Question

Steam at \(6.0 \mathrm{MPa}\) and \(700 \mathrm{K}\) enters a converging nozzle with a negligible velocity. The nozzle throat area is \(8 \mathrm{cm}^{2} .\) Approximating the flow as isentropic, plot the exit pressure, the exit velocity, and the mass flow rate through the nozzle versus the back pressure \(P_{b}\) for \(6.0 \geq\) \(P_{b} \geq 3.0\) MPa. Treat the steam as an ideal gas with \(k=1.3\) \(c_{p}=1.872 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K},\) and \(R=0.462 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\).

Short answer

Answer: To analyze the flow and determine the relationship between exit properties and back pressure in a converging nozzle, follow these key steps: 1. Use isentropic relations with the given initial steam conditions to determine the exit properties for different back pressures. 2. Apply the ideal gas law for the given conditions and solve for the exit temperature, finding a relationship between exit temperature and pressure. 3. Calculate the exit pressure, mass flow rate, and exit velocity for different back pressures using the derived equations from step 2. 4. Plot the calculated exit properties against the back pressure within the given range (3.0-6.0 MPa) to show the relationship between the exit properties and back pressure in the converging nozzle.

Step-by-step solution

Isentropic relations

With the given initial conditions of steam, isentropic relations can be used to determine the exit properties of the steam for different back pressures. The isentropic relation we will use is: \(P_2 = P_1 \cdot \left( \frac{T_2}{T_1} \right)^{\frac{k}{k - 1}}\) Label the initial condition properties as \(P_1\), \(T_1\) and the exit properties as \(P_2\), \(T_2\). We will need to find a relationship between \(T_2\) and \(P_2\) to use this formula.

Ideal gas law

Apply the ideal gas law for the given conditions and solve for the exit temperature: \(P_1 V_1 = m R T_1\) \(P_2 V_2 = m R T_2\) Divide these two equations: \(\frac{P_1 V_1}{P_2 V_2} = \frac{T_1}{T_2}\) Since we have a converging nozzle, we can approximate the volume ratio: \(\frac{V_1}{V_2} \approx \frac{A_1}{A_2}\) Combining the equations: \(\frac{P_1}{P_2} \frac{A_1}{A_2} = \frac{T_1}{T_2}\) Now, we can establish the relationship between \(T_2\) and \(P_2\): \(T_2 = T_1 \cdot \frac{P_2}{P_1} \cdot \frac{A_2}{A_1}\)

Calculate exit properties

Plug in the calculated exit temperature into the isentropic relation, and calculate the mass flow rate and exit velocity, by adjusting the back pressure within given range (3.0-6.0 MPa). Using the ideal gas law for the mass flow rate: \(m = \frac{P_1 A_1}{R T_1}\) Using the isentropic relations to find the exit velocity: \(v_2 = \sqrt{2 c_p (T_1 - T_2)}\)

Plot results

For each value of back pressure within the given range (3.0-6.0 MPa), calculate the exit pressure, exit velocity, and mass flow rate using the derived equations, and plot them against the back pressure. The plots will show the relationship between the exit properties and back pressure in the converging nozzle with the given initial conditions.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, and temperature of an ideal gas. This law is typically written as
\(PV = nRT\),
where

• \(P\) is the pressure of the gas,
• \(V\) is the volume,
• \(n\) is the number of moles,
• \(R\) is the universal gas constant, and
• \(T\) is the temperature.

In the context of the isentropic nozzle flow, the Ideal Gas Law is used to understand how a gas behaves when it is forced through a nozzle. By assuming the gas is ideal, you can apply this equation to calculate essential properties such as exit temperature and mass flow rate, as highlighted in the textbook exercise. The law simplifies the complex interactions of particles within the gas to a manageable level, allowing for more straightforward calculations of state changes, which is invaluable for engineers and physicists. The equation is adjusted to account for mass flow by using mass-specific values, where \(R\) becomes the specific gas constant, and the amount of substance is represented by mass \(m\) instead of moles. This method simplifies the equation to
\(PV = mRT\),
which becomes crucial when analyzing the gas flow through nozzles.

Converging Nozzle

A converging nozzle is a device that accelerates a fluid to high speeds by narrowing its flow area. When a gas passes through a converging nozzle, its velocity increases as it moves towards the narrowest section, or the 'throat' of the nozzle, where the cross-sectional area is smallest.

During isentropic flow, which is a reversible adiabatic process, the gas remains constant in entropy, meaning no heat is transferred into or out of the system, and no entropy is generated within the system. In real-life applications, converging nozzles are commonly used in jet engines and rocket propulsion, where high-speed gas ejection is required to produce thrust.

According to the exercise problem, the volume flow rate is inversely proportional to the cross-sectional area of the nozzle when assuming steady, one-dimensional flow. This inverse relationship allows for simplifying the Ideal Gas Law to analyze the relationships between temperature, pressure, and velocity in such a nozzle. Understanding how the area affects these variables is vital for practical designs and optimizations in engineering systems.

Isentropic Relations

Isentropic relations are equations that describe the behavior of a fluid undergoing an isentropic process, which is an idealization where the process is both adiabatic and reversible, meaning no heat transfer occurs, and there are no losses due to friction or shock waves. These relations are critical to nozzle flow analysis since they give engineers the ability to predict the changes in temperature, pressure, velocity, and other properties as a gas expands and accelerates through the nozzle.

For instance, the isentropic relation that correlates the pressure and temperature of a gas is given by
\(P_2 = P_1 \left( \frac{T_2}{T_1} \right)^{\frac{k}{k - 1}}\),
where

• \(P_1\) and \(P_2\) are the initial and final pressures,
• \(T_1\) and \(T_2\) are the initial and final temperatures, and
• \(k\) is the specific heat ratio of the gas.

The key factor here is the specific heat ratio, which depends on the type of gas. This ratio is used in determining how a particular gas will behave under isentropic conditions. As shown in the exercise, employing isentropic relations provides a method to evaluate the performance and efficiency of nozzles in terms of pressure, velocity, and mass flow rate for varying back pressures. This process heavily relies on an understanding of thermodynamics and fluid mechanics, which are critical in various engineering applications such as power generation, propulsion, and refrigeration systems.