# Compression

Parameteric model for compression process: isentropic and isothermal

# Expansion

Parameteric model for expansion process: isentropic and isenthalpic

# Heating

Heating process at constant pressure

# Cooling

Cooling process at constant pressure

# Heat Exchanger (two streams)

Parameteric model for heat exchange between two streams

# Thermodynamic components

These components are used to build thermodynamic cycles (used in heat pumps, heat engines, liquefiers, etc.)

# Compressor

There are two compressor models: isothermal and isentropic. They represent the two extremes in terms of heat exchange with the environment.

An isentropic compressor compresses the fluid adiabatically (without exchanging heat with the environment). This model approximates a very quick compression model, where there is not enough time for significant heat exchange with the environment. The outlet state of an ideal isentropic heat exchanger has the same entropy as the inlet state. In reality, due to friction and other irreversible processes, the entropy increases and the real work is higher than the ideal work, which is given by:

\begin{equation*} w_{id} = h\left(p_2, s_1\right) - h\left(p_1, s_1\right) \end{equation*}

The isentropic efficiency of the compressor $$\eta$$ is a parameter of the compressor such that:

\begin{equation*} w_{r} = w_{id} / \eta \end{equation*}

where $$w_{r}$$ is the real specific work. A second parameter $$f_Q$$ defines the fraction of the work that is dissipated as heat in the environment:

\begin{equation*} q_{out} = w_{r} \cdot f_Q \end{equation*}

Finally, the real outlet state of the compressor can be computed by outlet pressure and specific enthalpy, where the specific enthalpy is determined by the energy balance:

\begin{equation*} h_{out} = h_{in} + w_{r} - q_{out} \end{equation*}

An isothermal compressor compresses the fluid (gas or liquid) at constant (ambient) temperature. It is the most efficient way of compressing fluid (minimal work), although it is not feasible. The compression process has to be very slow in order for the temperature to constantly equilibrate with the environment. To approach isothermal compression in practice, a multiple-stage compressor with intercoolers can be used.

The heat flow to the environment is linked to the entropy change:

\begin{equation*} q_{out} = T * \left( s_{out} - s_{in} \right) \end{equation*}

Then the work is given by:

\begin{equation*} w_{r} = h_{out} - h_{in} + q_{out} \end{equation*}

# Turbine

The ideal turbine is also isentropic. A real turbine produces less work:

\begin{equation*} w_{r} = \eta\cdot w_{id} \end{equation*}

The change of enthalpy is:

\begin{equation*} h_{out} = h_{in} + w_{r} - q_{out} \end{equation*}

# Throttle (Joule-Thompson) valve

Expansion through an orifice or capilary tube is isenthalpic (adiabatic and no work is produced). Therefore

\begin{equation*} h_{out} = h_{in} \end{equation*}

However, the temperature could decrease (near the two-phase region) or sometimes increase (at high pressures and temperatures).

# Evaporator / Condenser

Pressure drop is neglected. No work is done, so the change of enthalpy is equal to the heat input to the fluid.

# Two-stream heat exchanger

This is a general abstraction of a few different heat exchanger design concepts. The heat exchangers could be counter-flow, parallel-flow and cross-flow. The heat capacity flows are defined as:

\begin{equation*} {\dot Q}_1 = {\dot m}_1 \left( h(T_{2}^{in}, p_1) - h_{1}^{in} \right) \end{equation*}
\begin{equation*} {\dot Q}_2 = {\dot m}_2 \left( h(T_{1}^{in}, p_2) - h_{2}^{in} \right) \end{equation*}

The maximal possible heat flow rate, which can be achieved in an infinetely long countercurrent heat exchanger, is the lesser of the two:

\begin{equation*} {\dot Q}_{m} = \min \left( {\dot Q}_1, {\dot Q}_2 \right) \end{equation*}

Then the real heat flow rate is:

\begin{equation*} {\dot Q} = \varepsilon \cdot {\dot Q}_{m} \end{equation*}

where $$\varepsilon$$ is the effectiveness of the heat exchanger. In the $$NTU-\varepsilon$$ method [Wiki-NTU], the effectiveness is determined as a function of the number of transfer units and the heat capacity ratio:

\begin{equation*} \varepsilon = f\left( NTU, C_r \right) \end{equation*}
\begin{equation*} NTU = \frac{U \cdot A}{\dot{C}_{min}} \end{equation*}
\begin{equation*} C_r = \frac{\min\left(\dot{Q}_{1},\dot{Q}_{2}\right)}{\max\left(\dot{Q}_{1},\dot{Q}_{2}\right)} \end{equation*}

where $$\dot{C}_{min} = {\dot Q}_{m} / (T_1^{in} - T_2^{in})$$, $$U$$ is the overall heat transfer coefficient, and $$A$$ is the heat exhcange area

The effectiveness formula depends on the flow configuration, as shown in the table below:

Flow configuration Effectiveness correlation
parallel flow $$\varepsilon \ = \frac {1 - \exp[-NTU(1 + C_{r})]}{1 + C_{r}}$$
counter-current flow $$\varepsilon \ = \frac {1 - \exp[-NTU(1 - C_{r})]}{1 - C_{r}\exp[-NTU(1 - C_{r})]}$$
evaporation / condensation $$\varepsilon \ = 1 - \exp[-NTU]$$

# References

 [Wiki-NTU] Wikipedia, NTU method, http://en.wikipedia.org/wiki/NTU_method