Question

Steam enters a long, insulated pipe at \(1200 \mathrm{kPa}\) \(250^{\circ} \mathrm{C},\) and \(4 \mathrm{m} / \mathrm{s},\) and exits at \(1000 \mathrm{kPa}\). The diameter of the pipe is \(0.15 \mathrm{m}\) at the inlet, and \(0.1 \mathrm{m}\) at the exit. Calculate the mass flow rate of the steam and its speed at the pipe outlet

Short answer

Answer: 1. Find the density of steam at the inlet using steam table and given temperature and pressure. 2. Calculate the area of the pipe at the inlet and outlet using given diameters. 3. Calculate the mass flow rate at the inlet using inlet density, pipe area, and speed of steam. 4. Determine the mass flow rate at the outlet, which is the same as at the inlet due to the law of conservation of mass. 5. Find the density of steam at the outlet using the steam table and given outlet pressure and constant steam quality. 6. Calculate the speed of the steam at the outlet using mass flow rate, density, and pipe area at the outlet.

Step-by-step solution

Find the density of steam at the inlet

Go to the steam table and, using the given temperature and pressure, find the specific volume, \(v\), at the inlet. The specific volume is the inverse of density, so calculate the density, \(\rho_{inlet}\), as follows: \(\rho_{inlet} = \frac{1}{v}\). (Note: Water tables and steam tables are available on the internet. You may need to interpolate between the given values to find the density.)

Find the area of the pipe at the inlet and outlet

Use the given diameters to calculate the areas, \(A_{inlet}\) and \(A_{outlet}\), of the pipe at the inlet and outlet respectively. The area of a circle is given by: \(A = \pi\left(\frac{D}{2}\right)^2\).

Calculate the mass flow rate at the inlet

Using the inlet density, the area of the pipe, and the speed of the steam at the inlet, calculate the mass flow rate, \(\dot{m}_{inlet}\), as follows: \(\dot{m}_{inlet} = \rho_{inlet} \times A_{inlet} \times v_{inlet}\).

Determine the mass flow rate at the outlet

Since the steam flow is continuous and no heat is transferred, the mass flow rate at the outlet is the same as the mass flow rate at the inlet in accordance to the law of conservation of mass. Thus, \(\dot{m}_{outlet} = \dot{m}_{inlet}\).

Find the density of steam at the outlet

Go to the steam table again and, using the given outlet pressure and the fact that the steam quality doesn't change, find the specific volume at the outlet. Calculate the density, \(\rho_{outlet}\), as follows: \(\rho_{outlet} = \frac{1}{v}\). (Note: You can use the steam table to find the temperature corresponding to the given pressure and entropy.)

Calculate the speed of the steam at the outlet

Use the mass flow rate at the outlet, the density of the steam at the outlet, and the area of the pipe at the outlet to find the speed, \(v_{outlet}\), as follows: \(v_{outlet} = \frac{\dot{m}_{outlet}}{\rho_{outlet} \times A_{outlet}}\). Now, you've found the mass flow rate of the steam and its speed at the pipe outlet.

Key concepts

Steam Table Usage

Understanding and utilizing steam tables is essential when dealing with problems in thermodynamics involving steam, such as calculating the mass flow rate of steam through a pipe. Steam tables provide various thermodynamic properties of water and steam at different temperatures and pressures. To use a steam table, one must first identify the known properties of the steam—often the pressure and temperature.

With these two parameters, you can find other properties such as specific volume, enthalpy, and entropy. In the problem at hand, the temperature and pressure at the inlet and outlet were given, allowing us to look up the specific volume, which is inversely related to density. If the exact values are not listed in the table, interpolation between the closest available values is required to estimate the properties accurately.

Keep in mind that steam tables are based on the properties of water and steam at equilibrium; hence, they are most accurate for closed systems or steady-flow processes like the one described in our exercise.

Conservation of Mass in Thermodynamics

The conservation of mass is a fundamental concept in physics and engineering, particularly in thermodynamics, stating that mass cannot be created or destroyed in a closed system. This principle is critical when calculating mass flow rates in systems like pipes, turbines, and heat exchangers.

In the context of our steam pipe exercise, we assume the pipe is a control volume with no mass being added or lost along the way, meaning the mass flow rate should remain constant between the inlet and the outlet. This allows us to state that \( \dot{m}_{inlet} = \dot{m}_{outlet} \) irrespective of any changes in pressure, temperature, or pipe diameter. This simplification is widely used in engineering calculations and serves as the basis for continuous flow systems, making our calculations much more straightforward.

Fluid Density and Specific Volume

In thermodynamics, understanding the relationship between a fluid's density and its specific volume is crucial for solving problems involving fluid flow and properties. Density \( (\rho) \) is a measure of how much mass is contained within a unit volume of a substance and is typically expressed in kilograms per cubic meter \( (kg/m^3) \) for fluid dynamics problems.

Specific volume, on the other hand, is the inverse of density and represents the volume occupied by a unit mass of a substance, usually given in cubic meters per kilogram \( (m^3/kg) \) for steam and other gases. To calculate the flow rate of fluids, we need to know either the density or the specific volume along with the velocity and cross-sectional area through which the fluid is moving.

In our exercise, we calculated the mass flow rate by first finding the specific volume via the steam table and then determining the density, which then fed into our mass flow rate computation. It's a straightforward yet powerful relationship that underpins many fundamental calculations in fluid mechanics.