Question

Explain why the maximum flow rate per unit area for a given ideal gas depends only on \(P_{0} / \sqrt{T_{0}} .\) For an ideal gas with \(k=1.4\) and \(R=0.287 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K},\) find the constant \(a\) such that \(\dot{m} / A^{*}=a P_{0} / \sqrt{T_{0}}\).

Short answer

Answer: The maximum flow rate per unit area of an ideal gas depends only on the ratio \(P_{0} / \sqrt{T_{0}}\) because of the isentropic flow condition and the relationships between pressure, density, and temperature in an ideal gas. Through a series of substitutions, we derived the equation \(\frac{\dot{m}}{A} = \frac{P_{0}\left(\frac{2}{k+1}\right)^{\frac{k}{k-1}}\sqrt{k}}{\sqrt{T_{0}}}\), which shows that the flow rate per unit area depends on a constant and the ratio \(P_{0} / \sqrt{T_{0}}\). This relationship holds for maximum flow (choking condition) when the Mach number is equal to 1.

Step-by-step solution

Ideal Gas Law and Flow Characteristics

To start, we will make use of the Ideal Gas Law, which states: $$PV = nRT$$ Where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is the temperature. For the problem we are given, the gas constant is per unit mass. So, we can rewrite the Ideal Gas Law as: $$P = \rho RT$$ Where \(\rho\) is the density of the ideal gas. Now, let's look at the mass flow rate per unit area, which can be expressed as: $$\frac{\dot{m}}{A} = \rho u$$ Where \(u\) is the flow velocity and \(A\) is the cross-sectional flow area.

Express Density in Terms of Pressure and Temperature

By making use of the Ideal Gas Law, we can express the density as: $$\rho = \frac{P}{RT}$$ Which gives us: $$\frac{\dot{m}}{A} = \frac{Pu}{RT}$$

Apply Isentropic Flow Relation

For isentropic flow (no heat transfer and negligible friction), we have a relationship involving the Mach number \(M\) (ratio of flow velocity to the speed of sound \(a\)) and the specific heat ratio \(k\): $$\left(\frac{P}{P_{0}}\right) = \left(1 + \frac{k-1}{2}M^{2}\right)^{-\frac{k}{k-1}}$$ For maximum flow (choking condition) with Mach number \(M=1\): $$\left(\frac{P}{P_{0}}\right)_{max} = \left(\frac{2}{k+1}\right)^{\frac{k}{k-1}}$$

Derive the Flow Rate per Unit Area

Substituting the pressure relation into our mass flow rate equation and using the maximum pressure ratio, we have: $$\frac{\dot{m}}{A} = \frac{P_{0}\left(\frac{2}{k+1}\right)^{\frac{k}{k-1}}u}{RT_{0}}$$ The flow velocity is related to the speed of sound \(a\) by the Mach number \(M\): $$u = Ma = M \sqrt{kRT_{0}}$$ At choking condition (maximum flow), \(M=1\), so: $$u = \sqrt{kRT_{0}}$$ Now, substituting the flow velocity back into the flow rate per unit area equation, we get: $$\frac{\dot{m}}{A} = \frac{P_{0}\left(\frac{2}{k+1}\right)^{\frac{k}{k-1}}\sqrt{kRT_{0}}}{RT_{0}}$$

Simplify the Equation

Simplifying the equation, we get: $$\frac{\dot{m}}{A} = \frac{P_{0}\left(\frac{2}{k+1}\right)^{\frac{k}{k-1}}\sqrt{k}}{\sqrt{T_{0}}}$$

Compute the Constant \(a\)

For \(k = 1.4\) and \(R = 0.287\ \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\), we can compute the constant \(a\): $$a = \left(\frac{2}{k+1}\right)^{\frac{k}{k-1}}\sqrt{k} = \left(\frac{2}{1.4+1}\right)^{\frac{1.4}{1.4-1}}\sqrt{1.4} \approx 0.740$$ Therefore, the constant \(a\) for the given ideal gas is: $$a \approx 0.740$$ The flow rate per unit area for the given ideal gas can now be expressed as: $$\frac{\dot{m}}{A} = \frac{0.740 P_{0}}{\sqrt{T_{0}}}$$

Key concepts

Isentropic Flow

When we deal with the flow of gases, particularly in applications like jet engines or nozzles, isentropic flow is a common assumption that simplifies the analysis. Isentropic flow refers to an idealized process in which a fluid or gas flows in a manner that is both reversible and adiabatic. In simpler terms, it means that there is no heat exchange with the surroundings (adiabatic) and no energy is lost to friction or dissipation (reversible).

In the context of an ideal gas, such isentropic conditions imply that all changes happening in the flow are due to changes in the gas's pressure and density while maintaining entropy constant. The equations governing such flow are derived from the first law of thermodynamics and the ideal gas law, which are then used to understand and predict the behavior of the gas under varying conditions.

For instance, when a gas expands in a nozzle, if the process is isentropic, the temperature and pressure of the gas decreases while the velocity increases, all without losing energy. This is particularly critical when calculating the maximum flow rate, as the efficiency of the flow process in devices like turbines or compressors is directly linked to this assumption of isentropic change.

Mach Number

The Mach number is a dimensionless quantity used in fluid dynamics to characterize the speed of a flow relative to the speed of sound within the same medium. It is defined as the ratio of the object's speed to the speed of sound:\[M = \frac{u}{a}\]where \(u\) is the flow velocity and \(a\) is the speed of sound in the medium. For example, Mach 1 indicates the flow velocity is equal to the speed of sound, Mach 0.5 signifies half the speed of sound, and so on.

This number is critical in the study of gas dynamics because it helps to classify the type of flow. At Mach numbers less than 1, the flow is considered subsonic; at Mach 1, it's transonic; above Mach 1 but below Mach 5, it's supersonic; and above Mach 5, it is hypersonic.

In isentropic flow conditions, which are inherently ideal and without heat exchange or friction, the Mach number helps in determining how the properties of the gas, like pressure and temperature, change relative to its flow velocity. This is especially important for predicting the behavior of the gas under 'choked' conditions, where the Mach number will be equal to 1 and the flow rate reaches its maximum for a given cross-sectional area.

Mass Flow Rate per Unit Area

The mass flow rate per unit area, often referred to as the mass flux, is the amount of mass flowing through a particular cross-sectional area per unit time. It is denoted by the equation:\[\frac{\dot{m}}{A}=\rho u\]where \(\dot{m}\) is the mass flow rate, \(A\) is the flow area, \(\rho\) is the density of the fluid or gas, and \(u\) is the flow velocity.

When analyzing the flow of an ideal gas, especially under the assumption of isentropic conditions, the mass flow rate per unit area becomes a pivotal factor in the design and operation of various aerodynamic and thermodynamic systems. It allows engineers to estimate the capacity of jets, nozzles, and engines in terms of how much gas can pass through per unit time and area.

Furthermore, it is linked with the concept of choked flow – a condition where the flow rate does not increase with a further decrease in downstream pressure past a critical point. In such conditions, the mass flow rate per unit area achieves a maximum; this is crucial for the safe and efficient design of propulsion systems and industrial gas flow processes. Understanding this concept helps in optimizing such systems to achieve maximum throughput without crossing the limits imposed by physics.