Effect of water–air heat transfer on the spread of thermal pollution in rivers
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Abstract
While working on practical problems related to the spread of thermal pollution in rivers, we face difficulties related to the collection of necessary data. However, we would like to predict the increase in water temperature at the best accuracy to forecast possible threats to the environment. What level of accuracy is necessary and which processes that influence the water temperature change have to be taken into account are usually problematic. Those problems, with special stress on water–air heat exchange in practical applications in the socalled midfield region in rivers, which is very important for the environmental impact assessment, constitute the main subject of the present article. The article also summarises the existing knowledge and practice on water–air heat exchange calculations in practical applications.
Keywords
Thermal pollution modelling Water–air heat exchange Heat budget Mixing in rivers RivMix model Heat transport equationIntroduction
In the aquatic environment, thermal pollution is understood as a result of any process that changes ambient water temperature. In rivers, such change in water temperature that goes beyond the natural range of temperature variation may be caused by discharged heated water coming from different industrial facilities using water for the cooling purpose (thermal electric power plants, chemical plants, etc.). We are aware that relatively small changes in natural ambient temperature might create substantial environmental problems; for instance, they may influence the dissolved oxygen concentration (RajwaKuligiewicz et al. 2015), population of fish and other aquatic organisms and plants (Brett 1956; Coutant 1999; Currie et al. 1998; Murray 2002). The list of potential environmental impacts of thermal pollution is very long and widely discussed in the literature (see, e.g., Caissie 2006; Hester and Doyle 2011; Webb et al. 2008). Therefore, the prediction of possible water temperature increase and assessment of its environmental impact, for example, before constructing industrial facilities become mandatory.
Schematic description of characteristic regions for mixing in rivers
3D region nearfield  2D region midfield  1D region farfield 

Vertical mixing prevails  Horizontal mixing prevails  Longitudinal mixing prevails 
It begins at the discharge point and finishes after full vertical mixing. Rapid process with maximum dimensions of few tens of the water depth (Jirka and Weitbrecht 2005), of the order of 100 H (Endrizzi et al. 2002). Approximated distance to the location of complete vertical mixing: L_{mv} ≈ 50 H^{1}  It begins after complete mixing over the depth and finishes after full lateral mixing May continue as long as several hundreds of river width (Endrizzi et al. 2002). For typical river (B/H = < 10, 100 >), the complete mixing requires from 100 B to 1000 B. Approximate distance to the location of complete horizontal mixing: L_{mh} ≈ 8 (B/H)  It begins after complete mixing over the depth and width and stretches down the river 
3D heat transport equation: \(\frac{\partial T}{\partial t} = \nabla \left[ {\left( {{\mathbf{D}}_{{\mathbf{M}}} + {\mathbf{D}}_{{\mathbf{T}}} } \right) \cdot \nabla T} \right]  \nabla \left[ {{\mathbf{v}} \cdot T} \right] + Q\)  2D depthaveraged heat transport equation: \(h\frac{\partial T}{\partial t} = \nabla \left( {h{\mathbf{D}} \cdot \nabla T} \right)  \nabla \left( {h{\mathbf{v}} \cdot T} \right) + Q\)  1D crosssectionally averred heat transport equation: \(\frac{\partial T}{\partial t} = \frac{1}{A}\frac{\partial }{\partial x}\left( {AD\frac{\partial T}{\partial x}} \right)  v_{x} \frac{\partial T}{\partial x} + Q\) 
\({{\text{hight}}/{\text{slow}}}\xleftarrow{\hspace{6cm}{{\text{computational}}\;{\text{cost}}/{\text{simulation}}\;{\text{time}}}}{{\text{low}}/{\text{fast}}}\)  
\({\text{huge}} \xleftarrow{\hspace{10cm}{{\text{amount}}\;{\text{of}}\;{\text{input}}\;{\text{data}}}}\)  
Result: 3D temperature filed  Result: 2D depthaveraged temperature field  Result: 1D crosssectionally averaged temperate values 
The description of the heat exchange between river water and river surrounding is usually very complicated. The involved processes often depend on various local and temporal factors. It is often taken for granted that the more factors are taken into account in the model/report, the better this model/report is. At the same time, the exact results are expected despite the availability of an insufficient amount of input data. An obvious question arises what result is necessary and what accuracy of the solution is really required. In most cases, modelling of the whole 3D temperature field is not feasible and needed. On the other extreme, we may ask whether 1D models can be sufficient tools for environmental impact assessments for rivers. Usually, it is not the case since vertical mixing takes a lot of time and crosssectionally averaged temperature values may be much lower than the point values within the cross section. Nevertheless, due to the lack of input data allowing to run 2D models, 1D approaches are often used, but one has to realise that they may introduce serious errors. In case of applying 1D approach, the resulting predicted temperature increase at the beginning (crosssectionally averaged values) is very low compared to the predicted maximum temperature that may appear at comparable distances from the discharge point in case of 2D approach. The focus of the present paper is, therefore, the modelling of the spread of thermal pollutions in the river midfield zone, very crucial from the environmental point of view, using the 2D depthaveraged models. For such models within the relatively short timescale (compared to the 1D models), the issue related to the additional heat sources that eventually should be taken into account is of crucial importance. To be very accurate, one may obviously try to include all known possible processes that can potentially affect the river temperature (see, e.g., Evans et al. 1998; Hannah et al. 2004; Johnson 2004; Webb and Zhang 1999; Xin and Kinouchi 2013): heat exchange with the atmosphere and with the river bottom and banks; rainfall and groundwater flow, heat production from biological and chemical processes or friction. However, very often data necessary to calculate those heat fluxes are not easily, if at all, available. For instance, the interactions between riverbed and the stream water are complex and depend on many local and temporal factors like groundwaterstream water flux, shading, bed morphology, geologic heterogeneity. It is difficult to assess them reliably. Most of those heat fluxes usually insignificantly affect the final results in the midfield zone, especially when we are interested only in temperature difference caused by the artificial heat source (not the actual water temperature itself). Whether their omission is admissible or not strongly depends on the considered case. Note that in some cases, for example, in long timescale or specific situation, some of them may be significant. Generally, in the case of thermal pollution modelling, the issue of omitting different heat exchange terms with the environment such as heat exchange with the bottom, banks and sediment is very common, but the heat exchange with the atmosphere is usually taken into account in various heat transport models and desired by users (even if in 2D models other unavoidable errors committed during the calculation affect the final results to a much larger extent). Often its estimation is very problematic and doubtful, especially in practical applications for thermal pollution spreading. However, the issue of excluding the heat exchange with the atmosphere is usually a subject of discussion. The role of the heat exchange between water and air in case of thermal pollution modelling in rivers in the midfield region is the main subject of the presented study. This article also summarises existing knowledge on water–air heat exchange calculations, with the special emphasis on practical applications.
Water–air heat exchange
Concerning the heat exchange between a river and its environment, the heat exchange between water and atmosphere is the most significant (Evans et al. 1998; Webb and Zhang 1999). Its intensity depends on water temperature and external meteorological and hydrological conditions, and it is influenced by many processes (Chapra 2008; Edinger 1974; Rutherford et al. 1993) which may be divided into two groups:

Shortwave solar radiation—radiation emitted by the sun (also called shortwave radiation or solar radiation);

Longwave atmospheric radiation—radiation that water receives from the atmosphere (also called thermal radiation); sometimes, it is calculated together with other sources of the longwave radiation from surrounding terrain like, for example, radiation from vegetation;

Longwave water back radiation—radiation emitted by the water surface (also called longwave back radiation);

evaporation and condensation—there are processes with matter changes from one state to another; evaporation is the loss of water to the atmosphere in the form of water vapour, and it is associated with the heat loss from the water surface, while the condensation is the reverse process;

conduction and convection—processes that take place at the border of water and air if they have different temperatures.
The practice of calculations of water–air heat exchange
An estimate of the net heat flux at the water–air interface turns out to be a challenge, especially when it has to be based on historical data. The first problem is related to obtaining of necessary input data for the analysed site. Intuitively, the best source in case of meteorological data is the nearest meteorological station. Unfortunately, in many cases, the nearest station does not provide all the necessary data or is still too far from the considered river reach. Another problem is associated with the differences in data obtained from the neighbouring stations. Even in conducive situations when the necessary meteorological data are available in the vicinity of the river reach, we still may receive uncertain results related to the location of the station, for which conditions such as shading or wind speed are often considerably different from those at the river channel. The problem is widely discussed in the literature (see, e.g., Benyahya et al. 2010; Garner et al. 2014; Johnson 2004). Moreover, some measured quantities may vary along the river channel (or even across the river width). Finally, very different values of net heat flux may be obtained. Johnson (2004), for example, measured the heat fluxes for the same stream in case of different conditions. The results showed that the final sum of heat fluxes measured at the same time was 580 W m^{−2} towards the stream in full sun and 149 W m^{−2} away from the stream under the shade. Additionally, each term in Eq. (1) is sensitive to the chosen computation method; i.e., different formulae may lead to varying results since they often depend on not welldefined parameters, factors or coefficients that are site specific. Moreover, in the “competition” for the best formula, increasingly “more accurate” formulae take into account more and more factors and thus require more and more input data, which again in practical applications are rarely available or costly.
Study sites and input data
Considered rivers’ sections characteristics (during the measurement campaign in 2013)
Name  Width B [m]  Averaged depth H [m]  Slope S [−]  Hydraulic radius R [m]  Discharge Q [m^{3} s^{−1}]  Shear velocity U* [m s^{−1}]  Mean velocity U [m s^{−1}] 

Narew River  20  1.8  0.0001  1.16  11.0  0.03  0.3 
Świder River  20  0.3  0.0023  0.26  3.0  0.07  0.5 
List of input data sets mentioned in the paper
Data set  Measurements time range  Meteo station  Distance^{1} [km]  Data available 

Narew River  
SET I  From 16.10 00:00 to 17.10.2013 00:00 T_{w} (every 5 min)^{ 2}  Meteo NPN ^{ 3}  1.8  T_{a}, R_{h}, p_{a}, q_{si}, u 
Meteo Choroszcz ^{ 4}  12  T_{a}, R_{h}, p_{a}, q_{si}  
Meteo Waw ^{ 5}  159  T _{ a} , R _{ h} , p _{ a} , q _{ si} , u  
Meteo IGF UW ^{ 6}  168  T_{a}, R_{h}, p_{a}, q_{si}, q_{ai}, u  
Świder River  
SET II  From 26.09 00:00 to 27.09.2013 00:00 T_{w} (every 5 min)  Meteo Świder IGF PAN ^{ 7}  1.1  T_{a}, u 
Meteo IGF UW ^{ 6}  20  T_{a}, R_{h}, p_{a}, q_{si}, q_{ai}, u  
Meteo Waw ^{ 5}  26  T_{a}, R_{h}, p_{a}, q_{si}, u  
Meteo Belsk IGF PAN ^{ 8}  42  T_{a}, R_{h}, p_{a}, u  
SET III  From 02.05 00:00 to 03.05.2016 00:00 T_{w} (every 1 min)  Meteo Local ^{ 9}  0.1  T_{a}, R_{h}, p_{a}, q_{si}, u, C 
Meteo Prosiakowo ^{ 10}  2.5  T _{a} , R _{h} , p _{a} , u  
Meteo Waw ^{ 5}  26  T _{a} , R _{h} , p _{a} , q _{si} , u  
SET IV  From 30.07 00:00 to 31.07.2016 00:00 T_{w} (every 1 min)  Meteo Local ^{ 9}  0.1  T _{a} , R _{h} , p _{a} , q _{si} , u, C 
Meteo Prosiakowo ^{ 10}  2.5  T _{a} , R _{h} , p _{a} , u  
Meteo IGF PAN ^{ 10}  25  T _{a} , R _{h} , p _{a} , q _{si} , u  
Meteo Gusin ^{ 10}  21  T _{a} , R _{h}  
Meteo Konary ^{ 10}  26  T _{a} , R _{h}  
Meteo Waw ^{ 5}  26  T _{a} , R _{h} , p _{a} , q _{si} , u  
Meteo Wichradz ^{ 10}  36  T _{a} , R _{h} , p _{a} , u  
Meteo Szpital Zachodni ^{ 10}  42  T _{a} , R _{h} , p _{a} , q _{si} , u 
Results and discussion
Each term in Eq. (1) was analysed and computed taking into account various calculation methods and different available input data sets for both rivers, preceded by existing knowledge summary. Details of calculations, different options, various empirical or semiempirical formulae and the problems encountered are discussed below.
Processes independent on water temperature
Shortwave solar radiation

sun’s position—varies depending on the date and time of day, site location and elevation above the sea level;

scattering and absorption—some amount of solar radiation is absorbed and dissipated by the atmosphere, reflected by clouds or absorbed by atmospheric gases and dust;

reflection—upon reaching the water surface, a part of the solar radiation is reflected by the surface;

shading—for example, from the riverbank or the riparian vegetation.
The heat flux value may range between 800 and 1000 W m^{−2} for a sunny day and between 100 and 300 W m^{−2} for a very cloudy day.
Equation (2) may be additionally corrected, taking into account the shade, by (1 − S_{F}) term, where S_{F} is the shading factor. The S_{F} factor ranges from 0 (no shading) to 1 (complete shading). Many researchers try to consider the influence of the riverbank or riparian vegetation on the net shortwave solar radiation (see, e.g., Garner et al. 2017; Glose et al. 2017; Sinokrot and Stefan 1993), but then additional information about the parameters describing the shading effect is required. Since all those parameters may vary spatially and seasonally, the shading effect is usually not considered in practical applications. It is worth pointing out that not only the density of the vegetation but also the orientation of the channel plays an essential role in controlling solar radiation inputs; refer to Garner et al. (2017), Lee et al. (2012) and Li et al. (2012) for details.
In case q_{si} is not measured, it may be estimated for the given geographical location and time of the year and day, using one of the available formulae for the socalled theoretical incoming clearsky solar radiation, based on the value of the solar constant (see, e.g., Allen et al. 1998; Carmona et al. 2014; Flerchinger et al. 2009; Lhomme et al. 2007). During the calculations, it is crucial not to forget the difference between the local time and solar time (see, e.g., Allen et al. 1998; Khatib and Elmenreich 2015; Lhomme et al. 2007). Otherwise, an unnecessary error may be introduced. Figure 5 presents the results of theoretical q_{si} calculations for the Narew and Świder rivers for the selected data sets (SET I, SET III and SET IV) (the orange lines). However, such formulae do not include additional effects that reduce the shortwave solar radiation reaching the water surface like, for example, cloud cover. Therefore, a discrepancy between the results based on the theoretically calculated and measured values of q_{si} is readily seen. It may be huge for cloudy days. Indeed, the discrepancy may be reduced using the formula that includes cloud cover. An example is shown in Fig. 5c (pink line) for the Świder River (SET III). The result is much closer to the result determined based on the measurements from the local meteorological station (black line). However, one has to face the problem of rare availability of the information on cloud cover, and even if it is available, it may be a subject of the judgemental estimate of the observer. Note that cloud cover, also called cloudiness, is the portion of the sky cover that is attributed to clouds and measured in eighths (oktas) or per cents. More exact and devoid of subjunctives is the estimate of cloudiness with use of allsky cameras.
Consequently, the net shortwave solar radiation values obtained from measured or computed values of q_{si} may differ significantly. Obviously, the best will be the results based on the value of q_{si} measured directly on (or close to) the river site. In the case when the meteorological station is far distant from the site, it might be better to use the calculated value of q_{si}, but only if the cloudiness information is available. In other cases, all received values of the net flux will be very uncertain.
It is worth to note that other methods to estimate hourly solar radiation are also in use, pertaining to machine learning techniques or using a variety of empirical formulae. For example, Khatib and Elmenreich (2015) have proposed a model for predicting hourly solar radiation data using daily solar radiation averages. They have also made an overview of existing empirical formulae.
Longwave atmospheric radiation
The atmosphere as all terrestrial objects emits longwave radiation. The value of atmospheric longwave radiation mostly depends on the air temperature and varies between 30 and 450 W m^{−2} (Wunderlich 1972). It could be measured directly by pyrgeometer, but while the measurements of shortwave solar radiation are relatively easily available from various meteorological stations, the measurements of longwave atmospheric radiation are unique. For the Narew River analysed case, the nearest meteorological station with such measurements has been found only at a distance of about 159 km away from the measurements site (Meteo IGF UW). The pyrgeometers are also more expensive then pyranometers, and their measurements have usually more significant errors (Choi et al. 2008). The net longwave atmospheric radiation heat flux (q_{a}) is therefore generally calculated using one of the formulae that are based on more readily available measured meteorological data. However, a large number of the proposed formulae for calculation of q_{a} make the choice of the best one very difficult.
Selected formulae for atmospheric emissivity
Reference/source  Formula  Remarks 

Clearsky conditions  
Brunt (1932), Monteith (1961), Swinbank (1963), Sellers (1965), Berger et al. (1984) and Heitor et al. (1991)  \(\varepsilon_{a} = a_{1} + a_{2} \sqrt {e_{a} }\)  With different values for a_{1} and a_{2} coefficients presented in Table 5 
Swinbank (1963)  \(\varepsilon_{a} = a_{8} T^{2}\)  a_{8} = 9.365 × 10^{−6} 
Brutsaert (1975)  \(\varepsilon_{a} = a_{3} \left( {\frac{{e_{a} }}{{T_{a} + 273.15}}} \right)^{{a_{4} }}\)  a_{3} = 1.24, a_{4} = 0.14 
Allsky conditions  
Abramowitz et al. (2012)  \(\varepsilon_{a} = a_{5} e_{a} + a_{6} (T_{a} + 273.15)  a_{7}\)  a_{5} = 3.1, a_{6} = 2.84, a_{7} = 522.5 
Sridhar et al. (2002)  \(\varepsilon_{a} = a_{3} \left( {\frac{{e_{a} }}{{T_{a} + 273.15}}} \right)^{{a_{4} }}\)  a_{3} = 1.31, a_{4} = 0.14 
Cloudy sky conditions (corrections)  
Crawford and Duchon (1999)  \(\varepsilon_{a}^{cld} = (1  s) + s\varepsilon_{a}^{{}}\)  s = q_{si}/q_{si0}—solar index, c = (1 − s)—fractional cloud cover, q_{si}—measured solar radiation, q_{si0}—estimated theoretical solar radiation for clearsky condition 
Lhomme et al. (2007)  \(\varepsilon_{a}^{cld} = (1.37 + 0.34s)\varepsilon_{a}\)  
Wunderlich (1972)  \(\varepsilon_{a}^{cld} = (1 + 0.17c^{2} )\varepsilon_{a}^{{}}\) 
Additionally to the cloud sky correction, other information, for example, information about site elevation, may be considered (e.g., Deacon 1970; Marks and Dozier 1979), but it insignificantly affects the results (Flerchinger et al. 2009). Also, since all objects emit the longwave radiation, the longwave radiation from topography and vegetation (see, e.g., Leach and Moore 2010) may be taken into account. A detailed review of various relationships based on different input parameters, with various possible corrections, may be found in (Vall and Castell 2017). But it is worth to remember that more corrections and parameters are taken into account, more input data are necessary. Overall, if no measurements for incoming longwave radiation and no additional information are available, the clearsky or allsky condition algorithms are the first and the only possible assumption for incoming longwave radiation in practical applications.
Processes dependent on water temperature
Longwave water back radiation
Evaporation and condensation
Without taking into account the radiation fluxes, the evaporation and condensation heat flux term has the most significant contribution in the total heat budget. However, its value significantly varies among sites and seasons. At the same time, especially within a short time step (like the hourly rate of evaporation), it is difficult to measure evaporation from water in open channels. Commonly accepted methods use the evaporation pan to measure the evaporation rate, but since the hourly evaporation may be of the order of a millimetre or less, very accurate precision is required. Otherwise, the results may become uncertain (Tan et al. 2007). Finally, the evaporation and condensation term in Eq. (1) is usually estimated using one of many indirect methods available in the literature.
Various types of approaches to estimate evaporation are available, for example, radiationbased models, mass or momentum transfer models, temperaturebased models and models that used artificial intelligence or mixedtype methods. Comparisons of various approaches may be found, for example, in Ali et al. (2008), Rosenberry et al. (2007), Tan et al. (2007) and Winter et al. (1995). Most of them, however, have been derived or calibrated for long time steps (to estimate daily or monthly evaporation/condensation), and it should be noted that they may not work accurately enough for the hourly time step analysis.
Note that there are different ways to define the heat flux q_{e}. Depending on the definition, in some cases, the wind function must be divided by latent heat of water and water density to be the same as defined by Eqs. (16) and (18). Above all, it is again essential to notice units for f(u) function. Different units may be used for heat flux and vapour pressure which results in different units for f(u) and finally different values of the b_{0}, b_{1} and b_{2} coefficients. Similarly to the coefficients in formulae for atmospheric emissivity calculation, also for coefficients in function f(u), we may encounter mistakes derived from simple units calculation errors.
Coefficients in the wind speed function empirical formulae used in this paper
The wind speed function formula f(u)[W m^{−2} mb^{−1}]  b_{0} [W m^{−2} mb^{−1}]  b_{1} [W m^{−3} mb^{−1} s]  b_{2} [W m^{−4} mb^{−1} s^{2}] 

Brady et al. (1969), Ahsan and Blumberg (1999), Arifin et al. (2016) and Ji (2008)  6.9  0  0.34 
Miller and Street (1972)  7.42  0  0.49 
Czernuszenko (1990)  0  3.75  0 
Marciano and Harbeck (1952)  0  2.47  0 
Ryan (1973)  6.9  3.07  0 
Meyer (1928)  8.4  3.07  0 
According to Winter et al. (1995), errors in the input data may cause serious error in the final evaporation rate value. For example, a few degrees error in measured (or assumed) surface water temperature may cause 40% error in the evaporation rate calculation. In water temperature modelling, usually the depthaveraged values of water temperature (or crosssectionally averaged values in 1D models) are used and the surface water temperature for sure differs from the depthaveraged values. Although the difference for rivers is usually small, it strongly depends on the local conditions.
To conclude and to obtain accurate results, the best is to calibrate the selected wind speed formula for local conditions using data from local measurements. However, since evaporation measurements are troublesome, and local meteorological data are usually not available, we have to bear in mind that whatever formula we use for calculation of evaporation and condensation heat flux, the results will be very uncertain.
Conduction and convection
Conduction is the process of heat exchange between bodies of different temperatures, which are in direct contact. It consists in transmitting the kinetic energy of the chaotic movement of particles as a result of their collisions.
Heat flux with the omission of independent of water temperature terms
Conclusions
In the paper, for thermal pollution modelling applications, it is recommended to use models that compute the temperature change instead of the actual river temperature whenever it is possible. Such approach reduces the amount of necessary input data and finally the computation error. In most cases in the midfield zone, omission of all terms related to heat exchange with the environment including the heat exchange with the atmosphere is recommended. Not perfect input data may in some cases introduce much larger error to the final results than just simple omission of the heat fluxes terms. The problem is especially important in practical cases when we deal with limited and not ideal data. We of course fully realise that in some applications it is necessary to include the heat exchange with the atmosphere and/or other heat fluxes. For example, after full mixing between the heated water and the river water, water temperature change depends only on the heat exchange with the environment. Another example is the heat exchange with the riverbed and banks, which is negligible for most practical applications since it is small compared to the value of heat exchange between the water surface and atmosphere and moreover it is also very uncertain due to its variability and complexity. However, when the source of heated water is located near or on the riverbank, it may turn out to have significant value. Also, most of the possible heat sources will not be significant for large, deep rivers, but much more important for very shallow streams. Since the situation very much depends on the considered case and timescale and space scale, to decide whether 1, 2 or 3D approach is appropriate and which additional heat exchange processes should be taken into account, it is very important to make the proper analyses before calculation or applying of any model. First, the expected outcome together with the appropriate timescale and space scale should be defined. Next, all affecting processes should be analysed subject to their significance and the availability of necessary input data, but also taking into account all other errors that may be committed during the calculations. Since the necessary effort to be taken to provide detailed input data (even the perfect one) for computing the heat exchange with the atmosphere (or environment) is not worth the final outcome in many practical cases (while taking into account other unavoidable committed errors).
In the cases where the heat exchange with the atmosphere estimation is necessary, it is important to bear in mind that its estimation is based on empirical formulae that depend on many uncertain parameters. To be precise, it will be necessary to measure many parameters directly on the side taking into account their space and time dependence to adjust the applied empirical formula to the current conditions. Although such measurements are usually possible for research purpose, in practice, we cannot take advantage of specially planned experiments, whether laboratory or field, allowing to measure a sufficient amount of data. In most cases for various reasons, there are no enough data to perform all the necessary calculations and prediction must be done based on existing historical, often limited and incomplete, sometimes also inaccurate data, as has been shown using two case studies presented in the paper. Therefore, in practical applications, heat exchange with the atmosphere estimation is full of judgements and extremely subjective. The most problematic to estimate is the wind speed function and atmospheric emissivity formulae. The most fragile to local conditions are measured shortwave solar radiation and wind speed value. The analysis and detailed description of particular processes involved in heat exchange with the atmosphere provided in the paper may be useful for practitioners.
Notes
Acknowledgements
This study has been partly supported by the grant IP2012 028772 from the Polish Ministry of Science and Higher Education and supported within statutory activities No. 3841/E41/S/2017 of the Ministry of Science and Higher Education of Poland. Author would like to thank Paweł Rowiński for his valuable suggestions and comments; Agnieszka RajwaKuligiewicz for sharing water temperature data; Krzysztof Markowicz for sharing data from IGF UW Meteorological observatory; Małgorzata Krasowska and Piotr Banaszuk for sharing data from Meteo Choroszcz; and the authority of the Narew National Park for sharing data from Narew National Park Weather Station.
Compliance with ethical standards
Conflict of interest
The author states that there is no conflict of interest.
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