Prediction of the theoretical and semi-empirical model of ambient temperature

Abstract

It is well known that the ambient temperature is a sensitive parameter which has a great effect on biology, technology, geology and even on human behavior. A prediction is a statement about an uncertain event. It is often, but not always, based upon experience or knowledge. Although guaranteed accurate information about the future is in many cases impossible, prediction can be useful to assist in making plans about possible developments. As a result, temperature profiles can be developed which accurately represent the expected ambient temperature exposure that this environment experiences during measurement. The ambient temperature over time is modeled based on the previous T_min and T_max data and using a Lagrange interpolation. To observe the comprehensive variation of ambient temperature the profile must be determined numerically. The model proposed in this paper can provide an acceptable way to measure the change in ambient temperature.

Keywords

ambient temperature / environment / correlation / theoretical model / semi-empirical

Introduction

There are numerous methods for developing ambient temperature profiles. They range from empirical analyses, where the distribution lanes are mapped and monitored for temperature data and other relevant environmental to theoretical models, where the rationale is determined based on geography, history weather data, or other general information about the distribution environment. Profiles may differ as a result of geography. Currently, there is no standard for the development of ambient temperature profiles. Ma et al. [1] have consecrated the effects of crosswind speed or ambient temperature on the performance of the indirect dry cooling system similar under different unit loads. They have used the outlet water temperature of the tower with ambient temperature and crosswind under full load conditions.

Borinaga-Treviňo et al. [2] have described a new method to analyze and to mitigate the ambient temperature influence on the registered fluid temperature of a thermal response test (TRT) using the method based on two parameter equations to take the influence of the ambient temperature degree and the period of time which effect the registered fluid temperature. The parameters have been determined by minimizing the time dependent thermal conductivity oscillation. For that purpose, two different TRTs have been analyzed, one with equipment only insulated against solar radiation and the other with another enhanced insulation testing machine. For both tests performed, the influence of the ambient temperature on the heat injected has been analyzed, as well as its influence on the mean fluid temperature and the consequent thermal conductivity variation.

Firmanda Al Riza et al. [3] have predicted the hourly solar radiation data by both using a decision matrix from measured RH and ambient temperature data and by using a RH-clearness index, clearness index-beam atmospheric transmission and beam atmospheric transmission-RH correlation. The result shows that both methods perform well. Hamrouni et al. [4] have studied the influence of the solar radiation and ambient temperature variation on the performance of a standalone photovoltaic pumping system which composed of a PV generator, DC-DC adaptor, DC-AC inverter and an immersed group motor-pump.

Sanusi et al. [5] have captured the effect of ambient temperature on the performance of an amorphous silicon photovoltaic system (ASPS) in Ogbomoso, a tropical area in Nigeria by monitoring the variation in the power output of the system with ambient temperature of the area from 2006 to 2008. Some researchers have discussed the effect of temperature and light on the growth of algae species [6]. They conclude that the wavelengths of blue and red light are sensitive. Other studies have indicated the interactive effects of resin formulation and ambient temperature of cure on the percentage conversion, molar heat of cure, surface hardness and depth of cure of selected bonding agents. It has been found that when increasing ambient temperature of cure, the percentage conversion increases in all resins [7]. Human behaviors are closely related to the variation of ambient temperature. Some studies have discussed the effect of ambient temperature on gross-efficiency in cycling, and have concluded that the temperature-induced changes in gross-efficiency could account for about half of the well-established performance decrements during time trial exercise in the heat [8]. Some researchers have used the ambient temperature as inlet of solar collector. Therefore, it can be concluded that ambient temperature has a great effect on the thermal efficiency of solar collectors [9–18].

Modeling ambient temperature

In most cases, the ambient temperature is taken as the average value. Its evolution from sunrise to sunset is marked by minimum values at sunrise and sunset and a maximum value in the middle of the day. These are the only values given by weather stations. Modeling the temperature of the air is generally used as heat transfer fluid in the solar collectors, which is of paramount importance.

Declination (d)

The declination (d, in degrees), for any day of the year (N) can be calculated approximately by [19]

(1)\delta =\mathrm{23.45}\sin \left({\displaystyle \frac{\mathrm{360}}{\mathrm{365}}}(\mathrm{284}+N)\right).

Hour angle (w)

The hour angle (w), of a point on the earth’s surface is defined as the angle through which the earth would turn to bring the meridian of the point directly under the sun. The hour angle can also be obtained from the apparent solar time (AST); i.e., the corrected local solar time is [19]

(2)\omega =\mathrm{15}(\mathrm{AST}-\mathrm{12}).

Sunrise and sunset times and day length

The sun is said to rise and set when the solar altitude angle is 0. So, the hour angle at sunset, w_ss, can be found to be w when h= 0° [19]:

(3)\cos ({\omega }_{\mathrm{ss}})=-\tan (L)\tan (\delta ).

Since the hour angle at local solar noon is 0°, with each 15° of longitude equivalent to 1h, the sunrise and sunset time in hours from local solar noon is then

(4)t_{\mathrm{ss}}=\mathrm{12}-{\displaystyle \frac{{\omega }_{\mathrm{ss}}}{\mathrm{15}}}.

The hour angle w_ss at sunset is the opposite of the hour angle at sunrise, so w_ss =–w_sr and the duration of the day is

(5)t_{\mathrm{sr}}=\mathrm{12}+{\displaystyle \frac{{\omega }_{\mathrm{s}\mathrm{r}}}{\mathrm{15}}},

(6)\mathrm{\Delta }t=\mathrm{2}\times {\displaystyle \frac{{\omega }_{\mathrm{ss}}}{\mathrm{15}}}.

Validation models

Generally, the weather stations give the minimum and maximum values recorded on the day in question. This is not sufficient to follow the change, especially as much heat transfer phenomena are related thereto.

Ambient temperature is closely linked to the radiant temperature of the sky (sky temperature). This would have an influence. Indeed, for the sites located in altitudes, heat losses must be considered. The equivalent temperature of the ambient air can be calculated from [20]

(7)T_{\mathrm{eq}}={\displaystyle \frac{(T_{\mathrm{am}}\times h_{\mathrm{cve}}+T_{\mathrm{sky}}\times h_{\mathrm{rve}})}{H_{\mathrm{ve}}}},

where h_cve is the coefficient of thermal convection losses between the front of the sensor and the external environment, and h_rve is the coefficient of thermal losses by radiation between the front of the sensor and the external environment.

Theoretical model

The theoretical change in ambient temperature can be modeled by

(8)T_{\mathrm{e}}(t)={\displaystyle \frac{T_{\max }-T_{\min }}{\mathrm{2}}}+\left({\displaystyle \frac{T_{\max }-T_{\min }}{\mathrm{2}}}\right)\cos \left({\displaystyle \frac{\mathrm{2}\pi \mathrm{AST}}{\mathrm{\Delta }t}}\right).

In this model, the maximum temperature is reached at true solar noon when the flux density is a maximum. This model does not correspond to reality as it is considered that the maximum temperature reaches thermal solar noon corresponding to the solar midday instant real temperature plus 1/8 of the length of the solar day (Dt) [21]. This is mainly caused by the thermal inertia of the soil, the thermal equilibrium between the ambient medium and the ground.

Model corrected [20]

The theoretical model is replaced with a model that introduced the notion of “thermal solar midday” where the temperature reaches its maximum value. Given the nocturnal radiative exchanges, the ambient temperature reaches its minimum value at sunrise to sunset, and finally, in this model the ambient temperature is taken as the average temperature. And room temperature can be modeled by

(9)T_{\mathrm{e}}(t)={\displaystyle \frac{T_{\max }+T_{\min }}{\mathrm{2}}}+\left({\displaystyle \frac{T_{\max }-T_{\min }}{\mathrm{2}}}\right)\cos \left({\displaystyle \frac{\mathrm{2}\pi \left(\mathrm{AST}-\mathrm{12}-{\displaystyle \frac{\Delta t}{\mathrm{8}}}\right)}{\mathrm{\Delta }t}}\right).

Idliman model [22]

In this model, the ambient temperature is evaluated by

(10)T_{\mathrm{ab}}=T_{\mathrm{1}}+T_{\mathrm{2}}\cos ((\mathrm{14}-\mathrm{AST})\pi /\mathrm{12})+\mathrm{273.15},

with T₁ = (T_max +T_min)/2 and T₂ = (T_max –T_min)/2, T_max and T_min are the maximum and minimum ambient temperatures during the day.

Proposed theoretical model

The ambient temperature over time is modeled based on the temperature of sunset T_ss and the temperature of sunrise T_sr previously which means the duration of the light days, using a Lagrange interpolation which can be expressed as listed in Table 1.

Principle of Lagrange interpolation [23]

In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points.

In engineering and science, a lot has a number of data points, as obtained by sampling or experiment, and tries to construct a function which closely fits those data points. This is called curve fitting or regression analysis. Interpolation is a specific case of curve fitting, in which the function must go exactly through the data points. A different problem which is closely related to interpolation is the approximation of a complicated function of a simple function. Suppose the function is known, it is too complex to evaluate efficiently. Then a few known data points could be picked from the complicated function, creating a lookup table, and attempt can be made to interpolate those data points to construct a simpler function.

Lagrange base

Polynomial interpolation involves finding a polynomial of order ‘n’ that passes through the ‘n+1’ points. One of the methods to find this polynomial is called Lagrangian interpolation. Other methods include the direct method and the Newton’s divided difference polynomial method. Lagrangian interpolating polynomial [23] is given by

(11)f_n(t)={\displaystyle \sum _{i=\mathrm{0}}^n{L}_i(t)f(t_i)},

where ‘n’ in f(x) stands for the nth order polynomial that approximates the function y=f(x) given at (n-1) data points as (x₀, y₀), (x₁, y₁),…, (x_n-1, y_n-1), (x_n, y_n). L_i(x) is a weighting function that includes a product of (1-n) terms with terms of j=i omitted. The application would be clear using an example.

Based on this principle, the variation of the ambient temperature with time is modeled as

(12)T_{\mathrm{am}}=\zeta (\mathrm{AST},\omega )\times \psi (\mathrm{AST})\times {\displaystyle \frac{T_{\mathrm{ss}}}{\mathrm{2}}}+\upsilon (\mathrm{AST},\omega )\times T_{\mathrm{sm}}.

The values \zeta, \psi and \upsilon are expressed as

\zeta =(AST-t_{\mathrm{ss}})\times {\left({\displaystyle \frac{\mathrm{15}}{{\omega }_{\mathrm{ss}}}}\right)}^{\mathrm{2}}\times \left({\displaystyle \frac{\omega }{\mathrm{15}}}\right),

\psi =\mathrm{0.5}+{\displaystyle \frac{(\mathrm{AST}-t_{\mathrm{sr}})}{(\mathrm{AST}-t_{\mathrm{ss}})}},

\upsilon ={\displaystyle \frac{\omega }{\mathrm{60}}}\times {\left({\displaystyle \frac{\mathrm{15}}{{\omega }_{\mathrm{ss}}}}\right)}^{\mathrm{2}}\times (\mathrm{AST}-t_{\mathrm{ss}})-(\mathrm{AST}-t_{\mathrm{sr}}).

When AST is equal to t_ss, it implies that the ambient temperature is equal to T_ss.

This model is used to calculate the ambient temperature between sunset and sunrise, which means at daytime.

Proposed semi-empirical correlation

At this time, an attempt is made to model the relationship between the ambient temperature and the minimum and maximum temperatures corresponding to all hours of the day. Then the semi-empirical correlation can be proposed as

(13)T_{\mathrm{am}}=Y_{\mathrm{0}}+\left(\mathrm{97.275}/\left(w(X_c)\sqrt{{\displaystyle \frac{\pi }{\mathrm{2}}}}\right)\right)\exp \left(-\mathrm{2}\times {\left({\displaystyle \frac{(\mathrm{AST}-X_c)}{w(X_c)}}\right)}^{\mathrm{2}}\right).

The constants Y₀, X_c and w are written as

(14)Y_{\mathrm{0}}=-\mathrm{0.369}+\mathrm{0.854}\times \left({\displaystyle \frac{T_{\max }+T_{\min }}{\mathrm{2}}}\right),r=\mathrm{0.86993},

(15)X_{\mathrm{c}}=\mathrm{12.34}+\mathrm{0.328}\times (T_{\max }-T_{\min }),

(16)w(X_c)=\mathrm{97334.6}-\mathrm{23517.399}{X}_c+\mathrm{2127.83}{X}_c^2-\mathrm{85.438}{X}_c^3+\mathrm{1.2845}{X}_c^4.

Table 2 shows the results of the constants corresponding to the semi-empirical correlation for each month. It is known that each month it is determined by a maximum and a minimum ambient temperature.

Results and discussion

These data measured at Biskra, Algeria show that the region is characterized by high temperatures with strong seasonal variations in July and January, as illustrated in Fig. 1.

Figures 2 to 5 show the variations in ambient temperature as a function of true solar time. The profile of the ambient temperature has been illustrated from proposed correlation, and its effect depends on the maximum and minimum temperatures which are demonstrated in Figs. 2 to 5. For the early season that begins with the winter season are December, January and February, the ambient temperature is between 19°C in February and 17°C in January. For the spring season, the temperature is increased by 13°C than that of winter, and is between 33°C in May and 23°C in April. The height, ambient temperature is selected in the summer season that reaches its value of 43.5°C in July and low ambient temperature is 38°C in August. In the fall season, the temperature is between 33°C in September and 24°C in November.

Change in ambient temperature

The thermal conversion of solar energy for applications such as drying food products or space heating is based on the solar collectors using air as the coolant. The air at the inlet of these systems is that of the ambient medium. Typically, weather stations provide only the minimum and maximum values, which is not sufficient to follow the change from sunrise to sunset. This does not reflect reality and affects the accuracy of the results. The model proposed in this paper describes an acceptable way as a function of ambient temperature T_max and T_min. It can be expressed as

T_{\mathrm{am}}=\zeta \times \psi \times {\displaystyle \frac{T_{\min }}{\mathrm{2}}}+\upsilon \times T_{\max }.

Other models encountered in the analysis related to the ambient temperature are also simulated. The results are compared and shown in Figs. 6 to 9.

Note that there is a considerable gap between the values given by each model, especially early in the day as displayed in Figs. 6 to 9. The solar midday models give more or less similar values. The proposed theoretical model is in perfect agreement with that proposed by Idliman.

Experimental and simulated evolution of ambient temperature

Figures 10 to 13 show a comparison between the daily variation of the experimental and ambient temperatures after the proposed model.

The examination of these curves indicates that the proposed model translated satisfactorily the variation of the ambient temperature and that this model can be adopted to predict the change in ambient temperature.

Figure 14 shows a variation of ambient temperature measurement experimentally at a site in Biskra, Algeria. The tests were conducted in a period extending from January to December 2014 at the University of Biskra (34.8 °N, 5.73 °E). This region is characterized by a dry climate in summer. The tests were performed in clear days, free of disturbances (clouds).

Figure 15 shows the theoretical calculation of ambient temperature for one year as a function of AST. The higher evolution has shown in July with a value estimated by a maximum pick of ambient temperature of 42°C at 16:00, and the lower evolution in February with a value estimated by a maximum pick of ambient temperature of 18°C at 16:00.

Figure 16 represents the relative errors of ambient temperature for each month of the year. A maximum relative error was selected in February and December estimated by 15% to 25%, and a minimum value between 2% and 10%, corresponding to experimental measurements and the value calculated by the semi-empirical model.

Figure 17 represents an absolute error of ambient temperature for each month of the year. A maximum absolute error was selected in February and December estimated by 2.8 to 5, and a minimum value between 0.8 to 2, corresponding to experimental measurements and the value calculated by the semi-empirical model.

Conclusions

In this paper, prediction of ambient temperature was made by theoretical test, and mathematical logic was used to set the ambient temperature profile. A comparison was made between the theoretical models proposed in literature. Note that there is a considerable gap between the values given by each model, especially at the beginning of the day. The solar midday models give more or less similar values. The proposed model is in perfect agreement with that proposed by Idliman.

The data provided are used to assist the modeling and defining of a good mathematical model. From the results obtained, it can be noted that the ambient temperature profile made by the semi-empirical model proposed in this paper is the most approximate to that of the experimental data.

References

[1] Ma H, Si F, Li L, Yan W, Zhu K. Effects of ambient temperature and crosswind on thermo-flow performance of the tower under energy balance of the indirect dry cooling system. Applied Thermal Engineering, 2015, 78(5): 90–100

[2] Borinaga-Treviňo R, Norambuena-Contreras J, Castro-Fresno D. How to correct the ambient temperature influence on the thermal response test results. Applied Thermal Engineering, 2015, 82(5): 39–47

[3] Firmanda Al Riza D, Ihtsham ul Haq Gilani S, Shiraz Aris M. Hourly solar radiation estimation using ambient temperature and relative humidity data. International Journal of Environmental Science and Development, 2011, 2(3): 188–193

[4] Hamrouni N, Jraidi Mand Chérif A. Solar radiation and ambient temperature effects on the performances of a PV pumping system. Revue des Energies Renouvelables, 2008, l(11): 95–106

[5] Sanusi Y K, Fajinmi G R, Babatunde E B. Effects of ambient temperature on the performance of a photovoltaic solar system in a tropical area. Pacific Journal of Science and Technology, 2011, 12(2): 176–180

[6] Singh S P, Singh P. Effect of temperature and light on the growth of algae species: a review. Renewable & Sustainable Energy Reviews, 2015, 50: 431–444

[7] Vaidyanathan J, Vaidyanathan T K. Interactive effects of resin composition and ambient temperature of light curing on the percentage conversion, molar heat of cure and hardness of dental composite resins. Journal of Materials Science: Materials in Medicine, 1992, 3(1): 19–27

[8] Hettinga F J, de Koning J J, de Vrijer A, Wüst R C I, Daanen H A M, Foster C. The effect of ambient temperature on gross-efficiency in cycling. European Journal of Applied Physiology, 2007, 101(4): 465–471

[9] Chabane F, Moummi N, Benramache S. Experimental study of heat transfer and thermal performance with longitudinal fins of solar air heater. Journal of Advanced Research, 2014, 5(2): 183–192

[10] Chabane F, Moummi N, Benramache S, Tolba A S. Experimental study of heat transfer and an effect the tilt angle with variation of the mass flow rates on the solar air heater. International Journal of Science and Engineering Investigations, 2012, 1(9): 61–65

[11] Chabane F, Moummi N, Benramache S. Experimental performance of solar air heater with internal fins inferior an absorber plate: in the region of Biskra. Journal of Energy Resources Technology, 2012, 4(33): 1–6

[12] Chabane F, Moummi N, Brima A, Benramache S. Thermal efficiency analysis of a single-flow solar air heater with different mass flow rates in a smooth plate. Frontiers in Heat and Mass Transfer, 2013, 4(1): 013006

[13] Chabane F, Moummi N, Benramache S, Bensahal D, Belahssen O. Collector efficiency by single pass of solar air heaters with and without using fins. Engineering Journal, 2013, 17(3): 43–55

[14] Chabane F, Moummi N, Benramache S. Effect of the tilt angle of natural convection in a solar collector with internal longitudinal fins. International Journal of Science and Engineering Investigations, 2012, 1(7): 13–17

[15] Chabane F, Moummi N, Benramache S. Experimental analysis on thermal performance of a solar air collector with longitudinal fins in a region of Biskra, Algeria. Journal of Power Technologies, 2013, 93(1): 52–58

[16] Chabane F, Moummi N, Benramache S, Belahssan O, Bensahal D. Nusselt number correlation of SAH. Journal of Power Technologies, 2013, 93(2): 100–110

[17] Chabane F, Moummi N. Heat transfer and energy analysis of a solar air collector with smooth plate. European Physical Journal Applied Physics, 2014, 66(1): 10901

[18] Chabane F, Hatraf N, Moummi N. Experimental study of heat transfer coefficient with rectangular baffle fin of solar air heater. Frontiers in Energy, 2014, 8(2): 160–172

[19] Soteris Kalogiro. Solar energy engineering: processes and systems. Academic Press, 2009

[20] Benlahmidi S. Study of convective drying by solar energy red products. Dissertation for the Doctoral Degree. Biskra: University Mohamed Khider, 2013

[21] Moummi N, Moummi A, Belgacem A, Hachemi A, Le Ray M. Systematic performance prediction of solar air flat plate collectors in various different climates sites in Algeria. National Energy Conference. Neptur Roumani, Juillet, 1995 (in French)

[22] Idliman A. Theoretical study of a drying system leather skins for the Marrakech region, consisting of an agricultural greenhouse acting as a hot air solar generator and a conventional dryer. Dissertation for the Master’s Degree. National School of Marrakech Morocco, 1990

[23] Maufred G. Numerical methods. Department of Econometrics. University of Geneva, 2006

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