FORECAST OF WATER TEMPERATURE IN RESERVOIR BASED ON ANALYTICAL SOLUTION

507Available online at www.sciencedirect.comScienceDirect' HHDJoumal of Hydrodynamicswww. sciencedirect com/EL SEVIER2008,20(4):507-513science/joumal/10016058FORECAST OF WATER TEMPERATURE IN RESERVOIR BASED ONANALYTICAL SOLUTION"JI Shun-wen, ZHU Yue-ming, QIANG ShengCollege of Water Conservancy and Hydroelecric Enginering, Hohai University, Nanjing 210098, China, E-mail:jishunwen@hhu.edu.cnZENG Deng-fengWater Power Plant of Sanhe Shangyoujiang, Ganzhou 341212, China(Received June 4, 2007, Revised August 20, 2007)Abstract: The water temperature in reservoirs is dificult to be predicted by numerical simulations. In this article, a statitical modelof forecasting the water temperature was proposed. In this model, the 3-D thermal conductiondiffusion equations were convertedinto a system consting of 2-D equations with the Fourier expansion and some hypotheses. Then the stistical model of forecasingthe water temperature was developed based on the analytical solution to the 2-D thermal equations. The simplified stistial modelcan elucidate the main physical mechanism of the temperature variation much more clearly than the nuimerical simulation with theNavier Stokes equations. Finally, with the presented statistical model, the distribution of water temperature in the Shangyoujiangreservoir was determined.Key words: water temperature, quasi 3-D statistical model, thermal conduction difusion equation, analytical solution, regressionanalysis, factor of radiation1. Introductionparameters into account, in the 1-D statistical modelThe major methods of forecasting the watersome different parameters were introduced for thetemperatures in reservoirs generally include empiricalempirical formulas, and the water temperature wasformulas, statistical models and numerical simulations.calculated with12].Under the assumption of periodical boundaryconditions and with the 1-D thermal equations inT(z,l)=Ce"+C2e" cos(ox-C-c2) (1)hydrodynamics, the empirical formulas presented inRef.[1] can be employed to forecast the distribution ofFurthermore, with the development of the FDMthe water temperature along the depth. However, theand FEM， numerical simulations have beenuse of the empirical formulas is limited to the flows inextensively used to calculate the distributions of thereservoirs and the method may introduce errors ofwater temperature by solving the vortex motionaliasing. In order to overcome the shortcomings ofequations of incompressible viscous fluid. Comparedempirical formulas, recently, some researchers tried towith empirical formulas and statistical models, thestudy and solve this problem from different aspects. Liamount for the numerical work is quite large. Inet al proposed a 1-D statistical model 4. Throughaddition, when there is numerical dispersion ortaking the influences of the flow pattern and the otheroscillation due to some improper treatments about theconvection terms, the prediction quality would b* Project Supported by the National Natural Sciencemuch中国煤化工- rical formulas andFoundation of China (Grant Nos. 50539010, 50539020, andstatist50579080).THCN MH Gcle, a quasi 3-DBiography: JI Shun-wen (1981- ), Malc, Ph. D. Candidatestatistican Houel IU1 uc waci temperature in508reservoirs is established, and some mathematicaldifusion equations could be converted into thedescriptions of the factors, such as the solar radiationfollowing combination of a 2-D thermal equation andand diffusivity, are given. In its second part, somea 1-D thermal equation which can be solvedtests are outlined to validate the model.individually I7:oT. JT_l 子2T。。子2T2. Governing equations+1=K-+h-+Sz(3)2.1 Thermal convectiondiffusiort equationdy)zay2)z2The 3-D thermal convection-diffusion equationswith a source term are defined as [B-6)OT+n，AT_, dT(4)ddz2aT.. T. aT . aT_n 子2TAtdx dy0zdx^2.3 Radiation factorAssume that the source term in Eq.(2) isindependent of time, and then the water temperatureaT. ,子2TJz2-+S,can be considered to reach the equilibrium state ofDy"heat exchange and satisfy Eq.(3). Based on Eq.(3), avertical 2-D numerical model for the solar radiation isT=T。+T。cosor, z=0, t20(2)established, and some mathematical descriptions ofhe factors, such as the source term and the termsrelated to the convection-diffusion, are obtained.where T is the water temperature, I is time, u,VMoreover, it is found that about 80% of theand W are respectively the components of velocityradiation could be absorbed by the superficial layer ofvectors, S denotes the source term, K=v,/σ, isthe water, and only about 5% could reach the depth ofthe thermal conductivity, v, stands for the ceofficientm, that is, the solar radiation decreasesexponentially with the depth of the water. So theof eddy viscosity, and σ, denotes the turbulentshort-wave radiation possesses the penetratingPrandtl number, T。 denotes the annual averagecapacity in water included in Eq.(3) and the sourcetemperature of the water, T is the annual variableterm can be defined as "amplitude of the temperature, 0 is the frequency.In general, the water temperature varies veryS(y,z)=e"g(y)(5)slightly along the dam axis, which meansaT/dx= 0.Firstly, for the bottom of the water, appropriateSo formeeting the requirements of predictionprecision in engineering, the 3-D thermnalconditions have to be applied to solve Eq.(3):conduction-diffusionequationscouldbeapproximately described with a system of 2-Dlim T(y,z)=0，lim S(y,z)=0(6)I→+∞.2- ++∞problems.where both of T(y,z) and S(y,z) are someSource tem:Sze* 'g(v)periodic functions in the y direction. Then, they业Boundary: T=T。+T。cosoutfollowing equations can be obtained by expandingg(y) and T(y,z) into the Fourier series in the ydirection:Reservoirg(y)=2+2.A, cos nTY+B, sinhty，Fig.1 Conventional diagram for 2-D thermal cquations2.2 Decomposition of the thermal convection-diffusionnty +T(y,z)=+ Z[C,(z)cosAs is shown in Fig.l, the water temperature ismainly influenced by solar radiation and air中国煤化工temperature on the surface. Therefore, under theassumptions of the source term and the periodicMYHCNMHG(7)boundary condition, the 3-D thermal convection-509Secondly, substiuting Eq.(7) into Eq.(3) gives8-101.0C;(z)-f{C,(z)-fiC,(z)-fJsD,(z)=kμ2+ uWD。=0 (n=0)Finally, for simplifying the above analyticalAsolutions, the solutions of order one should also bekgiven, and then T(y,z) can be defined asD;(z)-. fD,(z)- fD,(z)+ fzfsC,(z)=nTynityT(y,z)=an cos一+ azn sinB,=0lyl)e-" (n≠0)(8)-)+where f=w/k， fi=m/l, and f.=v/k. UnderT,the assumption that v,w and k in Eq.(8) areconstants, the following 4th-order differentialequations can be resulted:a4n sin(b,z-)] .(11)C,"(z)-2fC"(z)+(f2-2f)C;(z)+where am， b。and c, are functionsof v,W and ,，2f$?"C.(z)+(fa* +f2f2)C,(z)=1, indicates the periodof T(y,z).2.4 Time evolutionSince the flow velocity at inlet is small and the(u2A, +HufA,-f2A, +ffB,)heat exchange along the depth is weak in reservoirs,the distribution of water temperature should belayered along the depth, namely,. the. waterD,"(z)-2fD,(z)+(f -2f5)D;(z)+temperature at the same depth varies consistently withtime. Therefore, the time evolution can be describedby the following equations under the considerations of2ff*D,(z)+(2* +fh52)D,(z)=the diffusion along the depth and the neglect of theradiation term:(uPB, +ufB-f3B-554.)元eH" (9)oT8T_。HTThe particular solutions of the above equationsT=T +T。cosot, z=0, t20(12)are C,(z)=C,e-" and D,(z)=D.el". And then byintroducing the far field conditions, the generalIn order to solve Eq. (12), T should be firstlysolutions can be expressed asrepresented asC,(z)=an,e"H +lw2-)*(asn cosb,z+T=T。+T'.e长别(13)a4n sinb,z),And then Eq. (12) can be simplified asD,(z)= az,e-"+elw2k-o)(asn cosb,z-dr'_ L 子TT' = T, cos wtoea3n sinb,z) (n≠0)中国煤化工TYHCNMHG(14)510Finally, with the Laplace transformation, Eq. (14)are solved, the coefficients of diffusion andcan be solved, and T can be expressed as 1-5convection are also assumed to be constant instead ofbeing the functions of y and z. Besides, due to theT(z,1)=T, +-2E1-1-101. .high correlation between these factors, somesimplification of the factors could also be given.Assumethat (w/2k-a,)z ，-wz/2k-cos ot -V2o(15)(2f(w,k,a)z/4k . And V2z/f(w,k,0) in Eq.(18)f(w,k,@)are the functions of z and can be expanded in theTaylor series, the simplified 3~D statistical model canf(w,k,o)=√w2 +vw*+16k'a3(16)reduce computing time. and avoid themulti-collinearity of factors [14. Finally, the statisticalmodel of water temperature can be represented asFurthermore, to ensure that T is related withhe variables time and y and z，the fllowingT(C,z,t)= C。+Ce-"* +Ce@+aqz+a?".statistical model is suggested:T(y,z,l)= aT(y,z)+ βT(z,t)(17)cosbz+Ce*+ax+ag2 sinb,z+where T(y,2) denotes the water temperature inCe*antar cos(bgz +C})+Ceagaxaiz".two-dimensional space influenced by solar radiation,T(z,1) the water temperature impacted by airsin(bz+C,)+Ce+atogr.temperature, and a,B the weighted coefficients.As a result, depending on the physical parametersin Eqs.(11) and (15), the statistical model ofcos(or+d, +dz+d2z2)(19)forecasting the water temperature can be defined asEq.(18) with considering the factors ofwhere C，a，b，C and d; are regressiveconvection-diffusion and radiation:coefficients.In order_ to get the cofficients in Eq.(19) andT=T+I2 coso(t-E),ensure the. efficiency of estimating the ceffcients,the cofficients in Eqs.(1) and (19) are revised throughT=Co+e-"(C +Cz cosOy+ C; sin6y)+the regression analysis based on the measurementsh'5].As a result, the statistical models are defined asCelw2k-a); cosbqz + Celw2k-n); sinb,z+T(z,t)= 24.33e 0148 -10.29e 0102.2cos(0.5231 -0.547 - 0.0355z)(20)sin(bz + y),T(C,z,t)=4.18+ 20.12e 0+78.5-6-5.1.959-0.06162 cos5.02z -√2z0.29-32840.72 000sin3.97z+(18)030-22.122 0013)2cos(2.42z -3.82)+3. Statistical model based on analytical solution中国煤化工:z - 3.83)+Under the assumption that y in Eq. (18) is aconstant, the 2-D thermal equations is thus turmed intoCNMHGa 1-D equations. Furthermore, when Eqs.(3) and (4)14.89e w.- w cos(0.523t -511relatively high density. So heat diffuses from the3.681-0.0428z + 0.0001z2)(21)surface to the bottom, and the water temperaturedecreases linearly or nonlinearly with the depth due toBased on Eq.(1)， both the average waterthe increase of diffusion coefficient. In addition,temperature and the amplitudes of water temperatureduring the warming and cooling periods, such asin Eq.(20) can be regarded as a Ist-order exponentialspring and autumn, the water temperature in reservoirsfunction of z . Similarly, by considering the effects ofhas a distribution in obvious stratification form. Thedistributions in the upper layers are obviouslyconvection and diffusion, the average waterdifferent from those in the bottom layers. And then thetemperature and amplitude of water temperature in Eq.buffer layer moves to the bottom as time passes 0.(21) can be expressed by a 2nd-order exponentialCormpared with the quasi 3-D model, thefunction.disadvantage of the 1-D model lies in neglecting theAs is ilustrated in Fig.2, the results from the 1-Dconvection in the y direction. That is why the curvesmodel and the measurement are compared with theof the 1-D model have large remnant values. Whileresults from the quasi 3-D model for 6 differentfor the quasi 3-D model, the remnant values betweenmonths. The results show that, in general, there is ane quasi 3-D model and the measured data arefairly good agreement between the quasi 3-D modelrelatively much smaller by taking into account theand the measurements.factors of diffusion- convection and radiation,especially in the buffer layer with acute convections.Furthermore，analyzing the regularities of190190[。中temperature change at different elevations leads to80-s0discovering two opposite phenomena. On one hand,星170although affected by air temperature and radiation, theresults of the quasi 3-D model in the upper water are60 t160well ftted with the measurements. As is shown in1501020 5 10152025 30Fig.3, the predicted values of water temperature in theupper layer fit a sinusoidal function with large(a) Feb.(b)Apr.amplitude. As is ilustrated in Fig.4, in the buffer layer200where diffusion and convection of heat are strong, the90results of the quasi 3-D model can also be welle180g180consistent with the measurements. On the other hand,E170二170mainly influenced by the temperature of the60ffoundation and water around, the temperature change50ks0Lof the bottom layer cannot be well expressed as a101520253035101520253035sinusoid, so the models only provide a relatively poor(c) June(d) Aug.fitting to the data measured (Fig.5).90卜804. Conclusions工170217Based on an analytical solution to the 3-Dthermal convection-diffusion equations, a quasi 3-Dstatistical model of predicting the10152025301020reservoirs has been established. Due to the restriction(n Dec.of solving conditions, the velocity of flows ar- 0 Mceasured个- T(21)一0 TC21)assumed known, which meansu,v and w in ththermal convection-diffusion equations are constants.Fig.2 Comparison between computational data andAt the same time, by considering the high correlationsmeasurementsbetween the velocity fields and coordinates, thevelocity variables can be expanded in the TaylorIn winter, warm water near the bottom flowsseries by assuming that they are the functions of theupwards, and cool water at the top runs downwards.coordinate variables. Then the coefficients and aSo the water temperature does slightly changestatistical model with high accuracy are obtainedvertically, which means the vertical convection makesthe boundary conditions obscure. As a result, the中国煤化工; 3-D model is bettertemperaturestratificationbecomesindistinct.con.Y片C N M H Gements than the 1-DContrarily, in summer, the convection coefficientsmodel, by introducing tactors ot convection-diffusiondecrease because water with low temperature has51240 [30 tξ 200Mar.JuneSepl.Dee.1997199900020020032004Date .Fig. 3 Variation of water temperaure at elevation of 180.0 m0r-0- - Measured- 0- 7(Cz0)20 tgf1o个日AAV1a0Sept.May2000DateFig. 4 Variation of water temperature at elevation of 170.0 m- o Masured- 0 - IC2)台BAaAgASept,Dec.19982001Fig. 5 Variation of water temperature at elevation of 160.0 mand radiation. The increase in the number of factorsgives a better agreement between the measurementsand the simulated results. Nevertheless, only thReferencesstatistical model with the factors of y,z and t isinvestigated because the cefficients of diffusion and[1ZHU Bo-fang. Thermal stresses and temperatureconvection have significant relations with y and z.control of mass concrete[M]. Bejing: China ElectricPower Press, 1999 (in Chinese).Finally, the comparisons prove that the application of[2] LI Lan-tao, LUO Wen-sheng and LI Jin-jing. Study onthe quasi 3-D statistical model should be able towater temperature forecasting of Danjiangkou reservoirachieve a better simulation effect than that of the 1-Dnear upstream dam face after dam height increasing [].model.中国煤化工-106, 24(1): 52-54(inYHCNMHG513[3] LU Chang- gen, CAO Wei-dong and QIAN Jian-hua. A58(4): 567-569.study on numerical method of N-S equations and[l0] MULLINS W. W.. SEKERKA R. J. Stability of a planarnonlinear evolution of the coherent structures in ainterface during solidification of a dilute binary alloy[]hon!neapoiralanrlaverlaminar boundary layer [J]. Journal of Hydrodynamics,J. Appl. Phys, 1964, 35(2): 444-451.Ser. B.2000 18(3; 37371.. si.nhe，nal[1] GAO Zhi qiu, BIAN Lin-gen and ZHANG Ya-bin et al.[4] DEND Yun, LI Jia and LUO Liration onStudy on analytical resolution to soil thermal conductivestratification of the river-like deep reservoir [J]. Journalequation and soil thermal diffusivity over nagqu aera[J].of Hydrodynamics, Ser. A, 2004, 19(5): 604-609 (inActa Meteorologica Sinica, 2002, 60(3): 352-359 (inChinese). .Chinese).[5] QIU Chen-xia, WANF De-guan and XU Xie-qing. A[12] GAO Zhi-qiu ,WANG Jing-xing. Calculation of near-3-D numerical method for stratified flow with k-E modelsurface layer turbulent transport and analysis of surface[J]. Journal of Hydraulic Engineering, 1997, (7): 7-12thermal equilibrium features in Naqu of Tibet [J]. Phys.Chem. Earth, Ser. B, 2000, 25(2): 135-139.hinese).[6] ZHANG Jing-xin, LIU Hua. A vertical 2-D numerical[13]LU De-tang, ZENG Yi-shan and GUO Y ong-cun.llbr andsimulation of suspended sediment transport [J]. JournalAnalytical solution of temperature in: anof Hydrodynamics, Ser. B, 2007, 19(2): 217-224.formation in multilayer[J]. Journal of Hydrodynamics,[7] LARDNER R. W.,. SONG Y. Algorithm for threeSer. A, 2002, 17(3): 382-390 (in Chinese).dimensional convection and diffusion with very different[14] WU Zhong-ru, SHEN Chang -song and RUAN Huan-horizontal and vertical scales[J]. International Journalxiang. Safety monitoring theory and its application infor Numerical Methods in Engineering, 1991,hydraulie structures[M]. Nanjing: Hohai University32(6):1303-1319.Press, 1990 (in Chinese).[8]SU Pei,SHI Ying-ying and SUN Ren-ji et al.[15] RICHARD A.， DEAN W. Applied multivariateTwo-dimensional temperature fluctuation model in astatistical analysis[M]. New Jersey, USA: Prentice Hall,molten liquid with heat source[J]. Journal of Yantai000.Normal University (Natural Science), 2006, 22(1): 3-9[16] JU Shi-quan, su Huai-zhi and HOU Yu-cheng et al.(in Chinese).Calculation methods of forecasting reservoir water[9IVANTSOV G. P. Temperature ficld around atemperature[J]. Water Resources and Power, 2004,spheroidal, cylindrical and acicular crystal growing in22(3): 74-77 (in Chinese).suppercoolied melt [J]. Dokl, Akad Nauk SSSR, 1947,中国煤化工MYHCNMHG