If heat transfer is into the half-space, the spreading resistance is defined as Car-slaw and Jaeger, 1959; Yovanovich, 1976c; Madhusudana, 1996; Yovanovich and Antonetti, 1988 R s =Tsourc
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Figure 4.2 Heat flow lines and isotherms for steady conduction from a finite heat source into
a half-space (From Yovanovich and Antonetti, 1988.)
doubly connected areas (e.g., circular or annular area) The free surface of the half-space is adiabatic except for the source area If heat enters the half-half-space, the flux lines spread apart as the heat is conducted away from the small source area (Fig 4.2);
then the thermal resistance is called spreading resistance.
Ifthe heat leaves the half-space through a small area, the flux lines are constricted
and the thermal resistance is called constriction resistance The heat transfer may be
steady or transient The temperature fieldT in the half-space is, in general,
three-dimensional, and steady or transient The temperature in the source area may be two-dimensional, and steady or transient
If heat transfer is into the half-space, the spreading resistance is defined as (Car-slaw and Jaeger, 1959; Yovanovich, 1976c; Madhusudana, 1996; Yovanovich and Antonetti, 1988)
R s =Tsource− Tsink
whereTsourceis the source temperature andTsinkis a convenient thermal sink tem-perature; and whereQ is the steady or transient heat transfer rate:
Q =
A
q n dA =
A
−k ∂T
∂n dA (W) (4.17)
whereq nis the heat flux component normal to the area and∂T /∂n is the temperature
gradient normal to the area Ifthe heat flux distribution is uniform over the area,
Q = qA For singly and doubly connected source areas, three source temperatures
have been used in the definition: maximum temperature, centroid temperature, and area-averaged temperature, which is defined according to Yovanovich (1976c) as
Tsource = 1
A
A
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whereA is the source area Because the sink area is much larger than the source area,
it is, by convention, assumed to be isothermal (i.e.,Tsink= T∞) The maximum and centroid temperatures are identical for singly connected axisymmetric source areas;
otherwise, they are different (Yovanovich, 1976c; Yovanovich and Burde, 1977);
Yovanovich et al., 1977) For doubly connected source areas (e.g., circular annulus), the area-averaged source temperature is used (Yovanovich and Schneider, 1977) If the source area is assumed to be isothermal,Tsource= T0
The general definition ofspreading (or constriction) resistance leads to the follow-ing relationship for the dimensionless spreadfollow-ing resistance:
k L R s =A L
whereθ = T (x,y) − T∞, the rise ofthe source temperature above the sink tem-perature The arbitrary characteristic length scale ofthe source area is denoted as
L For convenience the dimensionless spreading resistance, denoted as ψ = kL R s
(Yovanovich, 1976c; Yovanovich and Antonetti, 1988), is called the spreading
resis-tance parameter This parameter depends on the heat flux distribution over the source
area and the shape and aspect ratio ofthe singly or doubly connected source area The spreading resistance definition holds for transient conduction into or out of the half-space If the heat flux is uniform over the source area, the temperature is nonuniform;
and ifthe temperature ofthe source area is uniform, the heat flux is nonuniform (Carslaw and Jaeger, 1959; Yovanovich, 1976c) The relation for the dimensionless spreading resistance is mathematically identical to the dimensionless constriction re-sistance for identical boundary conditions on the source area For a nonisothermal singly connected area the spreading resistance can also be defined with respect to its maximum temperature or the temperature at its centroid (Carslaw and Jaeger, 1959;
Yovanovich, 1976c; Yovanovich and Burde, 1977) These temperatures, in general, are not identical and they are greater than the area-averaged temperature (Yovanovich and Burde, 1977)
The definition of spreading resistance for the isotropic half-space is applicable for single and multiply isotropic layers which are placed in perfect thermal contact with the half-space, and the heat that leaves the source area is conducted through the layer
or layers before entering into the half-space The conductanceh cannot be defined
for the half-space problem because the corresponding area is not defined
4.2.2 Spreading and Constriction Resistances in Flux Tubes and Channels
Ifa circular heat source ofareaA s is in contact with a very long circular flux tube
ofcross-sectional areaA t (Fig 4.3), the flux lines are constrained by the adiabatic
sides to “bend” and then become parallel to the axis ofthe flux tube at some distance
z = " from the contact plane at z = 0 The isotherms, shown as dashed lines, are
everywhere orthogonal to the flux lines The temperature in planesz = " √A t
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[273],(13) Flux Tube or Channel
Heat Source (Contact Area, )A s
z
z = "
z = 0
A t
Figure 4.3 Heat flow lines and isotherms for steady conduction from a finite heat source into
a flux tube or channel (From Yovanovich and Antonetti, 1988.)
“far” from the contact planez = 0 becomes isothermal, while the temperature in
planes nearz = 0 are two- or three-dimensional The thermal conductivity ofthe
flux tube is assumed to be constant
The total thermal resistanceRtotalfor steady conduction from the heat source area
inz = 0 to the arbitrary plane z = " is given by the relationship
QRtotal= T s − T z=" (K) (4.20)
whereT s is the mean source temperature andT z="is the mean temperature ofthe
arbitrary plane The one-dimensional resistance ofthe region bounded byz = 0 and
z = " is given by the relation
The total resistance is equal to the sum ofthe one-dimensional resistance and the spreading resistance:
Rtotal= R1D+ R s or Rtotal− R1D= R s (K) (4.22)
By substraction, the relationship for the spreading resistance, proposed by Mikic and Rohsenow (1966), is
R s= T s − T z=0
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whereQ is the total heat transfer rate from the source area into the flux tube It is
given by
Q =
A s
−k ∂T
∂z
z=0 dA s (W) (4.24)
The dimensionless spreading resistance parameter ψ = kL R s is introduced for convenience The arbitrary length scaleLis related to some dimension ofthe source area In general,ψ depends on the shape and aspect ratio ofthe source area, the
shape and aspect ratio ofthe flux tube cross section, the relative size ofthe source area, the orientation ofthe source area relative to the cross section ofthe flux tube, the boundary condition on the source area, and the temperature basis for definition of the spreading resistance
The definitions given above are applicable to singly and doubly connected source areas; however,A s /A t < 1 in all cases The source area and flux tube cross-sectional
area may be circular, square, elliptical, rectangular, or any other shape The heat flux and temperature on the source area may be uniform and constant In general, both heat flux and temperature on the source area are nonuniform Numerous examples are presented in subsequent sections
4.3 SPREADING AND CONSTRICTION RESISTANCES
IN AN ISOTROPIC HALF-SPACE 4.3.1 Introduction
Steady or transient heat transfer occurs in a half-spacez > 0 which may be isotropic
or may consist ofone or more thin isotropic layers bonded to the isotropic half-space
The heat source is some planar singly or doubly connected area such as a circular annulus located in the “free” surfacez = 0 ofthe half-space The dimensions ofthe
half-space are much larger than the largest dimension of the source area The “free”
surfacez = 0 ofthe half-space outside the source area is adiabatic Ifthe source
area is isothermal, the heat flux over the source area is nonuniform If the source is subjected to a uniform heat flux, the source area is nonisothermal
4.3.2 Circular Area on a Half-Space
There are two classical steady-state solutions available for the circular source area of radiusa on the surface ofa half-space ofthermal conductivity k The solutions are
for the isothermal and isoflux source areas In both problems the temperature field is two-dimensional in circular-cylinder coordinates [i.e.,θ(r, z)] The important results
are presented here
(Sneddon, 1966) in the free surface are
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z =
0 0≤ r < a θ = θ0
∂z = 0
(4.25)
and the condition at remote points is: As √
r2+ z2 → ∞, then θ → 0 The
temperature distribution throughout the half-spacez ≥ 0 is given by the infinite
integral (Carslaw and Jaeger, 1959)
θ = 2
πθ0
∞ 0
e −λz J0(λr) sin λa dλ
λ (K) (4.26)
whereJ0(x) is the Bessel function ofthe first kind oforder zero (Abramowitz and
Stegun, 1965) andλ is a dummy variable The solution can be written in the following
alternative form according to Carslaw and Jaeger (1959):
θ = π2θ0sin−1 2a
(r − a)2+ z2+ (r + a)2+ z2 (K) (4.27) The heat flow rate from the isothermal circular source into the half-space is found from
Q =
a 0
−k
∂θ
∂z
z=0
2πr dr
= 4kaθ0
∞ 0
J1(λa) sin λa dλλ
From the definition ofspreading resistance one finds the relationship for the spreading resistance (Carslaw and Jaeger, 1959):
R s =θ0
Q=
1
The heat flux distribution over the isothermal heat source area is axisymmetric (Car-slaw and Jaeger, 1959):
q(r) = Q
2πa2
1
1− (r/a)2 0≤ r < a (W/m2) (4.30) This flux distribution is minimum at the centroidr = 0 and becomes unbounded at
the edger = a.
surface are
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z =
0 0≤ r < a ∂θ ∂z= −q0
k
0 r > a ∂θ ∂z= 0
(4.31)
where q0 = Q/πa2 is the uniform heat flux The condition at remote points is identical The temperature distribution throughout the half-spacez ≥ 0 is given by
the infinite integral (Carslaw and Jaeger, 1959)
θ =q0a k
∞ 0
e −λz J0(λr)J1(λa) dλ
whereJ1(x) is the Bessel function ofthe first kind oforder 1 (Abramowitz and Stegun,
1965), andλ is a dummy variable The temperature rise in the source area 0 ≤ r ≤ a
is axisymmetric and is given by (Carslaw and Jaeger, 1959):
θ(r) = q0a
k
∞ 0
J0(λr)J1(λa) dλ
The alternative form of the solution according to Yovanovich (1976c) is
θ(r) = 2
π
q0a
k E
r
a
whereE(r/a) is the complete elliptic integral ofthe second kind ofmodulus r/a
(Byrd and Friedman, 1971) which is tabulated, and it can be calculated by means of computer algebra systems The temperatures at the centroidr = 0 and the edge r = a
ofthe source area are, respectively,
θ(0) = q0a
k and θ(a) =
2
π
q0a
The centroid temperature rise relative to the temperature rise at the edge is greater
by approximately 57% The values ofthe dimensionless temperature rise defined as
kθ(r/a)/(q0a) are presented in Table 4.1.
TABLE 4.1 Dimensionless Source Temperature
r/a kθ(r/a)/q0a r/a kθ(r/a)/q0a
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The area-averaged source temperature is
θ =πa12q0a
k
a 0
∞ 0
J0(λr)J1(λa) dλλ
2πr dr (K) (4.36) The integrals can be interchanged, giving the result (Carslaw and Jaeger, 1959):
θ =2q0
k
∞ 0
J2
1(λa) dλ
λ2 = 8
3π
q0a k
(K) (4.37)
According to the definition ofspreading resistance, one obtains for the isoflux circular source the relation (Carslaw and Jaeger, 1959)
R s =Qθ = 8
3π2
1
ka
The spreading resistance for the isoflux source area based on the area-averaged tem-perature rise is greater than the value for the isothermal source by the factor
(R s )isoflux
(R s )isothermal
= 32
3π2 = 1.08076
4.3.3 Spreading Resistance of an Isothermal Elliptical Source Area
on a Half-Space
The spreading resistance for an isothermal elliptical source area with semiaxesa ≥ b
is available in closed form The results are obtained from a solution that follows the classical solution presented for finding the capacitance of a charged elliptical disk placed in free space as given by Jeans (1963), Smythe (1968), and Stratton (1941)
Holm (1967) gave the solution for the electrical resistance for current flow from
an isopotential elliptical disk The thermal solution presented next will follow the analysis ofYovanovich (1971)
The elliptical contact area x2/a2 + y2/b2 = 1 produces a three-dimensional
temperature field where the isotherms are ellipsoids described by the relationship
x2
a2+ ζ+
y2
b2+ ζ+
z2
The three-dimensional Laplace equation in Cartesian coordinates can be transformed into the one-dimensional Laplace equation in ellipsoidal coordinates:
∇2θ = ∂ζ ∂ f (ζ) ∂θ ∂ζ
whereζ is the ellipsoidal coordinate for the ellipsoidal temperature rise θ(ζ) and where
f (ζ) = (a2+ ζ)(b2+ ζ)ζ (4.41)
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The solution ofthe differential equation according to Yovanovich (1971) is
θ = C2− C1
∞ ζ
dζ
√
The boundary conditions are specified in the contact plane(z = 0) where
θ = θ0 within x2
a2 +y2
b2 = 1
∂θ
∂z = 0 outside
x2
a2 +y2
b2 = 1
(4.43)
The regular condition at points remote to the elliptical area isθ → 0 as ζ → ∞ This
condition is satisfied byC2 = 0, and the condition in the contact plane is satisfied
byC1 = −Q/4πk, where Q is the total heat flow rate from the isothermal elliptical
area The solution is, therefore, according to Yovanovich (1971),
θ = Q
4πk
∞ ζ
dζ
(a2+ ζ)(b2+ ζ)ζ (K) (4.44)
Whenζ = 0, θ = θ0, constant for all points within the elliptical area, and when
ζ → ∞, θ → 0 for all points far from the elliptical area According to the definition
ofspreading resistance for an isothermal contact area, we find that
R s =θ0
Q =
1
4πk
∞ 0
dζ
(a2+ ζ)(b2+ ζ)ζ (K/W) (4.45)
The last equation can be transformed into a standard form by setting sint = a/
a2+ ζ The alternative form for the spreading resistance is
R s = 1
2πka
π/2 0
dt
{1 − [(a2− b2)/a2] sin2t}1/2 (K/W) (4.46)
The spreading resistance depends on the thermal conductivity ofthe half-space, the semimajor axisa, and the aspect ratio ofthe elliptical area b/a ≤ 1 It is clear that
when the axes are equal (i.e.,b = a), the elliptical area becomes a circular area and the
spreading resistance isR s = 1/(4ka) The integral is the complete elliptic integral
ofthe first kindK(κ) ofmodulus κ = (a2− b2)/a2 (Byrd and Friedman, 1971;
Gradshteyn and Ryzhik, 1965) The spreading resistance for the isothermal elliptical source area can be written as
R s = 1
2πka K(κ) (K/W) (4.47)
The complete elliptic integral is tabulated (Abramowitz and Stegun, 1965; Magnus
et al (1966); Byrd and Friedman, 1971) It can also be computed efficiently and very accurately by computer algebra systems
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TABLE 4.2 Dimensionless Spreading Resistance of an Isothermal Ellipse
4.3.4 Dimensionless Spreading Resistance of an Isothermal Elliptical Area
To compare the spreading resistances ofthe elliptical area and the circular area, it is necessary to nondimensionalize the two results For the circle, the radius appears as the length scale, and for the ellipse, the semimajor axis appears as the length scale
For proper comparison ofthe two geometries it is important to select a length scale that best characterizes the two geometries The proposed length scale is based on the square root ofthe active area ofeach geometry (i.e.,L=√A) (Yovanovich, 1976c;
Yovanovich and Burde, 1977; Yovanovich et al., 1977) Therefore, the dimensionless spreading resistances for the circle and ellipse are
(k√A R s )circle=
√ π
4
(k√A R s )ellipse= 1
2√ π
a
b K(κ)
whereκ = 1− (b/a)2 The dimensionless spreading resistance values for an iso-thermal elliptical area are presented in Table 4.2 for a range of the semiaxes ratioa/b.
The tabulated values ofthe dimensionless spreading resistance reveal an inter-esting trend beginning with the first entry, which corresponds to the circle The di-mensionless resistance values decrease with increasing values ofa/b Ellipses with
larger values ofa/b have smaller spreading resistances than the circle; however, the
decrease has a relatively weak dependence ona/b For the same area the
spread-ing resistance ofthe ellipse witha/b = 10 is approximately 74% ofthe spreading
resistance for the circle
4.3.5 Approximations for Dimensionless Spreading Resistance
Two approximations are presented for quick calculator estimations of the dimension-less spreading resistance for isothermal elliptical areas:
k√A R s=
√ πα
(√α + 1)2 for 1≤ α ≤ 5
1
2√
παln 4α for 5≤ α < ∞
(4.48)
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whereα = a/b ≥ 1 Although both approximations can be used at α = 5, the second
approximation is slightly more accurate, and therefore it is recommended
4.3.6 Flux Distribution over an Isothermal Elliptical Area
The heat flux distribution over the elliptical area is given by (Yovanovich, 1971)
q(x,y) = Q
2πab 1−
x
a
2
− y
b
2−1/2
(W/m2) (4.49)
The heat flux is minimum at the centroid, where its magnitude isq0= Q/2πab, and
it is “unbounded” on the perimeter ofthe ellipse
4.4 SPREADING RESISTANCE OF RECTANGULAR SOURCE AREAS 4.4.1 Isoflux Rectangular Area
The dimensionless spreading resistances ofthe rectangular source area−a ≤ x ≤
a, −b ≤ y ≤ b with aspect ratio a/b ≥ 1 are found by means of the integral method
(Yovanovich, 1971) Employing the definition ofthe spreading resistance based on the area-averaged temperature rise withQ = 4qab gives the following dimensionless
relationship (Yovanovich, 1976c; Carslaw and Jaeger, 1959):
k√A R s =
√ π
sinh−1 1 +1 sinh−1
3
1+ 13 −
1+ 12
3/2
(4.50)
where the dimensionless spreading resistance is obtained from the relationship (Carslaw and Jaeger, 1959)
k√A R s =
√ π
1 sinh−1 −11
(4.51)
Typical values ofthe dimensionless spreading resistance for the isoflux rectangle based on the area-average temperature rise for 1≤ a/b ≤ 10 are given in Table 4.3.
Table 4.3 Dimensionless Spreading Resistance of an Isoflux Rectangular Area
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byC1 = −Q/4πk, where Q is the total heat flow rate from the isothermal elliptical
area The solution is, therefore, according