Answered: Water is in steady fully developed…

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 Description:** Water is in steady fully developed laminar flow between two horizontal, very wide (W) and long (L) parallel surfaces separated by a distance $b$. The bottom surface at $y = 0$ moves in the negative x-direction at a speed $v_0$, while the top surface at $y = b$ is stationary. In addition, a constant pressure gradient $dP/dx$ is acting on the liquid in the x-direction. **Tasks:** - (a) Write the simplified form of the Navier-Stokes equation and the appropriate boundary conditions. - (b) Derive an equation for the velocity profile in terms of the parameters, $dP/dx$, $b$, $v_0$, and $\mu$. - (c) Obtain an equation for the volumetric flow rate. - (d) Assume that water has a density $\rho = 1000 \, \text{kg/m}^3$ and viscosity $\mu = 1.0 \times 10^{-3} \, \text{kg/m} \cdot \text{s}$. If the pressure gradient is $dP/dx = -1.5 \, \text{Pa/m}$, $b = 6 \, \text{mm}$, $W = 1.0 \, \text{m}$, and the speed of the bottom surface is $1.0 \, \text{cm/s}$, calculate the volumetric flow rate of water and specify the direction of flow (positive or negative x-direction).$

Transcribed Image Text:**Problem Description:** Water is in steady fully developed laminar flow between two horizontal, very wide (W) and long (L) parallel surfaces separated by a distance $b$. The bottom surface at $y = 0$ moves in the negative x-direction at a speed $v_0$, while the top surface at $y = b$ is stationary. In addition, a constant pressure gradient $dP/dx$ is acting on the liquid in the x-direction. **Tasks:** - (a) Write the simplified form of the Navier-Stokes equation and the appropriate boundary conditions. - (b) Derive an equation for the velocity profile in terms of the parameters, $dP/dx$, $b$, $v_0$, and $\mu$. - (c) Obtain an equation for the volumetric flow rate. - (d) Assume that water has a density $\rho = 1000 \, \text{kg/m}^3$ and viscosity $\mu = 1.0 \times 10^{-3} \, \text{kg/m} \cdot \text{s}$. If the pressure gradient is $dP/dx = -1.5 \, \text{Pa/m}$, $b = 6 \, \text{mm}$, $W = 1.0 \, \text{m}$, and the speed of the bottom surface is $1.0 \, \text{cm/s}$, calculate the volumetric flow rate of water and specify the direction of flow (positive or negative x-direction).

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