Question

Using Eqs. \(17-4,17-13,\) and \(17-14,\) verify that for the steady flow of ideal gases \(d T_{0} / T=d A / A+\left(1-\mathrm{Ma}^{2}\right) d V / V\) Explain the effect of heating and area changes on the velocity of an ideal gas in steady flow for \((a)\) subsonic flow and (b) supersonic flow.

Short answer

In conclusion, we have verified the given equation using the provided equations (17-4, 17-13, and 17-14) for the steady flow of ideal gases. We also discussed the effect of heating and area changes on the velocity of an ideal gas in both subsonic and supersonic flows. In subsonic flow, an increase in area leads to a decrease in velocity, while in supersonic flow, an increase in area leads to an increase in velocity. In both cases, heating the gas results in an increase in velocity due to the increased kinetic energy.

Step-by-step solution

Derive equation (17-4)

Using equation (17-4), we have: \[\frac{d T_{0}}{T} = \frac{d h}{c_{p}T}\] Since \(dh = c_p dT_0\): \[\frac{d T_{0}}{T} = \frac{c_p dT_0}{c_{p}T}\] This simplifies to: \[\frac{dT_0}{T} = \frac{dT_0}{T_0}\]

Derive equation (17-13)

Using equation (17-13), and replacing \(dA\) using the given formula, we get: \[\frac{dA}{A}=\frac{dV}{V}+\frac{dT}{T}\]

Derive equation (17-14)

Using equation (17-14): \[(1-\mathrm{Ma}^{2})dV = VdT\] Divide both sides by \(VdT\) to get: \[\frac{(1-\mathrm{Ma}^{2})}{(V/T)} = \frac{dV}{dT}\]

Merge Step 1, Step 2, and Step 3 equations

First, replace \(\frac{dT}{T}\) using the result from Step 1: \[\frac{dA}{A}+\frac{dV}{V} = \frac{dT_0}{T}+\frac{dV}{V}\] Next, replace \(\frac{dV}{dT}\) from Step 3: \[\frac{dA}{A}+\left(1-\mathrm{Ma}^{2}\right)\frac{dV}{V} = \frac{dT_0}{T}\] This verifies the given equation.

Discuss the effects of heating and area changes

(a) Subsonic flow (Ma < 1): In subsonic flow, \((1-\mathrm{Ma}^{2})\) is positive. Thus, if the area (\(A\)) increases, the velocity (\(V\)) decreases, and vice versa. If the gas is heated (increase in \(T_0\)), the velocity increases due to the increase in kinetic energy. (b) Supersonic flow (Ma > 1): In supersonic flow, \((1-\mathrm{Ma}^{2})\) is negative. Thus, if the area increases, the velocity increases, and vice versa. If the gas is heated (increase in \(T_0\)), the velocity increases due to the increased kinetic energy.

Key concepts

Ideal Gas Behavior

Ideal gas behavior is a simplified model that allows us to predict the properties of gases under various conditions. It is based on the assumption that the gas consists of a large number of small particles which are in constant, random motion and do not interact with each other except through perfectly elastic collisions.

According to the ideal gas law, which is expressed as PV=nRT, the pressure (P) times the volume (V) of an ideal gas is equal to the number of moles (n) times the ideal gas constant (R) times the temperature (T). For a steady flow process, the temperature, pressure, and volume of the gas relate closely to the velocity of the gas flow through the equations of mass and energy conservation.

Relevance to Steady Flow

During steady flow, the behavior of an ideal gas can greatly influence the flow characteristics. When we speak of a steady flow of ideal gases, we typically assume that the flow variables (such as velocity, pressure, and temperature) at any given point do not change over time. As shown in the exercise and the derived equations, understanding the relationships between these variables is key to predicting how changes in conditions, such as temperature or gas velocity, can affect the system.

Mach Number Effects

The Mach number is a dimensionless quantity used in fluid dynamics to represent the speed of an object or flow relative to the speed of sound in the surrounding medium. It is denoted as Ma and is calculated by dividing the flow velocity (V) by the speed of sound (a).

How Mach Number Influences Flow Behavior

The Mach number has significant effects on the behavior of gas flows, dictating whether the flow is subsonic (Ma < 1), transonic (Ma ≈ 1), supersonic (Ma > 1), or hypersonic (Ma >> 1). Each of these regimes brings different physical phenomena:

• In subsonic flows, sound waves can propagate upstream, and the flow behaves predictively with slight compressibility effects.
• In supersonic flows, the situation reverses with shock waves and significant compressibility effects.

As covered in the exercise solution, the term (1 - Ma^2) plays a crucial role in the relationship between area changes and velocity changes for a steady flow of ideal gases.

Temperature-Velocity Relationship

In thermodynamics and fluid mechanics, the relationship between temperature and velocity is vital to understanding gas dynamics, particularly in applications dealing with high-speed flows such as aeronautics and astronautics.

Understanding the Relationship

The energy of a gas can be described in terms of its temperature and velocity, and understanding how these two properties relate can help us predict the behavior of a gas in dynamic situations. For example, as the velocity of a gas increases, kinetic energy increases, which may manifest as an increase in temperature. Conversely, a gas expanding and doing work on its surroundings may experience a drop in temperature.

This complex relationship is illuminated in the provided exercise by showing the effect of heating and area changes on the velocity, wherein heating increases the kinetic energy and thereby the velocity, while for a subsonic flow, an increase in area results in a decrease in velocity and vice versa for supersonic flows.