NUMERICAL ANALYSIS OF HE IMAC OF HE INLE AND OULE JES FOR HE HERMAL SRAIFICAION INSIDE A SORAGE ANK A. Zachár I. Farkas F. Szlivka Deartment of Comuter Science Szent IstvÆn University Æter K. u.. G d llı H-03 Hungary +36-8-505 +36-8-40804 azachar@mszi.gau.hu Deartment of hysics and rocess Control Szent IstvÆn University Æter K. u.. G d llı H-03 Hungary +36-8- 5055 +36-8-40804 ifarkas@fft.gau.hu Deartment of Fluid Machinery Szent IstvÆn University Æter K. u.. G d llı H-03 Hungary +36-8-5047 +36-8-40804 szlivka.gurt@mgk.gau.hu Abstract In this aer a numerical analysis has been carried out in order to describe the velocity and temerature distribution inside a storage tank under various boundary conditions with inlet and outlet flows. he aim of the study is to evaluate more recisely the imact of the inlet jet for the stratification. he aer resents the changing of the thermal stratification of water from various initial temerature field conditions. Vertical inlet and outlet jets were studied to examine the imact of the jet for the temerature rofile of the water along the vertical direction. he initial temerature rofile was linear or constant everywhere and the temerature difference between the to and bottom art of the tank was about 40-50 o C. Keywords natural convection thermal stratification solar thermal system Boussinesq aroximation. INRODUCION he thermal stratification is an imortant factor of hot water storage tanks because the calculated system erformance can be imroved by 35-38% in comarison with a system having a fully mixed tank. Detailed descrition of the thermal stratification in solar water heating systems can be read in (Duffie and Beckmann 980). A comrehensive review can be found about the low-flow solar water heating systems in (Hollands and Lightstone 989). Comrehensive descrition of the numerical treatment of the roblem described below can be found in (orrance and Rocket 969) (orrance 968) and (Stoyan and ak 997). hermal stratification in storage tanks deends mainly on the flow rates of the entering and leaving fluid streams the size and location of the inlets and outlets and the volume of the tank. he main destratification factor in a thermal storage tank is the mixing introduced by the inlet and outlet flows of the fluid (Kleinbach et al. 993). he aim of the current research is to develo models to redict the evolution of the temerature field of the water inside a storage tank under symmetrical inlet and outlet flows and to find a flow rate limit to reserve the stratification inside the storage tank. he symmetrical means in this context that the inlet and outlet are located centrally on the to and bottom region of the tank. It makes it ossible to use a two dimensional aroach to model the fluid motion inside the tank. he fluid is rest initially in the entire calculation domain and there are not inlet and outlet fluid flows initially. Boussinesq fluid roerties were suosed to calculate the natural convection of the water. he hysical situation and the geometrical size and arrangement of the storage tank can be seen in Fig.. he following geometrical sizes and mass flow rate were secified: L=0.8 m d =0.4 m d in =0.05 m; d out =0.05 m; Q=0.000 m 3. Fig.. Geometrical scheme of the storage tank
. MAHEMAICAL FORMULAION wo-dimensional mathematical formulation was used to calculate the velocity and temerature field. he momentum equation of the fluid is the two dimensional Navier-Stokes equations and a no-sli boundary condition is suosed on the wall because of the viscosity. Vorticity streamline formulation is used to solve the Navier-Stokes equations and the continuity equation. his section includes the basic equations that must be solved to describe the velocity field and the temerature distribution inside a storage tank. he hysical roblem is an unsteady two dimensional axisymmetric cylindrical geometry of an enclosure. A schematic diagram of the system indicating the geometrical dimensions and the grid generated to discretize the comutational domain is shown in Fig.. According to the geometrical symmetry of the roblem the calculation was carried out between the centerline of the tank and the hysical boundary. he deendent variables that describe the resent flow situation are: the temerature the velocity comonent in the radial direction U and the velocity comonent in the vertical direction V... Conservation equations he following set of artial differential equations describes the flow and temerature field in storage tanks.... Continuity equation he continuity equation is formulated in the following manner in cylindrical coordinate system U + V U + r = 0. () Equation () can be alied to incomressible fluids it means that ρ( r y t) = const.... Momentum equations he following equation system is the reresentation of the momentum equations in cylindrical coordinate system where the r means the radial and y means the vertical direction. U U U + U + V = f r () t ρ ρ U U U U + ν + + r r V V V + U + V = f Y (3) t ρ ρ V V V + ν + +. r In equation () and (3) ν is the laminar kinematic viscosity of water and f X and f Y are the body forces...3. Heat transort equations wo dimensional convection-diffusion equation was solved to calculate the temerature field t λ ρ c + U + V = + + r y where λ f is the laminar thermal diffusivity of the fluid... Vorticity streamline formulation (4) First the vorticity equation is derived from equation () and (3). he first equation is derived by y and the second one is derived by r and the second was substracted from the first. he following notation will be introduced for the vorticity U := V r (5) to simlify the vorticity transort equation. he mathematical form of the vorticity is the same in both equations. U + U + V = (6) t r r y [ ] rot f + φ ν + + r r ρ Equation (6) is the vorticity transort equation but an additional equation is needed to calculate U and V in equation (6). It will be an ellitic artial differential equation for the stream function. U Ψ Ψ = V = r r U and V ut into equation (5) the following equation can be formulated Ψ Ψ Ψ + = r. (7) r
he Boussinesq aroximation was alied to exress the buoyancy term where β is the coefficient of the volumetric thermal exansion and g is the acceleration of gravity. he following density function is a simle formulation of the Boussinesq aroximation ρ( ) = βρ 0 ( i ) where i is an initial temerature. [ rot f ] ρ( ) = g = gβ ρ φ ρ x 0 0.3. Initial and boundary conditions Initial and boundary conditions of the conservation equations and the domain of discretization are detailed in this section. An 80 times 80 grid system is used to calculate the velocity and temerature field in the tank and 6 additional grid lines are used horizontally for the outlet ie. It can be seen in Fig.. x (8) calculate a more accurate velocity rofile near the outlet osition of the tank..3... Boundary conditions of the temerature field Aroximately insulation can be suosed at each side of the tank and to consider this fact a second kind boundary condition was secified. he heat flow across the wall of the tank deends uon the ambient temerature of the surroundings and the convective behaviour of the surrounding air but the cooling of the inner fluid is much slower because of the heat loss than the imact of the inlet jet for the temerature rofile. An additional exlanation is needed for the calculation of the temerature field along the centerline because singularity arises (Smith 978). If the roblem is symmetrical with resect to the origin the boundary condition will be = 0 at r=0. he boundary condition of the centerline is calculated the following manner to avoid the indeterminate form 0 0 that comes from r. By Maclaurins exansion of 978) at r=0 so the limiting value of r to zero is the value of r (Smith as r tends at r=0. Hence equation (4) at r=0 can be relaced by t λ U V + + = + ρ c y. Fig.. Grid system of the storage tank.3.. Initial conditions he initial velocity field is zero everywhere in the domain of interest. It means that the and Ψ are also zero everywhere. he velocity and the temerature of the inflow are secified from the first calculated time ste. In the first case the initial state of temerature field to be considered a constant everywhere inside the tank and ie and a linear initial rofile is used in the second case..3.. Boundary conditions Uniform constant velocity rofile was suosed at the inlet osition of the tank. A constant velocity rofile was also assumed at the end of the outlet ie. An outlet ie was formed in the geometrical domain of the model to lower right center uer n n + = 0 uer n lower center right = in = 0. where n is the number of the gridlines in the inlet osition in is the temerature of the inflow water and is the temerature of the fluid in the ie..3... Boundary conditions of the stream function (9)
he stream function is suosed to be constant everywhere around the inner side of the wall and the outlet ie to secify the zero velocity erendicular to the wall. Q is the volumetric flow of the fluid into the storage tank and it has negative value. Ψ = Q Ψ i { n... i } i0 i m + = Q + + V Ψ R i m + = 0 Ψ = Ψ 0 j n + j i {... n } Ψ 0 j = 0 Ψ n j = Q Ψ i 0 = Q = 0 0 (0) where i is the index to the r direction and j is to the y direction. n+ is the number of the vertical gridlines and m+ is the number of the horizontal gridlines..3..3. Boundary conditions of the vorticity function hom condition was used for the boundary condition of the vorticity function to calculate the vorticity on the inner side of the wall. It is secified by the assumtion that the velocity arallel to the wall is zero along the wall. = Q Ψ i0 i r i y = Q Ψ i m + i m r y i { n +... i + } = i m + i m R r = Q Ψ n + j n j = 0 0 j i0 = i i {... n } ( Q Ψ ) n j = n j R i out r () where R is the radius of the tank and R out is the radius of the outlet ie. he above equations () come from the aylors series exansion of the stream function Ψ (Roache 976) and (Stoyan and ak 997). A finite difference code is develoed in Matlab environment to solve the transort equations with the secified boundary conditions. he model was validate by comarision with the solutions of (Bandini 998) (orrance and Rocket 969) (orrance 968). he "U- Wind" method is used to avoid the numerical instabilities and each time ste was calculated with the Courant condition. he discretized equation for function Ψ was solved by the Matlab built-in solver for linear equations and the result of the velocity and temerature fields were resented with the Matlab built-in functions. 3. NUMERICAL RESULS Some ictures can be seen about the temerature field and the velocity field. hese results were calculated from the two different initial temerature field conditions. he interesting area of the flow field is the inlet and outlet osition of the tank. 3.. Initially constant temerature field Evolution of temerature rofiles inside the storage tank and velocity fields near the inlet osition can be seen on the following ictures. he initial temerature field was constant temerature 0 0 C everywhere. he hotter fluid transorted by the inflow makes a vertical temerature rofile. It is strongly emhasized that the turbulence of the fluid is very low because of the laminar hysical roerty of the water. he results can be seen in Figs. 3-6. Fig. 3. emerature field of the water at time t=7 s.4. Numerical solution of the transort equations
3.. Initially linear temerature field Changing the thermal stratification from an initially linear temerature field with higher inlet fluid temerature than the temerature of the uer ortion of the water in the tank can be seen in the following ictures Figs 7-. Fig. 4. Velocity field of the water at time t=7 s Fig. 7. emerature field of the water at time t=7 s Fig. 5. emerature field of the water at time t=55 s. Fig. 8. Velocity field of the water at time t=7 s Fig. 6. Velocity field of the water at time t=55 s he scales of the ictures of the velocity field in the vertical and the radial directions show the grid number. he left side of the icture is the centerline of the tank.
Fig. 9. emerature field of the water at time t=55 s Fig.. Velocity field of the water near the oulet osition at time t=55 s he largest value of the velocity at the centerline is 0.06 m/s and velocity rofile nearly arabolic inside the outlet ie. Fig. 0. Velocity field of the water near the inlet osition of the storage tank at time t=55 s o comare and see the size of the velocity vectors the value of the uniform velocity at the inlet osition is 0.05 m/s 4. CONCLUSIONS wo-dimensional mathematical model was used to analyse numerically the evolution of the temerature distribution of the water inside a storage tank. he mass flow rate was chosen very low to maintain or develo a stratified temerature distribution in the storage tank. In the first case when the initial temerature was constant everywhere the inlet flow makes vertical change of the temerature rofile. In case of the initially linear rofile the stratified temerature distribution is reserved because the hotter inflow water goes to the highest art of the tank. Finally it can be concluded that the temerature rofiles in the tank show good stratification starting from both initial temerature conditions but it is imortant to mention that the imact of the turbulence is very low because of the coarse grid system. he imact of the turbulence mixing is imortant for this reason alication of a turbulence model is need to calculate more recisely the mixing along the jet. NOMENCLAURE c secific heat of the water (J kg - K - ) f body force (N m -3 ) g gravity acceleration (m s - ) temerature function of the water inside tank ( o C) cs temerature function of the water inside ie ( o C) a ambient temerature ( o C) U velocity comonent to the r direction (m s - ) V velocity comonent to the y direction (m s - )
Greek symbols α heat transfer coefficient of the air around the tank (W m - K - ) β volumetric thermal exansion of the water (K - ) λ f thermal conductivity of the water (W m - K - ) λ W thermal conductivity of the wall (W m - K - ) ν kinematic viscosity of the water (m s - ) ρ density of the fluid (kg m -3 ) ρ 0 reference density of the fluid (kg m -3 ) Ψ vorticity function of the velocity field stream function of the velocity field ACKNOWLEDGEMENS he authors thank the suggestions of G. Stoyan in the toic of the numerical techniques of the comutational fluid dynamic. We also would like to thank to A. Asz di for the useful literature references and discussions. REFERENCES Bandini M. A. Vielmo H. A. Numerical analysis of velocity and temerature fields in storage tanks of solar thermal systems World Renewable Energy Congress (998). 080-083. BuzÆs J. Farkas I. B r A. NØmeth R. Modelling and simulation asects of a solar hot water system Mathematics and Comuters in Simulation 48 (998). 33-46. Duffie J. A. Beckmann W. A. Solar engineering of thermal rocesses Willey New York (980). Hollands K. G.. Lightstone M. F. A review of lowflow stratified-tank solar water heating systems Solar Energy 43 No.. 97-05 (989). Kleinbach E. M. Beckmann W. A. Klein S. A. erformance study of onedimensional models for stratified thermal storage tanks Solar Energy 50 (993). 55-66. alacio A. Fernandez J. L. Numerical analysis of greenhouse-tye solar stills with high inclination Solar Energy 50 No. 6. 469-476 (993). atankar S. V. Numerical heat transfer and fluid flow aylor & Francis (980). Roache. J. Comutational Fluid Dynamics Hermosa ubl. Albuquerque (976). Schlichting H. Boundary Layer heory Sixth edition McGraw-Hill New York (968). Smith G. D. Numerical Solution of artial Differential Equations: Finite Difference methods Clarendon ress Oxford (978). Stoyan G. Numerical Solution of ieline system roblems by monotone difference aroximations Second Euroean Symosium on Mathematics in Industry Stuttgart (987) 95-09. Stoyan G. ak G. Numerikus módszerek 3. ELE- yo E X Budaest (997). (In hungarian) Szlivka F. LohÆsz M. Flow attern calculation in an axial flow fan cascade Mezõgazdasági echnika XLI. (000) (In hungarian) orrance K. E. Comarision of Finite-Difference Comutations of Natural Convection Journal of Research of the National Bureau of Standards-B 7B No. 4. 8-30 (968). orrance K. E. and Rockett J. A. Numerical study of natural convection in an enclosure with localized heating from below-creeing flow to the onset of laminar instability J. Fluid Mech. 36 No.. 33-54 (969). Van Berkel J. Mixing in thermally stratified energy stores Solar Energy 58 No. 4-6. 03- (996).