The Physics of Energy, page 118
29.1.2 Local Characterization of a Fluid
The molecules in a fluid are so small that the smallest volume of interest to us still contains a vast number of molecules. A cubic millimeter of liquid water, for example, contains more than water molecules. This number is so huge that we can idealize a fluid as a continuous medium even at scales much smaller than a millimeter. We make this continuum assumption throughout our discussion of wind and water power. A property such as the mass density of the fluid is therefore naturally described as a continuous function taking values at every point x within the fluid at every time t. This continuous function should be understood as the limit of the mass in a finite volume of size centered at x, as the volume becomes small compared to any macroscopic scale of interest, but still large enough to contain an immense number of molecules,
(29.1)
Although all materials are to some degree compressible, it is often a good approximation to neglect small changes in density and replace by a constant and assume that a fluid is incompressible. We indicate when our results apply only to incompressible fluids and when they are more general.
The local properties ascribed to fluids should be understood in the same fashion as the density: their values at any point and time are defined as averages over a macroscopically small volume that nevertheless contains very many molecules. In addition to the mass density, there are several other properties of fluids that enter the description of wind and water power.
Velocity field The average velocity of the molecules in a fluid at a point x at time t is described by a velocity field υ(x, t). Like the electric and magnetic fields encountered in §3, υ(x, t) assigns a vector to each point in space and time.
Mass flux The flux of mass is the amount of mass crossing a unit area per unit time. Like the current density (3.27) in electromagnetism, which is a flux of charge, the mass flux is defined to be the product of the local density and velocity fields
(29.2)
In analogy to Figure 3.12, the mass flowing through a small surface in a time dt is given by .
Kinetic energy density Moving mass carries kinetic energy. In the case of a flowing fluid, the mass in a volume about a point x carries kinetic energy , leading, in the limit , to a kinetic energy density ,
(29.3)
Note that the velocity and kinetic energy density of a fluid depend on one’s frame of reference. A fluid flowing rapidly past a stationary observer may possess considerable kinetic energy, while to an observer moving along with the fluid, the fluid is at rest and possesses no kinetic energy. Generally we are interested in harvesting power from a fluid moving with respect to an observer stationary on Earth’s surface, so this is the frame in which kinetic energy density is relevant for wind power.
Kinetic energy flux and power density The flux of kinetic energy is defined in analogy to the mass flux, as the kinetic energy per unit time crossing a unit area normal to the fluid’s direction of motion,
(29.4)
The power density, introduced in the previous chapter, is the magnitude of J,
(29.5)
For a comparison of the power density in moving air and water, see Figure 29.2.
Figure 29.2 The power density in a fluid grows as the cube of the velocity. Wind speed ~12 m/s gives 1000 W/m2, comparable to insolation from overhead sun on a clear day. Water moving at a speed of 1.25 m/s has the same power density.
Pressure Within a moving fluid, the pressure at a point x is the force per unit area on a hypothetical surface in a reference frame that is at rest with respect to the fluid. This pressure is often referred to as static pressure in the context of fluid mechanics. Air and water, like most fluids, are isotropic: at any point in the fluid the static pressure is the same in all directions. It is the static pressure that is related to the fluid’s temperature and density by an equation of state. The static pressure in a moving fluid decreases and increases as the flow velocity increases and decreases according to Bernoulli’s equation (§29.2.2). For the fluids of interest here, the range of variation in the static pressure in the flowing fluid is generally much smaller than the absolute value of the pressure (see Example 29.3).
Temperature The intermolecular interactions that bring a fluid into local thermal equilibrium are typically rapid compared to the time scales over which the fluid moves macroscopically, so fluids that we are interested in maintain local thermal equilibrium as they flow and can be characterized by a temperature that is a local function of x. Note that a fluid contains thermal energy – a result of the random thermal motion of the molecules – that is distinct from the fluid’s bulk motion. We assume that the temperature of the fluid when at rest is fixed and equal to the temperature of the environment. The temperature of a moving fluid varies slightly, however, along with the pressure, as the velocity of flow changes (Example 29.3). Since moving wind and water are typically at or very close to the same temperature as their environment, these small fluctuations in thermal energy do not represent a resource that can be extracted for human use. Although frictional forces can dissipate the kinetic energy of fluid motion, the resulting change in temperature is often negligible as shown in Example 29.1, so we ignore the thermal energy content of flowing fluids here and in the study of wind turbines (§30) and water power (§31).
Example 29.1 Thermal Effects in the Flow of Wind and Water
Suppose the kinetic energy in a flowing fluid was dissipated as thermal energy. How much would the temperature of the fluid rise?
The kinetic energy per unit volume is . If this energy is converted to thermal energy, it warms the same volume of water by , where c is the specific heat (per unit mass). The resulting increase in temperature is
For wind, kJ/kg K, and
Thus, converting the kinetic energy in hurricane winds blowing at ~100 knots (~50 m/s) to thermal energy would raise the air’s temperature by only ~1 K. For water, where kJ/kg K, the temperature rise is roughly 1/4 as large. Note that the kinetic energy in fluid motion is much more useful than the thermal energy into which it could be converted (Problem 29.2).
The equation of state (§5) provides a relationship among the static pressure, density, and temperature of a fluid. For air we assume an ideal gas law, , where m is the average mass of a molecule of air. The thermal conductivity of air is small and the temperature fluctuations involved are also small in the regimes of interest to us. Therefore, as small fluctuations occur in the density during the motion of the fluid, heat flow can be neglected and the local change in pressure and temperature can be modeled adiabatically. In general, when a fluid is compressed adiabatically both the temperature and pressure increase. In particular, for adiabatic expansion or compression of air (see §10.2.2), the adiabatic gas law, (), implies ; this combined with the ideal gas law gives .
The dynamics of a flowing fluid can be viewed either from a fixed coordinate system (the Eulerian approach) or by using variables that move along with the fluid (the Lagrangian approach). These two approaches are essentially different methods of bookkeeping to keep track of the motion of the fluid. The dynamics of the fluid is, of course, independent of which method is used to describe it, although the dynamics may look quite different in these two formalisms. In this book we are primarily interested in how Earth-based devices harvest energy from wind or water as it flows by. Therefore, we generally use the Eulerian approach to the description of fluid motion, in which all variables, such as the density and velocity, are considered as functions in a fixed coordinate system. Thus υ(x, t), for example, measures the velocity at time t of the wind at a point x fixed on Earth’s surface. In §31.2.1, where we describe energy flow in ocean waves, however, it is convenient to switch to Lagrangian coordinates.
29.2Simplifying Assumptions and Conservation Laws
The dynamics of fluid flow can be quite complicated, as evinced for example by the complicated flow of water as it cascades down a stream bed or over a waterfall. Fortunately, this full complexity need not be confronted in order to give a satisfactory introductory description of wind (or water) power. For most of our purposes, it suffices to study ideal (i.e. frictionless) fluid flows that are steady, i.e. unchanging in time. We explain here first how the assumption of steady flow combines with conservation of mass to simplify the geometry of a fluid flow. We next make the further assumption that a fluid flow is ideal or inviscid, and derive Bernoulli’s equation, which expresses conservation of energy in the absence of friction. Finally we introduce vorticity, a measure of the angular momentum in fluid flow, which is conserved in the steady flow of an incompressible, ideal fluid.
29.2.1 Steady Flow of a Fluid
We first consider the implications of conservation of mass for fluid flow. The situation is analogous to the discussion of electric charge conservation and electric current in §3.2.2. If the mass of fluid in a small region R is changing, then conservation of mass requires that a net mass must be flowing through the surface that bounds R. The local form of this conservation law relates the time derivative of the mass density to the divergence of the mass flux in exact analogy to eq. (3.33),
(29.6)
In fluid dynamics, eq. (29.6) is known as the continuity equation; it applies to wind and water flow independent of the simplifying assumptions we introduce below.
The usefulness of the continuity equation is limited by the irregularities of fluid flow. At a given point x fixed in space, the velocity of the flowing fluid can vary with time in a highly irregular fashion. The situation is much simpler in the case of steady flow. In a steady flow, in the Eulerian description, all the characteristics of the fluid, , υ(x, t), , etc. are independent of time. Each individual element of fluid changes its position and velocity with time as it moves, but it simply replaces the element of fluid that formerly occupied the space that it moves into, leaving the pattern unchanged. Steady flow contrasts with turbulent flow, where the flow pattern changes over time and may appear chaotic. The flow of water from a tap (Figure 29.3(a)) is a familiar example of steady flow, which contrasts with the turbulent flow shown in Figure 29.3(b) The conditions under which steady or turbulent flow of a fluid is favored are explored in §29.3.
Figure 29.3 Steady (a) and turbulent (b) flow. (a) Water flowing steadily from a faucet. (b) Hot air rising from a cigarette, made visible by the smoke, is initially a laminar flow, but becomes turbulent. (Credit: (b) Jessie Murphy – http://www.uglyhedgehog.com)
The continuity condition simplifies in the case of steady flow because
(29.7)
at every point in space. When this is combined with the continuity equation, we obtain
(29.8)
(29.9)
where in the last line we used eq. (B.19).
Note that the continuity equation simplifies further when we specialize to incompressible fluids, for which is a good approximation over the relevant range of pressures. For water flowing at the range of pressures experienced on Earth’s surface, this is an excellent approximation. During steady flow of an incompressible fluid, the density does not change from point to point, so and eq. (29.9) reduces to
(29.10)
This is the same condition we obtained for the electric field in charge-free space (see §3). In electromagnetism, implies that electric field lines can never begin or end in a charge-free region of space. By analogy, the velocity field υ(x) that describes the steady flow of an incompressible fluid can be envisioned in terms of streamlines that neither begin nor end. The streamlines are the paths followed by individual fluid elements. For a compressible fluid in steady flow, the mass flux vector is divergenceless; the streamlines of this flow are the same as those of the field υ.
By definition, fluid flows along the streamlines. Since υ(x, t) has a unique value at each point, streamlines cannot cross. This enables us to define the useful concept of a flow tube, which is the domain bounded by the streamlines that cross a closed curve fixed in space (Figure 29.4). The material inside a flow tube, cannot – by definition – ever get out.
Figure 29.4 A flow tube showing some of the streamlines that form its boundary.
Conservation of Mass in Fluid Flow
Conservation of mass in fluid flow leads to the continuity equation, which relates the density and velocity fields,
If the flow is steady then the mass flux vector ϕ = ρυ is divergenceless,
This means that the streamlines – the paths followed by individual fluid elements – neither begin nor end.
29.2.2 Energy Conservation in an Ideal Fluid and Bernoulli’s Equation
In general, kinetic energy is not conserved in fluid flow, even when no external forces act on the fluid. Frictional forces between adjacent regions of fluid moving with slightly different velocities transform kinetic energy into thermal energy, in somewhat the same way that friction dissipates kinetic energy when solid surfaces slide by one another. This is how an agitated fluid comes to rest. Common experience suggests that different fluids dissipate energy at very different rates. Viscosity is the measure of the strength of dissipative forces in a liquid. We postpone further study of viscosity until §29.3.
When viscosity can be ignored, a flow is termed inviscid or ideal.2 In many aspects of the study of wind and water power, air and water may be regarded as ideal. Under these circumstances, conservation of energy provides a powerful relation, known as Bernoulli’s principle or Bernoulli’s equation, that relates the static pressure and the kinetic and potential energy along any streamline in the fluid.
The basic framework of Bernoulli’s equation was described already in Box 13.1 where we implemented energy conservation within a control volume as shown in Figure 13.2. In that context, we consider a flow tube as the control volume, as illustrated in Figure 29.4, and choose the flow tube narrow enough that the fluid’s static pressure and speed can be considered constant over the cross section. We can analyze the flow of the fluid, as done in Box 13.1, by following a small volume, or slug, of fluid passing along the flow lines within the tube. Assuming that no heat is added (adiabatic, ) and no work is done on any external system (), conservation of energy gives the result that
(29.11)
at any two points along the flow tube, where u represents the internal energy per unit mass. Thus
(29.12)
along a flux tube. This is Bernoulli’s equation for a compressible fluid. If the fluid is incompressible, ρ and u are constant, and we have Bernoulli’s equation for an incompressible fluid,
(29.13)
A principal consequence of Bernoulli’s equation is that in regions where the fluid velocity is higher, the (static) pressure is lower. The quantity is sometimes referred to as dynamic pressure, although it does not express a physical pressure exerted as a force per unit area.
Essentially the same relation as (29.13) holds when the flow is compressible but adiabatic, as long as the fluctuations , , in the pressure, density, and specific internal energy are small compared to their ambient values. To see this, expand p, ρ, and u in eq. (29.12) about their ambient values , , and ,
(29.14)
For adiabatic changes, the first law of thermodynamics for a slug of fluid of mass m reduces to
(29.15)
so the term in parentheses in eq. (29.14) vanishes. So for small adiabatic fluctuations in a compressible fluid, Bernoulli’s equation takes essentially the same form as for an incompressible fluid
(29.16)
This simplified form of Bernoulli’s equation is useful in describing air flow in situations such as around the blade of a wind turbine (see §30.1.1).
The analysis here shows that the quantity in eq. (29.16) is approximately constant along streamlines. If all streamlines for a given flow originate from points with a common value of and z, then the constant is uniform across the flow. Note that the analysis of flow around an object such as a turbine blade or airplane wing must be performed in a frame where the object is stationary; otherwise the flow cannot be described as steady (Question 29.5).
Example 29.2 A Venturi Flow Meter
Bernoulli’s equation (29.13) predicts that the faster a fluid flows, the lower its pressure. This, together with conservation of mass, makes it possible to construct a simple and accurate meter to measure the flow rate of an incompressible fluid, like water flowing through a pipe. The result, known as a Venturi flow meter, is shown in the figure at right. The object is to measure the mass per unit time of a fluid flowing in the pipe by measuring the difference in the fluid’s pressure in the sections of pipe with cross-sectional areas and . The pressure and velocity are assumed to be constant across the pipe. The pressures are measured by the height of a column of liquid (a simple application of Bernoulli’s equation) in the two sections of pipe. Bernoulli’s equation for the flow of an incompressible fluid states that
Combining this with mass conservation leads directly to
Bernoulli’s Principle
Applying conservation of energy to the steady flow of an ideal fluid leads to Bernoulli’s equation, which relates the pressure in the fluid to the fluid velocity and the potential energy in a (constant) gravitational field. For an incompressible fluid,
along any streamline in the fluid.
For a compressible adiabatic ideal fluid where the fluctuations in pressure are small compared to the ambient pressure , we have the analogous relation
