When hour-by-hour {or other short-time base) performance calculations for a system are to be done, it may be necessary to start with daily data and estimate hourly values from daily numbers. As with the estimation of diffuse from total radiation, this is not an exact process. For example, daily total radiation values in the middle range between clear*day and completely cloudy day values can arise from various circumstances, such as inter­mittent heavy clouds, continuous tight clouds, or heavy’ cloud cover for part of the day. There is no way to determine these circumstances from the daily totals. However, the methods presented here work best for clear days, and those are the days that produce most of the output of solar processes (particularly those processes that operate at tem­peratures significantly above ambient). Also, these methods tend to produce conservative estimates of long-time process performance.

Statistical studies of the time distribution of total radiation on horizontal surfaces through the day using monthly average data for a number of stations have led to gen­eralized charts of r„ the ratio of hourly total to daily total radiation, os a function of day length and the hour in question:


Figure 2.13.1 shows such a chart, adapted from Liu and Jordan (I960) and based on WhiUier (1956, 1965) and Hottel and Whillier (1958). The hours are designated by the time for the midpoint of the hour, and days are assumed to be symmetrical about solar noon. A curve for the hour centered at noon is also shown. Day length can be calculated from Equation 1.6.11 or it can be estimated from Figure 1.6.3. Thus, knowing day length (a function of latitude ф and declination 5) and daily total radiation, the hourly total radiation for symmetrical days can be estimated.

A study of New Zealand data by Benseman and Cook (1969) indicates that the curves of Figure 2.13.1 represent the New Zealand data in a satisfactory way. Iqbal (1979) used Canadian data to further substantiate these relationships. The figure is based on long­term averages and is intended for use in determining averages of hourly radiation. Whil – iier (1956) recommends that it be used for individual days only if they are clear days.


60 75 90 105 120

$vni*[ hour sns’e, tj,, degrees

Figure 2.13.1 Relationship between hourly and daily total radiation on a horizontal surface as a function of day length. Adapted form Liu and Jordan (i960).


Benseman and Cook (1969) suggest that it is adequate for individual days, with best results for clear days and increasingly uncertain results as daily total radiation decreases.

Подпись: (2. і 3,2a)


The curves of Figure 2. ІЗ. I are represented by the following equation from Collnres – Pereira and Rabl (1979a):

The coefficients a and b sue given by

Подпись: What is the fraction of the average January daily radiation that is received at Melbourne. Australia, in the hour between 8:00 and 9:00?
a = 0.409 + 0.5016 stn(w, ~ 60) (2.13.2b)

b ~ 0.6609 – 0.4767 sinfw, – 60) (2.13.2c)

In these equations со is the hour angle in degrees for the time in question (i. e., the midpoint of the hour for which the calculation is made) and *», is the sunset hour angle.

Example 2.13.1


For Melbourne, ф = -38*. From Table 1.6.1 the mean day for January is the 17th. From Equation 1,6.1 the declination is -20.9°. From Equation 1.6, H the day length is 14.3 h. From Figure 2.ІЗ. І, using the curve for 3.5 h from solar noon, nt a day length of 14.3 h, approximately 7.8% of the day’s radiation will be in that hour. Or Equation 2.13.2 can be used; with w, = Ш7* and <o = —52,5*. the result is r, = 0.076. Hi

The total radiation for Madison on August 23 was 31.4 MJ/ivr. Estimate the radiation received between 1 and 2 PM.


For August 23, 8 = 11* and ф for Madison is 43*. From Figure 1.6.3, sunset is at 6:45 PM and day length ts І3.4 h. Then from Figure 2.13.1, at day length of 13.4 h and mean of 1.5 h from solar noon, the ratio hourly total to daily total r, = 0.П8. The estimated radiation in the hour from 1 to 2 pm ts then 3.7 MJ/m1. (The measured value for that hour was 3.47 MJ/nF.) S

image143 Подпись: (2.13.3)

Figure 2,13.2 shows a related set of curves for the ratio of hourly diffuse to daily diffuse radiation, as a function of time and day length. In conjunction with Figure 2.11.2, it Cun be used to estimate hourly averages of diffuse radiation if the average daily total radiation is known:

These curves are based on the assumption that iJHd is the same as 1JH„ and are rep­resented by the following equation from Liu and Jordan (I960);

7Г COS CO — COS ft),

24 . тло.

Подпись: (2.13.4)image146

Подпись: / 60 76 50 105 120 Sviwt hour *rvfi4 <o,. Degree* Figure 2.13,2 Relationship between hourly diffuse and daily diffuse radiation on a horizontal surface as a function of day length. Adapted form Liu and Jordan (I960).

.Example 2ЛЗ. З

From Appendix G, the average daily June total radiation on a horizontal plane in Madison is 23.0 MJ/nr. Estimate the average diffuse, (he average beam, and the average total radiation for the hours 10 to U and 1 to 2.



The mean daily extraterrestrial radiation #, for June for Madison is 41.7 MJ/m* (from Table 1.10.1 or Equation 1.10.3 with » = 162), <ax “ J13", and the day length is 15.1 h (from Equation 1.6.11). Then (as In Example 2.12.1), Kr «= 0.55. From Equation 2.12.1,

2.14′ Radiation on Sloped Surfaces 85 V

Hj/H = 0.38, and the average daily diffuse radiation is 0.38 X 23.0 = 8.74 MJ/m1. Entering Figure 2.13.2 for an average day length of I5.L h and for 1.5 h from solar noon, we find rA — 0.102, (Or Equation 2.13.4 can be used with <w = 22,5* and ws ~ 113* to obtain rd = 0.102.) Thus the average diffuse for those hours is 0.102 x 8.74 = 0.89 MJ/m2.

From Figure 2.13.1 (or from Equations 2.13.1 and 2.13.2) from the curve for 1.5 h from solar noon, for’an average day length of 15.1 h, r, = 0.108 and average hourly total radiation is 0.108 X 23.0 = 2.48 MJ/m*. The average beam radiation is the dif­ference between the total and diffuse, or 2.48 – 0.89 = 1.59 Ml/m*. В

Updated: August 2, 2015 — 10:03 am