Temperature distributions in diamond heat sinks
Citation for published version (APA):Molenaar, J., & Staarink, G. W. M. (1985). Temperature distributions in diamond heat sinks. (WD report; Vol. 8501). Radboud Universiteit Nijmegen.
Document status and date: Published: 01/01/1985
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Report no WD 85-01
Temp~rature Distributions in
Diamond Heat Sinks.
J. Molenaar, G.W.M. Staarink February 1985
Project no 840601
Wiskundige Dienstverlening Faculteit der Wiskunde
en Natuurwetenschappen Katholieke Universiteit Toernooiveld
CONTENTS.
List of Symbols 1
Units 2
Introduction 3
Mathematical Physical Model 4
Computational Aspects 7
Green Function Method 8
Results 12
Conclusions 16
References 17
LIST OF SYMBOLS. See also figures 1 and 2.
DCB F Foes FlO Fo G(r,r') G;,i h IP k(T) ko L Q R Ro T TmaxDCB
u
v
diamond-copper boundaryflux(density), defined by equation (4) flux at the DCB
flow-in opening, i.e. device-diamond boundary
value of the total amount of heat across FlO per unit time Green function
weight factor corresponding with cornerpoint i of ti base of triangle t i
irreducible part of the DCB
(possibly temperature dependent) thermal conductivity k of copper (a constant)
constant proportional to the k of diamond height of the diamond cylinder I block orthogonal transformation
quadratic form
radius I half edge of the diamond cylinder I block radius I half edge of the FlO
temperature
maximum temperature in the diamond; this maximum always occurs in the centre of the
FlO
minimum temperature in the diamond; this minimum always occurs at the DCB most far from its centre
maximum temperature at the DCB; this relative maximum always occurs in the centre of the DCB
ph
triangle in IP; IP is partitioned into similar triangles ti quantity related to T by relation (2)d. · · d. h. . . b h (
a a a )
gra tent; m cartestan coor mates t ts operator ts gtven y t e vector
-,-a ,-a
ax
yz
--a
an
Units
.
.
.
.
.
. .
a2
a2
a2
Laplactan; m cartestan coordmates th1s operator ts g1ven by (--,
ax-
+
~+
~) oy-oz-derivative in a point on a surface along the outward normal; this oz-derivative is also denoted by a subscript n, e.g.
~~
=
TnThroughout this report we use the following units:
length : centimeter (em) power :Watt (W)
temperature : centigrade with respect to 300 K
Where confusion is not possible these units will not be mentioned.
--§ 1. INTRODUCTION.
In various microscale techniques a considerable amount of thermal energy is continuously produced within a small volume. This energy has to be removed rapidly, because otherwise the increasing tem-perature will cause malfunctioning or even destruction of the device. To that end a special type of dia-mond is applied with extremely useful thermal properties (see [1) and [2]). Though being an electric in-sulator its thermal conductivity in the range of 0-200°C is about four times as high as it is in copper. By mounting the device on a commodious diamond the energy is fast spread over a larger region and the temperature range may be limited. To save money, copper is used for the further heat transport to the surroundings.
The thermal conductivity of diamond is, contrary to that of copper, temperature dependent and de-creases with increasing temperature. The diffusion equation for the transport of heat is therefore a non-linear equation. Because of this it may happen that the produced amount of heat per unit of time is too large for the geometry under consideration. Then no stationary transport of heat is possible and the temperature will increase infinitely. Therefore the PURPOSE of the project is :
Calculation of the stationary temperature distribution in diamond-copper systems for several geometries as a function of the applied heat flux.
The report is organized as follows. In § 2 we present the model to be used and the mathematical formulation of the physical requirements. For the numerical evaluation of the model an iterative scheme has to be applied, which is dealt with in § 3. In this scheme use is made of the Green function method. The detailed application of this method is described in § 4. In § 5 we present and discuss the results. The conclusions are summarized in § 6. After the References we give in the appendix the results of the computations in the form of contour plots of the temperature distribution in the diamond.
In the next sections a considerable number of symbols and abbreviations are used. They are nearly all summarized in the List of Symbols, to which we refer the reader in case of confusion. The geometri· cal parameters are also illuminated in figures 1 and 2.
--§ 2. MATHEMATICAL PHYSICAL MODEL.
In this section we describe the mathematical physical model to be used. We consider two kinds of geometries (see fig. 1 and 2):
1) The system has axial symmetry; the device and the diamond are cylinders.
2) The system has block symmetry; the device and the diamond are blocks. For both geometries we make the following assumptions:
a) The copper block is so large compared to the diamond, that it may be considered as being infinitely large; at infinity the block is at room temperature.
b) The heat transport through the air is so small compared to the heat transport through diamond and copper that it may be neglected.
c) There is a continuous and homogeneous heat flux through the device-diamond boundary, denoted by FlO.
d) In the relevant temperature range the thermal capacities and the densities of the materials are con-stants.
e) Possible heat resistances at the device-diamond and the diamond-copper boundaries, denoted by FlO and DCB respectively, may be neglected.
Figure 1 :the case of axial symmetry.
The cylinder of. diamond is mounted on a large piece of copper. From the device a con-tinuous flow of heat F0 enters the diamond through the circular opening FlO. The
circu-lar diamond-copper boundary is denoted by DCB.
--fo
Figure 2 : the case of block symmetry.
In this case the FlO and the DCB are squares.
As we are only interested in the stationary solution of the problem, in both materials the diffusion equation (see [3] and [4]) reduces to:
V(k(T)VT(r)) = 0
For the thermal conductivity coefficient k(T) we have (see [1] and [2])
in diamond: k(T) = k1(T+300)-312 , k1 = 106 Watt. (centigrade)'1'/cm
(1)
in copper : k(T)
=
k0=
3.87 Watt/centigrade/emIntroducing a function U{T), with no actual physical meaning, by (2) U(T)
=
-2(7+300)-'1>
the diffusion equations in diamond and copper reduce to the Laplace equations for U and T respective-ly, i.e we have:
in diamond : A U = 0 (3)
in copper : A T = 0.
To complete the mathematical statement of the problem we have to specify the boundary conditions in terms of temperature distributions and heat flux densities. The heat flux in a point on a surface, normal to this surface, is given by:
(4) F == - k ( T ) -
aT
an
--The derivative is in the outward direction normal to the surface. From the assumptions we have the following conditions:
In diamond:
i) Across the FlO F is homogeneous and given by
(5)
Fo
rrRJ F
=
Fo4R6
for axial symmetry
for block symmetry
where F0 is the total amount of heat per unit time across the FlO.
ii) At the diamond-air boundaries it holds that F = 0. In copper:
i) At infinity it holds that T = 0.
ii) At the copper-air boundary it holds that F 0.
At the diamond-copper boundary DCB the conditions are:
i) The T distribution computed in copper and the U distribution computed in diamond should be related by equation (2). In contrast with the flux condition below this relation is nonlinear. It can easily be seen that this obstructs an analytical solution of the problem. It is possible to write the solution of the Laplace equations for U and T in terms of exponentials, Bessel functions and/or sines and cosines by means of separation of variables. The coefficients in the resulting series should follow from the boundary conditions at the DCB, bu~ in practice it is impossible to manipulate the infinite queues in the required way.
ii) The fluxes at the DCB computed in the diamond and the copper respectively must balance. Us-ing equation (1), (2) and (4) we may express this requirement as:
ar
au
k o - = - k t - ·
an
an
To obtain T, U and F at the DCB from these conditions we adopted a numerical iteration scheme, which is described in the next section.
--§ 3. COMPUTATIONAL ASPECTS.
As pointed out in the preceeding section there is no direct way to compute the T or F distribution at the DCB. In this section we give a description of the iterative scheme to be used. First we note that for diamond it is sensible to prescribe the T distribution. Otherwise the problem for diamond has only Neumann boundary conditions and no unique solution. For copper it is sensible to prescribe the F dis-tribution because F is also prescribed at the rest of the plane of which the DCB is a part. The iteration procedure has the following steps (the iterations cycles are numbered by i=1,2, ... ):
1) Start with an initial flux at the DCB.
2) Compute in copper the T distribution T; at the DCB, corresponding with F;.
3) Compute the distribution U; at the DCB from relation (2).
4) Compute the U distribution everywhere in the diamond.
5) Compute, using the results from step 4, a new flux
F;
at the DCB. 6) Compute F;+l at the DCB fromF;+t
=
yF;+
(1-y)F; , 0<
y<
1.7) IfF; and F;+l do not differ more than a given tolerance then stop , else go to step 2.
Remarks on the respective steps.
step 1:
The choice of the initial flux in step 1 is quite arbitrary, provided that the condition of flux neutrali-ty in diamond is satisfied. In physical terms this condition reflects that no heat is generated or lost in the diamond. In practice a homogeneous flux appeared to be a good choice.
step 2:
In this step we use the Green function method. The application of this method in our case will be explained in the next section.
step 4:
To solve the Laplace equation ll.U = 0 with the given boundary conditions, we used the appropiate subroutines from the FISHPACK library ([6]). These routines compute the solution on a given grid, using a finite difference method.
step 5:
Once the U distribution is known at some grid, the flux F across the DCB is computed by numeri-cal differentiation of U in the direction normal to the DCB.
step 6:
In step 6 a weighted mean from F; and
F;
is taken to obtain F; + 1• If y is taken too small thesuc-cesive F; showed an oscillating behaviour and the process did not converge. For each geometry and F0 value one can find an optimal value for y for which the convergence rate is greatest. In practice
a y value of 0.8 appeared to satisfy in all cases.
--§ 4. GREEN FUNCTION METHOD.
In this section we deal in more detail with the calculations in step 2 of the applied iteration scheme. The size and the boundary conditions of the copper block are such that application of the Green function method is very appropriate. We shall outline that in this method the calculations reduce to in-tegrating the product of the prescibed F distribution and the Green function over the DCB. For a gen-eral Green function approach we refer to [5].
For our purposes it suffices to start with the following identity of Green for the Laplace operator
a
within a volume V with surface S.The functions
f
and g are twice differentiable within V. The derivatives along the outward direction normal to S are denoted byf
n and gn .LetS consist of two complementary parts S1 and S2• In copper we have to solve the problem: llT = 0,
T prescribed at S 1 ,
Tn prescribed at S2 •
The corresponding Green function G is defined as the solution of: llG(r,r') = 6(r-r')
G=OatS1 ,
The boundary conditions hold for one of the arguments of G but, because G is symmetric in its argu-ments, we need not a more explicit notation. .
With the substitutions T
=
f
and G=
g the given identity of Green yields an explicit expression for Tin V:
T =
f
dS TGnSt.
This expression reduces further if we specify the boundary conditions more in detail. In copper S 1
corresponds with infinity, where Tis assumed to vanish. S2 corresponds with the bounding plane of the
copper block on which the diamond is mounted. From assumption b) we have that on this boundary the flux density vanishes outside the DCB. From relation ( 4) it follows that the same holds for Tn. Using this relation we may express T in terms of F.
(6) T(r) = +-k1
J
dr'G(r,r')F(r')0 DCB
It remains to find G. This problem is most easily solved by the method of images (see [3], [4], [5]), which yields
where r" is the mirror point of r' with respect to the boundary. Because we only need G with r' at
DCB and thus in this plane we may use G in the form:
G(r,r') = 1 1 r E DCB.
lr-r'
I'
--We shall apply these equations for both geometries under consideration. --We note that equation (6) in this form only holds for points r inside the copper block. Analytically it is possible to derive an analogous expression for points on the boundary (see [5]). Because the latter requires the calculation of the principal value of the integral, it is numerically more actractive to use equation (6). We then calcu-late T(r) with r in a plane inside the copper just below the DCB. Because F and thus the derivative of
T normal to the DCB are known, it is straightforward to obtain a good approximation for T on the
DCB.
Axial symmetry
In this case DCB is a disc (see figure 1). We use cylindrical coordinates (r ,q,,z ), choosing the centre of the DCB as origin and the positive
z
axis pointing into the copper and normal to the DCB. Due to the axial symmetry the $ dependence can be dropped everywhere. From equation (6) the temperature T(r ,z) with z>
0 is given by 2lt R T(r,z) = 2 1 kJ
d$J
dr'r'(a-f) cos q,)-'hF(r) :lt 0 0 0with a and f) defined by
f)
=
2rr'Reduction to a !-dimensional integral is obtained by introducing the famous elliptic integral I by
2lt
/(a,f})
=
J
d$(a-f} cos q,)-'h, a> O,f}<
a. 0For the calculation of I rapidly converging power series are available for llll allowed values of a and f) (see [7]). The integrand of the remaining !-dimensional integral
R
T(r ,z) = __l_k
J
dr' r' I ( a,f})F(r)2:rt 0 0
consists of smoothly varying functions and the integration is easily performed by e.g. Simpson's rule.
Block symmetry
In this case the DCB is a square (see figure 2). We use cartesian coordinates choosing the centre of the DCB as origin and the positive
z
axis pointing into the copper and normal to the DCB. As pointed out above, inside the copper block Tis given by the integral:T(r)
= -
1-J
dr' 1 , F(r')2:nko DCB
I
r-r IBecause of the symmetry F is invariant under the following group of operations Pi, which work on the x andy coordinates of some vector (x ,y ,z ).
P 1 (x ,y ,z)
=
(x ,y ,z) P3(x,y,z)=
(x,-y,z) P s(x ,y ,z) (y ,x ,z) P1(x,y,z)=
(y,-x,z) p 2(X ,y ,Z)=
(-X ,y ,Z) P4(x,y,z)=
(-x,-y,z) P6(x,y,z)=
(-y,x,z) P 8(x ,y ,z)= (-
y , - x ,z) 9--By means of these orthogonal transformations the integration domain DCB, given by {- R :::;; x ,y :::;; R}, can be reduced to an irreducible part IP. A possible choice for IP is the triangle {0:::;; x:::;; R, 0:::;; y:::;; x}. Using the P1 we may write
8
T(r) = - 1-""' Jdr' 1 F(P (r')) 21tkoi~ IP lr-Pt(r')l 1
= - 1
-f
J
dr' 1 F(r')21tkot~liP !Pt-1(r)-r'l
So T(r) is obtained by integrating over the smaller domain IP but with respect to the 8 vectors P1-1(r).
We want to calculate T(r) with r dose to the boundary. This means that the integrand may be very steep. To deal with this kind of integrand we adopted the following approach. Let us divide the trian-gle IP into N similar smaller ones ti. If we define
/j(r) =
J
dr'I
r~r'
I
F(r') ,I
then the temperature is given by the double sum - 1 N 8 -1
T(r) - 21tko
~~ ~~/i(P;
(r)) .It remains to calculate /i(r). Because F is a smooth function we expand it linearly over each ti. The coordinate system and the ti can be chosen such that the bases of all ti are along the x and y axis. Denoting the value ofF at the rectangular point (xi,Yi) of ti by F1 and at the other points by F2 and F3
the expansion of F over ti is given by
where h is the base length.
For some point r = (x ,y ,z =e) inside the copper the integration over ti is given by Ii(x ,y ,e)
=
J
J
dx'dy '[e2+(x -x')2+(y -y ')2r'
12F(x' ,y')ti
where F(x ',y ') is by the approximation given above. So we arrive at
Gt,j(x,y,e) =
J
Jdx'dy'(e
2+(x~x')
2+(y-y'fr~[l+ x'~xi +y'~Yi]
ti
G3,j(X ,y ,E) =
J
J
dx 'dy '[e2+(x-x ')2+(y -y ')2]-lfz[ Y'~Yi
]. tjFor small values of E it is not attractive to evaluate the integrals numerically. Fortunately they can be
evaluated analytically. For each ti and (x ,y ,€) they are of the same kind. In terms of these weight fac-tors the Ii are given by the simple relation
3
/j(r) =
L
Gk,j(r)Fk k=lWe shall omit here the analytical expressions. To derive them use is made of the following undefined integrals (see [8] ). Denoting by Q the quadratic form
--Q =a+ bx
+
cx2with a ,b ,c some constants, it holds that
5
dx Q-I!z= -
1-ln(2YCQ +2cx+b), c > 0Vc
5
dx Q1;,=
(2cx+b)VQ + (4ac-b2)5
dx Q_I!z 4c 8c Also we need: 11--§ 5. RESULTS.
In this section we present and discuss calculated T distributions in diamond as a function of the relevant parameters. The four parameters to be varied are the geometrical ones R ,R0 and L and the
applied amount of heat per unit time F0 (see figures 1 and 2).
Scaling properties.
Before turning to the results of the calculations we remark that on beforehand much insight can be obtained by studying the scaling properties of the problem. Looking at the diamond-copper system as a whole, we observe that T is determined by the Laplace equations (3) and the boundary conditions dealt with in the preceeding section. All boundary conditions are homogeneous, except for those at the FlO.
There it holds from equations (2), (4) and (5) that
(7) k(T)an
oT ,.
=
ktau
an == Fo/ Fo/ (nRof (2Ro)2for cylinder symmetry for block symmetry
The Laplace equations and the homogeneous boundary conditions are invariant for a scaling of the coordinates or T by a constant multiplicative factor. So all scaling properties can be derived from equa-tion (7). This equation is e.g. invariant for a scaling of both the coordinates (and thus of R ,R0 and L)
and Fo by the same constant factor. This means that one of the parameters R ,R0,L and F0 may be
kept constant in the calculations. In view of this we keep in the following the R value fixed.
Important Quantities.
The calculated T distributions are presented in the form of contourplots in the appendix. These plots are particularly useful to study the global features of the results.
In the analysis of the results some quantities are of special interest. They are :
a) The maximum temperature T max in the centre of the FlO. In practice this quantity may not exceed
some upper limit ( a usual value is :::::: 200 ), because otherwise the device, which is to be cooled, sustains damage.
b) The minimum temperature T min on the DCB most far away from its centre.
c) The temperature TmaxDCB in the centre of the DCB. This quantity gives the maximum T at the
DCB. An upper limit for T maxDCB is determined by the way in which the diamond and the copper
are sticked together.
d) The flux F0c8 at the DCB. Also for Fvc8 an upper limit holds for the same reason as under c.
Axial Symmetry.
In the following results are presented for some particular geometries and F0 values. The chosen
examples are representative for all cases.
--1. Dependence of T max and T min on F O·
In figures la-c and 2a-c in the appendix contour plots are given for two rather different geometries and various F0 values.
In series l a frequently applied geometry is used (Rn
=
O.lR. L = 0.5R).In series 2 a little realistic geometry is chosen with a very long cylinder (L = 2R) and the FlO covering the whole cylinder (R0
=
R).The latter geometry strongly resembles the !-dimensional case.
Fo Tmax Tmin 10 2.3 0.6 100 24.4 6.0 200 51.6 12.0 400 116.5 24.0 600 199.7 35.8
Table L Results for T rna~ and T min in case of
cylinder symmetry for various values of F 0• The geometrical parameters used in the computations are R
=
1., R0 = 0.1 andL
=
0.5. Fo Tmax Tmin 100 10.0 6.1 200 20.3 12.3 500 53.1 30.7 1000 114.9 61.0 1500 187.6 91.0-Table 2. Results for T ma~ and T min in case of
cylinder symmetry for various values of F0•
The geometrical parameters are R = 1.0, R0 = 1.0 and L = 2.0. The corresponding
geometry resembles the case of a !-dimensional rod.
Corresponding T max and T min values are summarized in tables 1 and 2. In both tables it is seen that
T max and Fo are linearly related for small T ma~ values. At higher temperatures the calculated T max value
increasingly exceeds the linearly extrapolated one. This behaviour can be understood as follows from equation (7). If k(T) in diamond would be independent of T, just as it is in copper. then it would hold that T (and thus T max) and F0 are linearly related. As seen in equation (1) k(T) indeed approaches a
constant value for vanishing T but increasingly deviates from it with increasing T. This qualitatively ex-plains the calculated interdependence of Fo and T max· It is, however, a striking feature that T min
depends linearly on F0 over a much larger range than found for T max· This difference reflects the
influ-ence of the rather irregular geometry.
Another important conclusion from comparison of tables 1 and 2 is that within the relevant T range (0 ~ T ~ 200) the dependence on F 0 is nearly independent of the geometry. So in the following
we shall restrict ourselves to results with Fo fixed at F0 = 400. This choice is rather arbitrary but not un-realistic.
2. Dependence of T max and T min on Ro and L.
In table 3 we present T max values for various values of R0 and L. The corresponding T min values
are nearly all the same (~24.0) and therefore omitted. Contour plots corresponding with the P1 ,3'd and
51
h row of table 3 are given in the appendix in the respective figure series 3a,3b and 3c.
From the rows of table 3 we see that T max merely weakly depends on L in the range
0.1 ~ L ~ 1.0.
The columns of table 3 show the interdependence of T max and R0 • From equation (5) we see that
the flux at the FlO inversely scales with R0 squared. From the table it is clear that this does not hold
for T max• due to the T dependence of k(T) in diamond.
--Ro \ L 1.0 0.5 0.25 0.1 0.0 1.0 34.0 30.5 29.3 29.6 32.7 0.75 36.3 32.9 32.5 34.7 49.1 0.5 41.4 38.2 38.8 44.4 65.4 0.25 58.4 55.1 56.9 69.2 81.8 0.1 120.9 116.5 119.3 141.3 327.0
Table 3. T max values in case of cylinder
sym-metry, computed for various values of R 0 and L. The values of R and F0 are kept fixed at
R
=
1.0 and F0=
400.0. The last columncoresponds with a system without diamond and only copper present. All T min values. computed
with L in the range 0.1 ~ L ~ 1.0 , are ::::::24.0. Ru\ L 1.0 0.5 0.25 0.1 0.0 1.0 27.3 27.4 27.8 29.0 32.7 0.75 27.5 28.3 29.9 33.6 49.1 0.5 27.6 29.5 33.7 42.0 65.4 0.25 21.9 30.9 39.5 60.9 81.8 0.1 28.1 33.1 43.1 85.0 327.0
Table 4. Results for TmaxDCB corresponding with the T max data in table 3.
The last column of table 3 requires special attention. In this column results are given for L
=
0,i.e. a heat sink consisting of only the copper block. In this case T max inversely scales with R0 • This can
directly be understood from the scaling properties of equation (7) if one substitutes k(T)
=
k0 andkeeps F0 fixed.
The effectiveness of the diamond to reduce the T max value is measured by comparing the last
column of table 3 with the preceeding ones. Then it is seen that for r ~ O.lR the application of the di-amond is very useful , however thin it may be, but for R0 values closer to R its effect rapidly decreases.
3. Dependence of TmaxDCB on Ro and L.
In table 4 we give the TmaxDCB results corresponding to the data in table 3. As expected they con-siderably increase if both R0 and L become small. From this table it can be determined whether a
given geometry will lead to an admissible T maxDCB value.
4. Dependence of F oc8 on L.
In figures 4a-d in the appendix FocB is given as a function of the radius of the diamond cylinder. Because we are interested in the qualitative dependence of FocB on L the parameters R0 and Fo are
kept fixed. All figures show a sharp peak at the edge of the DCB. From geometrical reasons this is to be expected because only there the heat flux may diffuse sideways into the copper.
For decreasing L a second peak appears in the centre of the DCB. For small L and R0 values
(L ,R0::::::0.lR) the latter peak becomes dominating.
Block Symmetry.
After the extensive presentation and discussion of the axisymmetric results the case of block sym-metry can be dealt with much shorter. It is found that the qualitative behaviour of both geometries is the same. Even the the .quantitative results are very similar if the the cylinder and block are equal in height and volume. In figures Sa-c and 6a-c in the appendix we give contourplots for T as a function of
--Fo. Corresponding T max and T min values are summarized in tables 5 and 6. In these calculations L and
F0 run over the same values as in tables 1 and 2. Comparision of tables and figures 1.2 and 5.6 suffi.
ciently illustrates the above conclusion.
Fo T max T min 10 2.3 0.5 100 23.7 5.3 200 50.1 10.6 400 112.7 21.1 600 192.2 31.5
Table
5.
Results for T max and T min in case ofblock symmetry for various values of F0 •
The geometrical parameters used are R
=
V1t[4,
Ro=
0.1Y1iJ4 and L :::: 0.5. The height and volume of the diamond block are the same as for the cylinder used in table 1. •• 15 •• Fn Tmax Tmin 100 9.85.5
200 19.9 11.0 500 52.1 27.2 1000 112.8 53.8 1500 184.2 79.8Table 6. Results for T ma.x and T min in case of block symmetry for various values of F 0•
The geometrical parameters used are R
=
R0 =ViTJ4
and L = 2.0. The heightand volume of the diamant block are the same as for the cylinder used in table 2.
§ 6. · CONCLUSIONS.
The discussion in the previous section leads to the following conclusions.
1) The temperature distribution Tis a function of the geometrical parameters R .R0 and L and the ap· plied amount of heat per unit time F0 (see fig.l and 2). The equations to be used are invariant for
a scaling of the these four parameters by the same multiplicative factor. Therefore one of the parameters may be kept fixed.
2) The results for geometries with cylinder and block symmetry are very smilar if the cylinder and block are equal in height and volume. In particular the quantitative results for T max• T min• T maxDCB
and F DCB hardly differ.
3) The dependence of T max on F 0 is nearly independent of the geometry.
For small T max values this quantity depends linearly on F0• With increasing temperature an
increas-ing, positive deviation from linearity is found in a way nearly independent of the geometry. So it suffices to obtain results for only one F0 value.
4) The absolute value of T min is nearly independent of the geometry. This does not hold for T max (see
tables 1 and 2).
5) T max• calculated for fixed R0 value, shows a weak dependence on L in the range O.lR ~ L ~ R
(see the rows of table 3).
6) If no diamond is present (L = 0) T max (at the top of the copperblock) scales inversely with R0 (see
the last column of table 3). Moreover, in this special case T max scales linearly with F0 (see equation
(7)).
7) The effectiveness of the diamond to reduce the T max value is considerable as long as R0 ~ O.lR.
For larger R0 values its useful effect reduces rapidly ( see the rows of table 3).
8) TmaxDCB and T min differ, for given geometry and F0 value, only appreciably if both Ro and L are
small with respect toR (Ro,L"'='O.lR).
9) FDcB shows in all cases a high,sharp peak near to the edge. With decreasing L a second, broad peak appears in the centre. The latter peak exceeds the first one in height if L
<
0.25R and R0=
O.lR (see figures 4a-e in the appendix).10) Combining conclusions 5,7,8 and 9 above we formulate the following suggestions. The most effec-tive geometry is obtained if Ro ~ O.IR and L ""'0.25R. The value of L, and thus the amount of used diamond, may still be reduced to L = O.IR but at the cost of a small increase of T max (see
table 3), an increase of T mm:DCB with a factor 2 (see table 4) and a considerable increase of F DCB
(see figures 4a-e in the appendix). Whether these quantities in a specific geometry take on accept-able values (within the system tolerances) can on beforehand be determined from the results presented in this report.
--REFERENCES.
1 Berman R., Diamonds as heat sinks-a review. Diamond Research 1970, Industrial Diamond Infor-mation Bureau, London, 1970.
2 Buergemeister E.A. and Seal M., Thermal conductivities of diamonds with absorption at 3.22 mum, Nature, Vol. 279, No. 5716, p.785. 1979.
3 Mackie A.G., Boundary Value Problems, Oliver and Boyd, London, 1965.
4 Farlow S.J., Partial differential equations for scientists and engineers, J.Wiley & Sons, Chicester, England, 1982.
5 Roach G.F., Green's Functions, Van Nostrand Reinhold Co., London, 1965.
6 Adams J.,Swarztrauber P. and Sweet R., FISHPACK, Library for the solution of separable partial differential equations, Boulder, U.S.A., 1980.
7 Byrd P.F. and Friedman M.D., Handbook of Elliptic Integrals for Engineers and Scientists, Springer Verlag, Berlin, 1971.
8 Gradshteyn I.S. and Ryzhik I.M., Table of Integrals, Series and Products, Academic Press, New York, 1980.
--APPENDIX.
In the appendix we present (contour) plots of the T and F0c8 distributions in diamond for various
geometries and F0 values.
The contour plots are given on sections containing the vertical axis of the diamond cylinder or block. The origin in the figures corresponds with the center of the FlO at the top of the diamond. So. compared to figures 1 and 2 in the report, the sections are drawn upside down. The horizontal axis gives the distance to the vertical axis of the cylinder or block, on the vertical axis the distance beneath the top side of the cylinder or block is drawn.
In case of cylinder symmetry presentation of results on one section is sufficient.
In case of block symmetry two sections are chosen, one parallel to a horizontal axis and the other parallel to a horizontal diagonal of the block.
The radius of the cylinder is fixed at R
=
1.0, which yields a horizontal area equal to :rt. For the block the same horizontal area is chosen, which implies that R = (:rt/4)'h.The serial number of the contour plots refer to the corresponding tables in the report.
Figures la-c
Figures 2a-c
Contour plots ofT in the diamond cylinders, corresponding with the F1
, 3'd and 51h row
of table 1 resp .. F0 runs over the values 10, 200, 600 and the other parameter values
are R = 1.0, R0
=
OA, L=
0.5.Contour plots of T in the diamond cylinders. corresponding with the F1
, 3'd and 51h row
of table 2 resp .. F0 runs over the values 100, 500, 1500 and the other parameter values
are R
=
1.0, R0=
1.0, L=
2.0.Figures 3al-a4 Contour plots of T in the diamond cylinders. corresponding with the pr row of table 3. R = 1.0, R0
=
1.0, F0 = 400.0 and L runs over the values 1.0, 0.5. 0.25, 0.1.Figures 3bl-b4 Contour plots of T in the diamond cylinders, corresponding with the 3'd row of table 3.
R
=
1.0, R0=
0.5, F0 = 400.0 and L runs over the values 1.0, 0.5, 0.25, 0.1.Figures 3cl-c4 Contour plots of Tin the diamond cylinders, corresponding with the 5111 row of table 3.
Figures 4a-e
Figures 5a-c
Figures 6a-c
R
=
1.0, R0=
0.1, F0=
400.0 and L runs over the values 1.0, 0.5. 0.25, 0.1.FocB (in units of W/(cmf) as a function of the DCB radius in the diamond cylinder. R = 1.0, R0 = 0.1, F0 = 400.0 and L runs over the values 1.0, 0.5, 0.25, 0.1.
Contour plots of T in the diamond blocks, corresponding with the 1st, 3'd and 5rh row of table 5 resp .. R
=
1.0, R0=
0.1, L=
0.5 and F0 runs over the values 10, 200, 600.Contour plots ofT in the diamond blocks, corresponding with the 1st, 3'd and 51
h row of
table 6 resp.. R = 1.0, R0
=
1.0, L=
2.0 and F0 runs over the values 100, 500, 1500.--R X RO • L • FLU)( • TI~AX • TM I !'I • !NCR • 1 • 00,00 0.100~~ O.SIJ00 1C.00 2. 3229 0.6026 0.0500
o.s
O.i...
-
._ .. 0.3--.
0.2 O.to.o
o.o
0.1-
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\ I I.
'.
O.i 0.5 0.6 0.7 o.e 0.9 l.O
R • R0 • L • FLUX • TMAX • TM!N • INCR • 1.0000 0. 111!00 0.5M0 200.00 51.6178 12.0214 1 .0000 O.ST---~---~ 0.4
...
__
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·a.o
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I ' ' ' ' ' ' ' ' O.i ' ' ' \ \.
\ '.
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.
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o.s
Fig. lb'·.
---0.6 0.7 o.e 0.9 1.0
R • R0 • L • FLUX • TMAX • TMIN • IHCR • 1.01!100 a.1000 0.5000 600.00 199.6534 35.8226 5.0000 0.5~---~---~---, \ \ 0.4 0.3 ...
...
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--...___
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·--
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o.oJo·.-o
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R • R\1 • t.. • FLU>< • TMAX • TMIN • INCR • 1.0000 1.00M 2.0000 1500.00 1S7.5564 91.0124 $.0000 2.0---~---,-_-
__
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o.o-1---r---....,---.,....--..,..---t
O.ll 0.2 0.4 D.&
o.e
t.llR • RG • L •
nux·
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t - - - 3 2 . 0- - - 3 . 2 . 0 - - - -
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0.0
0.1
0.20.3
0.1 0.5 0.6 0.1 0.11 0.9 1.0 Fig. 3alR • R<') • t • F'tUX • TM~X • nliN • INCR • 1.00<'10 1. 001)0 0.5000 400.<'10 30.4894 24.6342 0.2000
D.O 0.1 o.z 0.3 0.1
o.s
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0,1 0.~ 0.3 0.4o.s
0.8 0.7 0.8 0.9 1.0 Fig. 3a3R • R0 • l. • FLUX • TMAX • TMIN • INCR • 1.0000 1.0000 0.1000 400.00 e9.59?9 24.1793 0.2000
o.o
0.1 0.2 0.3 O.io.s
0.6 0.7 0.8 0.9 1.0R • P.l) • L • FLUX • TMAX • TMlN • INCR • 1.000~ 0.5000 1.0000 400.00 41.4335 24.S72.t! 1.0000
1.0,---~-~---~---~~
0.9 0.8...
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Fig. 3bl ' ' ' \ ' ' 0.6 \' '
',
.
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o.e.
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---
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a. a
0.1o.a
' ' 0.3 0.4o.s
Q.6 0.7 Fig. 3b2.
' ' ' ' '·---
....
' ' '.
0.8·-.
...-.
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R • RO • t • F'tUX • n1AX • Tr1IN • INCR • 1.011)00 0.5000 \). l000 40CLI/l0 44.4212 21.91:)65 1.0001()
.
\ \ \.
.
' I o.o+===::~--~~~~~~~~~~~~~~~-l~_j--l-~--~~_j~--~~·----~o.o
0.1 0.2 o.J 0.~o.s
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s.eeee
1.0o.t
0.8 0.7 0.1o.s
---·-·-··---
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o.o
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l.O
R • R0 • L • FLUX • TMAX • TMIN • INCR • 1.0000 0.1000 0.5000 400.00 116.5141 aJ.sst9 5.0000 Fig. 3c2
R • R0 • t • FLUX • !MAX • TMIN • INCR • 1.0000 ~ .1000 0.2500 400.00 119.3243 23.0139 5.0000 0.2 0.1
o.o
0.1 0.2 I •.
•.
•
.
.
.
o.:s ' I I • I I I ' I I I I I 0.4 ' \ \ \ ' \ \ ' \ 0.5 0.8 0.7 0.8 Fig. 3c3 \ \ \ \ \ \ \•
'
\ \ '.
\•
0.9R • R0 • L • FLUX • iM~X • iMII'I • INCR • 1.0000 0.10;;)0 e.1000 4('0.00 141.3223 21.2141 5.0000
o.o
0.1o.a
0.3o.t
.
• ~,....:"'
•
I I 0.5 0.8 0.7 0.8 0.9 Fig. 3c4R • 1.0000 R0 • 0.1000 810.0 L • 1.0000 Fl.UX • 400.00 TM~X • 120.9"141 TMIN • 24.46"15 549.0 188.0 427.0 368.0 305.0 244.0 1113.0 122.0 11.0 0.04---~---~---r----~~----~----~---~---r----~~----~
o.a
a.1
o.a
o.J
o.t
o.s
0.1 ' · 'o.e
o.t
1 .o
R • R0 • l • FlUX • TMAX • TMII'I • 1.0000 0.1000 0.5000 400.00 116.5141 23.9619 510.0 513.0 456.0 399.0 342.0 2115.0 228.0 111.0 llt.O 57.0 0.04---T---~---.---,---~----~---~---.----~r---,
o.o 0.1 o.a o.3 o.t o.s 0.1 o.7 0.1 o.s 1.0
R • R0 • L • FLUX • TMAX • TMIN • 1.0000 0.1000 0.2500 400.00 119.32'13 23.0139 510.0 459.0 408.0 351.0
:soa.o
:155.0 204.0 153.0to:u
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.o.o
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o.:s
0.4o.s
0.8 0.1o.a
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.
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.
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.
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.
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.
1.2R• R0• L• FLUX• TMAX• TMIN• INCR•
o.zs
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I ' ' I I I I I..
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o.oo
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0.8.
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...
'•,...
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.
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-~---·-···
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---···----~~~---···-o.o-f---..,..--...,.---,...
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-
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..
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t.:a
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