Islamic Azad University Karaj Branch

Dr. M. Khosravy

Chapter 2 Conduction

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Chapter 2: ! Need to obtain detailed temperature profiles: Energy conservation written for a differential volume

Conservation of Energy

Can be written for control volume or control surface ! Control volume and control surface: Convenient, but do not give any information on actual temperature distributions

When there are no contributions of kinetic, potential, internal energy and work

Motivation Chapter 1:

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Conduction Heat Transfer •  Conduction refers to the transport of energy in a medium

(solid, liquid or gas) due to a temperature gradient. •  The physical mechanism is random atomic or molecular

activity •  Governed by Fourier’s law

•  In this chapter we will learn !  The definition of important transport properties and what governs thermal

conductivity in solids, liquids and gases !  The general formulation of Fourier’s law, applicable to any geometry and multiple

dimensions !  How to obtain temperature distributions by using the heat diffusion equation. !  How to apply boundary and initial conditions

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Thermal Properties of Matter •  Recall from Chapter 1, Fourier’s law for heat conduction:

!  The proportionality constant is a transport property, known as thermal conductivity k (units W/m.K)

•  Usually assumed to be isotropic (independent of the direction of transfer): kx=ky=kz=k

s  Is thermal conductivity different between gases, liquids and solids?

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Thermal Conductivity: Solids •  Solid comprised of free electrons and atoms

bound in lattice •  Thermal energy transported through

–  Migration of free electrons, ke

–  Lattice vibrational waves, kl

where

?  What is the relative magnitude in pure metals, alloys and non-metallic solids?

See Figure 2.4, Appendix Tables A.1, A.2 and A.3

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Thermal Conductivity: Gases •  Intermolecular spacing is much larger •  Molecular motion is random •  Thermal energy transport less effective than in solids;

thermal conductivity is lower ! Kinetic theory of gases:

where n the number of particles per unit volume, the mean molecular speed and l the mean free path (average distance travelled before a collision)

?  What are the effects of temperature, molecular weight and pressure?

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Thermal Conductivity: Liquids

•  Physical mechanisms controlling thermal conductivity not well understood in the liquid state

•  Generally k decreases with increasing temperature (exceptions glycerine and water)

•  k decreases with increasing molecular weight. •  Values tabulated as function of temperature. See Tables

A.5 and A.6, appendix.

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Thermal Conductivity: Insulators

–  Can disperse solid material throughout an air space – fiber, powder and flake type insulations

–  Cellular insulation – Foamed systems

! Several modes of heat transfer involved (conduction, convection, radiation)

! Effective thermal conductivity: depends on the thermal conductivity and radiative properties of solid material, volumetric fraction of the air space, structure/morphology (open vs. closed pores, pore volume, pore size etc.) Bulk density (solid mass/total volume) depends strongly on the manner in which the solid material is interconnected. See Table A.3.

?  How can we design a solid material with low thermal conductivity?

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Thermal Diffusivity Thermophysical properties of matter: •  Transport properties: k (thermal conductivity/heat transfer), n

(kinematic viscosity/momentum transfer), D (diffusion coefficient/mass transfer)

•  Thermodynamic properties, relating to equilibrium state of a system, such as density,r and specific heat cp.

–  the volumetric heat capacity r cp (J/m3.K) measures the ability of a material to store thermal energy.

•  Thermal diffusivity a is the ratio of the thermal conductivity to the heat capacity:

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The Conduction Rate Equation Recall from Chapter 1: •  Heat rate in the

x-direction

•  Heat flux in the x-direction

x

We assumed that T varies only in the x-direction, T=T(x)

Direction of heat flux is normal to cross sectional area A, where A is isothermal surface (plane normal to x-direction)

T1(high)

T2 (low)

qx”

x1 x2

A

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The Conduction Rate Equation In reality we must account for heat transfer in three dimensions •  Temperature is a scalar field T(x,y,z) •  Heat flux is a vector quantity. In Cartesian coordinates:

for isotropic medium

Where three dimensional del operator in cartesian coordinates:

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Summary: Fourier’s Law

•  It is phenomenological, ie. based on experimental evidence •  Is a vector expression indicating that the heat flux is normal to

an isotherm, in the direction of decreasing temperature •  Applies to all states of matter •  Defines the thermal conductivity, ie.

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The Heat Diffusion Equation

•  Objective to determine the temperature field, ie. temperature distribution within the medium.

•  Based on knowledge of temperature distribution we can compute the conduction heat flux.

!  Reminder from fluid mechanics: Differential control volume.

We will apply the energy conservation equation to the differential control volume

CV

Element of volume: dx dy dz

T(x,y,z)

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Heat Diffusion Equation Energy Conservation Equation

where from Fourier’s law

(2.1)

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Heat Diffusion Equation •  Thermal energy generation due to an energy source:

–  Manifestation of energy conversion process (between thermal energy and chemical/electrical/nuclear energy)

!  Positive (source) if thermal energy is generated !  Negative (sink) if thermal energy is consumed

•  Energy storage term –  Represents the rate of change of thermal energy

stored in the matter in the absence of phase change.

is the time rate of change of the sensible (thermal) energy of the medium per unit volume (W/m3)

is the rate at which energy is generated per unit volume of the medium (W/m3)

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Heat Diffusion Equation Substituting into Eq. (2.1):

Heat Equation

Net conduction of heat into the CV rate of energy generation per unit volume

time rate of change of thermal energy per unit volume

" At any point in the medium the rate of energy transfer by conduction into a unit volume plus the volumetric rate of thermal energy generation must equal the rate of change of thermal energy stored within the volume

(2.2)

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Heat Diffusion Equation- Other forms •  If k=constant

is the thermal diffusivity

•  For steady state conditions

•  For steady state conditions, one-dimensional transfer in x-direction and no energy generation

!  Heat flux is constant in the direction of transfer

(2.3)

(2.4)

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Heat Diffusion Equation

•  In cylindrical coordinates:

•  In spherical coordinates:

(2.5)

(2.6)

Steady versus Transient Heat Transfer

•  Steady implies no change with time at any point within the medium

•  Transient implies variation with time or time dependence

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Fourier-Poisson Equation

Assumptions

Geometry

Boundary and/or initial conditions

Solve the Equation for T Analytical or Numerical

Obtain Temperature Profile

Fourier’s Law of Heat Conduction

Obtain rate of heat transfer

TYPICAL METHOD FOR SOLVING CONDUCTION PROBLEMS

COORDINATE SYSTEM

Chapter 2 20 Dr. M. Khosravy

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Example (Problem 2.23 textbook)

The steady-state temperature distribution in a one-dimensional wall of thermal conductivity 50 W/m.K and thickness 50 mm is observed to be T(°C)=a+bx2, where a=200°C, b=-2000°C/m2, and x is in meters.

a)  What is the heat generation rate in the wall? b)  Determine the heat fluxes at the two wall faces. In what manner are these

heat fluxes related to the heat generation rate?

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Boundary and Initial Conditions •  Heat equation is a differential equation:

–  Second order in spatial coordinates: Need 2 boundary conditions –  First order in time: Need 1 initial condition

Boundary Conditions

1)  B.C. of first kind (Dirichlet condition):

x T(x,t)

Ts

Constant Surface Temperature At x=0, T(x,t)=T(0,t)=Ts

x=0

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Boundary and Initial Conditions

2)  B.C. of second kind (Neumann condition): Constant heat flux at the surface

1.  Finite heat flux

x T(x,t)

x T(x,t)

qx” = qs”

2.  Adiabatic surface qx”=0

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Boundary and Initial Conditions

3)  B.C. of third kind: When convective heat transfer occurs at the surface

T(x,t)

T(0,t)

x

= q”conduction q”convection

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Example 2.3, textbook A long copper bar of rectangular cross section, whose width w is much greater than its thickness L, is maintained in contact with a heat sink (for example an ice bath) at a uniform initial temperature To. Suddenly an electric current is passed through the bar, and an airstream of temperature is passed over the top surface, while the bottom surface is maintained at To. Obtain the differential equation, the boundary and initial conditions that can be used to determine the temperature as a function of position and time inside the bar.

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Example (Problem 2.39, textbook) Passage of an electric current through a long conducting rod of radius ri and thermal conductivity kr results in uniform volumetric heating at a rate of . The conducting rod is wrapped in an electrically nonconducting cladding material of outer radius ro and thermal conductivity kc, and convection cooling is provided by an adjoining fluid. For steady-state conditions, write appropriate forms of the heat equations for the rod and cladding. Express appropriate boundary conditions for the solution of these equations.

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Example (Problem 2.26 Textbook)

One dimensional, steady-state conduction with uniform internal energy generation occurs in a plane wall with a thickness of 50 mm and a constant thermal conductivity of 5 W/m.K. For these conditions, the temperature distribution has the form T(x)=a+bx+cx2. The surface at x=0 has a temperature of T(0)=To=120°C and experiences convection with a fluid for which and h=500 W/m2.K. The surface at x=L is well insulated. (a) Applying an overall energy balance to the wall, calculate the internal energy generation rate. (b) Determine the coefficients a,b, and c, applying the appropriate boundary conditions.

Example of Cylindrical Coordinate

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Summary •  Generalized form of Fourier’s law, to take into account conduction in 3

dimensions.

•  Physical meaning of thermal conductivity, k, and thermal diffusivity, a.

•  Modification of the energy balance, for cases involving conduction heat transfer – THE HEAT EQUATION

!  Simplified forms of the heat equation !  Heat equation in cylindrical and spherical coordinates !  Identification of boundary conditions