Specific heat ratio: Difference between revisions

Revision as of 10:02, 15 April 2009

Ideal gas relations

In thermodynamic terminology, $C_{p}$ and $C_{V}$ may be expressed as:

C_{p}=\left({\frac {\partial H}{\partial T}}\right)_{p}

and

C_{V}=\left({\frac {\partial U}{\partial T}}\right)_{V}

where $T$ stands for temperature, $H$ for the enthalpy and $U$ for the internal energy. For an ideal gas, the heat capacity is constant with temperature. Accordingly, we can express the enthalpy as $H=C_{p}\,T$ and the internal energy as $U=C_{v}\,T$ . Thus, it can also be said that the specific heat ratio of an ideal gas is the ratio of the enthalpy to the internal energy:^[3]

\kappa ={\frac {H}{U}}

The specific heats at constant pressure, $C_{p}$ , of various gases are relatively easy to find in the technical literature. However, it can be difficult to find values of the specific heats at constant volume, $C_{v}$ . When needed, given $C_{p}$ , the following equation can be used to determine $C_{v}$ :^[3]

C_{v}=C_{p}-R

where $R$ is the molar gas constant (also known as the Universal gas constant). This equation can be re-arranged to obtain:

C_{p}={\frac {\kappa R}{\kappa -1}}\qquad {\mbox{and}}\qquad C_{v}={\frac {R}{\kappa -1}}

Relation with degrees of freedom

The specific heat ratio ( $k$ ) for an ideal gas can be related to the degrees of freedom ( $f$ ) of a molecule by:

\kappa ={\frac {f+2}{f}}

Thus for a monatomic gas, with three degrees of freedom:

\kappa ={\frac {5}{3}}=1.67

and for a diatomic gas, with five degrees of freedom (three translational and two rotational degrees of freedom, the vibrational degree of freedom is not involved except at high temperatures):

\kappa ={\frac {7}{5}}=1.4

.

Earth's atmospheric air is primarily made up of diatomic gases with a composition of ~78% nitrogen (N₂) and ~21% oxygen (O₂). At 20 °C and an absolute pressure of 101.325 kPa, the atmospheric air can be considered to be an ideal gas. This results in a value of:

\kappa ={\frac {5+2}{5}}={\frac {7}{5}}=1.4

The specific heats of real gases (as differentiated from ideal gases) are not constant with temperature. As temperature increases, higher energy rotational and vibrational states become accessible to molecular gases, thus increasing the number of degrees of freedom and lowering $\kappa$ .

For a real gas, $C_{p}$ and $C_{v}$ usually increase with increasing temperature and $\kappa$ decreases. Some correlations exist to provide values of $\kappa$ as a function of the temperature.

Isentropic compression or expansion of ideal gases

The specific heat ratio plays an important part in the isentropic process of an ideal gas (i.e., a process that occurs at constant entropy):^[3]

(1)

p_{1}{V_{1}}^{\kappa }=p_{2}{V_{2}}^{\kappa }

where, $p$ is the absolute pressure and $V$ is the volume. The subscripts 1 and 2 refer to conditions before and after the process, or at any time during that process.

Using the ideal gas law, $pV=nRT$ , equation (1) can be re-arranged to:

(2)

{\frac {p_{1}}{p_{2}}}=\left({\frac {V_{2}}{V_{1}}}\right)^{\kappa }=\left({\frac {T_{2}}{T_{1}}}\right)^{\kappa }\left({\frac {p_{1}}{p_{2}}}\right)^{\kappa }

where $T$ is the absolute temperature. Re-arranging further:

(3)

\left({\frac {T_{2}}{T_{1}}}\right)^{\kappa }=\left({\frac {p_{1}}{p_{2}}}\right)\left({\frac {p_{2}}{p_{1}}}\right)^{\kappa }=\left({\frac {p_{2}}{p_{1}}}\right)^{\kappa -1}

we obtain the equation for the temperature change that occurs when an ideal gas is isentropically compressed or expanded:^[3]^[4]

(4)

{\frac {T_{2}}{T_{1}}}=\left({\frac {p_{2}}{p_{1}}}\right)^{(\kappa -1)/\kappa }

Equation (4) is widely used to model ideal gas compression or expansion processes in internal combustion engines, gas compressors and gas turbines.

Determination of $C_{v}$ values

Values for $C_{p}$ are readily available, but values for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v} are not as available and often need to be determined. Values based on the ideal gas relation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p - C_v = R} are in many cases not sufficiently accurate for practical engineering calculations. If at all possible, an experimental value should be used.

A rigorous value can be calculated by determining Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v} from the residual property functions (also referred to as departure functions)^[5]^[6]^[7] using this relation:^[8]

C_{v}=C_{p}+T{\frac {{\;\left({\frac {\partial p}{\partial T}}\right)}_{V}^{2}}{\left({\frac {\partial p}{\partial V}}\right)_{T}}}

Equations of state (EOS) (such as the Peng-Robinson equation of state) can be used to solve this relation and to provide values of $C_{v}$ that match experimental values very closely.

References

↑ Frank M. White (1999). Fluid Mechanics, Fourth Edition. McGraw-Hill. ISBN 0-07-0697167.
↑ Norbert A. Lange (Editor) (1969). Lange's Handbook of Chemistry, 10th Edition. McGraw-Hill, page 1524.
↑ ^3.0 ^3.1 ^3.2 ^3.3 ^3.4 Stephan R. Turns (2006). Thermodynamics: Concepts and Application, First Edition. Cambridge University Press. ISBN 0-521-85042-8.
↑ Don. W. Green, James O Maloney and Robert H. Perry (Editors) (1984). Perry's Chemical Engineers' Handbook, Sixth Edition. McGraw-Hill, page 6-17. ISBN 0-07-049479-7.
↑ K.Y. Narayanan (2001). A Textbook of Chemical Engineering Thermodynamics. Prentice-Hall India. ISBN 81-203-1732-7.
↑ Y.V.C. Rao (1997). Chemical Engineering Thermodynamics, First Edition. Universities Press. ISBN 81-7371-048-1.
↑ Thermodynamics of Pure Substances Lecture by Mark Gibbs, University of Edinburgh, Scotland.
↑ Isochoric heat capacity (pdf page 61 of 308)

[1] Frank M. White (1999). Fluid Mechanics, Fourth Edition. McGraw-Hill. ISBN 0-07-0697167.

[2] Norbert A. Lange (Editor) (1969). Lange's Handbook of Chemistry, 10th Edition. McGraw-Hill, page 1524.

[Turns-3] 3.0 ^3.1 ^3.2 ^3.3 ^3.4 Stephan R. Turns (2006). Thermodynamics: Concepts and Application, First Edition. Cambridge University Press. ISBN 0-521-85042-8.

[4] Don. W. Green, James O Maloney and Robert H. Perry (Editors) (1984). Perry's Chemical Engineers' Handbook, Sixth Edition. McGraw-Hill, page 6-17. ISBN 0-07-049479-7.

[5] K.Y. Narayanan (2001). A Textbook of Chemical Engineering Thermodynamics. Prentice-Hall India. ISBN 81-203-1732-7.

[6] Y.V.C. Rao (1997). Chemical Engineering Thermodynamics, First Edition. Universities Press. ISBN 81-7371-048-1.

[7] Thermodynamics of Pure Substances Lecture by Mark Gibbs, University of Edinburgh, Scotland.

[8] Isochoric heat capacity (pdf page 61 of 308)

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

@@ Line 106: / Line 106: @@
 The '''specific heat ratio''' of a [[gas]] is the ratio of the [[specific heat]] at constant [[pressure]], <math>C_p</math>, to the specific heat at constant volume, <math>C_v</math>. It is sometimes  referred to as the '''adiabatic index''' or the '''heat capacity ratio''' or the '''isentropic expansion factor''' or the '''adiabatic exponent''' or the '''isentropic exponent'''.
-Either <math>k</math> ([[K (letter)|Roman letter k]]), <math>\gamma</math> ([[Gamma (Greek letter) |gamma]]) or <math>\kappa</math> ([[Kappa (Greek letter) |kappa]]) may be used to denote the specific heat ratio:
+Either <math>\kappa</math> ([[Kappa (Greek letter) |kappa]]), <math>k</math> ([[K (letter)|Roman letter k]]) or <math>\gamma</math> ([[Gamma (Greek letter) |gamma]]) may be used to denote the specific heat ratio:
-:<math>k = \gamma = \kappa = \frac{C_p}{C_v}</math>
+:<math>\kappa = k = \gamma = \frac{C_p}{C_v}</math>
 where:
@@ Line 125: / Line 125: @@
 where <math>T</math> stands for [[temperature]], <math>H</math> for the [[enthalpy]] and <math>U</math> for the [[internal energy]]. For an [[Ideal gas law|ideal gas]], the [[heat capacity]] is constant with temperature. Accordingly, we can express the enthalpy as <math>H = C_p\, T</math> and the internal energy as <math>U = C_v\, T</math>. Thus, it can also be said that the specific heat ratio of an ideal gas is the ratio of the enthalpy to the internal energy:<ref name=Turns/>
-:<math> k = \frac{H}{U}</math>
+:<math>\kappa = \frac{H}{U}</math>
 The specific heats at constant pressure, <math>C_p</math>, of various gases are relatively easy to find in the technical literature. However, it can be difficult to find values of the specific heats at constant volume, <math>C_v</math>. When needed,  given <math>C_p</math>, the following equation can be used to determine <math>C_v</math> :<ref name=Turns/>
@@ Line 133: / Line 133: @@
 where <math>R</math> is the [[molar gas constant]] (also known as the [[Universal gas constant]]). This equation can be re-arranged to obtain:
-:<math> C_p = \frac{k R}{k - 1} \qquad \mbox{and} \qquad C_v = \frac{R}{k - 1}</math>
+:<math> C_p = \frac{\kappa R}{\kappa - 1} \qquad \mbox{and} \qquad C_v = \frac{R}{\kappa - 1}</math>
 === Relation with degrees of freedom ===
@@ Line 139: / Line 139: @@
 The specific heat ratio ( <math>k</math> ) for an ideal gas can be related to the [[degrees of freedom (physics)|degrees of freedom]] ( <math>f</math> ) of a molecule by:
-:<math> k = \frac{f+2}{f}</math>
+:<math> \kappa = \frac{f+2}{f}</math>
 Thus for a [[monatomic]] gas, with three degrees of freedom:
-:<math> k = \frac{5}{3} = 1.67</math>
+:<math> \kappa = \frac{5}{3} = 1.67</math>
 and for a [[diatomic]] gas, with five degrees of freedom (three translational and two rotational degrees of freedom, the vibrational degree of freedom is not involved except at high temperatures):
-:<math> k = \frac{7}{5} = 1.4</math>.
+:<math> \kappa = \frac{7}{5} = 1.4</math>.
 Earth's atmospheric [[air]] is primarily made up of [[diatomic]] gases with a composition of ~78% [[nitrogen]] (N<sub>2</sub>) and ~21% [[oxygen]] (O<sub>2</sub>). At 20 °C and an absolute pressure of 101.325 kPa, the atmospheric air can be considered to be an ideal gas. This results in a value of:
-: <math> k = \frac{5 + 2}{5} = \frac{7}{5} = 1.4</math>
+: <math> \kappa = \frac{5 + 2}{5} = \frac{7}{5} = 1.4</math>
-The specific heats of real gases (as differentiated from ideal gases) are not constant with temperature. As temperature increases, higher energy rotational and vibrational states become accessible to molecular gases, thus increasing the number of degrees of freedom and lowering <math>k</math>.
+The specific heats of real gases (as differentiated from ideal gases) are not constant with temperature. As temperature increases, higher energy rotational and vibrational states become accessible to molecular gases, thus increasing the number of degrees of freedom and lowering <math>\kappa</math>.
-For a real gas, <math>C_p</math> and <math>C_v</math> usually increase with increasing temperature and <math>k</math> decreases. Some correlations exist to provide values of <math>k</math> as a function of the temperature.
+For a real gas, <math>C_p</math> and <math>C_v</math> usually increase with increasing temperature and <math>\kappa</math> decreases. Some correlations exist to provide values of <math>\kappa</math> as a function of the temperature.
 == Isentropic compression or expansion of ideal gases ==
@@ Line 161: / Line 161: @@
 The specific heat ratio plays an important part in the [[isentropic]] process of an ideal gas (i.e., a process that occurs at constant [[entropy]]):<ref name=Turns/>
-:(1) &nbsp; &nbsp;<math> p_1{V_1}^k = p_2{V_2}^k </math>
+:(1) &nbsp; &nbsp;<math> p_1{V_1}^\kappa = p_2{V_2}^\kappa </math>
 where, <math>p</math> is the absolute pressure and <math>V</math> is the volume. The subscripts 1 and 2  refer to conditions before and after the process, or at any time during that process.
@@ Line 167: / Line 167: @@
 Using the ideal gas law, <math>pV = nRT</math>, equation (1) can be re-arranged to:
-:(2) &nbsp; &nbsp;<math>\frac{p_1}{p_2} = \left(\frac{V_2}{V_1}\right)^k = \left(\frac{T_2}{T_1}\right)^k  \left(\frac{p_1}{p_2}\right)^k</math>
+:(2) &nbsp; &nbsp;<math>\frac{p_1}{p_2} = \left(\frac{V_2}{V_1}\right)^\kappa = \left(\frac{T_2}{T_1}\right)^\kappa  \left(\frac{p_1}{p_2}\right)^\kappa</math>
 where <math>T</math> is the absolute temperature. Re-arranging further:
-:(3) &nbsp; &nbsp;<math>\left(\frac{T_2}{T_1}\right)^k = \left(\frac{p_1}{p_2}\right)\left(\frac{p_2}{p_1}\right)^k = \left(\frac{p_2}{p_1}\right)^{k-1}</math>
+:(3) &nbsp; &nbsp;<math>\left(\frac{T_2}{T_1}\right)^\kappa = \left(\frac{p_1}{p_2}\right)\left(\frac{p_2}{p_1}\right)^\kappa = \left(\frac{p_2}{p_1}\right)^{\kappa-1}</math>
 we obtain the equation for the temperature change that occurs when an ideal gas is isentropically compressed or expanded:<ref name=Turns/><ref>{{cite book|author=Don. W. Green, James O Maloney and Robert H. Perry (Editors)|title=Perry's Chemical Engineers' Handbook|edition=Sixth Edition|publisher=McGraw-Hill|pages=page 6-17|year=1984|id=ISBN 0-07-049479-7}}</ref>
-:(4) &nbsp; &nbsp; <math>\frac{T_2}{T_1} = \left(\frac{p_2}{p_1}\right)^{(k-1)/k}</math>
+:(4) &nbsp; &nbsp; <math>\frac{T_2}{T_1} = \left(\frac{p_2}{p_1}\right)^{(\kappa-1)/\kappa}</math>
 Equation (4) is widely used to model ideal gas compression or expansion processes in [[internal combustion engine]]s, [[gas compressor]]s and [[gas turbines]].
@@ Line 188: / Line 188: @@
 [[Equations of state]] (EOS) (such as the [[Peng-Robinson equation of state]]) can be used to solve this relation and to provide values of <math>C_v</math> that match experimental values very closely.
 == References ==
 <references/>

Specific heat ratio of various gases^[1]^[2]^[3]
Gas	°C	k	Gas	°C	k
H₂	−181	1.597	Dry Air	20	1.40
	−76	1.453		100	1.401
	20	1.41		200	1.398
	100	1.404		400	1.393
	400	1.387	CO₂	0	1.310
	1000	1.358		20	1.30
	2000	1.318		100	1.281
He	20	1.66		400	1.235
N₂	−181	1.47	NH₃	15	1.310
N₂	15	1.404	CO	20	1.40
Cl₂	20	1.34	O₂	−181	1.45
Ar	−180	1.76		−76	1.415
Ar	20	1.67		20	1.40
CH₄	−115	1.41		100	1.399
	−74	1.35		200	1.397
	20	1.32		400	1.394

Specific heat ratio: Difference between revisions

Revision as of 10:02, 15 April 2009

Contents

Ideal gas relations

Relation with degrees of freedom

Isentropic compression or expansion of ideal gases

Determination of $C_{v}$ values

References

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Specific heat ratio: Difference between revisions

Revision as of 10:02, 15 April 2009

Ideal gas relations

Relation with degrees of freedom

Isentropic compression or expansion of ideal gases

Determination of C v {\displaystyle C_{v}} values

References

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Determination of $C_{v}$ values