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Chapter 11: Problem 13

The mass flow rate of superheated steam \((\gamma=1.3)\) is to be measured by a venturi-meter having inlet and throat diameters \(100 \mathrm{~mm}\) and \(50 \mathrm{~mm}\) respectively. If the upstream stagnation pressure and stagnation temperature are respectively \(200 \mathrm{kPa}\) (absolute) and \(150^{\circ} \mathrm{C}\), what is the maximum mass flow rate that can be measured reliably in this way? Effects of friction and heat transfer may be neglected. (Thermodynamic tables will be needed.)

Short Answer

Expert verified
Calculate the mass flow rate using the properties from steam tables and continuity and Bernoulli’s equation in a venturi meter under maximum (choked) flow conditions.

Step by step solution

01

Determine relevant properties from tables

Identify the properties of superheated steam at the given stagnation conditions. From the steam tables, for superheated steam at a pressure of 200 kPa and temperature of 150°C, obtain the specific volume \(v_1\), specific gas constant \(R\), and the ratio of specific heats \((\text{gamma} = \gamma)\).
02

Apply continuity equation in the venturi meter

Use the continuity equation, which states \(A_1 v_1 = A_2 v_2\). Calculate the areas \(A_1\) and \(A_2\) for the inlet and throat using \(\text{Area} = \pi \frac{d^2}{4}\). From here, obtain the relationship between the velocities \(v_1\) and \(v_2\).
03

Use Bernoulli’s equation to connect pressure and velocity

Apply Bernoulli’s equation between the inlet and the throat of the venturi meter: \[\frac{p_1}{\rho_1} + \frac{v_1^2}{2} = \frac{p_2}{\rho_2} + \frac{v_2^2}{2}\]. With negligible heat transfer and friction, this simplifies to relate the velocities and pressures at these points.
04

Calculate Mach number and mass flow rate

Since the question asks for maximum mass flow rate, use the fact that maximum flow occurs at choked flow condition, where the Mach number at the throat is 1. Calculate the mass flow rate \( \text{mass flow rate} = A_2 \rho_2 v_2 \) at this condition. Use the relationship \(\text{mass flow rate} = \frac{A_2 p_t}{\text{specific volume} \sqrt{\text{gamma} R T_t}}\) where \(p_t\) and \(\rho_t\) are the stagnation pressure and density, respectively.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Venturi Meter
A Venturi meter is a device used to measure the flow rate of fluid through a pipe. It uses the principle of pressure difference created by the change in cross-sectional area of the pipe. At the throat of the Venturi meter, where the diameter is smallest, the fluid velocity increases, causing a drop in pressure. This pressure difference is used to calculate the flow rate.

To understand it better:
  • The Venturi meter consists of three parts: the inlet, the throat, and the outlet.
  • Fluid flow is streamlined, and the pressure difference gives insight into the flow rate.
  • It’s ideal for measuring high flow rates due to minimal pressure losses.

For example, in the given problem, we have inlet and throat diameters, which we can use to find the areas and velocities using the continuity equation and Bernoulli’s equation.
Superheated Steam
Superheated steam is steam that is heated beyond its boiling point without increasing pressure. At this state, it is not in equilibrium with water. It is often used in power plants because it carries more energy, improving efficiency.

Key points about superheated steam:
  • It is less prone to condensation, which can be crucial in preventing damage in turbines.
  • Its properties depend heavily on pressure and temperature.
  • For the given exercise, steam tables are used to find specific properties like specific volume and ratio of specific heats (gamma).

For instance, at 200 kPa and 150°C, the steam has certain specific properties that are essential for calculating the mass flow rate using the Venturi meter.
Bernoulli’s Equation
Bernoulli’s equation is a fundamental principle in fluid dynamics. It states that the total mechanical energy of the flowing fluid remains constant if the fluid flow is steady, incompressible, and there are no frictional losses.

The equation is expressed as:

\[ \frac{p_1}{\rho} + \frac{v_1^2}{2} + gz_1 = \frac{p_2}{\rho} + \frac{v_2^2}{2} + gz_2 \] \ where \( \frac{p}{\rho} \) is the pressure energy, \( \frac{v^2}{2} \) is the kinetic energy, and \( gz \) is the potential energy per unit volume.

For the Venturi meter problem:
  • Potential energy change \( (gz) \) is usually neglected in horizontal flow.
  • We simplify the equation to relate pressure and velocity at two different points (inlet and throat).

This simplified Bernoulli’s equation helps determine the flow velocity at different points, which is crucial for finding the mass flow rate.
Continuity Equation
The continuity equation ensures the conservation of mass in fluid flow. It states that the mass flow rate must remain constant from one cross-section of a pipe to another if the flow is steady.

It is expressed as:

\[ A_1 v_1 = A_2 v_2 \] \ where \( A_1 \) and \( A_2 \) are the cross-sectional areas at points 1 and 2, \( v_1 \) and \( v_2 \) are the fluid velocities at these points.

For the Venturi meter:
  • The cross-sectional areas are calculated using the formula \( A = \pi \frac{d^2}{4} \).
  • By using the continuity equation, we relate velocities between the inlet and throat.

This relationship, coupled with Bernoulli’s equation, allows us to solve for unknown velocities and mass flow rate.

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Most popular questions from this chapter

Determine the mass flow rate of air under adiabatic conditions through a \(50 \mathrm{~mm}\) diameter pipe \(85 \mathrm{~m}\) long from a large reservoir in which the pressure and temperature are \(300 \mathrm{kPa}\) and \(15^{\circ} \mathrm{C}\). The pressure at the outlet of the pipe is \(120 \mathrm{kPa}\). Assume a negligible pressure drop at the pipe inlet and a mean value of \(0.006\) for \(f\). To what value would the inlet pressure need to be raised to increase the mass flow rate by \(50 \%\) ?

A normal shock wave forms in front of a two-dimensional blunt-nosed obstacle in a supersonic air stream. The pressure at the nose of the obstacle is three times the static pressure upstream of the shock wave. Determine the upstream Mach number, the density ratio across the shock and the velocity immediately after the shock if the upstream static temperature is \(10^{\circ} \mathrm{C}\). If the air were subsequently expanded isentropically to its original pressure what would its temperature then be?

An aircraft flies at \(8000 \mathrm{~m}\) altitude where the atmospheric pressure and temperature are respectively \(35.5 \mathrm{kPa}\) and \(-37^{\circ} \mathrm{C}\). An air-speed indicator (similar to a Pitot-static tube) reads \(740 \mathrm{~km} \cdot \mathrm{h}^{-1}\), but the instrument has been calibrated for variable-density flow at sea-level conditions (101.3 kPa and \(15^{\circ} \mathrm{C}\) ). Calculate the true air speed and the stagnation temperature.

A Pitot-static tube in a wind-tunnel gives a static pressure reading of \(40.7 \mathrm{kPa}\) and a stagnation pressure \(98.0 \mathrm{kPa}\). The stagnation temperature is \(90{ }^{\circ} \mathrm{C}\). Calculate the air velocity upstream of the Pitot-static tube.

A convergent-divergent nozzle originally designed to give an exit Mach number of \(1.8\) with air is used with argon \((\gamma=5 / 3)\). What is the ratio of the entry stagnation pressure to the exit pressure when a normal shock is formed just inside the nozzle exit?
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