Question

Quiescent carbon dioxide at \(1200 \mathrm{kPa}\) and \(600 \mathrm{K}\) is accelerated isentropically to a Mach number of \(0.6 .\) Determine the temperature and pressure of the carbon dioxide after acceleration.

Short answer

Question: Determine the final temperature and pressure of CO₂ after isentropic acceleration given an initial temperature of 600 K, initial pressure of 1200 kPa, and a Mach number of 0.6. Answer: To find the final temperature and pressure of CO₂ after isentropic acceleration, follow the steps below: 1. Calculate the initial specific volume (ν) using the ideal gas law: ν = RT/P, where R = 188.92 J/(kg·K), T = 600 K, and P = 1200 kPa. 2. Use the isentropic flow relations to find the final specific volume (ν₂) and specific enthalpy (h₂), considering the Mach number (M = 0.6) and adiabatic index for CO₂ (γ = 1.3). 3. Find the final temperature (T₂) using the final specific volume (ν₂) and ideal gas law, and the final pressure (P₂) using the final specific enthalpy (h₂). Calculate the values and solve for the final temperature and pressure of the CO₂ after acceleration.

Step-by-step solution

Determine the initial specific volume and specific enthalpy of the CO₂ using the ideal gas law.

Using the ideal gas law, the specific volume (ν) can be determined as: $$ ν = \frac{RT}{P} $$ Where: R = 188.92 J/(kg·K) (specific gas constant for CO₂) T = the initial temperature (600 K) P = the initial pressure (1200 kPa) Calculate the initial specific volume.

Use the isentropic flow relations to find the final specific volume and specific enthalpy.

Using the isentropic flow relations, we can relate the initial and final states of the flow with the Mach number (M): $$ \frac{ν_{2}}{ν_{1}} = \left(\frac{T_{2}}{T_{1}}\right)^{1/(\gamma-1)} \left(\frac{\gamma + 1}{2}\right)^{\frac{\gamma+1}{2(\gamma-1)M^2}} $$ $$ \frac{h_{2}}{h_{1}} = \frac{T_{2}}{T_{1}} $$ Where: ν₁ and ν₂ are the initial and final specific volumes h₁ and h₂ are the initial and final specific enthalpies γ = 1.3 (adiabatic index for CO₂) M = 0.6 (Mach number) Solve these equations for the final specific volume and specific enthalpy.

Find the final temperature and pressure using the final specific volume and specific enthalpy.

Using the final specific volume (ν₂) and ideal gas law, we can find the final temperature (T₂): $$ T_{2} = \frac{P_{2}ν_{2}}{R} $$ Using the final specific enthalpy (h₂) and the relation between specific enthalpy and temperature, we can find the final pressure (P₂): $$ h_{2} = h_{1} + R(T_{2} - T_{1}) $$ Solve these equations to find the final temperature and pressure of the CO₂ after acceleration.

Key concepts

Mach number

The Mach number is a dimensionless quantity used in fluid dynamics to describe the speed of an object in relation to the speed of sound in the surrounding medium. It is defined as the ratio of the object's speed to the speed of sound in the fluid. Mathematically, Mach number (M) is expressed as:
\[\begin{equation}M = \frac{v}{a}\end{equation}\]
where \(v\) is the velocity of the object and \(a\) is the speed of sound in the medium. In our exercise, the Mach number of 0.6 implies that the carbon dioxide is moving at 60% of the local speed of sound after it is accelerated isentropically. Understanding Mach number is crucial when analyzing high-speed flows and their effects on objects, such as pressure and temperature changes.

Ideal gas law

The ideal gas law is a fundamental equation in thermodynamics that relates the pressure, volume, and temperature of an ideal gas. It is given by the equation:
\[\begin{equation}Pv = nRT\end{equation}\]
For a fixed mass of gas, this can be rewritten in terms of specific volume (\(v\)) as:
\[\begin{equation}Pv = RT\end{equation}\]
where \(P\) is the pressure, \(v\) is the specific volume, \(R\) is the specific gas constant, and \(T\) is the temperature. It assumes that the gas behaves ideally, meaning the molecules exert no forces upon each other except during elastic collisions. This law is used in the exercise to calculate the initial state of carbon dioxide before it undergoes isentropic acceleration.

Specific volume

Specific volume is a property of materials, defined as the volume occupied by a unit mass of a substance. In thermodynamics, it's especially relevant for gases where it inversely relates to density. The specific volume (\(v\)) is given by:
\[\begin{equation}v = \frac{V}{m}\end{equation}\]
where \(V\) is the volume and \(m\) is the mass. It is expressed in cubic meters per kilogram (\(m^3/kg\)). For an ideal gas, the specific volume can be directly calculated from the ideal gas law, as shown in the exercise. It is a critical parameter in the equation since it changes with temperature and pressure during the isentropic process.

Specific enthalpy

Specific enthalpy is the energy content per unit mass of a system. It is a concept in thermodynamics that represents the total heat content, and it's often used in the analysis of energy transfer systems. For a constant-pressure process, specific enthalpy (\(h\)) can be described in terms of the specific heat and temperature change as follows:
\[\begin{equation}h = c_p(T_2 - T_1)\end{equation}\]
where \(c_p\) is the specific heat at constant pressure, and \(T_1\) and \(T_2\) are the initial and final temperatures. In our context, the specific enthalpy is involved in calculating the final state of the carbon dioxide, along with the specific volume, using the relation between specific enthalpy and temperature.

Adiabatic index

The adiabatic index, also known as the heat capacity ratio or the isentropic expansion factor, is denoted by \(\gamma\) (gamma). It's the ratio of the heat capacity at constant pressure to the heat capacity at constant volume:
\[\begin{equation}\gamma = \frac{c_p}{c_v}\end{equation}\]
For an ideal gas, it relates pressure and volume during an adiabatic process, where no energy as heat is transferred to or from the system. In terms of the adiabatic process, it's used to relate initial and final thermodynamic states through isentropic flow relations, as seen in step 2 of the solution. In carbon dioxide, with a \(\gamma\) of 1.3, it reflects the gas's resistance to changes in volume at constant heat capacity.