Ch9.1 Heat & Mass Transport Intro

Chapter 9 Overview

  • This chapter focuses on modeling heat conduction.
  • We formulate differential equations for heat and temperature of systems, and emphasize difference between them.

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From Physics

Heat

  • Form of energy
  • Measured in Joules
  • Not determined directly by an instrument
  • Is a derived quantity

Temperature

  • How hot or cold a body is
  • Measured in F, C, K, etc
  • Directly determined by thermometer
  • Is a fundamental quantity

From Science

  • Heat is a form of energy.
  • It is the amount the molecules in a substance are moving.
  • At absolute zero, there is no motion at all.
  • In a the solid state, at any temperature above absolute zero, the molecules will be vibrating.
  • In a liquid they are flowing freely around and are not constrained.
  • In a gas they are flying around and to such an extent as to completely fill whatever container they are in and the resulting impacts on said container can be measured as pressure.

Different types of heat (energy flow)

  • We consider heat conduction

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Heat Conduction

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Heat Conduction

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Chapter 9 Overview

  • In the absence of heat conduction, we use the input- output principle, or the balance law introduced in Chapter 2, to determine the heat content of the object we are modeling.
  • When conduction is involved, we apply the input-output approach to an arbitrary thin section of the object.
  • The analytic solutions for these models in Chapter 9 are derived in the subsequent Chapters 10 and 11; numerical solutions are also obtained.

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Ch9.1 Introduction

  • We begin with problems that involve heating and cooling.
  • It is of great importance to understand whether or not heat conduction is an important factor in any particular application.
  • We begin with problems that involve heating and cooling.
  • We consider a variety of problems, some of which require consideration of heat conduction and others which do not.

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Cooling of a Cup of Coffee

  • Suppose the temperature of coffee is 60C, how long will it take to cool down to 40C?
  • We distinguish between temperature and heat, the two most important physical quantities in this problem.
  • If temperature of coffee is uniform throughout cup (homogeneous), then temperature will be a function of time alone (and not location as well).

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Cooling of a Cup of Coffee

  • The temperature of the coffee drops because heat energy is transferred to the surrounding air.
  • This is due to surrounding air at lower temperature.
  • Heat is a form of energy, so measured in Joules (SI system).

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Cooling of a Cup of Coffee

  • To answer the question about the cup of coffee, we need to formulate a suitable equation for the temperature.
  • Thus need to determine what is happening to heat.
  • So we construct a compartmental diagram for heat input and output of cup of coffee.

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Cooling of a Cup of Coffee

  • Equations relating heat energy to temperature are derived in Ch9.2, using the concept of specific heat.
  • Equations relating heat transfer to surroundings are derived in Ch9.2, as is Newton's law of cooling.

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Complex Models for Coffee Cup

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Complex Models for Coffee Cup

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Hot Water Heater

  • We model the time taken for body of water in a hot water tank to heat to a specified temperature.
  • In this application we also need to account for the heat supplied to system by heating element.

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Heat Conduction Through Wall

  • Model rate at which heat is transferred through wall.
  • Useful for insulation and energy efficiency.
  • As opposed to the coffee cup wall, temperature is a function of distance.

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Insulated Water Pipe

  • Model heat conduction through insulated pipe.
  • Insulating a hot water pipe is sometimes less efficient (since more heat escapes) than not insulating it at all.

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Cooling Fins

  • We model the cooling effect of conduction in heat fins, like those used computer chips and motorcycles.

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Diffusion

  • Model process by which particles disperse by random motion from regions of higher to lower concentration.
  • Similar to heat flow from higher to lower temperatures.
  • Can use this for refining our lake pollution models.

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