Question

Figure 16-29 shows liquid discharging from an orifice in a large tank at a distance \(h\) below the liquid surface. The tank is open at the top. (a) Apply Bernoulli's equation to a streamline connecting points 1,2, and 3, and show that the speed of efflux is $$ v=\sqrt{2 g h} $$ This is known as Torricelli's law. (b) If the orifice were curved directly upward, how high would the liquid stream rise? ( \(c\) ) How would viscosity or turbulence affect the analysis?

Short answer

The speed of liquid efflux from the tank is \(v = \sqrt{2 g h}\). If the orifice were curved upwards, the liquid would rise to the height of the original liquid surface level in the tank. Viscosity and turbulence would decrease the speed and height of the liquid flow.

Step-by-step solution

Apply Bernoulli's equation to points 1 and 2

Bernoulli's equation can be stated as \(P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2\), where, \(P\) represents pressure, \(\rho\) is fluid density, \(v\) is velocity, \(h\) is height and \(g\) is acceleration due to gravity. At the liquid surface (point 1), pressure \(P_1\) is atmospheric pressure \(P_0\), velocity \(v_1\) is 0, and height \(h_1\) is \(h\). At the orifice (point 2), pressure \(P_2\) is also \(P_0\), height \(h_2\) is 0, and velocity \(v_2\) is \(v\). Substituting these conditions into Bernoulli's equation yields \(P_0 + \rho g h = P_0 + \frac{1}{2} \rho v^2\).

Solve Bernoulli's equation for velocity

Rearrange the equation derived in step 1 to solve for velocity \(v\). First, cancel the atmospheric pressure term \(P_0\) from both sides because it's same at points 1 and 2. You get \(\rho g h = \frac{1}{2} \rho v^2\). We can cancel out \(\rho\) from both sides, leaving \(gh = \frac{1}{2}v^2\). Rearranging to solve for \(v\) and taking square root of both sides, you get \(v = \sqrt{2 g h}\).

Apply Bernoulli's equation to find the rise height when the orifice is curved upwards

To find how high the liquid will rise if the orifice were curved directly upward, we use Bernoulli's equation similarly to step 1, but set \(v = 0\) (the point of highest rise). Solving for \(h\), we find that \(h = \frac{v^2}{2g}\). Substituting the result of Step 2 for \(v\) gives \(h = \frac{((\sqrt {2gh})^2)}{2g} = h\), stating that the liquid would rise to the original liquid surface level

Analyze the effect of viscosity and turbulence

Bernoulli's equation assumes ideal fluid flow (inviscid and streamline flow). Introducing viscosity would decrease the speed of efflux due to frictional forces between the fluid layers. Turbulence causes mixing of fluid, deviating the flow from ideal streamline flow and decreasing velocity, thus reducing the height of liquid rise.

Key concepts

Torricelli's Law

Torricelli's Law provides a simple way to determine the speed at which a fluid will exit an orifice. This law is derived from Bernoulli's equation, which relates the different components of fluid flow. Bernoulli’s equation essentially balances pressure, kinetic energy, and potential energy of a fluid across different points in a flow system.

To find the efflux speed using Torricelli’s Law, we use the scenario of liquid leaving a tank through an opening. In such cases, the speed, denoted as \( v \), can be calculated using the equation: \( v = \sqrt{2gh} \), where \( g \) is the acceleration due to gravity and \( h \) is the height of the fluid column above the outlet.

This equation implies that the speed of efflux is related directly to the square root of the height of the liquid above the opening. The simplicity of Torricelli's law makes it an essential concept for applications like calculating flow rates from tanks or predicting the behavior of fluids in various engineering systems.

Fluid Dynamics

Fluid dynamics is the study of how fluids move, covering both liquids and gases. This area of physics analyzes fluid motion under various forces, pressures, and different conditions. Bernoulli's equation, which forms the basis of Torricelli's Law, is a fundamental principle used in fluid dynamics.

In Torricelli's scenario, we consider fluid moving from a region of higher pressure to one of lower pressure – for instance, from the larger tank to the outside atmosphere through an orifice. Key variables in Bernoulli’s equation include:

• Pressure – How force is applied on the fluid.
• Kinetic Energy – Related to the speed of the fluid.
• Potential Energy – Depending on the height of the fluid.

Fluid dynamics helps us understand and predict the behavior of fluids in motion, which is crucial in designing systems like pipelines, water distribution networks, and even understanding atmospheric phenomena.

Viscosity and Turbulence Analysis

When analyzing real-world fluid scenarios, additional factors, such as viscosity and turbulence, can significantly influence fluid movement. Viscosity refers to a fluid's resistance to flow – similar to friction in solid objects. More viscous fluids tend to flow more slowly and may require more force to move.

Turbulence, on the other hand, describes chaotic fluid movements, often resulting in swirling flows. It contrasts with laminar flow where the fluid moves in smooth and orderly layers. The presence of turbulence in a flow system can affect velocity and pressure distribution, impacting the application of Bernoulli's equation.

While Bernoulli's equation assumes ideal conditions with inviscid (non-viscous) flow and no turbulence, real scenarios seldom match these assumptions perfectly. Increased viscosity slows down the efflux speed, while turbulence can create variations that make the flow less predictable. Understanding these factors is essential for engineers to design and manage systems involving fluid dynamics.