91Ó°ÊÓ

The Steady Flow Energy Equation (SFEE) is vital in analyzing energy changes within a flow system like a nozzle. It simplifies the energy conservation principle for a fluid flowing steadily, meaning the properties at any fixed point in the system do not change over time.
In its simplest form for an adiabatic (well-insulated) nozzle, and neglecting potential energy changes, it can be expressed as:

\[ h_1 + \frac{V_1^2}{2} = h_2 + \frac{V_2^2}{2} \]

Here, \(h\) represents specific enthalpy, and \(V\) is velocity. This equation balances enthalpy and kinetic energy at two different points (inlet and exit) in the flow. This core principle helps us break down complex thermodynamic problems in a straight-forward manner.

The Ideal Gas Law relates the pressure, volume, and temperature of a gas and is given in the form:

\[ PV = nRT \]

Here, \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is temperature. In the context of our problem, we use the ideal gas law to find the density of helium at the nozzle's exit.
We rearrange the equation to density form:\[ \rho = \frac{P}{RT} \] and substitute the given values to calculate the exit density. This step is crucial for determining the mass flow rate through the nozzle.

The Specific Heat Ratio, denoted as \(k\) (or \( \gamma \)), is the ratio of the heat capacity at constant pressure \(c_p\) to the heat capacity at constant volume \(c_v\). For helium, \(k = 1.67\). This ratio plays a critical role in thermodynamics.
The formula to find the specific heat at constant pressure is:

\[ c_p = \frac{kR}{k-1} \]

Given that helium's gas constant \(R\) is 1.985 BtU/(lbm·°R), we convert \(R\) to consistent units:\[ R = 1.985 \times 25037 = 49743.045 \text{ft·lbf/(lbm·°R)} \]. By substituting \(R\) and \(k\) into the formula, we calculate\( c_p\), which is then used in the SFEE to bridge temperature and velocity.

The mass flow rate (\( \dot{m} \)) denotes the amount of mass passing through a cross-section of the nozzle per unit time. To calculate it, we use the continuity equation combined with the ideal gas law:\[ \dot{m} = \rho A V \], where \( \rho \) is density, \( A \) is the exit area, and \( V \) is the exit velocity. First, we determine the density using the rechill form of the ideal gas law: \[ \rho = \frac{P}{RT} \] substituting pressure (\(P\)), the gas constant (\(R\)), and temperature (\(T\)) at the exit.
After finding \( \rho \), we multiply it with the area (\( A \)) and the exit velocity (\(V\)) calculated in the final step, giving us the mass flow rate through the nozzle. This value is crucial for understanding the overall flow characteristics.