For the special cases mentioned previously, the expressions for hr are as follows: Case A:

Case B:

Case C:

It should be noted that the use of these linearized radiation equations in terms of hr is very convenient when the equivalent network method is used to analyze problems involving conduction and/or convection in addition to radiation. The radiation heat transfer coefficient, hr, is treated similarly to the convection heat transfer coefficient, hc, in an electric equivalent circuit. In such a case, a combined heat transfer coefficient can be used, given by hc hc + hr

In this equation, it is assumed that the linear temperature difference between the ambient fluid and the walls of the enclosure and the surface and the enclosure substances are at the same temperature.

Example 2.13

The glass of a 1 X 2 m flat-plate solar collector is at a temperature of 80°C and has an emissivity of 0.90. The environment is at a temperature of 15°C. Calculate the convection and radiation heat losses if the convection heat transfer coefficient is 5.1 W/m2K.

In the following analysis, the glass cover is denoted by subscript 1 and the environment by 2. The radiation heat transfer coefficient is given by Eq. (2.75):

= 0.90 X 5.67 X 10~8(353 + 288)(3532 + 2882) = 6.789 W/m2-K

Therefore, from Eq. (2.76), hcr = hc + hr = 5.1 + 6.789 = 11.889W/m2-K

Finally,

Q12 = A1hcr(T1 - T2) = (1 X 2)(11.889)(353 - 288) = 1545.6W

The amount of solar energy per unit time, at the mean distance of the earth from the sun, received on a unit area of a surface normal to the sun (perpendicular to the direction of propagation of the radiation) outside the atmosphere is called the solar constant, Gsc. This quantity is difficult to measure from the surface of the earth because of the effect of the atmosphere. A method for the determination of the solar constant was first given in 1881 by Langley (Garg, 1982), who had given his name to the units of measurement as Langleys per minute (calories per square centimeter per minute). This was changed by the SI system to Watts per square meter (W/m2).

When the sun is closest to the earth, on January 3, the solar heat on the outer edge of the earth's atmosphere is about 1400 W/m2; and when the sun is farthest away, on July 4, it is about 1330 W/m2.

Throughout the year, the extraterrestrial radiation measured on the plane normal to the radiation on the Nth day of the year, Gon, varies between these limits, as indicated in Figure 2.25, in the range of 3.3% and can be calculated by (Duffie and Beckman, 1991; Hsieh, 1986):

360N

where

Gon = extraterrestrial radiation measured on the plane normal to the radiation on the Mh day of the year (W/m2). Gsc = solar constant (W/m2).

The latest value of Gsc is 1366.1 W/m2. This was adopted in 2000 by the American Society for Testing and Materials, which developed an AM0 reference spectrum (ASTM E-490). The ASTM E-490 Air Mass Zero solar spectral

Jan Feb March April May June July Aug Sept Oct Nov Dec

Jan Feb March April May June July Aug Sept Oct Nov Dec

Day number

FIGURE 2.25 Variation of extraterrestrial solar radiation with the time of year.

Day number

FIGURE 2.25 Variation of extraterrestrial solar radiation with the time of year.

Wavelength (^m)

FiGURE 2.26 Standard curve giving a solar constant of 1366.1 W/m2 and its position in the electromagnetic radiation spectrum.

Wavelength (^m)

FiGURE 2.26 Standard curve giving a solar constant of 1366.1 W/m2 and its position in the electromagnetic radiation spectrum.

irradiance is based on data from satellites, space shuttle missions, high-altitude aircraft, rocket soundings, ground-based solar telescopes, and modeled spectral irradiance. The spectral distribution of extraterrestrial solar radiation at the mean sun-earth distance is shown in Figure 2.26. The spectrum curve of Figure 2.26 is based on a set of data included in ASTM E-490 (Solar Spectra, 2007).

When a surface is placed parallel to the ground, the rate of solar radiation, GoH, incident on this extraterrestrial horizontal surface at a given time of the year is given by goH = Gon cos($)

360N

The total radiation, Ho, incident on an extraterrestrial horizontal surface during a day can be obtained by the integration of Eq. (2.78) over a period from sunrise to sunset. The resulting equation is

360N

where hss is the sunset hour in degrees, obtained from Eq. (2.15). The units of Eq. (2.79) are joules per square meter (J/m2).

To calculate the extraterrestrial radiation on a horizontal surface by an hour period, Eq. (2.78) is integrated between hour angles, h1 and h2 (h2 is larger). Therefore,

360N

It should be noted that the limits hi and h2 may define a time period other than 1 h.

Example 2.14

Determine the extraterrestrial normal radiation and the extraterrestrial radiation on a horizontal surface on March 10 at 2:00 pm solar time for 35°N latitude. Determine also the total solar radiation on the extraterrestrial horizontal surface for the day.

Solution

The declination on March 10 (N = 69) is calculated from Eq. (2.5):

The hour angle at 2:00 pm solar time is calculated from Eq. (2.8):

h = 0.25 (number of minutes from local solar noon) = 0.25(120) = 30° The hour angle at sunset is calculated from Eq. (2.15):

hss = cos *[-tan(L) tan(6)] = cos *[-tan(35)tan(-4.8)] = 86.6'

Solar Radiation 91 The extraterrestrial normal radiation is calculated from Eq. (2.77):

1 + 0.033cos |
' 360Ns |
= 1366 |
1 + 0.033 cos |
' 360 X 69 ] |

365 1 |
The extraterrestrial radiation on a horizontal surface is calculated from Eq. (2.78): GoH = Goncos($) = Gon[sin(L)sin(6) + cos(L )cos(6)cos(ft)] = 1383[sin(35)sin(-4.8) + cos(35)cos(-4.8)cos(30)] = 911W/m2 The total radiation on the extraterrestrial horizontal surface is calculated from Eq. (2.79): 360N |

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