Development of mass-transfer evaporation model for Lake Nasser , Egypt

Evaporation from free water surface is considered a very important constituent in both the energy and hydrologic cycles. Precise measurement of evaporation from the free water surface is almost impossible. This is why we need a calculation model for free water evaporation. In this study, a simple mass-transfer evaporation model was developed to be applicable over Lake Nasser in the hyper-arid region located in the south of Egypt. Measured meteorological data (2011–2014) at two stations, Aswan and Abu-Simbel, were used to calculate free water surface evaporation using Priestly–Taylor equation. Priestly–Taylor equation was used because it is the most appropriate equation for Lake Nasser evaporation according to the literature. Results from this model were used to develop a simple mass-transfer evaporation model. The statistical analysis for both calibration and validation periods were very good. The slope of the regression line is about 0.9, with a coefficient of determination of 0.98. The t value is 0.6, at p value of 0.544, which is much greater than 0.05. The developed model could be used with confidence at Aswan meteorological station or on the average of the two meteorological stations, while it should be used carefully on Abu-Simbel meteorological station. doi: 10.2166/wcc.2019.116 ://iwaponline.com/jwcc/article-pdf/doi/10.2166/wcc.2019.116/640878/jwc2019116.pdf Mohamed El-Sayed El-Mahdy (corresponding author) Mohamed S. Abbas Hassan M. Sobhy Natural Resources Department, Faculty of African Postgraduate Studies, Cairo University, Giza, Egypt E-mail: m_elsayed50@cu.edu.eg


INTRODUCTION
The global warming phenomenon has attracted broad interest among the scientific community regarding evaporation and transpiration for their immense impact on the global hydrologic cycle. Traditionally, free water surface evaporation calculation measurements have been used for water resources planning and management (Elsawwaf et al. ; Li et al. ). For most of the man-made lakes, evaporation is the main cause of water losses (Balbag et al. ). Direct measurement of evaporation is too difficult (Wartena ).
In order to know how and where to record data and to be able to explicate the data of measurements, theoretical and practical developments are required (Tytler et al. ).
This provides another and very important reason for both development of theory and measurements of evaporation (Howard & Lloyd ). Many attempts have been made to find a precise evaporation model applicable for Lake Nasser, Egypt, which is a hyper-arid region. Given the nonlinear and complex behavior of the evaporation phenomenon, and because this parameter is not measured at some meteorological stations, and that the meteorological  Omar & El-Bakry () estimated monthly values of evaporation from Lake Nasser by the heat budget and bulk aerodynamic methods, using average monthly estimates of different meteorological elements over the lake.
The annual lake evaporation was found to be about 7.4 mm/d with maximum evaporation in June (10.9 mm/d) and minimum evaporation in January (3.8 mm/d). It was found that the average of heat budget and bulk aerodynamic methods gave good results, since the heat budget method only underestimated the evaporation, while the aerodynamic method overestimated the evaporation.
El Bakry () studied the net radiation over the water surface of Lake Nasser, Egypt. It was found that the outgoing radiation over the lake water surface during the cold season is higher than the warm season. An equation was developed to found net radiation by knowing the global solar radiation and the difference between water surface and air temperatures. It has been proven that the Swinbank formula is the best empirical formula to calculate the effective outgoing (long-wave) radiation over the lake.
Sadek et al. () used five methods to calculate evaporation from Lake Nasser, namely, water budget method, energy budget method, bulk aerodynamic method (Dalton), combination method (Penman), and complementary relationship lake evaporation (CRLE) model (Morton). The annual averages (in mm) were 5.9 for the water balance, 5.9 for the energy balance, 7.1 for the bulk aerodynamic, 6.6 for the Penman, and 5.7 for the CRLE method, respectively. The paper found that the best method representing the evaporation from the lake is the CRLE method.
Shaltout & El Housry () studied the evaporation from Lake Nasser. It was discovered that the lake evaporation ranged between 10 and 16 billion cubic meters (BCM) every year. The evaporation represented 20-30% of the Egyptian income from Nile water. Correlation analysis between ground station measurements for temperature, atmospheric infra-red, and water vapor content, on one side, and tile cloudiness observed by Meteosat in the infrared band (10.25-12.5 μm), on the other side, was performed at the lake's northern head near High Aswan Dam. Empirical relations for estimating the evaporation over the lake were developed and tested. The quantity of water evaporated every day was determined using Meteosat infra-red window observations and the developed empirical models.
Tolba () studied evaporation from Lake Nasser Badawy () studied climate change impacts on Lake Nasser; meteorological data for years  of three shore stations at the lake were used. In general, it was proved that the mean annual values of evaporation will not change much during the study period .
The study proved that the evaporation change for the three stations Aswan, Allaquy, and Abu-Simble were À0.47, 1.9, and À0.57%, respectively, and the total change for the whole lake is 0.29%.
Elsawwaf et al. () published a paper providing a comprehensive ten-year analysis of seasonal variations in lake evaporation using the Bowen ratio energy budget method (BREB) and six traditional methods. Evaporation rates were obtained ranging from 2.5 to 11.2 mm day À1 and averaged 5.90 mm day À1 . It was found that combination methods provide the best comparisons with the BREB evaporation.
El-Mahdy () studied High Aswan Dam Lake evaporation rate using water budget method, energy budget method (Priestly-Taylor model, 1972), mass transfer method (Harbeck model, 1962;Vikulina model, 1962;and Hyvarinen model, 1973), radiation method (Turc model, 1970), temperature-based method (Ivanov model, 1970), combination method (Penman model, 1948;and Borrelli-Sharif model, 1989). Statistical analysis has been done to showed that the accuracy of the CANFIS model in evaporation prediction was higher compared to the other AI models. It was also found that CANFIS was able to model evaporation from relative humidity and mean temperature only, with a NSE of 0.93, which was the highest among all other models.
From the literature, it is found that the most convenient evaporation model for Lake Nasser is the Priestly-Taylor 1972 model (Tolba ; El-Mahdy ). The main problem in application of the Priestly-Taylor, 1972 model is the shortage of data, since that the model requires a great deal of data such as solar radiation data, cloud cover data, albedo data, and others (Darwish ). There are many methods to calculate evaporation such as water budget method, energy budget method, mass-transfer method (aerodynamic method), radiation method, temperature-based method, combination method, and pan evaporation method. The available literature investigated the evaporation from Lake Nasser using the existing models. None of them tried to develop a model specifically designed for the lake.
The current study attempts to fill the gap by having a simple model with limited data, that could be applied to Lake Nasser and may be used for other lakes in arid regions.

STUDY AREA
Construction of the High Aswan Dam (HAD) across the River Nile during 1960-1971 was a landmark in recent Egyptian history. It is located 6 km south of Aswan city, Egypt. Its total length is about 3,600 m and its height above the river bed is about 111 m (Abu-Zeid & El-Shibini ).
High Aswan Dam Lake (Lake Nasser) ( Figure 1) is an artificial lake created behind the dam, and it has attracted much interest by many researchers to study the whole topological, environmental, ecological, hydrological aspects of the lake. Lake Nasser is composed of two lakes: Lake Nasser in Egypt and Lake Nubia in Sudan. Lake Nasser is located in southern Egypt and northern Sudan between latitudes from 20 27 0 N to 23 58 0 N and longitudes from 30 07 0 E to 33 15 0 E (Sadek et al. ). Its surface area extends up to 6,500 km 2 , with a volume of 162 BCM at an elevation of 182 m above mean sea level (amsl). The lake's total length is about 500 km (two-thirds of it in Egypt and the rest in Sudan).

DATA
The data used in this paper mainly consist of meteorological and radiation data. Temporal resolution of these data is daily  Here it is decided that the data should be based upon two meteorological stations, one on the downstream of the lake and the other at the upstream border of Lake Nasser in Egypt. Then, the data average for the two stations is used in the model development, calibration, and validation. The data average for the two stations is used to introduce values for the different parameters expressing the whole lake area (Tolba ). This approach is adopted to reach the most precise model, that is not only applicable over Aswan meteorological station impact area, but also applicable over the entire area of Lake Nasser. The main challenge to estimate free water surface evaporation is the lack of data. The standard meteorological records (e.g., air temperature, wind speed, relative humidity, radiation, air vapor pressure, and atmospheric pressure) might be available for some sites in a specific period of time. However, the data of water profile temperatures, which are essential to calculate the energy budget equation, are usually unavailable. Early researchers suggested the 1972 Priestly-Taylor model to calculate the evaporation from Lake Nasser, but the model is very demanding in terms of data. Thus, here it is suggested to develop a model, that requires lesser data, with acceptable precision. The model will be based upon mass-transfer equation.

METHODS
The constants of the model will be optimized to get the most

MODEL FORMULATION
The mass-transfer method is one of the simplest and oldest methods and is, till now, a widely used method for estimating free water surface evaporation because of its simplicity and reasonable accuracy (Valipour ). The mass-transfer methods are based on the Dalton equation, which for free water surface can be written as (Singh & Xu ): where E 0 is free water surface evaporation (mm/day), e s is the saturated vapor pressure (mb), e a is the actual vapor pressure in the air (mb), and C is aerodynamic conductance.
Although C depends on the horizontal wind speed, surface roughness, and thermally induced turbulence, it is normally assumed to be dependent on wind speed, u (Sarma et al. ). Therefore, Equation (1) can be expressed as: The general from of f(u) can be expressed as suggested by Singh & Xu () as: where A and B are constants and u is the wind speed at 2 m height above ground (m/s).
Substituting from Equation (3) into Equation (2) would result in: with considering that (e s À e a ) can be written as: where RH is the relative humidity (%).
Then, Equation (4) can be rewritten as: Thus, by knowing u, RH, and e s , the only unknowns will be the constants A and B.

MODEL APPLICATION
The current paper used spreadsheet software, Microsoft

MONTHLY MODEL CORRELATION
To correlate a model, it is recommended to use as much large time-series as available, so that the values of the con- April. The range of (B) factor is narrower than that of (A) factor, reflecting the relative importance of each one of them.  Table 3. The t value is 0.6, which is close to zero, at p value of 0.544.
Since the p value is much greater than 0.05, the null hypothesis is rejected and the alternate hypothesis is accepted, so it could be said that no significant difference exists between the means of the two models.

MODEL EFFICIENCY ON EACH STATION
It was interesting to test the model for each one of the two stations separately. Thus, we tested the performance of the developed model in both Aswan and Abu-Simbel meteorological stations.

Aswan meteorological station
The validation process was reimplemented on the data of year 2014 for Aswan meteorological station. The results of validation as shown in Figure 11 for daily validation and simplified to monthly results as represented in Figure 12, ensured that the values of A and B are good enough to be   19546 0.19528 applied with confidence on Aswan meteorological station.
The regression analysis of the validation period, as shown in Figure 13, clarified that the slope of the regression line is about 1.01, which is very close to 1, with a coefficient of determination of 0.98, reflecting the strength of the model. Also, the t-test was done on the validation period. The t value is 0.9, which is close to zero, at p value of 0.425.
Since the p value is much greater than 0.05, the null hypothesis is rejected and the alternate hypothesis is accepted, so it could be said that there is no significant difference between the means of the two models.

Abu-Simbel meteorological station
The validation process was reimplemented on the data of year 2014 for Abu-Simbel meteorological station. The results of validation as shown in Figure 14 for daily validation and simplified to monthly results as represented in Figure 15, showed that the values of A and B underestimated the evaporation on Abu-Simbel meteorological station by around 20%. The regression analysis of the validation period, as shown in Figure 16, clarified that the slope of the regression line is about 0.73, which is far from 1, with a coefficient of determination of 0.94, reflecting the relative weakness of the model. Also, the t-test was done on the validation period. The t value is 2.03, which is close to zero, at p value of 0.04. Since the p value is

CONCLUSIONS
In Lake Nasser, Egypt, measured meteorological data at Aswan and Abu-Simbel were used to calculate free water surface evaporation using the Priestly-Taylor ( Since the p value is much greater than 0.05, the statistical analyses for both calibration and validation periods were very good. Finally, we obtained a simple, less data requiring, and easy to apply evaporation model applicable over Lake