**Task 4: Simplification of Θ-Equation of Motion**Instructions: Simplify the Θ-equation of motion by removing zero terms and write the resulting equation below.(Note: There are no graphs or diagrams to explain in this image.) ### Annular Mixing Tank System**Diagram Overview:**- **Top View:** - Shows a cross-section of the annular mixing tank. - A smaller inner cylinder is centered within a larger circle representing the tank. - The fluid is present between the inner cylinder and the outer tank wall. - The inner cylinder spins at an angular velocity denoted by $ \Omega $. - Radius $ R $ represents the inner radius of the annular region, while $ k $ indicates the diameter of the spinning cylinder.- **Side View:** - Displays a vertical slice of the annular mixing tank. - The fluid fills the space between the inner, spinning cylinder and the tank wall. - Angular velocity $ \Omega $ signifies the rotation of the inner cylinder. - $ R $ is the distance from the center to the inner wall of the tank. - No flow occurs in vertical or radial directions.**Description:**1. **System Consideration:** - An infinitely tall, annular mixing tank is described. - The mixing process involves fluid between a rotating inner cylinder and a fixed outer wall. - An angular velocity $ \Omega $ (in $ \text{s}^{-1} $) enables mixing by spinning the inner cylinder. - Key dimensions include an inner radius $ R $ (in meters) of the tank, with the inner cylinder's diameter specified as $ k $ (in meters). - No net fluid flow occurs vertically or radially.**Questions for Discussion:**- Analyze specific aspects of this system with regard to fluid dynamics and mixing efficiency based on the given parameters.

Answered: 4. Simplify the e-equation of motion…

Introduction to Chemical Engineering Thermodynamics

8th Edition

ISBN:9781259696527

Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Chapter1: Introduction

Section: Chapter Questions

Problem 1.1P

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**Task 4: Simplification of Θ-Equation of Motion**

Instructions: Simplify the Θ-equation of motion by removing zero terms and write the resulting equation below.

(Note: There are no graphs or diagrams to explain in this image.)

### Annular Mixing Tank System

**Diagram Overview:**

- **Top View:**
- Shows a cross-section of the annular mixing tank.
- A smaller inner cylinder is centered within a larger circle representing the tank.
- The fluid is present between the inner cylinder and the outer tank wall.
- The inner cylinder spins at an angular velocity denoted by $ \Omega $.
- Radius $ R $ represents the inner radius of the annular region, while $ k $ indicates the diameter of the spinning cylinder.

- **Side View:**
- Displays a vertical slice of the annular mixing tank.
- The fluid fills the space between the inner, spinning cylinder and the tank wall.
- Angular velocity $ \Omega $ signifies the rotation of the inner cylinder.
- $ R $ is the distance from the center to the inner wall of the tank.
- No flow occurs in vertical or radial directions.

**Description:**

1. **System Consideration:**
- An infinitely tall, annular mixing tank is described.
- The mixing process involves fluid between a rotating inner cylinder and a fixed outer wall.
- An angular velocity $ \Omega $ (in $ \text{s}^{-1} $) enables mixing by spinning the inner cylinder.
- Key dimensions include an inner radius $ R $ (in meters) of the tank, with the inner cylinder's diameter specified as $ k $ (in meters).
- No net fluid flow occurs vertically or radially.

**Questions for Discussion:**
- Analyze specific aspects of this system with regard to fluid dynamics and mixing efficiency based on the given parameters.

Expert Solution

Step 1

Given:

Fluid is mixing in an annulus between inner cylinder and outer wall

Inner cylinder angular velocity= $𝞨 s - 1$

Inner radius of the tank= R m

Diameter of the inner cylinder= k m

It is required to simplify the $θ$ -equation of motion by removing zero terms using boundary conditions

Step 2

Equation of motion is described by momentum balance across the control volume in the annulus containing fluid

Equation of motion of r-component in cylindrical coordinates is given by the following equation

Assuming fluid is incompressible

$θ$ -component for incompressible fluid :

$𝞺 (\frac{\partial V_{θ}}{\partial t} + V r \frac{\partial V_{θ}}{\partial r} + \frac{V_{θ}}{r} \frac{\partial V_{θ}}{\partial θ} + V_{z} \frac{\partial V_{θ}}{\partial z} + \frac{V_{θ} V_{r}}{r}) = - \frac{1}{r} \frac{\partial P}{\partial θ} + 𝞵 (\frac{\partial}{\partial r} (\frac{1}{r} \frac{\partial}{\partial r} (r V_{θ})) + \frac{1}{r^{2}} \frac{\partial^{2} V_{θ}}{\partial θ^{2}} + \frac{\partial^{2} V_{θ}}{\partial z^{2}} + \frac{2}{r^{2}} \frac{\partial V_{r}}{\partial θ}) + 𝞺 g_{θ}$ Equation (1)

Boundary conditions:

$V_{r} = 0 (g i v e n n o n e t f l o w i n r a d i a l d i r e c t i o n) V_{z} = 0 (g i v e n n o n e t f l o w i n v e r t i c a l d i r e c t i o n) \frac{\partial V_{θ}}{\partial θ} = 0 (f r o m c o n t i n u i t y e q u a t i o n) A s s u m i n g s t e a d y s t a t e a n d g r a v i t a t i o n a l e f f e c t s a r e n e g l i g i b l e a n d n o c h a n g e i n p r e s s u r e g r a d i e n t i n a n g u l a r d i r e c t i o n \frac{\partial V_{θ}}{\partial t} = 0 & ρ g_{θ} = 0 & \frac{d P}{d θ} = 0$

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