On practical gas and liquid leakage diameter analytic estimation for
vacuum applications
JACOB NAGLER
NIRC, Haifa, Givat Downes
ISRAEL
Abstract: This paper presents analytical adaptive expressions for the two distinct cases of tank leakage
estimations for gas (sonic and subsonic) and liquid flows under specific measurements data that
assists to evaluate a circular hole/slit/orifice (crack) diameter and area. The analytic process is
performed by equalization between analytic reformulation of the traditional mass flow formulations
and the test formulation for mass flow dependent driven pressure differential over time multiplied by
volume. In case of uniform environment conditions, the slit diameter might also represent the total
sum of numerous exit holes/slits possible existence. Finally, a qualitative agreement was found
between literature and current results in the context of orifice diameter versus pressure differential.
Keywords: Gas leakage, Liquid leakage, Leakage orifice area, Leakage orifice diameter, Analytic
model and solution.
Received: June 17, 2021. Revised: June 24, 2022. Accepted: July 15, 2022. Published: September 19, 2022.
1 Introduction
The case of gas and liquid leakage through a
circular orifice (hole, crack, and pinhole), inside
a general shaped tank geometry, is represented
by mass flow equilibrium. In order to evaluate
the orifice diameter and area parameters, data
measurements of pressure difference through
time in the tank are required. The original mass
flow formulation is presented in number of
references as will elaborated here.
In 1966 Roth [1] has published an intensive
and comprehension book on vacuum sealing
techniques which introduces in the first chapter
the mass flow (for liquid and gases) dependent
on the driven pressure differential, divided by
the time difference and multiplied by the
chamber volume. The expression of the mass
flow might point on the leakage existence and
magnitude. During the same year, Amesz [2]
has also published an essay concerning flow
rates expressions, while the test measurement
expression is reported in his documental brief.
Moreover, he [2] has also developed
generalized expressions for the flow rate of
gases capillaries (or pinholes are small hole size
with diameter between a few micrometers and a
hundred micrometers alongside neglected
length size) using a combination between
Knudsen's law (molecular flow) and viscous
flow (dependent on the viscosities, pressure
differences,
temperature, molecular weight, capillary
diameter and capillary length). This formula can
be applied on large and small diameters
(d0.1), respectively. He also introduced
laminar liquid mass flow rate formulation.
Finally, he has presented capillary average
diameter depending internal and outside
pressure ratio among other flow properties
(temperature, molecular weight and gas
viscosity). In similar way, he developed the
cases of liquid and mixed gas-liquid capillaries
mass flow and average diameters expressions.
The current study will also suggest a
generalized modified approach for different
cases (including capillary).
In similar way to [2] (based on diffusive and
viscous flow), Davy [3] suggested analytic
formula to evaluate the gas leakage mass flow
and diameter parameters for micro hermetic
electronic packages. Based on aerosols
molecular behavior, Keller [4] has evaluated the
threshold-leak size of various packages loss of
sterility due to microorganism penetration.
Some modification improvement of the
mentioned mass flow dependent on the pressure
differential has been suggested and developed
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by Gahzi [5] in the context of particles transport
through narrow passages and particle
entrapment in laminar flow at the passage
entrance. In addition, Hou et al. [6] have
developed modified analytic solution dependent
on temperature difference, pressures and other
gas molecular properties for natural gas pipeline
mass flow, but without diameter estimations.
Later, Yoshida et al. [7] have calculated air
mass flow (including pinholes) using driven
pressure differential extended formula
compared to capillary samples and also
presented all types of theoretical mass flow
equations (laminar, turbulent, molecular,
compressible, Knudsen and modified Knudsen
equation), concentrated on capillary flows.
The test application of recognition leakage in
laminar and molecular flow is well presented by
Fojtášek et al. [8] and Leybold Catalogue [9].
Wu et al. [10] have performed numerical
method, presenting the two-phase gas-liquid
flow leakage.
Some alternatives methods based on finite
element analysis are also exist to forecast the
leakage behavior through crack in a pipe and
other geometrical elements (Ndalila et al. [11],
Moreira et al. [l2]). For liquid case, Ifran et al.
[13] have performed boundary layer numerical
analysis.
Alternative method to assess the orifice
diameter through droplet size analytic
estimation based on Weber number multiplied
by the characteristic contained length, has been
proposed by Plumecocq et al. [14].
An investigation considering plates with
different orifices diameters geometry were
examined experimentally regarding pressure
drop and mass flow rates by Mincks [15] and
Tomaszewski et al. [16]. Mincks [15] has
concentrated on three types of gas flow
(laminar, transition and turbulent) in relative to
Euler number and Reynolds number.
Tomaszewski et al. [16] have made
experimentally and numerically investigation
for a six-hole orifice flow meter with and
without obstacle, while comparing their results
to ISO 5167 single-hole orifice formulations.
Tomaszewski et al. have found good agreement
between all three methods (empirical, numerical
and ISO formulations). Note that Rahman et al.
[17] and Spaur [18] (who investigated also
irregular orifices) investigated the discharge
coefficient values versus the beta ratio
(diameter of orifice to pipe).
However, in the current essay we will
suggest an extension to the original work
performed by Guthrie and Wakerling [19] and
Yoshida et al. [7] by developing diameters
expressions using equalization between mass
flow expressions and test measurement mass
flow expressions for various types of flows: gas
(sonic and subsonic flows) and liquids,
including generalized approach.
The prominent advantage of diameter
expressions development might assist
evaluating the source of impermeability severity
in case where leaking occurs. In other words, an
evaluation of the hole or even number of holes
(sum of areas) could be estimated using these
expressions. Although the leaking occurrence
itself will be determined preceded by empirical
measurements evaluation (pressure difference).
In generalized analytic perspective, the
diameter of the leak hole is necessary to
calculate the Knudsen number (Kn - the ratio of
the mean free path to the diameter). Similarly,
the diameter (the area) of leak hole is necessary
to calculate the Reynolds number (Re) because
the flow velocity is calculated from the flow
rate Q divided by the area. In first view, this is a
kind of circular relationships due to the flow
parameters dependency (Kn and Re) which are
also dictating the flow regime type. However, in
the current essay, most applications might be
solved by evaluating the driven pressure
differential term multiplied by a given volume,
which is assumed to be given by experimentally
time dependent measurements that later
compared to analytic expressions based on
measured pressure ratio.
To sum it up, the motivation and importance
to investigate the leakage diameters is because
numerous industrial (reactor cooling) and
academic applications based on vacuum and
sealing elements.
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Figure 1. Gas flowing leakage illustration out of the container (tank) at flow rate Qg through a hole (slit/crack)
to the surrounding area.
Figure 2. Liquid flowing leakage illustration out of the container (tank) at flow rate QLiquid through a hole
(slit/crack) to the surrounding area.
Figure 3. Generalized Gas/Liquid control volume.
2 Problem Formulation – Gas leakage
orifice diameter/area estimation by
analytical method
Consider a given volume depicted by a
tank/container/package or a pipe geometry
surrounding by gas (usually air). Suddenly, a
leakage of internal gas with molecular weight
Mw, subjected to temperature T1 and
pressure P1 occurred through unknown
crack/orifice/hole inlet (with constant
geometry)
location discharging from the container to outlet
(outdoor ambient surrounding air (or other gas)
conditions) as shown in Fig. 1. Depending
temperature (inlet/outlet), gas pressure
(inlet/outlet), gas properties (density, discharge
and molecular weight) and section geometry,
mass flow might occurred for the following
cases; subsonic or sonic. Assuming one-
dimensional isentropic ideal gas steady flow.
During discussion the terms orifice, crack,
hole (similarly, as well as the terms container
(Surrounding ambient air)
Gas/Liquid
(Surrounding ambient air)
Liquid
(Surrounding ambient air)
Gas
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and tank), will be used or mentioned
interchangeably.
In order to distinguish between sonic and
subsonic flows, pressure conditions should be
measured and fulfill (inlet/outlet) within defined
time difference will be elaborated below for
each type of flow ([1]-[2], [20] – [21]).
Moreover, in order to discern between
viscous laminar flow, turbulent and molecular
flows; Knudsen (Kn) and Reynolds (Re)
number should be determined by measuring the
flow velocity (or Mach number), geometry and
fluid dynamic viscosity. Now, the differences
will be prescribed using references [2] and [7].
In case where Kn < 0.01, 1000 < Re <
2000 the flow is viscous laminar for long
circular pipe model.
In case where Kn < 0.01, Re > (2000 -
4000) the flow is turbulent for long
circular pipe.
In case where Kn > (0.5 – 1), the flow is
molecular.
The suitable relations that include the length
geometrical parameter will be developed using
literature references [2] and [7] in Sec. 4. In the
gas sonic, subsonic and liquid flows cases (Sec.
2.1, 2.2 and Sec. 3); it is assumed in those cases
that the slit length is small enough (leak point)
such as no dependency is exist between the
flow rate and the slit length.
2.1. Subsonic flow
Subsonic flow occurs for cases where Mach
number fulfils M < 1 and the pressure near the
outlet is equal to the outdoor pressure according
to the following condition [20] – [21]:
,
, (1)
with the appropriate gas mass flow (QSubsonic)
derivation [20] – [21]:
,
(2)
where
, are defined as the orifice
area (A) with the appropriate diameter (d) and
gas discharge coefficient (C) that intended to
include gas streamline flow losses through the
slit [22] - [23], respectively. A reasonable
value for the discharge coefficient is in between
the range of 0.7 – 0.9 for most
channel/tank/chamber vacuum/leakage
applications whereas. Also,
represents the molecular weight, – the inlet
temperature and is the universal gas constant.
The gas specific heat ratio () which defined by
the ratio ( – specific heat at a
constant pressure, - specific heat at a
constant volume), could also expressed by the
following pressure ratio ([20], [23] – [24]):
. (3)
Now, alternative formulation to gas flow ()
that derived from the continuity equation and
used in many pumping test measurements
methods, based on the pressure differential over
specific time difference [1] – [2] will be
brought as:
, (4)
whereas t parameters are the given tank
volume (V), internal tank volume pressure
difference () and the
representative time difference ()
between specified tank volume inlet pressures,
respectively. Note that there are two states
inside the tank chamber – before the leaking
() and after the leaking () which are
measured during the vacuum test.
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Next step, we will multiply the measured gas
flow (4) in the ratio to accommodate the
appropriate units as similar to [2]:
, (5)
where is the gas density that will be
determined by the simple linear average:
, (6)
that is divided by the internal tank pressure in
the first state () as appear in expression (5).
Alternative accurate methods to evaluate the
flow rate achieved by using the following
relation that considering the density difference
in the control volume (inside the tank – before
the slit minus after the slit, multiplied by the
appropriate pressure difference) as:
. (7)
Although the gas densities () before (inlet)
/after (outlet) the slit will be calculated by the
ideal-gas equation
. Equalization
between relations (2) and (7) yields the orifice
area:
.
(8)
Hence, the circular diameter will be given by:
(9)
2.2. Sonic flow
Sonic flow occurs for cases where Mach
number fulfils M = 1 as the gas flows out from
the hole (chock state) and the pressure near the
outlet will supply the following condition [1] –
[2]:
,
(10)
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with the appropriate gas mass flow (QSonic) derivation [6] – [7]:
. (11)
In similar way to the Sub - sonic case, the
following area and diameter parameters are
derived/obtained by making an equalization
between relations (11) and (7):
.
(12)
Such as the obtained hole diameter would be in
the form:
.
(13)
3 Problem Formulation – Liquid
leakage orifice diameter/area
estimation by analytical method
In similar way to the previous case, suppose
we have a tank or pipe filled with liquid that is
located at Z1 height, under pressure P1 and
velocity v1. Suddenly, the liquid exits the hole
with the appropriate pressure (P2) velocity (v2)
and height (Z2) as shown in Fig. 2. Then, by
using Bernoulli's equation (or by Torricelli's
law) without neglecting the height difference
(), we have [24]:
. (14)
Accordingly,
. (15)
whereas fluid Bernoulli's assumptions are
incompressibility, inviscid and steady flow
along a streamline. Symbolic representation:
– the liquid specific gravity, g – gravity
acceleration and – liquid mass density.
are the inlet and outer velocities before
and after the slit, respectively. Characteristic
discharge values are in the range 0.6 – 0.7. In
liquid, it also fulfilled that.
Equalization between relations (15) and (7)
leads to the following expressions for the slit
diameter and area parameters:
, (16)
. (17)
In cases where the height difference is small
() and the pipe/channel/chamber/tank
diameter are larger compared to the slit hole the
liquid velocity might be small enough ()
such as Eqs. (16) – (17) approximations are:
, (18)
. )19)
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It is assumed that length of leak hole is short
enough such as the viscosity of liquid might be
ignored in the flow rate calculation.
4 Problem Formulation – Generalized
formulation
In general, alternative derivation for the area
and diameters parameters provided by using
equation (7) and equalizing it to the general
analytic which might represent liquid, gas or
mixing as appear in many variations and cases
exhibiting by [2] and [7]. represents the mass
flow towards inside or outside container,
package, capillary or any other geometrical
volume, such as:
[kg/s]. (20)
Or alternatively as propose by [2] and [7],
[Pa m3/s]. (21)
Whereas (21) with the unit of [Pa m3/s] is
easily converted to (20) with [kg/s] by
multiplying with a ratio of .
Each formulation should be used according
to the measurement set that given in the
intended design. Now, in order to find the
diameter, one should extract it from the Q
equation which dependent on the diameter. In
cases where the mass flow dependent on the
diameter by polynomial form, one should solve
it numerically, approximately or analytic
quadratic equation.
Some examples to use the general
formulation (21) will be given in the context of
laminar viscous, turbulent and molecular flows
based on studies of Amesz [2] and Yoshida et
al. [7].
In case of viscous laminar gas mass flow for
long circular pipe model (Kn < 0.01, 1000 < Re
< 2000) [7] with (21):
;
(22)
where is the dynamic viscosity.
In case of turbulent mass flow for long circular
pipe model (Kn < 0.01, Re > (2000 - 4000)) [7]
with (21):
;
(23)
where the molecules arithmetic mean velocity
[7] is defined by .
In case of molecular gas mass flow Kn > (0.5 –
1):
;
(24)
whereas the diameter appearing in (23) will be
solved numerically. All other variations that
appear in [2] and [7] should be solved
(numerically implicit) similarly (Knudsen
modified equation, semi-empirical viscous
laminar & molecular flow equation).
For the compressible sonic and subsonic flows,
the appropriate forms that fit relation (21) units
will be using [7]:
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,
(25)
,
(26)
Note that analysis of length L crucial for the
viscous laminar, turbulent and molecular flows
since those relations (22) – (24) are dependent
on this parameter. Continually, this case will be
examined numerically for the viscous laminar
and turbulent flows. Although above sonic,
subsonic and liquid cases are not dependent on
the length since their model does not rely on the
pipe model. Technically, in many cases the
length L is constant and is not controllable.
5 Analytic examination
Examining Eqs. (8) – (9) (gas subsonic
flow), (12) – (13) (gas sonic flow), (16) – (19)
(liquid flow) and the generalized forms (20) –
(21) lead to the following comprehensions:
1. In case of uniform environment
conditions, the slit diameter might
represent the total sum of numerous
tiny or small sub-holes.
2. Otherwise, if the surrounding
conditions outside the tank/package
are non-uniform, then the problem
should be solved using different
control volumes regions dependent
on the outer conditions alongside
different flow rates that represent
different slits area (or total sum)
using the above relations.
3. In addition, a utilization could be
performed for liquid-gas mixtures
inside a container with appropriate
leakage holes/slits nature to separate
gas or liquid as shown in Fig. 3.
4. The case of mixing gas-liquid flow
rate through the same hole is not
concerned here and might occur in
various cases, i.e., if the gas contains
water vapor, freezing state might be
occurring during subsonic flow
through a hole.
5. Note that in all above formulations,
the slit diameters are dependent on
the volume square root ().
Therefore, in case of constant flow
rate, the increasing volume might
cause to the diameter increase.
6. In similar way to the previous
deduction, the same behavior also
communicated for the pressure
difference and the gas density
parameters under constant flow rate.
An inverse phenomenon in relating
the orifice diameter (square root
dependency) occurs for the time
difference (), the discharge
coefficient () and the internal
liquid pressure before the leaking
(). Although in the case of gas
the dependency ratio between the
orifice diameter and the pressure is
1/,
7. The case of capillary might be solved
using Amesz [2] and Yoshida et al.
[7] mass flow formulations equalized
to the measurement mass flow (21)
formulation as exemplified in Eqs.
(22) – (26).
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6 Results and comparison
In this section, an illustrative results
concerning Eqs. (8) – (9), (12) – (13) and (16) –
(17) will be plotted as shown in Figs. 4(a) – (d)
and compared qualitatively with literature Refs.
[6], [25]. Later, investigation of sonic, subsonic,
viscous - laminar and turbulent flows will be
exhibited through comparison between Figs.
4(a) – (c) and Figs. 5 (a) – (b). Finally,
formulations (22) – (26) (including the cases of
sonic and subsonic) will be compared with
Yoshida et al. numerical results [7] through Fig.
5(c).
Illustrations Fig. 4(a)-(b) present the driven
pressure differential () versus the orifice
diameter (d) and the orifice area for gas (sonic
and subsonic) and liquid flow, respectively.
Where the input data for Nitrogen and water,
respectively, is summarized in Table 1. Also,
the gas pressure function will be assumed to
behave according to the OCTAVE/MATLAB
program distributed function:
(27)
whereas the gas densities are calculated by
. In the case of liquid
at temperature and
.
One might observe that the liquid diameter
case is about 150 times greater than the average
values between the sonic and the subsonic cases
as appear in Figs. 4(a) – (c) and is mainly
derive due to the large liquid (water) density
. It was found that
the orifice diameter and area parameters
decrease with the pressure differential increase
for all type of flows as appear in Figs.4 (a) – (d)
which qualitatively fits with Hou et al. [6] (Fig.
9 there).
Also, since the flow rate () is proportional
to the area (A) for all liquid, sonic and subsonic
cases, then, the flow rate is parabolic
proportional to the squared diameter as
. The last conclusion also agrees qualitatively
with Hou et al. [6] (natural gas, Fig. 3).
Observing Fig. 3 at Hou et al. [6] teaches that
pipe model is different than small hole model
due to the length effect that is significantly
decreases the leakage rate effect regarding to
the current models (sonic and sub-sonic gas
flows). In order to understand its importance
one should also observe Fig. 6 at Hou et al. [6]
for the different lengths effect on the orifice
diameter. Similar behaviour might also been
observed at Mu and Zhang [25] Figs. 12 – 13.
Furthermore, it can be observed from
illustrations Figs. 4(a)-(c), that the subsonic
flows have larger diameter values in relative to
the sonic flows at the same pressure difference.
Logically, since sonic flows are required small
area in relative to subsonic flows for the same
mass flow rate.
Moreover, one might deduce from Figs. 5(a)
– (b) that turbulent gas flow requires the
smallest pipe diameter per same conditions (one
place before the viscous flow) as to the other
sonic and subsonic flows (compared to Figs.
4(a)-(c)) due to its fast Nitrogen gas molecules
velocity (v = 467 [m/sec]). The length
parameter is crucial in the laminar viscous and
turbulent flows, such as the orifice diameter
increases as the length parameter increasing per
pressure difference conditions. Qualitatively, on
the one hand, the case of sonic flow (Fig. 4(a))
has the same order of magnitude as the viscous
laminar flow (Fig. 5(a)), since they have similar
flow characteristics, on the other hand, the
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viscous laminar flow is dependent on the orifice
length (pipe model vs. leak point model).
Finally, analysing Table 2 based on Yoshida
et al. [7] numerical data (Fig. 9 there) alongside
the given pressures [7] ,
. Hence, substituting those values
into the modified Eqs. (22) – (26) yields for the
pinhole sonic, subsonic, viscous laminar (VL),
turbulent (TB) and molecular (M) flows
diameters at the size of ,
,
, as the maximum
value of the curves illustrated in Fig. 5(c) (the
molecular case was only calculated for the
maximum value without illustrating),
respectively. Hence, all results has the same
magnitude (5-50[]) as Yoshida et al. [7] result
(), but the sonic and turbulent
diameter values are most approaching to the
specified numerical value (8% and 32% errors,
respectively). In addition, it can be observed at
Fig. 5(c) that the orifice diameter maximum
value is achieved for the maximum pressure
difference, while the subsonic flow has the
highest profile values and the lower profile
values are connected with the viscous laminar
flow profile.
Note that the reason the flow profiles
represented in Fig. 5(c) increase alongside the
pressure difference while the opposite
phenomenon has been achieved in Figs. 4 and
5(a)-(b) is derived due different flow rate
formulation (21) [Pa m3/s] compared to (20)
[kg/s].
Remark that the reason that in some cases
only qualitatively comparison was made is
because the given data in the relevant literature
was insufficient to make a comparable
quantitative examination.
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Table 1: Gas and liquid data
Fluid Type/parameter
[m3]
[sec]
[mm]
C
[g/mole]
[K]
[Pa s]
S
Gas Nitrogen (N2)
0.2
200
--
0.85
1.4
28.0134
293
0.97
Liquid – water (H2O)
same
same
10
0.6
--
--
same
--
1
Table 2: Numerical data from Yoshida et al. [7] (Fig. 9 there) pinholes/capillaries
Fluid
Type/parameter
[m3]
[sec]
[Pa]
d*
[m]
[kg/
mole]
[K]
L
[m]
[Pa s]
Air Flow
(0.22O2+0.78N2)
1
30 -
1600
5-50
1.4
0.029
293
0.35
*d – Measured
(a)
(b)
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Figure 4. (a) Orifice diameter vs. the pressure difference for different gas flows. (b). Orifice area vs. the
pressure difference for different gas flows. (c) Orifice diameter vs. the pressure difference for water flow. (d)
Orifice area vs. the pressure difference for water flow.
Figure 5. (a) Orifice diameter vs. the pressure difference for viscous laminar gas flows for different length
values. (b). Orifice diameter vs. the pressure difference for turbulent gas flow for different length values. (c)
Orifice diameter vs. the pressure difference for sonic, subsonic, viscous laminar and turbulent type gas flows,
based on data given by [7] (Fig. 9 there).
(c)
(d)
(a)
(b)
(c)
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7 Conclusion
In this study, we present a general
framework for calculating orifice areas and
diameters inside containers/packages filled with
liquid or gas (sonic/subsonic) in various cases.
Analytical examination for the obtained
expression has been performed including
numerical analysis for the orifice area and
diameter parameters versus the pressure
difference.
It was found that the orifice diameter and
area parameters decreases with the driven
pressure differential increase for all type of
flows that agree with the relevant literature
references. In case of uniform environment
conditions, the slit diameter might also
represent the total sum of numerous exit
holes/slits possible existence. Otherwise, if the
surrounding conditions outside the tank are
non-uniform, then the problem should be solved
using different control volumes regions
dependent on the outer conditions alongside
different flow rates that represent different slits
area (or total sum) using the developed
formulations. In addition, a utilization could be
performed for liquid-gas mixtures inside a
container with appropriate leakage holes/slits
nature to separate gas or liquid. The slit
diameters relations are dependent on the
volume square root (). Therefore, in case of
constant flow rate, the increasing volume might
cause to the diameter increase. In similar way
to the previous deduction, the same behaviour
was found for the pressure difference and the
gas density parameters under constant flow rate.
An inverse phenomenon in relating the orifice
diameter occurs for the dependency on time
difference (), the discharge coefficient
() and the internal liquid pressure before
the leaking (), respectively. Although in
the case of gas the dependency ratio between
the orifice diameter and the pressure is ,
respectively. Specified formulations were
suggested to use for various cases (i.e.,
capillary, pinhole) using Yoshida et al. [7]
relations.
The fluid density was found to be
meaningful regarding the orifice diameter, e.g.,
for the liquid fluid, the obtained diameter is
about 150 times greater than the average values
between sonic and subsonic flows. In addition,
a qualitative agreement with literature was
found in relative to the orifice diameter and area
parameters decrease with the pressure
differential increase for all type of flows.
Another agreement was found in relative to the
flow rate () parabolic proportionality to the
squared diameter (d2) for all liquid, sonic and
subsonic cases.
Furthermore, examination has led to
conclusion that the subsonic flows have larger
diameter values regarding to the sonic flows at
the same pressure difference. Logically, since
sonic flows required small area in relative to
subsonic flows for the same mass flow rate.
Moreover, numerical deduction has shown that
turbulent gas flow requires the smallest pipe
diameter per same conditions (one place before
the viscous flow) as to the other sonic and
subsonic flows due to its fast Nitrogen gas
molecules velocity. The length parameter was
found to play main role in the laminar viscous
and turbulent flows, such as the orifice diameter
increases as the length parameter increasing per
pressure difference conditions. Qualitatively, on
the one hand, the case of sonic flow has the
same order of magnitude as the viscous laminar
flow, since they have similar flow
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.15
Jacob Nagler
E-ISSN: 2224-347X
159
Volume 17, 2022
characteristics; on the other hand, the viscous
laminar flow is dependent on the orifice length
(pipe model vs. leak point model).
Finally, numerical estimation and
comparison based on the data and relations
given by Yoshida et al. [7] concerning pinhole
orifice diameters values for various flow types
has shown same magnitude fitness.
In future, better leakage mechanism
approximations should be further study in the
context of suspensions as aerosols and
hydrogels including package sterility. In
addition, cooling processes models of reactors
based on orifices and pinholes, used for various
needs in the energy industry should be
developed.
Nomenclature
[m/sec]
Liquid velocity before the slit (inlet)
[m/sec]
Liquid velocity after (outer) the slit
[kg/m3]
Gas densities before (inlet)/at the slit throat/after (outlet) the slit.
[kg/m3]
Gas density
[kg/m3]
Water liquid density
[Pa-sec]
Dynamic viscosity
[m2]
Orifice area
--
Gas/Liquid discharge coefficient
[Jkg−1K−1]
Specific heat at a constant pressure
[Jkg−1K−1]
Specific heat at a constant volume
--
Mach Number
[kg/kmol]
Gas molecular weight
[Pa]
Internal gas pressure before the leaking
[Pa]
Internal gas pressure after the leaking
[Pa]
Outlet gas pressure
[Pa]
Driven pressure differential
[kg/sec]
Mass flow
Liquid mass flow
[kg/sec]
Molecular gas mass flow
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Jacob Nagler
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Volume 17, 2022
[kg/sec]
Sonic mass flow
[kg/sec]
Subsonic mass flow
[kg/sec]
Gas mass flow continuity formulation
[kg/sec]
Accurate mass flow expression
[kg/sec]
Average mass flow
[kg/sec]
Viscous laminar gas mass flow
[kg/sec]
Turbulent gas mass flow
8.3144
[JK−1mol-1]
Universal gas constant
--
Gas or Liquid specific gravity
[K]
The inlet temperature
[m3]
The container volume
[m]
Liquid height inside the container
[m]
Liquid height outside the container or slit height
[m]
Orifice diameter
[m/sec2]
Gravity acceleration
--
The gas specific heat ratio
[sec]
Time difference
References
[1] A. Roth, Vacuum sealing techniques,
Pergamon Press Ltd., (1966).
[2] Amesz, J., Conversion of leak flowrates
for various fluids and different pressure
conditions, EURATOM, (1966).
[3] Davy, J. D., Model calculations for
maximum allowable leak rates of
hermetic packages, Journal of Vacuum
Science & Technology 12, 423 (1975).
[4] Keller, S. W., Determination of the leak
size critical to package sterility
maintenance, Phd Thesis, Faculty of
Virginia Polytechnic Institute and State
University (1998).
[5] Ghazi, C., Measurement of fluid and
particle transport through narrow
passages, Graduate College
Dissertations and Theses 297 (2014)
[6] Hou, Q., Yang, D., Li, X., Xiao, G.,
Chun, S., Ho, M., Modified leakage rate
calculation models of natural gas
pipelines, Mathematical Problems in
Engineering, 2020, Article
ID 6673107, 1-10, (2020).
[7] Yoshida, H., Hirata, M., Hara,
T., Higuchi, Y., Comparison of
measured leak rates and calculation
values for sealing packages, Packag
Technol Sci., 34: 557– 566. (2021).
[8] Fojtášek, K., Dvořák, L., Krabica, L.,
Comparison and mathematical
modelling of leakage tests, EPJ Web
Conf. 213 02019 (2019).
[9] Rottländer, H., Umrath, W., Voss, G.,
Fundamentals of leak detection, Leybold
Catalogue (2016).
[10] Wu, X., Li, C., He, Y., Jia, W., Dynamic
modeling of the two-phase leakage
process of natural gas liquid storage
tanks, Energies 10, 1399 (2017).
[11] Nadalia, P., Li, Y., Liu, C., Modelling
flow behavior of gas leakage from
buried pipeline, EJERS, 5 (11) 1-6
(2020).
[12] Moreira, G., Magalhães, H. L. F., de
Almeida Tavares, D. P. S., de Brito
WSEAS TRANSACTIONS on FLUID MECHANICS
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Jacob Nagler
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Correia, B. R., Leite, B. E., da Costa
Pereira, A. B., de Farias Neto, S. R., de
Lima, A. G. B., Fluid Leakage in
Submerged Offshore Pipeline: An
Analysis of Oil Dispersion in Seawater,
Open Journal of Fluid Dynamics, 10 95-
121 (2020).
[13] Irfan, M., Farooq, M. A., Iqra, T., A
new computational technique design for
EMHD nanofluid flow over a variable
thickness surface with variable liquid
characteristics. Front. Phys. 8 (66) 1-14
(2020).
[14] Plumecocq, W., Audouin, L., Joret, J.
P., Pretrel, H., Numerical method for
determining water droplets size
distributions of spray nozzles using a
two-zone model. Nuclear Engineering
and Design, Elsevier, 324 67-77 (2017).
[15] Mincks, L. M.0, Pressure drop
characteristics of viscous fluid flow
across orifices, Retrospective Theses
and Dissertations, 20171 (2002).
[16] A. Tomaszewski, T. Przybylinski, M.,
Lackowski, Sensors 2020, 1-20, 7281
(2020). DOI: 10.3390/s20247281
[17] M. M. Rahman, R., Biswas, W. I.
Mahfuz, J. of Agric. Rural Dev. 2010, 7
(1), 151–156. DOI:
10.3329/jard.v7i1.4436
[18] P. J. Spaur, Investigation of discharge
coefficients for irregular orifices, Msc.
Thesis, West Virginia University 2011.
[19] A. Guthrie, R. K. Weakerling, Vacuum
equipment and techniques, McGRAW-
Hill Book Pub. Inc 1949.
[20] R. L., Daugherty, J. B. Franzini, E. J.
Finnemore, Fluid mechanics with
engineering applications, 8th ed. p. 275,
McGraw-Hill 1985.
[21] J. C., Kayser, R. L., Shambaugh,
Chemical Engineering Science 1991, 46
(7). 1697-1711.DOI: 10.1016/0009-
2509(91)87017-77
[22] A. H. Shapiro, The dynamics and
thermodynamics of compressible fluid
flow, 83-100, The Ronald Press Co.
1953.
[23] B. R. Munson, D. F. Young, T. H.
Okiishi, Fundamentals of fluid
mechanics, 3rd ed., John Wiley and
Sons 1998.
[24] White, F. M. Fluid mechanics, 5th ed.,
McGraw-Hill 2003.
[25] Z. Mu, H. G. Zhang, Journal of
Vibroengineering, 2017, 19 (7), 5434-
5447. DOI: 10.21595/jve.2017.17389
Conflict of interest statement
On behalf of all authors, the corresponding
author states that there is no conflict of interest
with his own fully creation, whereas NO
scientific funding is involved.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the Creative
Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_US
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