estimación de la radiación solar global mensual media como una función de la temperatura

7/27/2019 Estimacin de la radiacin solar global mensual media como una funcin de la temperatura

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0168-1923/00/$see front matter 2000 Published by Elsevier Science B.V. All rights reserved.

PII: S0168-1923(99)00090-8

Estimation of mean monthly solar global radiation as

a function of temperature

Francisco Mezaa,*, Eduardo Varasb,1aDepartamento de Ciencias de Recursos Naturales, Pontificia Universidad Catlica de Chile, Casilla 306, Correo 22, Santiago, Chile

bDepartamento de Ingeniera Hidrulica y Ambiental, Pontificia Universidad Catlica de Chile, Casilla 306, Correo 22, Santiago, Chile

Received 14 December 1998; received in revised form 11 August 1999; accepted 13 August 1999

Abstract

Solar radiation is the primary energy source for all physical and biochemical processes that take place on earth. Energy

balances are a key feature of processes such as temperature changes, snow melt, carbon fixation through photosynthesis in

plants, evaporation, wind intensity and other biophysical processes. Solar radiation level is sometimes recorded, but generally

it needs to be estimated by empirical models based on frequently available meteorological records such as hours of sunshine

or temperature.

This paper evaluates the behavior of two empirical models based on the difference between maximum and minimum

temperatures and compares results with a model based on sunshine hours. This work concludes that empirical models based

on temperature have a larger coefficient of determination than the model based on cloud cover for the normal conditions of

Chile. These models are easy to use in any location ifthe parameters are correctly adjusted. In addition, probability distribution

functions and confidence intervals for solar radiation estimates using stochastic modeling of temperature differences were

calculated. 2000 Published by Elsevier Science B.V. All rights reserved.

Keywords: Solar radiation; Temperature; Random variable; Fourier series

1. Introduction

In some cases a record of global solar radiation (RG)

using instruments such as pyranometers or actinome-

ters is available, however, there are many meteorolog-

ical stations which do not measure solar radiation, but

do register other variables such as precipitation, pres-

sure and temperature. For this reason, this paper eval-

*

Corresponding author. Fax: +56-2-553-92-31.

E-mail addresses:[email protected](F. Meza),[email protected]

(E. Varas).

1 Fax +56-2-686-58-76.

uates proposed mathematical models to estimate solar

radiation as a function of temperature differences and

compares their performance with models based on

sunshine hours.

Solar radiation is the principal energy source for

physical, biological and chemical processes, such as,

snow melt, plant photosynthesis, evaporation, crop

growth and is also a variable needed for biophysical

models to evaluate risk of forest fires, hydrological

simulation models and mathematical models of natu-

ral processes. Hence, in many occasions, a record ofobserved solar radiation or an estimate of radiation is

required.
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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232 F. Meza, E. Varas/Agricultural and Forest Meteorology 100 (2000) 231241

RA =

RG

2. Model description

Extra-terrestrial solar radiation, also known as An-

got radiation (RA, MJ m2 day1) can be calculated as

a function of the distance from the sun to earth (d,

km), the mean distance sunearth (dm, km), latitude

( , rad), solar declination (8, rad) and solar angle at

sunrise (sunset) (Hs, rad) using the following expres-

sion (Romo and Arteaga, 1983):

(86400) (1360) (dm 2 i r d

x [(Hs)sin sin(8 ) + cos( cos(8 )sin(Hs)](1)

Using the preceding relationship, solar radiation can be

calculated for any point in the earths outer atmo-

sphere for each day of the year as a function of latitude

and solar declination. However, gases and clouds in-

troduce changes to both magnitude and spectral com-position of solar radiation.

2.1. Angstrm model, 1924

Since the beginning of the century, efforts have

been made to estimate solar radiation as a function of

extra-terrestrial solar radiation and the state of the

atmosphere (Castillo and Santibez, 1981). The pa-

rameter most commonly used is hours of sunshine.

Usually, the ratio of global solar radiation to Angot

ra-diation is correlated to the ratio of effective

sunshine hours to total possible sunshine hours.

Effective sunshine hours (n) are measured with aheliograph (Martnez-Lozano et al., 1984). Although

this instrument has a threshold, under which sunshine

is not recorded, this error is not significant when esti-

mating daily solar radiation.

Angstrm (1924), suggested a simple linear re-

lationship to estimate global solar radiation (RG,

MJm2 day1) as a function of Angot radiation, actual

sunshine hours (n) and potential or theoretical

sunshine hours (N).

= a+ b n (2)RA N

Angstrm suggested values of 0.2 and 0.5 forempirical coefficients a and b respectively. Other

authors, such as Bennett (1962), Davies (1965),

Table 1

Angstrm coefficients (a and b) recommended for Chilean locali-

ties. (Castillo and Santibaez, 1981)

Locality a b Latitude Longitude Altitude

(S) (W) (m)

Arica 0.28 0.57 18.29 70.19 035

Iquique 0.23 0.47 20.13 70.09 008

Antofagasta 0.23 0.47 23.28 70.20 122Copiapo 0.26 0.51 27.21 70.20 283

Vallenar 0.22 0.46 28.35 70.46 469

La Serena 0.29 0.57 29.54 71.15 032

La Paloma 0.22 0.46 30.41 71.02 320

Quintero 0.22 0.45 32.47 71.32 002

Valparaiso 0.22 0.55 33.01 70.38 041

Santiago 0.22 0.44 33.27 70.42 520

Curico 0.23 0.47 34.58 71.13 227

Constituci on 0.22 0.45 35.20 72.26 007

Chillan 0.23 0.47 36.36 72.02 124

Concepci on 0.26 0.51 36.47 73.07 009

Temuco 0.23 0.47 38.46 72.39 114

Osorno 0.23 0.47 40.35 73.09 027

Puerto Montt 0.26 0.51 41.28 72.56 110

Ancud 0.26 0.51 41.54 73.48 020Puerto Aysen 0.26 0.51 45.24 72.42 010

Balmaceda 0.26 0.51 45.54 71.43 520

Punta Arenas 0.26 0.52 53.10 70.54 008

Monteith (1966), Penman (1948), and Turc (1961)

have calibrated this expression for different places.

Coefficients can vary significantly as Doorenbos and

Pruitt (1975) show. In Chile, Castillo and San-tibez

(1981), have recommended the values given in Table

1.

2.2. BristowCampbell model, 1984

Incoming solar radiation is determined by the state

of the atmosphere. However, the dynamics of the

atmosphere is very difficult to predict. Considering

transformations experienced by solar radiation, one

can expect to find a relationship to express solar

radiation as a function of meteorological variables

commonly registered at climatological stations. When

solar radiations reaches the earth surface, part of it is

reflected and part is absorbed. The same occurs with

long-wave radiation that each body emits as a

function of its temperature. As Chang (1968), reports,

there is usually a good relation between net radiationand global solar radiation, since the latter one is the

principal source of energy.


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F. Meza, E. Varas/Agricultural and Forest Meteorology 100 (2000) 231241 233

RG

RA

RG

RA

Furthermore, if the heat flow towards the soil is

neglected, one can find the ratio of sensible heat to

latent heat or Bowen ratio, on a daily basis (Campbell,

1977). Sensible heat is responsible for temperature

variations, so it is possible to obtain a relationship

between temperature differences and solar radiation,

being temperature a reflection of radiation balance.

Using this argument, Bristow and Campbell (1984),

suggested the following relationship for daily RG, as a

function of daily RA and the difference between

maximum and minimum temperatures (AT, C):

[ ]=A1 exp(BATC) (3)

Athough coefficients A, B and C are empirical,

they have some physical meaning. Coefficient A

represents the maximum radiation that can be

expected on a clear day. Coefficients B and Ccontrol

the rate at which A is approached as the temperature

difference increases. Values most frequently reported

for these coefficients are 0.7 forA, the range 0.004 to

0.010 forB and 2.4 forC.

Since clear days present large temperature differ-

encesA tends to be the ratio between global solar radi-

ation and Angot radiation, hence the sum of Angstrm

coefficients a and b tends to be similar toA.

2.3. Allen model, 1997

Allen (1997), suggested the use of a self-calibrating

model to estimate mean monthly global solar radiation

following the work of Hargreaves and Samani (1982).

He suggested that the mean dailyRG can be estimatedas a function ofRA, mean monthly maximum(TM, C)

and minimum temperatures (Tm, C).

= Kr(TMTm)0.5 (4)

Previously, Allen (1995), had expressed the

empiri-cal coefficient (Kr) as a function of the ratio of

atmo-spheric pressure at the site (P, kPa) and at sea

level (P0, 101.3kPa) as follows:

P 0 .5Kr= Kra (5)

P0

In his work, Allen suggested values of 0.17 forinterior regions and 0.20 for coastal regions for the

empirical coefficientKra.

3.Climatic dataIn order to compare the behavior of the different

models, monthly climatological data of 21 stations

representing different climatic regions of Chile were

collected. Data ranged from Arica (latitude 18.3S) to

Punta Arenas (53.1S) and was registered between the

years 1971 and 1992.

Selected meteorological variables were TM, Tm, P,

mean monthly degree of cloud cover (x) andRG.

For the locations mentioned in Table 1, monthly val-

ues of maximum and minimum temperatures, cloud

cover and atmospheric pressure for each year in the

period 1971 to 1992, were available. Unfortunately, for

global solar radiation only the average value for each

month in that period was available and monthly

radiation values for each year were impossible to ob-

tain from Direccin Meteorolgica de Chile.

In addition to the above, data from La Paloma sta-

tion was collected to compare the behaviour of modelsbased on temperature differences when they are ap-

plied to estimate monthly global solar radiation. The

selected meteorological variables in this case were TM,

Tm,P, andRGbetween the years 1971 and 1978.Finally, data from Santiago station was used to

compare the behaviour of BristowCampbell and

Allen models when they are applied to estimate daily

global solar radiation. The meteorological variables

were daily TM, Tm,P, andRG.

4.Models applied to mean monthly dataThe extension of the reviewed models to apply them

to monthly averages requires some explanation. The

Angstrm model was originally derived for daily solar

radiation and hours of sunshine. Nonetheless, be-ing a

linear function it can be readily applied to mean

monthly data since the expected value of a sum is

equal to the summation of the expected values. Allens

model was derived for monthly data so it can readily

be used. However, the BristowCampbell model is de-

fined for daily data and has no evident extrapolation to

mean monthly values. For this reason, one can ex-pect

to find a new set of coefficients when the same

expression is applied to monthly data.With the values of temperature, atmospheric pres-

sure and sunshine hours, mean monthly global solar


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radiation was calculated at each site, using the ex-

pressions and empirical coefficients suggested by

Angstrm (1924), Bristow and Campbell (1984), and

Allen (1995). Results show that models using the

coefficients proposed in the literature do not esti-mate

correctly the historical average in each location. The

slope of the relationship between calculated and

observed radiation is significantly different from

unity. This is especially notorious in the case of the

BristowCampbell model, although this result was

ex-pected since the coefficients suggested by the

authors are applicable to daily data.

Given the results, it was necessary to change the

Allen and BristowCampbell model coefficients to

obtain a better fit, following the idea suggested by

Castillo and Santibez (1981) for the Angstrm

model. Least squares coefficients, which minimize

the sum of square errors for each location were

calculated and included in Table 2.

Due to the fact that monthly solar radiation values,were not available for each year, as mentioned in the

section about climatic data, the A and Ccoefficients

of BristowCampbell model were assumed fixed and

theB coefficient was adjusted to minimize the square

Table 2

Adjusted coefficients (Kra and B) of Allen and BristowCampbell

models

Locality Kra B

Arica 0.3354 0.01354

Iquique 0.2854 0.01619

Antofagasta 0.4717 0.01944

Copiapo 0.2577 0.00203

Vallenar 0.3457 0.00200

La Serena 0.2697 0.00677

La Paloma 0.1589 0.00347

Quintero 0.2731 0.00589

Valparaso 0.0114 0.01144

Santiago 0.2593 0.00202

Curico 0.4348 0.00152

Constitucion 0.2423 0.00555

Chillan 0.2316 0.00159

Concepci on 0.3402 0.00242

Temuco 0.2583 0.00154

Osorno 0.3756 0.00150

Puerto Montt 0.3252 0.00290

Ancud 0.2820 0.00493

Puerto Aysen 0.2870 0.00463Balmaceda 0.3058 0.00348

Punta Arenas 0.3471 0.00389

Table 3

Regression between calculated and observed mean monthly global

solar radiation using adjusted parameters of 20 Chilean localities

Model Slope Upper

limit. (95%)

Lower

limit. (95%)

R2

Angstrm 0.959 0.970 0.939 0.892Allen 0.999 1.010 0.990 0.961

BristowCampbell 1.152 1.170 1.138 0.928

errors. The available data made it impossible to study

the contribution of coefficients A and C. However, A

represents the maximum radiation on a clear day and

its value represents the observed data reasonably well.

Moreover, a change in coefficient C does not affect

significantly the calculated global solar radiation.

Observed and calculated values for different

locations and models are shown in Fig. 1. In this

figure the improvement in the relationships when us-

ing locally calibrated coefficients can be appreciated.

The Angstrm model results using the coefficientsproposed by Castillo and Santibez (1981) are also

included for comparison. Slopes of the different mod-

els and the coefficients of determination are given in

Table 3.

Allens model presents the best relationship. It has

a higher coefficient of determination and the slope is

equal to unity with 90% confidence interval. The

BristowCampbell model tends to under-estimate

global solar radiation but explains a large proportion

of sample variance. The Angstrm model fit the data

poorer than the other two.

4.1. Models applied to monthly data.

Since the available data of global solar radiation for

most stations is only the average value for each month,

it was necessary to examine ifthe relationships with the

adjusted coefficients represent accurately the monthly

values for each year. One station available with

monthly global solar radiation data, is La Paloma. In

this case the models with the adjusted coefficients

derived with the average monthly values were used to

estimate monthly global solar radiation for each year. A

comparison between estimated monthly values for La

Paloma, compared to observed monthly values is

shown in Table 4.Results show that monthly global radiation for each

year can be adequately estimated with the derived


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Fig. 1. (a) Comparison between observed and measured mean monthly global solar radiation using Angstrm parameters from the literature

(see text); (b) Comparison between observed and measured mean monthly global solar radiation using Angstrm adjusted parameters; (c)

Comparison between observed and measured mean monthly global solar radiation using Allen parameters from the literature; (d) Comparison

between observed and measured mean monthly global solar radiation using Allen adjusted parameters; (e) Comparison between observed and

measured mean monthly global solar radiation using Bristow-Campbell parameters from the literature; and (f) Comparison between observed

and measured mean monthly global solar radiation using BristowCampbell adjusted parameters.


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Fig. 1 (Continued).


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AT j =AT + C cos

Table 4

Regression between calculated and observed monthly global solar

radiation at La Paloma station

Model Slope Upper

limit. (95%)

Lower

limit. (95%)

R2

Allen 1.000 1.010 0.990 0.97

BristowCampbell 0.994 1.006 0.982 0.96

models. Allens model presents the best relationship

between observed and calculated monthly solar global

radiation because it explains a large proportion of the

sample variance. In both models the slope is equal to

unity with 90 % confidence interval. This verifies that

the models can be used to estimate monthly values for

different years.

4.2. Global solar radiation distributionfunctions

A probability distribution function for global solar

radiation was obtained as a derived distribution, when

radiation is expressed as a function of temperature dif-

ferences and temperature differences are expressed as a

Fourier series with a random component. This ran-dom

error was found to be a random variable with normal

distribution. This hypothesis was tested in both for the

BristowCampbell and the Allen models using the

AndersonDarling test for normal distribution. Once a

distribution model for solar radiation is calculated,

confidence intervals for estimates can be computed.

4.3. Temperature difference modelling.

Temperature has a marked seasonal variation due to

periodicity in the earths orbit about the sun. For this

reason temperature variations can be represented us-

ing mathematical cyclic functions. In this paper, dif-

ferences between maximum and minimum tempera-

tures were modelled using a Fourier series once the

stationary component was removed, as suggested by

Van Wijk and De Vries (1966) and Campbell and Nor-

man (1997). These authors applied Fourier series with

one term to represent air temperatures.

TheATin location i and monthj (ATj, C) can beexpressed as a function of mean annualATin locationi (AT, C), Fourier series coefficients atlocation i (C,D ) and an error or residual in location i and monthj (E

j, C) as follows:

Table 5

Average temperatures (ATi) and Fourier series coefficients CandD of 20 Chilean localities

Locality AT C D

Arica 06.344 0.722 1.412Iquique 05.629 0.629 1.162Antofagasta 06.489 0.269 0.945

Copiapo 14.545 0.640

0.292Vallenar 13.193 1.264 0.177La Serena 07.856 0.220 0.044La Paloma 14.143 0.330 0.345Quintero 08.366 0.670 0.495Valparaso 05.549 0.804 0.330Santiago 13.917 2.539 1.830Curico 14.612 4.235 2.851Constituci on 08.397 0.723 0.188Chillan 13.802 3.991 2.910Concepci on 10.073 2.134 1.365Temuco 11.494 3.052 2.111Osorno 11.031 3.140 1.592Puerto Montt 08.592 1.862 0.773Ancud 07.255 1.636 1.051

Puerto Aysen 06.823 1.427 0.345Balmaceda 09.078 2.130 1.151

Punta Arenas 07.019 1.829 0.665

(2 i r j

1 2 f2 i r j)

+D sin _____ + E j (6)12

The coefficients CandD are given in Table 5 for the

sites used in this work.

4.4. Probability distributionfunctions

IfX is a continuous random variable with a proba-

bility density functionf(x) and Yis a monotonic func-

tion ofX, then the probability function of Y can be

obtained multiplying the inverse function by the

abso-lute value of the Jacobian of the transformation

(J) or determinant of the first derivative of w(y) with

respect toX(Walpole and Myers, 1992):


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g(y) = f[w(y)]|J| (7)

Using this procedure probability density and proba-

bility distribution functions forRG estimated by Allen

and BristowCampbell models were derived.


9/13


4.5. BristowCampbell model

In this case the distribution function is calculated

using Eq. (3) and replacing Tij for its expression in

terms of annual T in each location and the corre-

sponding Fourier series coefficients. Combining both

the expressions, an equation for the residuals is ob-

tained. Residuals were found to be well represented by

a normal distribution model, so the probability dis-

tribution of the errors was assumed known. The distri-

bution hypothesis was tested using AndersonDarling

test.

The probability density function for solar radiation

following Eq. (7), is equal to the product of the normal

density function evaluated at the residuals for location i

and month j and the absolute value of the transfor-

mation Jacobian (Eq. (8)). The residuals are given in

this case by Eq. (9) and the first derivative by Eq. (10).

g(RGij) = [J]fL(RGij)j (8)

The residuals are given by the following equation ex-

pressed as a function of terms already defined:

1/2.4

ln (1 RGij/0.7RAij)Eij= i

Bi

122Tj Disin 122rj

Cicos ____12) 12) (9)

The first derivative is:

|J| = 1\ 1)6.8 1 /2.4 1 1 1 RGij))1.4/2.4

ln 0.7RAij

1 1 RG i j/0 .7RA i j RA i j

1)1 ______________ (10)The cumulative distribution function (CDF) is ob-

tained by integrating the probability density function.The CDF was evaluated numerically, using very smallintervals and the trapezoidal integration method, todefine confidence intervals for global solar radia-tion.Results for two locations Arica and Vallenar areshown graphically in Fig. 2 (a,b).

4.6. Allens model

Similarly, for Allens model, 1997, the probability

density function is obtained using Eq. (4) and replac-

ing Tijfor its expression in terms of annual Tin


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1Eij = RG i j

|J| = (12)

each location and the corresponding Fourier series co-

efficients (Eq. (6)). Residuals in this case were also

found to be well represented by a normal distribution

model, so the probability distribution of errors was

assumed known.

The probability density function for solar radiation

is shown in Eq. (8).The residuals are given in this

case by Eq. (11) and the first derivative by Eq. (12):

) i

(RAij)Kra i(P /P0)0 .5

Cicos 1

21r12) Disin 122r12)(11)

The Jacobian is:

2RGij (P0)

K2ra i(RAij)2P

The CDF is obtained integrating the probability den-

sity function. It was evaluated numerically to define

confidence intervals for global solar radiation. Re-

sults for Arica and Vallenar are shown graphically

in Fig. 2(c,d). The expected value for global solar

radiation given by the CDF using Allens model are

higher than the Angot radiation because the limits of

integration derived in this case were zero and

infinite. On the other hand, the CDF using Bristow-

Campbell model have clear and defined limits which

are zero and A times the Angot radiation. For this

reason the CDF obtained with BristowCampbell

model is more accurate and has smaller confidenceintervals.

5. Models applied to daily data

5.1. Allens model

Allens model, 1997 includes a correction term for

barometric pressure which in fact represents the alti-

tude of the station above sea level, since the pressure as

a function of elevation can be expressed in terms of the

pressure at sea level, the temperature gradient, thetemperature at station elevation and the Avogadro air

constant. This correction term is small compared to the

influence of the temperature difference on radiation.


11/13


Fig. 2. (a) Expected values and confidence limits (5 and 95%) of daily mean global radiation using Bristow-Campbell model and Angot

radiation for Arica; (b) Expected value and confidence limits (5 and 95 %) of daily mean global radiation using Bristow-Campbell model and

Angot radiation for Vallenar; (c) Expected value and confidence limits (5 and 95 %) of daily mean global radiation using Allen model and

Angot radiation for Arica; (d) Expected value and confidence limits (5 and 95%) of daily mean global radiation using Allen model and Angot

radiation for Vallenar.

This model tends to over estimate global solar ra-

diation in a daily basis, and frequently estimates radi-

ation in excess of the extra-terrestrial radiation, since

the condition expressed by Eq. (13) is fulfilled. This

model does not have a limit for the estimated solar

radiation.

P 0(13)

(Kra)2P

This condition is frequently true when the model isapplied to points located in interior regions whichusually experience large daily temperature variations.

Even though Allens model has a larger coeffi -cient

of determination, the slope is clearly less than unity,

indicating that the model over-estimates solar

radiation.

5.2. BristowCampbell model

This model is defined solely in terms of

temperature differences and is thus simpler to apply.

The value for A coefficient is 0.7, which is areasonable value for clear days. This type of day

usually is associated to large temperature differences.


12/13


Fig. 2 (Continued).

Table 6

Regression between calculated and observed daily global solar

radiation at Santiago station

Model Slope Upper

limit. (95%)

Lower

limit. (95%)

R2

Allen 0.561 0.549 0.571 0.85BristowCampbell 1.090 0.979 1.202 0.79

The behavior of the BristowCampbell model is

more consistent and reliable, since it has an upper

limit given by parameter A. The regression analysis

shows that the BristowCampbell model performs

better (Table 6). On the other hand, BristowCampbellmodel gives consistently a better estimate when

applied to daily data.

6. Conclusions

Empirical models to estimate global solar radia-tion

are a convenient tool if the parameters can be

calibrated for different locations. These models have

the advantage of using meteorological data which are

commonly available.

For Chile, the models proposed by Allen and

BristowCampbell are adequate and allow estimates of

mean average global solar radiation as a function of air

temperature variation. Allens model has a larger coef-

ficient of determination but requires both atmospheric

pressure and temperature variation measurements.

Models were calibrated for 20 locations in Chile whichrepresent a wide variation in climatic characteristics and

hence the procedure described is considered to be


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of general application. Temperature variation can be

modelled by Fourier series and confidence intervals for

global solar radiation estimates can be obtained using

derived distribution procedures. Both the models have

limitations when applied to daily data. Solar radiation at

locations with large temperature differences are not

correctly modelled using Allen procedure and the

BristowCampbell model had a better performance.

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estimación de la radiación solar global mensual media como una función de la temperatura

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