Question

Steam enters a nozzle with a low velocity at \(150^{\circ} \mathrm{C}\) and \(200 \mathrm{kPa}\), and leaves as a saturated vapor at \(75 \mathrm{kPa}\). There is a heat transfer from the nozzle to the surroundings in the amount of \(26 \mathrm{kJ}\) for every kilogram of steam flowing through the nozzle. Determine ( \(a\) ) the exit velocity of the steam and (b) the mass flow rate of the steam at the nozzle entrance if the nozzle exit area is \(0.001 \mathrm{m}^{2}\)

Short answer

Answer: To find the exit velocity of the steam (a) and the mass flow rate of the steam at the nozzle entrance (b), follow these steps: 1. Use steam tables to find the initial enthalpy (h1) and final enthalpy (h2) at the given initial temperature and pressure, as well as at the final pressure when the steam leaves the nozzle as a saturated vapor. 2. Apply the energy balance equation in the nozzle, considering the heat transfer from the nozzle to the surroundings and the low initial velocity of the steam (V1 ≈ 0). 3. Calculate the exit velocity of the steam (V2) using the enthalpy values obtained in Step 1 and the energy balance equation obtained in Step 2. 4. Find the volume flow rate at the exit of the nozzle (V2A2) using the specific volume (v2) of the steam at the exit and the exit area (A2). 5. Calculate the mass flow rate of the steam at the nozzle entrance (m) using the volume flow rate at the exit and the specific volume of the steam at the exit. Following these steps will give you the exit velocity of the steam (a) and the mass flow rate of the steam at the nozzle entrance (b).

Step-by-step solution

Find the initial and final enthalpy of the steam

We can use the steam tables to find the initial enthalpy \(h_{1}\) and final enthalpy \(h_{2}\) at the given initial temperature and pressure, as well as at the final pressure when the steam leaves the nozzle as a saturated vapor.

Apply energy balance equation in the nozzle

Using the energy balance equation, we can write: \(h_{1} + \frac{V_1^2}{2} = h_{2} + \frac{V_2^2}{2} + q_{out}\) Where \(V_1\) and \(V_2\) are the initial and final velocities of the steam and \(q_{out}\) is the heat transfer from the nozzle to the surroundings. The steam enters the nozzle with a low velocity, so \(V_1 \approx 0\). Plug in the given value of heat transfer rate per kilogram of steam and rearrange the equation to find the exit velocity \(V_2\).

Calculate the exit velocity of the steam

Using the enthalpy values obtained in Step 1 and the energy balance equation obtained in Step 2, calculate the exit velocity of the steam, \(V_2\).

Find the volume flow rate at the exit of the nozzle

We can find the specific volume \(v_{2}\) of the steam at the exit using the steam table at the saturated vapor condition. Then, we can calculate the volume flow rate \(\dot{V_{2}}\) at the exit of the nozzle using the exit area \(A_{2}\): \(\dot{V_{2}} = V_{2}A_{2}\)

Calculate the mass flow rate of the steam at the nozzle entrance

Finally, we can calculate the mass flow rate \(\dot{m}\) of the steam at the nozzle entrance using the volume flow rate at the exit and the specific volume of the steam at the exit: \(\dot{m}=\frac{\dot{V_{2}}}{v_{2}}\) At the end of these steps, you will have found the exit velocity of the steam (\(a\)) and the mass flow rate of the steam at the nozzle entrance (\(b\)).

Key concepts

Enthalpy of Steam

Enthalpy, often denoted as 'h', is a measure of the total energy of a thermodynamic system, and it includes both the system's internal energy and the energy required to make room for it by displacing its environment. In the context of steam, enthalpy is crucial because it helps determine how much energy steam contains.

Steam can exist in different phases – as a liquid, a saturated mixture, or a superheated vapor – and each phase has its own enthalpy value. For the saturated vapor phase, enthalpy is often referenced from tables known as steam tables, which provide values for specific temperatures and pressures. When steam enters a nozzle and is accelerated, its enthalpy undergoes changes, which is essential to solving problems related to nozzle thermodynamics.

Understanding the enthalpy of steam is necessary for many engineering applications, such as turbines, condensers, and nozzles, where the energy transfer in the form of work or heat depends significantly on the enthalpy change of the steam flowing through the system.

Energy Balance Equation

The energy balance equation is a statement of the conservation of energy principle, which asserts that energy cannot be created or destroyed, only transformed from one form to another. In the case of nozzles, the energy balance is applied to the flowing steam to determine how its energy changes as it passes through the nozzle.

The general form of the energy balance for a steady-flow process is: \[h_1 + \frac{V_1^2}{2} = h_2 + \frac{V_2^2}{2} + q_{out}\]This equation reflects that the sum of enthalpy (\(h\)) and kinetic energy (\(\frac{V^2}{2}\)) is the same at the beginning and the end of the process, accounting for any heat loss or gain (\(q_{out}\)). For the problem at hand, the steam initially has low velocity, which simplifies the energy balance by minimizing the initial kinetic energy term, causing the enthalpy change and heat loss to become the primary focus for calculating the exit velocity of the steam.

Saturated Vapor

A saturated vapor is a state where a substance exists as vapor at its boiling point under a given pressure. At this condition, the vapor is in equilibrium with its liquid phase – meaning that there's no net evaporation or condensation occurring. In our exercise, the steam leaves the nozzle as a saturated vapor at a lower pressure of 75 kPa.

The significance of being 'saturated' is that the temperature and pressure are no longer independent variables. For steam, this means that if you know the pressure of the saturated vapor, you can determine its temperature and vice versa using the steam tables. The enthalpy of the saturated vapor (also available from steam tables) is a crucial parameter for engineers, as it allows them to determine the thermal energy content of the steam at this specific phase without needing to perform complex calculations.

Mass Flow Rate

Mass flow rate, usually denoted by \(\dot{m}\), is the mass of a substance that passes through a given surface per unit time. It is a critical factor in the analysis of fluid flow in various engineering systems, especially when looking at nozzles or any other device where fluid is in motion.

Mass flow rate can be calculated by dividing the volumetric flow rate by the substance's specific volume (\(v\)) as shown in the equation \(\dot{m} = \frac{\dot{V}}{v}\) where \(\dot{V}\) is the volumetric flow rate. The specific volume is another term from thermodynamics, indicating the volume occupied by a unit mass of a substance. In the case of steam, knowing the mass flow rate is essential for understanding how much steam is passing through the nozzle, which subsequently influences the generation of thrust or power in applications such as steam turbines or jet engines.