Aaron Johnson

National Institute of Standard and TechnologyGaithersburg, MD 20899

CFV Measurement Conference

September 20, 2013

Poitiers, France

The Critical Flow Function and Beyond(Real Gas Corrections for CFVs)

Objectives

1) To suggest the use of REFPROP Thermodynamic Database for calculating C*

2) To introduce real gas corrections for large = d/D applications

3) To present results from measuring C* experimentally

OutlineOutline

• Background on CFV Theory & Definitions• Listing of Most Common Methods for computing C*

Approximate Analytical Techniques Tables and Curve Fits Thermodynamic databases

• Overview of REFPROP• Evaluation of various methods used for computing C*• Real Gas Effects in large = d/D Applications• Measurement of C*• Discussion

Baseline CFV Flow Model

0TuR40P

2d M

thm*C i

Three Basic Assumptions1) One-dimensional flow

2) Isentropic flow

3) Perfect Gas (Z=1, cP=const)

Steady Navier-Stokes Flow Model

*C i

12

1

2

1

321 dddth

d CCC1mm

C

321 dddd CCCC

321 dddd C1C1C1C +

ErrorFactorization

ideal criticalflow function

k = 1, 2, or 3

Cd,k = 1 – Cd,k

Higher Order CFV Models

Three Different Models based on DifferentSimplifications of the Navier-Stokes Equations

= f2(Re,,)

= f1(,) = r/rcd1C 1m

thm

1) Correction for Sonic Line Curvature (Kliegel)• Inviscid Flow (no B.L.)• Perfect Gas (Z=1 & cP=const)

Assumptions 2 & 3 enforced

d2C 2m

thm

2) Correction for B.L. (Tang, Geropp)

• 1D flow (flat sonic line)• Perfect Gas (Z=1 & cP=const)

Assumptions 1 & 3 enforced

d3C 3m

thm

3) Correction for Real Gas Effects (Johnson)• 1D flow (flat sonic line)• Inviscid Flow (no B.L.)

Assumptions 1 & 2 enforced

0TuR4

*C0P2

d Mthm i

*CR

*C i

= f2(Re,,)d2C 2m

thm

2) Correction for B.L. (Tang, Geropp)

• 1D flow (flat sonic line)• Perfect Gas (Z=1 & cP=const)

Assumptions 1 & 3 enforced

Basic Equationss = const

h0 = h + ½ u2 = const

= f1(,) = r/rcd1C 1m

thm

1) Correction for Sonic Line Curvature (Kliegel)• Inviscid Flow (no B.L.)• Perfect Gas (Z=1 & cP=const)

Assumptions 2 & 3 enforced

3m4

2d u

0TuR4

*C0P2

d M R

*

*R

d Ci

CC

3

• All Real Gas Behavior Accounted for in 3dC

• Divide by 3dC to eliminate Real Gas Effects M

0*i0

2d4 TR

CPd

mC u

How to Define Cd Independent of Real Gas Effects?

21 dd CC

321 dddd CCCC

3d

dd C

CC

M0u

*R0

2

TR

CPd

m4

= f1(,)d1C• (Correction for Sonic Line Curvature)

= f2(Re,,)d2C• (Correction for B.L.)

,Re,Cd φ

• Physically Cd < 1

• CR eliminates real gas effects (if sufficiently accurate)(*

NumerousNumerous C* Models are Used by End Users

• Accuracy between different models can vary significantly

• Many C* models are tailored for a specific gas type End user must acquire different models for each gas type

• The numerous C* models can be confusing to end users Functional expressions for C* based on approximate analytical solutions Tables and Curve fits provided in CFV Standards (ISO 9300 and ASME) Various published C* values for different gases

• N2, Air, CO2, Ar, He, and others (R.C. Johnson 1965, NASA TND-2565 )• Steam (Owen & Amini, 1994 and 1997)• wet air (Aschenbrenner, 1983, Britton et. al. 1998)• dry & humid air, natural gas, methane and other gases (Sullivan, 1980’s)• N2, Ar, CH4, dry & humid air, and natural gas (Stewart et. al. 1999, 2000)

• Tables of Thermodynamic and Transport Properties (Hilsenrath, 1960) Thermodynamic Databases

• GERG (2004)• AGA 8 (1992)• AGA 10 (based on AGA 8, 2003)• REFPROP 9.1

Overview of REFPROP• REFPROP is an acronym for REFerence fluid PROPerties• Based on the most accurate pure fluid and mixture models currently available

Maintained by NIST (Eric W. Lemmon) Continuously updated (next version is being developed) More than 50 pure fluids Flexibility to create your own mixture (e.g., wet air, natural gas)

• REFPROP 9.1 Includes multiple Thermodynamic Databases GERG-2004 Model (Prof. Dr. Wolfgang Wagner and Dr. Oliver Kunz)

• 22,000 experimental natural gas data and natural gas like multicomponent data Modified GERG-2004 Model (Default Model in REFPROP)

• NIST modified the original GERG model making it more accurate • Mixture parameters are identical to GERG Model• Pure fluid equations of state are more complex and more accurate

AGA8 (1992)

• REFPROP Platform and Interface Capabilities Stand alone graphical user interface (GUI) Compatible with Excel, Fortran, Visual Basic, C++, MatLab, LabVIEW, Delphi

• Computes over 75 Thermodynamic properties (gas, liquid, and two phase) Density, Specific Heat, Enthalpy, Compressibility Factor, etc. C* is one of the properties applicable to the gas phase for CFV flows

Rapidly Converging C* Computations

0u

0d

TR

A*C*PCm

M

Pres. Temp. Entropy EnthalpyGas P 0 T 0 s 0 h 0

[ ] [MPa] [K] [kJ/kg-K] [J/kg]methane 10 293.15 3.990158 794321.95

n T* n h* n a* ne

n (d e /dT* )n T* n+1* n C* n

[ ] [K] [J/kg] [m/s] [%] [1/K] [K] [kg/m3] [ ]

1 216.843 672448 351.814 -7.6E+00 N/A N/A 35.100733 0.481342 293.150 794322 437.870 1.2E+01 0.257129 246.213 78.322379 1.336763 246.213 718355 380.322 -4.6E-01 0.266905 247.933 49.497274 0.733764 247.933 721058 382.129 -3.2E-02 0.248427 248.060 50.42673 0.751095 248.060 721260 382.264 6.9E-05 0.248963 248.060 50.49625 0.752396 248.060 721259 382.263 -5.1E-09 0.249000 248.060 50.4961 0.752397 248.060 721259 382.263 2.7E-09 0.248999 248.060 50.4961 0.752398 248.060 721259 382.263 2.0E-09 0.117680 248.060 50.4961 0.75239

M0

0u

P

TRρ*a*C*

M0

0u

P

TRρ*a*C*

1D Steady Navier-Stokes Flow Model Isentropic flow: s( , ) = s( , ) Isoenergetic flow: h( , ) = h( , ) + a( , )2/2

T0 P0 T* P*

T* P* T* P*T0 P0 s* s*

0u

0d

TR

A*C*PCm

M

1

2100

0

2nn

n h

ah **e

nnn1n

*

**dT

dεεTT

*** TT

εε

dT

dε

1-nn

1-nn

n

Numerical Method

UpdatedTemperature

CFVs are Used to Determine FlowCFVs are Used to Determine FlowPT

Flow

CFV (Critical Flow Venturi)

CdC*

T0Ru4

P0d 2m M

C* – Critical flow function used during CFV applicationCd – based on flow calibration of CFV

C*

m

P0d 2 M

T0Ru4Cd

• Uncertainty in m depends on level of correlation between Cd & C* m

- used during calibration

M0

0utt*

P

TRaC

M0

0utt*

P

TRaC

1D Steady Navier-Stokes Flow Model Isentropic flow: s( , ) = s( , ) Isoenergetic flow: h( , ) = h( , ) + a( , )2/2

N/ATables

N/ACurve Fits

Real Gas

Polytropic Process

Ideal Gas

Critical Flow Function Formula

Thermo.ExpressionsModel

constc

Z

P

1

constc

Z

P

1

Definition of and Methods used for Computing C*

constn

s

P

Pn

x

xx

xxf12

1

2

1

),,( kxTPZZ

fC *i

0*p ZnfC

M0

0utt*r

P

TRaC

Fits of *rC

Tables of *rC

T0 P0 Tt Pt

Tt Pt Tt PtT0 P0

• REFPROP 8.0 Modified GERG (R8,NIST) GERG (R8,GERG) AGA 8 (R8,AGA)

• AGA 10 (AGA10)

• REFPROP 7.0 (R7)

• ISO 9300 CFV Standard (2005)

M0

0utt*r

P

TRaC

1100

*NISTR8,

*i

C

C

ref0 PP

-10

-5

0

5

10

0 50 100 150 200

1100

*NISTR8,

*i

C

C

1100

*NISTR8,

*i

C

C

Evaluation of the Ideal Gas Model for C*

12

1

*i 2

1C

1) = const 2) = (T) 3) = (T,P)

How to implement the method?

T0 = 293.15 KPref = 101.325 kPa

CH4

dry air

He

N2

H2

Ar

O2

CO2

+ 9.3%

+ 0.2%

+ 2.1%

+ 0.5%

+ 2.7%

- 2.1%

- 1.8%

- 2.2%

MaxError Ci*

Evaluation of the Polytropic Model for C*

How to implement the method?

CH4

dry air

He

N2

H2

Ar

O2

CO2

+ 4.9%

+ 2.7%

+ 0.5%

+ 2.7%

+ 0.8%

+ 2.1%

+ 2.6%

- 0.3%

1100

*NISTR8,

*p

C

C

-10

-5

0

5

10

0 50 100 150 200 ref0 PP

T0 = 293.15 KPref = 101.325 kPa

n

nn

Z

nC

12

1

0

*p 2

1

+ 0.2%

+ 2.1%

+ 0.5%

+ 2.7%

- 2.1%

- 1.8%

- 2.2%

+ 9.3%

MaxError Ci*

MaxError Cp

*

ZTR

a

P

aP

Pn

u

22

s

M

000 , PTZZ

1)

2)

1st Ed. 1990

Evaluation of ISO 9300 Tabulated Evaluation of ISO 9300 Tabulated CC* Values* Values

ISO 9300 C* Tables (2005)• Range of Gas Types and Conditions 7 Gas Types: (CO2, Ar, N2, Ar, CH4, Air, Steam)

T0 range: 200 K to 600 K P0 range: up to 20 MPa

• Uncertainty of C* = 0.1 % (k = 2) Generally good agreement with R8 NIST Interpolation Errors can be Significant

Table B.1: C* values for Methane

0.89018 2.7 %0.91403230

0.83585

0.99220

CISO*

2nd Ed. 2005

240

220

T0 (K)

P0 = 8 MPaP0 = 8 MPa

0.83585

0.99220

0 %

0 %

CR8,NIST* % Diff.

R8NIST

ISOTable

• Interpolation Error ≈ 2.7 % C* Uncertainty• Limited gas types and P0 and T0 range• Not practical to Tabulate Mixtures

Wet Air Natural Gas

Objectives

1) To suggest the use of REFPROP Thermodynamic Database for calculating C*

2) To introduce real gas corrections for large = d/D applications

3) To present results from measuring C* experimentally

Real Gas Corrections for Large = d/D > 0.25

• CFV applications measure the recovery temp. (Tm) and the static pres. (P)

• Stagnation conditions T0 and P0 are necessary to computeo Critical flow function; Cr

*= Cr*(T0, P0)

o Mass flow;

• Stagnation conditions are based on 1) Ideal gas or 2) Polytropic gas

o Ideal Gas;

o Polytropic Gas;

Large errors can result for > 0.25 when real gas effects are significant

PTm

Flow

CFV (Critical Flow Venturi)

d

D

m0u

d*

0

4

2

TR

CCPd M

12i0 2

11

MPP

2i0 2

11 MTT

&

120 2

11p

nn

Mn

PP

20 2

11p M

nTT &

Required Inputs1) Diameter ratio: = d/D

2) Measured Temperature: Tm

3) Measured Pressure: P

How do you compute P0 and T0 for large ?

1D Steady Navier-Stokes Flow Model

1) Isentropic Flow: s(T0, P0) = s(T *, P*)

2) Isoenergetic Flow: h(T0, P0) = h(T *, P*) + a(T *, P*)2/2

3) Isentropic Flow: s(T0, P0) = s(T , P)

4) Isoenergetic Flow: h(T0, P0) = h(T , P) + u2/2

5) Mass Conservation: (T*, P*)a(T*, P*)2 = (T, P)u

6) Recovery Factor (RF): RF = (Tm – T )/ (T0 – T )

Evaluation of Ideal and Polytropic Gas Models

Comparison Parameters

1)% Difference T0

2)% Difference P0

3)% Difference C*

4)% Difference (Theoretical Mass Flux) 0u

0th

*

TR

CPm

M

1100 Rm xx% Diff x

Example: % Diff T0 1100 R0

i0 TT

% Diff T0 for Methane(Ideal Gas Model)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

% D

iffe

ren

ce T

o

Methane at T=295 K (Ideal Gas)P=0.1 MPaP=5 MPaP=10 MPaP=15 MPaP=20 MPa

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

% D

iffe

ren

ce P

o

Methane at T=295 K (Ideal Gas)P=0.1 MPaP=5 MPaP=10 MPaP=15 MPaP=20 MPa

% Diff P0 for Methane(Ideal Gas Model)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

% D

iffe

ren

ce C

*

Methane at T=295 K (Ideal Gas)P=0.1 MPaP=5 MPaP=10 MPaP=15 MPaP=20 MPa

% Diff C* for Methane(Ideal Gas Model)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

% D

iffe

ren

ce M

do

t F

lux

Methane at T=295 K (Ideal Gas)P=0.1 MPaP=5 MPaP=10 MPaP=15 MPaP=20 MPa

% Diff Mass Flux for Methane(Ideal Gas Model)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7% D

iffe

ren

ce T

o

Methane at T=295 K (n=const)P=0.1 MPaP=5 MPaP=10 MPaP=15 MPaP=20 MPa

% Diff T0 for Methane(Polytropic Gas Model)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

% D

iffe

ren

ce P

o

Methane at T=295 K (n=const)P=0.1 MPaP=5 MPaP=10 MPaP=15 MPaP=20 MPa

% Diff P0 for Methane(Polytropic Gas Model)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7% D

iffe

ren

ce C

*

Methane at T=295 K (n=const)P=0.1 MPa

P=5 MPa

P=10 MPa

P=15 MPa

P=20 MPa

% Diff C* for Methane(Polytropic Gas Model)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

% D

iffe

ren

ce M

do

t F

lux

Methane at T=295 K (n=const)P=0.1 MPaP=5 MPaP=10 MPaP=15 MPaP=20 MPa

% Diff Theoretical Mass Flux for Methane(Polytropic Gas Model)

Objectives

1) To suggest the use of REFPROP Thermodynamic Database for calculating C*

2) To introduce real gas corrections for large = d/D applications

3) To present results from measuring C* experimentally

A Technique for Measuring CR *• Measure CFV mass flow with low uncertainty standard in gas A

• Measure CFV mass flow with low uncertainty standard in gas B at the same Reynolds number:

• Determine CB using the following expression:*

o Gas A is selected so that it behaves closely to ideal gaso C* can be calculated by REFPROP at low uncertainty

At0

0uAd,

*

MAPC

TRmC

Bt0

0uBd,

*

MAPC

TRmC

B

A

A0

B0

B0

A0

A

B

Bd,

Ad,*A

*B M

M

T

T

P

P

m

m

C

CCC

*AC

o Gas B has significant real gas effects

A Technique for Measuring CR (Cont)*

Advantages•No geometric dependence

o it can be applied to small CFVs at high pressureso Analytical CFV theory can be used to estimated d

•Cd Ratio approaches unity at high Reynolds numbers

•All of the dependents can be measured at low uncertainty

B

A

A0

B0

B0

A0

A

B

Bd,

Ad,*A

*B M

M

T

T

P

P

m

m

C

CCC

Requirement •Low uncertainty primary standard capable of measuring multiple gas compositions

CFV (d = 0.387 mm) Calibrated using NIST 34 L PVTt std. (2002)•Nitrogen used for Gas A (P0 = 200 to 650 kPa)

•Argon used for Gas B (P0 = 200 to 650 kPa)

•CFV theory used to for Cd ratio

•Both gases calibrated over Reynolds numbers from 7 000 to 30 000

B

A

A0

B0

B0

A0

A

B

Bd,

Ad,*A

*B M

M

T

T

P

P

m

m

C

CCC

Comparison of Measured Cmeas to Computed CREFPROP* *

-0.10

-0.05

0.00

0.05

0.10

0 200 400 600 800

1100

*

*

meas

REFPROP

C

C

(kPa)0P

Argon

Helium

Some Disccussion Topics

1) How are you currently computing C*?

2) Does it make sense to standardize the software used to compute C*?

3) The ISO 9300 includes a multi-parameter curve fit for select gases to correct the CFV mass flux for large . Would it be useful to have REFPROP make these corrects?

4) Other questions and points of dissusion