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Heat Conduction EquationBMCG 2123 Faculty of Manufacturing Engineering, UTeM Taufik Week 21/28/2011 BMCG 2123 Heat Transfer 1

Learning Objectives

At the end of this chapter, students should be able to:Explain multidimensionality and time dependence of heat transfer. Obtain the differential equation of heat conduction in various coordinate systems. Solve one-dimensional heat conduction problems and obtain the temperature distributions within a medium and the heat flux, Analyze one-dimensional heat conduction in solids that involve heat generation.1/28/2011 BMCG 2123 Heat Transfer 2

Introduction Although heat transfer and temperature are closely related, they are of a different nature. Temperature has only magnitude it is a scalar quantity. Heat transfer has direction as well as magnitude it is a vector quantity. We work with a coordinate system and indicate direction with plus or minus signs.

Introduction

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The driving force for any form of heat transfer is the temperature difference. The larger the temperature difference, the larger the rate of heat transfer. Three prime coordinate systems: rectangular (T(x, y, z, t)) , cylindrical (T(r, J, z, t)), spherical (T(r, J, U, t)).

Introduction

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Classification of conduction heat transfer problems: steady versus transient heat transfer, multidimensional heat transfer, heat generation.

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

Multidimensional Heat Transfer Heat transfer problems are also classified as being: one-dimensional, two dimensional, three-dimensional.

In the most general case, heat transfer through a medium is three-dimensional. However, some problems can be classified as two- or one-dimensional depending on the relative magnitudes of heat transfer rates in different directions and the level of accuracy desired.

The rate of heat conduction through a medium in a specified direction (say, in the x-direction) is expressed by Fourier s law of heat conduction for one-dimensional heat conduction as:& ! kA dT Qcond dx (W) (2-1)

Heat is conducted in the direction of decreasing temperature, and thus the temperature gradient is negative when heat is conducted in the positive xdirection.

General Relation for Fourier s Law of Heat Conduction The heat flux vector at a point P on the surface of the figure must be perpendicular to the surface, and it must point in the direction of decreasing temperature If n is the normal of the isothermal surface at point P, the rate of heat conduction at that point can be expressed by Fourier s law as & ! kA dT Qn (W) (2-2) dn

General Relation for Fourier s Law of Heat Conduction-Continue In rectangular coordinates, the heat conduction vector can be expressed in terms of its components r r r r as (2-3) ! i j kn x y z

which can be determined from Fourier s law as & ! kA xT Q x x xx & ! kA xT (2-4) Qy y xy & ! kA xT Q z z xz

Heat Generation Examples: electrical energy being converted to heat at a rate of I2R, fuel elements of nuclear reactors, exothermic chemical reactions.

Heat generation is a volumetric phenomenon. The rate of heat generation units : W/m3 or Btu/h ft3. The rate of heat generation in a medium may vary with time as well as position within the medium. The total rate of heat generation in a medium of volume V can be determined from & & (2-5) E ! e dV (W)gen

gen

V

Exercise 1 The resistance wire of a 1200-W hair dryer is 80 cm long and has diameter of D = 0.3 cm. Determine: a) the rate of heat generation in the wire per unit volume, in W/cm3, and b) The heat flux on the outer surface of the wire as result of this heat generation.

1/28/2011

BMCG 2123 Heat Transfer

12

Solution 1a) The rate of heat generation: e gen ! gen

!

wire

D / 4 (0.3cm) / 4 cm L T 80 T2

gen 2

!

1200 W

! 212 W/cm 3

b) The heat flux on the outer surface: ! s E gen1/28/2011

!

E genT

ire

1200 W ! ! 15.9 W / c T (o.3c )(80c )

2

BMCG 2123 Heat Transfer

13

One-Dimensional Heat Conduction Equation - Plane WallRate of Rate of Rate of heat - heat + generation heat conductio conduction inside the at x+(x n element at x Rate of change = of the energy content of the element

& & Qx Qx (x & ,element gen(2-6)

(Eelement ! (t

( & Q & & Qx x (x gen ,element !

element

(t

(2-6)

The change in the energy content and the rate of heat generation can be expressed as Eelement ! Et (t Et ! mc t (t ( & & & E gen,element ! egenVelement ! egen A(x t

! Vc

(x

t (t

t

(2-7)(2-8)

Substituting into Eq. 2 6, we get& &x

& egen A(x ! VcA(x x (x

t (t

(t

t

(2-9)

Dividing by A(x, taking the limit as (x 0 and (t 0, and from Fourier s law:1 x xT kA xx A xx

xT & egen ! V c xt

(2-11)

The area A is constant for a plane wall the one dimensional transient heat conduction equation in a plane wall is (2-13) Variable conductivity: x k x egen ! V c x & Constantxx xx & x 2T egen 1 xT ! conductivity: 2 xx k E xt xtk ; E! (2-14) Vc

The one-dimensional conduction equation may be reduces to the following forms under special & d 2T egen conditions ! 0 (2-15) 2 1) Steady-state: dx k 2) Transient, no heat generation: d 2T 3) Steady-state, no heat generation: 2dx

x 2T 1 xT ! 2 xx E xt !0

(2-16) (2-17)

One-Dimensional Heat Conduction Equation - Long CylinderRate of Rate of Rate of heat - heat + generation heat conductio conduction inside the at r+(r n element at r Rate of change = of the energy content of the element

& Q & ,element ! ( & gen r r (r

element

(t

(2-18)

( &Q & & Qr r (r gen,element !

element

(t

(2-18)

The change in the energy content and the rate of heat generation can be expressed as Eelement ! Et (t Et ! mc Tt (t Tt ! VcA(r Tt (t Tt (2-19) ( & & & E (2-20) gen,element ! egenVelement ! egen A(r

Substituting into Eq. 2 18, we get& &r r (r

& A(r ! VcA(r Tt (t Tt egen (t

(2-21)

Dividing by A(r, taking the limit as (r 0 and (t 0, and from Fourier s law: 1 x xT xT & (2-23) kA egen ! V cA xr

xr

xt

Noting that the area varies with the independent variable r according to A=2TrL, the one dimensional transient heat conduction equation in a plane wall becomes 1 x xT xT (2-25) & rk xr egen ! V c xt Variable conductivity: r xr Constant conductivity:1 x xT r xr r xr & egen 1 xT k ! E xt (2-26)

The one-dimensional conduction equation may be reduces to the following forms under special & conditions 1 d dT egen (2-27) r dr k ! 0 1) Steady-state: r dr 2) Transient, no heat generation:1 x xT r xr r xr

1 xT ! E xt !0

(2-28) (2-29)

3) Steady-state, no heat generation:

d dT r dr dr

One-Dimensional Heat Conduction Equation - Sphere

Variable conductivity: Constant conductivity:

1 x 2 xT r k xr 2 r xr

xT & ! Vc egen xt

(2-30)

& 1 x 2 xT egen 1 xT r xr k ! E xt 2 r xr

(2-31)

General Heat Conduction Equation

& & & Q Q Qx (x Qy (y Qz (z Egen ,element ! ( Qx & & & y z

Rate of Rate of Rate of heat Rate of - heat + heat generation = change conductio conduction inside the of the at x+(x, n element energy y+(y, and at x, y, content of z+(z and z the elementelement

(2-36)

(t

Repeating the mathematical approach used for the one-dimensional heat conduction the threedimensional heat conduction equation is determined to be Twodimensional

Constant conductivity:

& x 2T x 2T x 2T egen 1 xT 2 2 ! 2 xx xy xz k E xtThreedimensional

(2-39)

1) Steady-state: 2) 3)

& x 2T x 2T x 2T egen 2 2 ! 0 (2-40) 2 xx xy xz k(2-41) (2-42)

x 2T x 2T x 2T 1 xT Transient, no heat generation: xx 2 xy 2 xz 2 ! E xt x 2T x 2T x 2T Steady-state, no heat generation: 2 2 2 ! 0 xx xy xz

Exercise 2 A 2-kW resistance heater with thermal conductivity k = 15 W/m.K, diameter D = 0.4 cm, and length L = 50 cm is used to boil water by immersing it in water. Assuming the variation of thermal conductivity of the wire with temperature to be negligible, obtain the differential equation that describes the variation of the temperature in the wire during steady state operation.

1/28/2011

BMCG 2123 Heat Transfer

23

AnalysisThe resistance wire of a water heater is considered to be a very long cylinder since its length is more than 100 times its diameter. Also heat is generated uniformly in the wire and the conditions on the outer surface of the wire are uniform. Therefore, it is reasonable to expect the temperature in the wire to vary in the radial r direction only and thus the heat transfer to be one-dimensional. Then we have T=T(r) during steady operation since the temperature in this case depends on r only.1/28/2011 BMCG 2123 Heat Transfer 24

Solution 1The rate of heat generation in the wire per unit volume: e gen ! E gen

!

E gen T 2

V

ire

/4

0.004m / 40.5m T2

!

2000W

! 0.318 x10 9 W/m 3

Nothing that the thermal conductivity is given

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