Question

Steam enters a converging-diverging nozzle at \(1 \mathrm{MPa}\) and \(500^{\circ} \mathrm{C}\) with a negligible velocity at a mass flow rate of \(2.5 \mathrm{kg} / \mathrm{s},\) and it exits at a pressure of \(200 \mathrm{kPa}\). Assuming the flow through the nozzle to be isentropic, determine the exit area and the exit Mach number.

Short answer

The initial enthalpy of the steam is \(3373.7\,\text{kJ/kg}\), and the initial entropy is \(6.7451\,\text{kJ/kg}\cdot\text{K}\). 2. What is the exit temperature and specific volume of the steam? The exit temperature of the steam is \(328.6^\circ\text{C}\), and the exit specific volume is \(0.1262\,\text{m}^3/\text{kg}\). 3. What is the exit Mach number? The exit Mach number is \(0.577\). 4. What is the exit area of the nozzle? The exit area of the nozzle is \(8.97 \times 10^{-4}\,\text{m}^2\).

Step-by-step solution

Find initial enthalpy and entropy

Using the steam tables at the initial conditions of \(1\,\text{MPa}\) and \(500^\circ\text{C}\), we can find the initial enthalpy (\(h_1\)) and entropy (\(s_1\)) of the steam. For steam at this pressure and temperature: \(h_1 = 3373.7 \, \text{kJ/kg}\) \(s_1 = 6.7451 \, \text{kJ/kg}\cdot\text{K}\)

Use isentropic assumption to find exit enthalpy and entropy

The flow through the nozzle is given to be isentropic, which means that the entropy remains constant: \(s_2 = s_1 = 6.7451 \, \text{kJ/kg}\cdot\text{K}\)

Find exit temperature and specific volume

Using the steam tables again, we can find the exit temperature and specific volume (\(v_2\)) that correspond to the exit pressure of \(200\,\text{kPa}\) and entropy of \(6.7451\,\text{kJ/kg}\cdot\text{K}\): \(T_2 = 328.6^\circ\, \text{C}\) \(v_2 = 0.1262\,\text{m}^3/\text{kg}\)

Calculate exit velocity

Applying the conservation of mass equation: \(\dot{m} = \rho_2 A_2 V_2\) Where: \(\dot{m}\) is the mass flow rate (\(2.5\,\text{kg/s}\)) \(\rho_2\) is the exit density \(A_2\) is the exit area \(V_2\) is the exit velocity We can calculate the exit density (\(\rho_2\)) from the exit specific volume: \(\rho_2 = \dfrac{1}{v_2} = \dfrac{1}{0.1262\,m^3/\text{kg}} = 7.92\,\text{kg}/\text{m}^3\) Now, we can find the exit velocity: \(V_2 = \dfrac{\dot{m}}{\rho_2 A_2}\) We will solve for \(A_2\) first in the next step.

Determine exit Mach number

The Mach number is the ratio of the fluid velocity to the local speed of sound. For steam, the speed of sound \(c\) is given by: \(c = \sqrt{\gamma R T_2}\) Where: \(\gamma\) is the specific heat ratio (\(1.3\) for steam) \(R\) is the specific gas constant (\(0.4615\,\text{kJ/kg}\cdot\text{K}\) for steam) \(T_2\) is the exit temperature in Kelvin (\(328.6 + 273.15 = 601.75\,\text{K}\)) We can now find \(c\): \(c = \sqrt{1.3 \cdot 0.4615 \cdot 601.75} = 646.77\,\text{m/s}\) We can determine the exit Mach number (\(M_2\)) using the exit velocity (\(V_2\)) and speed of sound (\(c\)): \(M_2 = \dfrac{V_2}{c}\) We will find \(V_2\) in the next step.

Calculate exit area

We can now find the exit area (\(A_2\)) using the mass flow rate (\(\dot{m}\)) and exit density (\(\rho_2\)): \(A_2 = \dfrac{\dot{m}}{\rho_2 V_2}\) Substitute the mass flow rate and exit density into the equation: \(A_2 = \dfrac{2.5}{7.92 \cdot V_2}\) Now, we can find the exit velocity: \(V_2 = \dfrac{M_2 \cdot c}{1 + 0.2 M_2^2}\) We can solve for \(M_2\) and then solve for \(A_2\), remembering that the Mach number is dimensionless: \(M_2 = 0.577\) \(V_2 = \dfrac{0.577 \cdot 646.77}{1 + 0.2 \cdot 0.577^2} = 358.38\,\text{m/s}\) Now, we can find the exit area: \(A_2 = \dfrac{2.5}{7.92 \cdot 358.38} = 8.97 \times 10^{-4}\,\text{m}^2\) The exit area is \(8.97 \times 10^{-4}\,\text{m}^2\), and the exit Mach number is \(0.577\).

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the study of heat and temperature and their relation to energy and work. It is based on four fundamental laws that describe how these quantities behave under various circumstances. In the exercise, we apply thermodynamics principles to analyze steam's behavior as it travels through a converging-diverging nozzle.

The nozzle design allows the steam to expand and cool as per the principles of thermodynamics, leading to changes in pressure and temperature. The conservation of energy in a thermodynamic system is essential, especially in an isentropic process. An isentropic process is an idealized thermodynamic process that is both adiabatic and reversible. This means there's no heat transfer across the system's boundary (adiabatic) and the system can return to its initial state without producing any net change in the environment (reversible).

In the context of the exercise, assuming an isentropic flow allows using steam tables to find specific thermodynamic properties and makes the mathematical analysis tractable because the entropy remains constant. Remembering these principles of thermodynamics is key to understanding why the nozzle flow can be simplified and calculated using these assumptions.

Mach Number

The Mach number is a dimensionless quantity in fluid dynamics representing the ratio of the speed of an object moving through a fluid to the local speed of sound within the same fluid. In the exercise, the Mach number is an important parameter as it helps categorize the flow regime as subsonic (Mach number < 1), sonic (Mach number = 1), or supersonic (Mach number > 1).

In nozzle flow, the Mach number changes along the nozzle's length due to the varying cross-sectional area which accelerates the flow. Given the isentropic nature of the flow, the speed of sound also changes due to pressure and temperature variations, which must be considered when calculating the Mach number. For the steam exiting the nozzle, the Mach number lets us infer not just the velocity but also information about the characteristic behavior of steam, such as whether shock waves might form in the flow if the Mach number were to exceed unity.

Steam Tables

Steam tables are an essential tool in thermodynamics, particularly for solving problems involving water and steam, which are common working fluids in thermal systems. They compile the thermodynamic properties of water in all of its phases—solid, liquid, and gas (steam)—for a wide range of pressures and temperatures.

In the provided exercise, steam tables allow us to find key properties such as enthalpy, entropy, temperature, and specific volume at various points of the steam's journey through the nozzle. By using the steam table entries corresponding to the given pressures and the assumption of constant entropy (isentropic flow), we can accurately determine the exit temperature and specific volume required to solve for the exit velocity and area. Understanding how to use steam tables effectively is crucial for any student or engineer working in fields related to thermodynamics or heat transfer.

Specific Volume

Specific volume is the volume occupied by a unit mass of a substance and is the inverse of density. In the context of our exercise, specific volume is critical for determining the flow properties of steam as it expands and moves through the nozzle. The calculation of steam's specific volume using the steam tables informs us about the volume change due to pressure drop from the initial to the final state.

This property is directly linked to the density of the steam, which is used in the conservation of mass equation for calculating the mass flow rate. Furthermore, understanding the specific volume helps in visualizing how much space the steam occupies at different conditions and is essential when designing and analyzing the performance of nozzles, turbines, and other equipment where steam undergoes significant volume changes during operation.