mfr-appnotes

©Smar Research Corporation 1 MFR-0902

HART®

Fieldbus Profibus Intrinsic Safety Configuration Tools Semiconductors Training Custom Design

SmarResearch TechnologySource

Mass Flow Rate

Determination for Multivariable Transmitter

(Application Notes)

V 1.0

Visit the SmarResearch technology center at: www.smarresearch.com


Topic Page Number I. Symbols and Definitions 3 II. Necessary Specifications 4 III. Parameters 5 IV. Mass Flow Equation 5 V. Determination of Diameters/Diameter Ratio/Velocity of Approach 6 VI. Determination of Expansion Factor 6

VII. Determination of Isentropic Exponent 7

VIII.Determination of Fluid Density 8

IX. Determination of Viscosity 9

X. Determination of Coefficient of Discharge

AGA3 Orifice Plates 10 ISO Orifice Plates 11 ASME Orifice Plates 12 Nozzle, ISA 1932, ISO 12 Nozzle, Long Radius Wall Taps, ISO 12 Nozzle, Long Radius Wall Taps, ASME 12 Venturi Nozzle, ISO 13 Venturi, Rough Cast Inlet, ISO 13 Venturi, Rough Cast Inlet, ASME 13 Venturi, Machined Inlet, ISO 13 Venturi, Machined Inlet, ASME 13 Venturi, Welded Inlet, ISO 13 Small Bore Orifice Plate, Flange Taps, ASME 14

Appendix A: Sample of AIChE/DIPPR Database 15 Appendix B: Formulation of Isentropic Exponent of Steam Approximation 16 Appendix C: Formulation of Steam Viscosity Approximation 18 Appendix D: Custom Liquid Calculations 20 Custom Gas Calculations 21 Appendix E: Natural Gas Compressibility Equations 22 Appendix F: Iterative Process Used to Solve for Coefficient of Discharge 31 Appendix G: References 32

Mass Flow Rate - Table of Contents

Mas

s F

low

Rat

e - T

able

of C

on

ten

ts


I. Symbols and Definitions

Symbol Definition Cd Coefficient of discharge Cp Specific heat at constant pressure d Throat diameter at flowing temperature davg Throat diameter at average temperature dr Throat diameter at reference temperature D Pipe diameter at flowing temperature Davg Pipe diameter at average temperature Dc Derivative of the correlation value of Cd Dr Pipe diameter at reference temperature Ev Velocity of approach factor Fc Correlation value of Cd Gi Ideal gas relative density k Isentropic exponent L1 Upstream tap position L2 Downstream tap position Mrair Molecular weight of air Mw Molecular weight Pc Critical pressure Pf Flowing pressure Qm Mass flow rate R Universal gas constant Re Reynolds number Tavg Average temperature/Median value of specified temperature range Tc Critical temperature Tf Flowing temperature Tr Reference temperature v Specific volume x Ratio of differential pressure to flowing pressure Y Fluid expansion factor Zf Compressibility at flowing conditions α1 Linear coefficient of thermal expansion of primary element α2 Linear coefficient of thermal expansion of pipe β Ratio of throat diameter to pipe diameter ∆P Differential pressure µf Absolute viscosity of flowing fluid ρf Density of fluid at flowing temperature

Mas

s F

low

Rat

e -

Sym

bo

ls a

nd

Def

init

ion

s


II. Necessary Specifications

Specify Fluid as: a. Gas (from AIChE/DIPPR database) b. Liquid (from AIChE/DIPPR database) c. Steam d. Natural Gas (3 options)

1. Gross Characterization Method - Mole fraction of all components

2. Detail Characterization Method #1 - Real gas relative density - Volumetric gross heating value - Mole fraction of carbon dioxide

3. Detail Characterization Method #2 - Real gas relative density - Mole fraction of carbon dioxide - Mole fraction of nitrogen

e. Custom Fluid - See Appendix D

Specify Primary Element as:

a. Orifice, Flange Taps (ISO, ASME, AGA3) b. Orifice, Corner Taps (ISO, ASME) c. Orifice, D & D/2 Taps (ISO, ASME) d. Small Bore Orifice, Flange Taps, ASME e. Nozzle, Long Radius Wall Taps (ISO, ASME) f. Nozzle, ISA 1932, ISO g. Venturi Nozzle, ISO h. Venturi, Rough Cast Inlet (ISO, ASME) i. Venturi, Machined Inlet (ISO,ASME) j. Venturi, Welded Inlet, ISO

Specify Pipe/Throat:

- Diameters at reference temperature - Materials

Specify Operating Range: - Temperature (Tmin, Tmax)

- Pressure (Pmin, Pmax) M

ass

Flo

w R

ate

- N

eces

sary

Sp

ecif

icat

ion

s


III. Parameters

Gas/Liquid Temperature Range: Tf = 300 - 1500 F (-184.4 - 815.6 C)

Steam Temperature Range: Tf = Tsat - 1500 F (Tsat - 815.6 C) Natural Gas Temperature Ranges: Detail Method Tf = -200 - 400 F (-128.9 - 204.4 C) Gross Methods Tf = 32 - 130 F (0- 54.4 C)

Absolute Pressure Range: Pf = 0.5 – 3626 psia

Pf = 3.447 - 25000.4 kPa Pf = 0.034 – 250 bar Differential Pressure Range: ∆P = 0 – 830 inH2O (0 - 206 kPa)

Pressure Ratio (x = ∆P/Pf) x < 0.25 Natural Gas Parameters Ideal gas relative density Gi = 0.554 - 0.87 Vol gross heating value: HV = 477 - 1150 Btu/ft3 (18.7 - 45.1 MJ/m3) Mole fraction CO2 xCO2 = 0 - 30 % Mole fraction N2 xN2 = 0 - 50 % Each primary element provides different parameters for the following:

Pipe diameter (D) Throat diameter (d) Diameter ratio (β) Reynolds number (Re)

Mas

s F

low

Rat

e -

Par

amet

ers/

Eq

uati

on

Cd = coefficient of discharge d = throat diameter at flowing temperature Ev = velocity of approach factor Qm = mass flow rate Y = fluid expansion factor ∆P = differential pressure ρf = density of fluid at flowing temperature

Qmπ

4Ev Cd d2 Y 2ρf ∆ P

IV. Mass Flow Equation


V. Determination of Diameters/Diameter Ratio/Velocity Approach

d = throat diameter at flowing temperature dr = throat diameter at reference temperature D = pipe diameter at flowing temperature

Dr = pipe diameter at reference temperature Tf = flowing temperature

Tr = reference temperature α1 = linear coefficient of thermal expansion of primary element α2 = linear coefficient of thermal expansion of pipe

davg = throat diameter at average temperature

Davg = pipe diameter at average temperature Tavg = average temperature/median value of specified temperature range

Ev = velocity of approach factor

Mas

s F

low

Rat

e - D

eter

min

atio

n o

f Dia

met

er/E

xpan

sio

n F

acto

r

For Orifice Plates:

k = isentropic exponent Pf = flowing pressure ∆P = differential pressure Y = fluid expansion factor For Nozzles or Venturi Tubes:

x = ∆P/ P f

Yk 1 x( )

2

k.

k 1

1 β4

1 β4

1 x( )

2

k

1 1 x( )

k 1

k

x

1

2

VI. Determination of Expansion Factor

βdavg

Davg

d dr 1 α 1 Tf Tr D Dr 1 α 2 Tf Tr

davg

dr

1 α1

Tavg

Tr

Davg

Dr

1 α2

Tavg

Tr

Ev1

1 β4

Y 10.41 0.35 β

4

k

∆ P

Pf


VII. Determination of Isentropic Exponent

For Gas:

Cp = specific heat at constant pressure R = universal gas constant A, B, C, D, and E are constants provided by AIChE/DIPPR database For Liquid: Y = 1 therefore determination of k is not necessary For Steam: k is estimated using the following linear approximation: Data used to formulate this approximation is contained within Appendix B. For Natural Gas: k is estimated according to AGA Report 3 Part 4:

k = 1.3

kCp

Cp R

Mas

s F

low

Rat

e - D

eter

min

atio

n o

f Ise

ntr

op

ic E

xpo

nen

t

Cp A B

C

Tavg

sinhC

Tavg

2

D

E

Tavg

coshE

Tavg

2

k 1.33544 6.24543 10 5 Tavg

.


VIII. Determination of Fluid Density

For Gas:

Mw = molecular weight Zf = compressibility at flowing conditions Zf is calculated using the Redlich-Kwong equations of state:

Pc = critical pressure Tc = critical temperature v = specific volume For Liquid:

A, B, C, and D are constants provided by AIChE/DIPPR database For Steam: ρf is calculated from Tables S-1 through S-5 from the ASME International Steam Tables. Density is equal to the inverse of specific volume (v). For Natural Gas:

Gi = ideal gas relative density Mrair = molecular weight of air Zf is calculated using the natural gas compressibility equations from AGA Report 8. These equations are contained in Appendix D.

ρ f

Pf Mw

Zf R Tf

Zfv

v b

a

R Tf1.5. v b( )

ρ fA

B1 1

Tf

C

D

ρ f

Pf Mrair Gi

Zf R Tf

Mas

s F

low

Rat

e - D

eter

min

atio

n o

f Flu

id D

ensi

ty

a .42748 R2 Tc2.5

Pc

b .08664 RTc

Pc

v3 R Tf.

Pf

v2 a

Pf Tf.5.

b2 b R. Tf.

Pf

va b.

Pf Tf.5.

0=


IX. Determination of Viscosity

For Gas:

µf = absolute viscosity of flowing fluid

A, B, C, and D are constants provided by AIChE/DIPPR database. For Liquid:

A, B, C, D, and E are constants provided by AIChE/DIPPR database. For Steam: µf is estimated using the following linear approximation: Data used to formulate this approximation is contained within Appendix C. For Natural Gas: µf is estimated according to AGA Report 3 Part 4: µf = 0.0000069 lbm/ft*sec or 0.010268 cP

µ f

A TfB

1C

Tf

D

Tf2

µ f e

AB

TfCln Tf D Tf

E

Mas

s F

low

Rat

e - D

eter

min

atio

n o

f Vis

cosi

ty

µf

7.51661 0.02249 Tf 10 6


X. Determination of Coefficient of Discharge

***For these primary elements, Cd is a function of Reynolds number, and Reynolds num-ber is a function of the unknown mass flow. Therefore, Cd can not be directly calculated and it is necessary to guess values for Cd and Reynolds number and use an iterative process to find the exact values. A detailed description of this iterative process is con-tained in Appendix D. AGA3 Orifice Plates: Flange taps Corner or D&D/2 taps 0.05m < D < 1m 0.05m < D < 1m d > 0.0125m d > 0.0125m 0.1 < β < 0.75 0.1 < β < 0.75 Re > 4000(β<0.5) Re > 4000(β<0.5) Re > 170,000Dβ2(β>0.5) Re > 16,000β2(β>0.5)

Assume an initial value of 4000 for the Reynolds number to get the following equations:

For D < .0711m add following term to Cd0:

Corner Pressure Taps: L1 = L2 = 0

D and D/2 Pressure Taps: L1 = 1 and L2 = 0.47

Flange Pressure Taps: L1 = L2 = .0254/Davg

Calculate for X:

.003 1 β( ) 2.8D

.0254

Mas

s F

low

Rat

e -

Det

erm

inat

ion

of C

oef

fici

ent

of D

isch

arg

e

Cd0 .5961 .0291 β2

.229 β8

.0433 .0712 e8.5 L1. .1145 e

6 L1. β4

1 β4

.0232L2

1 β.014853

L2

1 β

1.3

β1.1

Cd1 .0244 β0.7

Cd2 .145 β4

Cd3 .1177 β4.8

Cd4 .0346 .057 e8.5 L1. .0916 e

6 L1. β4.8

1 β4

.0113 L2

1 β7.232 10 3.

L2

1 β

1.3

β1.9

X4000 Dµ

Cd0 Ev. Yd2 2 ρ f ∆ P..

D

d


Use Cd0-4 and X to solve the following equations:

Cd

(1) = Cd0 - δCd

Recalculate X with new value of Cd

(1) in place of Cd. Use this value of X to recal-culate Fc, Dc, and δCd. Continue repeating this process until δCd < 5x10-6 .

ISO Orifice Plates***: Corner taps Flange or D&D/2 taps

0.05m < D < 1m 0.05m < D < 1m d > 0.0125m d > 0.0125m

0.2 < β < 0.75 0.2 < β < 0.75 Re > 5,000(β<0.45) Re > 1,260,000β2D(β<0.45) Re > 10,000(β>0.45) Re > 1,260,000β2D(β>0.45)

If L1 < 0.4333 then:

If L1 > 0.4333 then:

Corner Pressure Taps: L1 = L2 = 0

D and D/2 Pressure Taps: L1 = 1 and L2 = 0.47

Flange Pressure Taps: L1 = L2 = .0254/Davg

Cd 0.5959 0.0312 β2.1

0.184 β8

0.0029 β2.5 106

Re

.75

0.09 L1β

4

1 β4

0.0337 L2 β3

Cd 0.5959 0.0312 β2.1

0.184 β8

0.0029 β2.5 106

Re

.75

0.039 L1β

4

1 β4

0.0337 L2 β3

Mas

s F

low

Rat

e -

Det

erm

inat

ion

Co

effi

cien

t o

f Dis

char

ge

Fc Cd0 Cd1 X0.35 Cd2 Cd3 X0.8 X0.35 Cd4 X0.8

Dc 0.7 Cd1 X0.35 0.35 Cd2 1.15 Cd3 X0.8 X0.35 0.8 Cd4 X0.8

δCd

Cd0 Fc

1Dc

Cd0

D

d


ASME Orifice Plates***: Flange taps Corner or D&D/2 taps

0.0508m < D < 1m 0.05m < D < 1m d > 0.0125m d > 0.0125m

0.2 < β < 0.7 0.2 < β < 0.7

If L1 < 0.4333 then:

If L1 > 0.4333 then:

Corner Pressure Taps:

L1 = L2 = 0 D and D/2 Pressure Taps:

L1 = 1 and L2 = 0.47 Flange Pressure Taps:

L1 = L2 = .0254/Davg

Nozzle, ISA 1932, ISO***: 0.05m < D < 0.5m 0.3 < β < 0.8 7x104 < Re < 107(β < 0.44) 2x104 < Re < 107(β > 0.44)

Nozzle, Long Radius Wall Taps, ISO***: 0.05m < D < 0.63m

0.2 < β < 0.8 104 < Re < 107

Nozzle, Long Radius Wall Taps, ASME***: 0.1m < D < 0.75m

0.2 < β < 0.8 104 < Re < 6x106

Cd 0.5959 0.0312 β2.1

0.184 β8

0.0029 β2.5 106

Re

.75

0.09 L1β

4

1 β4

0.0337 L2 β3

Cd 0.5959 0.0312 β2.1

0.184 β8

0.0029 β2.5 106

Re

.75

0.039 L1β

4

1 β4

0.0337 L2 β3

Cd 0.9900 0.2262 β4.1

0.00175 β2

0.0033 β4.15 106

Re

1.15

Cd 0.9965 0.00653 β.5. 106

Re

0.5

.

Cd 0.9975 0.00653 β.5. 106

Re

0.5

.

Mas

s F

low

Rat

e -

Det

erm

inat

ion

of C

oef

fici

ent

of D

isch

arg

e

D

d

D

d

d

D

d

D


Venturi Nozzle, ISO:

0.065m < D < 0.5m d > 0.05m

0.316 < β < 0.775 1.5x105 < Re < 2x106

Venturi, Rough Cast Inlet, ISO:

0.1m < D < 0.8m 0.3 < β < 0.75 2x105 < Re < 2x106

Cd = 0.984 Venturi, Rough Cast Inlet, ASME:

0.1m < D < 1.2m 0.3 < β < 0.75 2x105 < Re < 6x106

Cd = 0.984 Venturi, Machined Inlet, ISO:

0.05m < D < 0.25m 0.4 < β < 0.75 2x105 < Re < 1x106

Cd = 0.995 Venturi, Machined Inlet, ASME:

0.05m < D < 0.25m 0.3 < β < 0.75 2x105 < Re < 2x106

Cd = 0.995 Venturi, Welded Inlet, ISO:

0.2m < D < 1.2m 0.4 < β < 0.7 2x105 < Re < 2x106

Cd = 0.985

Cd 0.9858 0.196 β4.5

Mas

s F

low

Rat

e -

Det

erm

inat

ion

of C

oef

fici

ent

of D

isch

arg

e

d

D

d

D

d

D

d

D

d

D

d

D


Small Bore Orifice Plate, Flange Taps, ASME***:

0.025 < D < 0.04m d > 0.006m

0.15 < β < 0.7 Re > 1000

Cd 0.598 0.468 β

410 β

121 β

40.87 8.1 β

4 1 β4

Re.

Mas

s F

low

Rat

e -

Det

erm

inat

ion

of C

oef

fici

ent

of D

isch

arg

e

D

d


Appendix A

Sample of AIChE/DIPPR Database

Mas

s F

low

Rat

e - A

pp

end

ix A

Liquid Database

Fluid Viscosity

A B C D A B C D E

Acetic Acid

Acetone

Acetonitrile

Acetylene

Acrylonitrile

Density

Fluid Mw Pc Tc Cp Viscosity

A B C D E A B C D

Acetic Acid

Acetone

Acetonitrile

Acetylene

Acrylonitrile

Gas Database


Appendix B

Formulation of Isentropic Exponent of Steam Approximation

According to the ASME Steam Tables, isentropic exponent is a function of both temperature and pressure. Its value can be found by plotting temperature and pres-sure on Figure 7 of the ASME Steam Tables and estimating the corresponding isen-tropic exponent. The complexity of the graph makes it impossible to create a direct formula for the calculation of isentropic exponent. Therefore, assumptions must be made in order to simplify the process. Various software was then analyzed to deter-mine what assumptions could be made. It can be assumed that the isentropic exponent does not vary with changing pressure, therefore it is only a function of temperature. To determine an equation for isentropic exponent as a function of temperature, various values of temperature were inputted into the software. The corresponding values of isentropic exponent were then graphed and a trend line was formulated. The equation of this trend line would provide values of isentropic exponent for all possible values of temperature. The following table is the values of isentropic exponent provided by the software at various temperatures:

Mas

s F

low

Rat

e - A

pp

end

ix B

Temp k Temp k Temp k Temp k 79.5855 1.32873 450 1.30801 825 1.28301 1200 1.26007

100 1.32806 475 1.30631 850 1.28141 1225 1.25863

125 1.32713 500 1.30461 875 1.27982 1250 1.25721

150 1.32608 525 1.30291 900 1.27824 1275 1.2558

175 1.32493 550 1.30121 925 1.27666 1300 1.25441

200 1.32368 575 1.29951 950 1.2751 1325 1.25304

225 1.32235 600 1.29783 975 1.27355 1350 1.25167

250 1.32093 625 1.29615 1000 1.272 1375 1.25033

275 1.31945 650 1.29447 1025 1.27047 1400 1.249

300 1.31792 675 1.29281 1050 1.26895 1425 1.24769

325 1.31633 700 1.29116 1075 1.26744 1450 1.24639

350 1.31471 725 1.28951 1100 1.26594 1475 1.24511

375 1.31306 750 1.28787 1125 1.26445 1500 1.24385

400 1.31139 775 1.28624 1150 1.26298

425 1.30971 800 1.28462 1175 1.26151


The following graph plots isentropic exponent of steam versus temperature. A linear trend line has been added to provide an equation for the value of k at any T. A linear function was chosen based on its simplicity and its accuracy to Figure 7 of the ASME Steam Tables.

Mas

s F

low

Rat

e - A

pp

end

ix B

Isentropic Exponent of Steam

k = -6.24543E-05T + 1.33544

1.23

1.24

1.25

1.26

1.27

1.28

1.29

1.3

1.31

1.32

1.33

1.34

0 200 400 600 800 1000 1200 1400 1600

Temperature (F)

Isen

tro

pic

Exp

on

ent

(k)


Formulation of Steam Viscosity Approximation

According to the ASME Steam Tables, steam viscosity is a function of both tem-perature and pressure. Its value can be found by applying temperature and pressure to Table 8 of the ASME Steam Tables. Various software was also analyzed to deter-mine what assumptions could be made.

After studying Table 8, it can be assumed that viscosity does not vary with

changing pressure. To determine an equation for viscosity as strictly a function of tem-perature, various values of temperature were inputted into the software, while main-taining a constant pressure. The corresponding values for viscosity were then graphed and a trend line was formulated. The equation of this trend line would provide values for viscosity for all possible values of temperature.

The following table is the values of steam viscosity provided by the software at

various temperatures and constant pressure:

Mas

s F

low

Rat

e - A

pp

end

ix C

Appendix C

Temp Vis(10-6 lb/ft-s)

Temp Vis(10-6 lb/ft-s)13.7

Temp Vis(10-6 lb/ft-s)

125 7.2 550 13.3 975 19.9

150 7.5 575 13.7 1000 20.2

175 7.8 600 14.1 1025 20.6

200 8.1 625 14.4 1050 21

225 8.5 650 14.8 1075 21.4

250 8.8 675 15.2 1100 21.7

275 9.2 700 15.6 1125 22.1

300 9.5 725 16 1150 22.5

325 9.9 750 16.4 1175 22.9

350 10.2 775 16.8 1200 23.2

375 10.6 800 17.2 1250 24

400 11 825 17.6 1300 24.7

425 11.4 850 17.9 1350 25.4

450 11.7 875 18.3 1400 26.1

475 12.1 900 18.7 1450 26.8

500 12.5 925 19.1 1500 27.5

525 12.9 950 19.5


The following graph plots steam viscosity versus temperature. A linear trend line has been added to provide an equation for the value of m at any T. A linear function was chosen based on its simplicity and its accuracy to Table 8 of the ASME Steam Tables.

Mas

s F

low

Rat

e - A

pp

end

ix C

Steam Viscosity

Viscosity = (0.01511T + 5.05093)*10-6

0

5

10

15

20

25

30

0 200 400 600 800 1000 1200 1400 1600

Temperature (F)

Vis

cosi

ty (1

0^-6

lb/f

t-s)


Custom Liquid Calculations If a liquid is used that is not listed in the database, then the following information must be provided in order to complete the necessary calculations: Critical Temperature (Tc) Viscosity (µ1) at Temperature (T1) Density (ρb) at 20 °C (68 °F) Use the following equation to solve for liquid density: The following method for solving liquid viscosity is derived from Figure 2.19 of the Flow Measurement Engineering Handbook: Solve for Y: Solve for T: Calculate T1: T1 = T + Tf - T1 Solve for Y1 :

Solve for µf:

Mas

s F

low

Rat

e - A

pp

end

ix D

Appendix D

µ f

Tf

T1

1.5 T1 0.9 Tc

Tf 0.9 Tc

µ 1

Y log µ 1 1

4.4 x10 15 T6 3.84 x10 12 T5 3.37 x10 11 T4 9.23 x10 7 T3 3.66 x10 4 T2 6.46 x10 2 T 6 Y 0

Y1 4.4 x10 15 T1 63.84 x10 12 T1 5

3.37 x10 11 T1 49.23 x10 7 T1 3

3.66 x10 4 T1 26.46 x10 2 T1 6

µ f log 1 Y1 1


Custom Gas Calculations If a gas is used that is not listed in the database, then the following information must be provided in order to complete the necessary calculations: Critical Temperature (Tc) Isentropic Exponent (k) Molecular Weight (Mw) Viscosity (µ1) at Temperature (T1) Compressibility (Z)

Use the provided isentropic exponent along with the equations from Section VI to solve for the expansion factor. Use provided molecular weight and compressibility along with the gas equation from Section VIII to solve for the gas density. Use the fol-lowing equation to solve for the gas viscosity:

Mas

s F

low

Rat

e - A

pp

end

ix D

Appendix D

µ f

Tf

T1

1.5 T1 0.9 Tc

Tf 0.9 Tc

µ 1


Natural Gas Compressibility Equations

There are three methods that can be used to solve for the compressibility factor of natural gas. The detail characterization method requires that the mole fraction of all elements of the natural gas be known. The gross characterization method has two op-tions. One option requires that the real gas relative density, volumetric gross heating value, and the mole fraction of carbon dioxide must be known; while the other option requires that the real gas relative density, mole fraction of carbon dioxide, and mole fraction of nitrogen must be known. Once a method is chosen use the corresponding equations to solve for the compressibility of natural gas: Detail Characterization Method

Z = compressibility factor B = second virial coefficient

C`n = coefficients which are functions of composition D = reduced density K = mixture size parameter T = absolute temperature bn, cn, kn, un = constants given in Table 4 (AGA Report 8)

xi = mole fraction of ith component Ki = size parameter of ith component (Table 5) Kij = binary interaction parameter for size (Table 6) N = number of components in the gas mixture

an = constant given in Table 4 Eij = second virial coefficient binary energy parameter B`nij = binary characterization coefficient Ei = characteristic energy parameter for ith component (Table 5) E`ij = second virial coefficient energy binary interaction parameter (Table 6)

Mas

s F

low

Rat

e - A

pp

end

ix E

Appendix E

Z 1D B.

K313

18

n

D C'n. T

un

13

58

n

C'n Tun bn cn kn D

kn Dbn exp cn D

kn

==

K5

1

N

i

xi Ki

5

2

=

2

2

1

N 1

i i 1

N

j

xi xj Kij5 1 Ki Kj

5

2

==

=

B

1

18

n

an Tun

1

N

i 1

N

j

xi xj Eij

un Ki Kj

3

2 B'nij

==

.

=

Eij E'ij Ei Ej

1

2


Gi j = binary orientation parameter Qi = quadrupole parameter for ith component (Table 5) Fi = high temperature parameter for ith component (Table 5) Si = dipole parameter for ith component (Table 5) Wi = association parameter for ith component (Table 5) gn, qn, fn, sn, wn = constants given in Table 4 Gi = orientation parameter for ith component (Table 5) G`ij = binary interaction parameter for orientation (Table 6)

G = orientation parameter Q = quadrupole parameter F = mixture high temperature parameter U = mixture energy parameter Uij = binary interaction parameter for conformal energy (Table 6)

d = molar density (moles per unit volume) Solve for d using following equation:

Mas

s F

low

Rat

e - A

pp

end

ix E

B'nij Gij 1 gn

gn Qi Qj 1 qn

qn Fi

1

2 Fj

1

2 1 fn

fn

Si Sj 1 sn

sn Wi Wj 1 wn

wn

Gij

G'ij Gi Gj

2

C'n an G 1 gn

gn Q2 1 qn

qnF 1 fn

fn Uun1 nan G n

D K3 d

P dRT 1 Bd D

13

18

n

C'n Tun

= 13

58

n

C'n Tun bn cn kn D

kn Db ecn D

kn

=

dRT

G

1

N

i

xi Gi

1

N 1

i i 1

N

j

xi xj G'ij 1 Gi Gj

===

xi

Q

1

N

i

xi Qi

=

F

1

N

i

xi2 Fi

=

U

1

N

i

xi Ei

5

2

=

2

2

1

N 1

i i 1

N

j

xi xj Uij5 1 Ei Ej

5

2

==

xi


Gross Characterization Method

Z = compressibility factor Bmix = second virial coefficient for the mixture Cmix = third virial coefficient for the mixture d = molar density (moles per unit volume) Bij = individual component interaction second virial coefficient N = number of components in gas mixture xi, xj, xk = mole fractions of gas components

Cijk = individual component interaction third virial coefficient Expansions of Bmix and Cmix are provided on page 30 of AGA Report 8. b0, b1, b2 = constants given in Table 7 T = temperature c0, c1, c2 = constants given in Table 7 HCH = molar gross heating value of the equivalent hydrocarbon i = 0, 1, 2 bi0, bi1, bi2, ci0, c i1, ci2 = constants given in Table 8

Mas

s F

low

Rat

e - A

pp

end

ix E

Z 1 Bmix d Cmix d2

Bmix

1

N

i 1

N

j

Bij xi xj

==

Cmix

1

N

i 1

N

j 1

N

k

Cijk xi xj xk

===

Bij b0 b1 T b2 T2

Cijk c0 ci T c2 T2

BCH CH B0 B1 HCH B2 HCH2

CCH CH CH C0 C1 HCH C2 HCH2

Bi bi0 bi1 T bi2 T2

Ci ci0 ci1 T ci2 T2


After Bmix and Cmix are calculated, use following equation to solve for d:

P = absolute pressure R = gas constant Use one of the following methods to solve for HCH

Mas

s F

low

Rat

e - A

pp

end

ix E

BN2 CH 0.72 1.875 x10 5 320 T( )2 BN2 N2 BCH CH

2

BCO2 CH 0.865 BCO2 CO2 BCH CH

1

2

CN2 CH CH 0.92 0.0013 T 270( )( ) CCH CH CH2 CN2 N2 N2

1

3

CN2 N2 CH 0.92 0.0013 T 270( )( ) CN2 N2 N22 CCH CH CH

1

3

CCO2 CH CH 0.92 CCH CH CH2 CCO2 CO2 CO2

1

3

CCO2 CO2 CH 0.92 CCO2 CO2 CO22 CCH CH CH

1

3

CCO2 N2 CH 1.10 CC02 CO2 CO2 CN2 N2 N2 CCH CH CH

1

3

P dRT 1 Bmix d Cmix d2


Method #1 (for determination of HCH) Necessary input: HV = volumetric gross heating value at reference conditions Th , Td, Pd Gr = relative density (specific gravity) of mixture xCO2 = mole fraction of carbon dioxide

HN0 = molar ideal gross heating value at 298.15K and 0.101325 MPa Pd = reference pressure for molar density R = gas constant, 8.31451 J/mol-K Td = reference temperature for molar density Th = reference temperature for heating value Z0 = compressibility factor at reference conditions (set Z0=1 for initial iteration)

Mr = molar mass (molecular weight) of the mixture Gr = relative density at reference conditions Tgr, Pgr Pgr = reference pressure for relative density

Tgr = reference temperature for relative density (ρ0)air = mass density of air at reference conditions Tgr, Pgr

Mr(air) = molar mass of air, 28.96256 g/mol

xCH = mole fraction of equivalent hydrocarbon G1 = -2.709328 G2 = 0.021062199 MrN2 = molar mass of nitrogen MrCO2 = molar mass of carbon dioxide xN2 = mole fraction of nitrogen

Mas

s F

low

Rat

e - A

pp

end

ix E

HN0 HV Z0. R. Td. 1 1.027 x10 4 Th 298.15

Pd

Mr

Gr Z0 R Tgr. ρ

0air

Pgr

ρ0

air Tgr Pgr,Mr air( )

R Tgr.

Pgr

Bair Tgr

Bair Tgr .012527 5.91 x10 4 Tgr 6.62 x10 7 Tgr2

xCH

Mr G2 HN0 MrN2 xCO2 MrN2 xCO2 MrCO2

G1 MrN2

xN2 1 xCH xCO2


HCH = molar gross heating value of the equivalent hydrocarbon

(Z0) new = compressibility factor for next iteration Bmix = second virial coefficient of mixture (calculated from previous equations) Repeat process, continuously replacing Z0 with (Z0) new, until (Z0/Z0

new-1) is less than the convergence criteria (5x10-11 in double precision or 5x10 -7 in single precision) M

ass

Flo

w R

ate

- Ap

pen

dix

E

HCHHN0

xCH

Z0new 1

Bmix Pgr

R Tgr.


Method #2 (for determination of HCH) Necessary input: Gr = relative density (specific gravity) of mixture xCO2 = mole fraction of carbon dioxide xN2 = mole fraction of nitrogen xCH = mole fraction of equivalent hydrocarbon

Mr = molar mass (molecular weight) of the mixture Gr = relative density at reference conditions Tgr, Pgr Pgr = reference pressure for relative density R = gas constant, 8.31451 J/mol-K

Tgr = reference temperature for relative density Z0 = compressibility factor at reference conditions (set Z0=1 for initial iteration)

(ρ0)air = mass density of air at reference conditions Tgr, Pgr Mr(air) = molar mass of air, 28.96256 g/mol

MrCH = molar mass of equivalent hydrocarbon MrN2 = molar mass of nitrogen MrCO2 = molar mass of carbon dioxide Mr = molar mass of mixture

Mas

s F

low

Rat

e - A

pp

end

ix E

xCH 1 xN2 xCO2

Mr

Gr Z0 R Tgr. ρ

0air

Pgr

ρ0

air Tgr Pgr,Mr air( )

R Tgr.

Pgr

Bair Tgr

Bair Tgr .012527 5.91 x10 4 Tgr 6.62 x10 7 Tgr2

MrCH

Mr xCO2 MrCO2 xN2 MrN2

xCH


HCH = molar gross heating value of the equivalent hydrocarbon G1 = -2.709328 G2 = 0.021062199 (Z0) new = compressibility factor for next iteration Bmix = second virial coefficient of mixture (calculated from previous equations) Repeat process, continuously replacing Z0 with (Z0) new, until (Z0/Z0

new-1) is less than the convergence criteria (5x10-11 in double precision or 5x10 -7 in single precision)

Mas

s F

low

Rat

e - A

pp

end

ix E

Z0new 1

Bmix Pgr

R Tgr.

HCH

MrCH G1

G2


Compressibility Symbols and Definitions Symbol Definition B Second virial coefficient Bij Individual component interaction second virial coefficient Bmix Second virial coefficient for the mixture B`nij Binary characterization coefficient Cijk Individual component interaction third virial coefficient Cmix Third virial coefficient for the mixture C`n Coefficient as a function of composition d Molar density (moles per unit volume) D Reduced density Ei Characteristic energy parameter for ith component (Table 5) Eij Second virial coefficient binary energy parameter E ij Second virial coefficient binary interaction parameter (Table 6) F Mixture high temperature parameter Fi High temperature parameter for ith component (Table 5) G Orientation parameter Gi Orientation parameter for ith component (Table 5) Gij Binary orientation parameter G`ij Binary interaction parameter for orientation (Table 6) Gr Relative density (specific value) of mixture HCH Molar gross heating value of the equivalent hydrocarbon HN0 Molar ideal gross heating value at 298.15K and 0.101325 MPa HV Volumetric gross heating value at reference conditions Th, Td, Pd K Mixture size parameter Ki Size parameter of ith component (Table 5) Kij Binary interaction parameter for size (Table 6) Mr Molar mass (molecular weight) of the mixture Mr(air) Molar mass of air, 28.96256 g/mol Mri Molar mass of ith component N Number of components in the gas mixture P Absolute pressure Pd Reference pressure for molar density Pgr Reference pressure for relative density Q Quadrupole parameter Qi Quadrupole parameter for ith component (Table 5) R Gas constant, 8.31451 J/mol-K r0air Mass density of air at reference conditions Tgr, Pgr Si Dipole parameter for ith component (Table 5) T Absolute temperature Td Reference temperature for molar density Tgr Reference temperature for relative density Th Reference temperature for heating value U Mixture energy parameter Uij Binary interaction parameter for conformal energy (Table 6) W i Association parameter for ith component (Table 5) xi Mole fraction of ith component Z Compressibility factor Z0 Compressibility factor at reference conditions Z0

new Compressibility factor for next iteration an, bn, cn, fn, gn, kn, qn, sn, un, wn Constants given in Table 4 b0, b1, b2, c0, c1, c2 Constants given in Table 7 bi0, bi1, bi2, c i0, c i1, ci2 Constants given in Table 8

Mas

s F

low

Rat

e - A

pp

end

ix E


Iterative Process Used to Solve for Coefficient of Discharge

1) Set Re equal to ∞ and solve for Cd 2) Multiply this value for Cd by the invariant A1 to obtain new value of Re: 3) Use new value of Re to solve for new value of Cd

4) Repeat process until:

For example:

If a long radius nozzle (ISO) had values: A1 = 100,000

β = 0.5 1) at Re = ∞: Cd = 0.9965 2) Cd * A1 = 99,650 2) Cd

(1) * A1 = 98,190 Re(1) = 99,650 Re(2) = 98,190 3) at Re(1) = 99,650 3) at Re(2) = 98,190 Cd

(1) = .9819 Cd(2) = .9818

4) (A1 – (Re/Cd))/A1 > 1* 10 -4 4) (A1 – (Re/Cd))/A1 = 1 * 10-4 Repeat from step 2 Therefore, Cd = .9818

Mas

s F

low

Rat

e - A

pp

end

ix F

Appendix F

A1

Ev Yd2 2 ρ f ∆ P..

µ f D

A1Re

Cd

A1

1 10 4.

Cd 0.9965 0.00653 β.5. 106

Re

0.5

.


References AGA 3, Orifice Metering of Natural Gas and Other Related Hydrocarbons, Part 1: General Equations and Uncertainty Guidelines, 3rd ed., American Gas Association, AGA Catalog No. XQ9210, Arlington, VA., 1990. AGA 3, Orifice Metering of Natural Gas and Other Related Hydrocarbons, Part 2: Specification and Installation Requirements, 4th ed., American Gas Association, AGA Catalog No. XQ0002, Arlington, VA., 2000. AGA 3, Orifice Metering of Natural Gas and Other Related Hydrocarbons, Part 3: Natural Gas Applications, 3rd ed., American Gas Association, AGA Catalog No. XQ9210, Arlington, VA., 1992. AGA 3, Orifice Metering of Natural Gas and Other Related Hydrocarbons, Part : Background, Development, Implementation Procedure, and Subroutine Documentation for Empirical Flange-Tapped Discharge Coefficient Equation, 3rd ed., American Gas Association, AGA Catalog No. XQ9211, Arlington, VA., 1992. AGA 8, Compressibility Factors of Natural Gas and Other Related Hydrocarbon Gases, Transmission Measurement Committee Report No. 8, AGA Catalog No. XQ 9212, Arlington, VA., 1992. ASME: ASME International Steam Tables for Industrial Use, American Society of Mechanical Engineers, New York, 2000. ASME Standard MFC-14M-2001, Measurement of Fluid Flow Using Small Bore Precision Orifice Meters, American Society of Mechanical Engineers, New York, 2001. ASME Standard MFC-3M-1989, Measurement of Fluid Flow in Pipes Using Orifice, Nozzle, and Venturi, American Society of Mechanical Engineers, New York, 1989. GPA Standard 2145-00, Table of Physical Constants for Hydrocarbons and Other Compounds of Interest to the Natural Gas Industry, Gas Producers Association, Tulsa, OK., 2000. ISO Standard 5167-1, Measurement of Fluid Flow by Means of Pressure Differential Devices, International Standards Organization, Geneva, 1991. Miller, R. W.: Flow Measurement Engineering Handbook, 3rd ed., McGraw-Hill, New York, 1996.

Mas

s F

low

Rat

e - A

pp

end

ix G

Appendix G


Smar Research Corporation 4250 Veterans Memorial Highway

Holbrook, NY USA 11741 Tel: 631.737.3111 Fax: 631.737.3892

[email protected] www.SmarResearch.com

Smar Research reserves the right to make changes to design and functionality of any product without notice. Smar Research does not assume any liability arising out of the application or use of any product. Smar Research , Technology Source, and the SRC logo are registered trademarks of Smar Research Corporation. The HART, Fieldbus, and Profibus Foundation logos are

trademarks of their respective owners.

mfr-appnotes

Documents