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MOVING BOUNDARY WORK

Moving boundary work (P dV work): The expansion and compression work in a piston-cylinder device.

Quasi-equilibrium process: A process during which the system remains nearly in equilibrium at all times.

Wb is positive for expansion

Wb is negative for compression

The work associated with a moving

boundary is called boundary work.

A gas does a differential amount of work Wbas it forces the piston to move by a differential amount ds.

The boundary work done during a process depends on the path followed as well as the end states.

The area under theprocess curve on a P-Vdiagram is equal, in magnitude, to the work done duringa quasi-equilibrium expansion or compression process of a closed system.

Polytropic, Isothermal, and Isobaric processes

Polytropic process: C, n (polytropic exponent) constants

Polytropic process

Polytropic and for ideal gas, n ≠ 1

When n = 1

Constant temp. process (isothermal )

Constant pressure process (isobaric)

What is the boundary work for a constant-volume process?

Schematic and P-V diagram for a polytropic process.

Boundary Work for a Constant-Volume Process

A rigid tank contains air at 500 kPa and 150˚C. As a result of heat transfer to the surroundings, the temperature and pressure inside the tank drop to 65˚C and 400 kPa, respectively. Determine the boundary work done during this process.

Wb= 0 kJ

Boundary Work for a Constant-Pressure Process

A frictionless piston-cylinder device contains 5 kg of steam at 400 kpa and 200˚C. Heat is now transferred to the steam until the temperature reaches 250˚C. If the piston is not attached to a shaft and its mass is constant, determine the work done by the steam during this process.

Wb= 122 kJ

For a constant-pressure expansion or compression process:

Energy balance for a constant-pressure expansion or compression processGeneral analysis for a closed system undergoing a quasi-equilibrium constant-pressure process. Q is to the system and W is from the system.

An example of constant-pressure process

SPECIFIC HEATS

Specific heat at constant volume, cv: The energy required to raise the temperature of the unit mass of a substance by one degree as the volume is maintained constant.

Specific heat at constant pressure, cp: The energy required to raise the temperature of the unit mass of a substance by one degree as the pressure is maintained constant.

True or False?

cp is always greater than cv

Constant-volume and constant-pressure specific heats cvand cp

(values are for helium gas).

The equations in the figure are valid for any substance undergoing any process.

- cvand cpare properties.
- cvis related to the changes in internal energy and cpto the changes in enthalpy.
- A common unit for specific heats is kJ/kg·°C or kJ/kg·K. Are these units identical?

Formal definitions of cvand cp.

INTERNAL ENERGY, ENTHALPY,AND SPECIFIC HEATS OF IDEAL GASES

Joule showed using this experimental apparatus that u=u(T)

Internal energy and enthalpy change of an ideal gas

For ideal gases, u, h, cv, and cpvary with temperature only.

At low pressures, all real gases approach ideal-gas behavior, and therefore their specific heats depend on temperature only.

- The specific heats of real gases at low pressures are called ideal-gas specific heats, or zero-pressure specific heats, and are often denoted cp0 and cv0.

- u and h data for a number of gases have been tabulated.
- These tables are obtained by choosing an arbitrary reference point and performing the integrations by treating state 1 as the reference state.

Ideal-gas constant-pressure specific heats for some gases (see Table A–2c for cpequations).

In the preparation of ideal-gas tables, 0 K is chosen as the reference temperature.

Internal energy and enthalpy change when specific heat is taken constant at an average value

(kJ/kg)

Three ways of calculating u and h

- By using the tabulated u and h data. This is the easiest and most accurate way when tables are readily available.
- By using the cvor cprelations (Table A-2c) as a function of temperature and performing the integrations. This is very inconvenient for hand calculations but quite desirable for computerized calculations. The results obtained are very accurate.
- By using average specific heats. This is very simple and certainly very convenient when property tables are not available. The results obtained are reasonably accurate if the temperature interval is not very large.

Three ways of calculating u.

Specific Heat Relations of Ideal Gases

The relationship between cp, cv and R

dh = cpdT and du = cvdT

On a molar basis

Specific heat ratio

- The specific ratio varies with temperature, but this variation is very mild.
- For monatomic gases (helium, argon, etc.), its value is essentially constant at 1.667.
- Many diatomic gases, including air, have a specific heat ratio of about 1.4 at room temperature.

The cpof an ideal gas can be determined from a knowledge of cvand R.

Evaluation of the ∆u of an Ideal Gas

Air at 300 K and 200 kPa is heated at constant pressure to 600 K. Determine the change in internal energy of air per unit mass, using (a) data from the air table (Table A-17), (b) the functional form of the specific heat (Table A-2c), and (c) the average specific heat value (Table A-2b).

- ∆u = 220.71 kJ/kg
- (b) ∆u = 222.5 kJ/kg
- (c) ∆u = 220 kJ/kg

Heating of a Gas in a Tank by Stirring

An insulated rigid tank initially contains 0.7 kg of helium at 27˚C and 350 kPa. A paddle wheel with a power rating of 0.015 kW is operated within the tank for 30 min. Determine (a) the final temperature and (b) the final pressure of the helium gas.

- T2= 39.4˚C
- (b) P2 = 364 kPa

Heating of a Gas at Constant Pressure

A piston-cylinder device initially contains air at 150 kPa and 27˚C. At this state, the piston is resting on a pair of stops, as shown in Fig. 4-32, and the enclosed volume is 400 L. The mass of the piston is such that a 350-kPa pressure is required to move it. The air is now heated until its volume has doubled. Determine (a) the final temperature, (b) the work done by the air, and (c) the total heat transferred to the air.

- T3= 1400 K
- (b) W13 = 140 kJ
- (c) Qin = 767 kJ

INTERNAL ENERGY, ENTHALPY, ANDSPECIFIC HEATS OF SOLIDS AND LIQUIDS

Incompressible substance: A substance whose specific volume (or density) is constant. Solids and liquids are incompressible substances.

Internal Energy Changes

Enthalpy Changes

The enthalpy of a compressed liquid

Usually a more accurate relation than

Cooling of an Iron Block by Water

A 50-kg iron block at 80˚C is dropped into an insulated tank that contains 0.5 m3 of liquid water at 25˚C. Determine the temperature when thermal equilibrium is reached.

T2= 25.6˚C

Heating of Aluminum Rods in a Furnace

Long cylindrical aluminum rods (ρ = 2700 kg/m3 and cp = 0.973 kj/kg.K) of 5-cm diameter are heat treated from 20˚C to an average temperature of 400˚C by drawing them at a velocity of 8 m/min through a long oven. Determine the rate of heat transfer to the rods in the oven.

Summary

- Moving boundary work
- Wb for an isothermal process
- Wb for a constant-pressure process
- Wb for a polytropic process
- Specific heats
- Constant-pressure specific heat, cp
- Constant-volume specific heat, cv
- Internal energy, enthalpy, and specific heats of ideal gases
- Specific heat relations of ideal gases
- Internal energy, enthalpy, and specific heats of incompressible substances (solids and liquids)

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