14.5.3 Two Applications

The speed of a flowing gas is often measured with a device called a Venturi tube. Venturi tubes measure gas speeds in environments as different as chemistry laboratories, wind tunnels, and jet engines.

FIGURE 14.31 shows gas flowing through a tube that changes from cross-section area $A_{1}$ to area $A_{2}$ . A U-shaped glass tube containing liquid of density $ρ_{liq}$ connects the two segments of the flow tube. When gas flows through the horizontal tube, the liquid stands height $h$ higher in the side of the U tube connected to the narrow segment of the flow tube.

The figure illustrates a venturi tube that measures gas-flow speeds.

FIGURE 14.31 Full Alternative Text

Figure 14.31 shows how a Venturi tube works. We can make this analysis quantitative and determine the gas-flow speed from the liquid height h. Two pieces of information we have to work with are Bernoulli’s equation

p_{1} + \frac{1}{2} ρ v_{1}^{2} + ρ g y_{1} = p_{2} + \frac{1}{2} ρ v_{2}^{2} + ρ g y_{2}

(14.29)

and the equation of continuity

v_{2} A_{2} = v_{1} A_{1}

(14.30)

In addition, the hydrostatic equation for the liquid tells us that the pressure $p_{2}$ above the right tube differs from the pressure $p_{1}$ above the left tube by $ρ_{liq} g h$ . That is,

p_{2} = p_{1} - ρ_{liq} g h

(14.31)

First we use Equations 14.30 and 14.31 to eliminate $v_{2}$ and $p_{2}$ in Bernoulli’s equation:

p_{1} + \frac{1}{2} ρ v_{1}^{2} = (p_{1} - ρ_{liq} g h) + \frac{1}{2} ρ {(\frac{A_{1}}{A_{2}})}^{2} v_{1}^{2}

(14.32)

The potential energy terms have disappeared because $y_{1} = y_{2}$ for a horizontal tube. Equation 14.32 can now be solved for $v_{1}$ , then $v_{2}$ is obtained from Equation 14.30. We’ll skip a few algebraic steps and go right to the result:

\begin{array}{l} v_{1} = A_{2} \sqrt{\frac{2 ρ_{liq} g h}{ρ (A_{1}^{2} - A_{2}^{2})}} \\ v_{2} = A_{1} \sqrt{\frac{2 ρ_{liq} g h}{ρ (A_{1}^{2} - A_{2}^{2})}} \end{array}

(14.33)

Equations 14.33 are reasonably accurate as long as the flow speeds are much less than the speed of sound, about $340 m / s$ . The Venturi tube is an example of the power of Bernoulli’s equation.

As a final example, we can use Bernoulli’s equation to understand, at least qualitatively, how airplane wings generate lift. FIGURE 14.32 shows the cross section of an airplane wing. This shape is called an airfoil.

The figure illustrates the generation of lift of a wing by airflow due to pressure differences.

FIGURE 14.32 Full Alternative Text

Although you usually think of an airplane moving through the air, in the airplane’s reference frame it is the air that flows across a stationary wing. As it does, the streamlines must separate. The bottom of the wing does not significantly alter the streamlines going under the wing. But the streamlines going over the top of the wing get bunched together. As we’ve seen, with the equation of continuity, the flow speed has to increase when streamlines get closer together. Consequently, the air speed increases as it flows across the top of the wing.

If the air speed increases, then, from Bernoulli’s equation, the air pressure must decrease. And if the air pressure above the wing is less than the air pressure below, the air will exert a net upward force on the wing. The upward force of the air due to the pressure difference across the wing is called lift. A full understanding of lift in aerodynamics involves other, more complicated factors, such as the creation of vortices on the trailing edge of the wing, but our introduction to fluid dynamics has given you enough tools to at least begin to understand how airplanes stay aloft.

14.5.3 Two Applications

FIGURE 14.31 A Venturi tube measures gas-flow speeds.

FIGURE 14.32 Airflow over a wing generates lift by creating unequal pressures above and below.