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Isentropic process

 
Sci-Tech Dictionary: isentropic process
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(¦īs·ən′träp·ik ′prä·ses)

(thermodynamics) A change that takes place without any increase or decrease in entropy, such as a process which is both reversible and adiabatic.

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Sci-Tech Encyclopedia: Isentropic process
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In thermodynamics, a process involving change without any increase or decrease of entropy. Since the entropy always increases in a spontaneous process, one must consider reversible or quasistatic processes. During a reversible process the quantity of heat transferred is directly proportional to the system's entropy change. Systems which are thermally insulated from their surroundings undergo processes without any heat transfer; such processes are called adiabatic. Thus during an isentropic process there are no dissipative effects and the system neither absorbs nor gives off heat. For this reason the isentropic process is sometimes called the reversible adiabatic process. See also Adiabatic process; Entropy.

Wikipedia: Isentropic process
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In thermodynamics, an isentropic process or isoentropic process (ισον = "equal" (Greek); εντροπία entropy = "disorder"(Greek)) is one during which the entropy of the system remains constant. [1][2] It can be proved that any reversible adiabatic process is an isentropic process.

Contents

Background

Second law of thermodynamics states that,

\delta Q \le TdS

where δQ is the amount of energy the system gains by heating, T is the temperature of the system, and dS is the change in entropy. The equal sign will hold for a reversible process. For a reversible isentropic process, there is no transfer of heat energy and therefore the process is also adiabatic. For an irreversible process, the entropy will increase. Hence removal of heat from the system (cooling) is necessary to maintain a constant internal entropy for an irreversible process in order to make it isentropic. Thus an irreversible isentropic process is not adiabatic.

For reversible processes, an isentropic transformation is carried out by thermally "insulating" the system from its surroundings. Temperature is the thermodynamic conjugate variable to entropy, thus the conjugate process would be an isothermal process in which the system is thermally "connected" to a constant-temperature heat bath.

Isentropic flow

An isentropic flow is a flow that is both adiabatic and reversible. That is, no energy is added to the flow, and no energy losses occur due to friction or dissipative effects. For an isentropic flow of a perfect gas, several relations can be derived to define the pressure, density and temperature along a streamline.

Derivation of the isentropic relations

For a closed system, the total change in energy of a system is the sum of the work done and the heat added,

dU = dW + dQ\,\!

The work done on a system by changing the volume is,

dW = -pdV\,\!

where p is the pressure and V is the volume. The change in enthalpy (H = U + pV\,\!) is given by,

dH = dU + pdV + Vdp = n C_p dT\,\!

Since a reversible process is adiabatic (i.e. no heat transfer occurs), so dQ = 0, dS=0\,\!. This leads to two important observations,

dU = -pdV\,\!, and
dH = Vdp\,\! or dQ = dH - Vdp = 0\,\!
dQ = TdS\,\! => dS = (1/T) dH - (V/T) dp\,\!

The heat capacity ratio can be written as,

\gamma = \frac{C_p}{C_V} = -\frac{dp/p}{dV/V}\,\!

For an ideal gas \gamma\,\! is constant. Hence on integrating the above equation, assuming a perfect gas, we get

pV^{\gamma} = \mbox{constant} \, i.e.
\frac{p_2}{p_1} = \left(\frac{V_1}{V_2}\right)^{\gamma}

Using the equation of state for an ideal gas, p V = n R T\,\!,

TV^{\gamma-1} = \mbox{constant} \,
\frac{p^{\gamma -1}}{T^{\gamma}} = \mbox{constant}

also, for constant Cp = Cv + R (per mole),

\frac{V}{T} = \frac{nR}{p} and p = \frac{nRT}{V}
S_2-S_1 = nC_p ln\left(\frac{T_2}{T_1}\right) - nRln\left(\frac{p_2}{p_1}\right)
\frac{S_2-S_1}{n} = C_p ln\left(\frac{T_2}{T_1}\right) - Rln\left (\frac{T_2V_1}{T_1V_2} \right ) = C_vln\left(\frac{T_2}{T_1}\right)+R ln\left(\frac{V_2}{V_1}\right)

Thus for insentropic processes with an ideal gas,

T_2 = T_1\left(\frac{V_1}{V_2}\right)^{(R/C_v)} or V_2 = V_1\left(\frac{T_1}{T_2}\right)^{(C_v/R)}

Table of isentropic relations for an ideal gas

\frac {p_2} {p_1} =\,\! \left (\frac{T_2}{T_1} \right )^\frac {\gamma}{\gamma-1} =\,\! \left (\frac{\rho_2}{\rho_1} \right )^{\gamma} =\,\! \left (\frac{V_1}{V_2} \right )^{\gamma}
\frac {T_2} {T_1} =\,\! \left (\frac{p_2}{p_1} \right )^\frac {\gamma-1}{\gamma} =\,\! \left (\frac{\rho_2}{\rho_1} \right )^{(\gamma - 1)} =\,\! \left (\frac{V_1}{V_2} \right )^{(\gamma-1)}
\frac {\rho_2} {\rho_1} =\,\! \left (\frac{T_2}{T_1} \right )^\frac {1}{\gamma-1} =\,\! \left (\frac{p_2}{p_1} \right )^\frac {1}{\gamma} =\,\! \frac{V_1}{V_2}
\frac {V_2} {V_1} =\,\! \left (\frac{T_1}{T_2} \right )^\frac {1}{\gamma-1} =\,\! \frac{\rho_1}{\rho_2} =\,\! \left (\frac{p_1}{p_2} \right )^\frac {1}{\gamma}

Derived from:

pV^{\gamma} = constant \,\!
pV = m R_s T \,\!
p = \rho R_s T\,\! \,\!
Where:
p\,\! = Pressure
V\,\! = Volume
\gamma\,\! = Ratio of specific heats = C_p/C_v\,\!
T\,\! = Temperature
m\,\! = Mass
R_s\,\! = Gas constant for the specific gas = R/M\,\!
R\,\! = Universal gas constant
M\,\! = Molecular weight of the specific gas
\rho\,\! = Density
C_p\,\! = Specific heat at constant pressure
C_v\,\! = Specific heat at constant volume

References

  • Van Wylen, G.J. and Sonntag, R.E. (1965), Fundamentals of Classical Thermodynamics, John Wiley & Sons, Inc., New York. Library of Congress Calatog Card Number: 65-19470

Notes

  1. ^ Van Wylen, G.J. and Sonntag, R.E., Fundamentals of Classical Thermodynamics, Section 7.4
  2. ^ Massey, B.S. (1970), Mechanics of Fluids, Section 12.2 (2nd edition) Van Nostrand Reinhold Company, London. Library of Congress Catalog Card Number: 67-25005

See also


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