Modeling Instantaneous Values of Direct and Diffuse Solar Radiation for Solar Thermal Systems

Hourly peaking solar radiation models for flat plate collector solar thermal systems.

The following paper is based largely on the work of the late Charles K. Alexander, Jr, PhD. I worked with Chuck and David D. Bailey to implement the methodology described below in software for use in a set of solar thermal simulators that also made use of the packed particle bed storage model and the solar collector/heat exchanger models described in other postings. The method described here was created for use with solar thermal systems and to my knowledge has not been applied to solar PV systems and I do not know if it would be of value there or not.

I have reworked the original paper to add additional explanation and correct a typographical error in one of the equations.

Modeling Instantaneous Direct And Diffuse Solar Radation Values

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Modeling Instantaneous Values

of Direct and Diffuse Solar Radiation for Solar Thermal Systems

Jeffrey D. Taft, PhD

Background

A method is presented which converts monthly average daily solar radiation into hourly direct radiation and diffuse radiation for a monthly average day for each month of the year. This method differs from previous techniques in that it results in an instantaneous peaking model rather than just average values. This significantly increases the number of different solar thermal systems which can be evaluated using computer simulation.

One of the most difficult parameters to model is the incoming solar radiation seen by a flat plate collector. Obviously the most desirable case is to know, on a minute-by-minute basis, the amounts of direct and diffuse solar radiation expected over the next 10 to 30 years (the period during which the solar system would be expected to operate). It is impossible to do this type of predictive analysis.

The second-best approach may be to use average performance data and use these data to predict performance. Liu and Jordan [1] made a significant contribution in the development of a solar radiation model. Using data collected at a variety of locations around the country, they were able to predict daily average direct and diffuse radiation (total energy) on a horizontal surface.

This paper uses Liu and Jordan's results and the Meinels' development of Laue's US observations [2] to develop a technique for modeling instantaneous values of direct and diffuse radiation for a monthly average day for each month of the year.

The Approach

Liu and Jordan's technique predicts monthly average daily direct radiation on a horizontal surface (HDIR) and monthly average daily diffuse radiation on a horizontal surface (HDIF). These data plus the effect of the earth's atmosphere (see Figure 1) on solar radiation allow the calculation of the instantaneous values of direct [DIR(t)] radiation and diffuse [(DIF(t)] radiation. Using DIR(t) and DIF(t) and the solar equation [see equation (1)] the direct and diffuse radiation which falls on any surface can be calculated.

cos(i) = sin(δ)sin(L)cos(b) - sin(δ)cos(t-)sin(b)cos(A)

+ cos(δ)cos(L)cos(b)cos(θ)

+ cos (δ) sin(L)sin(b)cos(A)cos(θ) 

  + cos(δ)sin(b)sin(A)sin(θ)                        (1)

where angles are expressed in degrees and

b = the angle between the horizontal and the collector surface ("tilt")

δ = declination (Earth axis tilt) (north positive)

δ = 23.45 sin[360 x (284+d)/365]

d= day of the year

L = latitude (north positive)

A = azimuth angle of collector surface normal relative to due south (east positive)

i = angle of incidence of direct solar radiation relative to the collector surface          
     normal

θ = 15 ∙ t where t is the time of day relative to solar noon (afternoon positive)
      (t in hours, 15 ∙ 24 = 360 degrees)

Figure 1 Solar Radiation vs. Zenith Angle for Desert, Standard, and Urban Areas

If the sky is clear and the location is known, the magnitude of the direct radiation and diffuse radiation can be read directly from the graph shown in Figure 1. We need the zenith angle, which can be derived from equation (1) by letting the collector tilt equal 0 degrees, yielding equation (2).

cos(Z) = sin(L)sin(δ) + cos(L)cos(δ)cos(H)                        (2)

where Z = angle of incidence of the direct solar radiation relative to the normal of the of the earth's surface.

Let

CDIR(I) = f(Z)                                                                (3)

CDIF(t) = g(Z)                                                               (4)

where f(Z) represents the value of direct radiation from Figure 1 as a function of the zenith angle Z, and g(Z) represents the value of diffuse radiation from Figure 1 as a function of Z.

(Note: this development works-for all the curves.)

CDIR(t) and CDIF(t) represent the absolute value of direct radiation on a clear day and the value of diffuse radiation on a horizontal surface on a clear day respectively.

where

t_sr = sunrise time

t_ss = sunset time

i_h = solar angle with respect to a horizontal surface.

If there is no cloud cover, equations (5) and (6) will be equalities. In reality, the days and hours of sunshine and cloudiness vary unpredictably. The solution is to create a model average day for each month of the year and use this for each of the days of that month in simulations. We model each monthly average day in such a way that each day has enough hours of sunshine to satisfy HDIR.

Figure 2 is a graphical representation of an average day model.

Figure 2 Direct and Diffuse Radiation Model vs Time of Day

On a clear day t₁ = t_sr and t₂ = t_ss and the areas shown in Figure 2 yield:   

In general, t₁ ≠  t_sr and t₂ ≠ t_ss The next step in this procedure is to set t₂ = solar noon and vary t₁ toward t_sr.

If equation (11) is satisfied before t₁ reaches t_sr then D takes on the shape shown in Figure 2. If equation (11) is not satisfied by the time t₁ = t_sr then t₁ is set equal to t_sr and t₂ is varied from solar noon toward t_ss .  Thus, t₂ increases until equation (11) is satisfied. This means the value of direct radiation is received (in the model) during a fixed part of the daylight hours. This also results in a value for diffuse radiation, C, during that time. C will be less than HDIF. The difference is taken care of by solving equation (12) for K.

A + B = HDIF – C which (with scaling parameter K) is

Once t₁ and t₂ are found, instantaneous solar radiation can be modeled for any location in the US where monthly average daily solar radiation data are available. Adaptations for other parts of the would only require replacing the curves of Figure 1 with ones appropriate to the locale involved.

Final Comments

This solar radiation model can provide two very important outcomes. First it can allow more accurate performance prediction of standard flat plate solar energy systems especially when they are operating under adverse conditions. Secondly, it can enable accurate analysis of solar thermal systems which use concentrators. The previously mentioned methods do not support such analyses.

The model can work because solar thermal systems are essentially low-pass filters, so that a radiation model that is quasi-static on a monthly basis can be used to obtain monthly average performance estimates for the full solar thermal system over a year.

A measure of the accuracy of the approach is to compare the value of t₂-t₁ with the measured values of the monthly average daily hours of sunshine. Two trial studies have shown excellent correlation between t₂-t₁ and the number of hours of sunshine.

References

1.      Liu, B.Y.H. and Jordan, R.C., "The Interrelationship and Characteristic Distribution of Direct, Diffuse and Total Solar Radiation", So1ar Energy, 4, No.3, pp. 1-19, l960.

2.      Meinel, A.B. and Meinel, M.P., Applied Solar Energy, Addison-Wesley, p. 47, 1976.