The Effect of Unsteady Water Discharge Through Dams Of

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Article The Eﬀect of Unsteady Water Discharge through Dams of Hydroelectric Power Plants on Hydrodynamic Regimes of the Upper Pools of Waterworks

Tatyana Lyubimova 1,2,*, Yanina Parshakova 1, Anatoly Lepikhin 3,4, Yury Lyakhin 4 and Alexey Tiunov 4

1 Institute of Continuous Media Mechanics UB RAS, 1, Koroleva Str., Perm 614013, Russia; [email protected] 2 Theoretical Physics Department, Perm State University, 15, Bukireva Str., Perm 614990, Russia 3 Hydrology and Water Resources Protection Department, Perm State University, 15, Bukireva Str., Perm 614990, Russia; [email protected] 4 Mining Institute UB RAS, 78A, Sibirskaya, Str., Perm 614007, Russia; [email protected] (Y.L.); [email protected] (A.T.) * Correspondence: [email protected]

Received: 30 March 2020; Accepted: 5 May 2020; Published: 8 May 2020

Abstract: The hydrological regimes of surface water bodies, as a rule, are unsteady. However, accounting for the non-stationarity substantially complicates the hydrodynamic calculations. Becauseof this, the scenario approach is traditionally used in the calculations. Characteristic scenarios are set with constant hydrological characteristics throughout the time covered in the calculations. This approach is fully justified if the characteristic time of the change in water flow rate is much longer than the calculation time. However, nowadays, tasks are becoming more and more urgent when accounting for flow variability during calculation period becomes crucial. First of all, such a problem arises when assessing the effect of non-stationary water discharge through hydroelectric power plant dams on the hydrodynamic regime of both the upper and lower pools of the reservoir. In the present paper, the effect of the intraday variability of the Kamskaya Hydroelectric Power Plant (Kamskaya HEPP) operation on the peculiarities of the hydrodynamic regimes of the near-dam part of the upper pool of the Kama reservoir is described. The importance of the problem is determined by the location of the main drinking water intake of Perm city and one of the largest thermal power plants (TPP) in Europe, Permskaya TPP, in this part of the reservoir. This TPP uses a direct-flow cooling system from the Kama reservoir, which is very sensitive to the peculiarities of the hydrodynamic regime of the reservoir. The computational experiments based on the combined hydrodynamic models in 2D/3D formulations have shown that the intraday oscillations of the discharge flow rate through the dam of the HEPP have a very significant effect on the hydrodynamic regime of the reservoir in the vicinity of the Permskaya TPP; therefore, these effects must be taken into account when minimizing the risks of thermal effluents entering the intake channel of the Permskaya TPP.

Keywords: reservoirs; models; back ﬂows; non-stationary discharge; hydroelectric power plant

1. Introduction When operating the power units of power plants, a large amount of water is used to cool their components. Cooling is carried out by means of reverse water supply, by using cooling towers, columns for cooling water, a pond-cooler, or by the direct-ﬂow method using water from natural reservoirs. When implementing the second cooling method, the discharge of heated water is carried

Water 2020, 12, 1336; doi:10.3390/w12051336 www.mdpi.com/journal/water Water 2020, 12, 1336 2 of 22 out into the environment at a lower temperature. The negative environmental impact of heated water has been described in numerous publications (see, for example [1–5]). An analysis of the impact of climate change and socio-economic factors on global water scarcity is presented in [6]. This work extends previous global water use research by analyzing not only the effects on livelihoods and climate change, but also electricity production, water use efficiency and other external factors for water scarcity. The temperature standards contained in the European directive [7] on freshwater fish are presented in [6]. The water temperature downstream of the discharge point should not exceed 21.5 ◦C and 28 ◦C (or 1.5 ◦C and 3 ◦C higher than the background temperature of the reservoir) for salmon and cyprinid fish species, respectively. The works [8–11] deal with the investigation of variations in the river water temperature due to the climate change. When solving the problems of thermal pollution, it is necessary to take into account the location of the source of discharges [12–19]. Based on the location and size of the water intake facilities, it is necessary to ensure the optimum temperature difference between the water of the reservoir and the heated wastewater with minimal damage to the environment [19]. It is also necessary to prevent the entry of heated water into the intake zone, which can significantly reduce the efficiency of the power plant cooling system [20]. These problems must be solved within the framework of the development of mathematical and numerical models. Initially, numerical modeling was carried out in the framework of two-dimensional models by averaging velocity over depth and considering temperature propagation in open channels [21,22]. At present, the apparatus of the three-dimensional numerical simulation of the distribution of thermal pollution is developed [5,13,14,23–25]. This simulation is necessary for an adequate description of processes, since thermal processes in water bodies are essentially three-dimensional. The account of the hydrological regime of reservoirs in the described works is carried out either for the average values of the flow velocity or for the time interval at which the reservoir velocity regime is unchanged. However, the hydrological regime of surface water bodies is usually unsteady. Accounting for non-stationarity fundamentally complicates the hydrodynamic calculations. That is why the scenario approach is conventionally used in the calculations. Typical scenarios are set with constant hydrological characteristics throughout the calculation time [25,26]. This approach is fully justified if the characteristic time of the change in water flow is much longer than the calculation time. However, nowadays, tasks are becoming more and more urgent when accounting for flow variability during calculations becomes crucial [27,28]. The aim of this work is to solve the problem of assessing the impact of unsteady water discharge through a Hydroelectric Power Plant (HEPP) on the hydrodynamic regimes of near-dam sections of the upper pulls of large reservoirs. The importance of this study is determined, on the one hand, by the significant intra-daily variability of electricity consumption and, accordingly, by the significant variations in water discharge through HEPP dams, and, on the other hand, by the location of large water consumers that are sensitive to changes in the hydrodynamic regime in the upper pools of reservoirs. This task is considered through the example of the Kamskaya HEPP and the Permskaya Thermal Power Plant (Permskaya TPP) located in the zone of its influence, which is the fourth in Russia in terms of electricity generation. The use of a direct-flow cooling system at this TPP determines its sensitivity to the peculiarities of the hydrodynamic regime of the reservoir.

2. Materials and Methods To simulate heat-affected zones in the influence area of Perm State District Power Station, a combined scheme was used, built on the basis of model conjugation in 2D and 3D settings. The need to use such a calculation scheme is due to the significant heterogeneity of the considered temperature fields both in the water area and the depth. At the same time, the capabilities of even the most powerful of the available clusters do not allow for accurate 3D calculations for large sections of reservoirs because. when performing calculations in 3D setting for the entire reservoir, a very large amount of information is obtained, which is then not used in full in the future. A two-dimensional model Water 2020, 12, 1336 3 of 22 of thermal pollution was constructed for an 82-km stretch of reservoir. For the considered problem, the results of 2D modeling play an auxiliary role, they are used to set the boundary conditions when building 3D models. The performed calculations in [25] showed that this model allows for a qualitative assessment of the distribution of current velocity fields for different directions and wind speeds in the Kama reservoir near Perm State District Power Station. A 3D hydrodynamic model was constructed for the Kama Reservoir section with linear dimensions of 11 km adjacent to the Perm State District Power Plant and includes a source of water intake and a source of wastewater discharge. This calculation scheme was previously used in another paper of ours [25]. In this paper, the issues of modeling temperature fields created in the Kama reservoir as a result of the discharge of thermal effluents from the Permskaya TPP were considered. The morphometry of the considered section of the reservoir was set on the basis of materials from a detailed bathymetric survey performed by us. Additionally, the model used was calibrated by selecting the most effective heat transfer coefficients of the water surface with the atmosphere and setting the optimal bottom roughness coefficients. Based on a comparison of the measured and calculated characteristics of the water temperature field, it was shown that the difference in the calculated and observed values did not exceed 15–20%. Since the models used in the present paper do not include parameters requiring additional calibration for a substantially unsteady hydrological regime, we believe that a model that works correctly and efficiently for stationary hydrological conditions should reflect the main features of the flow in unsteady regime.

2.1. Development of a Hydrodynamic Model in a Two-Dimensional Formulation A two-dimensional (in the horizontal plane) model of the Kama river, the upper pool of the Kama reservoir was built on the basis of the licensed, specialized, hydrological software package Surface Water Modeling System v.11.1 (SMS) from Aquaveo LLC (Provo, UT, USA), designed for modeling in two-dimensional approximation of hydrodynamics, the distribution of pollution and transport of suspended and entrained sediments in a water body. The model is based on several models (ADH, RMA2, RMA4 (developed by U.S. Army Corp of Engineers), FESWMS (developed by U.S. Federal Highways Administration), TUFLOW (developed by BMT WBM, Teddington, UK), RiverFlow2D (developed by Hydronia LLC, Pembroke Pines, FL, USA), and others) that allow for the solving of various problems. The choice of the licensed model in two-dimensional approximations is due to the signiﬁcant successful experience of its use for solving a wide range of applied problems, including for calculating the thermal pollution zones created by the Perm TPP in the Kama reservoir [25]. This software product allows for fast and accurate hydrodynamic non-stationary calculations using GPU (graphics processing unit) parallelization.

2.2. Model of Unsteady Flow in a Two-Dimensional Formulation of RiverFlow2D, Basic Equations Shallow water ﬂows can be mathematically described by depth-averaged equations of conservation of mass and momentum with all relevant assumptions [29]. The set of “shallow water” equations has the form: ∂ ∂ ∂ (h) + (uh) + (vh) = 0 (1) ∂t ∂x ∂y ! ! ∂ ∂ 2 1 2 σxx ∂ σxy ∂ZD (uh) + u h + gh h + uvh h + gh + ghSx = 0 (2) ∂t ∂x 2 − ρ ∂y − ρ ∂x ! ! ∂ ∂ σyx ∂ 2 1 2 σyy ∂ZD (vh) + uvh h + v h + gh h + gh + ghSy = 0 (3) ∂t ∂x − ρ ∂y 2 − ρ ∂y Water 2020, 12, 1336 4 of 22

where ρ is the water density; g is the gravity acceleration; ZD is the bottom level coordinate; h is the depth; u and v are x and y velocity components in Cartesian coordinate system; σ is the Reynolds stress tensor caused by turbulent and molecular stresses: ! ∂u ∂u ∂v ∂v σxx = 2ρv , σxy = σyx = ρv + , σyy = 2ρv (4) ∂x ∂y ∂x ∂y

S is the slope friction which can be calculated in two forms:

 n2 √u2+v2   n2 √u2+v2   u 3   v 3   C2 √h   C2 √h  S =  D  and S =  D  (5) x  C  y  C   f √ 2 2   f √ 2 2  gh u u + v gh v u + v where C f is the friction coefficient; n is the Manning roughness coefficient and CD is the dimension conversion coefficient (1 for SI units and 1.486 for US units). The numerical solution is carried out by the finite-volume method [30].

2.3. Construction of a Two-Dimensional Hydrodynamic Model of a Section of the Kama Reservoir and Initial Data Assignment A hydrodynamic model of the upper pool of the Kama reservoir was built in the SMS v.11.1software package using the RiverFlow2D GPU model of Hydronia LLC [30], which allows for hydrodynamic calculations of various hydrological conditions. A model is needed to determine changes in the velocity and level regimes of the Kama river, the upper pool of the Kama reservoir, due to the changing discharge flow from the Kamskaya Hydroelectric Power Plant and the assessment of the impact of its work in the area of the water intake channel of the Permskaya TPP. The hydrodynamic model built for the upper pool of the Kama reservoir is shown in Figure1. Characteristic dimensions of the model are the following: the length along the Kama river from the Kamskaya Hydroelectric Power Plant is 82 km, the width varies from 500 to 7000 m. The distance from the dam of the Kamskaya Hydroelectric Power Plant to the thermal pollution study area is ~55 km. To correctly set the morphometry of the water body (building a digital model of the relief of the reservoir), we used official maps of the Kama reservoir and materials of detailed bathymetric surveys of individual sections of the reservoir, performed by us. The regime of water discharge through the dam of the Kamskaya HEPP was used as a model and made up of the characteristic discharge flow rates. The discharge flow rate of 3750 m3/s is the maximum allowed through hydraulic units during routine operation of the HEPP and 300 m3/s is the minimum sanitary pass established for the Kamskaya HEPP. Material processing was carried out in the ArcGIS v.10.4 software package. The data obtained in the form of Esri shapefiles (coasts, contours, depth points) were finally processed and converted into a digital elevation model in the form of TIN (Triangulated Irregular Network). Then, in the SMS v.11.1 software package in a special module “Map”, the obtained data were converted into the internal format of the program for further use in creating the model. Hydrological and meteorological data were obtained from official sources of the Federal State Budgetary Institution Perm TsGMS of Roshydromet and the Kama Hydroelectric Power Plant, a branch of PJSC RusHydro. Their processing was carried out using Microsoft Excel. For the most complete and effective specification of the morphometry of the Kama river area, the upper pool of the Kama reservoir, an unstructured triangular grid was constructed in the computational region using the triangulation method, consisting of 975,157 elements with an average rib length of 30 m. In addition, the initial and boundary conditions were set on the computational region. An initial water level of 108.5 m BS (Baltic Sea) was used as the initial condition for the entire Kama reservoir. Water 2020, 12, 1336 5 of 22 Water 2020, 12, x FOR PEER REVIEW 5 of 22

FigureFigure 1. 1.Morphometry Morphometry ofof thethe consideredconsidered sectionsection ofof thethe KamaKama reservoir.reservoir. TheThe brownbrown rectanglerectangle showsshows thethe area area of of interest interest in in the the vicinity vicinity of of Dobryanka Dobryanka town. town. 2.4. Initial Data for 2D Modeling 2.4. Initial Data for 2D Modeling The following conditions were accepted as the hydrological regime of the Kama river and the The following conditions were accepted as the hydrological regime of the Kama river and the operation of the Kamskaya Hydroelectric Power Plant: operation of the Kamskaya Hydroelectric Power Plant: - The- The flow flow rate rate of waterof water entering entering the the Kama Kama reservoir reservoir (Kama (Kama River—1675 River—1675 m3/s, m Chusovaya—2263/s, Chusovaya—226 m3/s m3/s andand Sylva—159 m 33/s)/s) was was considered considered constant constant at at 2060 2060 m3 m/s3 for/s for 5 days. 5 days. This This is taken is taken on the on basis the of thebasis average of the averageannual annualsummer summer inflow inflow of water of water entering entering the theKama Kama reservoir reservoir along along its mainmain tributaries,tributaries, the Kama, the Kama, Chusovaya Chusovaya and andSylva Sylva rivers; rivers; - The- The discharge discharge flow flow rate rate of of water water at at the the Kamskaya Kamskaya HEPP HEPP (Kama (Kama river) river) for for 5 5 days dayswas wasvaried varied accordingaccording to the to following the following scheme: scheme: the initial the initial discharge discharge flow flow rate rateof 350 of 350m3/s m for3/s 10 for hours, 10 h, thenthen afterafter 2 hours2 h thethe dischargedischarge flowflow raterate roserose toto 3770 3770 m m3/3s,/s, then then within within 10 10 h hours constant constant flow rateflow of rate 3770 of m 37703/s, m3/s,then then decrease decrease in in 2 2 h hours of discharge of discharge flow rate flow to rate 350 to m 3503/s, thenm3/s, againthen again after 10after h rise 10 hours to 3770 rise m3 to/s 3770 and m3/s soand on so within on within 5 days. 5 Adays. two-hour A two-hour drop in drop water indischarge water discharge flow rate flow occurred rate occurred in the time in intervalthe time intervalfrom from 7 to 7 9 to p.m., 9 p.m., and and a rise a rise in water in water discharge discharge flow flow rate rate occurred occurred in the in the time time interval interval from from 7 to 7 to 9 9a.m. a.m. These These flow flow rates rates were were taken taken from from the the conditions conditions of ma maximumximum and minimumminimum loadload onon thethe KamaKama hydroelectric hydroelectric station station in summer in summer conditio conditions.ns. The maximum The maximum and minimum and minimum peaks peaks were weretaken as constants.taken as constants.In real conditions, In real conditions, the discharge the discharge flow rate flow of water rate of during water duringthe day the can day vary can varyquite strongly.quite The strongly. schedule The of schedule water discharge of water dischargeflow rates flowby tributaries rates by tributaries of the Kama of thereservoir Kama and reservoir water dischargeand water from dischargethe Kamskaya from HEPP the Kamskaya is shown HEPPin Figure is shown 2, the inbeginning Figure2 ,of the the beginning day (0 a.m.) of the is taken day as "0";(0 a.m.) is taken as “0”; - The- The flow flow rates rates of of water water intake intake and and discharge discharge at at the the Permskaya Permskaya TPP TPP(the (the KamaKama River,River, citycity ofof Dobryanka)Dobryanka) were were considered considered constant constant and and equal equal to to42.5 42.5 m m3/s3/ sfor for 5 5 days. days. This This value waswas obtainedobtained fromfrom the average the average long-term long-term data data of ofthe the Permskay Permskayaa TPP TPP in in the the summer summer period period with the continuouscontinuous operationoperation of two of twopower power units. units.

Water 2020, 12, 1336 6 of 22 Water 2020, 12, x FOR PEER REVIEW 6 of 22

FigureFigure 2.2.Change Change inin thethe dischargedischarge flowflow rate of water from the Kamskaya Hydroelectric Hydroelectric Power Power Plant Plant (HEPP)(HEPP) and and the the flowflow raterate ofof waterwater inflowinflow intointo thethe KamaKama reservoir. 2.5. Three-Dimensional Hydrodynamical Model 2.5. Three-Dimensional Hydrodynamical Model A three-dimensional numerical simulation of the dynamics of thermal pollution in the process of A three-dimensional numerical simulation of the dynamics of thermal pollution in the process discharge of waste water from the Permskaya TPP, taking into account the variable backwater from of discharge of waste water from the Permskaya TPP, taking into account the variable backwater the Kamskaya Hydroelectric Power Plant, was carried out for a section 11-km long. The calculation from the Kamskaya Hydroelectric Power Plant, was carried out for a section 11-km long. The area included the inlet and outlet channels, which are the main components of the Permskaya TPP calculation area included the inlet and outlet channels, which are the main components of the cooling system. The boundaries of the domain for which three-dimensional modeling was carried out Permskaya TPP cooling system. The boundaries of the domain for which three-dimensional are shown in Figure3 by red lines. modeling was carried out are shown in Figure 3 by red lines. The problem was solved in the framework of a non-stationary non-isothermal approach using the The problem was solved in the framework of a non-stationary non-isothermal approach using k-epsilonthe k-epsilon model model to describe to describe turbulent turbulent pulsations. pulsations. As a comparative As a comparative analysis showed,analysis thisshowed, turbulence this modelturbulence is optimal. model To is evaluate optimal. the To e ffievciencyaluate ofthe its efficiency application, of testits calculationsapplication, weretest calculations performed usingwere a higherperformed order using model—the a higher Reynolds order model—the stress model. Reynolds It was stress found model. that theIt was diff founderence that in the the results difference is no morein the than results 5%, andis no therefore more thethan k-epsilon 5%, and model therefore was usedthe k-epsilon for further model calculations. was used for further calculations.

Water 2020,, 12,, 1336x FOR PEER REVIEW 7 of 22

Figure 3. A map of the morphometric peculiarities of the bottom of the Kama reservoir in the region Figureof the Permskaya3. A map of Thermal the morphometric Power Plant peculiarities (TPP). Red of lines the showbottom the of boundariesthe Kama re ofservoir the computational in the region ofdomain the Permskaya for three-dimensional Thermal Power modeling. Plant (TPP). Red lines show the boundaries of the computational domain for three-dimensional modeling. 2.6. Model of Unsteady Flow in a Three-Dimensional Formulation, Basic Equations 2.6. Model of Unsteady Flow in a Three-Dimensional Formulation, Basic Equations → The equations for the Reynolds averaged velocity uin tensor form were written as [31]: The equations for the Reynolds averaged velocity u in tensor form were written as [31]: ∂ρ ∂ +∂∂ρ (ρui) = 0, ∂t ∂x+=i ρ ()0ui , ∂∂ u ∂ ∂ tx∂p i ∂ ∂ui ∂ j 2 ∂ul (ρui) + (ρuiuj) = + µ + δij + ∂t ∂xj ∂xi ∂xj ∂xj ∂xi 3 ∂xl (6) ∂∂ ∂∂−∂∂∂u − p uuilj 2 ρρ+=−++−+ μ∂ δ∂ui ∂uj 2 ∂ul ()uuuiij ( )  ij+ k + ∂∂tx ∂∂∂∂∂ xxxxx3∂x µt ∂x ∂x 3 ρ µt ∂x δij , (6) jijjilj j i − l

∂ ∂∂uu∂u =  where ρ is the density, ui are the components of the++−+ velocityμρμ vectoril (j i 21, 2, 3), µ is theδ kinematic ttijk , viscosity. Turbulent viscosity µ is a function of kinetic∂∂∂xxx turbulent energy3k and its dissipation ∂ x rate ε: t jji l 2 µt = ρCµk /ε, Cµ is a constant.

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The equations for turbulent kinetic energy and its dissipation rate were written as follows: " ! # ∂ ∂ ∂ µt ∂k (ρk) + (ρkvi) = µ + + Gk + Gb ρε (7) ∂t ∂xi ∂xj σk ∂xj −

" # 2 ∂ ∂ ∂ µt ∂ε ε ε (ρε) + (ρεvi) = µ + + C1ε (Gk + C3εGb) C2ερ (8) ∂t ∂xi ∂xj σε ∂xj k − k where Gk—the generation of turbulent kinetic energy due to the average velocity gradient, Gb—the generation of turbulent kinetic energy due to buoyancy, C1ε, C2ε are constants, σk and σε are the turbulent Prandtl numbers k and ε, respectively. Stratification density effects in the field of gravity due to changes in water temperature were taken into account in the term Gb, having the form: ! µt ∂T Gb = gi β (9) Prt ∂xi

2 where µt is the turbulent viscosity, which is determined as follows: µt = ρCµk /ε where Cµ is the constant. Turbulent heat transfer was modeled using the Reynolds model, in a similar way to turbulent momentum transfer. The equation for energy was as follows: ! ∂ ∂ ∂ ∂T ( ) + [ ( + )] = + ( ) ρE ui ρE p ke f f ui τij e f f (10) ∂t ∂xi ∂xj ∂xj

p where E = ch + ρ is the total energy, h = CpT system enthalpy, ke f f is the effective thermal conductivity, ( ) and τij e f f is the stress tensor deviator, defined as ! ∂uj ∂u 2 ∂u ( ) = + i k τij e f f µe f f µe f f δij (11) ∂xi ∂xj − 3 ∂xk where µe f f = ke f f Pr/Cp is the effective viscosity, effective thermal conductivity is defined as κe f f = κ + Cpµt/Prt, κ is the thermal conductivity coefficient. The applicability of the k ε turbulence model was evaluated in [25,26]. It was found that the − difference in the obtained data is no more than 5%, and therefore, the k ε model was used for − further research. The dependence of density on temperature was described in the framework of the Boussinesq approximation. As initial conditions, uniform distributions of temperature and velocity in the entire liquid volume were set. The values of the parameters of the problem Prt, Sct, G1ε, C2ε, Cµ, σk, σε were taken as follows [31]: Prt = 0.85, Sct = 0.7, C1ε = 1.44, C2ε = 1.92, Cµ = 0.09, σk = 1.0, σε = 1.3. The kinematic viscosity was taken as equal to µ = 9.34 10 7m2/s, the coefficient of molecular diffusion D = 1.0 10 9m2/s. × − × − To discretize the equations in space, a second-order accuracy scheme was used. The description of the dynamics of processes in time was carried out according to an explicit second-order scheme.

2.7. Boundary Conditions for 3D Modeling At the boundaries of the computational domain, the following boundary conditions were imposed: - at the bottom of the river and on its banks, the no-slip conditions and constant temperature were set u1 = u2 = u3 = 0, T = T0; - at the input of the computational domain, the time-variable velocity of the main ﬂow, determined in two-dimensional modeling was set the temperature was set as equal to the background temperature of the river ui = Vi(t), T = T0; Water 2020, 12, x FOR PEER REVIEW 9 of 22

At the boundaries of the computational domain, the following boundary conditions were imposed: - at the bottom of the river and on its banks, the no-slip conditions and constant temperature === = were set uuu1230, TT 0; - at the input of the computational domain, the time-variable velocity of the main flow, Waterdetermined2020, 12, 1336in two-dimensional modeling was set the temperature was set as equal to9 ofthe 22 == background temperature of the river uVtTTii(), 0 ; - in places of water intake and discharge, a constant water velocity and a constant temperature - in places of water intake and discharge, a constant water velocity and a constant temperature == were set: at the inlet of the working channel uVTTi 10, and at the outlet of the working channel were set: at the inlet of the working channel ui = V1, T = T0 and at the outlet of the working uVTT==, ; i channel22ui = V2, T = T2; - -the the upper upper boundary boundary of the of fluid the wasfluid considered was considered free, the windfree, e fftheect waswind taken effect into was account—shear taken into 2 τρ= 2 account—shearstresses were stresses set in were accordance set in withaccordance the formula with τthe= formulaρairCW presentedair CW in presented [32], where inρ air[32],is ρ wherethe airair isdensity, the airC density,is the dimensionless C is the dimensionless coefficient ofcoeffi windcient stress of wind and Wstressis the and wind W velocity is the wind at a velocitydistance at a distance of 10 m fromof 10 them from water the surface. water According surface. According to [32], for to wind [32], velocities for wind from velocities the range from 1 mthe/s 0.5 range< W1 m/s< 15 < m /Ws, the < dimensionless15 m/s, the dimensionless parameter has parameter the form C =has0.0005 the Wform. TheCW= calculations0.0005 0.5 . wereThe 3 made for wind velocity W = 8 m/s, so the= value C = 1.11 10− was used.− For3 the temperature calculations were made for wind velocity Wms8/, so the value× C =×1.11 10 was used. For the on the surface of the water, a linear law of heat transfer was set; taking into account the heating of temperature on the surface of the water, a linear law of heat transfer was set; taking into account the the surface from the surrounding air, the heat transfer coefficient was selected on the basis of heating of the surface from the surrounding air, the heat transfer coefficient was selected on the basis field measurements. of field measurements. 2.8. Construction of a Three-Dimensional Hydrodynamic Model of a Section of the Kama Reservoir 2.8. Construction of a Three-Dimensional Hydrodynamic Model of a Section of the Kama Reservoir The calculations were performed using the ANSYS Fluent computational fluid dynamics package. The calculations were performed using the ANSYS Fluent computational fluid dynamics The computational grid was built, taking into account the morphology of the bottom of the considered package. The computational grid was built, taking into account the morphology of the bottom of the area. The number of grid nodes in the vertical direction was 21, located non-uniformly and taking considered area. The number of grid nodes in the vertical direction was 21, located non-uniformly into account the non-uniformity of the river bottom. In horizontal directions, the grid consisted of and taking into account the non-uniformity of the river bottom. In horizontal directions, the grid quadrangular cells uniformly distributed over the entire domain, with a characteristic linear size of consisted of quadrangular cells uniformly distributed over the entire domain, with a characteristic 20 m. The total dimension of the grid was about four hundred thousand nodes. A volumetric image linear size of 20 m. The total dimension of the grid was about four hundred thousand nodes. A of the constructed mesh is shown in Figure4, where the vertical size for su fficient visualization is volumetric image of the constructed mesh is shown in Figure 4, where the vertical size for sufficient increased forty times. visualization is increased forty times.

Figure 4. Scheme of the computational domain. Computational grid. For sufficient visualization, Figure 4. Scheme of the computational domain. Computational grid. For sufficient visualization, the the vertical size is increased by 40 times. vertical size is increased by 40 times. To solve the problem of the adaptation of existing morphological data in the “coordinate- depth” To solve the problem of the adaptation of existing morphological data in the "coordinate- format to the capabilities of the grid builder, the procedure was used to “divide” the bottom morphology depth" format to the capabilities of the grid builder, the procedure was used to "divide" the bottom into simple geometric objects with some specified resolution. A code was written that allows us, morphology into simple geometric objects with some specified resolution. A code was written that using an array of data describing the morphology of the bottom of a water body, to output a command file for a task for a mesh generator, which is part of the ANSYS Fluent computational package. The created code has a general character and is applicable to the construction of similar geometries in other problems. The methodology for obtaining an acceptable numerical solution with a change in the time step was preliminarily worked out, a solution was obtained starting from a step of 0.001 s and, upon reaching Water 2020, 12, x FOR PEER REVIEW 10 of 22

allows us, using an array of data describing the morphology of the bottom of a water body, to output a command file for a task for a mesh generator, which is part of the ANSYS Fluent computational Water 2020, 12, 1336 10 of 22 package. The created code has a general character and is applicable to the construction of similar geometries in other problems. The methodology for obtaining an acceptable numerical solution with a change in the time step the convergence of the solution with an error of 0.001, the time step increased gradually, reaching a was preliminarily worked out, a solution was obtained starting from a step of 0.001 sec and, upon value ofreaching 10 s. The the total convergence calculation of the time solution for each with variantan error wasof 0.001, five the days. time step increased gradually, reaching a value of 10 sec. The total calculation time for each variant was five days. 3. Results 3. Results 3.1. The Results of Two-Dimensional Numerical Simulation 3.1. The Results of Two-Dimensional Numerical Simulation The results of two-dimensional calculations are presented in Figures5–8. Figures5 and6 show velocity vectorThe fields results in of the two-dimensional area of Dobryanka calculations town are at timepresented 4 days in andFigures 1 h 5–8. from Figures the start 5 and of 6 calculations, show velocity vector fields in the area of Dobryanka town at time 4 days and 1 hour from the start of which corresponds to 1 h at night, and at 4 days and 13 h from the beginning of calculations, calculations, which corresponds to 1 hour at night, and at 4 days and 13 hours from the beginning of which correspondscalculations, which to 1 o’clock corresponds in the to 1 afternoon. o'clock in the As af oneternoon. can As see, one in can the see, upper in the pool upper of pool the of Kamskaya the HEPP, changesKamskaya in HEPP, the magnitude changes in and the directionmagnitude of and the direction flow velocity of the areflow observed, velocity are a backward observed, wavea is formed,backward which aff waveects theis formed, Kama riverwhich evenaffects near the DobryankaKama river even town near (Permskaya Dobryanka TPP).town (Permskaya TPP).

Figure 5.FigureThe velocity5. The velocity vector vector ﬁeld field in in the the vicinity vicinity of Do Dobryankabryanka town town at 4 atdays, 4 days, 13 hours 13 hfrom from the the start start of calculations,Water 2020of calculations,, 12 which, x FOR correspondsPEER which REVIEW corresponds to 1 h to of 1 the hour day of the (stock day (stock ﬂow, calmflow, calm conditions). conditions). 11 of 22

Figure 6.FigureThe 6. velocity The velocity vector vector ﬁeld field in in the the vicinity vicinity of Do Dobryankabryanka town town at 4 at days, 4 days, 1 hour 1 hfrom from the thestart start of calculations,of calculations, which corresponds which corresponds to 1 h to at 1 nighthour at (back night ﬂows,(back flows, calm calm conditions). conditions).

From a comparison of Figures 5 and 6, due to the nonuniform discharge of water through the dam of the Kamskaya hydroelectric power plant, a drastic change in the velocity field is observed in the region of the Permskaya Thermal Power Plant. The flow fields shown in Figure 5 are typical for continuous discharge through dam, the presence of a steady flow from the upper part of the reservoir to the lower near-dam part. With sharp changes in water discharge at the Kamskaya hydroelectric power plant, back flows are observed (Figure 6), which reach up to Dobryanka town, to the water intake channel of the Permskaya TPP. It is interesting to analyze the change in the velocity module in the area of the water intake channel of the Permskaya TPP (Figure 7 (blue line)), in comparison with the change in the discharge flow rate of water through the Kamskaya Hydroelectric Power Plant (green line), and especially in comparison with the velocity module at a constant discharge of water through the Kamskaya HEPP (red line). As follows from Figure 7, three maximums and three minimums are observed during the day. This is due to the fact that the velocity modulus shows both the forward and backward flow of water at the measurement site; therefore, over a wave period of 24 hours, we see three periods of oscillations of the velocity modulus and observe the forward (Figure 5) and backward (Figure 6) flows. Sharp changes in discharge flow rate through the dam of the Kamskaya HEPP, as follows from Figures 5–7, can generate significant back flows in the area of the Permskaya TPP. It is evident that these back flows can have a significant impact on the distribution of temperature fields associated with the discharge of thermal waters from the Permskaya TPP.

Water 2020, 12, 1336 11 of 22

From a comparison of Figures5 and6, due to the nonuniform discharge of water through the dam of the Kamskaya hydroelectric power plant, a drastic change in the velocity field is observed in the region of the Permskaya Thermal Power Plant. The flow fields shown in Figure5 are typical for continuous discharge through dam, the presence of a steady flow from the upper part of the reservoir to the lower near-dam part. With sharp changes in water discharge at the Kamskaya hydroelectric power plant, back flows are observed (Figure6), which reach up to Dobryanka town, to the water intake channel of the Permskaya TPP. It is interesting to analyze the change in the velocity module in the area of the water intake channel of the Permskaya TPP (Figure7 (blue line)), in comparison with the change in the discharge flow rate of water through the Kamskaya Hydroelectric Power Plant (green line), and especially in comparison with the velocity module at a constant discharge of water through the Kamskaya HEPP (red line). As follows from Figure7, three maximums and three minimums are observed during the day. This is due to the fact that the velocity modulus shows both the forward and backward flow of water at the measurement site; therefore, over a wave period of 24 h, we see three periods of oscillations of the velocity modulus and observe the forward (Figure5) and backward (Figure6) flows. Sharp changes in discharge flow rate through the dam of the Kamskaya HEPP, as follows from Figures5–7, can generate significant back flows in the area of the Permskaya TPP. It is evident that these back flows can have a significant impact on the distribution of temperature fields associated with theWater discharge 2020, 12, x ofFOR thermal PEER REVIEW waters from the Permskaya TPP. 12 of 22

FigureFigure 7. 7.Change Change inin thethe velocityvelocity modulemodule in the area of the water intake channel channel of of Permskaya Permskaya TPP TPP (blue(blue line), line), in in comparison comparison withwith thethe changechange inin discharge flow ﬂow rate of of water water from from the the Kamskaya Kamskaya HEPP HEPP (green(green line) line) and and in in comparison comparison withwith thethe velocityvelocity module at a a consta constantnt discharge discharge of of water water through through the the KamskayaKamskaya HEPP HEPP (red(red line).line).

3.2.3.2. ResultsResults ofof 3D3D NumericalNumerical ModelingModeling InIn our our work work [25 [25],], a three-dimensionala three-dimensional numerical numerical simulation simulation of theof the propagation propagation of the of heatthe heat spot spot from thefrom Permskaya the Permskaya TPP was TPP performed was performed for various for meteorological various meteorological conditions andconditions various and technological various parameterstechnological of parameters the TPP at aof constant the TPP ﬂowat a rateconstant of water flow dischargerate of water at thedischarge Kamskaya at the Hydroelectric Kamskaya PowerHydroelectric Plant over Power the entire Plant computationover the entire time. computation In the present time. work, In the we present performed work, three-dimensional we performed three-dimensional numerical simulation of the propagation of heat spot from the Permskaya TPP at time-periodical discharge flow rate at the Kamskaya HEPP in conditions of summer low water. Two variants of wind direction similar to those considered in [25] and additional third variant of calm conditions were studied. For the first variant, the wind is directed opposite to the river flow, the second variant corresponds to the wind in the direction of the river flow and the calm conditions correspond to the absence of wind. The value of the wind velocity modulus at a distance of 10 m from the water surface in the first two variants was 8 m/s. The discharge flow rate of heated wastewater was considered equal to 63.0 m3/s, the temperature of the discharge water was 32.4 °C, and the temperature of the water of the reservoir receiver was 21.8 °C. The calculation time was five days. Based on the simulation results, an assessment was made of the zone of influence of heated water masses on the reservoir at a change in the hydrological regime of the river. The average monthly values of air temperature according to observations at a meteorological station in Dobryanka are presented in Table 1.

Table 1. The average monthly values of air temperature according to observations at a meteorological station in Dobryanka.

Month Value I II III IV V VI VII VIII IX X XI XII Cp. T, °C −13,0 −9,6 −3,0 2,6 10,4 14,8 16,7 15,6 9,5 1,7 −6,0 −9,7 The calculation results for the first variant (southeasterly wind directed against the river flow; the wind velocity is 8 m/s) are presented in Figure 8, which shows the temperature field and the

Water 2020, 12, 1336 12 of 22 numerical simulation of the propagation of heat spot from the Permskaya TPP at time-periodical discharge flow rate at the Kamskaya HEPP in conditions of summer low water. Two variants of wind direction similar to those considered in [25] and additional third variant of calm conditions were studied. For the first variant, the wind is directed opposite to the river flow, the second variant corresponds to the wind in the direction of the river flow and the calm conditions correspond to the absence of wind. The value of the wind velocity modulus at a distance of 10 m from the water surface in the first two variants was 8 m/s. The discharge flow rate of heated wastewater was 3 considered equal to 63.0 m /s, the temperature of the discharge water was 32.4 ◦C, and the temperature of the water of the reservoir receiver was 21.8 ◦C. The calculation time was five days. Based on the simulation results, an assessment was made of the zone of influence of heated water masses on the reservoir at a change in the hydrological regime of the river. The average monthly values of air temperature according to observations at a meteorological station in Dobryanka are presented in Table1.

Table 1. The average monthly values of air temperature according to observations at a meteorological station in Dobryanka.

Month Value I II III IV V VI VII VIII IX X XI XII Cp. T, C 13,0 9,6 3,0 2,6 10,4 14,8 16,7 15,6 9,5 1,7 6,0 9,7 ◦ − − − − −

The calculation results for the first variant (southeasterly wind directed against the river flow; the wind velocity is 8 m/s) are presented in Figure8, which shows the temperature field and the velocity vector field in the surface layer at different time moments. Calculations showed that under the indicated conditions, throughout the entire calculation time (5 days), the warm plume of wastewater moves near the riverbank from the discharge channel to the intake channel and does not change its direction. The wind duration of one direction (over 3 days) was chosen based on the inertia of the hydrodynamic processes under consideration. When the wind lasts for a shorter period, the hydrodynamics of the considered section of the reservoir do not have time to completely rebuild or adapt to a given direction of the wind. After 10 h from the start of the discharge, the warm wastewater enters the intake channel of the TPP cooling system. The movement of the water against the direction of the river flow is observed approximately in a six-meter layer from the surface; in the deeper layers, the water moves downstream (see, Figure9). Thus, the movement of water has a three-dimensional vortex structure, which justifies the need for a three-dimensional approach and limits the solution of the problem within the two-dimensional approach. Water 2020, 12, x FOR PEER REVIEW 13 of 22

velocity vector field in the surface layer at different time moments. Calculations showed that under the indicated conditions, throughout the entire calculation time (5 days), the warm plume of wastewater moves near the riverbank from the discharge channel to the intake channel and does not change its direction. The wind duration of one direction (over 3 days) was chosen based on the inertia of the hydrodynamic processes under consideration. When the wind lasts for a shorter period, the hydrodynamics of the considered section of the reservoir do not have time to completely rebuild or adapt to a given direction of the wind. After 10 hours from the start of the discharge, the warm wastewater enters the intake channel of the TPP cooling system. The movement of the water against the direction of the river flow is observed approximately in a six-meter layer from the surface; in the deeper layers, the water moves downstream (see, Figure 9). Thus, the movement of water has a three-dimensional vortex structure, Water 2020, 12, 1336 13 of 22 which justifies the need for a three-dimensional approach and limits the solution of the problem within the two-dimensional approach.

The place of water intake

The place of water discharge

t = 20 hours Temperature, Time from the start of calculations (t) = 2 hours (increase in discharge at Kamskaya HEPP ° C (at Kamskaya HEPP minimum discharge). within 1 hour).

The place of water intake

The place of water discharge

Temperature, t = 48 hours (2 days from the start of t = 95 hours (3 and 23 hours from the start

° C calculations). of calculations).

Figure 8. Cont.

Water 2020, 12, 1336 14 of 22 Water 2020, 12, x FOR PEER REVIEW 14 of 22 Water 2020, 12, x FOR PEER REVIEW 14 of 22

The place of water The placeintake of water intake

The place of water The place of water dischargedischarge

t = 97 hourst = 97 hours t = 112 hourst = 112 hours Temperature,Temperature, (4 days(4 and days 1 hourand 1 from hour the from start the of start of (4 days (4and days 16 hours and 16 from hours the fromstart ofthe start of ° C ° C calculations).calculations). calculations).calculations).

FigureFigureFigure 8. 8.TemperatureTemperature 8. Temperature fields ﬁelds in fields the in thenear-surface in near-surfacethe near-surface layer layer fo rlayer southeasterly for southeasterlyfor southeasterly wind; wind; the wind; time the from timethe time the from startfrom the startthe start of of calculations (t). The calculations were carried out for the period starting from midnight. calculationsof calculations (t). The (t). calculations The calculations were carriedwere carried out for out the for period the period starting starting from from midnight. midnight.

Velocity, m/s (a) (b) Velocity, m/s (a) (b) FigureFigure 9. 9.TheThe velocity velocity vector vector field field in the in thehorizontal horizontal plane plane near nearthe surface the surface (a) and (a )at and a depth at a depth of 6 of 6 m meters(b) forFigure (b the) for southeasterly 9.the The southeasterly velocity wind. vector wind. The fieldThe time timein after the afte thehor therizontal start start of ofplane calculations calculations near the is is 20surface 20 h.hours. (a ) and at a depth of 6 meters (b) for the southeasterly wind. The time after the start of calculations is 20 hours. FigureFigure 10 10 shows shows the the temperature temperature fields fields for the for case the caseof wind of windat a speed at a speedof 8 m/s, of directed 8 m/s, directed from from thethe northwest. northwest.Figure In In10this thisshows case, case, with the with temperaturestationary stationary backwa fields backwaterter for from the fromthecase Kamskaya of the wind Kamskaya at HEPP a speed HEPP(see, of [25]), 8 (see, m/s, the [ 25 directedheat]), the heatfrom spotspot thepropagates propagates northwest. downstream downstream In this case, of withthe of theriver. stationary river. WithWith backwaunsteady unsteadyter backwater from backwater the Kamskayafrom the from Kamskaya HEPP the Kamskaya(see, HEPP, [25]), the HEPP, heat vortexvortexspot structures structures propagates arise, arise, downstreamwhich which lead lead to of the tothe themove river. movementment With of warm unsteady of warm water backwater watermasses masses against from against thethe river Kamskaya the flow, river HEPP, flow, the area of the heat spot propagation increases, occupying the entire river bed. At some time the areavortex of thestructures heat spot arise, propagation which lead increases, to the move occupyingment of the warm entire water river masses bed. At against some timethe river intervals, flow, intervals, as with the southeasterly wind, the movement of warm wastewater against the river flow as withthe thearea southeasterly of the heat wind,spot propagation the movement increases, of warm occupying wastewater the against entire the river river bed. flow At is some observed. time is observed. The direction of the flow changes in an oscillatory manner. Theintervals, direction as of thewith flow the changessoutheasterly in an wind, oscillatory the move manner.ment of warm wastewater against the river flow is observed. The direction of the flow changes in an oscillatory manner.

Water 2020, 12, 1336 15 of 22 Water 2020, 12, x FOR PEER REVIEW 15 of 22

The place of water intake

The place of water discharge

Temperature, t = 20 hours t = 2 hours ° C (increase in discharge at Kamskaya HEPP (at Kamskaya HEPP minimum discharge). within 1 hour).

The place of water intake

The place of water discharge

Temperature, t = 95 hours t = 48 hours ° C (3 days and 23 hours from the start of (2 days from the start of calculations). calculations).

Figure 10. Cont.

WaterWater2020, 202012, 1336, 12, x FOR PEER REVIEW 1616 of of 22 22 Water 2020, 12, x FOR PEER REVIEW 16 of 22

Temperature, tt = = 97 97 hours hours t =t =112 112 hours hours

°° CC (4(4 daysdays and and 1 1 hour hour from from the the start start of of (4(4 days days and and 16 16 hours hours from from the the start start of of calculations).calculations). calculations).calculations).

FigureFigure 10. Temperature 1010.. TemperatureTemperature ﬁelds fields fields in thein in the the surface surface surface layer layer layer for for for northwesterly no northwesterlyrthwesterly wind; wind; wind; thethe the timetime time from from the the start start start of of of calculationscalculationscalculations (t). The(t).(t). TheThe calculations calculations calculations were were were carried carried carried out out out for for for the the the period period period starting starting starting from from from midnight.midnight. midnight.

FigureFigure 11 shows 1111 showsshows the results thethe resultsresults of three-dimensional ofof three-dimensional three-dimensional calculations calculations calculations for calmfor for calm conditions.calm conditions. conditions. As canAs As becan can seen, be be the vortexseen, the ﬂowsthe vortexvortex obtained flowsflows in two-dimensionalobtainedobtained inin two-dimensionaltwo-dimensional calculations are calculations alsocalculations observed areare in also three-dimensionalalso observedobserved in in calculationsthree-dimensional for the same calculationscalculations time moments. forfor thethe Moreover, samesame time timethe moments. moments. zone of Moreover,warm Moreover, water the the covers zone zone theof of warm entirewarm water water water areacovers corresponding thethe entireentire to waterwater the computational areaarea correspondingcorresponding domain. to to th th Ase ecomputational computational for the southeasterly domain. domain. wind,As As for for therethe the southeasterly issoutheasterly an intense movementwind, therethere of warm isis anan wastewater intenseintense movementmovement in the direction ofof warm warm of wa thewastewaterstewater Permskaya in in the the TPP direction direction intake channel.of of the the Permskaya Permskaya TPP TPP intake channel.channel.

TheThe place place of of water water intakeintake

The place of water The place of water discharge discharge

Temperature, t = 2 hours Temperature, t = 2 hours t = 48 hours t = 48 hours ° C (at Kamskaya HEPP minimum ° C (at Kamskaya HEPP minimum (2 days from the start of calculations). discharge). (2 days from the start of calculations). discharge).

Figure 11. Cont.

Water 2020, 12, 1336 17 of 22 Water 2020Water, 122020, x ,FOR 12, x PEER FOR PEER REVIEW REVIEW 17 17of of22 22

The placeThe of place water of water intake intake

The placeThe ofplace water of water dischargedischarge

Temperature, Temperature, t = 97 thours = 97 hours t = 112t = 112hours hours ° C ° C (4 days(4 anddays 1 andhour 1 hourfrom fromthe start the startof of (4 days(4 days and and16 hours 16 hours from from the thestart start of of calculations). calculations). calculations). calculations). FigureFigure 11. Temperature 11. Temperature ﬁelds fields in the in near-surfacethe near-surface layer layer for fo calmr calm conditions; conditions; the the timetime fromfrom the start start of of Figure 11. Temperature fields in the near-surface layer for calm conditions; the time from the start of calculationscalculations (t). The (t). calculationsThe calculations were were carried carried out out for for the the period period starting starting from from midnight. midnight. calculations (t). The calculations were carried out for the period starting from midnight. FigureFigure 12 shows 12 shows the distributionthe distribution of waterof water temperature temperature over over depth depth at at didifferentﬀerent verticals for for calm calm Figure 12 shows the distribution of water temperature over depth at different verticals for calm conditionsconditions at time at time (t) = (t)97 = h. 97 As hours. can beAs seen,can be at seen, allverticals, at all verticals, the temperature the temperature distributions distributions over over the conditions at time (t) = 97 hours. As can be seen, at all verticals, the temperature distributions over the depth are substantially inhomogeneous. depththe depth are substantially are substantially inhomogeneous. inhomogeneous.

0 H, m

-4

-8 The place of 1 water intake -12 0 H, m 1 -16 4 21.6 22 22.4 22.8 23.2 -4 The place of water T, oC discharge

2 -8 5 0 3 H, m 4 3 -2 -12 5 22.4 22.8 23.2 23.6 24 24.4 -4 2 T, oC

-6

-8

-10 22.4 22.6 22.8 23 23.2 23.4 23.6 Figure 12. The temperature distributions in depth at different verticalsT, oC for calm conditions at t = 97 hours. FigureFigure 12.12.The The temperaturetemperature distributionsdistributions in depth at didifffferent verticals for calm conditions at at t t = 9797 h. hours. To illustrate the effect of the discharge flow rate variation at the Kama HEPP, we present, in To illustrate the effect of the discharge flow rate variation at the Kama HEPP, we present, Figure 13, the temperature fields at 95 hours from the start of the discharge for the case of a variable To illustrate the effect of the discharge flow rate variation at the Kama HEPP, we present, in in Figuredischarge 13, the change temperature in accordance fields atwith 95 Figure h from 7, the and start for ofa constant the discharge discharge for flow the caserate ofequal a variable to the Figure 13, the temperature fields at 95 hours from the start of the discharge for the case of a variable dischargeaverage change flow inrate accordance in Figure 7, with2000 Figurem3/s. As7, can and be for seen a constantfrom the figu dischargeres, under flow all ratewind equal conditions, to the averagedischarge flow change rate inin Figureaccordance7, 2000 with m 3 Figure/s. As can7, an bed seenfor a fromconstant the figures,discharge under flow allrate wind equal conditions, to the average flow rate in Figure 7, 2000 m3/s. As can be seen from the figures, under all wind conditions,

Water 2020, 12, 1336 18 of 22 at a variable discharge flow rate, the thermal pollution spot covers a larger area than at a constant discharge flow rate equal to 2000 m3/s. Figure 13a,b corresponds to calm conditions. It can be seen that in these conditions, under a non- stationary discharge flow rate, a thermal spot propagating against the flow in the reservoir covers the entire computational domain and warm water enters the water intake channel intended for cooling Permskaya TPP blocks. With a constant discharge flow rate of 2000 m3/s, the thermal spot propagates Water 2020, 12, x FOR PEER REVIEW 19 of 23 to a smaller distance upstream, but it does not reach the water intake channel.

The place of water intake

The place of water discharge

(b) (a)

Temperature, Temperature fields in the near-surface layer for calm conditions. ° C

The place of water intake

The place of water discharge

Temperature, Temperature fields in the near-surface layer for southeasterly wind 8 m/s. ° C Figure 13. Cont.

Water 2020, 12, 1336 19 of 22 Water 2020, 12, x FOR PEER REVIEW 20 of 23

The place of water intake

The place of water discharge

(e) (f)

Temperature, Temperature fields in the surface layer for northwesterly wind 8 m/s. ° C

FigureFigure 13. Temperature 13. Temperature fields fields for Wind for Wind SE 8 SE m/ 8s m/s for 95for H. 95 (H.a,c (,ae,)—under c, e)—under the the influence influence of aof dischargea discharge from HEPP,from HEPP, regular regular mode mode in a changing in a changing discharge discharge of water; of water; (b, d(b,f,)—under d, f)—under minimum minimum influence influence of of a a dischargedischarge from HEPP, from HEPP, continuous continuous discharge discharge mode. mode.

At4. aConclusions constant wind exposure in the opposite direction to the flow in the reservoir (Figure 13c,d for the southeasterly wind 8 m/s), both at a non-stationary discharge flow rate and a constant discharge Sharp changes in discharge flow rate through hydroelectric station dams significantly affect the flow rate of 2000 m3/s, the thermal spot propagates against the river flow and warm water enters the hydrodynamic regime of not only the lower, but also the upper pools of the reservoirs. Using, as an intake canal. At a non-stationary discharge flow, the area of the thermal spot is larger; it reaches the example, the Permskaya Thermal Power Plant located in the backwater zone of the Kamskaya intake channel earlier. Hydroelectric Power Plant and using a direct-flow cooling system, the effect of unsteady water At a constant wind exposure in the direction of the flow in reservoir (Figure 13e,f for the northwesterly discharge flow rate through the hydroelectric station dam on the hydrodynamic regime of the upper 3 wind 8pool m/s), of both the reserv with anoir unsteady was studied. discharge rate and a constant discharge rate of 2000 m /s, a thermal spot propagatesTo conduct downstream computational to the Kamskaya experiments, HEPP a combined dam and model warm was water used does and not built enter on the basis water of intakecombination canal. of 2D and 3D models. In the framework of two-dimensional modeling, data were obtained on the evolution of the direction and velocity of flow averaged over the depth at the 4. Conclusions changing flow-rate of the discharged water at the Kamskaya Hydroelectric Power Plant. Vortex Sharpstructures changes of the in velocity discharge field flow near rate the throughPermskaya hydroelectric TPP were found. station Three-dimensional dams significantly numerical affect the hydrodynamicexperiments carried regime out of notvia onlya non- theisothermal lower, but approach, also the taking upper into pools account of the density reservoirs. stratification Using, as an example,effects, showed the Permskaya that a significant Thermal amount Power Plantof warm located water in can the backwaterbe created zonein the of upper the Kamskaya part of the Hydroelectriccooling pond, Power the Plant temperature and using of which a direct-flow is several coolingdegrees higher system, than the the eff backgroundect of unsteady temperature. water discharge flowThe rateaim throughof the present the hydroelectric work was not station a detaile damd onassessment the hydrodynamic of the velocity regime and of temperature the upper pool offields, the reservoir but the study was studied.of the influence of such a factor, which was not previously considered as related to the significant intraday variability of the hydroelectric power plant operation in the To conduct computational experiments, a combined model was used and built on the basis of hydrodynamic regime of the upper pool of a large reservoir. The field expeditionary estimation of combination of 2D and 3D models. In the framework of two-dimensional modeling, data were obtained the spatial–temporal structure of the processes under consideration required a correct comparison on the evolution of the direction and velocity of flow averaged over the depth at the changing flow-rate with the calculated values, which was very difficult due to inertial effects and was characterized by of the discharged water at the Kamskaya Hydroelectric Power Plant. Vortex structures of the velocity significant errors. At the same time, the performed daily observations of flow rates on separate field near the Permskaya TPP were found. Three-dimensional numerical experiments carried out via a measuring verticals located in the upper pool of the reservoir under consideration reflect significant non-isothermaloscillations approach, not only takingin the module, into account but also density in th stratificatione direction of etheffects, flow, showed determined that a significantby the daily amountvariability of warm of water the HEPP can be operation created in and the reproduced upper part ofby thethese cooling models. pond, In connection the temperature with the of whichformer, is severalwe degreesbelieve that higher the thanproposed the background scheme is acceptable temperature. for assessing the possibility of the influence of Theintraday aim of variability the present of workthe HEPP was operation not a detailed on the assessment hydrodynamic of the regime velocity of andthe upper temperature pool of fields,a large but thereservoir, study of and the the influence peculiarities of such of athe factor, water which use by was large not energy previously complexes considered located asthere. related to the significant intraday variability of the hydroelectric power plant operation in the hydrodynamic regime

Water 2020, 12, 1336 20 of 22 of the upper pool of a large reservoir. The field expeditionary estimation of the spatial–temporal structure of the processes under consideration required a correct comparison with the calculated values, which was very difficult due to inertial effects and was characterized by significant errors. At the same time, the performed daily observations of flow rates on separate measuring verticals located in the upper pool of the reservoir under consideration reflect significant oscillations not only in the module, but also in the direction of the flow, determined by the daily variability of the HEPP operation and reproduced by these models. In connection with the former, we believe that the proposed scheme is acceptable for assessing the possibility of the influence of intraday variability of the HEPP operation on the hydrodynamic regime of the upper pool of a large reservoir, and the peculiarities of the water use by large energy complexes located there. We assessed the role of intraday variability of discharge through the Kamskaya Hydroelectric Power Plant dam on the hydrodynamic regime of the dam section, including the location of the Permskaya Thermal Power Plant, located 55 km from the dam. It was shown that the role of this discharge variability is significant, it even increases the probability of such a limiting phenomenon as the entry of warm wastewater into the freshwater intake channel of Permskaya TPP. In the formation of this phenomenon, neither air temperature nor wind speed are decisive factors, since it can be observed quite clearly even under calm conditions. At the same time, with a constant discharge from Kamskaya HEPP, as shown by calculations in our previous work, this phenomenon can occur only with a certain combination of wind direction, speed and duration. This is the principal conclusion of the paper. Such phenomena can also occur in winter, when neither wind speed nor air temperature play any role. In the present paper, this situation was not considered since, in the winter period, the arrival of warm water into the freshwater intake channel of Permskaya TPP does not play a fundamental limiting role. The direction of the flow near the water surface is determined by wind exposure. If the wind is directed against the flow of the river and the duration of its impact is more than a day, then three- dimensional vortices arise, leading to a flow in the direction opposite to the main flow of the river in a 6-meter-deep layer from the surface. In this case, warm water may enter the intake channel designed to cool the thermal power plant systems. It was found that, during the unsteady operation of a hydroelectric station, the arrival of warm water to the intake channel of a power plant can be observed even under calm conditions. Thus, the performed computational experiments showed the significant effect of the non-stationary nature of the hydroelectric power plant operation on the hydrodynamic regime in the upper pool of the reservoir and the peculiarities of water use by the facilities located near it.

Author Contributions: Conceptualization—T.L. and A.L.; methodology—T.L. and A.L.; numerical calculations—Y.P. and A.T.; analysis of the results—T.L., A.L., Y.P. and Y.L.; visualization—Y.P. and A.T.; writing—T.L., A.L., Y.P. and Y.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by RUSSIAN SCIENCE FOUNDATION, grant No. 17-77-20093. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Abbaspour, A.H.J.; Moghimi, P.; Kayhan, K. Modeling of thermal pollution in coastal area and its economical and environmental assessment. Int. J. Environ. Sci. Technol. 2005, 2, 13–26. [CrossRef] 2. Madden, N.; Lewis, A.; Davis, M. Thermal effluent from the power sector: An analysis of once-through cooling system impacts on surface water temperature. Environ. Res. Lett. 2013, 8, 035006. [CrossRef] 3. Hester, E.T.; Bauman, K.S. Stream and retention pond thermal response to heated summer runoff from urban impervious surfaces. J. Am. Water Res. Assoc. 2013, 49, 328–342. [CrossRef] 4. Raptis, C.E.; Pfister, S. Global freshwater thermal emissions from steam-electric power plants with oncethrough cooling systems. Energy 2016, 97, 46–57. [CrossRef] 5. Issakhov, A.; Mashenkova, A. Numerical study for the assessment of pollutant dispersion from a thermal power plant under the different temperature regimes. Int. J. Environ. Sci. Technol. 2019.[CrossRef] Water 2020, 12, 1336 21 of 22

6. Alcamo, J.; Flörke, M.; Märker, M. Future long-term changes in global water resources driven by socio-economic and climatic changes. Hydrol. Sci. J. 2007, 52, 247–275. [CrossRef] 7. Directive 2006/44/EC of the European Parliament. 2006. Available online: https://eur-lex.europa.eu/legal- content/EN/ALL/?uri=uriserv:OJ.L_.2006.264.01.0020.01.ENG (accessed on 4 May 2020). 8. Van Vliet, M.T.H.; Franssen, W.H.P.; Yearsley, J.R.; Ludwig, F.; Haddeland, I.; Lettenmaier, D.P.; Kabat, P. Global river discharge and water temperature under climate change. Glob. Environ. Chang. 2013, 23, 450–464. [CrossRef] 9. Ptak, M.; Sojka, M.; Kału˙za,T.; Choiński,A.; Nowak, B. Long-term water temperature trends of the Warta River in the years 1960–2009. Ecohydrol. Hydrobiol. 2019, 19, 441–451. [CrossRef] 10. Webb, B.W.; Nobilis, F. Long-term changes in river temperature and the influence of climatic and hydrological factors. Hydrol. Sci. J. 2007, 52, 74–85. [CrossRef] 11. Ptak, M.; Sojka, M.; Choiński,A.; Nowak, B. Effect of environmental conditions and morphometric parameters on surface water temperature in Polish lakes. Water 2018, 10, 580. [CrossRef] 12. Ling, F.; Foody, G.M.; Du, H.; Ban, X.; Li, X.; Zhang, Y.; Du, Y. Monitoring thermal pollution in rivers downstream of dams with Landsat ETM+ thermal infrared images. Remote Sens. 2017, 9, 1175. [CrossRef] 13. Issakhov, A. Mathematical modeling of thermal process to aquatic environment with different hydro meteorological conditions. Sci. World J. 2014, 2014, 678095. [CrossRef][PubMed] 14. Issakhov, A. Mathematical modeling of the discharged heat water effect on the aquatic environment from thermal power plant. Int. J. Nonlinear Sci. Numer. Simul. 2015, 16, 229–238. [CrossRef] 15. Miara, A.; Vorosmarty, C.J. A dynamic model to assess tradeoffs in power production and riverine ecosystem protection. Environ. Sci. Process. Impacts 2013, 15, 1113–1126. [CrossRef] 16. Stewart, R.J.; Wollheim, W.M.; Miara, A.; Vörösmarty, C.J.; Fekete, B.; Lammers, R.B.; Rosenzweig, B. Horizontal cooling towers: Riverine ecosystem services and the fate of thermoelectric heat in the contemporary Northeast US. Environ. Res. Lett. 2013, 8, 025010. [CrossRef] 17. Miara, A.; Cohen, S.M.; Macknick, J.; Vorosmarty, C.J.; Corsi, F.; Sun, Y.; Tidwell, V.C.; Newmark, R.; Fekete, B.M. Climate-Water Adaptation for Future US Electricity Infrastructure. Environ. Sci. Technol. 2019, 53, 14029–14040. [CrossRef] 18. Issakhov, A.; Zhandaulet, Y. Numerical Study of Technogenic Thermal Pollution Zones’ Formations in the Water Environment from the Activities of the Power Plant. Environ. Model. Asess. 2020, 25, 203–218. [CrossRef] 19. Durán-Colmenares, A.; Barrios-Piña, H.; Ramírez-León, H. Numerical Modeling of Water Thermal Plumes Emitted by Thermal Power Plants. Water 2016, 8, 482. [CrossRef] 20. Hester, E.T.; Doyle, M.W. Human Impacts to River Temperature and Their Effects on Biological Processes: A Quantitative Synthesis. J. Am. Water Resour. Assoc. 2011, 47, 571–587. [CrossRef] 21. Issakhov, A. Mathematical modeling of the discharged heat water effect on the aquatic environment from thermal power plant under various operational capacities. Appl. Math. Model. 2016, 40, 1082–1096. [CrossRef] 22. McGuirk, J.J.; Rodi, W. A depth-averaged mathematical model for the near field of the side discharge into open-channel flow. J. Fluid Mech. 1978, 86, 761–781. [CrossRef] 23. Park, S.W.; Chung, M.K. Prediction of 2-dimensional unsteady thermal discharge into a reservoir. J. Korean Soc. Mech. Eng. 1983, 7, 451–460. 24. Lyubimova, T.; Lepikhin, A.; Konovalov, V.; Parshakova, Y.; Tiunov, A. Formation of the density currents in the zone of confluence of two rivers. J. Hydrol. 2014, 508, 328–342. [CrossRef] 25. Lyubimova, T.; Lepikhin, A.; Parshakova, Y.; Lyakhin, Y.; Tiunov, A. The modelling of the formation of technogenic thermal pollution zones in large reservoirs. Int. J. Heat Mass Transf. 2018, 126 Pt A, 342–352. [CrossRef] 26. Lyubimova, T.; Lepikhin, A.; Parshakova, Y.; Lyakhin, Y.; Tiunov, A. Application of hydrodynamic modeling in 2D and 3D approaches for the improvement of the recycled water supply systems of large energy complexes based on reservoirs-coolers. Int. J. Heat Mass Transf. 2019, 140, 897–908. [CrossRef] 27. Faulkner, A.; Bulgin, C.E.; Merchant, C.J. Coastal Tidal Effects on Industrial Thermal Plumes in Satellite Imagery. Remote Sens. 2019, 11, 2132. [CrossRef] 28. Pivato, M.; Carniello, L.; Viero, D.P.; Soranzo, C.; Defina, A.; Silvestri, S. Remote Sensing for Optimal Estimation of Water Temperature Dynamics in Shallow Tidal Environments. Remote Sens. 2020, 12, 51. [CrossRef] Water 2020, 12, 1336 22 of 22

29. Vreugdenhil, C.B. Numerical methods for shallow-water ﬂow. In Water Science and Technology Library; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1994; Volume 13. 30. Reference Manual “RiverFlow2D Two-Dimensional River Dynamics Model” August, 2016, Hydronia LLC. Available online: http://www.hydronia.com/riverﬂow2d (accessed on 4 May 2020). 31. Launder, B.E.; Spalding, D.B. Lectures in Mathematical Models of Turbulence; Academic Press: London, UK; New York, NY, USA, 1972; 169p. 32. Wu, J. Wind stress and surface roughness at sea interface. J. Geophys. Res. 1969, 74, 444–453. [CrossRef]

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