Chapter 3

Integral Relations for a Control Volume

table of content

One-Dimensional Flow


Overview

The third chapter in the book deals with flows that can be assumed to be one dimensional, i.e. flow quantities will only vary in one direction. Although this may seem to be of very limited use, the theory covered and equations derived in this section will be of great help in the coming chapters. The specific model problems analyzed in detail in chapter three are: normal shocks, one-dimensional flow with heat addition, and one-dimensional flow with friction.

Sections

3.2 One-dimensional Flow Equations

The one dimensional flow equations are derived starting from the conservation equations on integral form introduced in Chapter 2. With variation of flow quantities in one direction only, a simplified set of governing equations is derived using a control volume approach.

conservation of mass

$$ \rho_1 u_1 = \rho_2 u_2$$

conservation of momentum

$$ p_1 +\rho_1 u_1^2 = p_2 + \rho_2 u_2^2 $$

conservation of energy

$$ h_1 +\frac{u_1^2}{2} = h_2 + \frac{u_2^2}{2} $$

3.3 Speed of Sound and Mach Number

Using the above derived equations and choosing a control volume such that it includes the wave front of an acoustic wave the speed of sound is investigated. Using the fact that an acoustic wave is a very weak wave we can make assumptions about changes in flow quantities and losses that leads us to a relation between the speed of sound and the compressibility

$$ a^2=\left(\frac{\partial p}{\partial \rho}\right)_s $$

Finally, by assuming that we can treat the gas as calorically perfect, we end up with the following relation between speed of sound and temperature

$$ a=\sqrt{\gamma RT} $$

Physical implications of speed of sound
Figure 1: Physical implications of speed of sound


Physical implications of speed of sound
Figure 2: Effects of speed of sound in supersonic flows


3.4 Some Conveniently Defined Flow Parameters

This section introduces total and characteristic flow properties. The total properties represent the condition that we would have if we could slow the flow down to zero velocity isentropically (without losses). The characteristic properties represent the condition that we would have if we could accelerate or decelerate the flow adiabatically to sonic conditions.

3.5 Alternative Form of the Energy Equation

Using the definition of speed of sound for calorically perfect gases and the relations derived in the previous section one can rewrite the energy equation to obtain ratios of total and local flow properties and ratios of characteristics and local flow properties, respectively.

3.6 Normal Shock Relations

Applying the control volume approach to a one-dimensional flow with a normal shock situated within the control volume leads to the normal shock relations. The normal shock relations are expressions relating ratios of flow quantities from regions upstream of the shock and corresponding properties on the downstream side to the upstream Mach number. The relation between the characteristics Mach number on the upstream side \(M^*_1\) and the characteristic Mach number on the downstream side of the normal shock \(M^*_2\) tells us that behind a normal shock the flow is always subsonic.

Analysing the energy equation and the second law of thermodynamics shows that there is a direct relation between entropy increase and total pressure drop.

3.7 The Hugoniot Equation

The Hugoniot equation is an alternative normal shock relation based on thermodynamic quantities only.

3.8 One-Dimensional Flow with Heat Addition



One-dimensional flow with heat addition
Figure 3: Investigating one-dimensional flow with heat addition using a control volume approach

With heat addition in our control volume we end up with a modified energy equation. It is shown that the direct effect of the added heat is an increase of total temperature.

$$ q=C_p(T_{o2}-T_{o1}) $$

It is shown that adding heat will bring the flow Mach number closer to unity. If the flow is subsonic, adding heat will increase the flow Mach number until the flow reaches sonic conditions. After that adding more heat will not be possible without changing the inflow conditions. For a supersonic flow, adding heat will lower the Mach number until the flow is sonic.

In this section a new set of starred flow quantities are introduced. In this case the star denotes the flow properties that we would get if enough heat was added to reach sonic conditions. The starred quantities are constants in a given flow and can thus be used as a reference state when solving problems.



One-dimensional flow with heat addition
Figure 4: Starred quantities at Mach=1.0 for one-dimensional flow with heat addition


Rayleigh flow
Figure 5: Rayleigh curve - Mollier diagram for one-dimensional flow with heat addition

3.9 One-Dimensional Flow with Friction

The governing equations for one-dimensional flow with friction are derived. The introduction of friction gives results in the addition of an integral in the momentum equation and the equations are transformed to differential form. In the same way as adding heat lead to changed flow properties, the effect of the friction will be that the Mach number of a subsonic flow will be increased with increased pipe length until sonic conditions are reached and the Mach number of a supersonic flow will be decreased.



One-dimensional flow with friction
Figure 6: Investigating one-dimensional flow with friction using a control volume approach

A new set of starred quantities are introduced. For one-dimensional flow with friction, the starred flow properties represent the condition that we would have if the pipe was long enough to reach sonic conditions at the end. When sonic conditions are reached \(L=L^*\), the pipe cannot be made longer without modifying the inlet boundary conditions.



Fanno flow
Figure 7: Fanno curve - Mollier diagram for one-dimensional flow with friction



Study Guide

The questions below are intended as a "study guide" and may be helpful when reading the text book.


  1. Rewrite Newton's second law using the momentum of a system. What is the name of this relation?
  2. Define the angular momentum of a system
  3. Volume flow and massflow:
    1. Show how the volume flow \(Q\) and massflow \(\dot{m}\) over a control volume surface can be calculated in a general way
    2. How are volume flow \(Q\) and massflow \(\dot{m}\) related if the density is constant
    3. How is the volume-averaged mean velocity over a surface defined for a fluid with constant density?
  4. Give examples of when it is appropriate to use fixed control volume, moving control volume, and deformable control volume, respectively.
  5. Reynolds transport theorem:
    1. In Reynolds transport theorem, \(B\) and \(\beta\) denotes extensive and intensive properties respectively. Explain the difference between \(B\) and \(\beta\).
    2. If an intensive property, \(\beta\), is known, how is the corresponding extensive property, \(B\), calculated?
    3. Give two examples of intensive and extensive properties
    4. Explain the physical meaning of each of the terms in Reynolds transport theorem: $$\dfrac{d}{dt}\left(B_{syst}\right)=\dfrac{d}{dt}\left(\int_{cv}\beta \rho d\mathcal{V}\right)+\int_{cs}\beta \rho \left(\mathbf{V}_r\cdot \mathbf{n}\right)dA$$
    5. How can the generic form of Reynolds transport theorem (above) be be simplified for a fix control volume?
    6. What does it mean that inlets and outlets are one-dimensional?
    7. How can we simplify Reynolds transport theorem for one-dimensional inlets and outlets?
  6. The continuity equation:
    1. Derive the continuity equation on integral form for a fixed control volume using Reynolds transport theorem
    2. Explain the physical meaning of each of the terms in the continuity equation on integral form
    3. How can we simplify the continuity equation on integral form under the following circumstances (assuming that the control volume is fixed)? $$\int_{cv}\dfrac{\partial \rho}{\partial t}d\mathcal{V}+\int_{cs}\rho\left(\mathbf{V}\cdot\mathbf{n}\right)dA$$
      1. inlets and outlets can be assumed to be one-dimensional
      2. steady-state flow
      3. incompressible unsteady flow
  7. The momentum equation:
    1. Derive the momentum equation on integral form starting from Reynolds transport theorem
    2. Explain the physical meaning of each of the terms in the momentum equation on integral form
    3. How can we simplify the momentum equation on integral form under the following circumstances? $$\dfrac{d}{dt}\left(m\mathbf{V}\right)_{syst}=\sum\mathbf{F}=\dfrac{d}{dt}\left(\int_{cv}\mathbf{V}\rho d\mathcal{V}\right)+\int_{cs}\mathbf{V}\rho\left(\mathbf{V}_r\cdot\mathbf{n}\right)dA$$
      1. fixed control volume
      2. fixed control volume and one-dimensional inlets and outlets
      3. fixed control volume, one-dimensional inlets and outlets, and steady-state flow
  8. What is gauge pressure?
  9. The Bernoulli equation:
    1. Derive the Bernoulli equation for steady-state, incompressible flow along a streamline $$\dfrac{p_1}{\rho}+\dfrac{1}{2}V_1^2+gz_1=\dfrac{p_2}{\rho}+\dfrac{1}{2}V_2^2+gz_2=const$$
    2. What assumptions are made in the derivation of the Bernoulli equation?
  10. The energy equation:
    1. What does \(Q\) and \(W\) in the energy equation represent?
    2. The energy per unit mass, \(e\), and the work are both divided into parts, what parts are the terms divided into?
    3. The Bernoulli equation is a simplified form of the energy equation. $$\dfrac{p_1}{\rho}+\dfrac{1}{2}V_1^2+gz_1=\dfrac{p_2}{\rho}+\dfrac{1}{2}V_2^2+gz_2=const$$ In what ways are the Bernoulli equation above more limited than the energy equation on the form given below? $$\left(\dfrac{p_1}{\gamma}+\dfrac{V_1^2}{2g}+z_1\right)=\left(\dfrac{p_2}{\gamma}+\dfrac{V_2^2}{2g}+z_2\right)+\dfrac{\hat{u}_2-\hat{u}_1-q}{g}$$
    4. Why is the kinetic energy korrection factor introduced?
    5. Show that the kinetic energy correction factor is \(\alpha=2.0\) for laminar, incompressible pipe flow.
  11. Explain how to measure velocity using a Prandtl tube (Pitot-static tube) and derive the relation needed to estimate the velocity
  12. Explain how a venturi meter works and derive the relation needed to estimate the velocity

Document Archive
MTF053_C03.pdf Lecture notes chapter 3
MTF053_Formulas-Tables-and-Graphs.pdf A collection of formulas, tables, and graphs



Videos


The Bernoulli Equation