ES 202Fluid and Thermal Systems

Lecture 19:Models Versus General Substances

(1/27/2003)

Lecture 19 ES 202 Fluid & Thermal Systems 2

Assignments

• Homework:– 7-62, 7-63 in Cengel & Turner

• Reading assignment– 7-4, 7-5 and 7-6 in Cengel & Turner– ES 201 notes

Lecture 19 ES 202 Fluid & Thermal Systems 3

Announcements

• Homework due today at 5 pm in my office

• Check the revised course syllabus on the course web page again. There are more changes.

Lecture 19 ES 202 Fluid & Thermal Systems 4

Future Quieter Airplane!!

Sawtooth geometry (chevron) in engine exhaust nozzle is shown to reduce engine noise.

primarystream

secondary(bypass)stream

Lecture 19 ES 202 Fluid & Thermal Systems 5

Road Map of Lecture 19• Quiz on Week 6 materials

• Real gas versus ideal gas– notion of reduced coordinate– definition of compressibility factor– Z-chart

• Ideal gas model – change in specific internal energy and specific enthalpy– change in specific entropy– Gibbs equation and its interpretation– variation of specific heats

Lecture 19 ES 202 Fluid & Thermal Systems 6

Quiz on Week 6 Materials• Indicate in the following cases whether the given information is sufficient or

insufficient in fully determining the thermodynamic state of the substance:– pressure and temperature in compressed liquid (YES)– pressure and temperature in superheated vapor (YES)– pressure and temperature in saturated mixture (NO)– pressure and temperature in saturated vapor (YES)– pressure and specific volume in saturated liquid (YES)– pressure and specific entropy in saturated mixture (YES)– temperature and specific enthalpy in superheated vapor (YES)– quality and temperature in saturated mixture (YES)

• Given the following limited data from a property table of water at a pressure of 2 MPa:

– h = 3023.5 kJ/kg at T = 300 deg C– h = 3137.0 kJ/kg at T = 350 deg C

What is h at T = 330 deg C and P = 2 MPa? (h = 3091.6 kJ/kg)

Lecture 19 ES 202 Fluid & Thermal Systems 7

Quiz on Week 6 Materials (Cont’d)• According to the Compressed Liquid Approximation, how are the following

thermodynamic properties approximated in the compressed liquid region:

• Sketch two constant pressure curves (P = P1, P = P2 with P1 < P2) on the T-v diagram. Indicated clearly their behavior in the two-phase region and label them clearly.

• Sketch two constant temperature curves (T = T1, T = T2 with T1 < T2) on the P-v diagram. Indicated clearly their behavior in the two-phase region and label them clearly.

)(),( TuPTu f

)(),( TvPTv f

)(),( TsPTs f

)()()()()(),( TvTPPThTPvTuPTh fsatfff

Lecture 19 ES 202 Fluid & Thermal Systems 8

Quiz on Week 6 Materials (Cont’d)

P 2

P 1

T-v diagram P-v diagram

T2

T1

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Real Gas Versus Ideal Gas• Recall ideal gas as a simplified (yet powerful) model for real gas

behavior

• Its original derivation assumes negligible mutual interaction between gas molecules. Hence, it is expected to work well for gases under low pressure.

• But, the next logical question will be: “How low is low?” or “Against what standard is low pressure measured with respect to?”

• To answer this question, we need to recall the phase diagrams of a general substance.

Lecture 19 ES 202 Fluid & Thermal Systems 10

Two-phasedome

Superheatedvapor

Comp.liquid Saturated vapor line

Satu

rate

d liq

uid

line

Critical Point

Vapor

Liquid

Solid

Sublimati

on

Vaporiz

ation

Mel

ting Critical Point

Critical State and Reduced Coordinate

• Recall the phase diagrams of a general substance:

• Base on the thermodynamic properties associated with the critical point, a non-dimensional reduced coordinate (a group) can be defined for each substance:

crR P

PP reduced pressure:

crR T

TT reduced temperature:

,

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Compressibility Chart

(Taken from Figure 3-56 in Cengel & Turner)

TR

vPZ

Compressibility Factor:

Ideal Gas:

1Z

Good for:• low pressure• high temperature

critical point

Ideal Gas

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• In general,

• For an ideal gas, the specific internal energy (u) , hence, specific enthalpy (h) are functions of temperature only.

• For an ideal gas, the change in specific internal energy and specific enthalpy can be simplified as:

Revisit Ideal Gas Specific Heats

),( PThh dPP

hdT

T

hdh

T

c

P

p

),( vTuu dvv

udT

T

udu

T

c

v

v

Definition of cv

Definition of cp

dTcdu v dTcdh p ,

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Entropy Variation in Ideal Gas• Introduce the Gibbs equation for a general substance:

• Interpretation:

– for a simple compressible system,

• For an ideal gas, the Gibbs equation reduces to a simpler form.

PdvduTds

vdPdhTds or Pvuh

revint,

T

qds

revint,qTds

revint,wPdv

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Variation in Specific Heats• In general, the specific heats (cv, cp) are NOT true constants. They vary

(increase) slightly with temperature even for ideal gases.

• Afterall, it is the change in properties that matters (their absolute values depend on the chosen reference state.)

• For an ideal gas with finite temperature change:

• Different ways to approximate the integrals:– direct integration (cv and cp as functions of T)– divide and conquer– “average” specific heats

dTcduu v dTcdhh p ,

v

dvR

T

dTcdss v

P

dPR

T

dTcs p or

} geometrical interpretation