Pressure drop characteristics of viscous fluid flow across

Retrospective Theses and Dissertations Iowa State University Capstones, Theses and Dissertations

1-1-2002

Pressure drop characteristics of viscous fluid flow across orifices Pressure drop characteristics of viscous fluid flow across orifices

Leo Michael Mincks Iowa State University

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Pressure drop characteristics of viscous fluid flow across orifices

by

Leo Michael Mincks

A thesis submitted to the graduate faculty

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

Major: Mechanical Engineering

Program of Study Committee: Srinivas Garimella, Major Professor

Richard Pletcher Leroy Sturges

Iowa State University

Ames, Iowa

2002

Copyright© Leo Michael Mincks 2002. All rights reserved

11

Graduate College

Iowa State University

This is to certify that the Master's thesis of

Leo Michael Mincks

has met the thesis requirements of Iowa State University

Signatures have been redacted for privacy

iii

TABLE OF CONTENTS

LIST OF FIGURES

LIST OF TABLES

NOMENCLATURE

ABSTRACT

CHAPTER 1 INTRODUCTION

1.1 Background

1.2 Orifice Terminology

1.3 Scope of Current Research

1.3.1 Research Objectives

1.3 .2 Thesis Organization

CHAPTER 2 LITERATURE REVIEW

2.1 Incompressible Flow

2.2 Compressible Flow

2.3 Summary

CHAPTER 3 EXPERIMENTAL SET-UP AND PROCEDURES

3.1 Test Section Fabrication

3.2 Test Loop Description

3.3 Instrumentation

3 .4 Experimental Procedures

CHAPTER 4 ANALYSIS AND DISCUSSION OF RESULTS

4.1 Data Analysis

4.2 Experimental Results

Vl

x

Xl

Xlll

1

1

2

4

4

4

6

6

18

21

30

30

34

38

39

41

41

47

lV

4.2.1 Effect of Fluid Temperature 47

4.2.2 Effect of Orifice Thickness 51

CHAPTER 5 ORIFICE MODELING 55

5.1 Effect of Aspect Ratio on Euler Number 55

5.2 Comparisons with Previous Work 56

5.2.1 Small Aspect Ratios 56

5.2.2 Large Aspect Ratios 60

5.3 Model Development 64

CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS 74

6.1 Conclusions 74

6.2 Recommendations 76

BIBLIOGRAPHY 78

APPENDIX A DATA ANALYSIS 82

A.1 Sample Calculations 82

A.2 Error Analysis 87

APPENDIX B DENSITY AND VISCOSITY UNCERTAINTIES 91

B.1 Uncertainties Due to Pressure and Temperature Measurement 91

B.2 Uncertainty in the FORTRAN Subroutine Property Data 96

B.3 Overall Uncertainty Calculation 96

APPENDIX C EFFECT OF TEMPERATURE AND THICKNESS ON THE 0.5 AND 3 mm ORIFICES 99

C.l Effect of Temperature and Thickness on the 0.5 mm Orifices 99

C.2 Effect of Temperature and Thickness on the 3 mm Orifices 103

v

APPENDIX D ORIFICE BACK-CUT CALCULATIONS 107

VI

LIST OF FIGURES

Figure 1. Orifice Geometry 2

Figure 2. Geometries of Orifice Plates: (a) Square-Edged; (b) ASME Standard (Square-Edged with 45° Back-cut); (c) Sharp-Edged; (d) Streamlined-Approach (Rouse and Jezdinsky (1966)); (e) Sloping-Approach (Zhang and Cai (1999)); (e) Quadrant-Edged 3

Figure 3. Pressure Tapping Arrangements: (a) Flange Taps; (b) Flange Corner Taps; ( c) Vena Contracta Taps; ( d) Pipe Taps 4

Figure 4. Photograph of Test Section 30

Figure 5. Photograph of Flange Face 31

Figure 6. Test Section Dimensional Drawing 32

Figure 7. Orifice Plate Cross-Sectional Dimensions 33

Figure 8. Orifice Plate Photograph 34

Figure 9. Photograph of Test Loop 36

Figure 10. Test Loop Schematic 37

Figure 11. Flow Area Schematic For Pressure Drop Calculations 42

Figure 12. Schematic of Test Section With and Without The Orifice Installed 43

Figure 13. Loss Coefficient for a Sudden Contraction (Munson et al 1998) 46

Figure 14. Experimental and Calculated Pressure Drops for the Test Section with the Orifice Plates Removed, Shown in Comparison With the 1 mm Diameter, 3 mm Thick Orifice Data 47

Figure 15. Relative Contributions of Test Section Plumbing to Total Measured Pressure Drop 48

Figure 16. Effect of Temperature on Pressure Drop - Flow Rate Characteristics for the 1 mm Diameter, 1 mm Thick Orifice Plate 49

Figure 17. Effect of Temperature on Pressure Drop -Flow Rate Characteristics for the 1 mm Diameter, 2 mm Thick Orifice Plate 50

Vll

Figure 18. Effect of Temperature on Pressure Drop-Flow Rate Characteristics for the 1 mm Diameter, 3 mm Thick Orifice Plate 50

Figure 19. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics for the 1 mm Diameter Orifice, T ~ 20°C 52



Figure 22. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics, for the 1 mm Diameter Orifice T ~ 50°C 54

Figure 23. Effect of Aspect Ratio on Euler Number, T ::::: 20°C 55




Figure 27. Euler Number Variation for lid= 0.33 58

Figure 28. Euler Number Variation for l/d = 0.66 59

Figure 29. Euler Number Variation for lid= 1 60





Figure 34. Comparison of the Predicted Eu Numbers to Experimental Eu Numbers 66

Figure 35. Comparison of Measured and Predicted Eu for the 0.5 mm Diameter, 1 mm Thick Orifice 68


Vlll


Figure 38. Comparison of Measured and Predicted Eu for the 1 mm Diameter, 1 mm Thick Orifice 69






Figure 44. Model Predictions and Trends for a Diameter Ratio of 0.0231 72



Figure 47. Effect of Temperature on Euler Number for l/d ~ 3 75

Figure Bl. Effect of Pressure and Temperature on p 91

Figure B2. Effect of Pressure and Temperature onµ 92

Figure B3. Effect of Pressure and Temperature on 8p/ 8T 94

Figure B4. Effect of Pressure and Temperature on 8p/ 8P 95

Figure B5. Effect of Pressure and Temperature on 8µ/ aT 95

Figure B6. Effect of Pressure and Temperature on 8µ/ 8P 96

Figure Cl. Effect of Temperature on Pressure Drop - Flow Rate Characteristics for the 0.5 mm Diameter, 1 mm Thick Orifice Plate 99

lX

Figure C2. Effect of Temperature on Pressure Drop - Flow Rate Characteristics for the 0.5 mm Diameter, 2 mm Thick Orifice Plate 100

Figure C3. Effect of Temperature on Pressure Drop - Flow Rate Characteristics for the 0.5 mm Diameter, 3 mm Thick Orifice Plate 100

Figure C4. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics for the 0.5 mm Diameter Orifice, T "" 20°C 101



Figure C7. Effect of Temperature on Pressure Drop-Flow Rate Characteristics for the 3 mm Diameter, 3 mm Thick Orifice Plate 103

Figure CS. Effect of Temperature on Pressure Drop - Flow Rate Characteristics for the 3 mm Diameter, 3 mm Thick Orifice Plate 104

Figure C9. Effect of Temperature on Pressure Drop- Flow Rate Characteristics for the 3 mm Diameter, 3 mm Thick Orifice Plate 104

Figure ClO. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics for the 3 mm Diameter Orifice, T "" 20°C 105

Figure Cl 1. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics for the 3 mm Diameter Orifice, T "" 30°C 105

Figure C12. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics for the 3 mm Diameter Orifice, T ::::: 50°C 106

x

LIST OF TABLES

Table 1. Summary of the Literature 23

Table 2. Orifice Cross-Sectional Dimensions 33

Table 3. Flow Meter Specifications 38

Table 4. Coefficients for Equation (10) 41

Table 5. Constants for Equation (22) 65

Table Al. Raw Data Used in Calculations for Point 11-50-20 82

TableA2. Raw Data for Point 11-50-20 (S.I. Units) 82

Table A3. Test Section Dimensions and Results (1 mm Diameter, 1 mm Thick Orifice Plate) 83

TableA4. Property Values for Hydraulic oil at Pressures and Temperatures Listed 84

TableA5. Dimensions and Results for Minor Losses 86

Table A6. Uncertainties in Temperature, Pressure, and Flow Rate Measurements for the 1 mm Data 88

Table A7 Range of Reynolds Number and Euler Number Uncertainties for the Three Orifice Diameters in the Current Study 90

Table Bl. Uncertainty in Viscosity based on Pressure and Temperature Measurements 97

Table B2. Density and Viscosity Uncertainties from SWRI 97

Table B3. Overall Uncertainty in Density and Viscosity at 5 MPa 98

Table Dl. Orifice Back-Cut Diameters 108

XI

NOMENCLATURE

A cross sectional area (m2)

c energy dissipation constant

Cc contraction coefficient

Cd orifice discharge coefficient

C<lu ultimate orifice discharge coefficient

Cp velocity profile coefficient

Cv viscosity coefficient

Cve vena contracta coefficient

Cµ viscous coefficient

D pipe diameter (m)

d orifice diameter (m)

Eu Euler number (2~P/pV2)

f friction factor

g gravitational constant (9.81 m/s2)

K1 loss coefficient for sudden expansion

L length (m)

p Pressure (kPa)

Pr Prandtl number

Q flow rate (m3/s)

Re Reynolds number (pVd/µ)

St Strouhal Number

T temperature (°C)

Xll

v average velocity (mis)

Greek Letters

~ diameter ratio ( d/D)

/),, differential

E roughness (m)

µ dynamic viscosity {kg/m-s)

p density (kg/m3)

(J surface tension (Nim)

~ two-phase multiplier

Sub-scripts and Super-scripts

eff effective

f frictional

fo fluid only

g gas

liquid

lam laminar

m measurement

0 orifice

s SWRI properties

seg segment

t turbulent

tp two-phase

turb turbulent

Xlll

ABSTRACT

An experimental study of the flow of highly viscous fluids through small diameter

orifices was conducted. Pressure drops were measured over a wide range of flow rates for

each of nine different orifices, including orifices of 0.5, 1 and 3 mm nominal diameter, with

three thicknesses (nominally 1, 2 and 3 mm) tested for each diameter. The data were non-

dimensionalized to obtain Euler numbers and Reynolds numbers for the aspect ratio range

0.32 < l/d < 5.72, and orifice-to-pipe diameter range 0.023 < p < 0.137. It was found that in

the laminar region, increases in aspect ratio resulted in an increase in Euler number at the

same Reynolds number, while increases in diameter ratio resulted in an increase in Euler

number for a similar aspect ratio. In the transition region, the Reynolds number was less

significant in determining Euler number, tending toward a constant value dictated by the

diameter ratio and aspect ratio as the flow became progressively turbulent. . The data were

correlated using different expressions for the laminar and turbulent regions, which were then

combined to yield one continuous function for the Euler number as a function of Reynolds

number and the geometric parameters for the entire range of data. The model predicted

84.4% of the data to within ± 25% and is valid for the following range of conditions: 0.32 <

l/d < 5.72, 0.023 < p < 0.137, 8 <Re< 7285, and 0.028 < µ < 0.135 (kg/m-s).

1.1 Background

1

CHAPTER 1

INTRODUCTION

In current automotive and hydraulic applications, oil flowing to some components is

channeled through small openings, which can be simulated by small-diameter, square-edged

orifices. Because the oil is highly viscous, flow through the orifice tends to remain laminar

even at large flow rates and pressure drops. Currently, the most commonly available orifice

flow relationships are those developed for thin, large-diameter orifices such as those used in

flow meters. In real world applications, however, oil must flow through orifices of varying

thickness governed by considerations such as component strength and manufacturability.

The proposed research therefore addresses the problem of relating flow rate to pressure drop

across square-edged orifices of different thickness.

Although considerable research has been conducted in the study of orifice flow

characteristics, the majority has been devoted to applications involving flow meters. These

orifices typically have diameter ratios (~) in the range of 0.2 to 0.75 and aspect ratios (l/d)

less than 1. Figure 1 shows the orifice geometry and terms that will be used throughout this

discussion. For comparison, orifices of interest for the present study have diameter ratios of

0.022, 0.044 and 0.132 with aspect ratios ranging from 0.33 to 6. Additionally, the fluid used

in this investigation is highly viscous (0.023 kg/m-s < µ < 0.152 kg/m-s) in nature.

The standard convention for relating orifice flow rate to differential pressure is

through the use of the orifice discharge coefficient (Cct) as seen in Equation (1).

2

Q =C A ~2!1.P d or p (1)

Past research (Lichtarowicz et al. 1965 and Sahin and Ceyhan 1996) has also shown

that at low flow rates, Cd is generally considered to be a function of the aspect ratio, the

diameter ratio(~), and the orifice Reynolds number (Re) as shown in Equation (2). At high

Reynolds numbers, the effects of aspect ratio and Reynolds number decrease, with Cd

depending primarily on diameter ratio (Grose 1985).

Figure 1.

Orifice Geometry D = Upstream Pipe Diameter d = Orifice Diameter I = Orifice Thickness

Diameter Ratio (~) = d/D Aspect Ratio = ltd

Orifice Geometry

1.2 Orifice Terminology

D

(2)

Figure 2 shows the geometries for several different types of orifices that will be

discussed in subsequent chapters, while Figure 3 shows the standard tapping arrangements

generally used by orifice measuring devices (ASME 1990 Report: MFC-3M-1989). The

dimensions given in Figures 3c and 3d are based on characteristics of the flow meter and are

3

influenced by the installed geometry. For the pipe tap arrangement, the dimensions dl and

d2 are usually either equal to each other (dl = d2) or based on the pipe diameter such that dl

= D and d2 = D/2. The taps for vena contracta meters are somewhat different in that the

location of the downstream tap is based on the lowest pressure in the flow profile. For these

meters, the upstream tap is located at dl = D, while the downstream tap usually lies between

0.3D andD.

0 d 0 d 0 d

0.140 (a) (b) (c)

0--- d v r=d

0.140 (d) (e) (f)

Figure 2. Geometries of Orifice Plates: (a) Square-Edged; (b) ASME Standard (Square-Edged with 45° Back-cut); (c) Sharp-Edged; (d) Streamlined-Approach (Rouse and Jezdinsky (1966)); (e) Sloping-Approach (Zhang and Cai (1999)); (e) Quadrant-Edged

Figure 3.

4

Jl. ___ _ ' _J_ ,_ J (a) (b)

(c) (d)

Pressure Tapping Arrangements: (a) Flange Taps; (b) Flange Corner Taps; (c) Vena Contracta Taps; (d) Pipe Taps

1.3 Scope of Current Research

1.3.1 Research Objectives

Based on the above discussion, the research objectives of this study are as follows:

Determine the flow rate for a given pressure drop for several different orifice

plates.

Develop a model for the flow of viscous fluids through large contraction ratio

orifices that accounts for the effects of geometry and fluid properties.

1.3.2 Thesis Organization

The organization of this thesis is as follows:

5

hapter 2 provides a review of the literature on experimental and theoretical

s dies of orifice flow characteristics and discusses the need for further research

this area.

hapter 3 describes the experimental set-up used for this study along with the

p ocedures adopted for conducting the research.

presents experimental data and the resulting non-dimensional

hapter 5 compares the results of the present study with the literature and

velops a model for orifice flows.

inally, Chapter 6 summarizes the important conclusions of this study and

ovides some recommendations for further work in this area.

6

CHAPTER2

LITERATURE REVIEW

The literature available on the subject of small-diameter orifices can be categorized

into the following two general categories: incompressible flow and compressible flow.

2.1 Incompressible Flow

In 1929 Johansen ( 1930) constructed a test facility that allowed visual observation of

the flow characteristics in sharp-edged orifices. Using water, Castor oil (v = l .209 x 10-3

m2/s at 18 °C) and mineral oil (v = 1.14 x 10-4 m2/s at 18 °C) as the working fluid, tests were

conducted to determine the discharge coefficients for orifices with five different diameter

ratios (~ = 0.090, 0.209, 0.401, 0.595, and 0.794) over a range of Reynolds numbers from

less than 1.0 to 25,000. He tried to interpret the resulting plot of the discharge coefficients

based on the flow mechanisms observed in the dye injection test. He found that for

Reynolds numbers less than 10, Cd increases linearly with a constant slope and corresponds

to the steady flow conditions seen in the dye test. A further increase in Reynolds number up

to a value of 250 results in a non-linear increase in Cd up to its maximum, and corresponds to

the formation of a divergent jet in the flow patterns. Cd then begins to decrease as vortices

appear in the flow until it reaches a steady value of approximately 0.615 as the flow become

turbulent at Reynolds numbers above 2000. Johansen also notes that as the diameter ratio

increases, the Reynolds number at which these flow transitions occur is higher.

Shortly thereafter, Tuve and Sprenkle (1933) conducted over 500 experiments for the

Bailey Meter Company to establish an extensive plot of Cd versus Reynolds number. To

encompass the range of 4 <Re< 40,000, tests were conducted with water, light paraffin oil,

7

light motor oil, and heavy motor oil (v = 1.62 x 10·3 m2/s) as the working fluids. The eight

orifices used in the testing were constructed of brass, monel, and stainless steel, with

thicknesses of 0.794 mm (1/32 in) and diameter ratios ranging from 0.2 to 0.8. The orifices

were beveled at 45° on the downstream side to produce an orifice edge length of 0.397 mm

( 1/ 64 in). Based on the results of their experiments, the authors recommended that orifice

meters have diameter ratios between 0.2 and 0.5, and that they only be used for flow rates

corresponding to a Reynolds number of 100 or greater. They also compared their results

with data from authors such as Johansen (1930), Witte (1928), and Hodgson (1929) and

proposed that the slight differences in their results were due to a lack of similarity in

variables such as orifice bevel angle or pipe diameter.

Medaugh and Johnson (1940) constructed a test facility that could measure flow rate

and pressure drop across brass orifices at various conditions using water as the test fluid.

Orifices were constructed from 6.35 mm (0.25 in) brass sheet with diameters ranging from

6.35 to 50.80 mm (0.25 to 2 in) and pressure drops ranging from approximately 2.41 to 358.5

kPa (0.35 to 52 psi). It was observed that as the flow rate through the orifice increased, the

discharge coefficient dropped and that as the orifice diameter increased, the discharge

coefficient decreased for the same pressure drop. The authors determined that if the flow rate

was increased enough, the discharge coefficient would eventually drop to a value of 0.588

which was 6% lower that the data from Smith and Walker (1923) which was widely used at

the time. This was attributed to potential problems in the Smith and Walker data due to

bowing of the thin plate from the pressure, or from a depression that might have occurred

around the orifice opening during the drilling process.

By the early 1960s, investigators began to examine other effects that influence Cd

8

such as the aspect ratio. In 1965 Lichtarowicz et al. ( 1965) presented the results of

investigations by James (1961), Sanderson (1962), and Morgan (1963) who examined the

effects of aspect ratio on the discharge coefficients of square-edged orifices. Testing by

these three investigators was conducted on orifices with aspect ratios ranging from 0.5 to 10

with 1 < Re < 50,000. Lichtarowicz et al. (1965) then compared Cd values from these

investigations with data from previous investigations and found a correlation between the

aspect ratio and the maximum or ultimate value of the discharge coefficient (Cdu). As the

aspect ratio increases from 0 to approximately 1, Cdu rises linearly from 0.61 to 0.78, while in

the range of aspect ratios from 1 to 2, the increase is non-linear and achieves a maximum

value of 0.81. Further increases in aspect ratio result in a gradual linear decrease in Cdu to a

value of 0.74 at an aspect ratio of 10. Based on their results, the authors recommended

changes to the previously proposed equations for Cd and Cdu·

Al vi et al. ( 1978) compared the flow characteristics of nozzles and sharp-edged

orifices to those of quadrant-edged orifices. They conducted tests on these flow geometries

with diameter ratios of 0.2, 0.4, 0.6, and 0.8 for each geometry and orifice Reynolds numbers

in the range of 1 to 10000. They found that quadrant-edged orifices exhibit pressure drops

similar to those of sharp-edged orifices at low Reynolds numbers, while pressure drops at

high Reynolds numbers are closer to pressure drops in nozzles. They also suggest that the

flow characteristics of orifices can be divided into four regimes: Fully Laminar Region,

Critical Reynolds Number Region, Relaminarising Region, and Turbulent Flow Regime.

During the 1970s orifice meter pressure drop equations published by engineering

societies and meter manufactures received further scrutiny. Miller (1979) compared

laboratory flow data from different orifice-type flow meters with two commonly used

9

equations for predicting the flow characteristics of these flow meters: the ASME-AGA

(ASME 1971, AGA 1955) equation, and the IS0-5167 (1978) (or Stolz 1975) equation. By

using statistical analysis, he found that for flange tap orifice meters with 0.25 < P < 0.75 and

pipe diameters from 102 to 610 mm ( 4 to 24 in), these equations are accurate to ± 1 %, with

the Stolz (1975) equation being better. He also states that based on the work of Miller and

Kneisel (1974), it would be possible to further reduce these uncertainties to ± 0.5% with

better data.

Grose (1983) suggests that the orifice discharge coefficient is comprised of three

additional coefficients (the viscosity coefficient, the contraction coefficient, and the velocity

profile coefficient) such that Cd= Cc·Cv·Cp. He used the Navier-Stokes equations to model

an orifice and proposed a "viscosity coefficient." At low Reynolds numbers, the contraction

coefficient and the velocity coefficient tend to a value of one, resulting in the discharge

coefficient being a function of only the viscosity coefficient. He then compares viscosity

coefficients with experimentally determined discharge coefficients for Re < 16 and shows

excellent agreement between the two. Beyond this range, the viscosity coefficient over-

predicts the value of the discharge coefficient, which is most probably due to the contraction

coefficient beginning to decrease in value from one, which in tum causes a decrease in the

value of the discharge coefficient. In a subsequent paper (Grose (1985)), he develops

equations for the contraction coefficient (Cc). Using the Navier-Stokes equations with an

elliptical surface profile, the contraction coefficient is predicted solely as a function of

diameter ratio CP = d/D). He proposes that for purely inviscid flow (Re > 105), the effects of

the viscous and profile coefficients can be ignored, resulting in Cd= Cc. For diameter ratios

between 0 and 0.75, comparisons are made between the elliptical equation and modified

10

empirical equations derived from data from Stolz (1975)) and Miller (1979), and the ASME-

AGA orifice equation with vena contracta taps as presented by Miller ( 1979). The equations

agree quite well up to a diameter ratio of about 0.4, at which point the elliptical equation

begins to over-predict the results of the empirical equations. It is suggested that this occurs

because the empirical equations do not take into account the fact that the velocity profile

coefficient tends to unity as the Reynolds number tends to infinity. For contraction

coefficients, the empirical equations diverge from the theory and each other, with only the

Miller (1979) equation still moving in the direction suggested by the theory.

During the 1990s, continuous improvements in computer technology lead to a

greater number of orifice flow problems being solved numerically. Jones and Bajura (1991)

developed a numerical solution for laminar, pulsating flow through an orifice with Reynolds

numbers ranging from 0.8 to 64 and Strouhal numbers (St) ranging from 10-5 to 100, where:

S 2rrfD t=--v

where: f pulsation frequency D = pipe diameter V = average velocity in pipe

(3)

Two different orifice geometries with diameter ratios (~) of 0.5 and 0.2 were used in

their analysis. The plate thickness for each orifice was fixed at 0.2 times the pipe diameter

and a 45-degree bevel was introduced into the downstream side of each orifice to a depth of

50 percent. Navier-Stokes equations were used as the starting point in their analysis and

initial comparisons with steady flow data from Johansen (1930), Tuve and Sprenkle (1933),

and Keith (1971) as presented by Coder (1973) showed good agreement with the resulting

discharge coefficients. Results from the pulsating analysis were plotted as discharge

11

coefficient vs. time for every 45 degrees, and show that the discharge coefficient initially

oscillates around the discharge coefficient that would be expected for steady flow. As the

Strouhal number increases (increase in pulsation frequency), both the amplitude of the

oscillations and the time-averaged discharge coefficient begins to decrease. By plotting the

normalized discharge coefficient (mean discharge coefficient divided by steady flow

discharge coefficient) vs. the natural log of the Pulsation Product (Pp = ReSt) it is seen that

the normalized discharge coefficient remains fairly constant until ln(Pp) = -2.5. At this point,

the discharge coefficient begins to drop rapidly and becomes 40 percent of the steady flow

value at ln(Pp) = 2.5.

Sahin and Ceyhan (1996) used experiments and numerical analysis to examme

incompressible flow through orifices with diameter ratios of 0.5 and aspect ratios ranging

from 0.0625 to 1. A gear pump was used in their experiments to circulate oil through an

orifice at temperatures ranging from 30°C to 50°C with resulting Reynolds numbers ranging

from less than 1 to 150. The numerical analysis was conducted using two-dimensional

Navier-Stokes equations for axisymetric, viscous, incompressible flow through a square-

edged orifice in a circular pipe. The resulting equation for the discharge coefficient is shown

below:

(4)

where: V max is the velocity at the centerline of the pipe.

The numerical results were compared with their own experimental results and with those of

Nigro et al. (1978), Alvi et al. (1978) and Johansen (1930), and were found to agree within

5%.

12

Hasegawa et al. ( 1997) examined several thin orifices ranging from 1 mm to 10 µm in

diameter. Experiments were performed with distilled water (v = 1.00 x 10-6 m2/s), silicon

oils (v = 1.10 x 10-6, 2.22 x 10-6, and 5.13 x 10-6 m2/s), and glycerin solutions (v = 1.69 x

10-6, 2.39 x 10-6, 3.44 x 10-6, and 5.28 x 10-6 m2/s) as the working fluids. The resulting

pressure drop to flow rate relationship was examined for Reynolds numbers in the range of 1

to 1000. Additionally, numerical analysis was conducted for these same flow conditions.

The numerical solution compares quite well for orifices above 65 µm but under-predicts the

pressure drop for smaller ones. The under-prediction becomes worse as either the orifice

diameter or the fluid viscosity decreases. To explain this, the authors examined possible

causes such as material used in construction, burring that occurred in manufacturing, and

boundary layer thickness increases due to ionic effects of the liquid. They found that none of

these causes could produce the increases in pressure drop that were seen between the

experimental data and the numerical solution. It is unclear, however, if the increase in the

length-to-diameter ratio, which increased as the orifice diameter decreased for all orifices,

was ever examined as a possible cause by the authors.

Dugdale ( 1997) mathematically modeled the radial and angular velocity profiles of a

sharp-edged orifice. An experimental apparatus was also constructed to test molasses at flow

rates corresponding to Reynolds numbers on the order of 10-4. Two 0.082 mm thick orifice

plates were constructed from brass with diameters of 5.1 mm and 2.396 mm respectively.

For an applied pressure of 2.121 kPa, an energy dissipation constant (C) was experimentally

determined such that:

Eu = 4rrC(Re t1 (5)

These data were compared with Bond (1922) experimental results on mixtures of glycerin

13

and water. The energy dissipation constant calculated from Bond (3.21) was within the range

3.17 < C < 3.30 predicted from their data.

Zhang and Cai (1999) conducted an investigation to examme the pressure drop

characteristics of orifices with different profiles and contraction ratios. Of primary concern in

their investigation was the identification of the orifice geometry that produced the lowest

local downstream wall pressure for a given overall pressure drop. By minimizing excessive

pressure drops across orifices in flood conduits, cavitation and the resulting damage to

concrete tunnels and orifices can be reduced or eliminated. A model resembling a flood

conduit used in dam construction was fabricated for testing of orifices. Orifices with four

different diameter ratios ranging from 0.5 to 0.8 were tested with Reynolds numbers ranging

from 1.04 x 105 to 2 x 105• The authors gave a formula for the Euler number as a function of

the diameter ratio such that Eu ~ (1-P2) 2/P4 and found that for Euler numbers between 0.5

and 4, the sloping-approach type orifice worked best.

McNeil et al. (1999), interested in modeling small pressure relief valves, constructed

a test facility to measure the flow rate, pressure drop, and momentum effects in a nozzle and

an orifice. The nozzle and orifice both had a diameter ratio (p) of 0.491 and tests were

conducted with Reynolds numbers ranging from 40 to around 400 using a solution of

Luviskol K90 in water as the working fluid. The momentum results were determined from

the impingement of the fluid onto a balance plate as it was discharged from the test loop into

a catch tank. The data from the momentum test were used to calculate the actual momentum

correction factor which is the reciprocal of the contraction coefficient (Cc). The velocity

coefficient (Cv) was found by using the equation for the discharge coefficient (Cd = Cc·Cv)

from Massey (1975). The authors concluded that the contraction coefficient tends to unity at

14

low Reynolds numbers, and that the discharge coefficient is dependent on both the Reynolds

number and the flow geometry.

Valle et al. (2000) constructed a nozzle flow meter which used orifices with 45°

converging and diverging sections and diameters ranging from 0.6 to 3.0 mm. The flow

meter was initially tested with water and oil(µ= 0.08 and 1.62 kg/m-s), at room temperature

(:~25°C), and flow rates ranging from 2.0 to 75 ml/s. The results were presented as plots of

Euler number vs. Reynolds number for the two fluids. The authors conclude that "At low ·

Reynolds numbers, the flow is purely laminar and the pressure drop increases proportionally

with the viscosity. At high Reynolds numbers, the flow is dominated by inertia and the

pressure drop becomes independent of viscosity." The flow meter was then used to

investigate the extensional properties of a Boger (Boger and Walters 1993) fluid and a

Newtonian fluid with suspended solids. A Boger fluid is a fluid that exhibits significant

elastic properties while the viscosity remains independent of shear rate (Valle et al. 2000).

The authors showed that it was possible to determine the extensional viscosity (~) of

visceolastic fluids and suspensions from the following equation:

(6)

Where: 3µ = extensional viscosity of a Newtonian fluid. R = the vertical shift between the elastic fluid data and the

Newtonian fluid data on the Eu vs. Re plot.

They note that the extensional viscosity of the fluids was found to be about 45 times that of

the shear viscosity which was comparable to the findings of Sridhar ( 1990) for a different

Boger fluid.

Another area of interest involving orifices has been the concern that cavitation on the

downstream side of an orifice could affect the discharge coefficient or cause damage to the

15

system. Kim et al. ( 1997) investigated the effects of cavitation and plate thickness on the

orifice discharge coefficient by conducting tests on 3 orifices with diameter ratios of 0.10,

0.15, and 0.33. They found that cavitation occurred for pipe Reynolds numbers (Reo = PRe)

above 14000 for a p of0.10, 43000 for a p of0.15, and 100,000 for a p of0.33. It was seen

that for the three diameter ratios, cavitation did not affect the discharge coefficient for aspect

ratios less than or equal to 0.55 over the entire range examined (4000 <Re< 170,000).

Ramamurthi and Nandakumar (1999) examined the effects of aspect ratio and

cavitation on the discharge coefficients of square-edged orifices. Orifices with diameters of

0.3, 0.5, 1.0, and 2.0 mm and aspect ratios ranging from 1 to 50 were tested at flow rates with

Reynolds numbers in the range of 2000 to 100,000. They found that for flow conditions

exhibiting attached flow, the discharge coefficient was a function of both the aspect ratio and

the Reynolds number. When the flows became separated or exhibited cavitation, however,

they found that the discharge coefficient became a function of only orifice diameter. In the

separated flow region, it was noted that as the orifice diameter decreased, the discharge

coefficient went up. It was proposed that effects such as increased wetting of the orifice

walls and surface tension-induced pressure play an increasingly important role in the

discharge coefficient as orifice diameter decreases. It was also noted that cavitation has the

greatest effect on orifices with aspect ratios of approximately 5. It was proposed that this

occurs because bubbles formed during cavitation tend to collapse very near the exit of the

orifice causing the greatest disturbance to the flow patterns.

In recent years, studies on flow through constricted geometries have been conducted

by researchers interested in the use of orifices for component cooling. In addition to orifice

flow characteristics, these researchers have also studied the effects of two-phase flow

16

conditions and examined how free jets discharging from an orifice are affected by component

spacmg.

Kiljanski (1993) examined free jets from orifices and proposed that the discharge

coefficient can be related to the orifice Reynolds number by the equation: Cct = B .JRe, where B is an experimentally determined constant based on the aspect ratio. Four liquids

(ethylene glycol [µ= 0.02 kg/m-s], potato syrup [µ= 10 kg/m-s], and two glycerol solutions

[µ= 0.15 and 0.40 kg/m-s]) were tested using five different orifices over a flow range of 0.01

< Re < 500. Three orifices with aspect ratios of 0.5 and diameters of 2, 3, and 5 mm were

used along with two additional 3 mm diameter orifices having aspect ratios of approximately

0 (sharp-edged) and 1.0 respectively. It was shown from plots of Cct versus Reynolds number

that for Re< 10, all data followed lines with a slope of approximately 0.5. Additionally, the

value for the constant B increased as the aspect ratio increased. For Re> 10, the curves for

the different aspect ratios begin to converge, and become one curve near Re = 300. The

author suggests that this occurs because of the dominant effects of kinetic energy in this

region and that for Re > 300, the aspect ratio no longer affects the discharge coefficient.

Single and two-phase flow through thick and thin orifice plates was modeled by

Kojasoy et al. (1997). The mathematical models were based on the mechanical energy

equation and used to determine the pressure drop across the sudden expansion/contraction

and the resulting pressure loss coefficient. An experimental test loop was also constructed to

test refrigerant R-113 at various flow rates. The test section consisted of ten 2 mm thick

plates that were placed in a chamber with a spacing of either 2, 4, or 8 mm between each

plate. Two sets of plates were constructed, one with 48 holes and one with 50 holes, such

that the holes were offset from one plate to the next. The plates were tested with 1 mm holes

17

that were subsequently drilled out to 2 mm and 4 mm for additional testing. For single phase

testing, the Reynolds numbers ranged from 800 to 15000 for the thick plates, and from 1100

to 11000 for the thin plates. For the thick plate, loss coefficients, which were independent of

plate spacing, were calculated from the data and fitted as shown below for single phase flow:

k 3 456 R -o.os11 thick = . . e orifice (7)

For the thin plates, the authors felt more data were needed, but presented a value of 2.1 for

the thin plate loss coefficient. Experimentation was also done for two-phase flow resulting in

the determination of two-phase multipliers (<P:J, which were compared with the values

predicted by the models and found to be within 10.5% to 14.5%. Finally, their methodology

was applied to Janssen (1966) steam-water data and correlated with a mean error of 13.8%.

Morris et al. ( 1996) used experiments and numerical modeling to predict the

impinging jet heat transfer coefficients that could be obtained from 3.18 mm and 6.35 mm

diameter orifices with Reynolds numbers of 8500, 10000, and 13000. The numerical

modeling was conducted using the finite volume code FLUENT (1995) with a turbulent

Prandtl number (Pr1) of 1.2. Heat transfer coefficients obtained numerically and

experimentally were then compared with those obtained by using the turbulent Prandtl

number equations from Wassel and Catton (1972), Gibson and Launder (1976), Malhotra and

Kang (1984), and Kays (1994). They found that the numerical model under-predicted the

experimental values by 49 to 54% and that the experimental values compared to within 16 to

20% of the predictions of Gibson and Launder (1976).

Morris and Garimella ( 1998) extended their previous work and used the finite volume

code FLUENT (1995), to determine the length of the separation region in the orifice plate,

18

the pressure losses across the orifice plate, and the flow features in the confinement region.

Numerical results were presented for various area ratios (d2/D2) and aspect ratios (lid) for

Reynolds numbers in the range of 8500 to 23,000. The authors then compared the numerical

results for the 3.18 mm and 6.35 mm diameter orifices with the experimental data from

Ward-Smith (1971). The data from Ward-Smith (1971) were correlated to yield three

equations for the discharge coefficient that were based on the aspect ratio. The numerical

losses predicted by the authors were within 5% of the empirical correlations in all cases.

Morris et al. (1999) compared the flow fields generated from their numerical

simulation with photographs and laser-Doppler velocimetry (LDV) measurements taken from

a test loop constructed by Fitzgerald and Garimella (1997). They found good agreement

between the data from the numerical simulation and the experimental data at Re > 8500 but

not at Re of 2000 and 4000. They propose that this is due to not fully accounting for the

effects of laminar/semi-turbulent flow fields in their model.

2.2 Canpressible Flow

Kayser and Shambaugh ( 1991) investigated compressible flow of gases through small

diameter orifices (0.9 < d < 1.9 mm) with geometries such as knife-edged, square-edged

straight-bore, rounded-entry, and elliptical-entry. For the flows examined, orifice Reynolds

numbers ranged from 3,000 to 80,000 and pressure drops ranged from 100 to 350 kPa. They

found that for the knife-edged orifice plate, the discharge coefficient correlates poorly to

Reynolds number but correlates quite well to the dimensionless pressure drop (Pin -Pout)/(Pcnt

-Pout), and that the discharge coefficient showed virtually no dependence on fluid temperature

or orifice diameter. For the straight bore orifices, they found that the discharge coefficient

19

was a function of both the pressure ratio (Pin/Pout) and the aspect ratio and that as the aspect

ratio increased, so did the discharge coefficient. Finally, it was observed that both the round

and elliptical-nozzles performed similarly, and that they had the highest discharge coefficient

of any of the elements tested.

More recently Gan and Riffat (1997) constructed an experimental apparatus to

measure the pressure drop across an orifice plate in a square duct using air as the working

fluid. They also experimented with a perforated plate having the same area reduction as the

orifice plate. The plates were 2 mm thick with the orifice plate having an orifice diameter of

0.239 m and the perforated plate having 145, uniformly spaced, 20 mm diameter holes. Data

for both plates were compared for Reynolds numbers ranging from 1.6 x 105 to 3.7 x 105 and

showed that the orifice plate had a lower pressure drop than the perforated plate, which

contrasted earlier findings of ldelchik et al. ( 1986). A CFD analysis using FLUENT ( 1995)

was conducted to predict the pressure loss coefficients (Euler numbers) of the orifice plate

and the perforated plate with the results being within 8% of the experimental data. The CFD

program was then used to predict discharge coefficients for orifice plates of varying

thickness. The results show that for a constant free area ratio (area of orifice/area of duct),

the pressure loss coefficient decreases as the aspect ratio increases up to an aspect ratio of

approximately 1.5. As the aspect ratio increases beyond this value, the pressure loss

coefficient value shows a small but slightly increasing variation, which is similar to the

results of Stichlmair and Mersmann ( 1978) for Reynolds numbers of 400 to 106•

Emmons ( 1997) showed that the venting that occurs when holes are created during a

building fire and can be modeled as a nozzle or an orifice. He proposed that the discharge

coefficient (Cd) is comprised of two parts: Cµ which represents viscous effects and Cve which

20

corrects for the flow area change due to the vena contracta, such that Cd= Cµ·Cve· He also

showed that at low Reynolds numbers, Cµ tends towards zero and Cve tends towards unity as

is supported by data from Heskestad and Spaulding (1991) and Tan and Jaluria (1992) over

the range 800 <Re< 4000. Mathematical equations were then developed to determine the

mass flow rate through vertical, horizontal, and inclined vents based on Cd and for the

horizontal case, the Froude Number. Based on these equations, Emmons (1997) then

determined a theoretical Froude number equation for conditions where flow is due only to

differences in density. He found good agreement between his equation and data from

Heskestad and Spaulding ( 1991) and Epstein and Kenton ( 1989) but not with those from Tan

and Jaluria (1992).

Recently Samanta et al. ( 1999) investigated pressure drops resulting from the flow of

a mixture of a gas and a non-Newtonian pseudoplastic liquid through orifices of varying

diameter. An apparatus was constructed to measure this pressure drop using air as the gas,

and sodium salt of carboxy methyl cellulose (SCMC) as the non-Newtonian liquid at

temperatures of 31°C ± 1.5°C. Three orifice plates were used with diameters of 5.9 mm, 7.6

mm, and 9.0 mm and diameter ratios of 0.4646, 0.5984, and 0.7087 respectively. Single-

phase data were collected for both water and the sodium salt mixtures with pressure drops

ranging from approximately 1 kPa to 26 kPa. Two-phase data for the air-sodium salt mixture

were also collected in this same pressure range with liquid Reynolds numbers ranging from

45 to 2200 and gas Reynolds numbers ranging from 230 to 2200. From these data, formulas

were presented for both the liquid-only and the gas-liquid pressure drops in the terms of the

non-dimensional Euler number. The liquid-only Euler number was represented as a function

of Reynolds number and contraction ratio as shown in Equation (8), while the two-phase

21

Euler number was shown to be a function the liquid Reynolds number, the gas Reynolds

number, the contraction ratio, and the fluid properties of the liquid as shown in Equation (9).

( )-4.380±0.248

Eu, = l.202 Re~o.o48±0.044 ~

[ ( 4 J-0.125±0.021 0.205±0.166 ]

Eu = Eu 1 + O 003 Re -o.536±0.063 Re o.797±0.060 gµetT (_!!_) tp I • I g 3 D p,cr,

2.3 Sumey

(8)

(9)

Although considerable work has been done in the area of orifices, deficiencies still

exist. The majority of the work reviewed has been dedicated to the study of incompressible

flow through orifices with 0.2 < ~ < 0.8 as found in orifice flow meters. Additionally, much

of this work is for flow rates corresponding to Re > 1000, which is above the area of interest

for the present study. The data that are available for Re< 1000 are primarily for orifices with

~ > 0.1 or geometries other than square-edged.

Of the data that are available for square-edged orifices with ~ < 0.1 (or d < 1 mm) and

Re < 1000, few demonstrate the effects of varying aspect ratio on Cd. Lichtarowicz et al.

(1965) presented data from Morgan (1963) in which two sets of orifices were used with ~ =

0.044 and ~ = 0.071 respectively. The first set had aspect ratios of 0.5 and 2.0, while the

second set had aspect ratios of 1.0 and 4.0. Morgan's data are limited to Re< 100 but show

that for this region an increase in the aspect ratio results in a decrease in Cd for similar flow

rates. Additional data for this region are needed however to develop a comprehensive

understanding of the effect of aspect ratio on discharge coefficient.

Hasegawa et al. ( 1997) experimented with very small orifices and showed that as the

22

diameter of the orifice drops below 0.035 mm, the pressure drop is higher than the predicted

values. However, since all of the orifices they tested below 0.109 mm were of the same

thickness, it is unclear as to whether the resulting increase in aspect ratio was taken into

account as the orifice diameter decreased. This could explain why their data indicate larger

than expected pressure drops for small orifices, thus demonstrating the dependence of Cd on

aspect ratio.

Morris and Garimella (1998) have also shown that aspect ratio plays an important

part in determining Cd for Reynolds number in the turbulent regime. They proposed three

solutions for determining Cd based on small, medium, and large aspect ratios, which are valid

for p ~ 0.0635. In addition to expanding the range of covered aspect ratios up 9.5, they also

found that these equations predicted their data to within± 3.4 percent.

Although the above discussion of the literature (summarized in Table 1) shows that

orifice flow is affected by diameter ratio, aspect ratio, and orifice Reynolds number, it is still

not well understood how these interact in very small diameter orifices. It has also been

suggested that for very small diameter orifices, a fourth, as yet unidentified, parameter may

be required to further explain the increased pressure drop seen in these orifices (Hasegawa et

al. 1997). Additional research is needed to explain the effects of aspect ratio and diameter

ratio on Cd for orifices with p < 0.1, at low Reynolds numbers. Furthermore, fluid flow at the

high viscosities of interest in this study needs special attention. These interacting influences

of geometry, fluid properties, and flow rates are addressed in the present study.

Tabl

e 1.

S

f the

Lit

t A

utho

r G

eom

etry

O

rifi

ce

Dia

met

er

Asp

ect

ReR

ange

Fl

uids

D

iam

eter

Rat

ios

(fl)

Rat

io l/

D

v =

m2 /s

(m

m)

p=ke

/m-s

Jo

hans

en

Squa

re-

Vis

ualiz

atio

n:

Thin

<I

to 2

5,00

0 W

ater

, Cas

tor

(193

0)

Edge

d w

ith

0.1,

0.2

5, 0

.5,

(~ 0.

083)

oi

l (v

= 1.

21 x

45

° B

ack

0.75

10

-3 at

l 8°

C),

Cut

Pr

essu

re D

rop:

M

iner

al

0.20

9, 0

.401

, lu

bric

atin

g oi

l 0.

509,

0.7

94

(v =

1.14

x

10-4

at 1

8°C

) Tu

ve a

nd

Squa

re-

0.2

to 0

.8

Thin

4

to 4

0,00

0 W

ater

, Hea

vy

Spre

nkle

Ed

ged

with

(~ 0

.04)

M

otor

Oil

(v

(193

3)

45°

Bac

k =

1.62

x 1

0-3 )

,

Cut

Li

ght M

otor

O

il, L

ight

Pa

raff

in O

il

Med

augh

and

Sq

uare

-6.

35, 1

2.7,

~

1.0

30,0

00 to

W

ater

Jo

hnso

n Ed

ged

19.l,

25.

4,

350,

000

(194

0)

50.8

Lich

taro

wic

z Sq

uare

-0.

044,

0.0

54,

0.5,

1.0

, 2.0

, 0.5

to 5

0,00

0 W

ater

, Wat

er-

et al

. {l 9

65)

Edge

d 0.

071,

0.2

51,

4.0,

10.

0 G

lyce

rin

0.25

2 M

ixtu

re, O

il (v

ario

us

visc

ositi

es -

not g

iven

)

Tec

hniq

ue

Dat

a Pr

esen

tatio

n Fo

rmat

E

,V

Phot

os fr

om d

ye

test

. Plo

ts o

f Cd

vers

us s

quar

e ro

ot o

f Re

for

expe

rimen

ts.

E Pl

ots

of Cd

ve

rsus

log

Re

for t

heir

data

, Pl

ots

of Cd

vs.

P (c

onst

ant R

e)

com

paris

on w

ith

othe

r dat

a E

Plot

s of

Cd

vers

us p

ress

ure

drop

, plo

t of C

d V

S R

e fo

r hig

h R

e, ta

ble

of Cd

an

d D

P va

lues

R

Plot

s of

Cd v

s Re

for v

ario

us

aspe

ct ra

tios.

Find

ings

Cha

nges

in th

e Cd

ca

n be

cor

rela

ted

to

chan

ges

in f

low

pa

ttern

s as

obse

rved

by

dye

test

.

Foun

d cl

ose

agre

emen

t for

Re

> 15

0 an

d p <

0.5

with

pr

evio

us w

ork.

Com

pare

d to

Sm

ith

and

Wal

ker (

1923

), Ju

dd a

nd K

ing

(190

6), B

ilton

(190

8)

and

H. S

mith

(188

6).

good

agr

eem

ent w

ith

H. S

mith

(188

6) a

nd

Stric

klan

d (1

909)

C

ompa

red

expe

rimen

tal w

ork

done

by

Mor

gan

(196

3), J

ames

(196

1)

and

Sand

erso

n (1

962)

with

pre

viou

s w

ork.

Rec

omm

end

chan

ges

to C

d eq

uatio

ns su

gges

ted

by N

akay

ama

(196

1)

and

Asi

hmin

(196

1)

N

VJ

1 ao1

e 1.

:::

mm

mar

i 1 or

me

Lite

ratu

re (\

,;onu

nuea

J A

lvi e

t al.

Shar

p-0.

2, 0

.4, 0

.6,

Not

Giv

en

1to1

0,00

0 4

oils

wer

e E

(197

8)

Edge

d,

and

0.8

used

(v =

1.0

5 Q

uadr

ant-

(Qua

dran

ts: r

/d

x 10

·5 , 2

.0 x

Ed

ged

and

= 0.

08, 0

.125

, 10

·5 , 1

x 1

0·4 ,

Long

0.

18, 0

.25)

an

d 3.

5 x

104

Rad

ius

at 3

0°C

) Fl

ow

Noz

zles

Mill

er (

1979

) Sh

arp-

0.2

to 0

.7

Not

Giv

en

50,0

00 to

W

ater

A

Ed

ged,

4,

000,

000

Qua

dran

t-Ed

ged

and

Long

R

adiu

s Fl

ow

Noz

zles

G

rose

(198

3)

Shar

p-Le

ss th

an l

A

Ed

ged

Gro

se (

1985

) Sh

arp-

<0.7

>

100,

000

A

Edge

d

Plot

s of E

u vs

. Re

for d

iffer

ent

Orif

ices

bas

ed

on d

iam

eter

R

atio

s.

Perc

ent

devi

atio

n vs

Re

Plot

of C

d vs

Re

for e

xper

imen

tal

data

and

pr

edic

ted

valu

es

from

equ

atio

ns.

Plot

s of

Cd

vs ~

for G

rose

's eq

uatio

n, th

e St

oltz

(197

5),

ASM

E (1

971)

-A

GA

(195

5)

equa

tion,

M

iller

's eq

uatio

n ( 1

979)

, an

d av

aila

ble

data

Com

pare

d w

ell w

ith

Zam

pagl

ione

( 19

69)

and

Laks

hman

a R

ao

and

Srid

hara

n (1

972)

. Det

erm

ined

th

at q

uadr

ant-e

dge

orifi

ces

act s

harp

at

low

Re

and

like

nozz

les

at h

igh

Re.

Labo

rato

ry d

ata

are

com

pare

d w

ith

num

bers

pre

dict

ed

by A

SME-

AG

A a

nd

Stol

tz e

quat

ions

to

conc

lude

that

err

or

can

be re

duce

d w

ith

bette

r mea

sure

men

t. Pr

edic

ted

valu

es

com

pare

qui

te w

ell

with

dat

a fr

om

prev

ious

in

vest

igat

ors

for R

e <

16. C

d is

prop

ortio

nal t

o sq

uare

root

of R

e in

th

is ra

nge.

Th

e el

liptic

al

equa

tion

com

pare

s w

ell w

ith d

ata

from

Fl

ugge

(196

0) a

nd

pred

ictio

ns fr

om

othe

r equ

atio

ns fo

r be

ta<

0.4

but o

ver

pred

icts

abo

ve th

is.

Post

ulat

es th

at it

is

due

to p

rofil

e ef

fect

s.

N .i::.

1 ao1

e 1.

::s

umm

ar •

or tn

e Li

tera

ture

(t;o

ntm

ueal

Jo

nes

and

Squa

re-

0.2

and

0.5

Thin

0.

8 to

64

c B

ajur

a (1

991)

Ed

ged

with

(::

; 0.

25)

45°

Bac

k C

ut

Sahi

n an

d Sq

uare

-0.

5 0.

0625

, 0

to 1

50

Oil

E,C

C

eyha

n ( 1

996)

Edg

ed

0.12

5, 0

.25,

(v

isco

sitie

s 0.

5, a

nd 1

w

ere

not

give

n)

Has

egaw

a et

Sq

uare

-0.

01 t

o N

ot

1to1

000

Dis

tille

d E

,C

al. (

1997

) Ed

ged

1.0

cont

rolle

d w

ater

, bu

t ran

ges

glyc

erin

fr

om 0

.051

so

lutio

ns (v

=

up to

1.1

4 1.

7 x

10-6

, 2.

4 x

10-6 ,

3.4

x

10-6

, 5.

3 x

10-6

), a

nd

silic

on o

ils (v

=

1.1x

10-6

,

2.2

x 10

-6,

5.1

x 10

-6)

Dug

dale

Sq

uare

-2.

396

and

Thin

on

ord

er o

f 104

M

olas

ses

(µ =

E

,A

(199

7)

Edge

d 5.

100

(::;

0.03

4)

14 to

19.

7)

Tabu

lar d

ata

com

pare

d w

ith

prev

ious

re

sear

ch. P

lots

of

Cd

vs

Stro

uhal

num

ber

and

stre

amlin

e V

isua

lizat

ions

Pl

ot o

f Cd

vs

squa

re ro

ot o

f R

e fo

r th

eir

num

eric

al a

nd

expe

rimen

tal

data

and

dat

a fr

om A

lvi

(197

8) a

nd

Joha

nsen

(193

0)

Plot

s of

Eu

vs.

Re

for d

iffer

ent

orifi

ce

conf

igur

atio

ns

and

num

eric

al

solu

tion.

Als

o sh

ows

depa

rture

fr

om S

toke

s flo

w o

ccur

ring

at

app

roxi

mat

ely

Re=

10

Smal

l tab

le o

f da

ta g

ivin

g th

e flo

w ra

te, e

nerg

y di

ssip

atio

n co

nsta

nt (C

), te

mpe

ratu

re, a

nd

visc

ositv

Stea

dy fl

ow d

ata

pred

ictio

ns a

gree

d fa

irly

wel

l with

ex

peri

men

tal d

ata

for

this

rang

e.

St=

2nD

pipe

vpip

e Fo

und

that

as

Re

incr

ease

s in

this

ra

nge,

the

upst

ream

se

para

ted

flow

re

gion

shr

inks

and

do

wns

trea

m

leng

then

s. T

heor

y an

d ex

peri

men

ts

agre

e w

ell.

As

the

diam

eter

of

the

orif

ice

decr

ease

s be

low

0.0

65 m

m, t

he

num

eric

al s

olut

ion

unde

r pre

dict

s th

e pr

essu

re d

rop

that

oc

curs

. Bel

ow th

is

valu

e, th

e un

der

pred

ictio

n be

com

es

wor

se a

s ei

ther

the

orifi

ce d

iam

eter

or

the

fluid

vis

cosi

ty

decr

ease

s.

Com

pare

s w

ell w

ith

wor

k by

Bon

d (1

922)

but

is 5

to

10%

hig

her t

han

pred

icte

d by

theo

ry.

N

Vi

Tabl

e 1.

~ummari 1

ot th

e Li

tera

ture

(t,;o

ntin

uea)

Zh

ang

and

Cai

Squa

re-

51.0

, 61.

2, 0

.5, 0

.6, 0

.69,

N

ot G

iven

10

4,00

0 to

W

ater

E

(199

9)

Edge

d,

70.4

, and

an

d 0.

8 20

0,00

0 Sh

arp-

81.6

Ed

ged,

St

ream

-Li

ned,

Q

uadr

ant-

Edge

d an

d Sl

opin

g A

ppro

ach

McN

eil e

t al.

Shar

p-0.

491

Not

Giv

en

40 to

400

W

ater

and

E

(199

9)

Edge

d,

Solu

tions

of

Noz

zle

Luvi

skol

K90

in

Wat

er

Val

le e

t al.

45°

Bev

el

0.6

to 3

.0

Not

Giv

en

< l

to l

00,0

00

Wat

er, O

il(µ

E

,C

(200

0)

on in

let a

nd

= 0.

08 '1

.62)

, (P

OLY

2D)

outle

t C

om S

yrup

Kim

et a

l. Sq

uare

-10

, 15,

0.1,

0.1

5, 0

.33

0.2l

to0.

7 40

00 to

l 00

,000

Wat

er

E (1

997)

Ed

ged

with

an

d 33

ba

sed

on p

ipe

45°

Bac

k di

amet

er

Cut

Dim

ensi

onle

ss

wal

l pre

ssur

e vs

po

sitio

n in

pip

e,

dim

ensi

onle

ss

pres

sure

dro

p vs

p Fl

ow c

oeff

icie

nt

vs R

e, Fr

ictio

n fa

ctor

vs

Re, C

d vs

Re

Plot

s of E

u vs

Re

for

N

ewto

nian

flu

ids,

Bog

er

fluid

, and

su

spen

sion

flu

ids.

Cd v

s. Re

, C

avita

tion

in d

B vs

Re

Det

erm

ined

that

for

Eu b

etw

een

0.5

and

4, th

e sl

opin

g ap

proa

ch o

rific

e ge

omet

ry p

rovi

des

the

high

est p

ress

ure

drop

for t

he s

ame

dow

nstre

am m

inim

a as

com

pare

d to

the

othe

r orif

ices

. Fl

ow c

oeff

icie

nts

are

depe

nden

t on

Re a

nd

flow

geo

met

ry a

t low

R

e. T

he c

ontra

ctio

n co

effic

ient

tend

s to

un

ity a

s R

e go

es to

0.

Sh

ows

that

it is

po

ssib

le to

det

erm

ine

the

exte

nsio

nal

visc

osity

of

visc

eola

stic

flui

ds

and

susp

ensi

ons. ~

= 3µ

R w

here

R is

sh

ift in

plo

t and

3 µ

is th

e ex

tens

iona

l vi

scos

ity o

f a

New

toni

an fl

uid

by

the

Trou

ton

rela

tion.

D

eter

min

ed th

at

cavi

tatio

n di

d oc

cur

in o

rific

es a

nd th

at

Cd w

as o

nly

affe

cted

fo

r bet

a= 0

.1 an

d l/d

=

0.7.

Cav

itatio

n di

d no

t aff

ect C

d fo

r all

othe

r cas

es.

N

0\

1 ao1

e i.

~ummari ,

or m

e Li

tera

ture

(llo

nunu

ea)

Ram

amur

thi

Squa

re-

0.3,

0.5

, 0.

03, 0

.05,

0.1

, 1t

o50

2000

to 1

00,0

00 W

ater

E

,V

and

Edge

d 1.

0, a

nd

and

0.2

Nan

daku

mar

2.

0 (1

999)

Kilj

ansk

i Sh

arp-

2, 3

, and

5 0

.053

, 0.0

79,

0 an

d 1.

0 10

-2 to

500

Et

hyle

ne

E (1

993)

Ed

ged

and0

.132

fo

r 3 m

m

Gly

col(

µ=

orifi

ce. 0

.5

0.02

), fo

r oth

ers

Gly

cero

l So

lutio

ns (µ

=

0.15

, 0.4

0),

Pota

to S

yrup

(u

= 1

0)

Koj

asoy

et a

l. M

ulti-

Hol

ed 1

.0, 2

.0,

0.5,

1.0

, and

800

to 2

1,00

0 R

-113

E

,A

(199

7)

Plat

es

and

4.0

2.0

for

indi

vidu

al

hole

s

Mor

ris e

t al.

Squa

re-

3.18

and

2

8500

, 100

00,

FC-7

7 E

,C

(199

6)

Edge

d 6.

35

and

1300

0 (F

LUEN

T) Cd

vs.

Re

plot

s fo

r diff

eren

t ge

omet

ries

Cd v

s R

e pl

ots

and

tabu

lar d

ata

for a

ll da

ta

poin

ts

Plot

s of l

oss

coef

ficie

nt (k

) vs

. Re

for

indi

vidu

al

orifi

ces.

Plot

s of

pr

edic

ted

valu

es

vs. e

xper

imen

tal

to s

how

err

or.

Hea

t tra

nsfe

r co

effic

ient

s vs

flo

w p

ositi

on

Det

erm

ined

that

or

ifice

s w

ith a

spec

t ra

tios

of 5

are

mos

t af

fect

ed b

y ca

vita

tion.

A

spec

t rat

io o

nly

affe

cts

Cd fo

r Re

< 30

0.

The

auth

ors

expe

rimen

t on

sing

le

phas

e to

det

erm

ine

the

prop

ertie

s of

thei

r tes

t loo

p. T

hey

then

exp

erim

ent o

n 2

phas

e an

d de

term

ine

2-ph

ase

mul

tiplie

rs.

Pred

ictio

ns a

gree

w

ith Ja

nsse

n ( 1

966)

to

13.

8%, a

nd w

ith

expe

rimen

tal d

ata

to

with

in 1

0.5

to 1

4.5%

C

ompa

red

expe

rimen

tal h

eat

trans

fer c

oeff

icie

nts

with

num

eric

al

solu

tions

that

und

er

pred

icte

d th

e ex

perim

enta

l dat

a by

49

to 5

4%. G

ibso

n-La

unde

r (19

76)

equa

tion

agre

ed

with

in 1

6 to

20%

N

-....)

1 ao1

e i.

::>

umm

ar 1

or m

e Li

tera

ture

(llo

nunu

eaJ

Mor

ris a

nd

squa

re-

3.18

and

l

to 6

85

00 to

23,

000

c G

arim

ella

ed

ged

6.35

(F

LUEN

T)

(199

8)

Mor

ris e

t al.

squa

re-

3.18

and

20

00 to

23,

000

E,V

,C

(199

9)

edge

d 6.

35

(FLU

ENT)

Kay

ser a

nd

Shar

p-0.

9 to

1.9

0

to 3

.56

3000

to 8

0,00

0 A

ir, C

02,

E Sh

amba

ugh

Edge

d,

Arg

on,

(199

1)

Squa

re-

Hel

ium

, He

Edge

d,

mix

es

Qua

dran

t-Ed

ged,

and

El

liptic

al-

Entry

G

an a

nd R

iffat

Squ

are-

145.

0 an

d 0.

71

160,

000

to

Air

E

,C

(199

7)

Edge

d 23

9.0

370,

000

(FLU

ENT)

O

rific

e an

d M

ulti-

Hol

ed

Plat

e

Plot

of C

d vs

as

pect

ratio

, ta

bula

r dat

a fo

r pr

essu

re d

rop

com

pute

d fro

m

equa

tions

and

ex

perim

enta

l da

ta

Vel

ocity

pro

files

fo

r var

ious

ar

rang

emen

ts

Cd v

s pr

essu

re

ratio

and

Cd

vs.

Re

Pres

sure

ratio

vs

posi

tion

in p

late

, Eu

vs.

Re, E

u vs

l/d

The

auth

ors

re-

corr

elat

ed th

e ex

perim

enta

l dat

a fr

om W

ard-

Smith

(1

971)

to d

evel

op C

d ex

pres

sion

s th

at w

ere

foun

d to

agr

ee w

ith

num

eric

al

pred

ictio

ns to

with

in

5%

Expe

rimen

tal d

ata

agre

ed w

ell w

ith

num

eric

al s

imul

atio

n fo

r Re

> 85

00, b

ut

not a

t 400

0 or

200

0.

Post

ulat

ed th

at th

is

was

due

to n

ot

acco

untin

g to

la

min

ar/s

emi-

turb

ulen

t flo

w f

ield

s R

ound

and

elli

ptic

al

nozz

les p

erfo

rm b

est.

Kni

fe e

dge

does

not

co

rrel

ate

wel

l with

Re

A p

erfo

rate

d pl

ate

has

a hi

gher

pre

ssur

e dr

op th

an a

n or

ifice

w

ith s

imila

r flo

w

area

. The

pre

ssur

e lo

ss c

oeff

icie

nt (E

u)

drop

s as

asp

ect r

atio

in

crea

ses

to 1

.5, t

hen

incr

ease

s sl

owly

N

00

1 ao1

e 1.

::s

umm

ar 1

or m

e Li

tera

ture

(l;O

ntm

ueaJ

Em

mon

s R

ound

A

ir A

Pl

ot o

f Cd

vs R

e D

eter

min

ed

(199

7)

Hol

es

for e

xper

imen

tal

equa

tions

for m

ass

data

and

flo

w ra

te th

roug

h pr

edic

ted

valu

es

hole

s ba

sed

on C

d fro

m e

quat

ions

. an

d Fr

oude

num

ber.

Foun

d go

od

agre

emen

t with

dat

a fr

om H

eske

stad

and

Sp

auld

ing

( 199

1)

and

Epst

ein

and

Ken

ton

(198

9) b

ut

not w

ell w

ith T

an

and

Jalu

ria (1

992)

Sa

man

ta e

t al.

Squa

re-

0.46

46,

liqui

d:

Sodi

um s

alt

E D

P vs

. Pr

esen

ted

Eu

(199

9)

Edge

d 0.

5984

, and

45

<R

e< 2

200

ofC

arbo

xy

volu

met

ric fl

ow

equa

tions

for l

iqui

d 0.

7087

ga

s: m

ethy

l ra

te, p

ress

ure

vs

only

and

two-

phas

e 23

0 <

Re <

220

0 ce

llulo

se

posi

tion

cond

ition

s. Th

e liq

uid

for t

his

case

is

non-

New

toni

an.

Not

e:

The

follo

win

g ab

brev

iatio

ns a

re u

sed:

A -

Ana

lytic

al A

ppro

ach,

C -

Com

puta

tiona

l Met

hods

, E -

Expe

rimen

tal

Inve

stig

atio

n, V

-V

isua

lizat

ion

N

\0

30

CHAPTER3

EXPERIMENTAL SET-UP AND PROCEDURES

3.1 Test Section Fabrication

The test section consisted of an orifice plate mounted between two flanges, as shown

in Figure 4. The flanges were manufactured by the Anchor Flange Company and mate to the

orifice plate using an o-ring seal. The flanges were supplied with 1" NPT female pipe

threads machined into the body of the flange. Two one-inch by six-inch long, schedule 160,

316L, stainless steel pipe nipples were threaded into the flanges. The pipe nipples were

machined to fit flush with the orifice side of the flange when fully threaded into place, as

shown in Figure 5. To prevent leakage from the threaded joint, the nipples were then welded

to the flange, on the sides that were away from the orifice. They were not welded on the

sealing side of the flange to allow for expansion and contraction, thus preventing additional

stresses.

The one-inch nominal pipe nipple provides a large contraction ratio between the inlet

flow passage and the orifice. The six-inch length of the nipple allows flow development and

Figure 4. Photograph of Test Section

31

Figure 5. Photograph of Flange Face

recovery upstream and downstream of the orifice, respectively. The details of this flow

geometry are shown in Figure 6.

Three Imm diameter orifice plates were manufactured from 4.62 mm thick, 316L

stainless steel and measured 7.6 cm on each side. Figure 7 shows a representative cross-

sectional view of these plates while Table 2 provides the corresponding dimensions. This

overall plate thickness of 4.62 mm was provided to withstand the large pressure drops across

the orifice under consideration. The desired orifice thicknesses within these plates were

achieved by milling holes of the appropriate depth into the plate on the downstream side.

Thus, the 1 mm thick orifice was fabricated by milling a 5 .2 mm diameter hole to a depth of

3.62 mm. Similarly, a 19 mm diameter hole was milled to a depth of 2.62 mm to create the 2

mm thick orifice, and a 25 mm diameter hole was milled to a depth of 1.62 mm for the 3 mm

thick orifice. Appendix D provides details of the back-cut dimension calculations.

To S

yste

m

Plum

bing

_.

_ r-2

19

00

1---

-10.

5870

J_ ==

========~_L___

J

1 ..

., I

0.48

00

~

£ ~::

Orifi

ce F

lang

e As

sem

bly c

-j

Flan

ge

0.33

40

0.41

00

0 72

00

1.16

00

0.89

55

Tl •I

• 1.

0225

1

---2

.00

00

0 05

60

Orif

ice~

~ 0.

6920

i-------------6

.0

00

0

Note

: all

dim

ensi

ons

are

in in

ches

Figu

re 6

. T

est S

ectio

n D

imen

sion

al D

raw

ing

w

N

33

Similarly, three 3 mm diameter orifice plates were manufactured from 3.05 mm thick,

316L stainless steel and three 0.5 mm diameter orifice plates were manufactured from 3.18

mm thick, grade A-2 tool steel. Dimensions for the 0.5 mm and 3 mm orifices are shown in

Table 2. Photographs of the upstream and downstream sides of the 1 mm diameter orifices

are shown in Figure 8.

Flow I

' Ld_J _L l_L A-+~~~~

--. I B----

Figure 7. Orifice Plate Cross-Sectional Dimensions

Table 2. Orifice Cross-Sectional Dimensions 0.5 mm Orifice Dimensions

Nominal d B 1 A lid d/D Thickness (mm) (mm) (mm) (mm) (D = 22.75)

Imm 0.5244 7.38 0.9952 2.9337 1.8978 0.0231 2mm 0.5249 21.52 1.9782 2.9479 3.7687 0.0231 3mm 0.5259 NIA 3.0099 3.0099 5.7233 0.0231

1.0 mm Orifice Dimensions Nominal d B 1 A lid d/D

Thickness (mm) (mm) (mm) (mm) (D = 22.75) Imm 1.0130 5.18 1.0290 4.423 1.0158 0.0445 2mm 1.0030 19.05 1.9561 4.623 1.9502 0.0441 3mm 1.0109 25.40 2.8859 4.623 2.8548 0.0444

3.0 mm Orifice Dimensions Nominal d B 1 A lid d/D

Thickness (mm) (mm) (mm) (mm) (D = 22.75) Imm 3.1187 7.35 1.0128 3.0068 0.3248 0.1371 2mm 3.1071 21.60 2.1275 2.9941 0.6847 0.1366 3mm 3.0792 NIA 2.9972 2.9972 0.9734 0.1353

Overall Range: 0.3248 <lid< 5.7233 0.0231<d/D<0.1371

34

Figure 8. Orifice Plate Photograph

3.2 Test Loop Description

A photograph of the overall test facility is shown in Figure 9, while a schematic is

shown in Figure 10. A Cat Pumps model 660, triplex plunger pump capable of delivering a

flow rate of 3 8 L/min at a maximum discharge pressure of 21.1 MP a, was used to pump the

hydraulic fluid around the test facility. This pump is belt driven by an electric motor and can

be configured to produce lower flow rates and pressures by changing the pulley sizes on the

motor and pump. For this test loop, a 10 hp motor was chosen, which delivered a nominal

flow rate of 34.2 L/min at a discharge pressure of 11.3 MPa. The pump head is constructed

of bronze and contains the suction and discharge valve assemblies. The ceramic plungers use

35

viton seals which are resistant to oil.

System pressure was controlled by two methods. The primary method of pressure

control was through the use of a backpressure control valve located at the discharge of the

pump. This method was used for system pressures above 800 kPa as this is the minimum

discharge pressure for this pump. For testing at lower system pressures, the pump discharge

valve was throttled shut and the test section bypass valve was opened.

In addition to the minimum discharge pressure requirement, the Cat pump also

required a minimum suction pressure of 170 kPa. This was accomplished by using an

Accumulators, Inc. (Model# AM4531003) 45 cubic inch accumulator on the suction side of

the pump. With the loop shutdown, the accumulator was pressurized to 500 kPa, which was

the maximum pressure required to maintain the 170 kPa at the pump suction during loop

operation.

A 2.3 gallon stainless steel reservoir was connected to the loop on the suction side of

the pump. Because this reservoir was at the highest point in the loop, besides acting as an oil

reserve for the system, it also provided an easy location for adding additional oil to the test

loop. Also, the reservoir could be pressurized with nitrogen, typically to a pressure of 400

kPa, which was useful for two reasons. First it allowed for filling the accumulator with the

proper amount of oil. Second, the loop was configured to operate with the reservoir inline

after it was opened for maintenance, such as changing the orifice plate. This allowed any

foreign material that may have been left in the loop to collect and settle out in the reservoir.

Under normal operation, the reservoir was isolated from the test loop and flow was bypassed

around it.

36

Figure 9. Photograph of Test Loop

-,

t R

eser

voir

Fillin

g an

d C

harg

ing

Port

Res

ervo

ir --t>

<J-----------,--------------1

--1

Res

ervo

ir In

let

~--~-~ Is

olat

ion

Valv

e 1 R

eser

voir ~

1 O

utle

t ~

Isol

atio

n 1

I 1

Valv

e I

Res

ervo

ir t

Test

1 l B

ypas

s an

d Se

ctio

n~

l Th

rottl

e By

pass

f --

1 1

Valv

e Va

lve

1

L____

____

: ~

Pum

p --

Rel

ief

Dis

char

ge

L V

I Va

lve

-----

ave

1 -1

Mot

or

Lb

Back

pres

sure

C

ontro

l Val

ve

Stra

iner

Figu

re 1

0.

Test

Loo

p Sc

hem

atic

Plat

e H

eat

Exch

ange

r

ZHM

-03

ZHM

-01

JVM

-60K

L Fl

owm

eter

s

City

W

ater

: O

ut

l I

City

Ac

cum

ulat

or

Wat

er

In

w

-.J

38

System temperature was maintained by a Tranter Inc. (Model# UX-016-UJ-21) plate

heat exchanger located on the suction side of the pump. Cold water was supplied to the heat

exchanger from a city water line. To eliminate the effect of city water line pressure

variations on the cold water flow rate, it was first supplied to an open 55 gallon tank with a

drain and an overflow. This helped maintain a constant inlet pressure to the Little Giant

(model 977458) magnetic drive pump, which supplied water from the tank to the heat

exchanger.

3.3 Instrumentation

Inlet and outlet temperatures of the oil were measured using 3-wire RTD's supplied

by Omega Engineering, with a nominal accuracy of± 0.6 °C. Flow rates were measured

using three different positive displacement flow meters supplied by AW Company, as shown

in Table 3. Pulses generated by the flow meters were captured by inductive pickups on each

meter and sent to a flow monitor. The flow monitor generated a 4-20 mA output signal that

was converted to a 1-5 V signal for use by the data acquisition system.

Table 3. Flow Meter S ecifications Model Rane Accurac

ZHM-01 0.001 - 0.25 ZHM-03 0.1 - 5.5

NM-60KL 2-20

Absolute and differential pressures were measured using Rosemount model 3051

pressure transducers. The absolute pressure transducers were capable of measuring pressures

in the range of 0 psia to 10,000 psia, with an accuracy of ±0.075% of span. The differential

pressure transducer was capable of measuring pressures in the range of ±2000 psid, with an

accuracy of ±0.075% of span for spans larger than 400 psid. For smaller spans, the accuracy

39

accuracy of ±0.075% of span for spans larger than 400 psid. For smaller spans, the accuracy

of the transducer is ±[0.025 + 1 O/span ]% of span.

A PC-based data acquisition system supplied by IO Tech was used to display and

record data during the test. The Tempscan/lOOOA with expansion unit EXP/llA interfaced

with the computer through the program TempView 4.1, which allowed real-time display and

recording of the temperatures, pressures, and flow rates.

3.4 Experimental Procedures

A strict set of test procedures was established to ensure the collection of repeatable

and accurate data for each orifice plate. Whenever the orifice plate was changed, it was

necessary to fill the test section with oil and ensure that air was removed from the system.

This was accomplished by flooding the test section with oil before the orifice plate was fully

bolted into position. With the bolts at the bottom of the flanged slightly tightened, the top of

the orifice plate was moved back and forth in the direction of each flange. This allowed a

gap to open between the o-ring and the orifice plate, which allowed air to escape and the test

section to be fully filled with oil. Once the air was removed by this method, the system was

run for approximately 15 minutes in the maintenance configuration. For this configuration,

the accumulator was isolated and flow was directed through the pressurized reservoir. This

allowed any particulate in the line to settle out in the reservoir and any gases to be vented out

of the system.

Typically, eleven data points were taken for each orifice plate, corresponding to a

differential pressure range of 100 kPa to 1.0 MPa. For pressures above 800 kPa, the

backpressure control valve was used to maintain system pressure. For pressures below this

40

value, the pump discharge and test section bypass valves were used simultaneously to control

system pressure. This allowed the pump discharge pressure to stay above 800 kPa as was

required by the pump manufacturer. Oil temperature was maintained at the desired value by

controlling the city water flow rate in the plate heat exchanger. Two needle valves, a half-

inch valve and a quarter inch valve, were used for this task. The half-inch needle valve was

generally used when large cooling water flow rates were required, as occurred when

conducting tests at 20°C, 30°C and at 40°C for the high pressure drop cases. The quarter-

inch needle valve was used for smaller flow rates, as occurred when testing at 50°C and at

40°C for the low pressure drop cases. Once both the temperature and pressure were set, the

operator waited until steady state was reached before taking data. The on-screen strip chart

function of the data acquisition system was used to monitor the approach to steady state

conditions, which for the low pressure data points, could take up to two hours.

Once the test loop reached steady state, the temperatures, absolute pressures, pressure

drop, and flow rate were uploaded to the computer. During each test, the data acquisition

system constantly monitored each enabled channel over 100 times a second. To get a good

sample of the data, the readings were taken at the rate of one reading per second for two

minutes. The average value of this set of 120 data points for each test case was then used for

subsequent data analysis.

41

CHAPTER4

ANALYSIS AND DISCUSSION OF RESULTS

4.1 Data Analysis

The experimentally determined pressure drop for each test case represents the

pressure drop due to the orifice and that due to the test section piping located between the

taps of the differential pressure transducer, as shown in Figure 11. To determine the pressure

drop due only to the test section, a 3 mm thick orifice plate was constructed with a hole of the

same diameter as the inlet pipe. Figure 12 shows the flow geometry for the test section with

and without the orifice installed. For each of the four temperatures, data were collected at six

different flow rates, which were chosen to encompass the range of values recorded for the 0.5

mm and 1 mm diameter orifice plates. These data were plotted and curve-fit to yield a

function based on flow rate that could be used to subtract these extraneous contributions from

the measured pressure drop, as shown in Equation ( 10), with coefficients a and b shown in

Table 4.

Where:

Q = flow rate through the orifice in m3 /s

~p = pressure drop in kPa

Table 4. Coefficients for Equation (10) Temperature a (°C)

20 2.892 x 106

30 3.679x 105

40 5.808 x 106

50 5.768 x 106

(10)

b

1.100 0.947 1.267 1.314

A=

Pip

e Fl

ow

8 =

Expa

nsio

n C

= C

ontra

ctio

n 82 8

3 C1

Not

e: T

he s

egm

ents

labe

led

as in

let a

nd o

utle

t tee

s ea

ch re

pres

ent t

hree

tees

that

hav

e be

en c

ombi

ned

into

one

sea

men

t as

an e

auiv

alen

t len

ath.

1" b

y 1/

2"

Red

ucin

g C

oupl

ing

84

C2 C

3

A1

Inle

t Pip

e

A6

Out

let P

ipe

A7

Inle

t Tub

ing

Expa

nsio

n R

atio

s 81

= 0

.663

6 83

= 0

.385

3 82

= 0

.324

3 84

= 0

.493

6

Expa

nsio

n an

d C

ontra

ctio

n ra

tios

(B*)

For

Orif

ice

Back

-Cut

1m

m:

Ex. r

atio

= 0.

0627

2m

m:

Ex. r

atio

= 0

.846

9 3m

m: C

ont.

ratio

= 0.

6642

1 /2"

Sw

agel

ok

Tubi

ng A

dapt

er

Area

s (s

q. in

ches

) A1

=0.

1452

A

7=0.

5217

A2

= 0

.087

6 A8

= 1

.056

8 A3

= 0

.132

0 A9

= 0

.407

2 A4

= 0

.407

2 A1

0 =

0.13

20

A5 =

1.0

568

A11

= 0.

0876

A6

= 0

.629

8 A1

2 =

0.14

52

Out

let T

ubin

g

Con

tract

ion

Rat

ios

C1 =

0.5

960

C3

= 0.

3243

C2

= 0

.385

3 C

4 =

0.66

36

Orif

ice

Area

s (s

q. in

ches

) O

rific

e A

01

A02

1 m

m:

0.00

125

0.03

27

2mm

: 0.

0012

2 0.

4418

3m

m:

0.00

124

0. 78

54

Figu

re 1

1.

Flow

Are

a Sc

hem

atic

For

Pre

ssur

e D

rop

Cal

cula

tions

(Orif

ice

Dim

ensi

ons a

re fo

r 1 m

m D

iam

eter

Orif

ices

)

~

N

43

Blank Orifice Plate

D72ZZZZ/'.iZZZZZZ~i7ZZZZZ/?ZZZZZZZJ Direction __ .... ~

of Flow .... ====-.==

Test Loop With Blank Orifice Plate Installed Orifice Plate

~::c~==:= :;~r==

Test Loop With Orifice Plate Installed

Figure 12. Schematic of Test Section With and Without The Orifice Installed

For the 3 mm diameter orifice plates, modifications were made to the test loop which

resulted in a change in the pressure drop characteristics of the test section piping. Because of

this, additional data were collected at flow rates which encompassed the range of values

recorded for the three, 3 mm diameter orifice plates at the corresponding temperatures.

These data were also plotted and curve-fit to yield functions based on flow rate as follows:

T = 20°C: ~P = (-3.83x10 15)Q4 + (5.10xl0 12)Q3 - (8.25xl08)Q2 + (l.Olxl06)Q - 3.65 (11)

T = 30°C: ~P = (-2.28x10 15)Q4 + (l.78x1012)Q3 - (8.78x108)Q2 + (3.99x105)Q - 1.98 (12)

T = 50°C: ~p = (4.59x107)Q1.56 (13)

44

An estimate of the contribution of the upstream and downstream plumbing to the

measured pressure drop was also obtained using familiar pipe flow and 'minor loss' pressure

drop expressions. Sample calculations for the data point 1-50-20 are shown in Appendix A

for the equations discussed in this section. The first step in this analysis was the calculation

of the Reynolds Number for each segment of the test section, as follows:

Re=pVD µ

(14)

The velocity for each segment was determined by dividing the measured volumetric

flow rate by the cross-sectional area of that segment. The values for density and viscosity

were determined from curve-fits to property data provided by John Deere Product

Engineering Center. The geometry and cross-sectional areas of the various segments of the

plumbing arrangement are also provided in Figure 11. It should be noted that the inlet

temperature and pressure were used for the evaluation of the fluid properties for the segments

upstream of the orifice, whereas the outlet temperature and pressure were used for the

downstream segments.

For the frictional loss component of the pressure drop, the Darcy friction factor

correlation by Churchill ( 1977), Equation ( 15), was used to calculate the friction factor for

each segment. Thus,

( 8 )l2 f =8· Re +

1 2.457·ln ---09----

[_!_J. +0.27 .~ Re D

16

l 3 12 2

(15)

45

In the above expression, the roughness of drawn tubing (c = 0.0015 mm) was used for

each segment. This was deemed adequate since the flow was laminar in every segment

making the roughness value an insignificant contributor to the value of the friction factor.

The frictional pressure drops due to piping losses were then calculated as follows:

1 L 2 Af>pipe =-f-pV

2 D (16)

Minor losses due to the sudden expansion and contraction between different segments

must also be taken into account when determining the total pressure drop due to the test

section. To determine this pressure drop, a loss coefficient must first be determined. The

loss coefficient for sudden expansion was determined as follows (Munson et al. 1998):

(17)

The loss coefficient for a sudden contraction was obtained from the following curve-

fit to a graph ?f the loss coefficient versus contraction ratio (Figure 13), which is available in

Munson et al (1998):

A 2 -0.705 A1 KL = -0.021+0.585 1 +exp ----

0.233

-2.29

Once the loss coefficient KL was determined, the pressure loss was calculated as

follows:

1 2 Af>minor =-KLpV

2

(18)

(19)

46

0.2

o--~~~~--~~~~--~~~~---~~~~---~~-=---.d

0 0.2 0.4 0.6 0.8 1.0

Figure 13. Loss Coefficient for a Sudden Contraction (Munson et al 1998)

The sum of these frictional and minor loss pressure drops provides an estimate of the

pressure drop that can be attributed solely to the test section plumbing. These estimates are

compared with the corresponding measured values in Figure 14 for the range of flow rates of

interest in this study. It can be seen that there is very good agreement between the measured

and estimated values for the four temperatures under consideration. (Note that the log scale

exaggerates the small differences at the low end of the scale; however, compared to the

orifice ~p, these differences are minimal.) Based on this agreement, the orifice pressure drop

was computed by subtracting the experimentally determined pressure drop without the orifice

from the total measured pressure drop. The contributions of these various plumbing elements

to the total test section pressure drop, based on this approach, are shown for three

representative data points in Figure 15.

104

"" c. 0 c 102

~ ::l en en 101 -~ a.

47

• 20 °C - Blank Data + 30 °C - Blank Data • 40 °C - Blank Data .& 50 °C - Blank Data

-o- 20 °C - Blank Estimated --<r- 30 °C - Blank Estimated

• -0- 40 °C - Blank Estimated - -= --6- 50 °C - Blank Estimated

-e- 20 °C - Orifice -0- 30 °C - Orifice -0- 40 °C -Orifice --&- 50 °C - Orifice

10-s Flowrate, m3/s 104

· · Figure 14. Experimental and Calculated Pressure Drops for the Test Section with the Orifice Plates Removed, Shown in Comparison With the 1 mm Diameter, 3 mm Thick Orifice Data

4.2 Experimental Results

Once the orifice pressure drop was obtained using the techniques described above, the

effect of the two significant variables, temperature and orifice thickness, on the pressure drop

was investigated.

4.2.1 Effect of Fluid Temperature

The orifice pressure drop as a function of flow rate is shown for each of the three 1 mm

diameter orifice plates in Figures 16 to 18 with fluid temperature as a parameter. Similar

plots for the 0.5 mm and 3 mm diameter orifice plates are shown in Appendix C. From the

lmm thick orifice plot, it can be seen that as the temperature of the fluid decreases, the flow

rate also decreases for a constant pressure drop. The effect of temperature becomes

48

6000 ..---------------~ CV

~ 5000 c: ~4000 Cl.

~3000 e ~ 2000 Ill e

D.. 1000

0 5 10 15 20 25 30 35 40 Position, x, in.

0 5 10 15 20 25 30 35 40 Position, x, in.

Point 11-50-20 (1 mm Dia., 1 mm Thick, 50 Bar Pressure Drop, 20°C)

900.----------------. 900..----------------. CV

D.. ..iii: 850 c: <l c: 800 e c I!! 750 ::s Ill

! 700 a.

12 CV

~10 c: <l 8 c: e 6 c e 4 ::s Ill Ill 2 e

D.. 0

0

0

5 10 15 20 25 30 35 40

Position, x, in.

c: --·· ---4· .. ·--r 0 850

rooJ Lj 650 ..__..__.__.____ I __.____...__,__' O O __.__..____.__,, O O

0 5 10 15 20 25 30 35 40

Position, x, in.


5 10 15 20 25 30 35 40 Position, x, in.

~ 11.0j a: •• ______ ,.___...

• ·- l ~10.5 c 1.0

I!! ::s Ill Ill I!! a.

j j

-· • • 0.5 .__._ _ _.____._ _ _.____. _ _.___...____.__...___.

0 5 10 15 20 25 30 35 40 Position, x, in.


Figure 15. Relative Contributions of Test Section Plumbing to Total Measured Pressure Drop

49

indistinguishable beyond an imposed pressure difference of 3 MPa. This same trend is also

observed in the plots for the 2 and 3 mm thick, 1 mm diameter orifice plates. In addition, it

can also be seen that as the thickness of the orifice plate increases, the difference in the flow

rate between the 20°C case and the 30°C becomes more dramatic for a given pressure drop.

While the effect of temperature becomes negligible beyond an imposed pressure drop of

about 3 MPa for the 1 mm and 2 mm thick orifices, the 3mm thick orifice plot shows this

occurring at a higher pressure of around 5 MPa. Above this imposed pressure drop, the flow

rate increases at an almost constant slope, for all of the orifices. Thus, the effect of fluid

properties (primarily viscosity) is not significant at higher flow rates, perhaps due to the

approach to turbulence, which is consistent with the literature on discharge coefficients.

ca 104 a. ~

-ca ~ c: cu .... ~ c 102

10-5 Flowrate, m3/s 10-4

- 20°c, µ = 0.130 kg/m-s, p = 877.7 kg/m3, P = 2.6 MPa

-+-- 30°C, µ = 0.074 kg/m-s, p = 877.6 kg/m3, P = 2.6 MPa

- 40°C, µ = 0.045 kg/m-s, p = 865.4 kg/m3, P = 2.5 MPa

--+--- 50°C, µ = 0.029 kg/m-s, p = 859.5 kg/m3, P = 2.7 MPa

Figure 16. Effect of Temperature on Pressure Drop - Flow Rate Characteristics for the 1 mm Diameter., 1 mm Thick Orifice Plate

50

10-5 Flowrate, m3/s 104

-- 20°c, µ = 0.130 kg/m-s, p = 877.6 kg/m3, P = 2.6 MPa

-+- 30°C, µ = 0.074 kg/m-s, p = 871.6 kg/m3, P = 2.6 MPa

-+- 40°C, µ = 0.045 kg/m-s, p = 865.4 kg/m3, P = 2.5 MPa

-A- 50°C, µ = 0.029 kg/m-s, p = 859.5 kg/m3, P = 2.7 MPa

Figure 17. Effect of Temperature on Pressure Drop - Flow Rate Characteristics for the 1 mm Diameter, 2 mm Thick Orifice Plate

~ 104

~

"' e :l tn tn (1) ....

CL 103

cu ;; c e ;E c 102

10-5 Flowrate, m3/s 104

-+- 20°c, µ = 0.130 kg/m-s, p = 877.6 kg/m3, P = 2.6 MPa

-+- 30°C, µ = 0.074 kg/m-s, p = 871.6 kg/m3, P = 2.6 MPa

___.__ 40°C, µ = 0.045 kg/m-s, p = 865.4 kg/m3, P = 2.5 MPa

__._ 50°C, µ = 0.029 kg/m-s, p = 859.5 kg/m3, P = 2.7 MPa

Figure 18. Effect of Temperature on Pressure Drop - Flow Rate Characteristics for the 1 mm Diameter, 3 mm Thick Orifice Plate

51

The plots for the 0.5 mm diameter orifices show effects of temperature similar to

those discussed above for the 1 mm diameter orifices. However, the pressure at which the

temperature effects become negligible for all 0.5 mm diameter orifices is about 7 MPa. The

3 mm diameter orifices also show some effects of temperature, although the data for the 1

mm thick orifice is somewhat scattered. The 2 mm thick orifice appears to have negligible

temperature dependence beyond an imposed pressure drop of 0.5 MPa, while the 3 mm thick

orifice plate experiences this at a higher pressure drop of 2 MPa.

4.2.2 Effect of Orifice Thickness

Figures 19 to 22 show the effects of orifice plate thickness on the pressure drop-flow

rate characteristics for the 1 mm diameter orifices. Similar plots depicting the effect of

orifice thickness are shown in Appendix C for the 0.5 mm and 3 mm diameter orifices. From

all four plots, it is apparent that as the orifice plate thickness increases, the flow rate across it

decreases for the low pressure drop range. As the pressure drop increases, the flow rates

across the orifice appear to become independent of thickness and seem to be influenced only

by the differential pressure. The differential pressure at which the orifice thickness becomes

irrelevant is lower at the higher fluid temperatures. This indicates that as the differential

pressure increases, the flow rate becomes more dependent on Reynolds number and less

dependent on orifice thickness.

Similar trends are also seen for the 0.5 mm diameter orifice. However, in the 0.5 mm

diameter case, the flow rates for the three thicknesses never converge to a single graph,

independent of thickness, as seen in the 1 mm diameter orifices. This is explained by the fact

that the flow rate at which this occurs in the 1 mm diameter orifice is approximately 5 x 10-5

52

m3 /s, which is considerably higher than the flow rates seen in the 0.5 mm diameter orifices.

In the 3 mm diameter orifices, the 2 mm and 3 mm thicknesses appear to have almost

identical flow rates over the entire range of data. The 1 mm thickness, however, exhibits a

lower flow rate for the same pressure drop experienced by the 2 mm and 3 mm thick orifices,

which becomes more apparent as the temperature increases.

10·5

• 1 mm • 2mm

3mm

Flowrate, m3/s 104

Figure 19. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics for the 1 mm Diameter Orifice, T ~ 20°C

ns 104 CL ~

-ns ·-.... c: ~ C1) ~ c 102

10·5

53

• 1 mm • 2mm

3mm

Flowrate, m3/s 104

Figure 20. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics for the 1 mm Diameter Orifice, T :::z 30°C

ns 104 CL ~

.... ~ :::s ti) C1)

n.. 103

ns ·-.... c: C1) .... C1) t: c 102

10·5

• 1 mm • 2mm

3mm

Flowrate, m3/s 104

Figure 21. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics for the 1 mm Diameter Orifice, T :::z 40°C

10-5

54

- Col 7 vs Col 8 __..,.Col 14 vs Col 15 __........ Col 21 vs Col 22

Flowrate, m3/s 10-4

Figure 22. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics for the 1 mm Diameter Orifice, T ~ 50°C

55

CHAPTER 5

ORIFICE MODELING

In this chapter, trends in the data from the present study are discussed in terms of the

relevant non-dimensional parameters, and compared with the results from previous studies.

Results from the present study are also modeled using regression techniques to obtain an

overall orifice flow model for the full range of data.

5.1 Effect of Aspect Ratio ~n Euler Number

The effects of aspect ratio on Euler number are shown in Figures 23 to 26. From

these plots, it can be seen that at low Reynolds numbers, increasing aspect ratio tends to

cause an increase in the Euler number for a similar Reynolds number. For the 3 mm

100 • L/D = 0.33 • L/D = 0.66 D. ... L/D = 1.0 (3 mm Dia.)

N e L/D = 1.0 (1 mm Dia.) > 0 .'6. . D L/D = 2.0 (1 mm Dia.) a. ......... . -0- L/D = 2.0 (0.5 mm Dia.) c.. D "[),.

<l A A L/D = 3.0 N 10

· D L/D = 4.0 0 , II [:;, . L/D = 6.0 ::s 0

w

1 10 100 1000 10000

Re= pVd/µ

Figure 23. Effect of Aspect Ratio on Euler Number, T ~ 20°C

56

diameter orifices, however, it is observed that in the transition and turbulent regions, as

aspect ratio decreases, the Euler number increases for a similar Reynolds number.

5.2 Comparisons with Previous Work

The data from the present study were plotted as Eu vs. Re graphs for the seven aspect

ratios investigated in this study (l/d = 0.33, 0.66, I, 2, 3, 4, and 6). These plots are also used

to compare the results of the present study with the corresponding literature. These graphs

and the associated discussion are subdivided into two groups based on geometry: small

aspect ratios (0.33 ~ l/d < 2) and large aspect ratios (2 ~ l/d ~ 6).

5.2.1 Small Aspect Ratios

Data for l/d = 0.33 are shown in Figure 27 with comparable data from previous

100 ....--------- • L/D = 0.33

N > a. ........ a. -<:] 10 N II :1 w

1

A

A D,

0

0

10

• L/D = 0.66 ... L/D = 1.0 (3 mm Dia.) e L/D = 1.0 (1 mm Dia.) D L/D = 2.0 (1 mm Dia.)

····0 - L/D = 2.0 (0.5 mm Dia.) A L/D = 3.0 0 L/D = 4.0 'A

"'A, 6 L/D = 6.0 '66.

100 1000 10000

Re= pVd/µ


N > Q.. ........ D.. <:] 10 N II ::l w

10

57

-e- LID = 1.0 (1 mm Dia.} -er- LID = 2.0 (1 mm Dia.) -A- LID = 3.0

100 1000 10000

Re= pVd/µ


N > a.. ........ c. <:] N II :J w

100 ..-------------1 - LID = 0.33 ~LID= 0.66 _._ LID = 1.0 (3 mm ·Dia.) --e--- LID = 1.0 (1 mm Dia.) -er- LID = 2.0 (1 mm Dia.}

o LID = 2.0 (0.5 mm Dia.) ---A- LID = 3.0 ·

10 -- -El- - LID = 4.0 ~

6.- ·- LID = 6.0

0 0

1 ....___...___ __________________ ____,

10 100 1000 10000

Re= pVd/µ


58

investigators. The Euler number resulting from using the fully developed pipe flow friction

factor is also plotted for reference. The data from the present study show very good

agreement with those of James (1961) and Kiljanski (1993) although both authors tend to

under predict the current data for the range of values compared, which is most probably

caused by the slightly larger thickness ratios used in their studies. The data also agree quite

well with the results of Tuve and Sprenkle (1933) for thin, small diameter ratio orifices used

in flow meters.

As the aspect ratio increases to 0.66, the data from the current study tend to flatten out

at higher Reynolds numbers with a slight increase for the 50°C case near the upper end of the

Re· range examined. Figure 28 shows the data for this aspect ratio and indicates that the data

from James (1961) and Kiljanski (1993) now over predict the data from the current study, as

• Present Study 20°c 10 • Present Study 30°C

\ • Present Study 50°C \ \ a Kiljanski (1993) ~ = 0.079, l/d = 0.5

7 \ \ 0 James (1961) ~ = 0.251, l/d = 0.5 \ > \ \ -·-·- Tuve and Sprenkle (1933) ~ = 0.2 a. 5 \ \ - ~ --- Eu = 64/Re x l/d ......... a.. \ \

<::] \ \ 08 N \ \ 0 II 3 \ ' ocP

0 ::l \ " ............... ~Q w \

2 \ ·,er_ a

\ \ .

101 102 103 104

Re= pVd/µ Figure 27. Euler Number Variation for lid= 0.33

10 -

7 N > a. 5 ........ a.. <:] N II 3 :J w

2

I

101

I

• • a 0

59

Present Study 20°c Present Study 30°C Present Study 50°C Kiljanski (1993) ~ = 0.079, 1/d = 0.5 James (1961) ~ = 0.251, l/d = 0.498 Tuve and Sprenkle (1933) ~ = 0.2 Eu = 64/Re x l/d

I I

102 103 104

Re= pVd/µ

Figure 28. Euler Number Variation for l/d = 0.66

do the results ofTuve and Sprenkle (1933).

Figure 29 shows the results for l/d = 1 and again shows very good agreement with the

data from James (1961). Please note that the present work includes two data sets with lid=

1: 3 mm diameter by 3 mm thick and 1 mm diameter by 1 mm thick. It is also observed that

the current data obtained for the 3 mm diameter orifice tends to have lower Eu numbers at

higher Re numbers than those obtained from the 1 mm diameter orifice. This slight

difference is most probably the result of the slight difference in geometries between the two

sets of orifice plates. The 3 mm diameter orifice in this case (nominally l/d = 1) has an actual

aspect ratio of 0.973 (1=2.997 mm, d = 3.079 mm) and a diameter ratio (dlD) of 0.135 while

the 1 mm diameter orifice has an aspect ratio of 1.016 (1 = 1.029 mm, d = 1.013 mm) and a

diameter ratio of 0.045. The data also agree quite well with that of Sahin and Ceyhan (1996),

\ 0¢

10 \ \ Q, . \ N \ \ 8¢ > 7 \ \ a. \ \ o~ ..._ a.. 5 \ \ oc:O'b <:] • .l:i,

\ \ a)i:i. N \\ <!o/:i. II -~ 6 :::s 3 \' ~ ,A~t::;. w \ ·,

2 \ ......... _. __ \ \ \

60

• • A

0 D !::;.

0 6

a 0 0

-·-·-· ----

Present Study 20°C (3 mm) Present Study 30°C (3 mm) Present Study 50°C (3 mm) Present Study 20°C (1 mm) Present Study 30°C (1 mm) Present Study 40°C (1 mm) Present Study 50°C (1 mm) James (1961) ~ = 0.071, ltd = 1 Kiljanski (1993) ~ = .079, l/d = 1 Hasegawa et al. (1997) l/d = 1.09 Sahin and Ceyhan (1996) l/d = 1 Tuve and Sprenkle (1933) ~ = 0.2 Eu = 64/Re x l/d

---·---·-·-·-·-·

101 102 103 104

Re= pVd/µ

Figure 29. Euler Number Variation for lid= 1

Hasegawa et al. (1997), and Kiljanski (1993), but now shows considerable deviation from the

results of Tuve and Sprenkle (1933).

A comparison of Figures 27-29 shows that the data from the present study are

primarily in the transition region where the Euler number reaches a minima. Also, the

minima moves toward higher Re values as the aspect ratio increases. This transition region is

inherently more prone to instabilities, which could be responsible for some of the deviation

between the present data and the literature.

5.2.2 Large Aspect Ratios

The data for the large aspect ratios (2 ::; lid ::; 6) are compared with the literature in

Figures 30-34. For the case with lid= 2, these data also show that the resulting Eu number

61

• Present Study 20°C (0.5 mm) 100 • Present Study 30°C (0.5 mm)

N ... Present Study 50°C (0.5 mm) > 50 0 Present Study 20°c (1 mm) 0... D Present Study 30°C (1 mm) ......... a. l::J. Present Study 40°C (1 mm)

<:] ¢ Present Study 50°C (1 mm) N Lichtarowicz et al. (1965) l/d = 2 II 10 -·-·-· Tuve and Sprenkle (1933) ~ = 0.2 ::::s 5 ---- Eu = 64/Re x l/d w

·-·-·-·-·

1 0.5 _______________ ...._____.

101 102 103 104

Re= pVd/µ Figure 30. Euler Number Variation for lid= 2

for a given Re number is larger for the 1 mm diameter orifice than the 0.5 mm diameter

orifice. This is again most probably attributable to the slight differences in aspect ratio

between the two orifices with the 1 mm diameter orifice (l/d = 1.950, d = 1.003 mm) having

both a larger aspect ratio and a larger diameter ratio than the 0.5 mm diameter orifice (lid=

1.898, d = 0.524 mm). For Re > 200, the curve fit by Lichtarowicz et al. (1965), which

includes data from James ( 1961) and others, again yields fairly good agreement with the

current data, although it appears to slightly under predict the data near the largest Re values

tested here. For Re< 200, the curve fit by Lichtarowicz et al. (1965) over predicts the data.

The results of Tuve and Sprenkle (1933), on the other hand, under predicts the current data at

low Re, and over predicts them for Re > 500. Thus, the data from the present study are

62

between the values predicted by these investigators.

Similarly for all other cases with l/d ~ 3, the data from the current study continue to

be bracketed by the equation of Lichtarowicz et al. ( 1965) and the results of Tuve and

Sprenkle (1933) as shown in Figures 31, 32, and 33. It should be noted that as the aspect

ratio increases in this range, these two curves intersect at higher Re numbers. Also in this

range of lid ratios, the data in the laminar region appear to approach the fully-developed pipe

flow friction factor expression given by:

Eu= 64 _!_ Red

(20)

This is to be expected because as l/d mcreases, the geometry more closely

approximates a circular tube.

100

N 50 > a.. ......... c.. <:] N 10 II :::s 5 w

1

0 D /::;.

<>

Present Study 20°C Present Study 30°C Present Study 40°C Present Study 50°C Lichtarowicz et al. (1965) 1/d = 3 Tuve and Sprenkle (1933) ~ = 0.2 Eu = 64/Re x l/d

0.5 --------------------101 102 103 104

Re= pVd/µ


1000

> ~ 100 a.. <l N II ::J 10 w

1

' ' '-;e

' "•

63

• • • Present Study 20°c Present Study 30°C Present Study 50°C Lichtarowicz et al. (1965) l/d = 4 Tuve and Sprenkle (1933) ~ = 0.2 Eu = 64/Re x 1/d

' ' ' , ... ''·-. '~'mt

---·--·--~~·-·

101

' ' ' 102 103

Re= pVd/µ 104


1000

N > ~ 100 a.. <l N II :::s 10 w

1

' ,. ' "-" '

• • • Present Study 20°c Present Study 30°C Present Study 50°C Lichtarowicz et al. (1965) 1/d = 6 Tuve and Sprenkle (1933) ~ = 0.2 Eu = 64/Re x l/d

, .. ..... " .__ ' ' '-9'1 '

'· ' .. •ti• '·-. ' . 101

-·-·-.-~.~.-.... '

102

' ' ' 103

Re= pVd/µ 104


64

5.3 Model Development

Insights obtained from the graphs discussed above and the Eu expressions available in

the literature were used to develop a model for orifice flow based on the data from the

present study.

Several authors (Hasegawa et al. 1997, Lichtarowicz et al. 1965, Ramamurthi and

Nandakumar 1999) have suggested that for laminar flow at low Reynolds numbers, the Euler

number is composed of a viscous term and a constant as shown below:

(21)

The first term represents viscous losses while the constant represents the additional

pressure drop resulting from changes in the velocity profile at the entrance and exit to the

orifice. Using this equation as the basis, the effects of the orifice Reynolds number, the

aspect ratio, and the diameter ratio were incorporated into a generic equation of the following

form for the low Re range:

Eu iam = a Re b ( 1 +pc ~r ) (22)

Here, constants a, b, c, and e are floating parameters to be determined through a regression

analysis of the data.

At high Reynolds numbers, the Euler number tends to a constant represented by the

following equation (Morris and Garimella 1998):

(23)

Data from Ward-Smith (1971) were re-correlated by Morris and Garimella (1998) to

obtain the following equations for the discharge coefficient that are valid for diameter ratios

65

less than 0.25:

0 <lid 0.9: [ r 11µ.195] o.356 cd =0.255 l+vaJ + (1+ Yctt14o (24)

0.9 <lid 2.5: c d = 0.876- 0.0139 }';;- 0~4 (25)

2.5<1/d 9.5: [ { 1/\-0.068] 0.292 Cd = 0.292 1 +\la} + (1 +fatso (26)

In the absence of a large number of data points for the fully turbulent region, these equations

were used to represent the turbulent Eu for the present study also.

Finally, the laminar and turbulent Euler number equations were combined to form an

overall equation representing the entire range of experimental data as follows:

(27)

SigmaPlot graphing software by SPSS Inc. was used to conduct the required

regression analysis and obtain the values of the correlation parameters shown in Table 5.

T bl 5 C a e . onstants ~ E or ;quat1on (22) Variable Value

a 96.352 b -0.861 c 1.791 e 0.586

The resulting correlating equation is as follows:

(28)

This equation has a correlation coefficient of R2 = 0.9863, and results in very good

66

agreement with the data. As shown in Figure 34, this correlation predicts 259 of the 307 data

points (84.4 % ) within ± 25 %.

In observing the data, it was noticed that for high pressure drop cases, the frictional

losses through the orifice resulted in temperature rises across the orifice of up to 5.1°C. This

rise in temperature caused a decrease in the viscosity across the orifice with resulting

viscosity ratios (inlet/outlet) as high as 1.6. In an attempt to capture this phenomenon,

Equation 28 was modified to include a viscosity ratio term. While this resulted in a slight

increase in the correlation coefficient, it was deemed to be insignificant. Additionally, the

resulting model would require a priori knowledge of the inlet and outlet temperatures, which

would make the use of the correlation impractical for real applications. Therefore, this

"'C Cl) ...... (.) ·-"'C

100 [ 3 ( 4 J3]x Eu= (96.352Re-0861 {kiJ

791

~0586 ) + ~~~'

I R2 = 0.98631

e 10 -a.. :::s w

1

• d = 0.5 mm, I = 1 mm • d = 0.5 mm, I = 2 mm .& d = 0.5 mm, I = 3 mm o d = 1 mm, I = 1 mm a d = 1 mm, I = 2 mm ll. d = 1 mm, I = 3 mm o d = 3 mm, I = 1 mm a d = 3 mm, I = 2 mm ~ d = 3 mm, I = 3 mm

Eu ±25%

1 10 100 Eu Experimental

Figure 34. Comparison of the Predicted Eu Numbers with Experimental Eu Numbers

67

additional term was not used. The range of applicability of this correlation is as follows:

0.32 <lid< 5.72

0.02 < B < o.137

7.9 <Re< 7285

0.028 < µ < 0.135 (kg/m-s)

More detailed comparisons between the data and the values predicted by this model

for specific geometries are shown in Figures 35 through 43. In general, the model agrees

well with the data at low Re, but tends to under predict the data in the transition region.

Good agreement is also seen with the relatively few points in the turbulent region.

The predictions of this model for the various geometries are further demonstrated in

Figures 44 through 46, where the predictions for different aspect ratios are plotted for each

diameter ratio. In each figure, it can be seen that in the laminar region, as the aspect ratio

increases, so does the Euler number. Also in this region, an increase in the diameter ratio

results in a larger Euler number for similar aspect ratios. In the transition region, the effect

of increasing Reynolds number diminishes as the slope of the Euler number graphs approach

zero. Finally, in the turbulent region, the Euler number tends to a constant which is dictated

solely by the diameter and aspect ratios. It is interesting to note that the constant Euler

number is lowest for an aspect ratio of approximately two. Above and below this aspect

ratio, the turbulent Euler number increases, although the increase is more noticeable as aspect

ratios decrease below two.

~ w 101

101

0 ... 2 g

68

102 Re

• Experimental o Predicted

103 104

Figure 35. Comparison of Measured and Predicted Eu for the 0.5 mm Diameter, 1 mm Thick Orifice

... Experimental 0 Predicted

0

~ go

w 101 (2)

0 ... ~ ... 4 ~~ ...

... ·~ ~~~~ .........

<DO CXX>

10° 101 102 103 104

Re


69

102

6 A Experimental

Qo 0 Predicted

::::s 101

p2 w 22o

~ % •• Q ••••

<tc,• A ~A A 4AA

Oo 000

10° 101 102 103 104

Re


::::s w 101

0

~· 0

10° 101 102

A


~~~~~

103

Re 104

Figure 38. Comparison of Measured and Predicted Eu for the 1 mm Diameter, 1 mm Thick Orifice

::l w 101

Q ~ ...

0

10° 101 102

70

.& Experimental o Predicted

~ ~~ ............... ~··· ~~&moo

103

Re 104


::l w 101

101 102

Re


103 104


71

~ 101

101 102

Re


103 104


~ 101

101 102

Re


103 104


72

~ 101

101 102

Re


103 104


:::J w 10

1 10 100

Re 1000

l/d = 2 1/d = 4 l/d = 6

Figure 44. Model Predictions and Trends for a Diameter Ratio of 0.0231

73

10

10 100 Re

l/d = 1 --- l/d=2 ------ l/d = 3

1000


l/d = 0.33 --- l/d=0.66 ------ l/d = 1

~ 10

~-..... ---------.... __ ---------------10 100 1000

Re


74

CHAPTER&

CONCLUSIONS AND RECOMMENDATIONS

6.1 Conclusions

In the present study, the flow characteristics of nine small diameter orifices were

determined experimentally. The orifice geometries included three different diameters (0.5, 1,

and 3 mm) and three different thicknesses (1, 2, and 3 mm). Euler numbers were found for

these orifices with Reynolds numbers ranging from 7 .9 to 7285 and pressure drops ranging

from 93.7 kPa to 10.0 MPa.

In the laminar range, an increase in the aspect ratio results in an increase in the Euler

number for similar Reynolds numbers. In the turbulent region, however, a minimum Euler

number tends to occur for aspect ratios around two. Aspect ratios increasing above or

decreasing below this value tend to result in increasing Euler numbers.

Oil temperature also affects the orifice flow characteristics. A representative graph

using the data for the lid= 3 case is shown in Figure 47. This figure shows that increasing

temperature in the laminar region results in an increase in the Euler number for similar

Reynolds numbers. Additionally, the extent of the laminar region increases as the

temperature increases, as evidenced by the location of the minima in the Eu-Re plots. As

temperature increases, the viscosity of the fluid decreases. Thus, to maintain the same

Reynolds number at a higher temperature, the velocity must also decrease. This lower

velocity, through its inverse square effect on Euler number (Eu ex: 1N2), will result in an

increased Euler number, which explains the trends seen in Figure 47. However, due to the

afore-mentioned increase in the extent of the laminar region at higher temperatures, this

75

effect is coupled with transition to turbulent behavior at increasing Re values. Thus, in the

transition region, the trends are not as clear, with additional data at higher Reynolds numbers

being needed for a better understanding of the corresponding dependence on temperature

(and therefore properties).

For comparison with the literature, the orifice data were divided into two sets: lid< 2,

and l/d 2 2. Data from the 3 mm diameter, 1 mm thick orifice (l/d = 0.33) tend to agree

quite well with that of James (1961) and the results of Tuve and Sprenkle (1933), but as the

aspect ratio increases to 0.66 for the 3 mm diameter orifice, both James (1961) and Tuve and

Sprenkle (1933) tend to over predict data from the current study.

Further increasing the aspect ratio to unity again results in very good agreement with

data from James (1961) and also shows good agreement with data from Hasegawa et al.

8 7 6

_._ 20°c

----- 30°C N > 5 -----+- 40°C Q. ____.,.__ 50°C ...._ a. 4 <I N II 3 :l w

2

101 102 103 104

Re= pVd/µ

Figure 47. Effect of Temperature on Euler Number for lid ~ 3

76

(1997) and Kiljanski (1993). The results of Tuve and Sprenkle continue to over predict the

current data in the transition region and under predict it in the laminar region. It also

becomes apparent that at low Reynolds numbers, the slope of the Euler graph approaches the

graph representing fully-developed pipe flow.

Further increases in aspect ratio from two up to six show that the data in the transition

region are between the predictions of the equation from Lichtarowicz et al. (1965) and the

results of Tuve and Sprenkle (1933) and that in the laminar region, the data approach the

graph representing fully-developed pipe flow.

Regression analysis was used to develop an Euler number equation covering the

entire range of data from the current study. This overall equation combined the two Euler

number equations representing laminar and turbulent flow and predicts 259 of 307 data

points to within ± 25 %.

6.2 Recommendations

Further study of orifice flow is required to fully understand the effects of fluid

properties on flow behavior in small diameter orifices, as it is apparent from the current study

that highly viscous fluids do not behave as predicted. To further assist this effort, more

orifice plates that increase the combinations of aspect and diameter ratios should be

fabricated and tested. Also, the Reynolds number range investigated here must be extended

until the data approach the expected fully turbulent values.

Currently, the temperature at the inlet and outlet of the orifice is measured at some

distance from the orifice plate itself. Although this was required to allow flow in the

upstream and downstream sections to fully develop, it has possibly allowed for inaccuracies

77

due to flow stratification. At very low flow conditions, it is possible for large temperature

gradients to appear as the temperature is either raised or lowered from room temperature. As

the RTD being used to measure these temperatures is quite long, it is possible that the RTD

will sag causing it to be positioned in a stratified layer that doesn't represent the actual

temperature of the fluid at the orifice. The pressure transmitters at the inlet and outlet of the

orifice were selected for both high accuracy and large range. Unfortunately, at low flow

conditions, the pressures at the inlet and outlet are quite low resulting in less accuracy than

was originally intended. Although this inaccuracy has very little effect on the outcome of the

data, it would be worth investigating further to ensure higher confidence in the results.

Also, the effects of temperature seen in this study appear to show an as-yet not

understood effect of viscosity on the Euler number. Further investigation at much lower

temperatures would extend the viscosity range, and document more clearly the effect of

increasing viscosities and potential non-Newtonian behavior at lower temperatures.

78

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Ward-Smith, A. J., 1971, "A Unified Treatment of the Flow and Pressure Drop Characteristics of Constructions Having Orifices With Square Edges," Part 4 in Pressure Losses in Ducted Flows, Butterworths, London.

Wassel, A. T. and Catton, I., 1973, "Calculation of Turbulent Boundary Layers Over Flat Plates with Different Phenomenological Theories of Turbulence and Variable Turbulent Prandtl Number," International Journal of Heat and Mass Transfer, Vol. 16, No.8,pp. 1547-1563.

Witte, R., 1928, "Flow Constants of the I. G. Measuring Devices for Water, Oil, Gas, and Steam," Zeitschrift V. D. I., Vol. 72, No. 42, pp. 1493-1502.

Zampaglione, D., 1969, "Sul Moto di Una Corrente Laminare Attraverso Diaframmi," L'Energia Electrica, Vol. 46, pp. 821-828.

Zhang, Z. and Cai, J., 1999, "Compromise Orifice Geometry to Minimize Pressure Drop," Journal of Hydraulic Engineering, Vol. 125, No. 11, pp. 1150-1153.

82

APPENDIX A

DATA ANALYSIS

A.1 Sample Calculations

This appendix describes the computation of the orifice pressure drop from the

measured pressure drop. The data point used for this demonstration is labeled 11-50-20,

which represents data for the 1 mm diameter, 1 mm thick orifice plate with an imposed

pressure drop of 50 bars and an inlet temperature of 20°C. 121 readings were taken for each

data point over a two-minute time period (one per second). Table Al shows the average

values for this data set while Table A2 shows the same values as in Table Al, but in S. I.

units.

Table A 1. Raw Data Used in Calculations for Point 11-50-20 Temperature Temperature Pressure Pressure Differential Flow

In Out In Out Pressure Rate (°C) (°C) (psia) (psia) (psid) (gpm)

20.00 22.17 828.30 107.90 724.14 0.9734

Table A2. Raw Data for Point 11-50-20 (S.I. Units) Temperature Temperature Pressure Pressure Differential Flow

In Out In Out Pressure Rate (°C) (°C) (kPa) (kPa) (kPa) (m3/s)

20.00 22.17 5710.95 743.98 4992.82 6.14E-05

The velocity through each segment was calculated by dividing the volumetric flow

rate by the cross-sectional area for each segment, as follows:

(Al)

83

The dimensions for each segment of Figure 9 and the resulting velocities are shown in

Table A3 while the oil properties for this test condition are shown in Table A4.

Table A3. Test Section Dimensions and Results (1 mm Diameter, 1 mm Thick Orifice Plate)

Segment Length Diameter Area Velocity Re f Af>r (m) (m) (m2) (mis) (kPa)

1 1.524 .0109 9.369E-05 0.656 45 1.424 37.55

2 0.152 .0085 5.653E-05 1.086 58 1.106 10.31

3 0.026 .0104 8.518E-05 0.721 47 1.358 0.77

4 0.015 .0183 2.627E-04 0.234 27 2.385 0.05

5 0.019 .0295 6.818E-04 0.090 17 3.843 0.01

6 0.152 .0228 4.063E-04 0.151 22 2.966 0.20

01 0.001 .0010 8.059E-07 76.205 548 0.117 302.21

02 0.004 .0052 2.109E-05 2.912 121 0.529 1.36

7 0.152 .0207 3.366E-04 0.183 30 2.112 0.23

8 0.019 .0295 6.818E-04 0.090 21 3.007 0.01

9 0.015 .0183 2.627E-04 0.234 34 1.866 0.04

10 0.026 .0104 8.518E-05 0.721 60 1.063 0.60 11 0.152 .0085 5.653E-05 1.086 74 0.866 8.03

12 1.524 .0109 9.369E-05 0.656 57 1.115 29.24 Note: Segment 01 refers to the orifice while segment 02 refers to the back-cut in the orifice plate.

The density and viscosity of the hydraulic oil were calculated from a Fortran

subroutine supplied by the John Deere Production Engineering Center. The properties for the

inlet and outlet segments were calculated at the inlet and outlet temperatures and pressures,

respectively. The properties at the orifice were calculated at the average pressure and

temperature. The Reynolds numbers for each segment were then calculated as follows:

(A2)

84

Table A4. Property Values for Hydraulic oil at Pressures and Temperatures Listed

Temperature Pressure Density Viscosity (oC) (kPa) (kg/m3) (kg/m-s)

Inlet 20.00 5711 879 0.140 Average 21.09 3227 877 0.124 Outlet 22.17 744 875 0.109

The total measured pressure drop therefore consists of the pressure drop due to the

orifice, pressure drops in small straight sections of upstream and downstream piping

(between the pressure taps), and losses due to expansions and/or contractions into and out of

these individual segments. It is clear that the flow through these small segments of piping is

not fully developed. However, since there are no readily available expressions for pressure

drops in such segments, the corresponding friction factor for fully developed flow through

straight circular tubes was used to provide an estimate of these losses. The accuracy of these

estimates is not very significant, because as will be shown later, the pressure drops in these

segments are extremely small fractions of the orifice pressure drop for most of the data.

Thus, the Darcy friction factor correlation by Churchill ( 1977) was used to calculate the

friction factor for each segment as follows:

1 2.457 · ln ---0 9-----

[_]_J . + 0.27 · _c Re dseg

( 8 )l2 f =8· Re +

16

1 3 12 2

(A3)

In the above expression, the roughness of drawn tubing (c= 0.0015 mm) was used for each

segment. This was deemed adequate as the flow was laminar in every segment making the

roughness value an insignificant contributor to the value of the friction factor. The frictional

85

pressure drops due to piping losses were calculated as follows:

(A4)

These frictional losses for each segment are shown in Table A3.

The loss coefficient for sudden expansion was determined as follows (Munson, et al.

1998):

(A5)

The loss coefficient for a sudden contraction was obtained from the following curve-

fit to a graph available in Munson et al (1998):

A 2 -0.705 A1 KL = -0.021+0.585 1 +exp ---=----

0.233

-2.29

Once the loss coefficient KL was determined, the pressure loss was calculated as

follows:

Afl = _!_K V2 minor 2 LP

(A6)

(A7)

Table A5 shows the area ratios used to calculate the loss coefficients and the resulting

pressure drops.

Finally, the estimated pressure drop due to the test section piping was calculated by

summing the frictional losses with the minor losses and subtracting out the pressure drop

calculated for the orifice itself (L'.1Pr,01). It should be noted that the pressure drops due to the

orifice back-cut (L'.1Pr,02 and L'.1Pminor,B*) are considered to be part of the system piping when

86

Table A5. Dimensions and Results for Minor Losses

Sudden Area Segment LiPminor

Change Ratio KL Velocity (mis) (kPa)

Bl 0.6636 0.1131 1.09 0.0587

B2 0.3243 0.4566 0.72 0.1044

B3 0.3853 0.3779 0.23 0.0091

Cl 0.5960 0.1711 0.15 0.0017

B* 0.0627 0.8786 2.91 3.2612

B4 0.4936 0.2564 0.18 0.0037

C2 0.3853 0.3281 0.23 0.0078

C3 0.3243 0.3684 0.72 0.0838

C4 0.6636 0.1241 1.09 0.0641

Note 1: B stands for a sudden expansion and C stands for a sudden contraction.

Note 2: B• is for the orifice plate back-cut and represents either a sudden contraction (1 mm diameter, 3 mm thick orifice) or a sudden expansion (all other orifices) into the downstream piping. For this example, it represents a sudden expansion.

calculating the pressure drop due only to the orifice. When comparing the estimated value of

the system pressure drop to the experimentally determined value however, these pressure

drops would also have to be subtracted. The estimated value for the test section pressure

drop was found to be 91.99 kPa while the experimental value was found to be 72.15 kPa

(including estimates for the back-cut losses.) To determine the pressure drop across the

orifice, the experimentally determined pressure loss of 72.2 kPa was subtracted from the

measured differential pressure drop of 4992. 7 kPa to yield a value of 4901 kPa. Thus, in this

case, the extraneous pressure drop is 1.45 % of the measured pressure drop.

Lastly, the Euler number for the orifice was calculated, as shown below, to be 1.933.

87

Eu= ~p }ipv~

(AS)

A.2 Error Analysis

Uncertainties in the dimensionless variables discussed in the previous section were

computed using an error-propagation approach. Equations for the Reynolds and Euler

numbers are shown below:

Re= 4Qp nDµ

n2D4~p Eu=---

8pQ2

The uncertainty in the Reynolds number for the orifice is given by:

u2 =(8Reu ) 2 +(8Reu ) 2 +(8Reu ) 2 +(aReu ) 2 Re 8Q Q 8D D 8p P 8µ µ

_aR_e =-4_p = 4(877) =S.92 x 106 _s_ n(.001x.124) m3 nDµ

_a_R_e =--4_Q_p = 4(6.14x102 -sXs11)=-S.4lxlosm-1 8D nD2µ n(.001) (.124)

8Re = 4Q = 4(6.14x10-5 )= 6.25 x 10-1 m3 8p nDµ n(. 001 X.124) kg

8Re 4Qp 4(6.14xl0-5 X8?7) =-4.43xl03 m·s 8µ = - nDµ 2 = n(.001X.124) . kg

(A9)

(AIO)

(All)

(A12)

(A13)

(A14)

(A15)

The measurement uncertainties for temperature, pressure, and flow rate are given in

Table A6. For the data point 11-50-20, the flow rate uncertainty was 1.84 x 10-7 m3/s.

88

Table A6. Uncertainties in Temperature, Pressure, and Flow Rate Measurements for the 1 mm Data

Uo ± 0.0025 mm for each orifice plate UT ± 0.6°C for all cases

UAP ± 10.342 kPa for all readings (Absolute Pressure) UDP 1 and 2 bar cases 5 bar and greater

(Differential Pressure) ± 0.948 kPa ± 7.757 kPa

UQ 20°c I 30°C, 40°C and 50°C 0.3% ofreading I 0.5% ofreading

Details of the calculation of uncertainties in density and viscosity are provided in

Appendix B. The resulting uncertainties, Up and Uµ, based on the conditions of this data

point were:

up = ~(o.366)2 + ((o.oo5X811))2 = ± 4.40 k~ m

(A16)

uµ =~(5.6o5x10-3 )2 +((o.0295Xo.124))2 =±6.69x10-3~ m-s

(A17)

Thus, the uncertainties in properties are ± 0.50% for density and ± 5.40% for

viscosity.

The other two uncertainties in the Reynolds number equation were:

3 UQ = 0.003(6.14xl0-5 )= ±1.84x10-7 m

s

U 0 = ±2.54x10--{) m

The resulting uncertainty in the Reynolds number is given by:

URe = ~(1.644)2 + (-1.352}2 + (2.751)2 + (-29.637)2 = ±29.85

Thus, the uncertainty in the Reynolds number is 5.45 %.

The uncertainty in the Euler number for the orifice is calculated as follows:

(A18)

(A19)

(A20)

89

u2 =(8Euu J2 +(8Euu ) 2 +(8Euu J2 +(8Euu ) 2

Eu 8Q_ Q 8D D 8p P 8~P DP

8Eu = _ rr2D4M = _ rr 2 (.001)4(4901xl03) = _6.2Sxl04 _s_ 8Q 4pQ3 4(877)(6.14x10-5 ) 3 m 3

8Eu = rr 2D3 M = rr 2 (.001) 3(490lxl03) = 7.62 x 103 m_1

8D 2pQ 2 2(877)(6.14xl0-5 ) 2

_aE_u = _ rr 2D4~P = rr 2 (.001)4(4901xl03) = _2.20xl0-3 m 3

8p 8p 2Q 2 8(877) 2 (6.14xl0-5 ) 2 kg

(A21)

(A22)

(A23)

(A24)

(A25)

For the uncertainty in differential pressure, it was necessary to also account for the

uncertainty in the value of the system loss subtracted from the measured differential pressure.

Because the system losses were measured using a blank orifice in place of an orifice plate, it

is possible that the system losses without the orifice plate would be somewhat different from

the losses in the presence of the orifice plate due to differences in the flow mechanisms

caused by the orifice plate. To account for these potential differences, a conservative

estimate of ± 25% was used for the uncertainty in the system losses. Therefore, the

uncertainty in the differential pressure was found by:

u DP = ~u~P.m + U~P.sys = ~(7.757)2 + ((o.25Xn.2))2 = ±19.64 kPa or 0.40 % (A26)

UEu = ~(-0.012)2 +(0.019)2 +(-0.010)2 +(0.004)2 = 2.57xl0-2 (A27)

The resulting uncertainty in the Euler number is 1.33 %.

Table A 7 gives the range of the Reynolds and Euler number uncertainties for the

current study. For the 3 mm orifice, the uncertainty in Euler number, at low pressure drops,

90

becomes as high as 18.4%. This is due to the large uncertainty in the system loss which now

accounts for a much larger portion of the measured pressure drop in these cases.

Table A7. Range of Reynolds Number and Euler Number Uncertainties for th Th 0 'f' D. t . th C t St d e ree r1 ice 1ame ers m e urren U IY

0.5 mm Diameter ReRange Uncertainty in Re Eu Range Uncertainty in Eu

7.9 <Re< 1994 0.45 < URe < 65.3 1.3 < Eu< 58.9 0.03 < UEu < 1.33 (3.07% < URe < 5.63%) (2.09% < UEu < 2.72%)

1 mm Diameter ReRange Uncertainty in Re Eu Range Uncertainty in Eu

39.6 <Re< 3261 2.2 < URe < 108.7 l.7<Eu<7.3 0.03 < UEu < 0.14 (3.27% < URe < 5.97%) (1.30% < UEu < 2.27%)

3 mm Diameter Re Range Uncertainty in Re Eu Range Uncertainty in Eu

214 <Re< 7285 11.4 < URe < 235 1.5 <Eu< 2.8 0.04 < UEu < 0.39 (3.19% < URe < 5.36%) {l.88% < UEu < 18.4%)

91

APPENDIX B

DENSITY AND VISCOSITY UNCERTAINTIES

B.1 Uncertainties Due to Pressure and Temperature Measurements

The effects of temperature and pressure on density and viscosity are shown in Figures

B 1 and B2. From these figures, it can be seen that both density and viscosity decrease as

temperature increases for a constant pressure. For a constant temperature, however, an

increase in pressure results in an increase in both the density and viscosity.

930

920 M

E 910 .._ C> ~ 900 .. ::t .. 890 ~ ..... tn 880 c: Cl)

870 c 860

-40 -20 0 20

-- 1 Bar ---- 10 Bar ........... 50 Bar -·-·- 100 Bar

40 Temperature, T, °C

Figure Bl. Effect of Pressure and Temperature on p

Hydraulic oil property values were determined using a FORTRAN subroutine

provided by the John Deere Production Engineering Center (JDPEC). To determine the

uncertainty in the property values provided by this program, the uncertainty in the measured

"' 100 I

E ....... C> 10 ~ ... ::t ... ~1 ·-"' 0 u "' 0.1 ·->

0.01 -40

92

-20 0

1 Bar ----- 10 Bar ·············· 50 Bar -·-·-· 100 Bar

20 40 Temperature, T, °C

Figure B2. Effect of Pressure and Temperature on µ

pressure and temperature must be taken into account. This was accomplished by creating

two representative data sets with pressure and temperature values that spanned the entire

range of recorded values. Both data sets contained eleven pressures for each of the four

temperatures measured (20, 30, 40, and 50°C). Additionally, for the data set used to

determine the uncertainty due to temperature, a 55°C case was added as this value was

slightly higher than the highest temperature in the recorded data.

The first data set was used to determine the uncertainty due to measured temperature.

To do this, the program was first run with the temperature uncertainty (0.6°C) subtracted

from the five temperatures. Next, the program was run with the temperature uncertainty

added to the five temperatures. Finally, the change in density and viscosity with respect to

temperature was determined as shown in Equations B 1 and B2.

93

Bp PT+llT,P -pT-llT,P = or (Bl)

Bµ µT+llT,P - µT-llT,P = or (B2)

These calculations were performed at each pressure in the data set.

Similarly, the second data set was used to determine the uncertainty due to the

measured pressure. In this case the value of the uncertainty in the pressure (10.342 kPa) was

first added and then subtracted from the pressures in the data set and the resulting change in

density and viscosity, with respect to pressure, calculated as shown in Equations B3 and B4.

Bp = PP+llP,T - PP-llP,T

BP 2Up (B3)

Bµ µP+llP,T - µP-llP,T =

BP 2Up (B4)

These calculations were conducted at all temperatures in the data set.

Finally, the uncertainty in the density and viscosity due to uncertainty in measured

pressure and temperature was found using Equations B5 and B6.

up,m = (B5)

uµ,m = (B6)

The respective changes in density and viscosity with respect to pressure and

temperature were then plotted as shown in Figures B3, B4, B5, and B6.

As is seen in Figures B3 and B4, the change in density with respect to both

temperature and pressure remains fairly constant over the range of values covered. The

94

maximum absolute values of both 8p /Of and 8p / 8P were used to determine a conservative

overall uncertainty in density (U p,m) due to measured temperature and pressure which was

then used to calculate the uncertainty for every data point. The values used for this

uncertainty are shown below in Equations B7, B8, and B9:

8p 3 - = 0.611 kg/m -K 8T

(B7)

(B8)

(B9)

The changes in viscosity with respect to pressure and temperature are shown in

Figures B5 and B6. These figures show that 8µ/8T and 8µ/8P are more dependent on

-0.585

-0.590

-0.595 1-"'C a. -0.600 "'C

-0.605

-0.610

-0.615 0 20 40 60

-- T=20°C ~ T=30°C - T=40°C __...._ T = 50 °C ~ T=55°C

80 100 Pressure, Bar

Figure B3. Effect of Pressure and Temperature on 8p/8T

120

6.4e-7 ..------ -- -------- -

6.2e-7

6.0e-7

a.. 5.Se-7 "'C ....... Q.

-c 5.6e-7

5.4e-7 --e- T = 20 °C ---+- T = 30 °C

5.2e-7 --- T = 40 °C ___._ T = 50 °C

95

5.0e-7 .___ _ __._ __ __.__ _ _____. ____ _..__ __ _.___ _ ___, 0 20 40 60 80 100 120

Pressure, Bar

Figure B4. Effect of Pressure and Temperature on Bpi BP

0.000 I IJII I I I I I

-0.002 •••• • • • • • I- -0.004 . . .. . • • • "'C "'3. • ... "'C -0.006 - T=20°C

--+-- T = 30 °C - T=40°C

-0.008 __..._ T = 50 °C ---..- T = 55 °C

-0.010 .._ _ ____.__ _ ___... __ ___._ _ _____._ __ _.___~ . 0 20 40 60 80 100 120

Pressure, Bar

Figure BS. Effect of Pressure and Temperature on Bµ/8T

------- T = 20 °C 5e-9 -+- T= 30 °C -e- T = 40 °C

96

4e-9

a. -c 3e-9 "'3..

--*-" T = 50 °C

/J .. ~ "C

2e-9

• •

1e-9 :;r_::_:_~ 0

0 20 40 60 80 Pressure, Bar

Figure B6. Effect of Pressure and Temperature on 8µ/8P

100 120

temperature than the corresponding parameters for density. For this reason, the maximum

absolute values of 8µ/0f and 8µ/8P at each of the four temperatures were used to calculate

the overall uncertainty in viscosity (U µ,m) due to measured temperature and _pressure. The

values for these uncertainties are given in Table B 1.

8.2 Uncertainty in the FORTRAN Subroutine Property Data

Property information returned by the FORTRAN subroutine was based on test data

supplied by the Southwest Research Institute (SWRI) of San Antonio, Texas to John Deere

Product Engineering Center. The property data supplied by SWRI included uncertainty

values for density and viscosity as shown in Table B2.

B.3 Overall Uncertainty Calculation

The overall uncertainties in the density and viscosity were calculated by combining

97

Table 81. Uncertainty in Viscosity based on Pressure and Temperature Measurements

20 °C 30 °C 40 °C 50 °C

aµ 3.959 x 10-9 -(kg/m-s-Pa)

BP 2.640 x 10-9 1.320 x 10·9 6.600 X 10-IO

aµ (kg/m-s-°K) BT

9.342 x 10-3 4.950 x 10-3 2.608 x 10-3 1.450 x 10-3

Uµ,m (kg/m-s) 5.605 x 10-3 2.970 x 10·3 1.565 x 10·3 8.700 x 10-4

T bl 82 D a e . "t ens1tv an dV" "t u 1scos1:v nee rt · r t am 1es rom SWRI Uo.s 0.50% of value

Uµ,s 20 °C 30 °C 40 °C 50 °C

2.95% of value 1.86% of value 0.76% of value 0.94% of value

the uncertainties in the properties due to measured uncertainties in temperature and pressure,

and the uncertainties in the calculation of (knowledge of) properties at any given condition as

follows:

up= J(up.m)2 +(upJ2

U µ = J(u µ,m )2 + (u µ,s )2

(BIO)

(Bl 1)

The resulting uncertainties for the four temperatures at 5 MPa are shown in Table B3.

From the data in Table B3, it can be seen that the uncertainty in the density due to

measurement error is quite small when compared to the uncertainty in the property

information received from JDPEC. The uncertainty in the viscosity due to measurement

error, however, is slightly larger than the uncertainty from the property information. In this

case, the two uncertainties are of roughly the same magnitude such that the uncertainty in

viscosity due to measurement error makes up a much larger portion of the overall viscosity

uncertainty.

98

Table 83. Overall Uncertainty in Densi1y and Viscosity at 5 MPa 20 °C 30 °C 40°C 50 °C

p (kg/m3) 878.9 872.9 866.9 860.9

Up,m (kg/m3) 0.366 0.366 0.366 0.366

Up,s {kg/m3) 4.395 4.365 4.335 4.305

Up (kg/m3) 4.410 (0.50%) 4.380 (0.50%) 4.350 (0.50%) 4.321 (0.50%)

µ (kg/m-s) 1.376 x 10·1 7.882 x 10·2 4.750 x 10·2 3.053 x 10·2

Uµ,m {kg/m-s) 5.605 x 10·3 2.970 x 10-3 1.565 x 10·3 8.700 x 104

(4.07%) (3.77%) (3.29%) (2.85%)

U µ,s (kg/m-s) 4.059 x 10-3 1.466 x 10-3 3.610 x 104 2.870 x 104

(2.95%) (1.86%) (0.76%) (0.94%)

Uµ, (kg/m-s) 6.920 x 10-3 3.312 x 10·3 1.606 x 10·3 9.161x104

(5.03%) (4.20%) (3.38%) (3.00%)

99

APPENDIX C

EFFECT OF TEMPERATURE AND THICKNESS ON THE 0.5 AND 3 mm

ORIFICES

C.1 Effect of Temperature and Thickness on the 0.5 mm Orifices

10~ 10~ Flowrate, m3/s

10-4

----- 20°c, µ = 0.130 kg/m-s, p = 877.7 kg/m3, P = 2.6 MPa

-+- 30°C, µ = 0.07 4 kg/m-s, p = 877 .6 kg/m3, P = 2.6 MPa

_..__ 50°C, µ = 0.029 kg/m-s, p = 859.5 kg/m3, P = 2.6 MPa

Figure Cl. Effect of Temperature on Pressure Drop - Flow Rate Characteristics for the 0.5 mm Diameter, 1 mm Thick Orifice

~ 104

~

Q) ... ::::J

"' "' Q) a: 103

ca ;:; c: e ~ c 102

100

10~ 10~ 104

Flowrate, m3/s -- 20°c, µ = 0.130 kg/m-s, p = 877.6 kg/m3

, P = 2.6 MPa -+- 30°C, µ = 0.074 kg/m-s, p = 871.6 kg/m3

, P = 2.6 MPa ____,._ 50°C, µ = 0.029 kg/m-s, p = 859.5 kg/m3

, P = 2.7 MPa

Figure C2. Effect of Temperature on Pressure Drop - Flow Rate Characteristics for the 0.5 mm Diameter, 2 mm Thick Orifice

10~ 10~ 104

Flowrate, m3/s

-- 20°c, µ = 0.130 kg/m-s, p = 877.6 kg/m3, P = 2.6 MPa

-+- 30°C, µ = 0.074 kg/m-s, p = 871.6 kg/m3, P = 2.6 MPa

____,._ 50°C, µ = 0.029 kg/m-s, p = 859.5 kg/m3, P = 2.7 MPa

Figure C3. Effect of Temperature on Pressure Drop - Flow Rate Characteristics for the 0.5 mm Diameter, 3 mm Thick Orifice

ca a. .Jf::: ... Cl) ._ ::s "' "' ~ 103 a. . ca ;; c: Cl) ._ Cl) ~ ·-c 102

101

10~ 10~

Flowrate, m3/s

1 mm 2mm 3mm

10-4

Figure C4. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics for the 0.5 mm Diameter Orifice, T ~ 20°C

ca 104 a.. .Jf::: ... ~ ::s "' "' Cl) c: 103

-ca ·-..... c: ~ Cl) ~ c 102

10-6 10-5

Flowrate, m3/s

1 mm 2mm 3mm

10-4

Figure CS. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics for the 0.5 mm Diameter Orifice, T ~ 30°C

ca 104 _ c.. ~ .. ~ :l ti) ti) Q) .... 103 c.. ca ;:; c Q) .... Q) ~ c 102

102

10-5

Flowrate, m3/s

1 mm 2mm 3mm

104

Figure C6. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics for the 0.5 mm Diameter Orifice, T ~ 50°C

103

C.2 Effect of Temperature and Thickness on the 3 mm Orifices

ca c.. ~ .. e :::s U) U) e 103 c.. ca +:; c: e ~ c 102

104 Flowrate, m3/s -- 20°c, µ = 0.130 kg/m-s, p = 877.7 kg/m3

, P = 2.6 MPa -+- 30°C, µ = 0.074 kg/m-s, p = 877.6 kg/m3

, P = 2.6 MPa __.,.___ 50°C, µ = 0.029 kg/m-s, p = 859.5 kg/m3

, P = 2.6 MPa

Figure C7. Effect of Temperature on Pressure Drop - Flow Rate Characteristics for the 3 mm Diameter, 1 mm Thick Orifice

ca c.. ~

"' e :J tn ~ 103 ~ c.. ca .. c e ~ c 102

104

10-4 Flowrate, m3/s ---- 20°c, µ = 0.130 kg/m-s, p = 877.6 kg/m3

, P = 2.6 MPa --+- 30°C, µ = 0.07 4 kg/m-s, p = 871.6 kg/m3

, P = 2.6 MP a ___._ 50°C, µ = 0.029 kg/m-s, p = 859.5 kg/m3, P = 2.7 MPa


-ca .. c e ;E c 102

10-4 Flowrate, m3/s

---- 20°c, µ = 0.130 kg/m-s, p = 877.6 kg/m3, P = 2.6 MPa --+- 30°C, µ = 0.074 kg/m-s, p = 871.6 kg/m3

, P = 2.6 MPa ___._ 50°C, µ = 0.029 kg/m-s, p = 859.5 kg/m3

, P = 2.7 MPa


105

ns a. ~

"' ~ 103 ::s tn tn Q) L. a. ns 1 mm ·- • .... c: 2mm Q) • L. Q) ~

.. 3mm ·-c 102

10-4 Flowrate, m3/s

Figure ClO. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics for the 3 mm Diameter Orifice, T ~ 20°C

-ns ·-.... c: Q) L.

~ c 102

• •

10-4 Flowrate, m3/s

1 mm 2mm 3mm

Figure Cll. Effect of Orifice Thickness on Pressure Drop- Flow Rate Characteristics for the 3 mm Diameter Orifice, T ~ 30°C

ca a. ~ .. Cl) ~ ::s "' "' Cl) 103 ~ a. -ca ·-..... c: ~ Cl) ~ c 102

106

• •

10-4 Flowrate, m3/s

1 mm 2mm 3mm

Figure C12. Effect of Orifice Thickness on Pressure Drop - Flow Rate Characteristics for the 3 mm Diameter Orifice, T ~ 50°C

107

APPENDIX D

ORIFICE BACK-CUT CALCULATIONS

The orifice plates were designed using the following formula available in Mark's

Handbook (1978).

Where:

S = k wR2 m t2

Sm w t R k

= =

ultimate strength of the material load evenly applied to the surface (psi) thickness of the material (inches) radius of the disc (back-cut radius, inches)

(DI)

a constant based on the ratio of disk radius to orifice radius

During the planning stages of this research project, it was expected that the maximum

differential pressure tested would be 20 MPa (:::::: 2900 psid). Because of this, the orifices

were originally designed to withstand a maximum differential pressure of 3000 psid. In

addition to the differential pressure, other factors were also considered for the design of the

orifices. Of primary concern was the effect the back-cut would have on the strength of the

orifice plate due to the introduction of sharp edges and under-cutting. Additionally, the

above equation is for a constant, evenly applied load, whereas the actual loading on the

orifice plate is likely to be uneven. To account for these unknowns, it was assumed that the

milling process would reduce the orifice thickness by 0.05 mm and that the ultimate strength

of the material would be reduced by 15 % in the 2 mm thick orifices, and by 50% in the 1

mm thick orifices (based on a conservative ultimate tensile strength of 80,000 ksi).

The following correlation was developed for the constant, k, based on values given in

Mark's Handbook:

108

k = 0.646 + l.133(r/R) - 4.070(r/R)2 + 2.258(r/R)3 (D2)

where r is the orifice radius (0.5, 1, and 3 mm)

The back-cut diameters calculated iteratively from the above two equations are

reported in Table Dl, along with the actual back-cut diameters used. Please note that in

every instance, the actual diameters used are more conservative than the values suggested by

these calculations.

Table Dl. Orifice Back-Cut Diameters Back-cut Diameter Based on Nominal Orifice Thickness (mm)

Nominal Orifice 1 mm Thick 2 mm Thick 3 mm Thick Diameter (mm) Calculated Actual Calculated Actual Calculated Actual

0.5 8.29 7.38 22.70 21.52 NIA NIA

1.0 8.13 5.18 22.37 19.05 NIA 25.40

3.0 8.65 7.35 21.72 21.60 NIA NIA

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