Heat Transfer Handbook part 33 ppsx

From the tabulated values it can be seen that the spreading resistance values for the isothermal strip are smaller than the values for the isoflux distribution, which are smaller than the

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on the strip, which may have the following values: (1)µ = −1

2 to approximate an isothermal strip provided thata/c  1, (2) µ = 0 for an isoflux distribution, and (3)

µ = 1

2, which gives a parabolic flux distribution The three flux distributions are

q(x) =















Q Lπ

1

a2− x2 forµ = −1

2

Q

2Q Lπa2

a2− x2 forµ = 1

2

(4.119)

The dimensionless spreading resistance relationship based on the mean source temperature is

kLR s =Γ(µ + 3/2)

π2

∞

n=1

2 µ+1/2

sin

n2 J µ+1/2 n (4.120) where

ϕ n= nπ + Bi tanh nπτ nπ tanh nπτ + Bi n = 1, 2, 3,

and the three dimensionless system parameters and their ranges are

c < 1 0< Bi =

hc

k < ∞ 0< τ =

t

c < ∞

The general relationship gives the following three relationships for the three flux distributions:

kLR s =



















1

2

∞

n=1

sin

2

1

2π3

∞

n=1

sin2

2

2π3

∞

n=1

sin

n3 J1 n forµ = 1

2

(4.121)

The influence ofthe cooling along the bottom surface on the spreading resistance is given by the functionϕ n, which depends on two parameters, Bi andτ Ifthe channel is relatively thick (i.e.,τ ≥ 0.85), ϕ n → 1 for all values n = 1 · · · ∞, and the influence

ofthe parameter Bi becomes negligible Whenτ ≥ 0.85, the finite channel may be

modeled as though it were infinitely thick This special case is presented next

312 THERMAL SPREADING AND CONTACT RESISTANCES

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4.12 STRIP ON AN INFINITE FLUX CHANNEL

Ifthe relative thickness ofthe rectangular channel becomes very large (i.e.,τ → ∞), the relationships given above approach the relationships appropriate for the infinitely thick flux channel The dimensionless spreading resistance for this problem depends

on two parameters: the relative size ofthe strip parameterµ The three relationships for the dimensionless spreading resistance are obtained from the relationships given above withϕ n= 1

Numerical values forLkR sfor three values ofµ are given in Table 4.14 From the tabulated values it can be seen that the spreading resistance values for the isothermal strip are smaller than the values for the isoflux distribution, which are smaller than the values for the parabolic distribution for all values of

as

4.12.1 True Isothermal Strip on an Infinite Flux Channel

There is a closed-form relationship for the true isothermal area on an infinitely thick flux channel According to Sexl and Burkhard (1969), Veziroglu and Chandra (1969), and Yovanovich et al (1999), the relationship is

kLR s= π1 ln 1

Numerical values are given in Table 4.14 A comparison ofthe values corresponding

toµ = −1

2 and those for the true isothermal strip shows close agreement provided that

4.12.2 Spreading Resistance for an Abrupt Change in the Cross Section

Ifsteady conduction occurs in a two-dimensional channel whose width decreases

from 2a to 2b, there is spreading resistance as heat flows through the common

TABLE 4.14 Dimensionless Spreading ResistancekLR sin Flux Channels

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interface The true boundary condition at the common interface is unknown The temperature and the heat flux are both nonuniform Conformal mapping leads to a closed-form solution for the spreading resistance

The relationship for the spreading resistance is, according to Smythe (1968),

kLR s =2π1 1



ln1

4



(4.123)

where values reveals that they lie between the values forµ = −1

2 andµ = 0 The average value ofthe first two columns corresponding toµ = −1

2 andµ = 0 are in very close agreement with the values in the last column The differences are less than 1% for

4.13 TRANSIENT SPREADING RESISTANCE WITHIN ISOTROPIC SEMI-INFINITE FLUX TUBES AND CHANNELS

Turyk and Yovanovich (1984) reported the analytical solutions for transient spreading resistance within semi-infinite circular flux tubes and two-dimensional channels The circular contact and the rectangular strip are subjected to uniform and constant heat flux

4.13.1 Isotropic Flux Tube

The dimensionless transient spreading resistance for an isoflux circular source of radiusa supplying heat to a semi-infinite isotropic flux tube ofradius b, constant

thermal conductivityk, and thermal diffusivity α is given by the series solution

4kaR s= 16

∞

n=1

J2

1(δ n n √Fo)

δ3

n J2

where 2 > 0, and δ nare the positive roots ofJ1(·) = 0.

The average source temperature rise was used to define the spreading resistance The series solution approaches the steady-state solution presented in an earlier section when the dimensionless time satisfies the criterion Fo 2or when the real time satisfies the criteriont ≥ a2 2

4.13.2 Isotropic Semi-infinite Two-Dimensional Channel

The dimensionless transient spreading resistance for an isoflux strip of width 2a

within a two-dimensional channel ofwidth 2b, length L, constant thermal

conduc-tivityk, and thermal diffusivity α was reported as (Turyk and Yovanovich, 1984)

LkR s =π13

∞

m=1

Fo

314 THERMAL SPREADING AND CONTACT RESISTANCES

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where

is defined as Fo= αt/a2 There is no half-space solution for the two-dimensional channel The transient solution is within 1% ofthe steady-state solution when the dimensionless time satisfies the criterion Fo 2

4.14 SPREADING RESISTANCE OF AN ECCENTRIC RECTANGULAR AREA ON A RECTANGULAR PLATE WITH COOLING

A rectangular isoflux area with side lengths c and d lies in the surface z = 0 of

a rectangular plate with side dimensionsa and b The plate thickness is t1 and its thermal conductivity isk1 The top surface outside the source area is adiabatic, and all sides are adiabatic The bottom surface atz = t1is cooled by a fluid or a heat sink that is in contact with the entire surface In either case the heat transfer coefficient is denoted ash and is assumed to be uniform The origin of the Cartesian coordinate

system (x,y,z) is located in the lower left corner The system is shown in Fig 4.14.

Figure 4.14 Isotropic plate with an eccentric rectangular heat source (From Muzychka et al., 2000.)

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The temperature rise ofpoints in the plate surfacez = 0 is given by the relationship

θ(x,y,z) = A0+ B0z

+

∞

m=1

cosλx(A1coshλz + B1sinhλz)

+

∞

n=1

cosδy(A2coshδz + B2sinhδz)

+∞

m=1

∞

n=1

cosλx cos δy(A3coshβz + B3sinhβz) (4.126)

The Fourier coefficients are obtained by means of the following relationships:

A0= Q

ab

t1

k1

+ 1

h

 and B0= − Q

A1=2Q

 sin2X c +c

2 λm − sin2X c −c

2 λm 

abck1λ2

A2=2Q

 sin2Y c +d

2 δn − sin2Y c −d

2 δn 

abck1δ2

A3=16Q cos(λ m X c ) sin

1

2λm c cos(δ n Y c ) sin1

2δn d abcdk1βm,nλmδn φ(β m,n ) (4.130)

The other Fourier coefficients are obtained by the relationship

whereζ is replaced by λm , δ n, orβm,nas required The eigenvalues are

λm= mπ a δn= nπ b βm,n=λ2

m+ δ2

n

The mean temperature rise ofthe source area is given by the relationship

θ = θ1D+ 2∞

m=1

A mcos(λ m X c ) sin

1

2λm c

∞

n=1

A ncos(δ n Y c ) sin

1

2δn d

δn d

+ 4

∞

m=1

∞

n=1

A mncos(δ n Y c ) sin1

2δn d cos(λ m X c ) sin1

2λm c

where the one-dimensional temperature rise is

316 THERMAL SPREADING AND CONTACT RESISTANCES

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θ1D= Q

ab

t1

k1

+1

h



for an isotropic plate The total resistance is related to the spreading resistance and the one-dimensional resistance:

Rtotal=Qθ = R1D+ R s (K/W) (4.134) where

R1D= ab1

t

1

k1

+1h



4.14.1 Single Eccentric Area on a Compound Rectangular Plate

Ifa single source is on the top surface ofa compound rectangular plate that consists oftwo layers having thicknessest1 andt2 and thermal conductivitiesk1 andk2, as shown in Fig 4.15, the results are identical except for the system parameterφ, which now is given by the relationship

φ(ζ) = (αe4ζt1− e2ζt1) + 1(e2ζ(2t1+t2)− αe2ζ(t1+t2))

(αe4ζt1+ e2ζt1) + 1(e2ζ(2t1+t2)+ αe2ζ(t1+t2)) (4.136)

where

1 = ζ + h/k2

ζ − h/k2

and α = 1− κ

withκ = k2/k1andζ is replaced by λm , δ n, orβm,n, accordingly The one-dimensional temperature rise in this case is

Figure 4.15 Compound plate with an eccentric rectangular heat source (From Muzychka

et al., 2000.)

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θ1D= Q

ab

t1

k1

+ t2

k2

+1

h



4.14.2 Multiple Rectangular Heat Sources on an Isotropic Plate

The multiple rectangular sources on an isotropic plate are shown in Fig 4.16 The sur-face temperature by superposition is given by the following relationship (Muzychka

et al., 2000):

T (x,y,0) − T f =

N

i=1

θi (x,y,0) (K) (4.139)

whereθi is the temperature excess for each heat source by itself andN ≥ 2 is the

number ofdiscrete heat sources The temperature rise is given by

Figure 4.16 Isotropic plate with two eccentric rectangular heat sources (From Muzychka et al., 2000.)

318 THERMAL SPREADING AND CONTACT RESISTANCES

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θi (x,y,0) = A i

0+

∞

m=1

A i

mcosλx +

∞

n=1

A i

ncosδy

+

∞

m=1

∞

n=1

A i

whereφ and A i

0 = θ1Dare defined above for the isotropic and compound plates

The mean temperature ofan arbitrary rectangular area ofdimensionsc j andd j

located atX c,jandY c,j, may be obtained by integrating over the regionA j = c j d j

θj = A1

j



A j

θi dA j = A1

j



A j

N

i=1

θi (x,y,0) dA j (4.141)

which may be written as

θj =

N

i=1

1

A j



A j

θi (x,y,0) dA j =

N

i=1

The mean temperature ofthe jth heat source is given by

T j − T f =N

i=1

where

θi = A i

o+ 2

∞

m=1

A i m

cos(λ m X c,j ) sin1

2λm c j

∞

n=1

A i n

cos(δ n Y c,j ) sin1

2δn d j

δn d j

+ 4∞

m=1

∞

n=1

A i mn

cos(δ n Y c,j ) sin1

2δn d j cos(λ m X c,j ) sin1

2λm c j

Equation (4.143) represents the sum ofthe effects ofall sources over an arbitrary location Equation (4.143) is evaluated over the region ofinterestc j , d j located at

X c,j,Y c,j, with the coefficients A i

0, A i

m , A i

n, andA i

mn evaluated at each ofthe ith

source parameters

4.15 JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS

The elastoconstriction and elastogap resistance models (Yovanovich, 1986) are based

on the Boussinesq point load model (Timoshenko and Goodier, 1970) and the Hertz

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2a

D

2

0

␦

r

E2 2, v

E1 1, v

z

Figure 4.17 Joint formed by elastic contact of a sphere or a cylinder with a smooth flat surface (From Kitscha, 1982.)

distributed-load model (Hertz, 1896; Timoshenko and Goodier, 1970; Walowit and Anno, 1975; Johnson, 1985) Both models assume that bodies have “smooth” sur-faces, are perfectly elastic, and that the applied load is static and normal to the plane ofthe contact area In the general case the contact area will be elliptical, having semimajor and semiminor axesa and b, respectively These dimensions are much

smaller than the dimensions ofthe contacting bodies The circular contact area pro-duced when two spheres or a sphere and a flat are in contact are two special cases ofthe elliptical contact Also, the rectangular contact area, produced when two ideal circular cylinders are in line contact or an ideal cylinder and a flat are in contact, are special cases ofthe elliptical contact area

Figure 4.17 shows the contact between two elastic bodies having physical proper-ties (Young’s modulus and Poisson’s ratio):E1, ν1andE2, ν2, respectively One body

is a smooth flat and the other body may be a sphere or a circular cylinder having ra-diusD/2 The contact 2a is the diameter ofa circular contact area for the sphere/flat

contact and the width of the contact strip for the cylinder/flat contact A gap is formed adjacent to the contact area, and its local thickness is characterized byδ

Heat transfer across the joint can take place by conduction by means of the contact area, conduction through the substance in the gap, and by radiation across the gap if the substance is “transparent,” or by radiation ifthe contact is formed in a vacuum

The thermal joint resistance model presented below was given by Yovanovich (1971, 1986) It was developed for the elastic contact of paraboloids (i.e., the elastic contact formed by a ball and the inner and outer races of an instrument bearing)

4.15.1 Point Contact Model

Semiaxes of an Elliptical Contact Area The general shape ofthe contact area

is an ellipse with semiaxesa and b and area A = πab The semiaxes are given by

the relationships (Timoshenko and Goodier, 1970)

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a = m 3F∆

2(A + B)

1/3 and b = n 3F∆

2(A + B)

1/3

(4.145)

whereF is the total normal load acting on the contact area, and ∆ is a physical

parameter defined by

∆ = 1 2

1− ν2 1

E1

+1− ν22

E2



when dissimilar materials form the contact The physical parameters are Young’s modulusE1andE2and Poisson’s ratioν1andν2 The geometric parametersA and

B are related to the radii ofcurvature ofthe two contacting solids (Timoshenko and

Goodier, 1970):

2(A + B) = 1

ρ1

+ 1

ρ

1

+ 1

ρ2

+ 1

ρ

2

= 1

where the local radii ofcurvature ofthe contacting solids are denoted asρ1, ρ

1, ρ2, andρ

2 The second relationship betweenA and B is

2(B − A) =



1

ρ1

−ρ1

1

2

+

1

ρ2

−ρ1

2

2

+ 2

1

ρ1

− 1

ρ

1


1

ρ2

− 1

ρ

2

 cos 2φ

1/2

(4.148)

The parameter φ is the angle between the principal planes that pass through the contacting solids

The dimensionless parametersm and n that appear in the equations for the

semi-axes are called the Hertz elastic parameters They are determined by means ofthe

following Hertz relationships (Timoshenko and Goodier, 1970):

m =

 2 π

Ek

k2

1/3 and n = π2kEk 1/3

(4.149)

where Ek is the complete elliptic integral ofthe second kind ofmodulus k (Abramowitz and Stegun, 1965; Byrd and Friedman, 1971), and

k= 1− k2 with k = m n = b a ≤ 1 (4.150) The additional parametersk and kare solutions ofthe transcendental equation (Tim-oshenko and Goodier, 1970):