EFFECTS OF NOZZLE GEOMETRY ON
FLAPPER VALVE CHARACTERISTICS
By
ROY F. MENNELL
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE BACHELOR OF SCIENCE DEGREE
AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY
January 1956
Submitted by_
Certified by_
Thesis Advisor

Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge 39, Massachusetts
January 16, 1956
Professor Leicester P. Hamilton
Secretary of the Faculty
Massachusetts Institute of Technology
Cambridge 39, Massachusetts
Dear Professor Hamilton:
In partial fulfillment of the requirements of the degree of Bachelor of Science in Mechanical Engineering at the
Massachusetts Institute of Technology, I herewith submit my thesis entitled "EFFECTS OF NOZZLE GEOMETRY ON FLAPPER VALVE
CHARACTERISTICS."
Respectfully submitted,
L/
Roy F. Mennell
RFM:em
I

ACKNOWLEDGMENT
The author wishes to express his appreciation to
Professor Lowen Shearer for his guidance and interest in this undertaking, to the personnel of the Dynamic Analysis and Control
Laboratory for their encouragement and continued interest in the project, and to my wife for her secretarial services.
Appreciation is also extended to the Dynamic Analysis and Control Laboratory for its sponsorship of this thesis.

TABLE OF CONTENTS
LETTER OF TRANSMITTAL
ACKNOWLEDGMENT
TABLE OF CONTENTS
LIST OF FIGURES
ABSTRACT
DEFINITION OF SYMBOLS
INTRODUCTION
I. ANALYSIS AND DETERMINATION OF MODEL VALVE
Model Valve
Nozzle Analysis
II. TEST SET-UP AND EQUIPMENT
Fluid Supply
Pressure Measurement
Measurement of Valve Opening
Volumetric Rate of Flow Measurement
Temperature Measurement
III. RESULTS AND CONCLUSIONS
Test Results and Observations
Conclusions and Recommendations
10
10
15
15
17
17
18
Page
5
6
3
4
1
2
7
19
26
-A

LIST OF FIGURES
1.
2.
3.
4.
5.
6.
7.
Model Valve
Flapper Nozzles
Test Set-Up
Discharge Coefficient as a Function of
Flapper Openings for a Sharp-Edged Nozzle with 600 Outside Chamfer
Discharge Coefficient as a Function of
Flapper Opening for Several Nozzles with
Flats with an Inside Chamfer at 450 and
Height c/d
=
0.15
Discharge Coefficient as a Function of
Flapper Opening for Two Nozzles with a
Flat of f/d =
0.10 and Chamfers at 450
Reynolds Number as a Function of Flapper
Opening for Several Flapper Nozzles
Page
11
12
16
22
23
24
25
4

ABSTRACT
A two-dimensional model of a flapper valve was constructed with transparent sides, using Reynolds Number as a basis for dimensional similitude. A large scale model was used to determine the effects of geometry on flapper-valve characteristics.
The three problems associated with the geometry of existing
iency variance at small flapper openings, and inability to maintain a sharp-edged nozzle. In an attempt to design a more favorable nozzle, flats on the leading edge of the nozzle and flow reliefs on the inside of the nozzle were tested.
It is concluded that chamfering the flapper nozzle greatly increases the coefficient of discharge without any adverse effects upon flapper-valve characteristics. It is possible to use a small flat (one-tenth the nozzle diameter) on the flapper nozzle without decreasing the range of flapper openings over which efficiency is constant. There is more radical variance of coefficient of discharge for the flatted nozzle than for the sharp-edged nozzle in the range of small flapper openings where both nozzles do not have constant discharge efficiency. No combination of flat and chamfer on the flapper nozzle was found which would extend the range of constant discharge coefficient to smaller flapper openings.
5

DEFINITIONS OF SYMBOLS c
Cd d f
Pd
Q
Qmax.
Rem
Rev vd x
V)
Chamfer height
Coefficient of discharge
Diameter of nozzle
Height of flat
Pressure in nozzle (gauge)
Actual volumetric rate of flow
Ideal volumetric rate of flow
Reynolds Number for model valve
Reynolds Number for actual valve
Average velocity of fluid discharge
Flapper opening
Viscosity
Kinematic Viscosity
Density

INTRODUCTION
Flapper valves have been used to advantage as a servomechanism in hydraulic and pneumatic systems. As a power amplifier it has the advantages of being small in size and weight and of having no friction surfaces other than fluid friction surfaces.
The flapper valve is essentially a plate positioned by a servomotor in front of a nozzle which ejects a fluid under high pressure. The valve is designed so that in its system a pressure output is theoreticly a linear function of a flapper displacement input. At the present time the flapper nozzle has a sharp-edged opening chamfered on the outer circumference.
There are several problems associated with the geometry of the present flapper-valve nozzle.
a. The efficiency of discharge is not constant over the full range of flapper displacements. At small openings the efficiency varies, and the output pressure is no longer a linear function of flapper displacement input.
b. Sharp edges on the flapper nozzle are difficult to machine accurately because the nozzle in the hydraulic valve is about 0.020 inches inside diameter; high pressures, up to 3000 pounds per square inch, cause wear on this sharp edge.
c. The efficiency of discharge over the full range of
7

flapper openings is relatively low (the coefficient
This thesis is a study of the means by which variations in nozzle geometry determine flapper-valve discharge characteristics. It is the objective of the following investigations to reveal the limit to which a practical design of the flapper nozzle can improve upon the discharge characteristics of the existing flapper nozzle.
8

I. ANALYSIS AND DETERMINATION OF MODEL VALVE
Dimensional similitude between an actual flapper valve and a model requires that the Reynolds Number for the flow in the actual valve equal Reynolds Number for the flow in the model valve. For flapper valves the appropriate value for the length term in the Reynolds
Number is the distance between flapper and flapper-valve nozzle.
By applying Bernoulli's formula across the valve, the average velocity of the discharging flow may be determined.
It is assumed for this application that there is orifice flow through the valve, that the fluid is non-compressible, and that the gravity effect is negligible.
From the equation of continuity it is seen that the maximum discharge from the valve is
The actual value of the volumetric rate of flow will be somewhat less and by definition
Therefore, the actual flow from the valve is given by
9

Model Valve
To facilitate visual inspection of streamlines, a twodimensional model with transparent sides was chosen to represent the actual three-dimensional valve (Figure 1). Because the two-dimensional model was symetrical about its centerline, only one-half of the full valve was used; flow discharged from one nozzle edge, and a plate ran along the opposite side of the centerline.
The model valve was controlled by equation (1) using the Moog flapper valve as an example of an actual valve. For a model using water as a fluid and having a nozzle area of one-half square inch, it was evident that a suitable maximum nozzle pressure would be 4.5 pounds per square inch for water at 700 F. The nozzle was made one inch thick to ensure that the effects of fluid friction on the walls were negligible.
To ensure laminar flow in the nozzle of the model valve, a flow convertor was molded from clear plastic potting compound. The flow convertor was designed to give a smooth transition between flow from a circular cross-sectional supply pipe to the rectangular cross-sectional valve. The transparent flow convertor and valve side plates have the advantage of revealing streamlines in the flow when small air bubbles are injected into the supply fluid.
Nozzle Analysis
As a basis for comparison the first nozzle was designed to simulate the actual valve (Figure 2, Nozzle a). Next, an ideal nozzle was designed to determine the maximum efficiency that could be obtained
10

'/g a aI0
, / 0
C f/dsc2/6'
112
I
I a.
c.
8.

from a flapper valve (Figure 2, Nozzle b). An elliptical contour for the ideal nozzle was chosen for several reasons. First, it changes slope 900 as does the flow discharging from the flapper valve. Second, its rate of change of slope is always increasing from the minor axis to the major axis. Nozzles with chamfers of several sizes were experimented with to determine their effect on the discharge coefficient level.
It was believed that chamferring would be a practical method of increasing the valve efficiency.
Because of the complications arising from the use of a sharpedged nozzle, two sizes of flats were made on nozzles to determine their effects on discharge coefficients at small flapper openings. It has been suggested that a 0.002-0.003 of an inch flat might be of value on the actual valve. The model valve is about twenty-five times as large as the actual valve. Therefore, nozzles with flats of 0.050 and 0.075 of an inch were designed and tested (Figure 2, Nozzles c, d and e).
The purpose of the flat and chamfer experimentation was to determine the degree to which each influenced the discharge characteristics of the flapper valve. Some combination of a flat and chamfer should increase valve efficiency, obviate the need for a sharp-edged nozzle, and optimisticly, improve the flapper valve discharge characteristics at small flapper openings.
As a measure of the efficiency of a given nozzle, curves were plotted of discharge coefficient versus the non-dimensional ratio of flapper opening to nozzle diameter. This ratio was used to facilitate comparison with an actual flapper valve. In addition, sample curves of
13
*7

Reynolds Number versus flapper opening were plotted to compare various nozzles and to allow the reader to determine applicability of these data to any specific valve.

II. TEST SET-UP AND EQUIPMENT
Fluid Supply
A six-inch diameter city water main was used as a source of water supply. Two valves were used to control this supply. For flows over four gallons per minute, a needle valve in the line between the supply and valve regulated flow. When smaller flows were used, it was necessary to use a by-pass valve to obtain and maintain pressure in the test valve. This set-up with the fluid source (Figure 3) maintained a constant pressure. The maximum pressure change during any test run was
0.1 inches of mercury.
Pressure Measurement
A pressure tap of 1/32 inch diameter was drilled in the test valve wall to measure static pressure. The tap was small enough not to interfere with the flow in the valve and close enough to the valve opening (approximately two and one half inches from the opening on the opposite wall) to measure the actual pressure drop across the valve by minimizing the losses between the pressure tap and valve opening.
The pressure was measured on a mercury manometer. Equation (1) indicated that 9.2 inches of mercury should be the pressure drop across the test valve throughout the test runs. The error in the pressure reading was estimated to be 0.1 inches of mercury.

Model Valve
To facilitate visual inspection of streamlines, a twodimensional model with transparent sides was chosen to represent the actual three-dimensional valve (Figure 1). Because the two-dimensional model was symetrical about its centerline, only one-half of the full valve was used; flow discharged from one nozzle edge, and a plate ran along the opposite side of the centerline.
The model valve was controlled by equation (1) using the Moog flapper valve as an example of an actual valve. For a model using water as a fluid and having a nozzle area of one-half square inch, it was evident that a suitable maximum nozzle pressure would be 4.5 pounds per square inch for water at 700 F. The nozzle was made one inch thick to ensure that the effects of fluid friction on the walls were negligible.
To ensure laminar flow in the nozzle of the model valve, a flow convertor was molded from clear plastic potting compound. The flow convertor was designed to give a smooth transition between flow from a circular cross-sectional supply pipe to the rectangular cross-sectional valve. The transparent flow convertor and valve side plates have the advantage of revealing streamlines in the flow when small air bubbles are injected into the supply fluid.
Nozzle Analysis
As a basis for comparison the first nozzle was designed to simulate the actual valve (Figure 2, Nozzle a). Next, an ideal nozzle was designed to determine the maximum efficiency that could be obtained
17
A

Measurement of Valve Opening
Valve opening was measured with a Federal Dial Gauge, which is calibrated in increments of 0.0001 of an inch. The gauge was fixed to the test valve with the feeler rod against the back face of the flapper plate and in a line with the nozzle tip. Valve opening was varied by loosening the screws clamping the flapper plate and then tapping the flapper into position.
Because of thermal effects upon valve opening and opening measurement, care was taken to ensure that the test valve was at the fluid temperature when test runs were taken. With the precaution of maintaining constant temperature, the error in measurement of valve opening was estimated to be 0.0002 of an inch.
Volumetric Rate of Flow Measurement
For flows in excess of four gallons per minute, a Fisher and
Porter Rotameter was used to measure flow. The Rotameter was first calibrated at fluid temperature and pressure that was to be used in the actual testing. The estimated error in flow measurement by this method was t 0.1 gallons per minute.
Flows less than four gallons per minute were measured by a direct weight-time method. The valve discharge was collected in a container on a balance scale. After the proper valve pressure had been regulated, a stop watch was started when a known weight of water had been collected. The scale was then set to another, greater known weight.
When the balance lifted for the second time, the watch was stopped, and

the difference between the first and final weight was the amount of water collected in a measured time. The measurement of the weight of the flow was accurate to about 0.05 pounds. The stop watch was calibrated in increments of 0.01 minutes. Taking into consideration reaction time, the estimated error in time measurement is t 0.01 minutes.
Temperature Measurement
Temperature was measured by a laboratory thermometer calibrated to 10 F. The average temperature for a run was measured in the collecting container. The average temperature for all runs was approximately 390 F.
18-b

_1r_
III. RESULTS AND CONCLUSIONS
Test Results and Observations
Figure 4 shows a plot of discharge coefficient versus x/d for small openings of the flapper. Flapper valves normally operate in a range of flapper opening between x/d
=
0 to x/d
=
0.1. Over most of the range of x/d the discharge coefficient remained about 0.65.
This value held, although not shown on the plot, for large openings and was checked as far out on the curve as x/d a 0.8. At very small openings (x/d 0.01) the discharge coefficient increased markedly.
This increase can be explained by the fact that at very small openings the velocity of the flow in the nozzle near the discharge opening becomes small so that the vena contracta, which makes the effective area of discharge smaller than the actual area at large openings, does not fully develop.
The elliptical nozzle meets its design objective by increasing the coefficient of discharge to a maximum value. At flapper openings in excess of x/d Z 0.1 the discharge coefficient is equal to or greater than 0.95. Data has not been plotted for this nozzle. In the range of openings of actual flapper valves (x/d
=
0 to x/d n 0.10), no reliable data could be taken because of excess leakage due to taper in the elliptical nozzle tip. The taper had the effect of keeping one side of the nozzle tip from closing against the flapper plate.
Three nozzles with a chamfer-type relief at 450 and of height c/d = 0.15 were tested and the results plotted on Figure 5. In all
19

three cases the discharge coefficient over a wide range of flapper openings was 0.83. The nozzle with a sharp edge at openings less than x/d
=
0.01 behaved in a similar manner to the sharp-edged nozzle with no chamfer. The range of constant coefficient of discharge is slightly extended to smaller flapper openings by the chamfer, and the efficiency of discharge increased at very small flapper openings.
Flats effected discharge coefficients negligibly over a wide range of openings. At small openings, depending on the height of the flat, the efficiency of the valve decreased rapidly. In the case in which f/d
=
0.10, the efficiency dropped at about x/d : 0.01. For the f/d = 0.15 flat efficiency dropped at x/d
=
0.015. At very small flapper openings as the friction area caused by the flats increased, valve efficiency decreased. Experimentation with smaller flats could conceivably reveal an optimum discharge curve at the Cd =
0.83 level, but a flat of such a small size would be impractical to manufacture on the actual valve. In addition, the errors in measurement of flapper opening and discharge increase in this range, and the model valve could not be tested to any acceptable degree of accuracy for openings less
=
Additional nozzles with larger chamfers were tested. It was hoped that a combination of chamfer with a flat of f/d : 0.10 would reveal a balance at which the discharge coefficient would remain constant nearer the fully closed position. With larger chamfers it was felt that the discharge coefficient would be higher than 0.83.
The results of tests with a nozzle of c/d = 0.50 are compared with data from tests with the c/d
=
0.15 nozzle on Figure 6. These
20

curves show that increasing the chamfer on the nozzle increases the coefficient of discharge. The nozzle with a chamfer of c/d = 0.50
has a coefficient of discharge of 0.89 over a wide range of flapper openings. Because of the possible error in measurement at small openings, it is not possible to conclude that an increase in chamfer with a constant flat decreases the range of flapper openings over which the coefficient of discharge remains constant. It is possible to infer from Figure 6 that an increase in chamfer from c/d = 0.15 to c/d
=
0.50 with a constant flat of f/d = 0.10 does not increase the range of flapper openings over which the coefficient of discharge remains constant. A sample test run was taken with a nozzle with c/d
=
0.80, but data were too erratic to be included. Further testing should be performed on this nozzle as well as the nozzle with c/d x 0.50 to determine the effect of chamfering upon the point of flapper opening at which the discharge coefficient begins to decrease.
The error in measurements at small flapper openings (as is witnessed by the spread in data), gives curves which are qualitative rather than quantitative in the region of flapper openings less than x/d
=
0.01. For greater flapper openings it is believed that the curves of Figures 4, 5 and 6 give an acceptable discharge coefficient for the flapper valve under the conditions tested.
Figure 7 shows the Reynolds Number for two nozzles as a function of flapper opening. A plot is included of maximum efficiency of discharge for the flapper valve at the previously stated test conditions.

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II

Conclusions and Recommendations
The use of a small flat does not diminish the range of flapper openings over which discharge coefficient is constant. In the range of flapper openings in which both sharp-edged and flatted nozzles reveal a varying coefficient of discharge, the variance is considerably greater with the flatted nozzle than with the sharp-edged nozzle. If the extreme variance of efficiency over a small range of openings can be tolerated, a flat on the nozzle corrects the disadvantages of a sharp-edged nozzle.
Chamfering of the flapper nozzle increases the discharge coefficient. The full extent to which this is so has not been tested.
It is recommended that a nozzle with a larger chamfer (c/d
=
0.8 to c/d
=
1.0) be tested so that the maximum effect of chamfers and the relationship between size of chamfer and discharge coefficient be determined. A large chamfer on the flapper nozzle may result in a widerrange of flapper openings which have a constant discharge coefficient.
No combination of flat and chamfer on the flapper nozzle was found which would extend the range of constant discharge efficiency to smaller flapper openings. Tests with a nozzle, which has a small flat and a larger chamfer (c/d
=
0.8 to c/d = 1.0), would be valuable.
Together with data from previous testing, curves could be plotted which would determine the effect of a chamfer-flat combination upon the range of flapper openings over which the discharge coefficient remains constant.
26