Answered: Consider the system shown below in…

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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$### Problem Statement: Consider the system shown below in which a rod is contained within a coaxial cylinder. The system is aligned vertically. The rod has a radius $ R_1 $ and the outer cylinder has a radius of $ R_2 $. The region between the rod and outer cylinder contains an incompressible Newtonian liquid that has a density $ \rho $ and viscosity $ \mu $. The rod is being moved in the positive z-direction at a constant velocity $ V $ while the outer cylinder is stationary. Assume the liquid is in steady fully developed laminar flow, and there is no pressure gradient acting on the liquid in the direction in which the rod is moving. #### Tasks: (a) Write the simplified forms of the Continuity and Navier-Stokes Equations in cylindrical coordinates that will be used for solving the velocity profile in the liquid. (b) Write the applicable boundary conditions that will be used to solve for the velocity profile. (c) Solve for the velocity profile $ v_z $ in terms of $ r $, $ R_1 $, $ R_2 $, $ V $, $ \rho $, $ g $, and $ \mu $. (d) Assume the rod has a radius $ R_1 = 3 \, \text{cm} $ and the outer cylinder has a radius $ R_2 = 6 \, \text{cm} $. At steady-state, the rod is being moved in the positive z-direction at a constant velocity $ V = 2.0 \, \text{cm/s} $. The liquid has a density $ \rho = 1260 \, \text{kg/m}^3 $ and viscosity $ \mu = 1.8 \, \text{Pa s} $. Calculate the magnitude and direction of the viscous force per unit length in units of N/m acting on the rod surface. ### Diagram Explanation: The diagram depicts a vertical coaxial system consisting of a rod and an outer cylinder. Key components include: - **Rod**: Positioned centrally with radius $ R_1 $. - **Outer cylinder**: Surrounds the rod with radius $ R_2 $. - **Liquid Region**: The space between the rod and the outer cylinder filled with an incompressible Newtonian liquid. - **Gravity ($ g $)**: Acts downwards along the length of the system. - **$

Transcribed Image Text:### Problem Statement: Consider the system shown below in which a rod is contained within a coaxial cylinder. The system is aligned vertically. The rod has a radius $ R_1 $ and the outer cylinder has a radius of $ R_2 $. The region between the rod and outer cylinder contains an incompressible Newtonian liquid that has a density $ \rho $ and viscosity $ \mu $. The rod is being moved in the positive z-direction at a constant velocity $ V $ while the outer cylinder is stationary. Assume the liquid is in steady fully developed laminar flow, and there is no pressure gradient acting on the liquid in the direction in which the rod is moving. #### Tasks: (a) Write the simplified forms of the Continuity and Navier-Stokes Equations in cylindrical coordinates that will be used for solving the velocity profile in the liquid. (b) Write the applicable boundary conditions that will be used to solve for the velocity profile. (c) Solve for the velocity profile $ v_z $ in terms of $ r $, $ R_1 $, $ R_2 $, $ V $, $ \rho $, $ g $, and $ \mu $. (d) Assume the rod has a radius $ R_1 = 3 \, \text{cm} $ and the outer cylinder has a radius $ R_2 = 6 \, \text{cm} $. At steady-state, the rod is being moved in the positive z-direction at a constant velocity $ V = 2.0 \, \text{cm/s} $. The liquid has a density $ \rho = 1260 \, \text{kg/m}^3 $ and viscosity $ \mu = 1.8 \, \text{Pa s} $. Calculate the magnitude and direction of the viscous force per unit length in units of N/m acting on the rod surface. ### Diagram Explanation: The diagram depicts a vertical coaxial system consisting of a rod and an outer cylinder. Key components include: - **Rod**: Positioned centrally with radius $ R_1 $. - **Outer cylinder**: Surrounds the rod with radius $ R_2 $. - **Liquid Region**: The space between the rod and the outer cylinder filled with an incompressible Newtonian liquid. - **Gravity ($ g $)**: Acts downwards along the length of the system. - **

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