Guide 15-1. Bernoulli's Equation and Conservation of Energy
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,
E._{sys}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, A,
and moves at velocity _{1}v in a particular portion of a
pipe. The pipe continuously narrows. At some later time, a slice of the same
mass has density _{1}
ρ_{2},
cross-sectional area A_{2} and velocity v.
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._{2}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 Δ y.
Due to this change in height,_{2}we include the Earth in the system. The slices have equal volume Δ_{ }V as a result of the fact that they have
the same mass and density. The widths of the slices are Δx and Δ_{1}x_{2}.
Since we're taking the slices to be cylinders, then ΔV
= AΔ_{1}x =
_{1 }AΔ_{2}x_{2}.
The pressures on the left side of the first slice and on the right side of
the second slice are P and _{1}P
respectively. The corresponding forces on the slices are _{2}F and _{1}
= P_{1}A_{1}F.
Note that _{2} = P_{2}A_{2}F points in the direction of fluid
displacement, while _{1}F 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. _{2}
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 Now let's apply conservation of energy. We start with the same equation
as always:
E.
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 _{sys}F is
_{1}W = _{1}FΔ_{1}xcos0°
= _{1}FΔ_{1}x,
while that done by _{1}F_{2} is W = _{2}
F_{2}Δx_{2}cos180°
= -F_{2}Δx_{2}.
Therefore, the net external work done on the system is:Now we equate the external work to Δ We should point out the reason why we use Δ After substituting Δ 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 v = 0. In that case, Bernoulli's equation reduces to
.
This can be rearranged to give Equation 15-7 in text:_{2}, where the depth y._{2}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 |