mtx311 lecture 3 - real gas volume

MTX 311 Thermodynamics Dr Axel Lexmond Lecture 4 2013

MTX 311 Thermodynamics Dr Axel Lexmond Lecture 3

Real Gases


ideal gas

Molecules have finite mass but NO finite volume No intermolecular forces

P~n/V mv² T~mv²

P~nT/V (P=nRT/V) (PV=mRT)


ideal gas


P~n/V mv² T~mv²

P~nT/V (P=nRT/V) (PV=mRT)

Δ𝑕 = Δ(𝑢 + 𝑃𝑣)=Δ 𝑢 + 𝑅𝑇 = 0

Intermolecular forces are 0: internal energy is not a function of pressure or volume

Constant temperature Compression:

Δ𝑢 = − 𝐹𝑖𝑛𝑡𝑒𝑟𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟𝑑𝑥

Compression:


ideal gas


Constant temperature Compression:

S~n lnV s~ lnV ~ − ln P

PV=RT


“ideal” liquid

Molecules have finite mass and finite volume Packing density depends on temperature but independent from external forces

Δ𝑕 = Δ 𝑢 + 𝑃𝑣 = Δ 𝑃𝑣 = 𝑣Δ𝑃

Compression: Δ𝑢 = − 𝐹𝑖𝑛𝑡𝑒𝑟𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟𝑑𝑥 = −𝐹𝑖𝑛𝑡𝑒𝑟𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 ∙ 0 = 0

Incompressible fluid: V= V(T) ≠ V(P)


What happens in a real gas?

Molecules have finite volume (results in larger specific volume than predicted by ideal gas) Molecules attract each other (results in smaller specific volume than predicted by ideal gas

Small molecules at ambient conditions (air, water…) Average distance is about 10x diameter of the molecules


What happens in a real gas?

Finite volume of molecule

P(V-V0)=RT

s~ ln(V − V0)

Higher pressures (at given T,V)

Lower entropy (at given T,V)

Molecular attraction

h and u are not affected

Molecule decelerates when getting out of the gas (to the wall)smaller force on the wall lower pressure

Entropy is not affected

h and u will be smaller than h and u at 0 pressure

Δ𝑢 = − 𝐹𝑖𝑛𝑡𝑒𝑟𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟𝑑𝑥 < 0

At very high pressure, molecules will start touching (repelling) each other h and u will increase


real gases

Which relationships do NOT work any more: PV=RT Polytropic relationships u=u(T), h=h(T)

Real gas

V=V(P,T)

h=h(P,T) u=h-PV=u(P,T)

V=RT/p

h=h(T)=h0+Cp(T-T0)


How to describe a non-deal gas?

Experimental data Tables graphs Substance specific correlations Computer programs (e.g. Aspen)

Generic equations F=Fig+ Fcf (Pr , Tr) (enthalpy, entropy) F=F(Fig, Pr , Tr) (sp volume) ig = ideal gas cf correction factor Pr= reduced pressure (Pr=P/ Pc) Tr=reduced temperature (Tr=T/ Tc)


𝑃𝑣 = 𝑍𝑅𝑇

Z =𝑃𝑣

𝑅𝑇

Many ways to describe a real gas

• Tables of measured data • digital databases • Substance-specific correlations: virial equations General equations that use only critical temperature and pressure Redlich and Kwong compressibility factors (reduced coordinates) van der Waals

Typical errors

<±1%

Up to ±25%

• Build your own general equations based on reduced data from comparable substances

• “ interpolation” between substances by digital databases

±5%

Might be difficult to distinguish!

• •

• •

Limited accuracy

MTX 311 Thermodynamics Dr Axel Lexmond Lecture 3 Do you need correction?

You need correction when H, S and PVT-correlations deviate significantly for m ideal gas correlations IF one of these quantities deviates, usually all three do. Method: check PVT (because that is simple) If PVT data follows ideal gas, you can assume that enthaly and entropy also will follow ideal gas correlations


Method 1 use the reduced temperature and the reduced pressure

𝑇𝑟 =𝑇

𝑇𝑐

𝑃𝑟 =𝑃

𝑃𝑐

When Tr>2 OR Pr<0.2, you can assume ideal gas


Method 2 use the density / specific volume

Look up the density of the liquid Divide this by the actual density of the fluid.

𝜌𝑙𝜌> 25 → 𝑠𝑚𝑎𝑙𝑙 ∕ 𝑛𝑜 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛𝑠 𝑓𝑟𝑜𝑚 𝑖𝑑𝑒𝑎𝑙 𝑔𝑎𝑠

𝜌𝑙𝜌< 8 → 𝑚𝑖𝑔𝑕𝑡 𝑏𝑒 𝑙𝑎𝑟𝑔𝑒 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛𝑠

When T≈2Tc, positive deviation (from the size of the molecules) and negative deviations

(resulting from attractive forces) are equally large. This means that there is no net deviation This temperature is called the Boyle temperature of the gas. At higher temperatures, deviations from molecular volume dominate (Z>1, at lower temperatures, attractive forces dominate (Z<1)


Method 3 use the reduced temperature and the reduced pressure

𝑇𝑟 =𝑇

𝑇𝑐

𝑃𝑟 =𝑃

𝑃𝑐

Look up compressibility factor (Z) in table D1 When Z is between 0.95 and 1.05, you can assume small (no) deviations from ideal gas When Z is smaller than 0.9 or larger than 1.1, you can assume large deviations

MTX 311 Thermodynamics Dr Axel Lexmond Lecture 3 Compressibility factor

Z1 when Pr0 Z1 when Tr Tboyle (~ 2,5 )


Method 4 use PVT tables

Look up the specific volume in the table Estimate the specific volume suing ideal gas law:

Vig =𝑅𝑇

𝑃

Z=V/Vig

When Z is between 0.95 and 1.05, you can assume small (no) deviations from ideal gas When Z is smaller than 0.9 or larger than 1.1, you can assume large deviations


Z =𝑃𝑣

𝑅𝑇

Virial equation

Z = 1 +𝐵(𝑇)

𝑣 +𝐶(𝑇)

𝑣2+𝐷(𝑇)

𝑣3+⋯

Z = 1 + 𝐵 𝑇 𝑃 + 𝐶 𝑇 𝑃2 + 𝐷 𝑇 𝑃3 +⋯

Not very good near the critical point Constants are quite often presented in non-SI units (e.g. notes)


𝑷 +𝒂

𝒗𝟐𝒗 − 𝒃 = 𝑹𝑻

v/d Waals equation

Due to intermolecular forces Due to molecular volume

a and b can be calculated from critical point(𝑑𝑃

𝑑𝑣= 0;

𝑑²𝑃

𝑑𝑣²= 0):

𝑇𝑐 =8𝑎

27𝑅𝑏 𝑃𝑐 =

𝑎

27𝑏²

Advantages: intuitive model Disadvantage: limited accuracy


𝑷 +𝒂

𝒗𝟐𝒗 − 𝒃 = 𝑹𝑻

v/d Waals equation In reduced form

𝑇𝑐 =8𝑎

27𝑅𝑏

𝑃𝑐 =𝑎

27𝑏²

𝑉𝑐 = 3𝑏

Big advantage: only critical constant necessary Disadvantage: not very accurate

MTX 311 Thermodynamics Dr Axel Lexmond Lecture 3 Tomorrow

Handout, problems 10.6

We want to calculate the amount of air (nitrogen) which is present in a 50l diving tank at 250 bar, 30°C Do we need to account for non-ideality? What will be the specific volume?

mtx311 lecture 3 - real gas volume

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