Fundamentals of Heat Transfer - Fourier's Law

7/30/2019 Fundamentals of Heat Transfer - Fourier's Law

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ME353HEATTRANSFER1

LectureNotes

Prof.MetinRenksizbulutMechanicalEngineering


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Fouriers Law

and the

Heat Equation

Chapter Two

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A rate equation that allows determination of the conduction heat flux

from knowledge of the temperature distribution in a medium

Fouriers Law

Its most general (vector) form for multidimensional conduction is:

q k T

Implications:

Heat transfer is in the direction of decreasing temperature

(basis for minus sign).

Direction of heat transfer is perpendicular to lines of constant

temperature (isotherms).

Heat flux vector may be resolved into orthogonal components.

Fouriers Law serves to define the thermal conductivity of the

medium /k q T

2

Phenomenological Law


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Heat Flux Components

T T Tq k i k j k k

r r z

rq q zq

Cylindrical Coordinates: , ,T r z

sinT T Tq k i k j k k r r r

rq q q

Spherical Coordinates: , ,T r

Cartesian Coordinates: , ,T x y z

T T Tq k i k j k k

x y z

xq yq zq

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The Heat Equation A differential equation whose solution provides the temperature distribution in a

stationary medium.

Based on applying conservation of energy to a differential control volume

through which energy transfer is exclusively by conduction.

Cartesian Coordinates:

Net transfer ofthermal energy into the

control volume (inflow-outflow)

p

T T T T k k k q c

x x y y z z t

Thermal energy

generation

Change in thermal

energy storage

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THE CONDUCTION EQUATION

Energy balance:

Inflow - Outflow + Generation = Accumulation


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One-Dimensional Conduction in a Planar Medium with Constant Properties

and No Generation

2

2

1T T

tx

2thermal diffusivit of the medium my /sp

k

c

p

T Tk c

x x t

becomes

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Heat Equation (Radial Systems)

2

1 1p

T T T T kr k k q c

r r r z z t r

Spherical Coordinates:

Cylindrical Coordinates:

2

2 2 2 2

1 1 1sin

sin sinp

T T T T kr k k q c

r r tr r r

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THERMALCONDUCTIVITY

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Boundary and Initial Conditions For transient conduction, heat equation is first order in time, requiring

specification of an initial temperature distribution: 0

0t=

T x,t = T x,

Since heat equation is second order in space, two boundary conditions

must be specified. Some common cases:

Constant Surface Temperature:

0 sT ,t = T

Constant Heat Flux:

0x= s

T-k | = q

x

Applied Flux Insulated Surface

0 0x=T

| =x

Convection:

0 0x=T

-k | = h T - T ,t x

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Note that the temperature profilevaries with time but the surfacetemperature remains constant.

Note that the surface temperature varies with time.


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Problem: Thermal Response of Plane Wall

Problem 2.57 Thermal response of a plane wall to convection heat transfer.

KNOWN: Plane wall, initially at a uniform temperature, is suddenly exposed to convective heating.

FIND: (a) Differential equation and initial and boundary conditions which may be used to find the

temperature distribution, T(x,t); (b) Sketch T(x,t) for the following conditions: initial (t 0), steady-

state (t ), and two intermediate times; (c) Sketch heat fluxes as a function of time at the two

surfaces; (d) Expression for total energy transferred to wall

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Problem: Thermal Response (cont).

ASSUMPTIONS: (1) One-dimensional conduction, (2) Constant properties, (3) No internal

heat generation.

ANALYSIS: (a) For one-dimensional conduction with constant properties, the heat equation has the

form,

2

2

T 1 T=

a tx

0

Initial: 0 0 uniform temperature

Boundaries: 0 0 adiabatic surface

it T x, = T

x = T / x =

x = L

surface convectionL

- k T / x = h T L,t - T

and the

conditions are:

Note that the gradient at x = 0 is always zero, since this boundary is adiabatic. Note also that the

gradient at x = L decreases with time.

Fundamentals of Heat Transfer - Fourier's Law

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