11 th National Congress on Theoretical and Applied Mechanics, -5 Sept. 009, Borovets, Bulgaria A MATHEMATICAL MODEL OF DRIP EMITTER DISCHARGE DEPENDING ON THE GEOMETRIC PARAMETERS OF A LABYRINTH CHANNEL * NINA PHILIPOVA Institute of Mechanics, Bulgarian Academy of Sciences, Acad.G.Bonchev Str., Bl.4, 1113 Sofia, Bulgaria, e-mail: philipova@mech.bas.bg NIKOLA NIKOLOV Institute of Mechanics, Bulgarian Academy of Sciences, Acad.G.Bonchev Str., Bl.4, 1113 Sofia, Bulgaria e-mail: n.nikolov@imbm.bas.bg GEORGI PICHUROV Technical University of Sofia, Dept. Hydroaerodynamicsand Hydraulic Mashines, 8 Kliment Ochridsky buolevard., 1000 Sofia, Bulgaria e-mail: george@tu-sofia.bg DETELIN MARKOV Technical University of Sofia, Dept. Hydroaerodynamicsand Hydraulic Mashines, 8 Kliment Ochridsky buolevard., 1000 Sofia, Bulgaria e-mail: detmar@tu-sofia.bg ABSTRACT. The influence of the geometric parameters of emitter labyrinth channel on the emitter discharge is investigated in this work., The influence of the dentation angle, the dentation spacing, the dentation height is investigated among the geometric parameters. Numerical simulations of water flow movement are performed according to the cubic central composition design by using programs GAMBIT and FLUENT. As a result, a mathematical el of emitter discharge is derived depending on the above mentioned geometric parameters. The most important influence on the emitter discharge has the increase of the dentation spacing * The author would like to acknowledge European Social Fund, Operational programme for the financial support Grant BG051PO001/07/3.3-0-55/17.06.008.
N. Philipova, N. Nikolov, G. Pichurov, D. Markov, followed in times lower by this of the high, and finally the angle has the smallest influence. KEY WORDS: drip irrigation, emitter, labyrinth channel, pressure losses, mathematical el 1. Introduction The emitter labyrinth dissipates the energy along its length by means of the built in the channel obstacles. The geometric parameters of these obstacles define the the hydraulic characteristics of the labyrinth channel and maintains turbulent flow that helps in preventing clogging. The examined hydraulic characteristics of the labyrinth channel in this paper are the pressure losses per a dentation spacing of emitter labyrinth and the emitter discharge at the dripper oulet. The geometric parameters of the labyrinth channel are the following parameters: dentation angle, dentation spacing, dentation height, channel width and depth. There are two approaches in examinating how the geometric parameters influence on the emitter hydraulic parameters. The first approach is on the basis of Computational Fluid Dynamics (CFD) programs [1]- [4], by means of programs such as FLUENT, GAMBIT, ANSYS helping in creating numerical simulations of the flow movement in the labyrinth channel. The second approach is experimental one. It requires the production of a great number of moulds according to a statistical design [5]. The objective of the work presented in this paper is a mathematical el derivation of emitter discharge depending on the geometric parameters on the basis of numerical simulation of water flow movement in emitter labyrinth channel.. Model equations The Reynolds stress el (RSM) [6] - [8] was adopted, which can account for the effect of streamline curvature, swirl, rotation and rapid changes in strain rate..1. Continuity equation: (.1) ( ui ) = 0 x ρ i.. Reynolds Averaged Navier- Stokes Equations (RANS Eq.): i (.) ( ρ u ) = + u u iu j μ ρ ' x j x j p where: u i - the velocity vector, ρ - the water density, μ - the fluid molecular viscosity, and u i j - the Reynolds stress tensor. In the equation (.3) the left hand side term is the convection. The first term from the right hand side is the stress production term P ij. The second term is the pressure-strain term Φ ij. The third term u
A mathematical el of emitter discharge depending on geometric parameters.. is the sum of turbulent and molecular diffusion and the last term is the dissipation term. μ ui t j ui j (.3.) ( ρukuiu j ) = Pij + Φij + + μ ρεδij x k σ k xk x k 3 where: μ t - the turbulent viscosity, linked to the turbulent kinetic energy k and dissipation rate ε via the relation: (.7) k μt = ρcμ ε The differential transport equations for k and ε are: ( ρku μ i ) t k 1 (.8) = μ + + ρε σ Pij x j k x j ( ρεui ) μt ε 1 ε (.9) = μ + C1ε Pij Cε ρ x + j σ ε x j k where the el constants are the following: C.09, C = 1.44, C = 1.9, = 1, σ 1.3 μ = 0 1 ε ε σ k ε = Fig. 1. Geometric parameters of the labyrinth channel: θ- dentation angle, H- dentation hight, B- dentation spacing, W- channel width Boundary conditions at the inlet face are x = nlet, pinlet = p0 and at the outlet face is p outlet = 0. Boundary conditions for the wall standard log-law function was used to bridge the near-wall linear sublayer [3]. Initial conditions are: t = 0, p inlet = p 0 v inlet = v0. Boundary conditions for Reynolds stresses are: u η λ = 5.1 = 1.0 =.3 u u u η = 1.0
N. Philipova, N. Nikolov, G. Pichurov, D. Markov where: - the tangential co-ordinate, η - the normal co-ordinate, λ -binormal coordinate, u -the friction velocity 3. Simulation of the water flow movement in the labyrinth channel. The geometric parameters of the labyrinth channel can be seen in Fig.1.The water flow movement in the labyrinth channel was simulated by using the CFD programs GAMBIT and FLUENT. GAMBIT creates the geometry and the mesh of the labyrinth volume. The labyrinth volume is meshed with a hexahedral grid. The el equations and the boundary and the initial conditions were determined in FLUENT. The governing equations which have been described above were discretized by the control volume technique, and then the SIMPLE pressure-velocity coupling technique with a second-order upwind scheme was applied. 4. A mathematical el of pressure losses depending on the geometric parameters. exp The emitter discharge data Δ q used during the Regression analysis are calculated numerically according to Full factorial experiment and Cubic central composition design of experiments (Table 1). Fourteen planned numerical experiments are conducted following the Cubic central composition design and using the commercial codes GAMBIT and FLUENT. A mathematical el of the emitter discharge depending on the dentation angle θ, dentation spacing B and dentation high H of the labyrinth channels is derived through Regression analysis with two-rate polynomial. It has the type: Δq = b0 + b1 X1 + b X + b3 X 3 + b1 X1 X + b3x X 3 + b13x1 X 3 + (4.1) + b X + b X + b X where: Δq 11 -emitter discharge, bij -coeficients, 1 33 3 X j are the transformed parameters x i into non-dimensional parameters, belonging to the interval ( 1 1); i, j = 1,,3, av (4.) X j = max av where: av - the average value of x i, max - the mamum value of the x i. The el equation is the following: q = 3.445064 + 0.04θ + 0.699B + 0.3606H 0.004775θB + 0.0053θH (4.3) 0.0373BH 0.08564θ 0.044B + 0.0549H Looking at the coefficient in eq. (4.3), the next conclusions could be drawn about the influence of the dentition angle θ, the dentation spacing B and the dentation hight H on the emitter discharge. The self-depended increases of three parameters lead to increasing emitter discharge. Most important influence on the
A mathematical el of emitter discharge depending on geometric parameters.. Table 1. Cubic central composite design No θ B H exp q q exp q q [deg] [mm] [mm] [l/h] [l/h] % 1 8 1.0 3.316 3.559 0.74 36 1.0 3.439 3.11 1.014 3 8.5 1.0 4.6713 4.738 1.411 4 36.5 1.0 4.6778 4.6741 0.00079 5 8 1.6.5956.599 0.1385 6 36 1.6.6413.5743.536 7 8.5 1.6 3.8995 3.933 0.8341 8 36.5 1.6 3.918 3.8884 0.635 9 36.5 1.3 3.3806 3.5084 3.64 10 8.5 1.3 3.6808 3.559 3.474 11 3.5 1.3 4.401 4.1083 1.693 1 3 1.3.6985.770.588 13 3.5 1.6 3.084 3.1387 1.74 14 3.5 1.0 3.9147 3.8599 1.399 emitter discharge has the increase of dentation spacing B followed by this one of the hight H and angle θ. The simultaneous increase of B together with H has a negative influence on the emitter discharge whereas the simultaneous increase of θ together with H has a positive influence on it. The comparison between the el and the experimental data can be seen in Fig.3. The average difference between the experimental and the el data is 1.50135% (Table 1). Emitter Discharge q, [l/h] 5.00 4.50 4.00 3.50 3.00.50 E1 E Numerical experiments Mathematical el B=mm E9 E5 E6 E1 B=.5mm E10 E13 E14 B=.5mm 0.00 4.00 8.00 1.00 16.00 Experiments as function of step B E3 E4 E7 E8 E11 Fig.. Comparison between the el and the experimental data. Adequacy of the el (4.3) is checked through correlation coefficient R. R = 0.9957498 is obtained by calculation of the scattering of initial value around its average value [9]. The final decision about the adequacy is taken after calculation of the values F related to R : R R ( N k) F R = = 51.7 and check (1 R )( k 1) ( 1, ) ( ν 4, ν ) = F L ( 0.05,9, ) 6. 04 L R F R F >. The limit L R F α, 1 R = is determined from the Fisher s tables [9] after assigning significance level α = 0. 05, where ν 1 = k 1, ν = N k, and N is the number of experiments, k is the number of
coefficients in (4.3). N. Philipova, N. Nikolov, G. Pichurov, D. Markov 5. Conclusions. In this paper a mathematical el of hydraulic parameters is derived depending on the geometric parameters on the basis of numerical simulation of water flow movement in emitter labyrinth channel. The RSM has been used for the simulation of highly rotational turbulent water flow occurring in the labyrinth channel. A number of conclusions can be drawn from the derived el: 1) The self-depended increases of three parameters - the dentition angle θ the dentation spacing B and the dentation high H lead to increasing emitter discharge ) Most important influence on the emitter discharge has the increase of the dentation spacing B followed by this of the high H and the angle θ. 3) The simultaneous increase of B together with H has a negative influence on the emitter discharge whereas the simultaneous increase of θ together with H has a positive influence on it. R E F E R E N C E S [1] YUNKAI. L., Y. PEILING, R. SHUMEI, Hydraulic characterization of tortuous flow in path drip irrigation emitters, J. of Hydrodynamics, Ser. B, 18 (006), No 4, 449-457. [] WEI Q.,Y. SHI, W. DONG, G. LU, Study on hydraulic performance of drip emitters by CFD, Agr. Water Management, Elsevier, 84 (006), 130-136. [3] ZHANG J., W. ZHAO, Y. TANG, Z. WEI, B. LU, Numerical investigation of the clogging mechanism in labyrinth channel of the emitter, Int. J. for Num. Meth. in Eng., 70 (007), 1598-161. [4] DAZHUANG Y., Y.PEILING, R. SHUMEI, L.YUNKAI, Numerical study on flow property in dentate path of drip emitters, New Zealand J. of Agr. Research, 50 (007), 705-71. [5] LI, G., J. WANG, M. ALAM, Y. ZHAO, Influence of geometric parameters of labyrinth flow path of drip emitters on hydraulic and anti-clogging performance, Transactions of ASABE, 49 (006), No 3, 637-643. [6] VERSTEEG, H., W. MALALASEKERA, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Willey, New York, 1995. [7] HANJALIC, K., B. LAUNDER, A Reynolds stress el of turbulence and its application on thin shear flow, J. Fluid Mech., 5 (197), No 4, 609-638. [8] LAUNDER, B., G. REECE, W. RODI, Progress in the development of a Reynolds-stress turbulence closure, J. Fluid Mech., 68 (1975), No 3, 537-566. [9] VUCHKOV, I. S. STOYANOV, Mathematical elling and optimization of technologic objects, Technica, Sofia, 1980 (in Bulgarian).