**Educational Content: Water Flow Analysis in an Open Tank****Problem Statement:**The open tank contains water at 20 °C and is being filled through section 1. Assume incompressible flow. 1. Derive an analytical expression for the rate of change of the water level, \( \frac{dh}{dt} \), in terms of arbitrary volume flows (\( Q_1, Q_2, Q_3 \)) and tank diameter \( d \).2. If the water level \( h \) is constant, determine the exit velocity \( v_2 \) using the given data: - \( v_1 = 3 \, \text{m/s} \) - \( Q_3 = 0.01 \, \text{m}^3/\text{s} \) - Given diameter \( d = 50 \, \text{cm} \).**Diagram Explanation:**- The diagram shows a vertical open tank with a water inflow at section 1 (left side), an outflow at section 2 (right side), and an additional outflow marked as section 3 (top).- Inflow at section 1 has a diameter \( D_1 = 5 \, \text{cm} \).- Outflow at section 2 has a diameter \( D_2 = 7 \, \text{cm} \).- The top outflow, marked as section 3, has a fixed flow rate \( Q_3 = 0.01 \, \text{m}^3/\text{s} \).- The tank diameter \( d \) is given as 50 cm.**Select the correct exit velocity \( v_2 \) from the options:**- \( 5.38 \, \text{m/s} \)- \( 3.42 \, \text{m/s} \)- \( 2.56 \, \text{m/s} \)- \( 4.13 \, \text{m/s} \)**Solution Approaches:**To solve this problem, apply the continuity equation for incompressible fluid flow, and consider the conservation of mass in the system. By maintaining the water level constant (\( \frac{dh}{dt} = 0 \)), you equate the inflows and outflows to solve for \( v_2 \).

Flow incompressibleFlow Q1 and Q3 are entering the tank and flow Q2 is exiting the tankInlet…

The open tank contains water at 20 °C and is being filled through section 1. Assume incompressible flow. First derive an analytical expression for the water-level change dh/dt in terms of arbitrary volume flows (Q1, Q2, Q3) and tank diameter. Then, if the water level h is constant, determine the exit velocity v2 for the given data v1 = 3 m/s and Q3 = 0.01 m3/s. Given that d = 50 cm. D₁ = 5 cm 3 Q = 0.01 m³/s h Water D₂=7 cm

The open tank contains water at 20 °C and is being filled through section 1. Assume incompressible flow. First derive an analytical expression for the water-level change dh/dt in terms of arbitrary volume flows (Q1, Q2, Q3) and tank diameter. Then, if the water level h is constant, determine the exit velocity v2 for the given data v1 = 3 m/s and Q3 = 0.01 m3/s. Given that d = 50 cm. D₁ = 5 cm 3 Q = 0.01 m³/s h Water D₂=7 cm

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Concept explainers

Open Channel Flow

Open channel flow is a branch of fluid mechanics. It deals with the flow of fluid in an open conduit or free surface. In an open channel flow, gravity induces the flow of water whose surface is open to the air. This flow is free-falling rather than confined in rigid body conduits. The open channel flow depends upon two parameters that are the cross-section of the conduit and the velocity profile. Practical examples of open channel flow are streams, rivers, canals, and other drainage systems.In open channel flow, the cross-sectional area of at least one of the sides is exposed to air, and the surface of water faces atmospheric pressure. This flow is also known as free gravity flow. The cross-sectional area of the conduit can be either man-made or a natural channel. For instance, the flow in rivers and streams are natural conduits, and the flow in a canal and sanitary systems are man-made or engineered channels.

Question

**Educational Content: Water Flow Analysis in an Open Tank**

**Problem Statement:**

The open tank contains water at 20 °C and is being filled through section 1. Assume incompressible flow.

1. Derive an analytical expression for the rate of change of the water level, \( \frac{dh}{dt} \), in terms of arbitrary volume flows (\( Q_1, Q_2, Q_3 \)) and tank diameter \( d \).
2. If the water level \( h \) is constant, determine the exit velocity \( v_2 \) using the given data:
- \( v_1 = 3 \, \text{m/s} \)
- \( Q_3 = 0.01 \, \text{m}^3/\text{s} \)
- Given diameter \( d = 50 \, \text{cm} \).

**Diagram Explanation:**

- The diagram shows a vertical open tank with a water inflow at section 1 (left side), an outflow at section 2 (right side), and an additional outflow marked as section 3 (top).
- Inflow at section 1 has a diameter \( D_1 = 5 \, \text{cm} \).
- Outflow at section 2 has a diameter \( D_2 = 7 \, \text{cm} \).
- The top outflow, marked as section 3, has a fixed flow rate \( Q_3 = 0.01 \, \text{m}^3/\text{s} \).
- The tank diameter \( d \) is given as 50 cm.

**Select the correct exit velocity \( v_2 \) from the options:**

- \( 5.38 \, \text{m/s} \)
- \( 3.42 \, \text{m/s} \)
- \( 2.56 \, \text{m/s} \)
- \( 4.13 \, \text{m/s} \)

**Solution Approaches:**

To solve this problem, apply the continuity equation for incompressible fluid flow, and consider the conservation of mass in the system.

By maintaining the water level constant (\( \frac{dh}{dt} = 0 \)), you equate the inflows and outflows to solve for \( v_2 \).

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