91Ó°ÊÓ

The ideal gas law is a fundamental equation in thermodynamics. It relates the pressure, volume, and temperature of an ideal gas using the formula:
\[ PV = nRT \]
In this equation:
* \(P\) represents the pressure of the gas.
* \(V\) represents the volume of the gas.
* \(n\) is the number of moles of the gas.
* \(R\) is the universal gas constant.
* \(T\) is the absolute temperature of the gas.
For specific exergy calculations, we often deal with the specific volume (\(v\)). The specific volume is the volume per unit mass (\text{lb}). The ideal gas law can be adapted to include specific volume in the following way: \( v = \frac{RT}{P} \). For constant specific gas constant \(R = 0.06855 \frac{Btu}{lb \cdot R} \).
Understanding the ideal gas law allows us to interrelate changes in state properties such as pressure, volume, and temperature, key knowledge for calculating exergy.

Internal energy change is a crucial concept in thermodynamics, particularly for specific exergy calculations. For an ideal gas, the change in internal energy \((u - u_0)\) is directly related to the change in temperature, represented by:
\[ u - u_0 = c_v (T - T_0) \]
Here:
* \(c_v\) is the specific heat at constant volume.
* \(T\) is the initial temperature and \(T_0\) is the reference temperature, both in absolute units (Rankine).
For air, the specific heat at constant volume \(c_v\) is typically \(0.171 \frac{Btu}{lb \cdot R} \). This direct relationship simplifies exergy calculations by allowing internal energy changes to be derived solely from temperature differences.

The change in entropy \((s - s_0)\) in thermodynamic processes is significant for evaluating energy efficiency and exergy. For an ideal gas, the change in entropy comprises both temperature and pressure changes and is given by:
\[ s - s_0 = c_p \ln\left( \frac{T}{T_0} \right) - R \ln\left( \frac{P}{P_0} \right) \]
Where:
* \(c_p\) is the specific heat at constant pressure.
* \(R\) is the specific gas constant.
* \(ln\) denotes the natural logarithm.
In our example, \(c_p\) is \(0.240 \frac{Btu}{lb \cdot R} \) and \(R\) is \(0.06855 \frac{Btu}{lb \cdot R} \).
By calculating the entropy change for various pressures, we can better understand how pressure and temperature variations affect exergy.

Specific volume \((v)\) is defined as the volume occupied by a unit mass of a substance, and it plays a significant role in the ideal gas law and specific exergy calculations. It can be derived from the ideal gas law using:
\[ v = \frac{RT}{P} \]
For consistency, the reference specific volume \(v_0\) at the reference state is:
\[ v_0 = \frac{RT_0}{P_0} \]
Where:
* \(T_0\) and \(P_0\) are the reference temperature and pressure respectively.
* \(R\) is the specific gas constant, and \(T\) and \(P\) are the temperature and pressure of the gas.
The specific volume helps to complete the exergy balance as it translates pressure, temperature, and volume changes into energy terms.