91Ó°ÊÓ

In thermodynamics, the steady-state energy equation helps to analyze systems where conditions remain constant over time. This means that the energy entering the system equals the energy leaving it. For our diffuser problem, we consider kinetic and enthalpy changes since potential energy differences are negligible.

By writing the equation as: \[ h_1 + \frac{v_1^2}{2} = h_2 + \frac{v_2^2}{2} \] we are balancing the specific enthalpy and kinetic energy of the gas at the entry (subscript 1) and at the exit (subscript 2), with: \[ h = c_p \times T \] The specific heat capacity \( c_p \) is used to relate the temperature changes to the enthalpy. This equation gives us the tools to solve for the exit temperature when given the velocities and temperatures of the gas entering and exiting the diffuser.

The mass flow rate \( \dot{m} \) is a measure of the amount of mass flowing through a cross-section of a stream per unit time. For our exercise, we use the equation: \( \dot{m} = \rho v A \), where \( \rho \) is the density, \( v \) is the velocity, and \( A \) is the flow area.

First, we calculate the density at the inlet using the ideal gas law. With density in hand, we determine the mass flow rate using the inlet conditions. This value, calculated once, applies to both the inlet and outlet due to conservation of mass in a steady-flow system.

The ideal gas law links pressure \( P \), volume \( V \), temperature \( T \), and the amount of gas in moles \( n \). It is expressed as: \[ PV = nRT \] or in terms of density: \[ \rho = \frac{P}{\left( \frac{R_u}{MW} \right) T} \] Here, the universal gas constant \( R_u \) and the molecular weight \( MW \) of the gas are used to convert the pressure and temperature into density. This equation was critical in finding the densities needed for mass flow rate calculations and further analysis.

Specific heat capacity \( c_p \) is the amount of heat required to raise the temperature of one kilogram of a substance by one Kelvin. In our problem, it is used to transition between the temperatures and the enthalpies of the gas. For carbon dioxide, \( c_p \) is given as 844 J/(kg·K).
By applying this to our energy balance equation, we can solve for temperature changes: \[ c_p \left( T_1 - T_2 \right) = \frac{v_1^2 - v_2^2}{2} \]

The continuity equation states that the mass flow rate is the same at all points in a closed system. It is expressed as: \[ \dot{m} = \rho_1 v_1 A_1 = \rho_2 v_2 A_2 \] This equation asserts that the products of density, velocity, and cross-sectional area at the inlet and outlet remain equal because mass is conserved. Using the continuity equation, the densities at the exit can be derived from the calculated mass flow rates, incorporating the exit velocities and areas.