Temperature distributions in diamond heat sinks II

Temperature distributions in diamond heat sinks II

Citation for published version (APA):

Molenaar, J., & Staarink, G. W. M. (1985). Temperature distributions in diamond heat sinks II. (WD report; Vol. 8502). Radboud Universiteit Nijmegen.

Document status and date: Published: 01/01/1985

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Report no. 85-02

Temperature Distributions in Diamond Heat Sinks.

II.

J. Molenaar, G.W.M. Staarink June 1985

Project no 850301

Wiskundige Dienstverlening Faculteit der Wiskunde

en Natuurwetenschappen Katholieke Universiteit Toernooiveld

6525 ED Nijmegen

·-CONTENTS.

List of Symbols Units

Introduction

Mathematical Physical Model Computational Aspects Results Conclusions References Appendix 1 2 3 4 7 9 13 14

15

LIST OF SYMBOLS. See also figures 1 and 2.

CCB DCB F Fncs FlO

Fo

k(T) ko L R Ro T TIDJJX

u

v

a

an

copper-copper boundary diamond-copper boundary

flux( density), defined by equation (4) flux across the DCB

flow-in opening, i.e. the device-diamond boundary

value of the total amount of heat across the FlO per unit time (possibly temperature dependent) thermal conductivity

thermal conductivity of copper (a constant) constant proportional to the k of diamond

height of the diamond cylinder radius of the diamond cylinder radius ofthe FlO

temperature

maximum temperature in the diamond; this maximum always appears in the centre of the FlO

minimum temperature in the diamond; this minimum always appears on the DCB most far away from the FlO

quantity related to T by relation (2)

d. t · · d' h. · · b th (

a a a )

gra ten ; 1fl carteSian COOr mates t IS op~rator lS gtven Y e VeCtOr OX ,

ay , az

' 2 2 2

L l . ap actan; lfl . cartesian . COOf d. mates t h' lS operator IS . gtven Y . b ( OX2

a

+

ay2

a

+

az2

a )

derivative in a point on a surface along the outward normal; this derivative is also denoted by a subscript n, e.g.

~~

=

Tn

Units

Throughout this report we use the following units:

length power temperature

: centimeter (em) :Watt (W)

: centigrade with respect to 300 K ·

§ 1. INTRODUCTION.

In various microscale techniques a considerable amount of thermal energy is continuously produced within a small volume. This energy has to be rapidly removed, 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

[2]

and

[3]).

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.

Earlier we reported results of calculations for systems with the diamond sticked on top of the copper block ([7]). This geometry is widely used in practice. One of the conclusions was, that in order to obtain an optimal reduction of the temperature a rather specific ratio of diameter and height should be chosen.

In this report we study the configuration in which the diamond is sunken in the copper block and thus replaces a piece of copper.

It is of great practical importance to compare the efficiency of both geometries. Therefore we present in this report new data for both configurations. The technigues described in

[7]

are also applica-ble to the sunken geometry, provided that some modifications are introduced. These modifications are dealt with in § 2 and § 3. The results and conclusions are given in § 4 and § 5 respectively. The ap-pendix contains contourplots of temperature distributions and flux profiles across the diamond boun-daries, which visualize the results in a useful way.

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 diamond cylinder is sunken in a large copper block (geometry 1, fig 1).

2) The diamond cylinder is mounted on top of a large copper block (geometry 2, fig. 2). For both geometries we make the following assumptions:

a) The copper block is so big compared to the diamond, that it may be considered as being infinitely big; at infinity the block is at room temperature.

b) The heat transport through the air is so small compared to the heat transport via the diamond and the copper that it may be neglected.

c) There is a continuous and homogeneous heat flux across 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.

.

a1r

L

fo

II _I

II

l

_I

I'·'

_I I I I I . I

•

•

...

_FlO

---c

I

R

I

I

0

R

Figure 1 : Vertical intersection ofgeometry 1.

Diamond

/

L

Fo

It II

I

Ill

I a I I I I I

I+

I

•

•

...

...

FlO

---t=:

I

I

I

Ro

I

,

..

R

Figure 2 : Vertical intersection of geometry 2.

Diamond

/

.

Olr

..

The diamond cylinder is mounted on top of the copper block. The notation is as in fig.

1.

Since we are only interested in the stationary solution of the problem, in both materials the heat diffusion equation (see [5] and [6]) reduces to:

V{k(T)VT(r))

=

0

For the thermal conductivity coefficient k(T) we have (see [4] and [5])

in diamond: k(T)

=

k₁(T+300)-f3, k₁

=

27530 Watt. (centigrade)<P-lljcm ,~ = 1.26 (1)

in copper : k(T) = k0 = 3.87 Watt/centigrade/em

Introducing a function U(T) with no actual physical meaning by (2) U(T) = - 1-(T+300)l-fS

1-~

the diffusion equations in both diamond and copper reduce to the Laplace equations for U and T

respectively, i.e we have: in diamond : 6 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)-

ar

'It ~

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)

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.

ii) The fluxes at the DCB , computed in the diamond and the copper respectively , must balance. Using equations (1), (2) and (4) we may express this requirement as:

ar

au

ko-

_an

= - k t - ·

_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.

In the numerical computation of the T and F distributions on the DBC an iterative procedure is applied, in which the Laplace equations (3) are solved again and again. Because the library used (see [1]) can only deal with regularly shaped regions, the copper block is divided into the regions I and II, between which an artificial copper-copper boundary CCB is introduced. See fig. 3, wherein the diamond region is denoted by Ill.

Conner

-~~-

Diamond

/

Figure 3 : Division of the copper block into the regions I and II with artificial copper-copper boundary CCB. The DCB has a circular part DCBl and a cylindrical one DCB2.

In essence the iteration scheme is the same as described in [7], apart from the increased number of steps per cycle. During the i1

h cycle the approximations T; and F; for the T and F distributions on the

interfaces DCBl, DCB2 and CCB are improved. The procedure is started with: 1) Choose an initial flux distribution at the interfaces.

Assume that at the start of cycle i the flux distribution Fi is known. Then this cycle consists of the

fol-lowing steps:

2) Compute in I the T profile _T1 at the DCB 1 and the CCB.

3) Compute in II, given the _F1 at the DCB2 and _T1 at the CCB, the T distribution. This directly yields the T1 profile at the DCB2. Also compute a new flux

F

1 at the CCB.

4) Compute in III the U distribution, given the T; profile at the DCB1 (from step 2) and the DCB2

(from step 3).

5) Compute from the result of step 4 new fluxes

F;

at the DCB 1 and DCB2.

6) Compute the new approximation F;+t at the interfaces by the formula F;+t = yF;

+

(1-y)F;.

7) IfF; and Fi+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 at the DCB 1 and

DCB2. At the CCB a vanishing initial flux satisfied. step 2:

In this step we use the Green function method. The application of this method in our case is ex-plained in {7].

steps 3 and 4:

To solve the Laplace equations (3) in regions II and III with the given boundary conditions, we used the appropiate subroutines from the FISHPACK library ([1]). These routines compute the solution on a given grid, using a finite difference method. Region II extends to infinity. To avoid numerical problems we applied the transformation r-+1/r with r the radius of the cylinder in polar

coordinates. By this transformation region II is mapped into a region quite similar to III. The con-sequently transformed equations with the appropiate boundary conditions are solved using a routine from [1].

steps 3 and 5:

Once the U or T distribution is known at some grid, the flux F across the boundaries is computed by numerical differentiation of U or T in the direction normal to this boundary.

step 6:

In step 6 a weightt,:d mean from F; and

F;

is taken to obtain F; + 1• If y is chosen too small the

suc-cesive F; show an oscillating behaviour and the process does 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

§ 4. RESULTS.

In this section we present and discuss calculated T profiles in diamond as functions of the relevant parameters and geometries. Results for geometries 1 and 2 (see fig. 1 and 2) will be compared. In both geometries the four parameters to be varied are R, R0 , and L and the applied amount of heat per unit

time Fo.

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)ar

=

k au=

Fo

an 1

an

(nR₀₎2

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 equaequa-tion is e.g. invariant for a scaling of both the coordinates (and thus of R ,R0 and L)

and F0 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 Ro value fixed and use as

parameters the ratio's R/Ro, L/R0, and FO/R0•

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 in the diamond. c) The flux Fvcs across the DCB.

Dependence of T max and T min on F O·

In tables 1 and 2 results for T ma11 and T min are given as functions of Fo for geometries 1 and 2

respectively. In both tables the form of the diamond cylinder is the same. It is clear that in that case geometry l'yields the better temperature reduction for all F₀ values. We will discuss these differences later on. Here we want to emphasize that the general behaviour of T max and T min as functions of F₀ is quite similar.

At low temperatures (T<10) T max and F0 are linearly related. Of course it holds that T max=O if

F0=0 and the slope of the line depends on the configuration. At higher temperatures the computed

T max values increasingly exceed the linearly extrapolated one by the factors given between parentheses in the tables 1 and 2. This behaviour can be understood as follows from equation (7). If k(T) in dia-mond would be independent of T, just as it is in copper, then it would hold that T (and thus T max) and

Fo are linearly related. As seen in equation (1) k(T) indeed approaches a constant value for small T but increasingly deviates from it with increasing T. This qualitatively explains the calculated interdepen-dence of Fo and T max·

As for T min a linear dependence on F0 is found for all temperatures. This clearly reflects the fact

that T min is taken on on the boundary with the copper, which, contrary to diamond, behaves linearly. In [7] it is shown that the factors between parentheses in the tables 1 and 2 are only slightly dependent on the specific form of the diamond and are mainly determined by the temperature itself. Therefore we shall restrict ourselves in the following to calculations with the F0 value fixed. The results for other F0 values can easily be estimated from the data in tables 1 and 2. A rough, but in practice probably suffi-cient , approximation is obtained if it is assumed that both T max and T min depend linearly on F _0•

Fo Tmax Tmin 400 9.3 (1.00) 3.2 1200 29.0 (1.04) 9.6 2000 50.0 (1.08) 16.0 2800 72.5 (1.11) 22.4 4000 109.3 (1.18) 32.0 5000 143.0 (1.23) 39.9 6000 179.9 (1.29) 47.8 _' 6400 195.6 (1.31) 50.9 6800 211.9 (1.34) 54.1

Table 1. Results for T max and T min for vari-ous F0 values in geometry 1 (fig. 1). The parameters are R/R0=5 and L/R0=2.5. The

factors between parentheses give the devia-tion from linearity for the T max values.

Dependence of T max on R and L .

Fo Tmax Tmin 400 11.1 (1.00) 4.9 1200 34.6 (1.04) 14.6 2000 59.9 (1.08) 24.2 2800 87.0 (1.12) 33.8 4000 131.8 (1.19) 48.1 5000 173.1 (1.25) 60.0 6000 218.7 (1.31) 71.8 6400 238.2 (1.34) 76.5 6800 258.4 (1.37) 81.2

Table 2. Results for T max and T min for vari-ous F0 values in geometry 2 (fig. 2). The

parameters are R/R0=5 and L/R0=2.5. The

factors between parentheses give the devia-tion from linearity for the T max values.

In table 3 T max results in geometry 1 are presented for various values of R=R/R0 and L=L/R. Be-cause it is not realistic to study geometries withlextremely different L and R values, only L/R ratios are

studied in the order of one. Corresponding results for geometry 2 are given in table 4. The tables show some marked differences. In geometry 1 (table 3) T max decreases if the ratios R=R/Ro and/or L==L/R increase. This is easy to understand from geometrical considerations. The larger L and/or R the more copper is replaced by diamond, which reduces the temperature. It is clear, however, that a saturation ef-fect occurs, i.e. increasing the volume of diamond is little efef-fective if already much diamond is present.

R\L

0.0 0.25 0.5 1.0 2.0 1.0 327.0 258.8 230.1 206.3 189.3 2.5 327.0 167.4 143.4 128.7 120.9 5.0 327.0 123.6 109.3 101.3 96.3 10.0 327.0 100.3 92.7 88.6 85.8

Table 3. Results for T max for various values of the scaled parameters R=R/Ro and L=L/R in

geometry 1. The values of R0 and Fr/R0 are.

fixed at R0=1 and Fr/R0=4000.

R\L

0.0 0.25 0.5 1.0 2.0 1.0 327.0 318.5 345.8 425.3 644.1 2.5 327.0 191.6 187.8 206.4 260.3 5.0 327.0 136.1 131.8 139.4 164.1 10.0 327.0 106.7 104.1 107.7 123.1

Table 4. Results for T max for various values of the scaled parameters R=R/R0 and L=L/R in

geometry 2. The values of R0 and Fc{R0 are

fixed at R0=1 and Fr/R0=4000.

In geometry 2 ( table 4 ) the effects of two competing mechanisms can be recognized. In this geometry the diamond does not only spread the heat more rapidly than copper but it is also an extra thermal resistance. In [7] the resulting behaviour of T max has already been discussed. The main conclu-sion is that for each R/RO value an optimal L/R ratio can be found , which varies from L/R=0.2 if R/RO=l.O to LjR-:::::.0.5 if R/R0=10.0.

Summarizing we may conclude that in geometry 1 the temperature reduction mainly depends on the volume of the diamond. The form of the cylinder is not very important. This is consistent with the fact that in this geometry the area of the diamond-copper boundary is roughly the same for aU cylinders with some prescribed volume and which are not too extreme of form (i.e. L/R>>l or L/R<<1). In

geometry 2, however, both the diamond volume and the area of the diamond-copper boundary are determining factors.

The first columns of tables 3 and 4 require special attention. They contain results for L =0 , i.e. a heat sink consisting of only the copper block. The effectiveness of the diamond is measured by comparing the first and the following columns.

Dependence of T min on R and L .

In tables 5 and 6 the T min results corresponding to the data in tables 3 and 4 respectively are given.

The columns of both table shows a linear dependence of T min on R if L/R is fixed, thus if an isotropic

expansion of the diamond volume is applied. Increasing the L/R ratio causes in table 5 an considerable

decrease of T min• whereas in table 6 this has no effect. This is directly understood from the difference in

geometry. In geometry 2 the heat canno_t leave the diamond sideways, so changing L does not change the area of the DCB. On the contrary, in geometry 1 the DCB area is strongly determined by the L

value.

R\L

0.0 0.25 0.5 1.0 2.0 1.0 327.0 182.7 156.4 123.1 85.5 2.5 327.0 71.5 63.7 50.9 35.4 5.0 327.0 35.9 31.9 25.6 17.7 10.0 327.0 18.0 16.0 12.8 8.9

Table 5. Results for T min for various values of

the scaled parameters RsR/Ro and L=L/R in

geometry l. The values of R₀ and Fr/Ro are

fixed at Ro=l and FofR0=4000.

R\L

0.0 0.25 0.5 1.0 2.0

1.0 327.0 239.5 240.5 240.7 239.6 2.5 327.0 92.2 95.8 97.9 97.4 5.0 327.0 46.3 48.1 49.0 48.9 10.0 327.0 23.2 24.1 24.5 24.5

Table 6. Results of T min for various values of

the scaled parameters R=R/R0 and L=L/R in

geometry 2. The values of Ro and FofR₀ are fixed at R₀=1 and Fo/R₀==4000.

Dependence or F DCB on R and L.

In figures 9a-llb in the appendix some examples of F0c8 are given in geometry 1. All figures show

a sharp peak at the edge where DCBl and DCB2 touch each other. For geometrical reasons this is to be expected because there the heat flux can most easily diffuse into the copper. For ratios L/R >0.25

most of the heat leaves the diamond via the edge. For smaller values also a peak is found in the centre of the DCBl, which even exceeds the peak at the edge in height.

§ 5. CONCLUSIONS.

The discussion in the previous section leads to the following conclusions.

1) The temperature distribution in diamond is a function of the geometrical parameters R ,R0 and L.

and the applied amount of heat per unit time F0 (see fig.1 and 2). The equations to be used are

in-variant for a scaling of the these four parameters by the same multiplicative factor. Therefore one of the parameters may be kept fixed.

2) At low temperatures (T<10) T max depends linearly on F₀ (see tables 1 and 2). Of course T max=O

if

F

o=O

and the slope of the T max curve at low temperatures is determined by the geometry under co-n-sideration. With increasing temperature an increasing, positive deviation from linearity is found. This deviation, however, depends only slightly on the geometry ( the diamond in or on top of the copper ) and the diamond form ( determined by the ratio L/R ). So in practice it suffices to calcu-late results for one F0 value only. Reasonable estimates (with an inaccuracy< 10 % ) can then be

derived from the data in tables 1 and 2. If it is assumed that T max linearly depends on F0 , an error

of at most 35 %in the range 0<T<250 is introduced.

3) T min and F₀ are linearly related in both geometries (see tables 1 and 2).

4) Foes has always a sharp peak where the DCB has a sharp edge, because at these points t!Je heat flux can easily enter the copper block.

5) In geometry 1 (fig. 1) an increase of the diamond volume always leads to a reduction of the tem-perature, because the diamond replaces copper (see table 3). This reduction shows a saturation ef-fect, i.e. it is hardly useful to replace more than a certain amount of copper by diamond. The tem-perature reduction tends to depend only on the diamond volume and not on the specific shape, be-cause in this geometry the area of the diamond-copper boundary depends merely weakly on the shape in configurations of practical interest.

6) In geometry 2 (fig. 2) the shape of the diamond is highly important. Increasing the height of the di-amond cylinder has in general a negative effect on the reduction of the temperature, because then only extra thermal resistance is introduced. Expansion of the diamond in radius is, on the contrary, highly effective because then the DCB area is increased.

Geometry 2 is most effective if R/RO>>l. and LjR-:::::;0.25.

7) Comparing the effectiveness of both geometries with respect to temperature reduction we may con~ elude the following.

If in a system with prescribed R₀,F0 and Tmax values it is necessary to invest so much diamond

that the conditions R/R 0>> 1. ( at least about a factor of 5 ) and L/R """ 0.25 can be fulfilled, then the two geometries are nearly equal in effectiveness. We note that this is the case in most of the applied heat sink systems. The choice between the two possibilities has in these cases to be deter-mined on account of other reasons (e.g. production costs).

If, however , application of a smaller diamond volume is sufficient ( e.g. because only a rela-tively small ~emperature reduction is required ) , then geometry 1 is clearly the more effective one.

REFERENCES.

1 Adams J.,Swarztrauber P. and Sweet R., FISHPACK, Library for the solution of separable partial differential equations, Boulder, U.S.A., 1980.

2 Berman R., Diamonds as heat sinks-a review, Diamond Research 1970, Industrial Diamond Infor-mation Bureau, London, 1970.

3 Buergemeister E.A. and Seal M., Thermal conductivities of diamonds with absorption at 3.22 mum, Nature, Vol. 279, No. 5716, p.785, 1979.

4 Enckevort W.J.P. van, private communication.

5 Farlow S.J., Partial differential equations for scientists and engineers, J.Wiley & Sons, Chicester, England, 1982.

6 Mackie A.G., Boundary Value Problems, Oliver and Boyd, London, 1965.

7 Molenaar J., Staarink G.W.M., Temperature Distributions in Diamond Heat Sinks, Report no WD 85-01, Wiskundige Dienstverlening, Toernooiveld, 6525 ED Nijmegen, 1985.

APPENDIX.

In the appendix we want to illustrate how the T distribution in the staedy state depends on the con· figuration. We present contourplots for T and F0c8 profiles in the diamond of geometry 1 (see fig. 1), varying the ratios R/R₀ and L/R. The FO/RO value is kept fixed because computations for various F₀ values in the same configuration (thus with R, Ro and L fixed) yields very similar plots.

The contour plots are given on vertical sections of the diamond cylinder. The origin in the figures corresponds with the center of the FlO at the top of the diamond. So, compared to fig. 1 in the report, the sections are drawn upside down. The horizontal axis gives the distance to the axis of the cylinder, while the upper edge of the plots corresponds with DCB2 in fig. 3.

Figures 1·4 Contour plots of T in the diamond of geometry 1, corresponding with the first row of table 3. F0=4000, R/R0=1 and L/R runs over the values 0.25, 0.5, 1.0, and 2.0 respec-tively.

Figures 5-8 Contour plots of T in the diamond of geometry 1, corresponding with the last row of table 3. F0=4000, R/R0=10 and L/R runs over the values 0.25, 0.5, 1.0, and 2.0

respec-tively.

Figures 9a-llb Fluxes FDCB across the boundaries DCB1 ( fig. 9a,l0a,lla ) and DCB2 ( fig. 9b,l0b,llb ) in geometry 1 corresponding with some entries in the second rows of tables 3 and 5. The parameter values are FO/R0=4000, R/R0=2.5, while the value of L/R runs over 0.25,0.5 and 1.0.

~

.

_1.0000 R0 • 1.0000 L • 0.2509 FLUX • 4000.00 TMAX • 258.8020 TMII't • 182.6555 lHCR

..

2.0000 ? .

..

.

o.a

L---2ii----'--L_-__ -.-•• -... -.-•• -•• -••

~~-.-.:-

•• -•• - .. -... -•• ...:

o.t

o.a

0.3

.:,·· R •

Itt• ' I. • F't.U>C • Tl'fAI< • Tl'llM • JHCA ' .1 O.f 0.2 0.1

R • 1.0000 _1.0 R0 • 1.0000 I. • 1.0000 Ft.UX • <4000.00 TI'IA)( • 206.2945 TMIN • 123.1174 INCR • 2.0000 0.1 ?

o.e

0.1 0 . 8 " ' 1

-a.s

0.4

R • R0 • L • FLUX • TMAX • TMIN • INCR • ? 1.0000 1.0000 2.0000 4000.00 189.3553 85.4902 2.5000 1.1

'·'t---1.4..,_ _ _ _ _ .1.2 1.0 0.1 0.4

R • R0 • L • FLUX • · T!1AX • TMIN • ItiCR • ? 10.0000 1.0000 2.5000 4000.00 100.3271 17.9591 2.0000

•

I

'

I

'

• I • • I

'

I I I \ I I I

.

;. ' ... R • R0 • 't.. FLUX • TI'IAX • TI1Itf • ltiCR •··

.,.

10.0000 1.0000 5.0000 <1000.00 92.7-428 15.9788 2.5000 5.0,---~---~---~---~

_....

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