Heat Conduction Presentation

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Chapter 2: Heat Conduction Equation Yoav Peles Department of Mechanical, Aerospace and Nuclear Engineering Rensselaer Polytechnic Institute Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Transcript of Heat Conduction Presentation

Chapter 2: Heat Conduction EquationYoav PelesDepartment of Mechanical, Aerospace and Nuclear Engineering Rensselaer Polytechnic Institute

Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

ObjectivesWhen you finish studying this chapter, you should be able to: Understand multidimensionality and time dependence of heat transfer, and the conditions under which a heat transfer problem can be approximated as being one-dimensional, Obtain the differential equation of heat conduction in various coordinate systems, and simplify it for steady one-dimensional case, Identify the thermal conditions on surfaces, and express them mathematically as boundary and initial conditions, 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, and Evaluate heat conduction in solids with temperature-dependent thermal conductivity.

2.1. 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.

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

cylindrical (T(r, , z, t))

spherical (T(r, , , t)).

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

Multi-dimensional 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 onedimensional 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 Fouriers law of heat conduction for one-dimensional heat conduction as:dT = kA dx

Qcond

(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 x-direction.

General Relation for Fouriers 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 Fouriers law asdT Qn = kA dn (W) (2-2)

In rectangular coordinates, the heat conduction vector can be expressed in terms of its components as

Qn = Qx i + Qy j + Qz k

(2-3)

which can be determined from Fouriers law as T Qx = kAx x T Qy = kAy y T Qz = kAz z

(2-4)

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

Egen = egen dVV

(W)

(2-5)

onedimensional problem !

2.2 One-Dimensional Heat Conduction Equation - (i) Plane WallRate of heat conduction at x

-

Rate of heat conduction at x+x

+

Rate of heat generation inside the element

=

Rate of change of the energy content of the element

Qx Qx +x + Egen,element(2-6)

Eelement = t

Qx Qx+x + Egen,element

Eelement = t

(2-6)

The change in the energy content and the rate of heat generation can be expressed asEelement = Et +t Et = mc (Tt +t Tt ) = cAx (Tt +t Tt ) (2-7) (2-8) Egen,element = egenVelement = egen Ax

Substituting into Eq. 26, we getQx Qx+x +egen Ax = cAx

Dividing by Ax, taking the limit as x and from Fouriers law:1 T kA A x x T + egen = c t

Tt +t Tt t

(2-9)

0 and t

0,

(2-11)

The area A is constant for a plane wall the one dimensional transient heat conduction equation in a plane wall is Variable conductivity: Constant conductivity: T k x x T + egen = c t

(2-13)k c

2T egen 1 T + = 2 k x t

; =

(2-14)

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

(2-15) (2-16) (2-17)

One-Dimensional Heat Conduction Equation (ii) Long CylinderRate of heat conduction at r

-

Rate of heat conduction at r+r

+

Rate of heat generation inside the element

=

Rate of change of the energy content of the element

Qr Qr +r + Egen,element

Eelement = t(2-18)

Qr Qr +r + Egen,element

Eelement = t

(2-18)

The change in the energy content and the rate of heat generation can be expressed asEelement = Et +t Et = mc (Tt +t Tt ) = cAr (Tt +t Tt ) (2-19) (2-20) Egen,element = egenVelement = egen Ar

Substituting into Eq. 218, we getQr Qr +r +egen Ar = cAr

Dividing by Ar, taking the limit as r and from Fouriers law:1 T kA A r r T + egen = c t

Tt +t Tt t

(2-21)

0 and t

0,

(2-23)

Noting that the area varies with the independent variable r according to A=2rL, the one dimensional transient heat conduction equation in a plane wall becomes Variable conductivity: Constant conductivity:1 T rk r r r T + egen = c t 1 T egen 1 T = r + t r r r k

(2-25) (2-26)

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

1 T r r r r

1 T = t =0

(2-28) (2-29)

3) Steady-state, no heat generation:

d dT r dr dr

One-Dimensional Heat Conduction Equation - (iii) Sphere

Variable conductivity: Constant conductivity:

1 2 T r k 2 r r r 1 2 T r 2 r r r

T + egen = c t egen 1 T = + t k

(2-30) (2-31)

One-dimensional steady-state heat conduction equation:

d 2T =0 dx 2

2.3 General Heat Conduction Equation

Rate of heat conduction at x, y, and z

Rate of heat Rate of heat - conduction + generation at x+x, y+y, inside the and z+z element

=

Rate of change of the energy content of the elementEelement (2-36) = t

Qx + Qy + Qz

Qx +x Qy +y Qz +z + Egen ,element

Repeating the mathematical approach used for the onedimensional heat conduction the three-dimensional heat conduction equation is determined to beTwo-dimensional

Constant conductivity:

2T 2T 2T egen 1 T + 2 + 2 + = 2 x y z t kThree-dimensional

(2-39)

1) Steady-state: 2) Transient, no heat

2T 2T 2T egen + 2 + 2 + = 0 (2-40) 2 x y z k 2T 2T 2T 1 T generation: x 2 + y 2 + z 2 = t 2T 2T 2T generation: 2 + 2 + 2 = 0 x y z

(2-41) (2-42)

3) Steady-state, no heat

Cylindrical Coordinates

1 T rk r r r

1 T + 2 k r

T + z k z

T + egen = c t

(2-43)

Spherical Coordinates

T T T 1 2 T 1 1 + 2 2 + 2 + egen = c kr k k sin r r sin r sin t r 2 r

(2-44)

2.4 Boundary and Initial Conditions (i) Specified Temperature Boundary Condition (ii) Specified Heat Flux Boundary Condition (iii) Convection Boundary Condition (iv) Radiation Boundary Condition (v) Interface Boundary Conditions (vi) Generalized Boundary Conditions

(i) Specified Temperature Boundary ConditionFor one-dimensional heat transfer through a plane wall of thickness L, for example, the specified temperature boundary conditions can be expressed as T(0, t) = T1 T(L, t) = T2(2-46)

The specified temperatures can be constant, which is the case for steady heat conduction, or may vary with time.

(ii) Specified Heat Flux Boundary ConditionThe heat flux in the positive xdirection anywhere in the medium, including the boundaries, can be expressed by Fouriers law of heat conduction asdT q = k = dx

Heat flux in the positive x-direction

(2-47)

The sign of the specified heat flux is determined by inspection: positive if the heat flux is in the positive direction of the coordinate axis, and negative if it is in the opposite direction.

Two Special Cases Insulated BoundaryInsulated boundary Thermal symmetry

T (0, t ) k =0 x

or

T (0, t ) =0 x

(2-49)

T ( L / 2, t ) = 0 (2-50) x

(iii) Convection Boundary ConditionHeat conduction at the surface in a selected direction

=

Heat convection at the surface in the same direction

T (0, t ) k = h1 [T1 T (0, t ) ] xand

(2-51a)

T ( L, t ) k = h2 [T ( L, t ) T 2 ] x

(2-51b)

(iv) Radiation Boundary ConditionHeat conduction at the surface in a selected direction

=

Radiation exchange at the surface in the same direction

T (0, t ) 4 Tsurr ,1 T (0, t ) 4 k = 1 xand

(2-52a)

T ( L, t ) 4 k = 2 T ( L, t ) 4 Tsurr ,2 x

(2-52b)

(v) Interface Boundary ConditionsAt the interface the requirements are: (1) the same temperature at the area of contact, (2) the heat flux on the two sides of an interface must be the same.

TA(x0, t) = TB(x0, t)and

(2-53)

TA ( x0 , t ) TB ( x0 , t ) k A = kB (2-54) x x

(vi) Generalized Boundary ConditionsIn general a surface may involve convection, radiation, and specified heat flux simultaneously. The boundary condition in such cases is again obtained from a surface energy balance, expressed as

Heat transfer to the surface in all modes

=

Heat transfer from the surface in all modes

2.5 Solution of Steady One-Dimensional Heat Conduction Problems

(a)

OK !

!

2.6 Heat Generation () in Solids

The quantities of major interest in a medium with heat generation are (i) the surface temperature Ts (ii) the maximum temperature Tmax that occurs in the medium in steady operation.

Heat Generation in Solids- (i) The Surface TemperatureRate of heat transfer from the solid

=

Rate of energy generation within the solid

(2-63)

For uniform heat generation within the medium Q = egenV (W) (2-64) The heat transfer rate by convection can also be expressed from Newtons law of cooling as

-

Q = hAs (Ts T )Ts = T +

(W)

(2-65) (2-66)

egenV hAs

(a) For a large plane wall of thickness 2L (As=2Awall and V=2LAwall)

Ts , plane wall = T +

egen L h

(2-67)

(b) For a long solid cylinder of radius r0 (As=2r0L and V=r02L)

Ts ,cylinder = T +

egen r0 2h

(2-68)

(c) For a solid sphere of radius r0 (As=4r02 and V=4/3r03)

Ts , sphere = T +

egen r0 3h

(2-69)

Heat Generation in Solids (ii)The maximumTemperature in a Cylinder (the Centerline)The heat generated within an inner cylinder must be equal to the heat conducted through its outer surface.dT kAr = egenVr dr(2-70)

Substituting these expressions into the above equation and separating the variables, we getegen dT 2 k ( 2 rL ) = egen r L dT = rdr 2k dregen r02 4k

(

)

Integrating from r =0 where T(0) =T0 to r=roTmax,cylinder = T0 Ts =(2-71)

Variable Thermal Conductivity, k(T) - The thermal conductivity of a material, in general, varies with temperature. An average value for the thermal conductivity is commonly used when the variation is mild. This is also common practice for other temperaturedependent properties such as the density and specific heat.

Variable Thermal Conductivity for OneDimensional CasesWhen the variation of thermal conductivity with temperature is known, the average value of the thermal conductivity in the temperature range between T1 and T2 can be determined from

kave

=

T2

T1

k (T )dT

(2-75)

T2 T1

The variation in thermal conductivity of a material with can often be approximated as a linear function and expressed as (2-79) k (T ) = k0 (1 + T )

the temperature coefficient of thermal conductivity.

Variable Thermal Conductivity For a plane wall the temperature varies linearly during steady onedimensional heat conduction when the thermal conductivity is constant. This is no longer the case when the thermal conductivity changes with temperature (even linearly).