Thermodynamics I 1

Chapter 4

Energy Analysis of Closed Systems

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SPECIFIC HEATSSpecific 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.

Specific heat is the energy required to raise the temperature of a unit mass of a substance by one degree in a specified way.

Constant-volume and constant-

pressure specific heats cv and cp

(values are for helium gas).

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• The equations in the figure can be proven using 1st law of thermodynamics

• They valid for any substance undergoing any process.

• cv is related to the changes in internal energy and cp to the changes in enthalpy.

• cv and cp are properties.• A common unit for specific heats is

kJ/kg · °C or kJ/kg · K. Are these units identical?

• The specific heats are sometimes given on a molar basis. They are then denoted by and and have the unit kJ/kmol · °C or kJ/kmol · K

The specific heat of a substance changes with temperature.

True or False? cp is always greater than cv.

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Important note:The above equations are property relations and such are independent of the type of the process.

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INTERNAL ENERGY, ENTHALPY,AND SPECIFIC HEATS OF IDEAL GASES

It has been demonstrated mathematically and experimentally that for an ideal gas the internal energy is a function of the temperature only.

We also know for an ideal gas:

Thus for an ideal gas the enthalpy is a function of the temperature only:

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INTERNAL ENERGY, ENTHALPY,AND SPECIFIC HEATS OF IDEAL GASES

𝐶𝑣=𝑑𝑢𝑑𝑇

𝐶𝑣=𝐶𝑣 (T )

𝑑𝑢=𝐶𝑣dT

∆𝑢=𝐶𝑣 ∆TIf Cv is constant:

𝐶𝑝=𝑑 h𝑑𝑇

𝐶𝑝=𝐶𝑝 (T )

h𝑑 =𝐶𝑝dT

∆ h=𝐶𝑝∆TIf Cp is constant:

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INTERNAL ENERGY, ENTHALPY,AND SPECIFIC HEATS OF IDEAL GASES

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

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

• 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.

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• 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.

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

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(kJ/kg)

For small temperature intervals, the specific heats may be assumed to vary

linearly with temperature.

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

The relation u = cv T is valid for any kind of process, constant-volume or not.

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1. By using the tabulated u and h data. This is the easiest and most accurate way when tables are readily available.

2. By using the cv or cp relations (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.

3. 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 and h

Three ways of calculating u.

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Specific Heat Relations of Ideal Gases

The cp of an ideal gas can be determined from a knowledge of cv and R.

On a molar basis

The relationship between cp, cv and R

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.

dh = cpdT and du = cvdT

Example

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Discuss Example 4-7 in class:

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)(c) the average specific heat value (Table A-2b)

Example

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Discuss Example 4-10 in class:

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INTERNAL ENERGY, ENTHALPY, ANDSPECIFIC HEATS OF SOLIDS AND LIQUIDS

The specific volumes of incompressible substances remain constant during a process.

It can be shown that the cv and cp values of incompressible substances are identical which are denoted by c.

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

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Internal Energy Changes

Enthalpy Changes

The enthalpy of a compressed liquid

A more accurate relation than

Example

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Discuss Example 4-12 in class:

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Summary• Moving boundary work

Wb for an isothermal process Wb for a constant-pressure process Wb for a polytropic process

• Energy balance for closed systems Energy balance for a constant-pressure expansion or

compression 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)

Homework #4 Due: Thursday, March 27

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Problems: 4.5E 4.8 4.24 4.25 4.34 4.42 4.46

4.55 4.59E 4.70E 4.74 4.81 4.89