Essay on Pressure Decay Leakage Test

7/17/2019 Essay on Pressure Decay Leakage Test

http://slidepdf.com/reader/full/essay-on-pressure-decay-leakage-test 1/23

Essay on leakage test by pressure decay

Matteo Ageno – February 24, 2014 – Rev. 1

Index

Introduction p. 2

Part 1: Physical model of the leakage test by pressure decay p. 3

Clause 1.1: Physical model of the pressurized container p. 3

Clause 1.2: Physical model of the fluid p. 4Case 1.2.1: Compressible fluid gas! p. 4

Case 1.2.2: Incompressible fluid li"uid! p. #

Clause 1.3: Physical model of the orifice p. #

Case 1.3.1: $rifice leaking gas p. #

Case 1.3.2: $rifice leaking li"uid p. %

Clause 1.4: Calculation of the pressure decay la& p. '

Case 1.4.1: (eformable container and gas p. '

Case 1.4.2: (eformable container and li"uid p. )

Case 1.4.3: *igid container and gas p. 11Case 1.4.4: *igid conatiner and li"uid p. 12

Part 2: (efinition of a practical leakage dimension p. 14

Clause 2.1: +se of e,ponential decay p. 14

Clause 2.2: +se of linear appro,imation for decay la& p. 1#

Clause 2.3: -uantification of error introduced by linear appro,imation p. 1'

Clause 2.4: -uantification of effects of testing temperature p. 21

otes p. 23

Page 1 of 23



Introduction

/he present essay collects in a single document se0eral notes released in the past by the uthor to

the I$/C 21567 4 test methods &orking team8 upon the theoretical frame&ork of the leakage

test by pressure decay8 and the true meaning of the dimension introduced by 9"uation .1 of

I$I9C (I '3;)<28 and sometimes inproperly referred as =leakage rate=.dditional scope of present essay is to "uantify the capability of 9"uation .1 of I$I9C (I

'3;)<2 to pro0ide a 0alid estimate of the leakage characteristic of a couple of connectors under

test8 independent from the test parameters and initial conditions.

/he essay starts &ith the description of a reasonable physical model for the intrepretation of the

fluid leakage by pressure decay test method.

/he test is modeled considering three interacting parts: the pressurized container8 either ideally rigid

or deformable8 the fluid8 either li"uid or gas that means8 in this conte,t8 either incompressible or

compressible fluids! and the leaking orifice.

/he constituti0e models of the interacting parts are then assembled considering the mutual

interactions and pro0iding the constituti0e e"uations of the leakage phenomenon./he mathematical result8 in terms of relation bet&een the internal pressure and the elapsed time

from start8 is then computed for the four possible combinations of fluid and container.

tarting from the pressure 0s. time function obtained in case of gas in a ideally rigid container8 the

9"uation .1 in >ustified and e,plained in its physical meaning. /herefore the underlying

appro,imation is "uantified and a possible temperature correction factor8 coherent &ith the physical

assumptions already adopted8 is proposed.

/he reader is kindly in0ited to inform the uthor about any editorial or technical error herein

detected8 at the mail address matteo.ageno?cedicbio.com.

Page 2 of 23

mailto:[email protected]

mailto:[email protected]



Part 1: Physical model of the leakage test by pressure decay.

Clause 1.1: Physical model of the pressurized container.

/he pressurized container can be considered either infinitely rigid or deformable./he t&o beha0iours can be summarized in the same approach as the rigid container formulation

follo&s by placing the elastic stiffness of the container approaching to infinity.

@or sake of simplicity the follo&ing reasonable &orking hypotheses are made:

Ayp. B! the problem is isothermal8 at least in the container.

Ayp. B1! the container is cylindrical &ith diameter D 8 length L and internal 0olume V . /his

assumption is reasonably generic as different shapes can be managed by introducing

appropriate dimensionless shape factors that modify the basic cylindrical shape by

multiplying the internal 0olume 1!.

Ayp. B2! the length is 0ery greater than the diameter: L≫ D . /he edges plays negligible role andthe problem can be treated in plane deformation conditions. 6ith negligible effect the

initial length can be considered e"ual to the deformed length: L≃ L . gain different

shapes can be considered by introducing appropriate dimensionless multiplicati0e factors.

Ayp. B4! the container has thin &all: D≫b . /his assumption simplifies the manual calculations.

ny&ay it does not affect the underlying principles that &ill be disclosed 2!.

Ayp. B#! the structural problem can be formulated in small strains and displacements field.

Ayp. B;! the problem is linear elastic &ith parameters E and ν . /his assumption is fully

reasonable as the problem is formulated in small strains and displacements field.

Figure 1.1.1 (left! t"e #i$%lifie& e'%eri$ent layout.

Figure 1.1.2 (rig"t! t"e Mariotte# a##u$%tion.

+nder these assumptions the problem8 as sho&n in @igure 1.1.18 can be approached according to

ariotteDs formulation illustrated in @igure 1.1.2. @rom simple e"uilibrium considerations8 the

circumferential stress is calculated as follo&s:

2⋅σ⋅b= %⋅ D

&here σ is the stress along the thickness since the &all thickness is 0ery small8 the stress is

assumed constant along it! and % is the internal pressure8 gi0en the outside pressure e"ual to zero.

ote that % is the 0alue of the relati0e pressure if the outside pressure corresponds to the free

atmosphere.

Page 3 of 23



In plane strain conditions the circumferential strain is the follo&ing:

ϵ=σ⋅1−ν2

E

7i0en D4 is the undeformed diameter8 the deformed diameter is:

D= D⋅(1+ϵ)

6ith some manipulations8 and placing b≃b as 0alid appro,imation in small strains and

displacements field8 the deformed 0olume is:

V =π D

2

4 L

=

V

(1−)%)21.1.1!

&here:

V =π D

2

4 L

is the starting 0olume8 and

) = D

2 b

⋅1−ν2

E

1

*a =

$#2

) 1.1.2! is the characteristic elastic compliance parameter.

9"uation 1.1.1! is the V =V ( %) constituti0e relation for the container.

Clause 1.2: Physical model of the fluid

/he fluid can be either compressible a gas! or incompressible a li"uid like &ater!.

+a#e 1.2.1 +o$%re##ible flui& (ga#.

/he follo&ing &orking hypothesis is assumed:

Ayp. B%! /he gas follo&s the la& of ideal gases 3!: %V =nR, 1.2.1.1!

&here: % is the absolute pressure [ *a ]=[)g /$#

2]

V is the 0olume [$3]

n is the number of moles on the 0olume [a&i$.] R is the constant of gases [ R='8314 - /$ol ⋅ ] is the absolute temperature [ ]

9"uation 1.2.1.1! can be re&ritten in the follo&ing form:

%V = $

M

R , 1.2.1.2!

Page 4 of 23



&here:$ is the mass of the gas in0ol0ed in the process [ g ] M is the molar mass of the gas [ g /$ol ]

In terms of relati0e pressure 9"uation 1.2.1.2! can be re&ritten as follo&s:

( %+ %at$)V = $ M

R, 1.2.1.3!

+a#e 1.2.2 /no$%re##ible flui& (liui&.

/he follo&ing &orking hypothesis is assumed:

Ayp. B'! /he li"uid like &ater! is incompressible8 that means the density is constant regardles of

the applied pressure 4!.

In mathematical terms it means:∂ρ∂ %

= 1.2.2.1!

&here ρ is the density of the li"uid [)g / $3]

Clause 1.: Physical model of the orifice.

/he orifice can leak either compressible a gas! or incompressible li"uids like &ater! fluids.

@or sake of simplicity the follo&ing &orking hypotheses are made:

Ayp. B)! /he orifice is circular &ith cross<section area A and diameter & as dra&n in @igure

1.3.1 ;!.

Ayp. B1! /he container near the orifice has thin &alls: the head loss is mainly due to inlet and

outlet terms. /he friction head loss along the orifice channel is negligible &ith respect to

the inlet and outlet terms. /his assumption is usually made for applied fluid dynamics

studies of leaking reser0oirs in technical applications #!.

+a#e 1..1 3rifie lea)ing ga#.

dditional &orking hypotheses:

Ayp.B11! /he leakage process is adiabatic. /his assumption is usually done in technical

applications #! for e0aluation of gas discharge from leaks in reser0oirs.

Ayp. B12! /he gas follo&s the la& of ideal gases described in e"uations 1.2.1.1! and 1.2.1.2!.

Page # of 23



Figure 1..1! t"e orifie $o&el.

+nder these assumptions the general formula for an orifice leaking gas from a pressurized container

is the follo&ing #8;8%!:

4=&$

&t =− A√2⋅ % in#ρ( γ γ−1)⋅[( % out

%in# )(2/ γ)

−( % out

% in# )(γ+1) γ ] 1.3.1.1!

&here:4=&$/ &t is the mass flo& rate lea0ing the container hence the minus sign! [)g / # ] .

A=π & 2/ 4 is the area of the leaking orifice [ $

2] .

%in# is the internal absolute pressure [ *a ] .

%out is the outside absolute pressure8 usually the atmospheric pressure %at$ [ *a ] .

γ= %/v is the ratio bet&een specific heats at constant pressure and constant 0olume [a&i$.] .ρ is the actual density of the gas [)g / $

3] .

@ormula 1.3.1.1! is 0alid for subsonic flo& through the orifice. onic flo& also referred as choked

flo&! condition occurs &hen the ratio bet&een outside and inside pressure satisfies the follo&ing

e"uation '!:

%out

%in#

=( 2

γ+1) γ

(γ −1 ) 1.3.1.2!

@or air in common en0ironments γ≃184 . /his 0alue is 0alid through a &ide range of

temperature8 pressure and humidity )!.

s a conse"uence the sonic flo& condition in air occurs &hen:

%out =2

γ+1

γ( γ−1) % in#≃8#2% %in# 1.3.1.3!

Pro0ided that usually the outside absolute pressure is e"ual to the atmospheric pressure8 the sonic

flo& condition occurs &hen % in#=18')# %out that means 1)1) $bar absolute8 or ); $bar

relati0e pressure at common laboratory conditions8 gi0en 113 $bar is the atmospheric pressure.

6hen the flo& that leaks into free atmosphere becomes sonic8 the mass flo& rate depends on the

inside pressure only and is go0erned by the follo&ing formula 1!:

4=&$

&t =− A√ γ %in# ρ⋅( 2

γ+1) γ+1

γ−1 1.3.1.4!

Page ; of 23



/he follo&ing &orking hypothesis is no& assumed:

Ayp. B13! since the pressure decay leakage test in I$ '3;)<2 can reach about 3 bar relati0e

pressure about 4 bar absolute pressure!8 the sonic condition is al&ays assumed as

simplifying assumption.

s a conse"uence the leaking orifice can be considered in sonic conditions for our practical purposes.

(ensity in 9"uation 1.3.1.4! can be &ritten in terms of the e"uation of gases 1.2.1.2!:

ρ=$

V =

%M

R, → 4=

&$

&t =− A %in# √

1

,

γ M

R ⋅( 2

γ+1) γ+1

γ−1 1.3.1.#!

/he same formula may be &ritten in term of relati0e pressure and after splitting the effects of

temperature and constituti0e nature of the gas:

4= &$&t

=− A( %+ %at$)√ 1

√ γ M

R ⋅(

2

γ+1) γ+1

γ−1 1.3.1.;!

&here: M is the molar mass of the gas [)g /$ol ] . is the internal actual absolute temperature [ ] .

@inally the follo&ing &orking hypothesis &ill be usually assumed in ne,t calculations:

Ayp. B14! /he container internal temperature 0ariations during discharge are assumed negligible: (t )≃

#tart . /o be precise the gas becomes colder during e,pansion. ince the typicalflo& rates in this kind of test are 0ery small &ith 0ery small temperature decrease8 and the

temperature is e,pressed in Eel0in scale8 the difference bet&een initial and current

temperature can be neglected. /he escaped gas is no more meaningful8 so its temperature

is not needed.

+a#e 1..2 3rifie lea)ing liui&.

In case of incompressible fluid8 the follo&ing assumption replace the corresponding one of pre0ious

clause:

Ayp. B12! /he fluid is incompressible as per e"uation 1.2.2.1!8 like &ater in common pressure and

temperature conditions.

/he follo&ing hypothesis is additionally assumed:

Ayp. B1#! /he container is small8 that means the hydraulic head 0ariation inside the 0olume is

negligible. $ne single pressure 0alue can represent the li"uid state inside the container.

/he general formula for an orifice leaking &ater from a pressurized container is the follo&ing 11!:

Page % of 23



4=&$

&t =− A+ D √ 2ρ( % in#− %out ) 1.3.2.1!

&here:4=&$/ &t is the mass flo& rate lea0ing the container hence the minus sign! [)g / # ] .

A=π & 2/ 4 is the area of the leaking orifice [ $

2] .

%in# is the internal absolute pressure [ *a ] . %out is the outside absolute pressure8 usually the atmospheric pressure %at$ [ *a ] .

+ D is the discharge coefficient [a&i$.] 12!.

fter a simple manipulation the constituti0e e"uation for the orifice leaking &ater to free

atmosphere can be &ritten in terms of relati0e pressure:

( %)=&$

&t =− A+ D √ 2ρ√ % 1.3.2.2!

Clause 1.!: Calculation of the pressure decay la"

@our cases may occur:

1. /he container is deformable and the fluid is compressible gas!.

2. /he container is deformable and the fluid is incompressible li"uid!.

3. /he container is rigid and the fluid is compressible gas!.

4. /he container is rigid and the fluid is incompressible li"uid!.

+a#e 1.4.1! "e ontainer i# &efor$able an& t"e flui& i# a ga#

Fet assemble the constituti0e e"uations:

• deformable container: V = V

(1−)%)21.1.1! ) =

D

2 b

⋅1−ν2

E 1.1.2!

• leaking orifice: 4=&$

&t =− A( %+ %at$)√ 1, √

γ M

R ⋅( 2

γ+1) γ+1

γ−1 1.3.1.;!

• ideal gas: ( %+ %at$)V =$

M R 1.2.1.3!

/he constituti0e e"uation of the deformable container may be introduced in the la& of ideal gas and

then8 after some maniplations:

$=( %+ %at$) V

(1−)%)2 ⋅

M

R 1.4.1.1!

/he mass flo& rate can be e0aluated by deri0ing 9"uation 1.4.1.1!:

&$

&t =

V M

R ⋅

&%

&t ⋅

[1

(1+)% )2

−2) ( %+ %at$)

(1+)%)3

]1.4.1.2!

Page ' of 23



9"uation 1.4.1.2! is then compared &ith 9"uation 1.3.1.;!8 obtaining the differential e"uation that

sol0es the leaking phenomenon in terms of %= %(t ) relati0e pressure function.

6ith the appropriate initial condition the problem becomes:

{V

4 M

R,

⋅&%

&t

⋅

[ 1

(1+)%)2−

2) ( %+ %at$)

(1+)%)3

]=− A( %+ %at$)

√ 1

,

√

γ M

R

⋅

( 2

γ+1

)

γ+1

γ−1

% (t =4)= %t

1.4.1.3!

fter a fe& manipulations the problem becomes:

{V √ M

R

1

√ ⋅

&%

&t ⋅[ 1

(1+)%)2−

2 ) ( %+ %at$)

(1+)%)3 ]=− A( %+ %at$)⋅√ γ( 2

γ+1) γ+1

γ−1

% (t =)= %t

1.4.1.4!

/his e"uation has no solutions in closed form. It can be sol0ed only by numerical methods.

+a#e 1.4.2! "e ontainer i# &efor$able an& t"e flui& i# liui&

Fet assemble the constituti0e e"uations:

• deformable container: V = V

(1−)%)21.1.1! ) =

D

2 b

⋅1−ν2

E 1.1.2!

• leaking orifice: 4 ( %)=

&$

&t =− A+ D √ 2

ρ√

% 1.3.2.2!

• incompressible li"uid:∂ρ∂ %

= 1.2.2.1!

@ollo&ing from the incompressibility hypothesis8 the mass inside the container is:

$=ρ⋅V 1.4.2.1!

regardles of the internal pressure.

/he constituti0e e"uation of the deformable container may be introduced in 9"uation 1.4.2.1! and

then8 after some maniplations:

$=ρ V

(1−)%)21.4.2.2!

9"uation 1.4.2.2! can be deri0ed &it respect to time8 obtaining the follo&ing:

&$

&t =

2) ρ V

(1−)%)3&%

&t 1.4.2.3!

/his formula is already &ritten in terms of relati0e pressure8 like the leakage function of the leaking

orifice in the free atmosphere8 and hence can be directly compared &ith it:

Page ) of 23



&$

&t =

2) ρ V

(1−)%)3

&%

&t =− A + D√ 2 ρ√ % 1.4.2.4!

*earranging the constant terms the follo&ing form may be found:

&%

&t =− A+ D

√ 2 ρV

(1−)%)3

) √ % 1.4.2.#!

/his form is 0ery useful because it allo&s the "ualitati0e study of the %= %(t ) function.

In small<bore connector field usually )% ≪1 because the aplied internal pressure is 0ery smaller

than the elastic modulus of the material adopted for the components.

/herefore: <(1−)%)≪1 → 4<(1−)%)n≪1 for n⩾1 .

7i0en the initial relati0e pressure % ()= %t 8 the follo&ing results can be obtained:

(

&%

&t

)t =

=− A + D

√ 2ρ V

(1−)% t )3

)

√ %t < 1.4.2.;!

lim % → G(&%

&t )=− A + D

√ 2ρV

(1−) ⋅G)3

) √ G≈−

A+ D

√ 2ρ V ⋅

1

) ⋅

G=<

1.4.2.%!

/he sign of the second deri0ati0e can be e0aluated considering that8 since )% ≪1 8 then1−%⋅)% is still positi0e:

& 2 %

& t 2 =

A2+ D

2

4ρ V 2

1

) 2 (1−)%)# (1−%⋅)% )>∀ %> 1.4.2.'!

@inally the shape of the function %= %(t ) is "uite understood:

• it starts at a generic point % ()= % A &here % A= % t initial relati0e pressure applied for

testingH

• it decreases and the decreasing rate 0aries from the ma,imum 0alue recorder for starting

time! − A+ D

√ 2 ρV

(1−)%t )3

) √ % t < to zero8 approaching zero from negati0e 0alues

decreasing slopes! for time approaching infinityH

• the con0e,ity is al&ays do&n&ard as &

2 %

& t 2 >4 .

Figure 1.4.2.1! t"e ualitative a#%et of %= %(t ) urve aor&ing to euation (1.4.2.5

Page 1 of 23



+a#e 1.4.! "e ontainer i# rigi& an& t"e flui& i# a ga#

/his case is obtained by placing E → ∞ in the constituti0e model of the container. /hat means

9"uation 1.1.2! becomes ) → . /his result can be inserted in system 1.4.1.4! obaining the

follo&ing simplified system:

{ V √ M

R

1

√ ⋅

&%

&t =− A( %+ %at$)⋅√ γ( 2

γ+1) γ+1

γ−1

%(t =)= % t

1.4.3.1!

/he differential e"uation can be re&ritten in the follo&ing form:

V 4

√

M

R

1

√ , ⋅

&%

&t =− A %⋅√ γ( 2

γ+1) γ+1

γ−1− A %at$⋅√ γ( 2

γ+1) γ+1

γ−1 1.4.3.2!

/his e"uation is8 despite the repulsi0e aspect8 a simple first order linear e"uation &ith constant

coeffcients &ith internal relati0e pressure as unkno&n function8 in the form:

+ 1⋅&%

&t ++ 2 %=+ 3

&here:

+ 1=V

4√ M

R

1

√ ,

+ 2= A⋅√ γ(

2

γ+1) γ+1

γ−1

+ 3=− A %at$⋅√ γ( 2

γ+1) γ+1

γ−1

Introducing the initial condition % (t =)= %t the solution is the follo&ing 13!:

% (t )=( %t + %at$)e−

A

V √

R

M √ √ γ(

2

γ+1) γ+1

γ−1 t

− %at$

1.4.3.3!

/he same result can be obtained &ith a 0ariable replacement in system 1.4.3.1!. In fact the absolute

pressure is %ab#= %+ %at$ and therefore &% /&t =&%ab#/&t and hence the system 1.4.3.1!

becomes:

{ V √ M

R

1

√ ⋅

&%ab#

&t =− A %ab#⋅√ γ( 2

γ+1) γ+1

γ−1

%ab#(t =)= % t + %at$

1.4.3.4!

/he e"uation is e0en simpler because it is a separable e"uation.

Page 11 of 23



olution 1.4.3.3! defines an e,ponential decay of the internal pressure starting from the initial

0alue.

Please note that the 9"uation 1.4.3.3! is an appro,imation 0alid for small time duration. In fact:limt →∞

%( t )≈( % t + %at$)e−∞− %at$≈( % t + %at$)⋅4− %at$=− %at$ 1.4.3.3a!

It seems that the relati0e pressure for long times goes to&ard − %at$ that means the absolute

0acuum. $f course8 this is not possible because8 &hen the outside pressure is assumed to be the

atmospheric pressure8 the orifice seems to discharge e0en beyond the outside conditions. It is

ob0ious that the orifice cannot discharge &hen the inside absolute pressure is smaller than the

outside absolute pressure.

$ne shall remind the orifice is modeled assuming sonic flo&8 in &hich the mass flo& rate depends

on the inside absolute pressure only. /hat means &hen the inside pressure decreases under the

threshold for sonic flo& as per 9"uation 1.3.1.2!8 the constituti0e e"uation of the orifice shall be

changed using the subsonic flo& assumption and the 9"uation 1.4.3.3! is no more 0alid.

@inally the 9"uation 1.4.3.3! is re&ritten collecting all the terms that depend on the nature of the

gas only8 obtaining:

% (t )=( %t + %at$)e−

A

V √ ω t

− %at$1.4.3.#!

&here:

ω=√ R

M √ γ( 2

γ+1) γ+1

γ−1 [√( -

$ol )/( )g

$ol )√ a&i$.]=[$

#

1

√ ] 1.4.3.;!

It is easy to 0erify that A

V

√ ω is dimensionally the reciprocal of a time8 making the e,ponent

adimensional.

/he pressure decay can be &ritten as follo&s:

Δ %( t )=∣ %(t )− %()∣=( %t + %at$)[1−e−( A

V

√ ω)t ] 1.4.3.%!

It can be noted that:

• ω=√ R

M ( 2

γ+1) γ+1

γ−1 depends only on the gas nature.

• √ depends on the absolute temperature during test.

• A/V is the ratio bet&een the leaking area and the internal 0olume.

+a#e 1.4.4! t"e ontainer i# rigi& an& t"e flui& i# ater

/his case is obtained by placing E → ∞ in 9"uation 1.1.2! that becomes ) → .

/he case is sol0ed by studying the "ualitati0e beha0iour of %= %(t ) starting from starting time

by means of 9"uations 1.4.2.#! and 1.4.2.;!.

t starting time the limit for ) → of 9"uation 1.4.2.;! pro0ides the slope of the cur0e:

lim) →

(

&%

&t

)t =

=−lim) →

A + D

√ 2ρ V

(1−)% t )3

) √ %t ≈−

A + D

√ 2ρV

(1−)3

√ %t →−∞ 1.4.4.1!

Page 12 of 23



/he solution for rigid container and incompressible fluid degenerates in an Aea0iside step function

centered at t = and step amplitude % ()= % A= %t 8 as dra&n in @igure 1.4.2.1! in red color.

@or this reason the leakage test by pressure decay made on rigid container &ith incompressible fluid

becomes in principle not meaningful regardless of the leaking orifice dimensions.

ny potential detectable %= %(t ) cur0e is strongly biased by artifacts due to container elasticity

and follo&s 9"uations from 1.4.2.#! to 1.4.2.'!./he true effect of the leaking orifice cannot be detected.

Page 13 of 23



Part 2: #efinition of a practical leakage dimension

leakage dimension for practical purposes can be defined in case of rigid container and gas only.

In fact gi0en a test done &ith li"uid the result may be either not recordable in case of rigid

container as per Case 1.4.4! or an artifact of the elastic compliance of the testing de0ice as per Case

1.4.2!. /he adoption of a deformable container in a leak test &ith gas as per Case 1.4.1! is any&aynot suggested because the mathematics of the problem becomes 0ery comple, and because it needs

in principle the precise kno&ledge of the elastic compliance of the testing de0ice.

Clause 2.1: $se of exponential decay from Clause 1.!.

In this clause the e,ponential la& for the pressure decay o0er time as per 9"uation 1.4.3.#! is

applied in order to pro0ide a suitable practical formula for the "uantification of the leakage from

de0ices under test.

In case of rigid container and gas8 the pressure decay can be &ritten as follo&s:

Δ %( t )=∣ %(t )− %()∣=( %t + %at$)[1−e−( A

V

√ ω)t ] 1.4.3.%!

&here:

• ω=√ R

M √ γ( 2

γ+1) γ+1

γ−1 [$/ # √ ] 1.4.3.;! depends only on the gas natureH

• √ [√ ] depends on the absolute temperature during testH

• A/V [1/ $] is the ratio bet&een the leaking area and the internal 0olumeH

• % t is the starting relati0e pressure.

/he leakage test pro0ides the follo&ing e,perimental data:

• the 0olume V H

• the initial applied pressure % t that is usually slightly different to nominal pressure % #

defined by the rele0ant tandard or inernal procedure! due to regulation accuracyH

• the time dutation Δ t of the testH

• the pressure decay Δ % after Δ t .

6ith these data8 and gi0en the mathematical function 1.4.3.%! that defines the pressure decay

during time8 one &ould e,pect a suitable definition of a leakage dimension that is:

a! independent from the pressurized 0olume V of the apparatusH

b! normalized to the nominal test pressure % # or8 in other &ords8 referred to the same

nominal pressue e0en if the true initial pressure % t is slightly different due to regulation

accuracyH

c! dependent on the size A of the leaking orifice8 increasing &hen A increases.

/he ratio bet&een pressure leak and elapsed time8 properly called the decay rate8 can be &ritten as

follo&s:

Page 14 of 23



Δ %

Δ t =

% t + %at$

Δ t [1−e

−( A

V 4

√ , ω)Δ t ] [ *a / #] 2.1.1!

@ormula 2.1.1! is not a good candidate for a leakage dimension because it is not independent from

the 0olume of the testing apparatus. In fact Δ % /Δ t depends from V in a strong non<linear

form. In addition it is not normalized to the nominal pressure %

# to be applied.

properly defined leakage rate8 in terms of leakage of 0olume o0er time8 can be &ritten as follo&s:

Δ %

Δ t ⋅

V

% ref

=V

%ref

%t + %at$

Δ t [1−e

−( A

V

√ ω)Δ t ] [$3/ # ] 2.1.2!

&here %ref is a reference pressure introduced for dimensional homogenization to 0olume o0er

time dimension.

either formula 2.1.2! is a good candidate for a leakage dimension because it is e0en more

dependent from the 0olume of the testing apparatus8 and it is still not normalized to the nominal pressure % # to be applied.

9"uation .1 from I$I9C (I '3;)<2 is no& tested &ith 9"uation 1.4.3.%!. It introduces a

leakage dimension L improperly called =leakage rate=. In fact its dimension is not a proper leak

rate in $3/ # . $ne may call it =leakage inde,=.

L= % #

% t

V Δ %

Δ t [ *a $

3/ # ]=[)g $2/ #3] 2.1.3!

Introducing the pressure decay from 9"uation 2.1.1! L becomes:

L= % #

% t

V 4

%t + %at$

Δ t ⋅[1−e

−( A

V 4

√ , ω)Δ t ] 2.1.4!

@ormula 2.1.4! is another bad candidate for a leakage dimension because it is not independent from

the 0olume of the testing apparatus. Δ % /Δ t depends again from V in a strong non<linear

form. In addition it is not normalized to the nominal pressure % # to be applied8 because the

1/ % t does not take into account that the analogous multiplying into the numerator factor uses the

absolute pressure instead of the relati0e one.

(espite such shortcomings8 the no& called =leakage inde,= L &as used in a &ide range ofapplications &ith satisfactory results 14!.

Clause 2.2: $se of linear approximation for decay la"

Fet us linearize the e,act formulation in the follo&ing se"uence:

% (t )=( %t + %at$)e− A

V

√ ω)t

− %at$1.4.3.#!

%a(Δ t )= % ()+ &%

&t t =

⋅Δ t 2.2.1!

Page 1# of 23



(&%

&t )=−( A

V 4

√ , ω)( % t + %at$)e− A

V 4

√ , ω t

2.2.2!

(&%

&t )t =

=− A

V

√ ω ( %t + %at$) 2.2.3!

%a(Δ t )= % t −

A

V √

ω ( %t + %at$)⋅Δ t 2.2.4!

@rom clause 1.4.3 as e,pected A

V

√ ω is dimensionally the reciprocal of a time.

/he 9"uation 2.2.4! pro0ides the linear appro,imate la& %a(t ) for pressure 0s. time cur0e.

/he follo&ing figure 2.2.1 sho&s the comparison bet&een e,act presure decay % (t ) from

1.4.3.#! and appro,imate pressure decay %a(t ) from 2.2.4!. Please remember that =e,act= means

it is correctly applied &hen the orifice leaks in sonic flo& conditions.

Figure 2.2.1! o$%ari#on beteen e'at for$ulation % (t ) fro$ Euation (1.4..5 (re&

ontinuou# line an& lineari6e& for$ulation %a(t ) (blue ontinuou# line fro$ Euation (2.2.4.

7ote t"at bot" % (t ) an& %a(t ) are vali& for #oni flo on&ition# t"roug" orifie. 8"en t"e

&erea#ing %re##ure rea"e# % #oni evaluate& aor&ing Euation (1..1., t"e %re##ure &eay

la #"all be refor$ulate& ta)ing into aount t"e #ub#oni flo on&ition# a# %er Euation(1..1.1 an& obtaining a #ub#oni #olution % ## (t ) t"at a%%roa"e# to 6ero relative %re##ure for

ti$e a%%roa"ing to infinity. /n fat, a# alrea&y note&, Euation (1.4..5 %rovi&e# an artifat

#olution, a%%roa"ing to ab#olute vauu$, un&er % #oni beau#e t"e #oni flo t"roug" t"e orifie

&e%en&# on t"e in#i&e ab#olute %re##ure only.

9"uation 2.2.4! allo&s to &rite the linear appro,imated formula for the pressure decay:

Δ %= %a()− %a(Δ t )= %t −[ %t −( A

V √ ω)( %t + %at$)⋅Δ t ]

Page 1; of 23



Δ % = A

V

√ ω ( % t + %at$)⋅Δ t 2.2.#!

/he ratio bet&een pressure leak and elapsed time8 properly called the pressure decay rate8 can be

no& &ritten as follo&s:

Δ %

Δ t =( %t + %at$)

A

V

√ ω [ *a / #] 2.2.;!

(espite the ma>or non<linearity of 9"uation 2.1.1! ha0e been sol0ed8 9"uation 2.2.;! is still not

independent from the 0olume of the testing apparatus and not normalized to the nominal pressure % # to be applied.

/he properly defined leakage rate8 in terms of leakage of 0olume o0er time as per pre0ious

9"uation 2.1.2!8 no& can be &ritten as follo&s:

Δ %Δ t

⋅ V

% ref

= %t + %at$

%ref

( A√ ω ) [$3/ # ] 2.2.%!

&here %ref is a reference pressure introduced for dimensional homogenization to 0olume o0er

time dimension. +nfortunately it is still not normalized to the nominal pressure % # to be applied.

@inally let us resume the proposed e"uation .1:

L= % #

% t

V Δ %

Δ t [ *a $

3/ # ]=[)g $2/ #3] 2.1.3!

y replacing the e,pression for Δ % /Δ t it can be &ritten:

L= % #

%t + %at$

% t

( A√ ω) 2.2.'!

In that case L is independent from the pressurized 0olume V . oreo0er8 it is independent

from any 0olume definition8 and depends on the size A of the leaking orifice8 linearly increasing

&hen A increases.

$n the other hand the term ( % t + %at$)/ % t still pre0ent full normalization to the nominal pressure./he numerator sho&s the starting pressure in absolute terms. /he denominator sho&s the same

starting pressure but in relati0e terms.

y replacing % t in 9"uation 2.1.3! &ith its absolute 0alue8 the full normalization is achie0ed:

L= % #

%t,ab#

V Δ %

Δ t = % #

% t + %at$

% t + %at$

( A√ ω )= % # A√ ω [ *a $3/ # ]=[)g $

2/ #3] 2.2.)!

ince the use of a relati0e pressure for the numerator and an absolute pressure for the denominator

may seem uncomfortable8 &ithout loss in generality8 9"uation 2.2.)! may be re&ritten introducing

the absolute 0alue for both pressures:

Page 1% of 23



L= % #,ab#

%t,ab#

V Δ %

Δ t = % #,ab# A√ ω [ *a $

3/ # ]=[)g $2/ #3] 2.2.1!

/herefore 9"uation .1 of I$I9C (I '3;)<2 as per formula 2.1.3! represents a consistent

leakage inde,8 pro0ided that the pressures are introduced in their absolute 0alues8 as per formula

2.2.1!.

Clause 2.: %uantification of error introduced by linear approximation

/he linear appro,imation 2.2.1! &as assumed in order to find a consistent leakage inde,8

formulated according to 9"uation 2.2.)!.

/o be acceptable8 any&ay such simplified formulation shall be 0erified to result:

J accurate for small leakages. In fact if the tested components are satisfactory the leakage

must be small8 and then the simplified formula becomes reliable.

J o0erestimating the error in order to a0oid false positi0e results.

/he t&o pressure decay la&s to be compared are the follo&ing:

% (t )=( %t + %at$)e− A

V √ ω t

− %at$1.4.3.#! is the e,act pressure decay la&H

%a(Δ t )= % t − A

V

√ ω ( %t + %at$)⋅Δ t 2.2.4! is the appro,imate pressure decay la&.

bo0e mentioned condition J is already satisfied because8 as sho&n by @igure 2.2.1!8 %a(t )o0erestimates the pressure decay assuming a constant slope that is the absolute ma,imum slope of

% (t ) .

/he follo&ing dimensionless 0ariables can be introduced in order to simplify the calculations:

t = %t

%at$

2.3.1!

#oni= % #oni

%at$

2.3.2!

= %( t ) %at$

2.3.3!

!= A

V √ ω t 2.3.4!

/he t&o pressure decay la&s can be re&ritten in dimensionless form:

(!)=(t +1)e−!−1 2.3.#! is the e,act pressure decay la&H

a(!)=t −!(t +1) 2.3.;! is the appro,imate pressure decay la&.

Fet us define the pressure decay in dimensionless terms:

Page 1' of 23



Δ =∣Δ %

%at$∣=∣ % (t )− %t

%at$ ∣=t − 2.3.%!

/hen let us define the discrepancy as follo&s:

ϵ=

−a

Δ 2.3.'!

/he physical meaning of 9"uation 2.3.'! is illustrated in @igure 2.3.1.

Figure 2..1.

@or a gi0en elapsed time t from start8 that correspond to the dimensionless time ! through

9"uation 2.3.4!8 the pressure has dropped from starting 0alue % t to actual 0alue % (t ) 8 that

corresponds to pressure drop Δ in dimensionless terms. /he pressure decay is e,pected to

follo& the e,ponential la& 1.4.3.#!8 or 2.3.#! in dimensionless terms. t the same time the linear

appro,imation pro0ides a different pressure calculated using 9"uations 2.2.4! or 2.3.;!.@inally8 for a gi0en time three meaningful numbers are collected: the pressure drop Δ 8 the

actual pressure and the corresponding pressure a that is estimated by means of the linear

appro,imation.

/he difference bet&een actual pressure and appro,imated estimate a is the error introduced

by the appro,imation. uch error is compared &ith the elapsed pressure decay8 obtaining the

relati0e discrepancy defined by 9"uation 2.3.'!.

9"uation 2.3.'! can be re&ritten using 9"uations 2.3.#! and 2.3.;! and highlighting Δ as

independent 0ariable. In such &ay the follo&ing formula represents the relati0e error introduced on

the pressure decay for a gi0en elapsed pressure decay:

ϵ=ϵ(Δ)=t +1

Δ ⋅ln[ (t +1)

(t +1)−Δ]−1 2.3.)!

In order to study the function in 9"uation 2.3.)! the notation can be further simplified introducing

the follo&ing terms:

t +1= 2.3.1!

Δ = ' 2.3.11!

/herefore:

Page 1) of 23



ϵ( ')=

'⋅ln[

− ' ]−1 2.3.12!

@unction ϵ( ') is mathematically defined for < '" . (ue to sonic condition assumption

through the orifice8 the physically meaningful upper bound of ' corresponds to the 0alue ' #oni

of pressure decay at &hich the subsonic transition occurs.

/he first and second deri0ati0es are:

ϵ ( ' )=−

'2⋅ln[

− ' ]+

' ( − ' )2.3.13a!

ϵ ( ( ( ')=2 .

'3 ⋅ln[ .

. − ' ]+ .

' ( . − ')[ 1

. − '−

2

' ] 2.3.13b!

@unction ϵ( ') and its deri0ati0e ϵ ( ' ) are sho&n in @igure 2.3.2. /he follo&ing meaningful

results can be obtained by sol0ing some undetermined forms:

ϵ(4)=4

ϵ ( (4)= 1

2

lim ' →

ϵ( ')=+∞

lim '→

ϵ ( ( ' )=+∞

ϵ ( ')> ∀ ' :< '<

s e,pected8 the function ϵ( ') increases &hen '=Δ increases.

Figure 2..2.

Fet us apply the formulation herein de0eloped to a practical case.

s e,ample8 let us analyze the leakage test by pressure decay on a standard Fuer connector as per

I$ #)4<1:1)'; and ne& I$ '3;)<% under de0elopment.

Fet us imagine the test conducted in free atmosphere8 starting from the starting pressure %t =3 %at$ 8 a little bit bigger than the nominal % #=3 bar . Aence t =3 and =4 . /he

discrepancy function becomes:

Page 2 of 23



ϵ( ')=4

'⋅ln[ 4

4− ' ]−1 2.3.14!

/he follo&ing table sho&s some numerical 0alues of ϵ( ') and the corresponding 0alues of

pressure decay8 inside pressure and appro,imation errors8 assumed that %at$=113 *a .

'=Δ 81 82 83 84 8# 8; 8% 8' 8) 81 82

ϵ 8# 81 81# 821 82; 831 83% 842 84' 8#4 811;

Δ % [ *a ] 113 22; 33) 4#2 #;# ;%' %)1 '14 )11% 113 22;

Δ % [$bar ] 1 2 3 41 #1 ;1 %1 '1 )1 11 23

% (t ) [$bar ] 32) 31) 3) 2))' 2)'' 2)%' 2);' 2)#' 2)4' 2)3' 2'3;

−a=ϵ Δ #1<#21<4

4,51<48,41<4

1,31<31,91<3

2,61<33,41<3

4,31<35,41<3 823

%− %a [$bar ] 8# 82 84; 8'# 183 18) 28; 384 484 #8# 24

@or a test performed in a laboratory in free atmosphere8 from 9"uation 1.3.1.3! the sonic flo&

condition occurs until % in#=18')# %out =18')# %at$ &ritten in absolute terms8 that means

% #oni=8')# %at$ &ritten in relati0e terms. /hat means8 in present case8 the formulation is 0alid

until the inside relati0e pressure reaches % #oni=8')#⋅113 $bar =)% $bar .

Clause 2.!: %uantification of effects of testing temperature

Fet us obser0e the appro,imate decay la& that pro0ides the rationale for mentioned e"uation

9"uation .1:

%a(Δ t )= % t − A

V

√ ω ( %t + %at$)⋅Δ t 2.2.4!

Δ % = A

V

√ ω ( % t + %at$)⋅Δ t 2.2.#!

In this formulation the displayed temperature represents the actual one during testing8

t 8 thatmay be slightly different from the 0alue specified for nominal testing contition # .

s a result 9"uation 2.2.1! should be re&ritten as follo&s:

L= % #,ab#

%t,ab#

V Δ %

Δ t = % #,ab# A√ t ω [ *a $

3/ # ]=[)g $2/ #3] 2.2.1!

In order to normalize the temperature8 one may include a normalization multiplying factor in such

form:

Page 21 of 23



L2=

% #,ab#

%t,ab#

V

Δ %

Δ t

# t

= % #,ab# A √ # ω 2.4.1!

s final comment8 let us take a look at the temperature correction factor √ # / t for se0eral

nominal and effecti0e temperatures in a reasonable test range.

/he follo&ing table sho&s the correction in the range from KC to #KC assuming as possiblenominal temperature # se0eral 0alues from KC to #KC.

t KCJ 1# 2 23 2# 3 #

# KCJ

18 182% 183; 1841 1844 18#3 18''

1# 8)%4 18 18) 1814 181% 182; 18#)

2 8);# 8))1 18 18# 18' 181% 18#

23 8); 8)'; 8))# 18 183 1812 1844

2# 8)#% 8)'3 8))2 8))% 18 18' 1841

3 8)4) 8)%# 8)'3 8)'' 8))2 18 182%

# 8)1) 8)44 8)#2 8)#% 8);1 8);) 18

/he relati0e correction is:

t KCJ 1# 2 23 2# 3 #

# KCJ

28%L 38;L 481L 484L #83L '8'L

1# <28;L 8)L 184L 18%L 28;L #8)L

2 <38#L <8)L 8#L 8'L 18%L #8L

23 <48L <184L <8#L 83L 182L 484L

2# <483L <18%L <8'L <83L 8'L 481L

3 <#81L <28#L <18%L <182L <8'L 28%L

# <'81L <#8;L <48'L <483L <38)L <381L

Page 22 of 23



&otes

1! In such kind of problems a 0olume of generic shape can be treated as an e"ui0alent cylindrical

0olume by means of a dimensionless transformation coefficient that can be e0aluated analitically

for simple shapes8 and numerically by meand of finite element analysis for comple, shapes. uch

transformation coefficient is constant in small strains and displacements field and linear elastic

constituti0e model. ee e.g. =*oarkMs @ormulas for tress and train=8 %th

edition8 Chapter 13.2! ee e.g. =*oarkMs @ormulas for tress and train=8 %th edition8 /able 13.#. In small strains and

displacements field and linear elastic constituti0e model8 the relation bet&een applied pressure and

strain is still linear e0en for a container &ith thick &alls.

3! (iscrepancies from the la& of ideal gases are not practically detectable &ithin the pressure and

temperature range of common mechanical testing laboratory en0ironment and e"uipment.

4! /rue incompressibility does not e,ist. Ao&e0er li"uids like &ater sho&8 &ithin the pressure and

temperature range of common mechanical testing laboratory en0ironment and e"uipment8 negligible

compressibility.

#! ee e.g. @9 =Aandbook of chemical hazard analysis procedures=8 ppendi, 8 paragraph

.3.

;! In present essay the @9 Aandbook formulation has been adopted. (ifferent formulations8 in&hich the cross<section area of the leaking orifice is multiplied by dimensionless shape factors to

keep into account the original shape8 are a0ailable in technical literature. ny&ay these

impro0ements do not introduce any different efefct on the final result.

%! @9 Aandbook e"uation .'.

'! @9 Aandbook e"uation .1.

)! /he ratio bet&een specific heats a.k.a. adiabatic inde, of air depends in principle on temperature

and humidity. @or dry air at 113 mbar it goes from 1843 at KC to 1841 at 1KC FangeDs

Aandbook of Chemistry8 1th ed. p. 1#24!. /he effect of humidity and temperature on adiabatic

inde, &as e,hausti0ely studied by medeo 0ogadro in his masterpiece =@isica deD corpi

ponderabili ossia trattato della costituzione generale deD corpi=8 1'418 Nol. 4 p. '4; and follo&ing!

and found to be negligible for common practical purposes.

1! @9 Aandbook e"uation .).

11! @9 Aandbook e"uation .%.

12! /he discharge coeffcient + D of an orifice leaking li"uid depends on the shape of the leaking

orifice. It is a dimensionless coeffcient that relates the effecti0e leaking cross<section area to the

circular area e"ui0alent in terms of leaking capability. In addition it takes into account the

irreco0erable losses associated &ith a certain shape &ith respect to the theoretical ones calculated

for the reference shape usually a cylindrical channel!. ee e.g. @rank . 6hite8 =@luid echanics=8

%th 9dition.

13! /he general solution of an e"uation in the form y +ay=b is y ( ' )=+ e−a'+

b

a

&ith

intergration constant + to be e0aluated upon one initial condition.

14! including the tandard I$ #)4<1:1)'; about Fuer fittings8 that are currently the most used

small<bore connectors for medical de0ices in the entire 6orld.

Page 23 of 23

Documents

Essay on Pressure Decay Leakage Test