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1. Heating of a Spherical Particle Consider a solid spherical particle of radius R and thermal diffusivity a. Initially, the particle is at temperature To. We then immerse the particle in a bath of a fluid whose temperature is T1 > To everywhere. (a) Write down the governing equation for the temperature inside the particle, i.e., unsteady-state conduction (or diffusion) in spherical coordinates, and specify the initial and boundary conditions. (b) Introduce the dimensionless variables, 0 = (T - To)/(T1 – To), E = r/R, and T = at/R2, and rewrite the governing equation and its initial and boundary conditions in dimensionless form. (c) Derive the solution of the governing equation. As discussed in class, you must first obtain the steady-state solution, and then use it to obtain an eigenvalue problem with homogeneous boundary conditions.

1. Heating of a Spherical Particle Consider a solid spherical particle of radius R and thermal diffusivity a. Initially, the particle is at temperature To. We then immerse the particle in a bath of a fluid whose temperature is T1 > To everywhere. (a) Write down the governing equation for the temperature inside the particle, i.e., unsteady-state conduction (or diffusion) in spherical coordinates, and specify the initial and boundary conditions. (b) Introduce the dimensionless variables, 0 = (T - To)/(T1 – To), E = r/R, and T = at/R2, and rewrite the governing equation and its initial and boundary conditions in dimensionless form. (c) Derive the solution of the governing equation. As discussed in class, you must first obtain the steady-state solution, and then use it to obtain an eigenvalue problem with homogeneous boundary conditions.

Introduction to Chemical Engineering Thermodynamics
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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1. Heating of a Spherical Particle Consider a solid spherical particle of radius R and thermal diffusivity a. Initially, the particle is at temperature To. We then immerse the particle in a bath of a fluid whose temperature is T1 > To everywhere. (a) Write down the governing equation for the temperature inside the particle, i.e., unsteady-state conduction (or diffusion) in spherical coordinates, and specify the initial and boundary conditions. (b) Introduce the dimensionless variables, 0 = (T – To)/(T – To), § = r/R, and T = at/R, and rewrite the governing equation and its initial and boundary conditions in dimensionless form. (c) Derive the solution of the governing equation. As discussed in class, you must first obtain the steady-state solution, and then use it to obtain an eigenvalue problem with homogeneous boundary conditions.
Transcribed Image Text:1. Heating of a Spherical Particle Consider a solid spherical particle of radius R and thermal diffusivity a. Initially, the particle is at temperature To. We then immerse the particle in a bath of a fluid whose temperature is T1 > To everywhere. (a) Write down the governing equation for the temperature inside the particle, i.e., unsteady-state conduction (or diffusion) in spherical coordinates, and specify the initial and boundary conditions. (b) Introduce the dimensionless variables, 0 = (T – To)/(T – To), § = r/R, and T = at/R, and rewrite the governing equation and its initial and boundary conditions in dimensionless form. (c) Derive the solution of the governing equation. As discussed in class, you must first obtain the steady-state solution, and then use it to obtain an eigenvalue problem with homogeneous boundary conditions.
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