Coefficients

D. Kulkarni S. Khaire S. Idem, PhDMember ASHRAE

D. Kulkarni and S. Khaire are research assistants and S. Idem is a professor in the Department of Mechanical Engineering, TennesseeTech University, Cookeville, TN.

CH-09-006 (RP-1319)

36 ASHRAE Transactions

(a) (b)

(c) (d)

(e)

Figure 1 Flat oval elbows: (a) 5-gore 90° easy bend, (b) mitered 90° easy bend without vanes, (c) mitered 90° easy bend withvanes, (d) mitered 90° hard bend without vanes, and (e) mitered 90° hard bend with vanes.

ASHRAE Transactions 37

micromanometers having a measurement accuracy of±0.025 mm (0.001 in.). The gage pressure upstream anddownstream of the test section was measured by means ofinclined liquid-filled manometers having a readability of±0.25 mm (0.01 in.). Static pressure in the nozzle chamber wasmeasured using an electronic manometer having a scale read-ability of ±0.25 mm (0.01 in.). The air temperature in thenozzle chamber was measured using a mercury thermometerhaving a scale readability of ± 0.5°C (1.0°F). The dry- andwet-bulb temperatures of the ambient air were measured usingan aspirated psychrometer with an accuracy of ±0.5°C (1.0°F).The test section temperature was not measured directly, butwas assumed to be the same as the temperature of the air insidethe nozzle chamber. Ambient pressure was measured with aFortin-type barometer, with an accuracy of ±0.25 mm(0.01 in.) of mercury. All measurements of temperature andpressure in this project were in compliance with ANSI/ASHRAE Standard 120-1999.

Initially straight duct tests were performed so as to estab-lish the tare pressure loss for subsequent evaluation of flat ovalelbow loss coefficients. Darcy friction factors were deter-mined by plotting the data on a Moody chart. The frictionfactor data were found to lie along a single relative roughnesscurve for each duct cross section that was tested, thereby vali-dating the Colebrook equation model.

Elbow pressure loss measurements were then performedby inserting the test fittings into the straight duct test appara-

tus. The test apparatus and procedures complied withASHRAE Standard 120-1999, except as noted below. In eachinstance the lengths of the upstream sections, the downstreamsection, and the tailpiece section are listed in Table 4 for eachtest configuration. In some instances the exit lengths for hardbend elbow tests did not conform to the values stipulated byANSI/ASHRAE Standard 120-1999, due to the limited widthof the laboratory. Those cases are marked by an asterisk inTable 4. The duct sections employed in most elbow tests wereconnected to the nozzle chamber by means of 305 mm (12 in.)diameter round to flat oval transitions, which did not conformprecisely to ANSI/ASHRAE Standard 120-1999. This testsetup is shown in Figure 2a. The exceptions were the pressureloss measurements performed on the 965 × 264 mm (38 ×10 in.) and 940 × 254 mm (37 × 10 in.) hard bend elbows withturning vanes. In those instances the elbow tests wereconducted with a plenum chamber and bellmouth combina-tion mounted upstream of the ductwork as depicted inFigure 2b. Likewise all 762 mm (30 in.) diameter elbow testswere performed using the plenum chamber and bellmouthmounted upstream of the test section.

DATA ANALYSIS

The loss coefficient is defined by Equation 1 as the ratioof the total pressure loss across an elbow to velocity pressure,where the cross sections (subscripts) are shown by Figure 2

Table 1. Elbows Tested

Fitting Geometry Fitting Sizes Test Conditions

CF3-2 Elbow, Flat Oval, 5 Gore, 90°, Easy Bend See Table 2 R/a = 1.5

CF3-3 Elbow, Flat Oval, Mitered, Easy Bend, without Vanes See Table 2

CF3-4 Elbow, Flat Oval, Mitered, Hard Bend, without Vanes See Table 2

CF3-5 Elbow, Flat Oval, Mitered, Easy Bend, with Vanes See Table 2 See Table 3 for Number of Vanes

CF3-6 Elbow, Flat Oval, Mitered, Hard Bend, with Vanes See Table 2 See Table 3 for Number of Vanes

Table 2. Nominal Elbow Dimensions

Nominal Aspect Ratio

A × a,mm × mm (in. × in.)

1.0 762 × 762 (30 × 30)

2.2356 × 152(14 × 6)

559 × 254(22 × 10)

787 × 356(31 × 14)

3.7381 × 102(15 × 4)

559 × 152(22 × 6)

940 × 254(37 × 10)

3.8584 × 152(23 × 6)

965 × 254(38 × 10)

5.5838 × 152(33 × 6)

Table 3. Number of Turning Vanes

Easy Bend with Vanes Hard Bend with Vanes

Minor Axis,mm (in.)

Number of Vanes

Major Axis,mm (in.)

Number of Vanes

102 (4) 2356 (14) 4

381 (15) 4

152 (6) 2559 (22) 5

584 (23) 5

254 (10) 3787 (31) 5

838 (33) 5

356 (14) 4940 (37) 5965 (38) 5


Figure 2 Elbow test apparatus: (a) with round to flat oval transition and (b) with plenum chamber.

(a)

(b)


(1)

The velocity pressure at the test fitting is calculated based on

the airflow and cross sectional area at section ‘7’, thus

(2 SI)

(2 IP)

The total pressure loss across an equal area non-junction

fitting (elbow) is calculated by Equation 3

(3)

The terms L7-1 and L2-8 represent the separation distance

between the upstream taps and the entrance plane of the elbow,

and the exit plane of the elbow and the downstream pressure

taps, respectively. In Equation 1, the pressure drop is due

solely to dynamic losses in the duct fitting, since the friction

pressure loss has been subtracted from the overall pressure

drop across the fitting. The pressure friction loss per unit

length is the duct tare pressure loss per unit length , as

calculated by Equation 4

(4 SI)

(4 IP)

The least squares method was employed in order to obtainan overall loss coefficient for each elbow. Equation 1 can bewritten as

(5)

The slope of the curve plotted against can beinterpreted as the zero length loss coefficient of the fitting,since a curve plotted through the data points is a straight-lineif the loss coefficient is a constant. Hence, let

(6)

where

(7)

and

(8)

The intercept ‘b’ can be forced to zero in the case of HVACfittings, since ideally the pressure drop across the fittingshould be zero when the velocity of the air through the fittingis zero. In that case it can readily be shown that

Table 4. Elbow Test Setup Dimensions

Duct Cross Section,mm × mm (in. × in.)

Dh ,mm (in.)

LZ-1 , m (ft)

L7-1 , m (ft)

L2-8 , m (ft)

Tail Duct Length,m (ft)

356 × 152 (14 × 6) 223.5 (8.8) 2.2 (7.2) 0.3 (1.1) 2.3 (7.7) 1.2 (4.0)

381 × 102 (15 × 4) 165.1 (6.5) 2.2 (7.3) 0.2 (0.7) 1.8 (5.8) 0.7 (2.3)

559 × 152 (22 × 6) 248.9 (9.8) 2.5 (8.3) 0.4 (1.2) 2.5 (8.2) 1.9 (6.3)

584 × 152 (23 × 6) 251.5 (9.9) 2.6 (8.4) 0.4 (1.2) 2.7 (8.7) 1.8 (5.8)

838 × 152 (33 × 6) 264.1 (10.4) 3.4 (11.0) 0.3 (1.0) 2.5 (8.3) 1.1 (3.7)

559 × 254 (22 × 10) 363.2 (14.3) 4.3 (14.0) 0.6 (2.0) 4.0 (13.2) 2.1 (6.8)

787 × 356* (31 × 14) 510.5 (20.1) 6.6 (21.5) 0.8 (2.5) 6.0 (19.6) 2.6 (8.4)

965 × 254* (38 × 10) 416.6 (16.4) 4.4 (14.3) 0.5 (1.7) 4.7 (15.3) 2.7 (8.7)

940 × 254* (37 × 10) 414.0 (16.3) 4.2 (13.7) 0.7 (2.3) 4.9 (16.2) 2.4 (7.8)

762 × 762* (30 × 30) 762.0 (30.0) 22.1 (72.5) 1.1 (3.5) 7.7 (25.3) 0.9 (2.8)

* Exit lengths for hard bend elbow tests did not conform to the values stipulated by ANSI/ASHRAE Standard 120-1999.

Cpt 1-2,Δpv7

----------------- .=

pv712---ρ

Q7 1000⁄

A7----------------------

⎝ ⎠⎜ ⎟⎛ ⎞2

12---ρV7

2= =

pv7 ρQ7 A7⁄1097

-----------------⎝ ⎠⎜ ⎟⎛ ⎞

2

ρV7

1097------------⎝ ⎠

⎛ ⎞2

= =

pt 1-2,Δ ps 7-8,Δ L7-1 L2-8+( )pf ΔL

---------⎝ ⎠⎛ ⎞ .–=

pfΔL

--------

pfΔL

--------pv1

f⋅

Dh1 1000⁄-------------------------=

pfΔL

--------pv1

f⋅

Dh1 12⁄-------------------.=

pt 1-2,Δ C pv7.⋅=

pt 1-2,Δ pv7

yi mxi b+=

yi pt 1-2, xi,Δ pv7= =

m C.=


(9)

Measured data may contain both bias and precision(random) errors. Bias errors will either tend to shift the entiredata set above or below the true line curve or change the slope.Precision errors will cause the data to scatter about the appar-ent line. The objective of curve-fitting is to average out theprecision errors by calculating a curve that follows the appar-ent central tendency of the scattered data. The independentvariable x or the dependent variable y may include both preci-sion and bias errors. The least squares curve- fitting methodimplicitly assumes the precision error in y is much greater thanthat in x. Least squares curve-fitting cannot reduce bias error.

The linear correlation coefficient is a measure of how thevariance in y is accounted for by a linear curve-fit. It is inter-preted as the ratio of the variation assumed by the fit to theactual measured variation in the data interpreted. Hence

(10)

In general for a data set comprised of n variables the correla-tion coefficient can be calculated per Bethea et al. (1995) as

(11)

where and are the mean values of x and y, respectively.They are defined by

(12)

and

(13)

The linear correlation coefficient is a dimensionless quantity.In general , where a ‘+’ sign indicates positivelinear correlation and a ‘–’ sign implies negative linear corre-lation.

Confidence intervals for the slope can be calculated underthe assumption that the precision error in yi satisfies the normaldistribution. The error in the slope of a least squares curve-fit(which in turn equals the uncertainty in the measured losscoefficient) is given by Beckwith et al. (1993) as

(14)

where ta/2,n – 2 is the student’s t-statistic with n – 2 degrees offreedom (n is the number of points in the data set) and a = (1 –c) is the level of significance (in the present work c = 95%).

(15)

where yi is the actual value at point i, and y(xi) was the valueobtained by the least squares fit at point i. The quantity Sxx inEquation 14 is found from the expression

(16)

where is the mean x value.

RESULTS

Loss coefficient measurements were performed for the flatoval elbows listed in Tables 1 through 3. Figures 3 through 12depict the flat oval and round elbow loss coefficient data, plot-ted in terms of total pressure loss through the elbow as a func-tion of velocity pressure. In each instance the slopes of theleast squares curve-fit lines through the data correspond to theloss coefficients. Figures 3 through 12 are dimensionallycorrect for SI units only.

Figures 10 and 11 include loss coefficient data for 965 ×254 mm (38 × 10 in.) and 940 × 254 mm (37 × 10 in.) hardbend elbows with vanes, respectively. In these cases the tarepressure loss was calculated using the relative roughness dataobtained when a plenum chamber and bellmouth weremounted in the flow apparatus per Figure 2b. All other zero-length flat oval elbow loss coefficients determined in this proj-ect were based on the relative roughness data measured whenan abrupt transition was used to connect the nozzle chamber tothe ductwork, as shown in Figure 2a. The resulting loss coef-ficient data measured in this project are summarized inTable 5. Values of the linear correlation coefficient r obtainedby means of Equation 11 are provided in Table 6. Similarlyestimates of the error in the slopes of the least squares curvescalculated per Equation 14 are included in Table 6.

m

xiyii 1=

n

∑

xi2

i 1=

n

∑-------------------.=

r2 explained variationtotal variation

----------------------------------------------.=

r

xi x–( ) yi y–( )

i 1=

n

∑

xi x–( )2 yi y–( )2

i 1=

n

∑i 1=

n

∑

12---

---------------------------------------------------------------------=

x y

x 1n--- xii 1=

n

∑=

y 1n--- yi .i 1=

n

∑=

1 r 1+≤ ≤–

mΔ CΔ t± a 2⁄( )n 2–sy x⁄Sxx----------= =

sy x⁄1n 2–------------ yi y– xi( )[ ]2

i 1=

n

∑⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

12---

=

Sxx2 xi x–( )2

i 1=

n

∑=

x


Figure 3 356 × 152 mm (14 × 6 in.) elbow loss coefficient data.




Figure 12 762 mm (30 in.) elbow loss coefficient data.


Table 5. Pressure Loss Coefficients of Elbows

A × a,mm × mm(in. × in.)

5-Gore Easy BendMitered Hard Bend—

With VanesMitered Hard Bend—

No VanesMitered Easy Bend—

With VanesMitered Easy Bend—

No Vanes

A/a C a/A C a/A C A/a C A/a C

356 × 152(14 × 6)

2.33 0.50 0.43 1.07 0.43 1.68 2.33 1.03 2.33 1.45

381 × 102(15 × 4)

3.75 0.82 0.27 1.79 0.27 2.44 3.75 1.93 3.75 2.17

559 × 152(22 × 6)

3.67 0.50 0.27 1.45 0.27 2.11 3.67 1.03 3.67 1.40

584 × 152(23 × 6)

3.83 0.54 0.26 1.42 0.26 1.51 3.83 1.22 3.83 1.81

838 × 152(33 × 6)

5.50 1.06 0.18 1.68 0.18 2.40 5.50 1.53 5.50 2.00

559 × 254(22 × 10)

2.20 0.51 0.45 0.91 0.45 1.66 2.20 0.89 2.20 1.65

787 × 356(31 × 14)

2.21 0.45 0.45 0.90 0.45 1.45 2.21 0.69 2.21 1.50

965 × 254(38 × 10)

3.80 0.32 0.26 1.10 0.26 2.61 3.80 0.51 3.80 1.26

940 × 254(37 × 10)

3.70 0.87 0.27 1.23 0.27 1.91 3.70 1.17 3.70 1.73

762 × 762(30 × 30)

5-Gore Mitered—With Vanes Mitered—No Vanes

A/a C A/a C A/a C

1.00 0.44 1.00 0.72 1.00 1.49

Table 6. Flat Oval Elbow Loss Coefficient Correlation Analysis

Cross Section,

mm × mm(in. × in.)

5-Gore Easy BendMitered Hard Bend—

With VanesMitered Hard Bend—

No VanesMitered Easy Bend—

With VanesMitered Easy Bend—

No Vanes

r2 r2 r2 r2 r2

356 × 152(14 × 6)

0.995 0.03 0.990 0.07 0.993 0.12 0.995 0.07 0.993 0.10

381 × 102(15 × 4)

0.998 0.03 0.998 0.06 0.998 0.08 0.998 0.07 0.997 0.09

559 × 152(22 × 6)

0.998 0.01 0.990 0.09 0.996 0.08 0.999 0.03 0.997 0.06

559 × 254(22 × 10)

0.999 0.02 1.000 0.02 0.999 0.05 1.000 0.01 1.000 0.02

584 × 152(23 × 6)

0.996 0.03 0.998 0.05 0.999 0.04 0.996 0.08 0.998 0.07

787 × 356(31 × 14)

0.997 0.02 0.999 0.02 0.997 0.08 0.999 0.02 0.999 0.07

838 × 152(33 × 6)

0.993 0.08 0.998 0.07 0.999 0.08 0.998 0.06 0.995 0.13

965 × 254(38 × 10)

0.991 0.03 0.999 0.04 0.996 0.16 0.997 0.03 0.998 0.06

940 × 254(37 × 10)

0.999 0.03 0.942 0.02 0.999 0.08 0.999 0.04 0.999 0.02

762 × 762(30 × 30)

5-Gore Easy Bend Mitered—With Vanes Mitered—No Vanes

r2 r2 r2

0.979 0.05 0.978 0.10 0.994 0.10

CΔ CΔ CΔ CΔ CΔ

CΔ CΔ CΔ


CONCLUSIONS

Pressure loss tests were performed on 5-gore 90° easybend flat oval elbows, mitered 90° easy bend flat oval elbowswith and without vanes, and mitered 90° hard bend flat ovalelbows with and without vanes. The measured total pressureloss across each elbow was plotted as a function of velocitypressure. A zero length loss coefficient was calculated byfitting a least squares curve to the pressure loss data. A losscoefficient table (loss coefficient as a function of aspect ratioand dimensionless hydraulic diameter) suitable for insertioninto the DFDB was prepared for each fitting; these tables areavailable in Kulkarni et al. (2008).

The 5-gore easy bend elbow consistently had the lowestloss coefficient of the various types of elbows tested, for everycross section. For hard bend and easy bend mitered elbows,those with turning vanes had a lower loss coefficient than thesame size elbow without turning vanes. In general for eachcross section the highest loss coefficients were associated withthe hard bend mitered elbows without turning vanes. The qual-ity of installation of turning vanes can impact the elbow losscoefficient, as improperly installed vanes can create moreturbulence and pressure loss. In this project the vanes weremounted in the elbows strictly according to the specificationsof the manufacturer.

The aforementioned sudden transitions used to connectthe nozzle chamber to the flat oval ducts per Figure 2a werechosen to minimize the overall extent of the test setups, in orderto accommodate their lengths in the laboratory. It becameapparent that test configurations having large major dimen-sions were particularly prone to flow-induced vibrations whensuch transitions were employed. Hence the test program wasinterrupted so as to increase the overall length of the laboratoryby the removal of several walls and internal obstructions. Thisallowed insertion of a plenum chamber and bellmouth into theapparatus per Figure 2b, thereby bringing the setup intocompliance with ANSI/ASHRAE Standard 120-1999. Thiswas very successful, in that no random or excessive pressurefluctuations were apparent during these tests. In the future it isrecommended that flat oval duct and elbow pressure loss testsbe performed in strict accordance with ANSI/ASHRAE Stan-dard 120-1999, particularly for flat oval geometries with largemajor spans. It is likewise suggested that flat oval elbowshaving additional aspect ratios and hydraulic diameters(including round elbows having an aspect ratio of unity) betested, in order to augment the database.

ACKNOWLEDGMENTS

The work reported in this paper is the result of cooperativeresearch between ASHRAE (RP-1319) and Tennessee TechUniversity. The project was sponsored by TC 5.2, DuctDesign, and their technical assistance is gratefully acknowl-edged. The authors are deeply indebted to Brad Thomas, Chairof TC 5.2 and Vice-President of Hamlin Sheet MetalCompany, Inc., who donated the ducts, elbows, and bell-mouths used in this study.

NOMENCLATURE

A = major duct dimension, mm (in.) or cross-sectional duct area, m2 (ft2)

a = minor duct dimension, mm (in.)b = intercept of fitted line to dataC = loss coefficient, dimensionlessc = confidence interval, %Dh = hydraulic diameter, mm (in.)f = friction factor, dimensionlessL = length, m (ft)m = slope of fitted line to datan = number of data points

= velocity pressure, Pa (in. wg)R = elbow turning radius, mm (in.)r = linear correlation coefficient, dimensionlessSxx = total squared variationsy/x = standard error of y-data about the curve fitV = average velocity, m/s (ft/min)xi, yi = variables

, = mean values= frictional pressure losses, Pa (in. wg)= static pressure differential, Pa (in. wg)= total pressure differential, Pa (in. wg)= loss coefficient uncertainty, dimensionless

ρ = air density, kg/m3 (lbm/ft3)

Subscripts

1 = plane 12 = plane 27 = plane 78 = plane 8

Acronyms

EB = easy bend HB = hard bend

REFERENCES

ANSI/ASHRAE Standard 120-1999. Method of Testing toDetermine Flow Resistance of HVAC Ducts and Fit-tings. Atlanta: American Society of Heating, Refrigerat-ing and Air-Conditioning Engineers, Inc.

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

Beckwith, T.J., R.D. Marangoni, J.H. Lienhard V. 1993.Mechanical Measurements, 5th Edition. New York:Addison-Wesley Publishing Company, Inc.

Bethea, R.M., B.S. Duran, and T.L. Boullion. 1995. Statisti-cal Methods for Engineers and Scientists, 3rd Edition.New York: Marcel Dekker, Inc.

Idem, S., D. Kulkarni, and S. Khaire. 2008. Laboratory Test-ing of Duct Fittings to Determine Loss Coefficients.Final Report, ASHRAE RP-1319. Atlanta: AmericanSociety of Heating, Refrigerating and Air-ConditioningEngineers, Inc.

Kulkarni, D., and Khaire, S. and Idem, S. 2008. Influence ofAspect Ratio and Hydraulic Diameter on Flat Oval ElbowLoss Coefficients. ASHRAE Transactions, In Press.

pv

x ypfΔpsΔptΔCΔ

flat oval elbow loss coefficients

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