In this case 1 → 0 and the spreading resistance depends on the four system parameters a, t1, k1, k2 and the heat flux parameterµ.. Ifwe set t1= 0 or k1= k2, the general solution goes to t
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For all values in the range 0< Bi < ∞ and for all values τ > 0.72, tanh δ nτ ≈ 1 for alln ≥ 1 Therefore, φ n ≈ 1 for values n ≥ 1.
Characteristics of B n Whenτ1> 0.72, tanh δ nτ = 1, φn = 1 for all 0 < Bi < ∞;
therefore,B n = 1 f or n ≥ 1.
4.6.1 Special Cases of the Compound Disk Solution
The general solution for the compound disk may be used to obtain spreading re-sistances for several special cases examined previously by many researchers These special cases arise when some ofthe system parameters go to certain limits The spe-cial cases fall into the following two categories: isotropic half-space, semi-infinite flux tube, and finite disk problems; and layered half-space and semi-infinite flux tube problems Figures 4.5 and 4.6 show the several special cases that arise from the gen-eral case presented above Results for sevgen-eral special cases are discussed in more detail in subsequent sections
4.6.2 Half-Space Problems
Ifthe dimensions ofthe compound disk (b, t) become very large relative to the radius
a and the first layer thickness t1, the general solution approaches the solution for the case of a single layer in perfect thermal contact with an isotropic half-space In this case 1 → 0 and the spreading resistance depends on the four system parameters (a, t1, k1, k2) and the heat flux parameterµ Ifwe set t1= 0 or k1= k2, the general solution goes to the special case ofa circular heat source in perfect contact with an isotropic half-space In this case the spreading resistance depends on two system parameters (a, k2) and the heat flux parameterµ The dimensionless spreading resistance is now defined asψ = 4k2aR s, and it is a constant depending on the heat
flux parameter The total resistance is equal to the spreading resistance in both cases because the one-dimensional resistance is negligible The half-space problems are
shown in Figs 4.5d and 4.6d.
4.6.3 Semi-infinite Flux Tube Problems
The general solution goes to the semi-infinite flux tube solutions when the system parameterτ2→ ∞ In this case the spreading resistance will depend on the system parameters (a, b, t1, k1, k2) and the heat flux parameterµ The dimensionless spread-ing resistance will be a function of the parameters ( 1, κ) and µ Ifone sets t1= 0
ork1 = k2 = k, the dimensionless spreading resistance ψ = 4kaRs depends on the system parameters (
problems are shown in Figs 4.5c and 4.6c.
4.6.4 Isotropic Finite Disk with Conductance
In this case, one putsk1 = k2 = k or κ = 1 The dimensionless spreading
re-sistanceψ = 4kaR s depends on the system parameters (a, b, t, k, h) and µ or the
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a
a
a
a
b
b
b
k1
k1
k1
k1
k1
t1
t1
t2
t1
t1
t1
k2
k2
k2
k2
k2 ( ) 0 < <a ⬁ ( ) — 0, — , 0 < <c ⬁ ⬁
( ) — 0, — , 0 < <d ⬁ ⬁ ( ) Bi — , — , 0 < <b ⬁ ⬁ ⬁
b
T = 0
T = 0
h z
= 0
⫺= 1/2
t r q
= /k k1 2 Bi = /hb k1
= /t b1 = /a b
= /tb
Figure 4.5 Four problems with a single layer on a substrate (From Yovanovich et al., 1998.)
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q
q
q
a
a
a
a
a
b
b
b
k1
k1
k1
k1
k1
t1
t2
k2
( ) — 0, = 1, —d ⬁ ( ) = 1, Bi —b ⬁
b
T = 0
T = 0
h z
= 0
⫺= 1/2
t r q
= /k k1 2 Bi = /hb k1
1= /t b1 = /a b
Figure 4.6 Four problems for isotropic systems (From Yovanovich et al., 1998.)
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dimensionless system parameters (
problem is shown in Fig 4.6b These special cases are presented in the following
sections
WITH CONDUCTANCE
The dimensionless spreading resistance for isotropic (κ = 1) finite disks (τ1< 0.72)
with negligible thermal resistance at the heat sink interface (Bi= ∞) is given by the following solutions (Kennedy, 1960; Mikic and Rohsenow, 1966; Yovanovich et al., 1998): Forµ = −1
2:
4kaR s = 8 ∞
n=1
J1(δ n n
δ3
n J2(δ n ) tanhδnτ (4.81)
Forµ = 0:
4kaR s = 16 ∞
n=1
J2
1(δ n
δ3
n J2
0(δ n )tanhδnτ (4.82)
Ifthe external resistance is negligible Bi→ ∞, the temperature at the lower face of the disk is isothermal The solutions for isofluxµ = 0 heat source and isothermal base temperature were given by Kennedy (1960) for the centroid temperature and the area-averaged contact area temperature
4.7.1 Correlation Equations
A circular heat source ofradiusa is attached to one end ofa circular disk ofthickness
t, radius b, and thermal conductivity k The opposite boundary is cooled by a fluid
at temperatureT f through a uniform heat transfer coefficienth The sides ofthe disk
are adiabatic and the region outside the source area is also adiabatic The flux over the source area is uniform The heat transfer through the disk is steady The external resistance is defined asRext= 1/hA, where A = πb2
The solution for the isoflux boundary condition and with external thermal resis-tance was recently reexamined by Song et al (1994) and Lee et al (1995) They nondimensionalized the constriction resistance based on the centroid and area-averaged temperatures using the square root ofthe source area (as recommended
by Yovanovich, 1976b, 1991, 1997; Yovanovich and Burde, 1977; Yovanovich and Schneider, 1977; Chow and Yovanovich, 1982; Yovanovich et al., 1984; Yovanovich and Antonetti, 1988) and compared the analytical results against the numerical results reported by Nelson and Sayers (1992) over the full range of independent parameters:
Bi root ofthe source area to report their numerical results The agreement between the analytical and numerical results were reported to be in excellent agreement
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Lee et al (1995) recommended a simple closed-form expression for the dimen-sionless constriction resistance based on the area-averaged and centroid temperatures
They defined the dimensionless spreading resistance parameter asψ = √π kaR c, whereR cis the constriction resistance, and they recommended the following approx-imations: For the area-averaged temperature
ψave= 1 2
and for the centroid temperature:
ψmax=√1
with
ϕ c= Bi tanh(δ cτ) + δc
The approximations above are within±10% ofthe analytical results (Song et al., 1994; Lee et al., 1995) and the numerical results ofNelson and Sayers (1992) The locations ofthe maximum errors were not given
4.7.2 Circular Area on a Single Layer (Coating) on a Half-Space
Integral solutions are available for the spreading resistance for a circular source of radiusa in contact with an isotropic layer ofthickness t1and thermal conductivityk1 which is in perfect thermal contact with an isotropic half-space of thermal conduc-tivityk2 The solutions were obtained for two heat flux distributions corresponding
to the flux parameter valuesµ = −1
2andµ = 0
Equivalent Isothermal Circular Contact Dryden (1983) obtained the solution for the equivalent isothermal circular contact flux distribution:
2πa2√
The problem is depicted in Fig 4.7
The dimensionless spreading resistance, based on the area-averaged temperature,
is obtained from the integral (Dryden, 1983):
ψ = 4k2aR s= 4
π
k2
k1
∞ 0
λ2 exp(ζt1/a) + λ1 exp(−ζt1/a)
λ2 exp(ζt1/a) − λ1 exp(−ζt1/a)
J1(ζ) sin ζ
ζ2 dζ (4.88) withλ1= (1 − k2/k1)/2 and λ2= (1 + k2/k1)/2 The parameter ζ is a dummy
vari-able ofintegration The constriction resistance depends on the thermal conductivity
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Figure 4.7 Layered half-space with an equivalent isothermal flux (From Yovanovich et al., 1998.)
ratio k1/k2 and the relative layer thicknesst1/a Dryden (1983) presented simple
asymptotes for thermal spreading in thin layers, t1/a ≤ 0.1, and in thick layers,
t1/a ≥ 10 These asymptotes were also presented as dimensionless spreading
resis-tances defined as 4k2aR s They are:
Thin-layer asymptote:
(4k2aR s )thin= 1 +π4 t1
a
k 2
k1 −k1
k2
(4.89) Thick-layer asymptote:
(4k2aR s )thick= k2
k1 −π2t a
1
k2
k1
ln 2
1+ k1/k2
(4.90)
These asymptotes provide results that are within 1% of the full solution for relative layer thickness:t1/a < 0.5 and t1/a > 2.
The dimensionless spreading resistance is based on the substrate thermal conduc-tivityk2 The general solution above is valid for conductive layers wherek1/k2> 1
as well as for resistive layers wherek1/k2< 1 The infinite integral can be evaluated
numerically by means ofcomputer algebra systems, which provide accurate results
4.7.3 Isoflux Circular Contact
Hui and Tan (1994) presented an integral solution for the isoflux circular source The dimensionless spreading resistance is
4k2aR s = 32
3π2
k 2
k1
2 +π8
1−
k 2
k1
2 ∞ 0
J2
1(ζ) dζ
[1+ (k1/k2) tanh (ζt1/a)]ζ2 (4.91)
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which depends on the thermal conductivity ratiok1/k2and the relative layer thick-nesst1/a The dimensionless spreading resistance is based on the substrate thermal
conductivity k2 The general solution above is valid for conductive layers, where
k1/k2> 1, as well as resistive layers, where k1/k2< 1.
4.7.4 Isoflux, Equivalent Isothermal, and Isothermal Solutions
Negus et al (1985) obtained solutions by application ofthe Hankel transform method for flux-specified boundary conditions and with a novel technique of linear superpo-sition for the mixed boundary condition (isothermal contact area and zero flux outside the contact area) They reported results for three flux distributions: isoflux, equivalent isothermal flux, and true isothermal source There results were presented below
Isoflux Contact Area For the isoflux boundary condition, they reported the result forψq = 4k1aR s
ψq = 32
3π2 +π82
∞
n=1 (−1) nαn I q (4.92)
The first term is the dimensionless isoflux spreading resistance ofan isotropic half-space ofthermal conductivityk1, and the second term accounts for the effect of the layer relative thickness and relative thermal conductivity The thermal conductivity parameterα is defined as
α = 1− κ
1+ κ withκ = k1/k2 The layer thickness–conductivity parameterI qis defined as
I q = 1
2π
2
2(γ + 1)E
2/(γ + 1) − π
2√
2γIγ− 2πnτ1
with
Iγ= 1 + 0.09375γ2 +0.0341797γ4 +0.00320435γ6
The relative layer thickness isτ1= t/a and the relative thickness parameter is
γ = 2n2τ2
1+ 1 The special function E(·) is the complete elliptic integral ofthe second kind
(Abramowitz and Stegun, 1965)
Equivalent Isothermal Contact Area For the equivalent isothermal flux boundary condition, they reported the result forψei = 4k1aR s
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ψei = 1 +π8
∞
n=1 (−1) nαn I ei (4.93)
where as discussed above, the first term represents the dimensionless spreading resis-tance ofan isothermal contact area on an isotropic half-space ofthermal conductivity
k1and the second term accounts for the effect of the layer relative thickness and the relative thermal conductivity The thermal conductivity parameterα is defined above
The relative layer thickness parameterI eiis defined as
I ei= 1− β−2
β − β−1 +1
2sin−1
β−1 − 2nτ1
withτ1= t/a and
β = nτ1+ n2τ2+ 1
Isothermal Contact Area For the isothermal contact area, Negus et al (1985) reported a correlation equation for their numerical results They reported thatψT =
4k1aR sin the form
ψT = F1tanhF2+ F3 (4.94) where
F1= 0.49472 − 0.49236κ − 0.0034κ2
F2= 2.8479 + 1.3337τ + 0.06864τ2 with τ = log10τ1
F3= 0.49300 + 0.57312κ − 0.06628κ2 whereκ = k1/k2 The correlation equation was developed for resistive layers: 0.01 ≤
κ ≤ 1 over a wide range ofrelative thickness 0.01 ≤ τ1 ≤ 100 The maximum relative error associated with the correlation equation is approximately 2.6% atτ1 =
0.01 and κ = 0.2 Numerical results for ψ q , ψ ei, andψT for a range of values of
τ1 andκ were presented in tabular form for easy comparison They found that the values forψq were greater than those forψei and thatψei ≤ ψT The maximum
difference betweenψqandψT was approximately 8% The values forψT > ψ eifor very thin layers,τ1 ≤ 0.1 and for κ ≤ 0.1; however, the differences were less than
approximately 8% For most applications the equivalent isothermal flux solution and the true isothermal solution are simililar
The problem offinding the spreading resistance in an semi-infinite isotropic cir-cular flux tube has been investigated by many researchers (Roess, 1950; Mikic and
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k1
b
t
k k1 2/ = 1
= / —t b ⬁
= /a b ≤0.9
Figure 4.8 Isotropic flux tube with an isoflux area (From Yovanovich et al., 1998.)
Rohsenow, 1966; Gibson, 1976; Yovanovich, 1976a, b; Negus and Yovanovich, 1984a, b; Negus et al., 1989) The system with uniform heat flux on the circular area is shown
in Fig 4.8
This problem corresponds to the case whereκ = 1 and τ → ∞, and therefore the spreading resistance depends on the system parameters (a, b, k) and the flux
distribution parameterµ The dimensionless spreading resistance defined as ψ =
4kaR s, whereR s is the spreading resistance, depends on several studies are given below
4.8.1 General Expression for a Circular Contact Area with Arbitrary Flux on a Circular Flux Tube
The general expression for the dimensionless spreading (constriction) resistance
4kaR s for a circular contact subjected to an arbitrary axisymmetric flux distribution
f (u) (Yovanovich, 1976b) is obtained from the series
4kaR s =1 8/π
0 uf (u) du
∞
n=1
J1(δ n
δ2
n J2
0(δ n )
1
0
uf (u)J0(δ n (4.95)
whereδnare the positive roots ofJ1 source area
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Flux Distributions of the Form (1 − u2)µ Yovanovich (1976b) reported the
following general solution for axisymmetric flux distributions of the formf (u) = (1 − u2)µ, where the parameter µ accounts for the shape of the flux distribution The
general expression above reduces to the following general expression:
4kaR s = 16
π(µ + 1)2µΓ(µ + 1)
1 ∞
n=1
J1(δ n µ+1(δ n
δ3
n J0(δ n )(δ n µ (4.96)
whereΓ(·) is the gamma function (Abramowitz and Stegun, 1965) and Jν(·) is the
Bessel function of arbitrary orderν (Abramowitz and Stegun, 1965)
The general expression above can be used to obtain specific solutions for various values ofthe flux distribution parameterµ Three particular solutions are considered next
Equivalent Isothermal Circular Source The isothermal contact area requires solution ofa difficult mathematical problem that has received much attention by numerous researchers (Roess, 1950; Kennedy, 1960; Mikic and Rohsenow, 1966;
Gibson, 1976; Yovanovich, 1976b; Negus and Yovanovich, 1984a,b)
Mikic and Rohsenow (1966) proposed use ofthe flux distribution corresponding
toµ = −1
2to approximate an isothermal contact area for small relative contact areas 0
4kaR s =π81
∞
n=1
J1(δ n n
δ3
n J2
An accurate correlation equation ofthis series solution is given below
Isoflux Circular Source The general solution above withµ = 0 yields the isoflux solution reported by Mikic and Rohsenow (1966):
4kaR s =16
π
1 ∞
n=1
J2
1(δ n
δ3
n J2(δ n ) (4.98)
An accurate correlation equation ofthis series solution is given below
Parabolic Flux Distribution Yovanovich (1976b) reported the solution for the parabolic flux distribution corresponding toµ = 1
2
4kaR s= 24π 1
∞
n=1
J1(δ n n
δ3
n J2
0(δ n )
1
(δ n 2 −δ 1
(4.99)
An accurate correlation equation ofthis series solution is given below
Asymptotic Values for Dimensionless Spreading Resistances The three series solutions given above converge very slowly as