Simplified Method for Night Sky Radiation Analysis in a Cool-Pool System

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where , and denote the incoming downward sky radiation , the ambient absolute temperature (K) and the percentage of relative humidity, respectively. Also, the correction factor of is defined for a cloudy sky. The dimensionless parameters k and c are used to account the effect of cloud on sky radiation (Table 1) and is a constant equal to .

The outgoing radiation from water body can be calculated using the Stefan-Boltzmann law given by Eq. (2).

(2)

where is the radiated power from cool-pool water, A is the surface area () and is the water absolute temperature (K), is the Stefan-Boltzmann constant and is its emissivity. Assuming a suitable value for sky emissivity, Eq. (2) can also be used to determine the night sky temperature. Generally, calculating the sky emissivity is important because of its application in radiation cooling as well as determining solar collector losses ^[9]. In some literatures, some methods have been investigated to correspond sky emissivity with other important parameters such as dew point ^{[7, 8]}, but in the present work, we assume to be 0.74. The sky emissivity () is defined in a way, so that Eq. (1) can be rewritten in the following form ^[10]:

(3)

where is the ambient absolute radiative temperature. It means that sky acts as a gray body with the absolute temperature equal to ambient and apparent emissivity of . According to the following equation, we conclude that can be found by ^[10]:

(4)

Here, is measured using a pyrgeometer then, by dividing the data by , the sky emissivity can be expressed by fitting a curve usually with least square lines ^[11]. The net rate of energy loss from water body with area A can be computed using Eq. (5), where is the downward radiation power on the surface with area A.

(5)

2.2. Radiation Cooling Time

For a case with just outgoing radiation from an object assuming no work done, the total energy loss or gain can be expressed by:

(6)

Where, E is the internal energy. If a pure translational kinetic energy theory can be utilized, according to the equipartition of energy, the object’s energy will be described as:

(7)

where is the Boltzmann constant, N is the number of mole, is the Avogadro’s number and m and M are mass and molar mass, respectively. By substituting Eq. (6) in Eq. (7), we obtain the required cooling time in order to reach the final temperature from the initial temperature as:

(8)

Luciuk ^[12] utilized Eqs. (7) and (8) using a scale of to find the real required cooling time for an object exposed to night sky. But our present work shows that the specific heat wasn’t considered in his research. For example, aluminum has a specific heat C of 900 ^[13] at 300 K, but using the above method shows an apparent specific heat of 460 with the defining of . The main reason is the allocating of one equal average kinetic energy to all of the body particles.

Assuming internal energy of a liquid or solid to be the product of mass m, specific heat C and temperature T, we obtain the exact governing differential equation for water temperature using Eq. (5) and Eq. (6).

(9)

where is a function of time. But, for a period of time with constant ambient temperature, is constant and the nonlinear differential Eq. (9) can be solved analytically by separating variables method. The final answer is:

(10)

where and are the initial and final temperatures of water,

(11a)

and

(11b)

The emissivity of water in Eqs. (11 a, b) is 0.95 ^[13]. The required cooling time is a strong function of mass and specific heat. The mass of water used in the cooling system can be found, if the cooling load is known.

Table 1. Values of parameters c and k in the modified Swinbank sky radiation model

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3. Results

Figure 2 and Figure 3 show the variations of with ambient temperature and relative humidity for clear sky.

Considering Figure 2 and Figure 3, we obtain that is a linear function of ambient temperature and relative humidity. In Figure 2, shows a linear trend with temperature at constant relative humidity; while Figure 3 demonstrates the linear trend of incoming sky radiation with relative humidity at constant ambient temperature (for , the difference between the line and the curve is less than 1 ). The linear behaviors are extracted as the following equations.

(12)

(13)

Figure 2. Variation of night sky radiation with ambient temperature and different relative humidity

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Figure 3. Variation of night sky radiation with relative humidity and different ambient temperatures

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where the superscript denotes constant variables. Then the total energy received from sky per unit area, , during period can be theatrically calculated as below when the ambient temperature and relative humidity change from to and to , respectively.

(14)

(15)

where and are the averages of slope and vertical-axis intersect of the lines, respectively, at the end points or midpoints according to their variation with time. Now we should decide choosing between the first or the latter integrals. The calculation is continued using the first one, because the function behavior is really more similar to straight line than the latter. Another reason appears when we use differential of temperature, instead of time. So, the following correlation may be achieved.

(16a)

Assuming linear change of T with t, we have:

(16b)

Where, is the mean temperature during night time. As the following real example shows, the variation of temperature during night can be considered linear but it is not true about relative humidity. So, the integral can be calculated with good approximation. For the surface area of A, we use the following relation.

(17)

The values of , , and can be found in Table 2 and Table 3. The numbers are generated using the least mean squares line method.

Table 2. Values of α₁ and β₁ in Eq. (15) and Eq. (16)

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Table 3. Values of α₂ and β₂ in Eqs. (15) and (16)

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Table 4. Specification of the considered cooling pool

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Table 5. Required data and radiated energy for the 1^st day of August 2013 in Isfahan, Iran

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The coefficients listed in Table 2 and Table 3 are used to measure the clear sky radiation. The correction factor of should then be introduced to the final result. To calculate the night sky radiation cooling potential, the following steps shall be taken:

• Preparing a datasheet of hourly ambient temperature, relative humidity and cloudiness degree.

• Finding the mean temperature as well as the coefficients of Eq. (15) using Table 2 and Table 3 based on the most occurred relative humidity.

• Calculating the total radiation energy on the surface, using Eq. (16).

• Correcting the total radiation energy by introducing the cloudiness degree, using Table 1.

• Using Eqs. (10) and (11) to find the required time or temperature.

In Table 4, the first three columns show the required meteorological data and radiated energy for the first of August, 2013 (a sample day in summer) in Isfahan, Iran ^[14]. We use the above simplified model and compare the results. For and (two typical relative humilities), we have , , and , hence , and is taken as the linear average of and . Using Eq. (16), the result is or equally, which is very close to the total radiated energy in Table 4. A cool-pool with the specifications in Table 5 is considered in this study. Using Eqs. (10) and (11), it is understood that the required time to reach the water temperature from to is nearly 7 hours, which is less than the available time (7 hours and 30 minutes).

A conventional 4000 W room cooler which works for 6 hours a day, consumes 86400 kJ of electricity. Our calculated cool-pool system with dimensions of 3 m 4 m has an ideal cooling capacity L of about

(18)

which seems that compensate all the day cooling demands. Figure 4, based on Eq. (10), shows the temperature variation with time using the aforementioned model. Because we used the mean radiation value, the required time will be more but still less than the available night time.

Figure 4. Variation of cool-pool water temperature with time

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