Steady Flow Devices

Entropy Balance for an Open System

$\frac{d{S}_{CV}}{dt}=\sum \frac{{\dot{Q}}_k}{T_k}+{\dot{m}}_{in}{s}_{in}-{\dot{m}}_{out}{s}_{out}+{\dot{S}}_{gen}$

Reversible

Note, if something is reversible then $S_{gen}$ is 0.

Isentropic Efficiency of Turbines

Turbines extract work by passing fluid over a set of spinning blades

Typically:

• Adiabatic
• $\Delta PE=\Delta KE=0$ ${\eta }_T=\frac{w_a}{w_s}=\frac{\text{Actual turbine work}}{\text{Isentropic turbine work}}$
  ${\eta }_T=\frac{m^{\prime }\left(h_1-h_{2a}\right)}{m^{\prime }\left(h_1-h_{2s}\right)}$ ${\eta }_T\cong \frac{h_1-h_{2a}}{h_1-h_{2s}}$
  Here a means “actual” and s means “isentropic”

Efficiency

Small turbines are usually below 70%
Large good ones are usually above 90%

The T-s curve has compatible characteristics to the T-v curve since s is just another independent intensive property

Isentropic Efficiency of compressors

Compressors use work input to increase the pressure of a gas
Typically:

• Negligible heat transfer (adiabatic)
• $\Delta PE=\Delta KE=0$ ${\eta }_C=\frac{w_a}{w_s}=\frac{\text{Isentropic compressor work}}{\text{Actual compressor work}}=\frac{w_s}{w_a}$
  ${\eta }_C=\frac{\left(h_{2s}-h_1\right)}{\left(h_{2a}-h_1\right)}$

Isentropic Efficiency of Nozzles

A nozzle increases the velocity of a fluid at the expense of pressure Typically:

• Adiabatic
• No work
• $\Delta PE=0$
• $\Delta KE\ne 0$ ${\eta }_N=\frac{w_a}{w_s}=\frac{\text{Actual KE at nozzle exit}}{\text{Isentropic KE at nozzle exit}}=\frac{V_{2a}^2}{V_{2s}^2}$
  ${\eta }_N=\frac{\left(h_1-h_{2a}\right)}{\left(h_1-h_{2s}\right)}$

Isentropic Process for an Ideal Gas

For an ideal gas, assuming constant specific heats, the following relationships hold for isentropic processes
$T{v}^{k-1}=const$
$T{P}^{\frac{\left(1-k\right)}{k}}=const$
$P{v}^k=const$

$$\begin{array}{rl}\left(\frac{T_2}{T_1}\right)& ={\left(\frac{v_1}{v_2}\right)}^{k-1}\\ \left(\frac{T_2}{T_1}\right)& ={\left(\frac{P_2}{P_1}\right)}^{\frac{k-1}{k}}\\ \left(\frac{P_2}{P_1}\right)& ={\left(\frac{v_1}{v_2}\right)}^k\end{array}$$

Where $k=\frac{c_p}{c_v}$