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Lec 7: Property tables, ideal and real gases

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• For next time:– Read: § 3-8 to 3-12– HW4 due

• Outline:– EES– Quality, internal energy, enthalpy– Real gases

• Important points:– How to use the quality to find properties of mixtures– How to evaluate a given process in a property

diagram– How to calculate and apply corrections to the IGL for

real gases

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TEAMPLAY

• Find, for water, the following properties: the saturation pressure at a saturation temperature of 100 F.

• and find the saturation temperature at a pressure of 6 MPa.

• Make sure everyone in your group understands how to do this.

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Quality

• We often represent the relative amount of vapor present by something called the quality x.

total

vapor

liquidvapor

vapor

m

m

m m

mx

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Quality is related to the horizontal differences of P-V and T-v

diagrams

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What is the new v?

m

V v

tot

vap

m

mx

tot

liq

m

mx1

m

V V vapliq

m

mvmv vapgliqf

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So...

v = (1-x)vf + xvg = vf + x(vg - vf)

or, writing

vfg vg – vf

v = vf + xvfg

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Obtaining u in the vapor dome

• What you do for v works for u (and for other things)

u = (1-x)uf + xug = uf + x(ug - uf)

= uf + xufg

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A new property: enthalpy, H

• Enthalpy is simply the sum of the internal energy, U, and the pressure volume product, pV

• H U + pV

• Now, pvu

m

Vp

m

Uh

m

H

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Enthalpy--the bottom line

• H = U + pV

• h = u + pv

h = (1-x)hf + xhg = hf + x(hg - hf)

= hf + xhfg

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The P-V or P-v plane

• For the next few lectures we will often look at the two dimensions P and v, or P and V.

• The P is always on the ordinate and the v is always on the abcissa, just opposite to the familiar x-y plane.

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P

v

Superheated Region - 100% Vapor

Two-Phase or Saturation Region - gas and liquid

coexist

Subcooled or Compressed Liquid Region

Sat. Vapor Line

Sat. Liquid Line

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p

v

Table Page

A-4 936/7

A-5 938/9

A-4E 982/3

A-5E 983/4

saturation region

P=Psat and T=Tsat

Table Page

A-6 940/43

A-6E 986/88

If T=Tsat, P Psat

If P=Psat, T >Tsat

superheated region

Table Page

A-7 944

A-7E 989

If T=Tsat PPsat.

If P=Psat, TTsat.

subcooled or compressed liquid region

For water

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Superheat tables--compressed liquid tables are similar

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TEAMPLAYTEAMPLAY

Complete the table below as a team. The substance is water. Make sure everybody understands how to do it!

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TEAMPLAYTEAMPLAY

A container holds 1 kg of liquid water and 1 kg of steam in equilibrium at 0.7 MPa.

1. What is the temperature of the mixture in C?

2. If we hold the pressure constant and increase the temperature to 320 C, what is the change in volume?

3. Show the process on the Pv diagram.

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Sample Problem

Compare the values of the specific volume of water at saturated liquid conditions and 100C with the values of specific volume at of water at saturated liquid conditions and 100C and the following pressures:

5, 20, and 30 MPa.

What conclusions can be drawn from the comparison?

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SOLUTION

From compressed liquid tables

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What can we learn?

• Specific volume is approximately constant over large changes in pressure if T=C

• Liquid does not change specific volume significantly as pressure is changed…it can’t be “compressed”

• When compressed liquid tables are not available, estimate property data at sat. liquid conditions at the same temperature as the compressed liquid.

IMPORTANT!!

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Consider R-134a (Refrigerant 134a)

• We can make a diagram for this as we did for water but there is no data in the compressed or subcooled liquid region.

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P

v

Table Page

A-11 (T) 948

A-11E(T) 993

A-12 (P) 949

A-12E(P) 994

saturation region

P=Psat and T=Tsat

Table Page

A-13 950/1

A-13E 995/6

If T=Tsat, PPsat

If P=Psat, T>Tsat

superheated region

For R-134a

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R-134a IS NOT AN

IDEAL GAS!

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TEAMPLAYTEAMPLAY

• What is the specific volume of saturated liquid R-134a @ 0 °C?

• What is the internal energy at -10 °C and 0.14 MPa?

• What is the pressure at x = 0.1 and 0 °F?

• What is h at the same conditions?

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TEAMPLAYTEAMPLAY

Under what conditions is it appropriate to apply the ideal gas equation of state?

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Ideal gas “law” is a simple equation of state

RTPv

M

RR u

Ru = universal gas constant

= 8.3144 kPa•m3/kmol•K

= 1.545 ft • lbf/lbmol•R

mRTPV

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Molar mass or molecular weight is sometimes confusing

Take air as an example:

units SIin kmol

kg97.28Mair

units USCSin lbmol

lb97.28M m

air

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m

u

lb 28.97

lbmolx

Rlbmol

Btu 1.986 =

MR = R

Rlb

Btu0686.0R

m

Specific gas constant

Universal gas constant given inside back cover of your textbook

Specific gas constant calculated by dividing universal gas constant by molar mass

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Percent error for applying ideal gas equation of state to steam

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IN GENERALIN GENERAL

STEAM should not beSTEAM should not betreated as AN IDEALtreated as AN IDEAL

GAS!GAS!

An exception is waterAn exception is watervapor in the atmospherevapor in the atmosphere

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Real gases

• Pv = ZRT, or

• Pv = ZRuT, where v is volume per unit mole.

Z is known as the compressibility factor

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Principle of corresponding states

“The compressibility factor Z is the same for all gases at the same values of the reduced temperature and reduced pressure.”

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Compressibility factor

• What is it really doing?

• It accounts mainly for two things:– Molecular structure– Intermolecular attractive forces

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Reduced properties

cR

cR T

TT and

P

PP

PR and TR are reduced values.

Pc and Tc are critical properties.

Where:

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Where do you find critical properties?

• Look in the appendices of your text book.• For the SI system they are on p. 930 in

Table A-1, along with molar mass.• For USCS system, they are on p. 976 in

Table A-1E

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Reduced properties

• This works great if you are given a gas, a P and a T and asked to find the v.

• However, if you are given P and v and asked to find T (or T and v and asked to find P), you can use the pseudoreduced volume.

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Reduced properties

• In those cases use the pseudoreduced volume:

c

c

ccR RT

vP

/PRT

vv

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Compressibility factor

• It is shown in Figure 3-56 (p. 100) in terms of actual experimental data

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Compressibility factor for ten substances

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TEAMPLAYTEAMPLAY

Use the compressibility factor to determine the error in treating oxygen gas at 160 K and 3 MPa as an ideal gas.