Study on the Cutter Suction Dredgers Productivity Model and Its Optimal Control

Transcription

1 Modelin, Simulation and Optimization Technoloies and Applications (MSOTA 016) Study on the Cutter Suction reders Productivity Model and Its Optimal Control Minhon Yao, Yanlin Wan, Jin Shan and Jianyon Zhan* Hohai University, Chanzhou Campus, Chanzhou, China * Correspondin author Abstract This Controllin the yield of cutter suction dreder is a hot spot for researchers at home and abroad, however, there is few strict mathematical models provin the existin research results, and most of the results are qualitative. In order to optimize the value of dreders main control variables, this paper puts forward a non-linear productivity optimization model based on Laranian Alorithm. Takin Tianshi-Caofeidian project as an example, this paper provides the optimal value ranes of dreders main control variables and also proves some existin experimental conclusions. After comparin experimental control values with this papers results, the experimental control values are supposed to be increased, and this result can be used for practical enineerin. Keywords-cutter suction dreder; non-linear productivity optimization model; Laranian I. INTROUCTION Improvin Cutter suction dreders production has a sinificant influence on actual enineerin. Optimization research of Cutter suction dreders productive conditions is a hot spot at home and abroad. However, there is few strict mathematical models provin the existin research results, and most of the results are qualitative [1]. This paper puts forward a non-linear productivity optimization model based on Laranian alorithm to provide the optimal value ranes of dreders main control variables and also proves some existin experimental conclusions. II. INTROUCTION OF CUTTER SUCTION REGERS MU PUMP SYSTEM AN PING SYSTEM A. Work rules of Cutter Suction reder Cutter suction dreder usually use spud carriae to work []. reder is fixed with pillar in the middle of channel, and dreder can o around the pillar to remove mud by cables pull. ue to the limited channel width, the dreder will chane its direction periodically. Fiure 1 is dreders GPS trajectory durin workin hours. FIGURE I. REGERS GPS TRAJECTORY URING WORKING HOURS. B. Pump System and Pipin System Cutter suction dreder is mainly consist of hull, spud system, slurry transport system which includes mud pump system and pipeline system, cutter cuttin system, transverse system, Electric hydraulic auxiliary system and etc. The characteristics of centrifual mud pump can be deduced by Euler equation [], [3]. Followin is relation between pumps theoretical pressure head and theoretical flow rate Q: H pump -kq k1 cot ku uq 1 (1) Where Hpump denotes theoretical pressure head, Q denotes theoretical flow rate, u represents tanential velocity of pump, represents anle of vane, b is width of impeller, is diameter of entrance, K1 and K is coefficient of correction, is acceleration of ravity. There are many empirical formulas of pipin systems resistance loss [4], [5]. This paper selects slidin bed and Heteroeneous suspension flow model as follows: H pipe 33L s (1 w / s )C f 8LQ Re0.5 s (0.314 Re ) m + j 1 8 j Q 4 H () Where Hpipe denotes pipin systems resistance loss, Q denotes theoretical flow rate, Cf denotes slurry concentration, vs represents terminal settlin velocity, s represents relative Copyriht 016, the Authors. Published by Atlantis Press. This is an open access article under the CC BY-NC license ( 81

2 density of slurry, w represents relative density of water, is acceleration of ravity, j is local loss coefficient, is diameter of mud pipe, H is altitude, L is lenth of pipeline, Re is Reynolds number. The parameters mentioned above are closely related to the specific machine and construction environment. There are no numerical values for reference. C. Introduction of Stable Operatin Condition Random disturbin factors usually affect cutter suction dreders operatin condition. Actually, cutter suction dreders operatin condition tends to stabilize radually. Enineers suppose that the operatin condition et stable. This paper will concentrate on those stable operatin conditions for further study. Cutter suction dreder stable operatin condition is determined by cross point of the mud pump characteristic curve and the intersection of the pipeline characteristic curve [6], and this conclusion is basis for selectin stable operatin conditions, i.e. Hpump = Hpipe, the physical meanin of this equation is that the pressure head provided by the pump for slurry is balanced with the head consumed by the pipeline. III. ETERMINATION OF THE UNKNOWN PARAMETERS OF SLURRY TRANSPORT SYSTEM BASE ON THE CLEMS LAW As mentioned above, the parameters of the mud transport system are closely related to the specific construction project. In this section, the unknown parameters of the drede slurry transport system under the construction environment of Tianshi-Caofeidian Project will be solved correctly Accordin to the control equation Hpump = Hpipe, we can et followin equation: b Q u Q C f u 1. By definin a parameter vector a A B C E a b vector as. and are vectors in the R5 space, i.e: A*Q B*uQ C*C f *u E*1 0 (7) The equation implies that the vector a and b are perpendicular to each other in R5 space. In this paper, we select multiple sets of operatin condition vector roup b i i 1,, 3 4 j and solve the correspondin parameter vector a by the followin equation: A B C E (Q )1* (uq)1* (C f )1* (u )1 (Q ) * A (u Q) * B (C ) * C E (u ) f (Q ) * A (u Q ) * B (C ) * C E (u ) 3 3 f 3 3 A B C E (Q ) 4 * (uq) 4 * (C f ) 4 * (u ) 4 (8) 33L s (1 w / s )C f m 8 j Q 8LQ s (0.314 Re ) Re0.5 j 1 H =-K Q - K1 cot Ku uq 1 (3) After combinin some unknown parameters, we can et followin equation: AQ BuQ CC f u E 0 (4) Where: m 8 j 8L A K s (0.314 Re ) j 1 4 (5) B 33L s 1 w / s Cf K1 cot C Re0.5 K1 E H (6) In all stable operatin conditions, a lare number of data of flow conditions, mud concentration and mud pump tanential flow velocity can be used to make up the operatin conditions FIGURE II. THE INNER PROUCT OF ATA USING FOR VERIFYING ( A / ) j (Q )1 ( B / ) (Q ) j = (C / ) j (Q )3 ( E / ) j (Q ) 4 (u Q)1 (u Q) (u Q)3 (u Q) 4 1 (C f )1 1 (u )1 (C f ) 1 (u ) * (C f )3 1 (u )3 (C f ) 4 1 (u ) 4 (9) After proper screenin, the determinant of the coefficient matrix of the system has a unique solution accordin to the Cramer s law. Consider the actual measurement error, many parameters are not all parallel to the supper plane spanned by 8

3 these vectors, formin a small deviation to each other. The max P QC f. This paper solution here is to take the averae of many uses Matlab to select 500 sets of data, calculate the averae of AQ BuQ CC f u E 0 s.t 0 u u max 0 C 1 f aj aj 500 a j a j 1 500, which is noted by. The calculation results are as follows: A j 500 ( 500 ) j 1 ( A / ) 500 B j ) ( ( B / ) j C (C / ) ( )j ( E / ) j E ( )j j (11) Construct the Laranian auxiliary function based on the Larane multiplier method as follows: L QCf (A/ Q ) (B/ u ) Q (C / )Cf u (E / ) (1) Where: ( A / ) ( B / ) (10) In order to verify the reasonableness of the parameter vector, we choose 1400 sets of data to construct 1400 independent operatin condition vectors, aain usin Matlab. And the calculated parameters vector is respectively calculated with 1400 independent operatin condition vectors. As is shown in fiure, the result of the product fluctuates up and down at value zero, which means that the solution of parameter vector has a stron rationality. (C / ) Resolvin L Q, C f, u, the ( E / ) stationary point of (13) the function by the followin equations: L Q, Cf, u, 0 L Q, Cf, u, 0 Cf L Q, Cf, u, 0 u Q *( A / ) uq*(b / ) Cf *(C / ) (E / ) u (14) FIGURE III. THE VALIATION OF RESULTS IV. PROUCTIVITY OPTIMIZATION MOEL BASE ON LAGRANGIAN ALGORITHM We are committed to discuss optimization of cutter-suction dreder s control variables for a hih-yield taret, i.e. what we want is, the most value of the objective function P=Q*Cf [7], [8], where P is the output per unit time, followin is mathematical statement of this problem: FIGURE IV. THE VALIATION OF RESULTS The equations have no real solution, i.e. the objective function has no extreme value point. It shows that the maximum point is the boundary point of the constraint, so we will confirm where to et the point by the followin analysis. 83

4 This paper makes the partial derivatives on both sides of the constraint condition Hpump=Hpipe, and by analyzin the numerical properties of related parameters [9], we et followin results: cot K1 Q u b 0 = K cot u H pipe K Q 1 u b C f * 58% Q*.54m3 / s u* 57.5rad / s (15) Pmax Q(umax, C f * )*C f * =.54*0.58=1.47m3 / s V. (16) Assumin the pipe slurry concentration is certain, the yield is max, when u=umax, i.e. drede pump s speed n is max within an allowable rane for equipment. And if the drede pump s speed is certain, we can conclude the reater the slurry concentration, the smaller the pipe flow. As we can see, the conclusion is consistent with that ot by the empirical optimization method in Wan Qison s paper [8]. However, we verify the conclusion with specific mathematical model from mathematical perspective. And we can find that Pm ax Q ( u m ax, C f * ) C f *, i.e. the relation of Pmax(Cf) and slurry concentration Cf is not the direct or inverse proportion. The cutter suction dreder system has the optimal solution to maximize the yield per unit time. The problem has been transformed into solvin the sinle variable extreme of slurry concentration. Equations are as follows: (19) APPLICATION OF THEORETICAL ANALYSIS We select the data of stable operatin conditions in specific project to compare the theoretical value with actual value of slurry density and flow rate in fiure 3 and fiure 4 We select the data of stable operatin conditions in specific project and fiure out the theoretical maximum yield per unit time, and then we compare its theoretical value with actual value in fiure 5. As we can see, the theoretical maximum is reater than the most actual data, so the result is reasonable. In practical enineerin, most of us chane the mud pump speed n, reamer transverse movin speed Vt, reamer rotatin speed Vr and dredin depth h to control the work [1]. The flow and the slurry density in tubes are observed quantities for reference. In the previous sections, in order to et the maximum yield per unit time, the upper limit value of drede pump nmax should be set. And we fiure out Pmax=1.47m3/s, select all the stable points which meet P [1.44,1.50] to form the set 1. Since the yield per unit time of points in 1 is max, it s feasible to determine the values of control variables by analyzin the points in P u, C f Q u, C f Cf 0 H pump H pipe u u max (18) With further solution we also et: H pipe 0 H pipe K cot u K Q 1 b By solvin the equations, we can et followin results: 1. Throuh the statistics of control variables numerical properties in 1, we conclude the control interval of main control variables as Table I: TABLE I: OPTIMAL VALUE RANGES OF VARIABLES (17) Control variable reamer transverse movin speed (m/s) reamer rotatin speed (rad/s) dredin depth(m) VI. FIGURE V. THE VALIATION OF THEORETICAL INSTANTANEOUS OUTPUT Value rane [0.16,0.1] [.76,.90] [15.43,15.79] CONCLUSION Improvin cutter suction dreder s production has a sinificant influence on actual enineerin. Cutter suction dreder s construction control is closely related to the specific machine and construction environment. At present, there is few strict mathematical models provin the existin research results at home and abroad, and most of them are qualitative. This paper puts forward the non-linear productivity optimization model based on Laranian alorithm to provide the optimal value ranes of dreder s main control variables and also proves some existin experimental conclusions. However, there is still a hue research space for improvin the control value s precision. 84

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