**Study of Liquid in a Rotating Cylindrical Tank****Problem Statement:**Consider a liquid in a cylindrical tank where both the tank and the liquid rotate as a rigid body. The free surface of the liquid is exposed to room air, and surface tension effects are negligible. Define the boundary conditions necessary to analyze this system.**Boundary Condition Requirements:**1. **Velocity Boundary Conditions:** - Express the velocity conditions in cylindrical coordinates \((r, \theta, z)\) with velocity components \((u_r, u_\theta, u_z)\). - Apply these conditions at all relevant surfaces, including the tank walls and the free surface.2. **Pressure Boundary Conditions:** - Determine the appropriate pressure boundary conditions for the flow field.**Diagram Explanation:**- **Diagram Description:** - The diagram represents a cross-section of the cylindrical tank. - The tank contains liquid, and both the tank and liquid rotate together.- **Key Features:** - **Free Surface:** The curved shape of the liquid’s free surface indicates rotation, with pressure \(P = P_{\text{atm}}\). - **Rotation:** Illustrated with an arrow and angular velocity \(\omega\), indicating the direction of rotation. - **Coordinates:** Labeled as \(z\) for vertical height, \(R\) for the tank’s radius, and \(r\) as the radial distance extending from the center. - **Fluid Density (\(\rho\)):** Denoted within the liquid.The goal is to use these conditions to understand the behavior of the fluid in such a rotating system, taking into account interactions at the surface and other interfaces.

Given consider liquid in a cylindrical tank. Both the tank and the liquid rotates as a rigid body. the free surface of the liquid is exposed to room air. surface tension effect are negligible. for forced vortex flow Because both the tank and liquid rotates as a rigid bodyradial velocity of the fluid Vr =0velocity in the direction of the fluid Vz=0radial velocity of the fluid Vθ=ω×r where ω is angular velocity , ω=2πN60where ω is angular velocity dp=ρω2r2rdr-ρgdzconsider two points 1 and 2 in the fluid having force vortex flowintegrating the above equation for points 1 and 2 we get ∫12dp=∫12ρω2rdr-∫12ρgdzp2-p1=ρω2r2212-ρgz12p2-p1=ρω22r22-r12-ρgz2-z1p2-p1=ρ2Vθ22-Vθ12-ρgz2-z1if points 1 and 2 lie on the free surface of the liquid, then p2=p1= patm0=ρ2Vθ22-Vθ12-ρgz2-z1ρgz2-z1==ρ2Vθ22-Vθ12z2-z1=12gVθ22-Vθ12if point 1 lies on the axis of rotation, then V1= ω×r1=ω×0=0the above equation becomes as z2-z1=12gVθ22z2-z1=ZZ=12gVθ22Z=ω2×r222g

Engineering

Mechanical Engineering

Consider liquid in a cylindrical tank. Both the tank and the liquid rotate as a rigid body. The free surface of the liquid is exposed to room air. Surface tension effects are negligible. Present the boundary conditions required to solve this problem. Specifically, what are the velocity boundary conditions in terms of cylindrical coordinates (r, 0, z) and velocity components (ur, ưo, u:) at all surfaces, including the tank walls and the free surface? What pressure boundary conditions are appropriate for this flow field? Explain. Free surface P-Pan Liquid

Consider liquid in a cylindrical tank. Both the tank and the liquid rotate as a rigid body. The free surface of the liquid is exposed to room air. Surface tension effects are negligible. Present the boundary conditions required to solve this problem. Specifically, what are the velocity boundary conditions in terms of cylindrical coordinates (r, 0, z) and velocity components (ur, ưo, u:) at all surfaces, including the tank walls and the free surface? What pressure boundary conditions are appropriate for this flow field? Explain. Free surface P-Pan Liquid

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

See similar textbooks

Related questions

Q: What is the relation of our hands being at the side and being raised in the concept of fluid…

A: To explain: Explain the relation of our hands being at the side and being raised in the concept of…

Q: Q5-Air at 27-C and 1 atm flows over a flat plate at a speed of 2 m/s. The viscosity of air at 27-C…

A: Step 1: Step 2: Step 3: Step 4:

Q: A cylindrical steel tank is 6 m in diameter and 9m high at -15°C is completely full of gasoline. If…

A: Given Data: Initial temperature, Ti=-15 C Final temperature, Tf=22 C

Q: Please don't provide handwritten solution.

A: The given parameters are:Freestream temperature, T∞ = 245 KFreestream pressure, p∞ = 4.35 × 10^4…

Q: 5) Consider a vessel of diatomic nitrogen gas at room temperature. (a) What is the most likely speed…

Q: A compressed air tank is designed to contain 50 standard cubic feet of air when filled to a gauge…

A: Given data Interior volume of the tank = 0.253 ft3 Interior length of cylinder = 1 ft We have to…

Q: 2. A unknown fluid at 20°C fills the following portion of a sleeve with an inner diameter of 11.8 cm…

Q: A solid sphere having diameter 5 mm is falling through oil of density 800 kg/m? and dynamic…

Q: (b) Consider a thin flat plate at zero angle of attack in an airflow at Poo = 1.225 kg/m³, T∞ = 288…

A: Given:To find:Friction drag

Q: 94: Repeat Prob. 9–93, but let the inner cylinder be stationary and the outer cylinder rotate at…

Q: What are constitutive equations, and to which fluid mechanics equation are they applied?

A: These are the equations which describes the relationship between two physical quantity.

Q: A fuel tank in wing has dimensions in Fig. 1, where the width is equal to the height and the tank is…

A: The height above sea level is The velocity of air craft is The radius is The free steam static…

Q: Under what conditions may one describe atmospheric processes using the incompressible form of the…

A: The condition need to find out for which it can be described that atmospheric process using the…

Q: How does the Navier-Stokes equation encapsulate the complexities of fluid behavior, and what…

A: The Navier-Stokes equation is a fundamental equation in fluid mechanics that describes the motion of…

Question

**Study of Liquid in a Rotating Cylindrical Tank**

**Problem Statement:**
Consider a liquid in a cylindrical tank where both the tank and the liquid rotate as a rigid body. The free surface of the liquid is exposed to room air, and surface tension effects are negligible. Define the boundary conditions necessary to analyze this system.

**Boundary Condition Requirements:**

1. **Velocity Boundary Conditions:**
- Express the velocity conditions in cylindrical coordinates \((r, \theta, z)\) with velocity components \((u_r, u_\theta, u_z)\).
- Apply these conditions at all relevant surfaces, including the tank walls and the free surface.

2. **Pressure Boundary Conditions:**
- Determine the appropriate pressure boundary conditions for the flow field.

**Diagram Explanation:**

- **Diagram Description:**
- The diagram represents a cross-section of the cylindrical tank.
- The tank contains liquid, and both the tank and liquid rotate together.

- **Key Features:**
- **Free Surface:** The curved shape of the liquid’s free surface indicates rotation, with pressure \(P = P_{\text{atm}}\).
- **Rotation:** Illustrated with an arrow and angular velocity \(\omega\), indicating the direction of rotation.
- **Coordinates:** Labeled as \(z\) for vertical height, \(R\) for the tank’s radius, and \(r\) as the radial distance extending from the center.
- **Fluid Density (\(\rho\)):** Denoted within the liquid.

The goal is to use these conditions to understand the behavior of the fluid in such a rotating system, taking into account interactions at the surface and other interfaces.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

Similar questions

Conduct thorough a research on Boundary layer and viscous sub layer, Shear stress in circular pipe as it relates to mechanics of fluids
A cylindrical tank shown below has a radius ? and a circular orifice at the bottom of radius ?. At time ? = 0 the liquid level in the tank is ho. (a) Use the Bernoulli equation to determine the relations between the orifice velocity ?o and the free surface velocity ?s (assume the surface velocity is not zero). (b) Use conservation of mass to determine the orifice velocity in terms of the free surface velocity. (c) Use the results in (a) and (b) to derive a differential equation for h(?). Note that ?s = − dh/dt. d) Separate variables and solve the equation for h(?) with the initial condition h(0) = ho (e) Determine how long it will take to empty the tank.
What role does computational fluid dynamics (CFD) play in optimizing the design and performance of aerodynamic systems, such as aircraft and wind turbines, and how has the advancement of CFD software and hardware influenced engineering practices in these domains?
Please help class is heat transfer, will rate question. Show all steps and equations used, thank you.