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General models of circum-global and quasi-global radiation on Hills of simple geometrical shapes

Allgemeine Modelle der Zirkumglobal- und Quasiglobalstrahlung auf Bergen mit einfacher geometrischer Form. I. Teil: Theoretische Betrachtungen

Modèles généraux pour le rayonnement solaire circumglobal et quasiglobal sur les collines avec simples formes géométriques. Part I: Traités théoretiques

Part I: Theoretical consideration

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Summary

Models for calculation of circum-global and quasi-global radiation at any location on a cone-shaped hill and on a ridge having a form of a laying prism, in all weather conditions, are presented. Applied equations that include all main parameters, viz. astronomical, meteorological and topographical, are derived from basic equations. Special emphasis is given to the treatment of circum-global radiation, as defined by Flach [11]. Computer models are made, peculiarities of which are described, the main one being their generality.

Zusammenfassung

Es werden allgemeine Berechnungsmodelle der Zirkumglobal- und Quasiglobalstrahlung, die auf eine beliebige Stelle auf einem kegelförmigen Berg oder auf einem Bergrücken, bei beliebigen Wetterverhältnissen fallen, entwickelt. Die angewendeten Gleichungen, welche die wesentlichen astronomischen, meteorologischen und topographischen Parameter beinhalten, werden aus den Grundgleichungen abgeleitet. Ausführlich wird die Zirkumglobalstrahlung (nach der Definition von Flach [11]) behandelt. Auf dieser Grundlage werden die Rechenmodelle entwickelt und ihre Eigenheiten werden beschrieben. Sie sind für den allgemeinen Gebrauch geeignet.

Résumé

Les modèles pour la calculation du rayonnement circumglobal et quasiglobal sur les locations quiconques sur la colline conique et sur la faîte sont présentés. Les équations applicables comprenant tous les paramètres astronomiques, météorologiques et topographiques sont développées des équations fondamentales. Le rayonnement circumglobal (défini par Flach [11]) est traité avec emphase. Les modèles numériques sont faits aux bases mentionnées plus haut; les particularités des modèles sont décrites, la principale ayant la généralité.

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Abbreviations

a :

albedo of the soil surface

dt :

time interval

dσ:

surface element on the soil surface on which radiation is reflected

h :

height of the receiver above the soil surface

H-h :

vertical distance of the receiver from the hill top

I o :

solar constant

j I :

flux density of direct solar radiation

j′ I :

flux density of direct solar radiation at partially cloudy sky

j D :

flux density of diffuse radiation at horizontal plane

j D, ε :

flux density of diffuse radiation at the slope

j′ D, ε :

flux density of diffuse radiation at the slope from cloudless region of the sky

j cl D, ε :

flux density of diffuse radiation at the slope from cloudy regions of the sky

j R :

flux density of reflected radiation

k(z) :

empirical coefficient, depending on zenith distance of the sun

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {L} \) :

vector field, vectors from soil surface elements to the receiver

m :

relative optic path

n :

duration of sunshine during time periodN

N :

time period (e. g. one hour)

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {n} \) :

normal unit vector from soil surface element

r r :

radius of the sphere presenting the receiver

r (z, gen):

empirical coefficient depending on zenith distance of the sun and cloud genera (it is used at calculation of diffuse radiation from cloudy regions of the sky)

R CG :

circum-global radiation obtained by the receiver — the sphere per unit time

R cl CG :

circum-global radiation obtained by the receiver per unit time in cloudy weather

R D :

diffuse radiation obtained by the receiver per unit time

R′ D, ε :

diffuse radiation gained by the receiver per unit time in cloudy weather from cloudless region of the sky

R cl D, ε :

diffuse radiation obtained by the receiver per unit time in cloudy weather, from cloudy regions of the sky

R I :

direct solar radiation obtained by the receiver per unit time

R′ I :

direct solar radiation obtained by the receiver per unit time at partially cloudy sky

R QG :

quasi-global radiation — sum of direct solar radiation and diffuse radiation obtained by a unit surface on a tilted plane per unit time

R cl QG :

quasi-global radiation — sum of direct solar radiation and diffuse radiation received by a unit surface on a tilted plane in cloudy weather per unit time

R R :

reflected radiation obtained by the receiver — the sphere per unit time

R′ R :

reflected radiation obtained by the receiver — the sphere per unit time, in cloudy weather

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {s} \) :

unit vector in direction of sun

t ar :

time of astronomic sunrise

t as :

time of astronomic sunset

t rr :

time of actual sunrise with regard to the receiver

t rs :

time of actual sunset with regard to the receiver

z :

zenith distance of the sun

q a :

partial transmission coefficient of the atmosphere with regard to absorption

q s :

partial transmission coefficient of the atmosphere with regard to scattering

ε:

inclination of the slope

ϑ:

zenith distance

π:

Ludolf number

ρ=r 2 m /r 2 :

wherer=distance between the center of the Sun and Earth at the actual time of the year andr m =the mean distance between the centers of the Sun and the Earth

Σ:

summation sign

σ:

surface

ϕ:

azimuth

ϕ r :

azimuth of the receiver on the hill

ϕ g :

latitude

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Author notes

  1. A. Hočevar

    Present address: Institute for Soil Science and Plant Nutrition, Biotechnical Faculty, University of Ljubljana, YU-61000, Ljubljana, Yugoslavia

  2. J. Rakovec

    Present address: Meteorology Department, Faculty of Sciences, University of Ljubljana, YU-61000, Ljubljana, Yugoslavia

Authors and Affiliations

  1. Institute for Soil Science and Plant Nutrition, Biotechnical Faculty, University of Ljubljana, YU-61000, Ljubljana, Yugoslavia

    J. Rakovec

  2. Meteorology Department, Faculty of Sciences, University of Ljubljana, YU-61000, Ljubljana, Yugoslavia

    A. Hočevar

Authors
  1. A. Hočevar
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With 2 Figures

The work was done with the support of Sklad Borisa Kidriča (Contracts SBK 481/33-73 and SBK 481/1538-74).

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Hočevar, A., Rakovec, J. General models of circum-global and quasi-global radiation on Hills of simple geometrical shapes. Arch. Met. Geoph. Biokl. B. 25, 151–164 (1977). https://doi.org/10.1007/BF02253697

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