College Physics

# 12.3The Most General Applications of Bernoulli’s Equation

College Physics12.3 The Most General Applications of Bernoulli’s Equation

### Torricelli’s Theorem

Figure 12.8 shows water gushing from a large tube through a dam. What is its speed as it emerges? Interestingly, if resistance is negligible, the speed is just what it would be if the water fell a distance $hh size 12{h} {}$ from the surface of the reservoir; the water’s speed is independent of the size of the opening. Let us check this out. Bernoulli’s equation must be used since the depth is not constant. We consider water flowing from the surface (point 1) to the tube’s outlet (point 2). Bernoulli’s equation as stated in previously is

$P1+12ρv12+ρgh1=P2+12ρv22+ρgh2.P1+12ρv12+ρgh1=P2+12ρv22+ρgh2. size 12{P rSub { size 8{1} } + { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{1} } rSup { size 8{2} } +ρ ital "gh" rSub { size 8{1} } =P rSub { size 8{2} } + { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{2} } rSup { size 8{2} } +ρ ital "gh" rSub { size 8{2} } } {}$
12.29

Both $P1P1 size 12{P rSub { size 8{1} } } {}$ and $P2P2 size 12{P rSub { size 8{2} } } {}$ equal atmospheric pressure ($P1P1 size 12{P rSub { size 8{1} } } {}$ is atmospheric pressure because it is the pressure at the top of the reservoir. $P2P2 size 12{P rSub { size 8{2} } } {}$ must be atmospheric pressure, since the emerging water is surrounded by the atmosphere and cannot have a pressure different from atmospheric pressure.) and subtract out of the equation, leaving

$12ρv12+ρgh1=12ρv22+ρgh2.12ρv12+ρgh1=12ρv22+ρgh2. size 12{ { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{1} } rSup { size 8{2} } +ρ ital "gh" rSub { size 8{1} } = { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{2} } rSup { size 8{2} } +ρ ital "gh" rSub { size 8{2} } } {}$
12.30

Solving this equation for $v22v22 size 12{v rSub { size 8{2} } rSup { size 8{2} } } {}$, noting that the density $ρ ρ$ cancels (because the fluid is incompressible), yields

$v22=v12+2g(h1−h2).v22=v12+2g(h1−h2). size 12{v rSub { size 8{2} } rSup { size 8{2} } =v rSub { size 8{1} } rSup { size 8{2} } +2g $$h rSub { size 8{1} } - h rSub { size 8{2} }$$ } {}$
12.31

We let $h=h1−h2h=h1−h2 size 12{h=h rSub { size 8{1} } - h rSub { size 8{2} } } {}$; the equation then becomes

$v22=v12+2ghv22=v12+2gh size 12{v rSub { size 8{2} } rSup { size 8{2} } =v rSub { size 8{1} } rSup { size 8{2} } +2 ital "gh"} {}$
12.32

where $hh size 12{h} {}$ is the height dropped by the water. This is simply a kinematic equation for any object falling a distance $hh size 12{h} {}$ with negligible resistance. In fluids, this last equation is called Torricelli’s theorem. Note that the result is independent of the velocity’s direction, just as we found when applying conservation of energy to falling objects.

Figure 12.8 (a) Water gushes from the base of the Studen Kladenetz dam in Bulgaria. (credit: Kiril Kapustin; http://www.ImagesFromBulgaria.com) (b) In the absence of significant resistance, water flows from the reservoir with the same speed it would have if it fell the distance $hh size 12{h} {}$ without friction. This is an example of Torricelli’s theorem.
Figure 12.9 Pressure in the nozzle of this fire hose is less than at ground level for two reasons: the water has to go uphill to get to the nozzle, and speed increases in the nozzle. In spite of its lowered pressure, the water can exert a large force on anything it strikes, by virtue of its kinetic energy. Pressure in the water stream becomes equal to atmospheric pressure once it emerges into the air.

All preceding applications of Bernoulli’s equation involved simplifying conditions, such as constant height or constant pressure. The next example is a more general application of Bernoulli’s equation in which pressure, velocity, and height all change. (See Figure 12.9.)

### Example 12.5Calculating Pressure: A Fire Hose Nozzle

Fire hoses used in major structure fires have inside diameters of 6.40 cm. Suppose such a hose carries a flow of 40.0 L/s starting at a gauge pressure of $1.62×106N/m21.62×106N/m2 size 12{1 "." "62" times "10" rSup { size 8{6} } "N/m" rSup { size 8{2} } } {}$. The hose goes 10.0 m up a ladder to a nozzle having an inside diameter of 3.00 cm. Assuming negligible resistance, what is the pressure in the nozzle?

Strategy

Here we must use Bernoulli’s equation to solve for the pressure, since depth is not constant.

Solution

Bernoulli’s equation states

$P 1 + 1 2 ρv 1 2 +ρ gh1 =P2+ 12 ρv22 +ρ gh2 , P 1 + 1 2 ρv 1 2 +ρ gh1 =P2+ 12 ρv22 +ρ gh2 , size 12{P rSub { size 8{1} } + { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{1} } rSup { size 8{2} } +ρ ital "gh" rSub { size 8{1} } =P rSub { size 8{2} } + { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{2} } rSup { size 8{2} } +ρ ital "gh" rSub { size 8{2} } } {}$
12.33

where the subscripts 1 and 2 refer to the initial conditions at ground level and the final conditions inside the nozzle, respectively. We must first find the speeds $v1v1 size 12{v rSub { size 8{1} } } {}$ and $v2v2 size 12{v rSub { size 8{2} } } {}$. Since $Q = A 1 v 1 Q = A 1 v 1 size 12{Q=A rSub { size 8{1} } v"" lSub { size 8{1} } } {}$ , we get

$v1=QA1=40.0×10−3m3/sπ(3.20×10−2m)2=12.4m/s.v1=QA1=40.0×10−3m3/sπ(3.20×10−2m)2=12.4m/s. size 12{v rSub { size 8{1} } = { {Q} over {A rSub { size 8{1} } } } = { {"40" "." 0 times "10" rSup { size 8{ - 3} } " m" rSup { size 8{3} } "/s"} over {π $$3 "." "20" times "10" rSup { size 8{ - 2} } " m"$$ rSup { size 8{2} } } } ="12" "." 4" m/s"} {}$
12.34

Similarly, we find

$v2=56.6 m/s.v2=56.6 m/s. size 12{v rSub { size 8{2} } ="56" "." 6" m/s"} {}$
12.35

(This rather large speed is helpful in reaching the fire.) Now, taking $h1h1 size 12{h rSub { size 8{1} } } {}$ to be zero, we solve Bernoulli’s equation for $P2P2 size 12{P rSub { size 8{2} } } {}$:

$P2=P1+12ρ v 1 2 − v 2 2 −ρgh2.P2=P1+12ρ v 1 2 − v 2 2 −ρgh2. size 12{P rSub { size 8{2} } =P rSub { size 8{1} } + { {1} over {2} } ρ $$v rSub { size 8{1} rSup { size 8{2} } } - v rSub { size 8{2} rSup { size 8{2} } }$$ - ρ ital "gh" rSub { size 8{2} } } {}$
12.36

Substituting known values yields

$P2= 1.62 × 106 N/m2 + 12 (1000 kg/m3 )[ (12.4m/s )2 −(56.6m/s )2] −(1000 kg/m3 )(9.80 m/s2 )(10.0 m) =0. P2= 1.62 × 106 N/m2 + 12 (1000 kg/m3 )[ (12.4m/s )2 −(56.6m/s )2] −(1000 kg/m3 )(9.80 m/s2 )(10.0 m) =0. size 12{P rSub { size 8{2} } =1 "." "62" times "10" rSup { size 8{6} } " N/m" rSup { size 8{2} } + { {1} over {2} } $$"1000"" kg/m" rSup { size 8{3} }$$ left [ $$"12" "." 4" m/s"$$ rSup { size 8{2} } - $$"56" "." 6" m/s"$$ rSup { size 8{2} } right ] - $$"1000"" kg/m" rSup { size 8{3} }$$ $$9 "." 8" m/s" rSup { size 8{2} }$$ $$"10" "." 0" m"$$ =0} {}$
12.37

Discussion

This value is a gauge pressure, since the initial pressure was given as a gauge pressure. Thus the nozzle pressure equals atmospheric pressure, as it must because the water exits into the atmosphere without changes in its conditions.

### Power in Fluid Flow

Power is the rate at which work is done or energy in any form is used or supplied. To see the relationship of power to fluid flow, consider Bernoulli’s equation:

$P+12ρv2+ρgh=constant.P+12ρv2+ρgh=constant. size 12{P+ { {1} over {2} } ρv rSup { size 8{2} } +ρ ital "gh"="constant"} {}$
12.38

All three terms have units of energy per unit volume, as discussed in the previous section. Now, considering units, if we multiply energy per unit volume by flow rate (volume per unit time), we get units of power. That is, $(E/V)(V/t)=E/t(E/V)(V/t)=E/t size 12{ $$E/V$$ $$V/t$$ =E/t} {}$. This means that if we multiply Bernoulli’s equation by flow rate $QQ size 12{Q} {}$, we get power. In equation form, this is

$P+12ρv2+ρghQ=power.P+12ρv2+ρghQ=power. size 12{ left (P+ { {1} over {2} } ρv rSup { size 8{2} } +ρ ital "gh" right )Q="power"} {}$
12.39

Each term has a clear physical meaning. For example, $PQPQ size 12{ ital "PQ"} {}$ is the power supplied to a fluid, perhaps by a pump, to give it its pressure $PP size 12{P} {}$. Similarly, $12ρv2Q12ρv2Q size 12{ { { size 8{1} } over { size 8{2} } } ρv rSup { size 8{2} } Q} {}$ is the power supplied to a fluid to give it its kinetic energy. And $ρghQρghQ size 12{ρ ital "ghQ"} {}$ is the power going to gravitational potential energy.

### Making Connections: Power

Power is defined as the rate of energy transferred, or $E/tE/t size 12{E/t} {}$. Fluid flow involves several types of power. Each type of power is identified with a specific type of energy being expended or changed in form.

### Example 12.6Calculating Power in a Moving Fluid

Suppose the fire hose in the previous example is fed by a pump that receives water through a hose with a 6.40-cm diameter coming from a hydrant with a pressure of $0.700×106N/m20.700×106N/m2 size 12{0 "." "700" times "10" rSup { size 8{6} } "N/m" rSup { size 8{2} } } {}$. What power does the pump supply to the water?

Strategy

Here we must consider energy forms as well as how they relate to fluid flow. Since the input and output hoses have the same diameters and are at the same height, the pump does not change the speed of the water nor its height, and so the water’s kinetic energy and gravitational potential energy are unchanged. That means the pump only supplies power to increase water pressure by $0.92×106N/m20.92×106N/m2 size 12{0 "." "92" times "10" rSup { size 8{6} } "N/m" rSup { size 8{2} } } {}$ (from $0.700×106N/m20.700×106N/m2 size 12{0 "." "700" times "10" rSup { size 8{6} } "N/m" rSup { size 8{2} } } {}$ to $1.62×106N/m21.62×106N/m2 size 12{1 "." "62" times "10" rSup { size 8{6} } `"N/m" rSup { size 8{2} } } {}$).

Solution

As discussed above, the power associated with pressure is

$power = PQ = 0.920×106N/m2 40.0×10−3m3/s. = 3.68×104W=36.8kW power = PQ = 0.920×106N/m2 40.0×10−3m3/s. = 3.68×104W=36.8kW .$
12.40

Discussion

Such a substantial amount of power requires a large pump, such as is found on some fire trucks. (This kilowatt value converts to about 50 hp.) The pump in this example increases only the water’s pressure. If a pump—such as the heart—directly increases velocity and height as well as pressure, we would have to calculate all three terms to find the power it supplies.