ideal and real discharge coefficients – using fundamental

© RUB, Marquardt

Ideal and Real Discharge Coefficients – Using Fundamental Equations of State in Mass-Flow Measurements with Sonic Nozzles

R. SpanSeptember 2021

R. Span | Ideal and Real Discharge Coefficients | 09/2021

Slide 2

Overview

What is an ideal gas and a perfect gas?

The ideal discharge coefficient

The real discharge coefficient

Fundamental equations of state – how they describe real gases

How to utilize fundamental equations of state to calculate the realdischarge coefficient

Hydrogen – a well-known pure fluid?

This talk addresses only thermodynamic effects – viscous / multidimensional flow effects are completely neglected


Slide 3

What is an Ideal Gas?

Definition: An ideal gas is a gas in which atoms / molecules do not interactwith each other and are mass points (do not have a volume). The only relevant interaction is by elastic hit against walls.

A (simplified) molecular dynamics approach yields the thermal equation of state of the ideal gas: p = pressure

vm = molar volume (= 1/rm)V = VolumeRm = molar gas constant

(8.314662 J/(mol K))n = amount of substance [mol]v = specific volume (= 1/r)R = specific gas constant

(R = Rm / MGas)m = mass

푝 푣 = 푅 푇푝 푉 = 푛 푅 푇

푝 푣 = 푅 푇푝 푉 = 푚 푅 푇


Slide 4

How about Caloric Properties of an Ideal Gas?

• Caloric properties are (among others) enthalpy (h), entropy (s), internal energy (u), heat capacities (cp, cv), in a certain way also speed of sound (w), ….

Caloric properties are essential for an isentropic expansion!• Caloric properties are related to the ability of molecules to store energy

• By definition, ideal gases cannot store energy by molecular interaction (their molecules do not interact)

• Molecules of ideal gases can store energy by translational movement (kinetic energy), rotation and internal vibration

• Translational movement has three degrees of freedom (three directions in 3D volume), contributes cv,trans = 3/2 R ½ R per degree of freedom


Slide 5

How about Caloric Properties of an Ideal Gas?

• Energy stored by rotation can be assigned to degrees of freedom as well• Rotational energy requires a moment of inertia; no rotational energy if there

is no distance between mass point and center of rotation (simplified! actually we are talking quantum physics…)

monoatomic gasno degree of freedom

cv,rot = 0

linear moleculestwo degrees of freedom

cv,rot = 2/2 R

non-linear moleculesthree degrees of freedom

cv,rot = 3/2 R


Slide 6

Isochoric and Isobaric Heat Capacity of an Ideal Gas

• Vibrational modes contribute 2/2 R (2 degrees of freedom) per mode

• As a general relation we find for molecules with n ≥ 2 atoms:

linear molecules:

non-linear molecules:

• A perfect gas is an ideal gas with constant heat capacity

푐 (푇) ≤ 52푅 + 3푛 − 5 푅

푐 (푇) ≤ 62푅 + 3푛 − 6 푅

vibrational contribution,depends on temperature


Slide 7

Isochoric and Isobaric Heat Capacity of an Ideal Gas

• Rule of thumb: The more atoms are contained in a molecule and the heavier the atoms are, the stronger is the temperature dependency of the heat capacity

• Keep this in mind if you deal with complex fluids,the perfect gas assumption is likely questionable!

푐 푇 = 푐 푇 + 푅


Slide 8

The Isentropic Exponent of Ideal and Real Gases

• Some definitions common for the description of flow processes are actually valid only for ideal or even perfect gases

• One important example is the isentropic exponent k

ideal gas

푝 푣 = 푝 푣 = 푐표푛푠푡. = −푐표푛푠푡. 휅 푣 = −휅

휅 =휕푝휕푣

푣푝

휅 =푐푐

real fluid휅 =

휌 푤푝

≠ 푐푐

Erroneous calculations for real fluids may be hidden deep in computer codes!

(= const. only for perfect gas)


Slide 9

CO2

Erroneous calculations for real fluids may be hidden deep in computer codes!

The Isentropic Exponent of Ideal and Real Gases


Slide 10

The Ideal Discharge Coefficient

• Ideal expansion of an ideal gas isentropic relation & constant total enthalpy

푝푣 = const. 푇 = 푇푝푝

ℎ = ℎ 푇 +12 푐 = const. 푐 = 푐 + 2 푐 푇 1 −

푝푝

• At throat conditions of a critical nozzle with c0 = 0 this results in:

푞 , = 휅2

휅 + 1 퐴 푝푅 푇

= 퐶 퐴 푝푅 푇

perfect gas푐 = const.

perfect gas휅 = const.


Slide 11


• The ideal isentropicexponent of (normal)hydrogen is a function of temperature

• Rotational modes are not fully excited at low temperature


Slide 12


• The resulting ideal discharge coefficient is a function of temperature as well

• To assume a constant value may imply an error of 0.5% in the relevant tempera-ture range


Slide 13


• Ideal expansion of an ideal gas isentropic relation & constant total enthalpy

푝푣 = const. 푇 = 푇푝푝

ℎ = ℎ 푇 +12 푐 = const. 푐 = 푐 + 2 푐 푇 1 −

푝푝

• At throat conditions of a critical nozzle with c0 = 0 this results in:

푞 , = 휅2

휅 + 1 퐴 푝푅 푇

= 퐶 퐴 푝푅 푇

perfect gas푐 = const.

h h(T,p) ideal gas

휌 = 푝푅푇 ideal gas

perfect gas휅 = const.


Slide 14

The Real Discharge Coefficient

• The real discharge coefficient is defined as:

푞 , = 퐶 퐴 푝푅 푇

= 휌 푐푅 푇푝

퐴 푝푅 푇

= 퐴 휌 푐

퐶 = 휌 푐푅 푇푝

t = at throat conditions, speed of sound and density at throat temperature and pressure


Slide 15


푇 = 푇푝 = 푝


Slide 16


푇 = 푇푝 = 푝


Slide 17




= 휌 푐푅 푇푝

퐴 푝푅 푇

= 퐴 휌 푐



• Density and speed of sound at throat conditions need to be known• Problem: Temperature and pressure at throat conditions need to be known

Iterative solution based on a consistent description of density, speed of sound, enthalpy and entropy as a function of temperature and pressure


Slide 18

What is a Fundamental Equation of State?

Hypothesis: The second law of thermodynamics yields acomplete description of equilibrium states

A complete description of equilibrium states should be possible in terms of the independent variables introduced by the second law of thermodynamics

For pure fluids this approach results in

• Closed systems:

• Flow processes:

• Entropy cannot be measured directly, has an arbitrary reference state Nobody wants an equation of state that uses entropy as an independent variable

푇푑푠 = 푑푢 + 푝푑푣 ⇒ 푑푢 = 푇푑푠 − 푝푑푣 ⇒ 푢 = 푢(푠, 푣)

푇푑푠 = 푑ℎ − 푣푑푝 ⇒ 푑ℎ = 푇푑푠 + 푣푑푝 ⇒ ℎ = ℎ(푠, 푝)


Slide 19

What is a Fundamental Equation of State?

• Legendre transformation allows for change of variables

• Closed systems:

with the Helmholtz (free) energy

• Flow processes:

with the Gibbs (free) enthalpy

• In engineering calculations typical combinations of independent variables are T,p direct calc. from g(T,p), one dim. iteration from a(T,v)p,h one dim. iteration from g(T,p), two dim. iteration from a(T,v)p,s one dim. iteration from g(T,p), two dim. iteration from a(T,v)

g(T,p) seems to be the right choice for engineering applications

푢 = 푢(푠, 푣) ⇒ 푎 = 푎(푇, 푣)푎(푇, 푣) = 푢 − 푇푠

ℎ = ℎ(푠, 푝) ⇒ 푔 = 푔(푇, 푝)푔(푇, 푝) = ℎ − 푇푠


Slide 20

Which one is the Right Fundamental Equation of State?

• To allow for consistent calculations across phase boundaries and between sub- and supercritical states, it is important to describe the whole fluid surface by a single mathematical expression

• In terms of temperature and pressure, there is a discontinuity at the phase boundary

• Gibbs energy allows for fast calculations but does not allow for a closed description of the whole fluid surface

• Closed description becomes possible in terms of temperature and density, saturated liquid and vapor density are distinguishable

Helmholtz energy (free energy)


Slide 21

Empirical Fundamental Equations of State• Empirical fundamental EOS are usually formulated in terms of the free (Helmholtz)

energy to be able to describe the whole fluid surface with a single formulation• Reduced properties are preferred• The ideal gas behavior can (in principle) be derived from spectroscopic data and quantum

mechanics• The residual behavior of the real fluid needs to be characterized by empirical fits to

experimental data The reduced free (Helmholtz) energy is usually written as a sum of ideal and residual

contribution:훼 훿, 휏 = 훼 훿, 휏 + 훼 훿, 휏 =

푎 푇, 푣푅푇

+푎 푇, 푣푅푇

with 훿 =푣red.푣

=휌

휌red. and 휏 =

푇red.푇

common definitions are 휌red. = 휌 and 푇red. = 푇


Slide 22

The Ideal Part of Helmholtz Equations

• The ideal part of the reduced free (Helmholtz) energy can be derived from an equation for cv

o(T) or cpo(T)

• Equations for cvo(T) or cp

o(T) usually contain polynomial and / or “Planck-Einstein” terms

푐 (푇)푅

= 푛translation

and rotation

+ 푛 휏

polynomials in

+ 푚 휗 휏푒

푒 − 1

Planck Einst. terms

= −휏휕 훼휕휏

• Twofold integration yields

푎 휏, 훿 = 푐 + 푐 휏integration constants

from and

+ 푛 ln 휏constant

in

−푛

푡 푡 − 1⇌ 휏

polynomials in

+ 푚 ln 1 − 푒

Planck Einst. terms

+ ln훿density dependence

of entropy


Slide 23

The Residual Part of Helmholtz Equations• The residual part of the reduced free (Helmholtz) energy is fitted to experimental data• The “functional form” of the residual part is essential!• Too few coefficients do not allow for accurate fits• Too many and intercorrelating terms result in numerical instability, poor extrapolation

• Most reference EOS introduce additional, more complex terms for an improved description of the critical region

훼 휏, 훿 = 푛 훿 휏 + 푛 훿 휏 exp −훿 + special terms

functional form

coefficients


Slide 24

Empirical Fundamental Equations of State

• Properties can be calculated by combining derivatives with respect to the reduced temperature and density

푧 =푝푣푅푇

= 1 + 훿훼

푢푅푇

= 휏 훼 + 훼

ℎ푅푇

= 1 + 휏 훼 + 훼 + 훿훼

푐푅

= −휏 훼 + 훼

푐∗

푅푇= 1 + 2훿훼 + 훿 훼 −

(1 + 훿훼 − 훿휏훼 )휏 훼 + 훼

푠푅

= 휏 훼 + 훼 − 훼 − 훼

with 훼 =휕훼휕훿

, . . .

푔푅푇

= 휏 훼 + 훼Phase equilibrium! 푔 푇, 휌 = 푔(푇, 휌 )


Slide 25




= 휌 푐푅 푇푝

퐴 푝푅 푇

= 퐴 휌 푐



• Solve

iteratively for s(Tt, pt) = s(T0, p0)

Tt, pt are known, rt, ct, and Cr can be calculated (or qm,s can be calculated directly)

ℎ 푇 ,푝 +12 푐

∗ 푇 ,푝 = ℎ 푇 , 푝 +12 푐


Slide 26




= 휌 푐푅 푇푝

퐴 푝푅 푇

= 퐴 휌 푐



• Solve

iteratively for s(Tt, pt) = s(T0, p0)

Tt, pt are known, rt, ct, and Cr can be calculated (or qm,s can be calculated directly)

ℎ 푇 ,푝 +12 푐

∗ 푇 ,푝 = ℎ 푇 , 푝 +12 푐

Ding et al. (2014) report good results for this method.However, comparisons mainly with CFD, only very fewexperimental results for comparison. Shown deviations

are too large for reference measurements.H. Ding, C. Wang, Y. Zhao: Flow characteristics of hydrogen through a

critical nozzle. Int. J. Hydrogen Energy, 39, 3947-3955 (2014).

= q m

/qm

,i

0.5%


Slide 27

Hydrogen – A Well-Known Pure Fluid?• Hydrogen is a mixture of para- and orthohydrogen

• Equilibrium composition is temperature dependentT > 200 K xpara 0.25 / xortho 0.75 normal hydrogenT < 20 K xpara 1.00 / xortho 0.00

• Equilibration is slow unless catalysts are used to accelerate it

• Even with catalysts equilibrium cannot be taken for granted in processes

• Enthalpy difference of para/ortho-conversion is larger than enthalpy of evaporation

A reference model for hydrogen should …

… accurately describe pure and arbitrary (equilibrium and non equilibrium) mixtures of para- and orthohydrogen

… consistently reflect the enthalpic effect of the para/ortho-conversion


Slide 28

Hydrogen – A Well-Known Pure Fluid?

parahydrogen carbon dioxide

Representation of selected density-data for

• Deviations to experimental data are at least one order of magnitude higher for H2


Slide 29


• Deviations to experimental data are at least one order of magnitude higher for H2

Representation of selected density-data for

nitrogennormal hydrogen


Slide 30


• The data base for hydrogen was established in the 1950’s to 1970’s

• Only few measurements in the 1980’s and 1990’s, mostly at pipelining conditions


Slide 31

Density and Speed of Sound Measurements

• Beside phase-equilibrium data, data for density and speed of sound are crucialfor the development of (highly) accurate thermodynamic property models

• None of the “state-of-the-art” measuring principles has been applied for hydrogen

• None of the existing apparatuses is able to measure at liquid-hydrogen temperatures

© RUB, Marquardt

Ideal and Real Discharge Coefficients – Using Fundamental Equations of State in Mass-Flow Measurements with Sonic Nozzles

Thank you for your patience!

R. SpanSeptember 2021

This presentation was part of the

Workshop on the state-of-the-art regarding

Critical Flow Venturi Nozzles

From the MetHyInfra Project

This project “Metrology infrastructure for high-pressure gas and liquified hydrogen flows” (MetHyInfra) has received funding from the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation programme under Grant agreement No [20IND11] .

MetHyInfra Team

______________________________________

Division 1 Mechanics & Acoustics

Dep. 1.4 Gas Flow

Physikalisch-Technische Bundesanstalt (PTB)

Bundesallee 100, DE-38116 Braunschweig (Germany)

https://www.methyinfra.ptb.de/

www.linkedin.com/in/methyinfra-an-empir-project-b8442321a

Tel: +49 531 592 1121/1335(Franziska/Hans-Benjamin)

Fax: +49 531 592 69 1121/1335

E-Mail: [email protected]

https://www.methyinfra.ptb.de/

http://www.linkedin.com/in/methyinfra-an-empir-project-b8442321a

mailto:[email protected]

ideal and real discharge coefficients – using fundamental

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