Applying control volume finite element for modelling. Direct injection boom spraying flow

Transcription

1 Applyng control volume fnte element for modellng Drect njecton boom sprayng flow Abdellah El Assaou Natonal Insttute of Agrcultural Research, Dryland Research Center, PO Box 589 Settat 6, Morocco Frederc Lebeau Gembloux Agrcultural Unversty, Mechancs and Constructon Unt, passage des Déportés, Gembloux 53, Belgum Mare-France Destan Gembloux Agrcultural Unversty, Mechancs and Constructon Unt, passage des Déportés, Gembloux 53, Belgum Karm Houmy Agronomc and eternary Hassan II Insttute, PO Box 6, Rabat-Insttutes, Morocco Abstract. Assessment of njecton lag transport and unformty of drect njecton boom sprayer s an mportant ssue for successful varable rate sprayng technology. To estmate the boom lag transport and pressure loss, a numercal model s formulated on the bass of flud hydrodynamc conservaton equatons. The software s mplemented n vsual basc. To solve the pressure veloctes equatons, control volume fnte element method (C) s used to delmt elementary volumes of the boom. Lnearzaton of the conservaton laws s ensured by consderng dscrete form of the equatons and calculatng velocty and pressure step by step throughout the whole boom. The flow behavour s smulated nto a boom secton dvded nto N elementary volumes, each of them ncludng one nozzle. To test the model, three boom dameters (5, 6 and 8 mm) and two chemcal vscostes ( -6 and -5 m /s) were used. Expermental trals are carred out on boom havng.5 m length (5 nozzles) for measurng pressure gradent and lag transport. Results showed that the model can predct the pressure losses and the lag transport accurately (error wthn 5%) to optmze boom desgns.. Keywords. Drect njecton, control volume method, lags transport, frcton losses, vscosty. Introducton Drect njecton system (DIS) can be very nterestng to apply pestcde accurately and safely. In fact, control of concentraton proportonally to forward speed s probably the best soluton for solvng msapplcaton that occurs wth conventonal sprayers. However, DIS advantage s stll condtoned by materal performance, especally the ablty to apply the desred pestcde rate and settng t n real tme. The performance can be evaluated manly through materal effcency to control and to mantan desred mxture concentraton. Practcally, adjustng appled concentraton to forward speed varaton or to a new set-pont needs to be carred at mnmal possble tme delay. The response tme s a crtcal parameter for evaluatng DIS accuracy. It has two man components. The frst one s related to the transport tme between the njecton pont and the nozzles. The second component s the response characterstc of the njecton meterng system (Pace & al., 995). The transport lag depends manly on optmzaton of hydraulc boom layout when njecton pont s located closely to the centre of the boom lne. The use of electrcal energy to supply DIS pump gves possblty to avod any sgnfcant complementary lag tme amplfcaton and lmts t only to boom scheme. Pace & al. (997) used two

2 transent characterstcs, tme constant and rse tme, to evaluate the dynamc response of a sprayer. The tme constant s the tme requred to reach 63. % of the step nput whle the rse tme s the tme requred to go from to 9 % of the step nput. The DIS boom desgn needs to be computed effcently, not only to maxmze flud flow for obtanng short lag tme and turbulent flow regme for mprovng onlne mxng, but also to contan frcton losses that cause pressure decrease and affects nozzles unformty along boom (ondrka & al, 7). Moreover, varablty of pestcde vscosty can potentally ncreases frcton losses and affects slghtly boom jet unformty when heavy pestcde s appled at low temperature condtons. In fact, lqud pestcdes formulatons have large magntude of vscosty from mpa.s to mpa.s. Although, the most commercalsed formulatons have dynamc vscosty under mpa.s (Hloben, 7). Studyng vscosty effect of sprayed mxture helps to approach potental hydraulc mpact on boom pressure drop. To study boom flow behavour for optmal DIS desgn, three ponts should be detaled: - The optmal hydraulc boom structure to obtan fast response to establsh concentraton equlbrum and mnmal pressure drop that keeps acceptable nozzles unformty. - The dscrete profle of lag tme and turbulence along standard boom layout of constant dameter. - The potental effect of sprayed mxture vscosty n relaton to frcton losses n DIS boom. The objectve of ths study was: - To develop numercal model based on fnte volume method to characterse mxture flow n DIS boom. - To desgn DIS laboratory bench based on two boom layouts of ten nozzles (seral and parallel tp nozzles scheme) n order to compare smulated results wth expermental data. Materal and methods Model Approach The model s developed to carry out dscrete scheme of DIS boom hydraulc flow by usng fnte volume method (Reddy, 993, Lakhdar & al, 6). It takes advantage of physcal and chemcal parameters of boom and the sprayed mxture; wdth, dameter, ppe materal roughness, number of nozzles, nozzle flow rate coeffcent, upstream boom pressure, downstream boom pressure, lnear and local frcton losses, densty and vscosty. As results, the model gves a numercal gradent scheme of pressure, flow speed, Reynolds number, frcton loss and lag tme for each control volume (C) and then for lateral boom secton. The used numercal method s based on teratve ncremental search (James & al., 993) for solvng non lnear algebrac equatons of pressure and flow rate. To compute frcton losses, the Darcy-Wesbach frcton losses model (Sullvan, 989; Gleen, 3) was used and the Newton-Raphson numercal method was appled to yeld frcton factor n Colebrook equaton. The dscrete computaton conssts of applcaton of mass and energy conservaton equatons on elementary C that contans one nozzle body of DIS boom sprayer (Fgure ). H() () Re() f() L + H(+) (+) Re(+) f(+) Fgure. Arbtrary control volume () and upstream and down stream parameters: pressure (H), flow speed (), Reynolds number (Re) and frcton factor (f) appled for numercal computaton Computaton of nozzle flow rate average q at control volume depends on upstream pressure H, downstream pressure H+, nozzle flow rate coeffcent (k) and pressure exponent (x):

3 q = kh x H + H + q = k x () Mass balance computaton at control volume depends on upstream and downstream flow rate (or on ppe secton area A and upstream () and downstream (+) flow speeds) and on nozzle flow rate: A. = A. + + q And A = πd / 4 () + = ( 4q / πd ) + ( / d ). x H + H+ = 4 π (3) Energy balance computaton at control volume s based on Bernoull theorem. It depends manly on hydrostatc pressure, knetc energy and nduced frcton loss between nput and output ponts of C (potental energy keeps zero for horzontal boom poston): g g g g + + H + = H Hf, + H + = H + Hf, + (4) Frcton loss computaton for C of L length s based on Darcy-Wesbach equaton and depends on upstream and downstream lnear frcton factors (f() and f(+)) and mnor loss factor ξ at nozzle body level : f ( ) + f ( + ) L + Hf ξ (5) g d +, + = + Frcton factor f() computaton depends on flow regme: o For lamnar and transtory flow (Re 3), f() depends only of Reynolds number: Re( ) = d µ 64 f ( ) = (6) Re( ) o For turbulent flow (Re>3), f() can be computed by Colebrook equaton and depends on Reynolds number and ppe absolute roughness (ε): f ( ) ε = log 3.7d.5 + Re( ) f ( ) (7) For solvng numercally the Colebrook equaton, the model was mplemented by Newton-Raphson teratve method subroutne. Ths method quckly yelds f () values for mnmal teraton number (James & al., 993). Lag transport computaton depends on nput and output flow speeds and C length:

4 L ( ) = + T lag (8) + Input parameters N: number of nozzles, L: boom wdth d: boom dameter, Ra: absolute roughness H : Upstream boom pressure H n, H c: Smulated and computed values of downstream pressure H nf: Maxmal smulated value of downstream pressure p: pressure ncrement, W : teraton number k: nozzle flow rate coeffcent, x : nozzle pressure exponent ρ: sprayed mxture densty µ: cnematc vscosty of sprayed mxture. Choce of sprayed mxture parameters (densty and cnematc vscosty). Loop for dameters boom choce (N dameters) 3. Choce of upstream boom pressure H (pressure value ndcated of nput flow) 4. Loop of W teratons to ncrement H n towards H nf by pressure ncrement p 5. Loop of N (nozzles) teratons to compute hydraulc parameters for each C: - nozzle flow rate from smulated pressure gradent - Flow speeds and Reynolds numbers n upstream and downstream C levels - Frcton losses coeffcents n upstream and downstream C levels (subroutne for solvng numercally Colebrook equaton based on Newton-Raphson teratve method) - Frcton losses n C (use of Darcy-Wesbach equaton) - Lag transport n C - Concludng computed pressure value from mass conservaton equaton 6. Computaton of hydraulc parameters for boom secton of N nozzles (C) 7. Convergence test of smulated pressure value toward computed one from mass conservaton equaton Output parameters - Data sheet of hydraulc parameters results and convergence for each C, each teraton and each boom case; - Trend of convergence test showng smulated pressure, effectve computed pressure, frcton losses gradents and convergence pont. Boom desgns smulaton Fgure. Scheme of computatonal algorthm For testng the model, three dameters (5, 6 and 8 mm) are smulated to study flow behavour of DIS boom secton of 5 seral nozzles. Moreover, two vscostes of -6 m²/s (water) and -5 m²/s ( tmes more than water) are taken nto account. The smulated cases resulted n numercal schemes of pressure gradent, nozzles unformty, flow regme and lag transport that helped to approach boom qualty applcaton. The convergence test was based on smulatng H n value and ncrementng t by step p toward H nf. The optmal H n for each boom dameter moved toward computed pressure H c to satsfy the chosen convergence rato (practcally less than -3 ). Ths rato s the absolute value obtaned by subtractng smulated pressure gradent H s (H -H n) from computed pressure gradent H c (H - H c). The nput parameters nto model were: For boom: N = 5, L=,5 m, d= 8 mm, d= 6 mm, d3=5 mm, Ra= um, Ho=3 m (3 bars), ξ=.3 For nozzle flow rate: k = -5, x=.5, q=. -5 m 3 /s (Teejet XR 83:. l/mn ~ 3 bars), For sprayed mxture: ρ = kg/ m 3, µ= -6 m²/s (water), µ= -5 m²/s ( tmes more).

5 Expermental desgn To assess model computaton, two cases of seral and parallel boom layouts were studed (fgure 3). Laboratory DIS equpped wth man daphragm pump (Flojet of Sherflo, 4 DC, l/mn~,8 bars ), and perstaltc pump (Marlow Watson TM 4D/E, two channels, 38 ml/mn (x )) to nject fluorescen tracer at upstream pont of the man pump. Pressure boom gradent was measured va two sensors (Sensorthechncs TM CTE 85GY7, Pmax=5bars, non-lnearty=., hysteress=.5) mounted upstream and downstream boom sdes. Lag transport was approached by fve fluorometrc sensors that were desgned and calbrated to sense fluorescen transmttance at 5 um at each nozzle body (El Assaou & al.., 7). To construct seral boom layout, commercal copper ppng (d=6 mm, roughness~ µm) was used to connect nozzles body mounted n tee junctons as shown n fgure 4. Parallel boom layout was formed of quck connect flexble (Festo TM, d= 4mm, roughness~ um) to attach each nozzle body to the collector as shown n fgure 3. A LabIEW I was mplemented to acqure data by DAQ NI-USB65 at samplng frequency of 5 Hz. Daphragm and perstaltc pumps were actuated by PWM (S of CJ Controls LTD) to adjust concentraton njecton rato, operatng pressure and/or to nduce step change. Fgure 3 DIS laboratory bench based on two boom layouts of parallel (A) and seral (B) schemes Results and dscusson Model computatonal results and dscusson

6 The calculated model results (table ) showed that 6 mm boom dameter could be satsfactory for keepng applcaton unformty up to 97% and short lag transport (dead tme wthn.5 s for.5 m boom secton length) n seral supply scheme. Predcton of vscosty effect showed that t kept non sgnfcant from -6 to -5 m /s to affect boom flow unformty (wthn %). Otherwse, the choce of small dameter (less than 6 mm) could be also satsfactory to supplyng nozzles separately from common collector n parallel scheme. The calculated lag transport tended to ncrease exponentally from 9% (nozzle ) to 43% (nozzle 5) as shown n fgure 4. The fourth and ffth nozzles took 65% of total lag because of the low flow speed occurrng at the boom end whch caused long dead tme and low gan to erase dead volume. The calculated dscrete profle of Reynolds number showed that the flow was kept turbulent (Re>3) for good mxng. Furthermore, the vscosty affected consderably flow regme by decreasng Reynolds number. Table : calculated results of seral boom layouts at 3 bars Nozzles number d (mm) vscosty (m²/s) Convergence H s- H c H c (m) (H -H n) H c/h (%) Hf (m) Lag transport (s) Unformty q 5/q (%) -6,35,64,5,935,4 99,33-5,49,99 3,3,3,4 98,9-6,7,958 6,53,664,5 97,4-5,6,675 8,9 3,38,48 97,8-6,4 4,999 6,66 6,46, 93,3-5,3 5,696 8,99 7,57,99 9, Nozzle 5 43% Nozzle 9% Nozzle % Nozzle 3 5% Nozzle 4 % Fgure 4 Dscrete profle of lag transport n seral nozzles at 3 bars (6 mm dameter)

7 Reynolds number E-6 m²/s E-5 m²/s Nozzle Nozzle Nozzle 3 Nozzle 4 Nozzle 5 Fgure 5 Dscrete profle of Reynolds number n seral boom layout at 3 bars (6 mm dameter) Expermental results and dscusson Pressure drop The test of boom secton of 5 serals nozzles (fgure 6) showed that measured pressure drop kept around the smulated values (7% ± ). The dvergence at bar can be explaned by mss adaptaton of the nozzle law at low pressure. Practcally, the dffculty to predct accurately pressure gradent kept n approachng rghtly mnor losses that can occur n dfferent junctons and n usng sophstcated dfferental pressure sensors. Furthermore, smulated trends showed that nozzles number can affect pressure gradent sgnfcantly. The test of the parallel boom layout (fgure 7) was carred out by sensng pressure gradent between collector and the nearest nozzle (.5 m) and between collector and the farthest nozzle (.5m). The pressure gradent between the upstream and downstream nozzles kept around 7% ±. Pressure decrease (%) nozzles (measured) 6 nozzles (smulated) 5 nozzles (smulated) 4 nozzles (smulated) 3 Boom pressure (bar) Fgure 6: Pressure drop n seral boom layout (dameter of 6 mm)

8 Pressure decrease (%) Boom pressure (bar) Nozzle (.5 m) Nozzle 5 (.5 m) Fgure 7: Pressure drop measured n parallel boom layout (dameter of 4 mm) Lag transport To characterze lag transport of seral and parallel boom layout, three parameters were used, dead tme, tme constant and rse tme. The frst tme s delay from start nput pont to the start of response whch depends on dead volume and flow speed n each C. The second one s tme requred to reach 63.% of concentraton change. The thrd one s tme needed to go from to 9% of step change. The measured lag transport of the fve nozzles mounted n seral layout was about 4.5 s as shown n fgure 8. The dead tme moved upward from.3 s (Nozzle ) to.8 s (Nozzle 5). The tme constant changed from s (Nozzle ) to.3 s (Nozzle 5). The rse tme ncreased slghtly to form dfferent S-shaped curves. The total lag transport of the fve parallel nozzles kept around 4 s. Dead tme stepped constantly from.4 s (nozzle ) to s (Nozzle 5). The tme constant kept constant for the fve nozzles around.9 s. The rse tme took the same value of. s for the fve nozzles, formng smlar S-shaped curves as shown n fgure 9. concentraton (%) tme (s) Nozzle Nozzle Nozzle 3 Nozzle 4 Nozzle 5 Fgure 8: Response of 5 seral nozzles to step change at 3 bars (dameter of 6 mm)

9 concentraton (%l) tme (s) Nozzle (.5m) Nozzle (m) Nozzle 3 (.5m) Nozzle 4 (m) Nozzle 5 (.5m) Fgure 9 Response of 5 parallel nozzles to step change at.7 bars (dameter of 4 mm) Concluson The ablty of the developed model to predct dscrete hydraulc profle by applyng control volume element method, helps desgn optmal boom scheme. The model accuracy keeps condtoned by the mss adaptablty of the Darcy-Wesbach model to compute frcton losses n transtory flow regme band flow and by the dffculty to approach accurately the mnor losses occurrng actually n boom lne. The comparson between seral boom layout and parallel boom layout showed how t was nterestng to consder the effect of many lags n seres and n parallels, coupled and uncoupled lags, on boom dynamc response. Acknowledgment Authors thank Belgan Techncal Cooperaton (BTC) for the research fund, and acknowledge techncan, Rudy Schwartz, for hs contrbuton to carryng out expermental desgn, and programmer Mokhtar El Ouad for the help to code the model software n B. References El Assaou A., Lebeau F., Destan M-F. 7: Development of an optcal sensor to measure drect njecton sprayng system performance.di-meg Agrcultural Engneerng Conference, Hannover, November 9-, 7. Hloben P., 7: Study on the Response Tme of Drect Injecton Systems for arable Rate Applcaton of Herbcdes, PhD thess, Unversty of Bonn. 7p. Gleen O. Brown, 3: The Hstory of the Darcy-Wesbach Equaton for Ppe Flow Resstance, Envronmental and Water Resource Hstory, Oklahoma State Unversty. James M.L., Smth G.M., Wolford J.C.; 993: Appled Numercal Method for Dgtal Computaton, Unversty of Nebraska. 4th Ed, ISBN , 7p. Lakhdar Z., Kettab A., Chasseraux G., 6: Desgn of mcro-rrgaton system based on the control volume method, Bothechn; Agron.Soc.Envron., (3), Pace M.E.R., Mller P.C.H and Bodle J.D, 995: An Expermental Sprayer for the Spatally Selectve Applcaton of Herbcdes. Agrculture Engneerng Research, 6, 995, p.7-6 Pace M.E.R., Mller P.C.H and Lane A.G, 997: The response characterstcs of a patch Sprayng system based on njecton meterng. Aspects of Appled Bology, 48, 997, p Reddy J.N, 993: An Introducton to the Fnte Element Method, Texas A&M Unversty, nd Ed, 679p. Sullvan J.A., 989: Flud Power: Theory and Applcatons, Southern Unversty Illnos. 3rd Ed, ISBN , 59p. ondrcka J., Peter H., Lammers P., 7: Optmzaton of Drect Nozzle Injecton System for ste specfc herbcde applcaton. ASABE Annual Meetng, Mnnesota, 7- jun 7.