The entropy of a given mass does not change during a process that is internally reversible and adiabatic. A process during which the entropy remains constant is called an isentropic process, written \Delta s=0 or s_1 = s_2 . Some examples of theoretically isentropic thermodynamic devices are
pumps,
gas compressors,
turbines,
nozzles, and
diffusers.
Isentropic efficiencies of steady-flow devices in thermodynamic systems Most steady-flow devices operate under adiabatic conditions, and the ideal process for these devices is the isentropic process. The parameter that describes how efficiently a device approximates a corresponding isentropic device is called isentropic or adiabatic efficiency. Isentropic efficiency of turbines: : \eta_\text{t} = \frac{\text{actual turbine work}}{\text{isentropic turbine work}} = \frac{W_a}{W_s} \cong \frac{h_1 - h_{2a}}{h_1 - h_{2s}}. Isentropic efficiency of compressors: : \eta_\text{c} = \frac{\text{isentropic compressor work}}{\text{actual compressor work}} = \frac{W_s}{W_a} \cong \frac{h_{2s} - h_1}{h_{2a} - h_1}. Isentropic efficiency of nozzles: : \eta_\text{n} = \frac{\text{actual KE at nozzle exit}}{\text{isentropic KE at nozzle exit}} = \frac{V_{2a}^2}{V_{2s}^2} \cong \frac{h_1 - h_{2a}}{h_1 - h_{2s}}. For all the above equations: : h_1 is the specific
enthalpy at the entrance state, : h_{2a} is the specific enthalpy at the exit state for the actual process, : h_{2s} is the specific enthalpy at the exit state for the isentropic process.
Isentropic devices in thermodynamic cycles Note: The isentropic assumptions are only applicable with ideal cycles. Real cycles have inherent losses due to compressor and turbine inefficiencies and the second law of thermodynamics. Real systems are not truly isentropic, but isentropic behavior is an adequate approximation for many calculation purposes. == Isentropic flow == In
fluid dynamics, an
isentropic flow is a
fluid flow that is both adiabatic and reversible. That is, no heat is added to the flow, and no energy transformations 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. Note that energy
can be exchanged with the flow in an isentropic transformation, as long as it doesn't happen as heat exchange. An example of such an exchange would be an isentropic expansion or compression that entails work done on or by the flow. For an isentropic flow, entropy density can vary between different streamlines. If the entropy density is the same everywhere, then the flow is said to be
homentropic.
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 = \delta W + \delta Q. The reversible work done on a system by changing the volume is :\delta W = -p \,dV, where p is the
pressure, and V is the
volume. The change in
enthalpy (H = U + pV) is given by :dH = dU + p \,dV + V \,dp. Then for a process that is both reversible and adiabatic (i.e. no heat transfer occurs), \delta Q_\text{rev} = 0, and so dS = \delta Q_\text{rev}/T = 0 All reversible adiabatic processes are isentropic. This leads to two important observations: : dU = \delta W + \delta Q = -p \,dV + 0, :dH = \delta W + \delta Q + p \,dV + V \,dp = -p \,dV + 0 + p \,dV + V \,dp = V \,dp. Next, a great deal can be computed for isentropic processes of an ideal gas. For any transformation of an ideal gas, it is always true that :dU = n C_v \,dT, and dH = n C_p \,dT. Using the general results derived above for dU and dH, then : dU = n C_v \,dT = -p \,dV, : dH = n C_p \,dT = V \,dp. So for an ideal gas, the
heat capacity ratio can be written as :\gamma = \frac{C_p}{C_V} = -\frac{dp/p}{dV/V}. For a calorically perfect gas \gamma is constant. Hence on integrating the above equation, assuming a calorically perfect gas, we get : pV^\gamma = \text{constant}, that is, : \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} = \text{constant}. (Proof: PV^\gamma = \text{constant} \Rightarrow PV\,V^{\gamma-1} = \text{constant} \Rightarrow nRT\,V^{\gamma-1} = \text{constant}. But
nR = constant itself, so TV^{\gamma-1} = \text{constant}.) : \frac{p^{\gamma-1}}{T^\gamma} = \text{constant} also, for constant C_p = C_v + 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) - nR\ln\left(\frac{p_2}{p_1}\right) : \frac{S_2-S_1}{n} = C_p \ln\left(\frac{T_2}{T_1}\right) - R\ln\left(\frac{T_2 V_1}{T_1 V_2}\right ) = C_v\ln\left(\frac{T_2}{T_1}\right)+ R \ln\left(\frac{V_2}{V_1}\right) Thus for isentropic 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 : Derived from : PV^{\gamma} = \text{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 = molar specific heat at constant pressure, : C_v = molar specific heat at constant volume. == See also ==