|Guide 15-1. Bernoulli's Equation and Conservation of Energy
Prerequisite: Study sections 15-6 to 8 of the text.
The text begins the discussion of Bernoulli's equation with the special cases of i) change in speed of fluid without change in height and ii) change in height of fluid without change in speed. Below, we will see that all such situations are governed by two relationships: a) the equation of continuity, and b) the general conservation of energy equation, Wext = ΔEsys.
The equation of continuity is presented in section 15-6. The equation simply states that the mass of fluid passing through any particular cross-sectional slice of a pipe per unit of time is constant. That is, ρAv = constant, where ρ is the density of the fluid, A is the cross-sectional area of the pipe, and v is the velocity of the fluid. If the fluid is incompressible, then the density of the fluid is also a constant. In this case, Av = constant. These results are also expressed by Equations 15-11 and 15-12 in the text, where the subscripts 1 and 2 indicate any two points in time. Note that while the examples in the text show a sudden change in cross-sectional area (see Example 15-7), the continuity equation applies just as well to situations where the cross-sectional area changes continuously. The diagram below illustrates this. A slice of fluid of mass Δm has density ρ1, cross-sectional area A1, and moves at velocity v1 in a particular portion of a pipe. The pipe continuously narrows. At some later time, a slice of the same mass has density ρ2, cross-sectional area A2 and velocity v2. Note that the slices in the diagram are exaggerated in size. Generally, we assume that the slices are so narrow that we can take the cross-sectional area from one side of a slice to the other to be uniform.
From this point on, we'll assume that the fluid is incompressible, and we'll drop the subscripts from the density.
Next we'll develop the conservation of energy equation. The general situation is illustrated in the diagram below. We take as our system the volume of fluid between and including the two slices of mass Δm. Actually, one should think of this as a slice of mass being raised from height y1 to height y2. Due to this change in height, we include the Earth in the system. The slices have equal volume ΔV as a result of the fact that they have the same mass and density. The widths of the slices are Δx1 and Δx2. Since we're taking the slices to be cylinders, then ΔV = A1Δx1 = A2Δx2. The pressures on the left side of the first slice and on the right side of the second slice are P1 and P2 respectively. The corresponding forces on the slices are F1 = P1A1 and F2 = P2A2. Note that F1 points in the direction of fluid displacement, while F2 points opposite the direction of fluid displacement. Note also that these forces are external to the system that we've selected. Hence, they do external work on the system. We need not consider forces exerted to the left on slice 1 or to the right on slice 2, as these forces are internal to the system we've selected. Likewise, we need not consider forces exerted on the mass of fluid between the slices.
It's important to note that we're assuming there are no frictional forces either between the fluid and the walls of the pipe or between different parts of the fluid. Such a situation is called non-viscous flow. For real fluids of low viscosity (water, for example) in pipes, the analysis we give below is reasonably correct as long as the flow is not too fast and the pipe is not too narrow.
Now let's apply conservation of energy. We start with the same equation as always: Wext = ΔEsys. The right-hand side has kinetic energy and gravitational potential energy terms, while the left-hand side has terms for the work done by the external forces. Let's look the latter next. The work done by F1 is W1 = F1Δx1cos0° = F1Δx1, while that done by F2 is W2 = F2Δx2cos180° = -F2Δx2. Therefore, the net external work done on the system is:
Now we equate the external work to ΔEsys.
We should point out the reason why we use Δm in the ΔUg term even though our system includes the fluid between the slices. One can easily imagine that the two slices are positioned adjacent to each other so that there is no mass of fluid between them. This makes no difference to the preceding analysis. So we needn't consider the mass between the slices.
After substituting ΔV = Δm/ρ, we can divide out a common term of Δm.
We arrange terms to place all terms related to slice 1 on the left-hand side of the equal sign and all terms related to slice 2 on the right-hand side.
This is the standard form of Bernoulli's equation. If there is no change in elevation of the fluid, then the equation reduces to
This is Equation 15-14 in the text. If there is a change in elevation but no change in cross-sectional area and hence velocity of the fluid, Bernoulli's equation reduces to
This is Equation 15-15 in the text.
One more example is that of a static fluid for which v1 = v2 = 0. In that case, Bernoulli's equation reduces to . This can be rearranged to give Equation 15-7 in text:
where the depth h = y1 - y2.
The textbook provides examples of the application of the general Bernoulli's equation as well as the special cases mentioned above. However, except for the frequently-used equation P2 - P1 = ρgh, we recommend starting with the general equation and then applying the specific conditions of the problem. Of course, one needs to identify the initial and final states first. Then the pressures, speeds, and elevations associated with those states are identified. One may also need to use the equation of continuity to determine how the speed of the fluid changes. If, for example, the cross-sectional area of the pipe does not change and the fluid is incompressible, then the speed will not change.