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Minimum mass design of a three-throw plunger pump
Int. J. Pres. Ves. & Piping 49 (1992) 35-59
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Minimum Mass Design of a Three-Throw Plunger Pump Jerzy Kowalski School of Engineering, Autonomous University of Zacatecas, 98000 Zacatecas, Mexico
& Konrad Pylak Department of Mechanical Engineering, Technical University of Lublin, ul. Nadbystrzycka 36, 20-618 Lublin, Poland (Received 14 February 1991; accepted 13 April 1991) ABSTRACT Because of problem complexity, the selection of optimum design features for a plunger pump exceeds even in the best designer's possibilities and abilities if he is only using traditional design methods. The goal of the present paper is to present a method for selecting the optimum design features of three-throw plunger pump with a given structure for minimum mass. The pump is treated as a system composed of two subsystems such as the main part and crank mechanisms. The 43-dimensional optimization problem has been reduced by decomposition-making to the 12-dimensional coordination problem and two decomposed problems with 16 and 15 dimensions each, resolved in parallel. The permissible region is created by 66 stress, stiffness, total life, vibration, noise, design and assembly inequality constraints. As an application example, an existing Polish pump has been optimized.
1 INTRODUCTION The problems of o p t i m u m displacement p u m p design, and impeller p u m p design are seldom investigated. The design optimization of the closed impeller for a centrifugal p u m p has been considered, minimizing the total loss of head within the impeller.1 The optimization model only contains two variables. A simplified optimization model of a gear p u m p 35 Int. J. Pres. Ves. & Piping 0308-0161/91/$03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland
Jerzy Kowalski, Konrad Pylak
36
including three variables has been also formulated. 2 As the objective function, the gear diametral section was assumed. A multiobjective approach to optimum design has been reported for a horizontal centrifugal pump. 3 The components of the preference function are: the pump mass, total efficiency, the equalization of ball bearing total life and noise intensity level. The optimization problem is reduced to finding the minimum of the preference function in a 16dimensional space bounded by 38 inequaIity constraints. The optimum design problem for a reciprocating motion piston pump leads us to mechanical system optimization. To formulate and solve the problem, it is necessary to decompose the optimization model. The goal of this paper is to present a method for selecting the optimum design features of a three-throw plunger pump with a given structure, for minimum mass.
2 PUMP O P T I M I Z A T I O N P R O B L E M D E C O M P O S I T I O N A N D ITS F O R M U L A T I O N
Figure 1 shows the pump kinematic diagram, but the graph of the pump structure design is given in Fig. 2. The pump is treated as a system
"I"
j:3
!
rstern 1
Fig. 1. Pump kinematic diagram.
Minimum mass design of a three-throw plunger pump
~-1~
. Subsystem 1
37
Subsystem 2 D.
Fig. 2. Graph of pump structure design. 1, plunger; 21, slider; 22, bearing bush of slider base; 3, slider pin; 41, slider tube; 421, screw of slider tube-and-cylinder joint; 5, force air chamber; 6, cylinder; 7, suction air chamber; 81, sleeve packing; 82, plunger packing; 83, packing gland; 841, packing gland screw; 91, multi-ring valve; 101, valve plate; 111, valve spring; 1211, valve pilot bar; 1221, nut of valve pilot bar; 1231, nut washer of valve pilot bar; 131, sleeve of connecting-rod small end; 132, connecting-rod; 133, sleeve of connecting-rod big end; 141, cover of connecting-rod big end; 142, cover sleeve of connecting-rod big end; 1431, connecting-rod screw; 15, crankshaft.
composed of two subsystems such as: the main part (subsystem 1) and the crank mechanism (subsystem 2) without power transmission including geared flywheel. As it is insensitive to the assumed objective function, the housing can be excluded from optimization. (The insensitivity is produced by certain free choice of forming the housing dimensions.) On the other hand, the flywheel will be selected after carrying out the optimization procedure. Based on the existing pump, 4 the geometric models of design elements have been created. For example, Fig. 3 shows the geometric model of pump cylinder. The geometric model of pump crankshaft is presented in Fig. 4. On the other hand, Fig. 5 gives the idea of pump optimization model decomposition. The primary problem is treated as the pump design optimization problem regarding minimum mass of the total system. It can be transformed to the following decomposed form: minm(~,/~) = min m(~, )7,/~) ~v
~x07)
min
r
[ min ~1(~[~1,~1, pl) _~_ min m2(~ 2, ;2,/~2) 1j
(1)
Therefore, the decomposition is based on the separation of the primary problem in two problems of the subsystem level I and one problem of
Jerzy Kowalski, Konrad Pylak
38
5
8~
/
Fig. 3.
Geometric models of pump cylinder assembly.
\
I,'\ -_'t s/ ~'~"
"~'
/ r',~~~.,'~ ~ Pig. 4.
Geometric model of pump crankshaft.
40
Jerzy Kowalski, Konrad Pylak System Three-throw plunger pump (43 variables) ~"
-Subsystem I Pump main part - -~,I (16variables)
System level ( I I ) (12 coordinative variables) Y
Subsystem 2 Subsystem level (I) -I Crank mechanism [~'2 (15 variables)
Fig. 5. Pump optimization model decomposition.
the coordinative level II. Moreover, primary level are based on finding: (a) (b)
particular problems for the
the vector set £~o0,07) • X l ~ ) , minimizing the function m'(21, yl) and the function m~mi.(21p,03), )71) for the subsystem 1; the vector set 22p,~) • X207), minimizing the function m2(22, )72) and the function rn2mi.(22op,(~),)72) for the subsystem 2.
On the other hand, the problem of the level II is based on finding the vector )7op,• y and determining the vector £ °0' • X03 = )7op,) which minimizes the following function m~mi.(2~opt07), )7') + m2i,(22pt07), )72) and the determination of m m~". The decomposition also encloses adequate decomposition of the variable vector fi given above. It is separated to the coordinative variable vector )7 for the system and the subsystem variable vector 2. If it is necessary to differ the variables for particular subsystems, )7 and 2 vectors are consequently separated to )71 and )72 vectors and 2 ~ and 22 vectors, respectively. The coordinative variable vector ;1 for the first subsystem is given by )71 = (D, S, gl, L2, B2, D20, 120, 0.2, d3, 13, d062 ^ d~62)
(2)
and for the second subsystem )72 = (Ik)
(3)
where the plunger stroke, S, inside diameter, D, and wall thickness gl, the slider base length, L2, and width, B2; the slider inside diameter, D2o, inner wall spacing, 12o, and guide diameter D~; the slider pin diameter, d3, and length, /3, the full diameter, do62, of packing gland
Minimum mass design of a three-throw plunger pump
41
screw (thread root diameter dr62=combine variable) and the connecting-rod length, lk, are treated as the coordinative variables. The subsystem variable vector ~1 for the first subsystem can be expressed as ~1 = (g4, io4, do4 ^ dr4, d6, g6, 16ol,/61, i062, h7, n9,
h~9, mzg, dii, Dll, nil, dzl2 A dr12)
(4)
and for the second subsystem ~2 = (1i3, gi3, Do13, B, H, gp, gs, Dt, dk, lsk, az,4 A aria, gr, hr, d9,, 191)
(5)
The subsystem variables are: the thickness, g4, of slider tube flange; the screw number, io4, and full diameter, d04, for slider tube-and-cylinder joint (thread root diameter dr4 --- combine variable); the multi-ring valve full diameter, d6; the cylinder wall thickness, g6; the cylindrical part length, 1601, for force air chamber; the suction air chamber length, 161; the packing gland screw number, i062; total length, h7, of suction air chamber; the valve ring number, n9; the rib height, h~9, and number, m~9, for multi-ring valve; the valve spring wire diameter, d~l, coil mean diameter, DH, and coil number, nil; the screw full diameter, dz12, for valve pilot bar (thread root diameter, dri2 = combine variable); and the connecting-rod small end length, 113, thickness, g13, arid inside diameter, D013; 1-section width, B, height, H, flange thickness, gp, and web thickness, gs, for connecting-rod shank; the connecting-rod big end inside diameter, Dt; the crank-pin diameter, dk; the connecting-rod big end length, lsk; the connecting-rod screw full diameter, dz14 (thread root diameter dr14=combine variable); the crankweb thickness, gr, and height, hr; the crankshaft main bearing journal diameter, d91, and length, 191. The subsystem parameter vector/~1 for the first subsystem is ffi = (flu, Ds ^ Dks, ls7, D* ^ Okt, It6, n, r/v, eg A bg ^ Sg, 3g, hmax, C~9); u = 1, 21, 22, 3, 41, 421, 5, 6, 7, 81, 82, 83, 841, 91, 101, 111, (6) 1211, 1221, 1231 and for the second subsystem
~2 = (Pv);
'/3 = 131, 132, 133, 141, 142, 1431, 15
(7)
where Po, = mass density of the wth design element material D~ = suction pipeline diameter (Dk~= suction pipe flange; combine parameter)
42
Jerzy Kowalski, Konrad Pylak
suction pipeline length for air chamber D* = f o r c e pipeline diameter ( D k t = f o r c e pipe flange; combine parameter) /t6 = force pipeline length for pump cylinder n = crank rotation speed r/v = pump volumetric efficiency eg = ring mean spacing for valve (bg= ring thickness; Sg = gland ring thickness; combine parameters) /3g = rib inclination angle for valve hmax = maximum valve lift a~9 = ratio of mean valve diameter to real delivery of single-cylinder pump. The following operation is the constraint set decomposition. Based on the recommendations for separating the constraint set in optimum mechanical system design 5,6 and the second author's idea, the set of 66 inequality constraints formulated for the total system has been separated into three subsystems: ls7 =
(a)
Constraints exclusively concerning the coordinative variable space W °'t >- 0; W 0"12~" 0;
(b)
(yl variables) (37 variables)
Constraints concerning the variables of the subsystem 1 W~, -> 0; W~ '1 >- 0; W~,'~2 >- 0;
(c)
i = 1 , . . . , 12 j = 1,... , 4
k = 1. . . . 4 (£1 variables) l = 1 . . . . 12 (£~ and )71 variables) n = 1, 2 (£1 and )7 variables)
Constraints concerning the variables of the subsystem 2
w~,--- o; W~q,' -> o; W 2,12 > ft. W 2,2 s >- O;
p = 1, 2, 3 (£z variables) q = 1 , . . . 13 (£z and )7~ variables) r = 1, . . . 15 (£2 and f variables) s = 1 (£2 and )72 variables).
The particular permissible regions (sets of permissible vectors) are determined as follows: (a)
for the subsystem level I Xlff)
= {.1~1:W l ( . ~ 1) ~ 0 ("] W:'l()~ 1, y a ) ~ 0 ('~ Wln'12(£ 1, ; ) ~ O}
X 2 f f ) = (..1~2"W 2 ( £ 2) ~ 0 ("l Wq'21(~32, y l ) ~ 0 ~ ('~ w~2,2( ~2 ~ , y~) - o}
x f f ) = x'(y) u x~(y)
~2 ;)--w~2,12(.~,
0
(8)
Minimum mass design of a three-throw plunger pump
43
(b) for the coordinative level II y = {37:Wi°,1071 ) _> 0 n .
W ~'~ > " ' j° ' 1 2 ",9'/ -- 0 n
1,12 *1 * n W n (Xopt(y) , y) ~on
~1 w , 1,1(~op,(Y), Yl)---0 -
2,1 *2 "~ W q (XoptQ~) , ~ 1 ) ~ 0
n w22(~op,(y), ~) - 0 n wv(~op, fy), D - 0}
(9)
The constraint list is as follows: 0,1 V¢ 1-3
= design conditions determining the plunger basic dimensions; = condition eliminating the possibility of vibration resonance of liquid column on the pump delivery side, 7 (ensuring that the work is within subcritical region); wo,1 = condition eliminating the possibility of vibration resonance of liquid column on the pump suction side; WO,1 = condition limiting the pump noise intensity level; 8 WO,1 = stress condition for plunger wall; W% = stress conditions for slider-pin-guide design pair; 0,1 W l o , ll = design conditions determining the pin dimensions; wo~ = design condition determining the slider ~guide diameter; WO,12 = stress conditions for slider-pin-guide design pair; 1,2 WO,12 = design conditions limiting the connecting-rod length factor; 3,4 = stress condition for cylinder wall; w] = design condition determining the height of suction air w~ chamber; = design condition determining the height of valve body rib; w~ = design condition determining the screw full diameter for w~ valve pilot bar; W~,1 = design condition determining the basic dimensions of force air chamber; = stress condition for bending the valve body rib; w~. 1 = design condition determining the basic dimensions of suction air chamber; w~,' = stress condition determining the screw number for packing gland; w~:~ = stress and stiffness conditions for valve spring; w~, ~ = design condition determining the flange diameter for packing gland; 1,1 Ws...lO = design conditions determining the valve body dimensions; Wh1 = stress condition for screw of the valve pilot bar; w L 1 = design condition limiting the conic part length of pump main part; W1,12 = stress conditions of slider tube including joint; 1,2 WO,1
44
Jerzy Kowalski, Konrad Pylak W~ W~,3 WE'1 W~:~ W42:~ W2'~ 2,1 Ws-lO 2,1 Wl1,12
W2~1 W 2 14 ,12 W72'~2 W2,12
8,9
W2,12 10-12 W2,12
13,~4
W2~~2
W 2,2
= design condition determining the spacing of connecting-rod screws; = design conditions determining the dimensions of connectingrod small end; =design condition determining the sleeve thickness for connecting-rod small end; = design conditions determining the dimensions of connectingrod small end; = connecting-rod stress conditions; =stress and heating conditions for slide bearing of the connecting-rod in top dead centre; p = crankshaft stress conditions for $15, E~5 and $15 sections in top dead centre; = stress and heating conditions for B15 crankshaft slide bearing in top dead centre; = design condition determining the crankweb height; = stress conditions for connecting-rod including cover and screws; = condition eliminating the collision between the slider tube and crankshaft; =stress and heating conditions for slide bearing of the connecting-rod in crank position corresponding to maximum crank pin effort; = crankshaft stress conditions in crank position corresponding to maximum crank pin effort; = stress and heating conditions for B15 crankshaft slide bearing in crank position corresponding to maximum crank pin effort; = condition eliminating the possibility of resonance of the crankshaft torsional vibration by Geiger method 9 (ensuring that the work is within subcritical region); = condition eliminating the possibility of resonance of the connecting-rod transverse vibration.
Figure 6 shows the solution chart for the pump decomposed optimization model. 3 TWO-LEVEL HIERARCHIC OPTIMIZATION MODELING SYSTEM F O R T H E PUMP To increase the efficiency of pump mathematical modelling, the idea of a two-level hierarchic optimization modelling system has been used. 6,1°
Minimum mass design of a three-throw plunger pump System level (II)
..................... ] rain
m (-~)
45
mmin -~opt ~'opt +
-~¢y T Iteration
of 7 se,ectio.
/ Sets of
/ Xop,<7); / mu, ¢x% *(7)
LYL_°P' Subsystem level (1) -
Fig. 6,
-
rain m u (-~u,-~u) -'-Pu u "-" J "x eX ( y )
u=1-2
Solution chart for pump decomposed optimization model.
The modelling system is based on the orderly sequence of the optimization models in which the quantity model is a minute detail of the analytic-structural model. The analytic-structural model of the design object is an effective means to determine structural relations, correlations amongst the material parameters, and the sets of nonmaterial parameters occurring in the constraints. The formulation of the analytic-structural model also enables the creation of the object quantity model, by which this model is directly systematized. The structural relations are given by the optimization criterion and constraint matrices. The objective function matrix (Table 1) determines the occurrence of coupling amongst particular components of the objective function and the variables, and their repeatability in the component set. It is sparse. Because of additive type of the optimization criterion, the matrix must have the block-diagonal form. The exact subsystem segregation so that the objective function is separable regarding variables, appears to be impossible. The approximation has been used based on the overestimation of mr31 sleeve mass for the connecting-rod small end. It is exact when the W~'1 design condition is equally satisfied. The possible inaccuracy of this component has very little effect on the total objective function. On the other hand, the constraint matrix is given in Table 2. It determines the structural relations amongst the constraints and particular variables. The constraint matrix is complex and sparse. The correlations amongst the material parameters are presented in the model by means of the network diagram and the matrix of outer material parameters. Let us limit to representing the vector of material
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Minimum mass design of a three-throw plunger pump
51
parameters occurring in the constraints. It can be expressed as: ffM = (krcu, krlc132, krjp,
P~y-e, P,~15b, k~oe, k~t, k'~i132, kcj132, ksi111,
~x~r132, /~yoe132, E132, G,~, A132, B132, C15_133, C15_b)
(10)
where = allowable stress for alternate tension and compres-
krcac
sion; a~ = 1, 41, 6 (kr'c132 for the connecting-rod small end); krjp = allowable pulsatory tensile stress; fl = 421, 841, 1221, 1431; allowable unit pressure; y-t~ = 2 1 - 3 , 22-41, 3-131, P ~v-6 = 15-133 (P~ls-b) for crankshaft and sleeve of crankshaft main bearing journal); allowable oscillatory bending stress; e =3, 15 (to kgoe compensate the effect of flywheel mass and belt tension, kgo15 value must be reduced at 20%); allowable pulsatory bending stress; ~ = 91, 141 (k~132 for the connecting-rod big end); kcj132 = allowable pulsatory compression stress for the connecting-rod shank; ksjll I = allowable pulsatory torsional stress for the valve spring; nxa¢132, nyor132 --'~ safety factors for buckling the connecting-rod shank; E132 -- connecting-rod elastic modulus; GZ = shear modulus; r/= 111, 15, A132, B132 = factors occurring in Tetmayer buckling formula; C15-133, C15-b = heat dissipation factors for crankshaft slide bearings. The sets of the nonmaterial parameters occurring in the constraints may be presented with the aid of nonmaterial parameter vector of the first kind (characterizing the significant parameters) and vector of the second kind (enclosing the parameters of less significance). The parameter vector of the first kind is given by :1 = [a, ~
A p, L*,/-/~, Ls*, n, r/v, (S/O)min, (S/O)max, LI~]
where Q = H~t = p = L* = =
pump calculation delivery static force lift positive gauge pressure in the pump cylinder force pipeline length static suction lift
(11)
Jerzy Kowalski, Konrad Pylak
52
L* = suction pipeline length (S/D)min , (S/D)max minimum and maximum values of the relation of plunger stroke to diameter; LI~ = admissible pump noise intensity level. =
On the other hand, the parameter vector of the second kind can be expressed as: 132 = [Kp, 6p,, 6ps, eg A bg A Sg, fig, a~9, (P)homin, (P)homax, ~9, ~'min, ~'max,ap, ap, e*]
(12)
Here Kp = overload factor; 6pt, 6ps=factors of pressure fluctuation for force and suction chambers; (P)homi.,(P)h . . . . = m i n i m u m and maximum loads for multi-ring valve; ~9 = calculation factor for valve weight including hydrostatic lift; Zmi,, Amax= minimum and maximum values of the connecting-rod length factor; ap, ap = casing allowances increasing theoretical wall thickness of the plunger and cylinder; e* = relation of delivery dynamic component to average delivery. Based on principles for the model transformation, 6'1° the quantity model for the pump has been formulated. In this model, one quantifies all kinds of relations as the mathematical formulas, i.e. equalities and inequalities. It contains the nonlinear objective function of 43 variables and 66 inequality constraints including 10 linear ones. Let us look at one of the typical elements of the objective function, i.e. the cylinder mass which strongly influences the mass, m 1, of the pump main part (subsystem 1). It may be presented as: f:r D2k6 - D62)gk6 + ffl( ~-[(D6 + 2g6) 2 - D 2] (0.5.16ol - gk6 + D* m6 = p6/~( + 16o2+ 16o3+ 16o4) + 2 . 4 [(D6 + 2g6) 2 - d~]h6 q- 4 [(D6 -t- 2g6)2-d21](h6-1 - 5) q-4 (D261-d21)gk61 + ~" [(d61 + 2g6) 2 - d21]1611- ~~ Dt,2g6 - ~ D23g6
53
Minimum mass design of a three-throw plunger pump 7f 2 7f -I- ~" (D2t - D* )g6 + ~- [(D* + 2g6) 2 - D*2](lt6-g6)
-t- ]'~/63[(D64 + 2g6) 2 4" (D64 + 2g6)(D63 + 2g6) + (063 -4- 2g6) 2]
~2 1 6 3 ( 0 2 + 0 6 4 0 6 3 "~ 0 2 3 ) -4- -~- (064 -q- 2g6)2166 flit
"~
2
"~ 2
•
ff~'
+ ~ (D65 + 2g6)2(/67 -t- 168) + ~ Dk4g4 -- ~- d~g4t04 +~- D 2 • 2 7f
Jt
2
"" - T 4f d 2 062/56 '~ i 062 - - 4 D64164-~D25165~4 "[- "4" D 2w62~6
J
(13)
where 16o2= Hnl2 -~ 5 -J¢-1121 + Hll + 0.6//9 - 0-5Dt*
(14)
with nut height for valve pilot bar Hn12 = 0-8dz12
(15)
1121 = hm~, + 2dll
(16)
H9 = 1.2bg + h~9 + 5
(17)
Hll = 4 x 1-2(1.2bg - 10) + n~(d~l + 1)
(18)
D64 = D + 2 V ~
(19)
063---- 1"6/94
(20)
163 = S + 0"50 - 2 - / 9 6 - 2g6
(21)
06 = d6 + 15
(22)
D6s = D + 4 V ~
(23)
/65 = 5 + Is2 + g6 = 5 + D + g6
(24)
/64 =/66 + 167 - 5 = 0 - 5
(25)
loaded spring height
multi-ring valve height
and valve plate height
Consequently
Let us examine one of the simple constraints which appeared, however, to be active in numerical calculations, i.e. the stress condition
JerzyKowaiski,KonradPylak
54
for cylinder wall. It is formed as:
x/kr~6+_0.4p W~=g6- [~ ( ¥kr~6__ l.3p
1)+ a~]-0
(26)
Other typical objective function components and constraints are presented in a previous paper. 6
4 NUMERICAL EXAMPLE Let us consider the existing pump with the following basic parameters: 4 Q = 0.05
m3/s,
/~t = 125 m,
Lt* = 23 0a,
n = 170 rpm
The design materials are assumed as follows: •
•
•
• •
• • • •
plungers, sliders, slider tubes, force air chambers, cylinders, suction air chambers are grey cast iron Z120 (equivalent GG-20 DIN); slider pins, screws of the slider tube-and-cylinder joints, packing gland screws, connecting-rod screws, crankshaft are constructional carbon steel St5 (St50 DIN); valve plates and pilot bars, nuts and nut washers of valve pilot bars, connecting-rods are constructional carbon steel St3 (St37 DIN); packing glands, covers of connecting-rod big ends are constructional Carbon steel St4 (St42 DIN); sleeve packings, sleeves of connecting-rod big ends, sleeves of crankshaft main bearing journal are tin-phosphor bronze CuSnl0P (GBz-10 DIN); bearing bushes of slider bases are Babbit metal SnSb11Cu6 (WM80 DIN); multi-ring valves are grey cast iron Z135 (GG-35 DIN); valve springs are silicon steel 50S2 (50Si2 DIN); plunger packings are packing cord.
By analysis, it was stated that the existing design fulfills all constraints. It enables us to determine the starting point for numerical computations. To solve the two-level optimization problem, the method of systematic searching is used based on optimization of a variable and constraint arrangement and progressive reduction of the searching step and variation block. Because of the large number of variables and their discrete distribution, as well as the large number of constraints including complex types of functional relations, it is recommended that
Minimum mass design of a three-throw plunger pump
55
TABLE 3
The Values of the Objective Function and Its Components for Existing and Optimum Pumps Objective function component
Existing pump
Optimum pump
m (kg) m I (kg) m2 (kg)
5 296-826 4 972.576 324.250
4 081-044 3 860.903 220.141
this m e t h o d be used. For that reason, the suitable systematizing of the constraint set is very important. A PC/AT-386 computer has been used. C P U time was equal to about 3 h for o p t i m u m and 26 suboptimum solutions, for the preliminary reduced variable block. Table 3 gives the values of the objective function and its components for existing and o p t i m u m pumps. The p u m p mass reduction at 22.95% (1215.782 kg) has been achieved compared with the existing design. It corresponds to a 23.25% material cost reduction (that means about $120). For lot production of pumps, it is possible to receive significant material cost reduction. On the other hand, the values of coordinative variables for existing and o p t i m u m pumps are presented in Table 4. The values of variables of both subsystems are given in Tables 5 and 6, respectively. The optimum p u m p is characterized by a smaller average piston speed at 4.29% (0-102 m/s) and a decrease in the noise intensity level at 0.44% TABLE 4
The Values of Coordinative Variables for Existing and Optimum Pumps Variable
Existing design
Optimum design
D (mm) S (mm) gl (mm) L2 (mm) B2 (mm) D20(mm) 12o(mm) D~2 (mm) d3 (mm) 13(mm) do~2(mm) lk (ram)
140 420 12 310 210 200 76 380 50 126 16 1 150
136 402 11 290 186 180 80 360 42 120 14 1 080
Jerzy Kowalski, Konrad Pylak
56
TABLE 5
The Values of Variables of Subsystem 1 for Existing and Optimum Pumps
Variable g4 (mm) io4 do4 (mm) d6 (mm) g6 (mm) 1~ol(mm) 161(mm) i6~ (ram) h7 (mm) n9 h~9 (mm)
Existingdesign
Optimum design
20 16 20 415 22 594 400 8 660 3 105
16 8 16 380 20 570 370 8 640 3 85
mz9
6
6
dll (mm) Dll (ram) nit d~lz (ram)
7 70 8 14
7 80 6 10
TABLE 6
The Values of Variables of Subsystem 2 for Existing and Optimum Pumps
Variable 113(mm) g13 (mm) Do~3(mm) B (ram) H (mm) gp (mm) gs (ram) Dt (mm) dk (mm) lsk (mm) dz14 (mm) g~ (mm) h r (mm)
d91 (mm) 194 (mm)
Existingdesign
Optimum design
72 35 58 65 120 10 10 200 160 90 20 35 400 150 80
60 30 54 54 105 8 8 180 130 80 16 28 380 135 65
Minimum mass design of a three-throw plunger pump
57
TABLE 7 The Absolute and Per Cent Margins of Active Constraints for Optimum Pump
Most active constraints
Optimum design Absolute margin
Per cent margin
W19J W~1
0.641 30 12-000 0 0"002 25 0.044 12 3 066 248 36 496"0 0"13685 5"901 5 15"649 4
3-31 1-80 4"50 1"47 2"86 0"04 1"99 4"16 0"21
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/
i :392
i
394
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396
398
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402
404
406
408
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Fig. 7. D - S plane of coordinative variable space: - - , constraint lines only depending on D.S. variables; . . . . , constraint lines also depending on variables of subsystem 1; [~77~, suboptimum solution region; (~), ith solution (the first is the optimum one.
58
Jerzy Kowalski, Konrad Pylak
(0.89 dB). Based on Ref. 11, the flywheel has been selected. Its mass is reduced at 11.94% (11-673 kg). Table 7 gives the absolute and percentage margins of active constraints for the optimum pump. Let us note that the other constraints, e.g. W~4, W 2'1, W 2'12 have limited the permissible region for the suboptimum solution area. Amongst 39 constraints depending on material parameters, 33 conditions have been improved in the direction of better utilization of material properties; W~i1, W216~, w2,~2 became rr 2,6 worse, but W2~12 and W~'2, practically, do not show any change. One of the more interesting sections of the variable space is the plane of D - S coordinative variable space (Fig. 7). Full lines show constraint lines only depending on S and D variables. On the other hand, the broken lines represent constraint lines also depending on the variables of the subsystem 1. The lining area shows the suboptimum solution region; i is the ith solution ( i - - 1 , 2 , . . . 27). The values of m ~ component for particular solutions are equal to 3860.903, 3861.205, 3861.506, 3861.400, 3861-416, 3861.731,... 3865.238kg, respectively. The solutions 1 , 2 , . . . 2 3 are given for h T = 6 4 0 m m . On the other hand, the solutions 24, 25, 26, 27 located out of W~,~ constraint are for h7=645 mm. The remaining variables have the same values. The component m E = 220.141 kg is the same for all solutions. Figure 7 points at flatness of the objective function in the neighbourhood of the minimum, at least in the D - S section.
5 CONCLUSIONS The presented optimization method can be easily adapted, e.g. to high pressure plunger pumps.
REFERENCES 1. Reddy, Y. R. & Kar, S., Optimum vane number and angle of centrifugal pumps with logarithmic vanes. J. Basic Engng. 93 (1971). 2. Sav~enko, V. S., Introduction into Optimum Machine Design. Nauka i Technika Publishers, Minsk, 1974 (in Russian). 3. Kowalski, J., Method of mathematical modeling the horizontal centrifugal pump for optimum design. La Ventana de la UAZ, No. 159, 4-5 (1989) (in Spanish). 4. Demeter, T., Collection of Design Problems. WSIP Publishers, Warsaw, 1974 (in Polish).
Minimum mass design of a three-throw plunger pump
59
5. Kowalski, J., Large object mathematical modelling in optimal design of machine construction. Elements of methodology of creating the optimization models. Strojarstvo, 24 (1982) 125-31. 6. Kowalski, J,, Modeling of Design Objects for the Optimum Design. W N T Publishers, Warsaw, 1983 (in Polish). 7. Brill, K. F., Berechnungen von Druckstoi3schwingungen in hydraulischen PreBanlagen", Konstruktion, 12 (1960) 31-5. 8. Harris, C., Handbook of Noise Control. McGraw-Hill, New York, 1957. 9. Rytel, Z., Piston machines. Mechanics of crankshaft assembly. In: Mechanical Engineer Handbook, vol. 2. WNT Publishers, Warsaw, 1970 (in Polish). 10. Kowalski, J., An appropriate strategy for mathematical modeling in optimum design of machine construction. J. Mech. Transmiss. Autom. Des., 107 (1985) 463-76. 11. Bouch6, Ch., Die kolbenpumpen. In: Dubbels Taschenbuch fiir den Maschinenbau, Bd. II. Springer-Verlag, Berlin/G6ttingen/Heidelberg, 1963.
r
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Minimum Mass Design of a Three-Throw Plunger Pump Jerzy Kowalski School of Engineering, Autonomous University of Zacatecas, 98000 Zacatecas, Mexico
& Konrad Pylak Department of Mechanical Engineering, Technical University of Lublin, ul. Nadbystrzycka 36, 20-618 Lublin, Poland (Received 14 February 1991; accepted 13 April 1991) ABSTRACT Because of problem complexity, the selection of optimum design features for a plunger pump exceeds even in the best designer's possibilities and abilities if he is only using traditional design methods. The goal of the present paper is to present a method for selecting the optimum design features of three-throw plunger pump with a given structure for minimum mass. The pump is treated as a system composed of two subsystems such as the main part and crank mechanisms. The 43-dimensional optimization problem has been reduced by decomposition-making to the 12-dimensional coordination problem and two decomposed problems with 16 and 15 dimensions each, resolved in parallel. The permissible region is created by 66 stress, stiffness, total life, vibration, noise, design and assembly inequality constraints. As an application example, an existing Polish pump has been optimized.
1 INTRODUCTION The problems of o p t i m u m displacement p u m p design, and impeller p u m p design are seldom investigated. The design optimization of the closed impeller for a centrifugal p u m p has been considered, minimizing the total loss of head within the impeller.1 The optimization model only contains two variables. A simplified optimization model of a gear p u m p 35 Int. J. Pres. Ves. & Piping 0308-0161/91/$03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland
Jerzy Kowalski, Konrad Pylak
36
including three variables has been also formulated. 2 As the objective function, the gear diametral section was assumed. A multiobjective approach to optimum design has been reported for a horizontal centrifugal pump. 3 The components of the preference function are: the pump mass, total efficiency, the equalization of ball bearing total life and noise intensity level. The optimization problem is reduced to finding the minimum of the preference function in a 16dimensional space bounded by 38 inequaIity constraints. The optimum design problem for a reciprocating motion piston pump leads us to mechanical system optimization. To formulate and solve the problem, it is necessary to decompose the optimization model. The goal of this paper is to present a method for selecting the optimum design features of a three-throw plunger pump with a given structure, for minimum mass.
2 PUMP O P T I M I Z A T I O N P R O B L E M D E C O M P O S I T I O N A N D ITS F O R M U L A T I O N
Figure 1 shows the pump kinematic diagram, but the graph of the pump structure design is given in Fig. 2. The pump is treated as a system
"I"
j:3
!
rstern 1
Fig. 1. Pump kinematic diagram.
Minimum mass design of a three-throw plunger pump
~-1~
. Subsystem 1
37
Subsystem 2 D.
Fig. 2. Graph of pump structure design. 1, plunger; 21, slider; 22, bearing bush of slider base; 3, slider pin; 41, slider tube; 421, screw of slider tube-and-cylinder joint; 5, force air chamber; 6, cylinder; 7, suction air chamber; 81, sleeve packing; 82, plunger packing; 83, packing gland; 841, packing gland screw; 91, multi-ring valve; 101, valve plate; 111, valve spring; 1211, valve pilot bar; 1221, nut of valve pilot bar; 1231, nut washer of valve pilot bar; 131, sleeve of connecting-rod small end; 132, connecting-rod; 133, sleeve of connecting-rod big end; 141, cover of connecting-rod big end; 142, cover sleeve of connecting-rod big end; 1431, connecting-rod screw; 15, crankshaft.
composed of two subsystems such as: the main part (subsystem 1) and the crank mechanism (subsystem 2) without power transmission including geared flywheel. As it is insensitive to the assumed objective function, the housing can be excluded from optimization. (The insensitivity is produced by certain free choice of forming the housing dimensions.) On the other hand, the flywheel will be selected after carrying out the optimization procedure. Based on the existing pump, 4 the geometric models of design elements have been created. For example, Fig. 3 shows the geometric model of pump cylinder. The geometric model of pump crankshaft is presented in Fig. 4. On the other hand, Fig. 5 gives the idea of pump optimization model decomposition. The primary problem is treated as the pump design optimization problem regarding minimum mass of the total system. It can be transformed to the following decomposed form: minm(~,/~) = min m(~, )7,/~) ~v
~x07)
min
r
[ min ~1(~[~1,~1, pl) _~_ min m2(~ 2, ;2,/~2) 1j
(1)
Therefore, the decomposition is based on the separation of the primary problem in two problems of the subsystem level I and one problem of
Jerzy Kowalski, Konrad Pylak
38
5
8~
/
Fig. 3.
Geometric models of pump cylinder assembly.
\
I,'\ -_'t s/ ~'~"
"~'
/ r',~~~.,'~ ~ Pig. 4.
Geometric model of pump crankshaft.
40
Jerzy Kowalski, Konrad Pylak System Three-throw plunger pump (43 variables) ~"
-Subsystem I Pump main part - -~,I (16variables)
System level ( I I ) (12 coordinative variables) Y
Subsystem 2 Subsystem level (I) -I Crank mechanism [~'2 (15 variables)
Fig. 5. Pump optimization model decomposition.
the coordinative level II. Moreover, primary level are based on finding: (a) (b)
particular problems for the
the vector set £~o0,07) • X l ~ ) , minimizing the function m'(21, yl) and the function m~mi.(21p,03), )71) for the subsystem 1; the vector set 22p,~) • X207), minimizing the function m2(22, )72) and the function rn2mi.(22op,(~),)72) for the subsystem 2.
On the other hand, the problem of the level II is based on finding the vector )7op,• y and determining the vector £ °0' • X03 = )7op,) which minimizes the following function m~mi.(2~opt07), )7') + m2i,(22pt07), )72) and the determination of m m~". The decomposition also encloses adequate decomposition of the variable vector fi given above. It is separated to the coordinative variable vector )7 for the system and the subsystem variable vector 2. If it is necessary to differ the variables for particular subsystems, )7 and 2 vectors are consequently separated to )71 and )72 vectors and 2 ~ and 22 vectors, respectively. The coordinative variable vector ;1 for the first subsystem is given by )71 = (D, S, gl, L2, B2, D20, 120, 0.2, d3, 13, d062 ^ d~62)
(2)
and for the second subsystem )72 = (Ik)
(3)
where the plunger stroke, S, inside diameter, D, and wall thickness gl, the slider base length, L2, and width, B2; the slider inside diameter, D2o, inner wall spacing, 12o, and guide diameter D~; the slider pin diameter, d3, and length, /3, the full diameter, do62, of packing gland
Minimum mass design of a three-throw plunger pump
41
screw (thread root diameter dr62=combine variable) and the connecting-rod length, lk, are treated as the coordinative variables. The subsystem variable vector ~1 for the first subsystem can be expressed as ~1 = (g4, io4, do4 ^ dr4, d6, g6, 16ol,/61, i062, h7, n9,
h~9, mzg, dii, Dll, nil, dzl2 A dr12)
(4)
and for the second subsystem ~2 = (1i3, gi3, Do13, B, H, gp, gs, Dt, dk, lsk, az,4 A aria, gr, hr, d9,, 191)
(5)
The subsystem variables are: the thickness, g4, of slider tube flange; the screw number, io4, and full diameter, d04, for slider tube-and-cylinder joint (thread root diameter dr4 --- combine variable); the multi-ring valve full diameter, d6; the cylinder wall thickness, g6; the cylindrical part length, 1601, for force air chamber; the suction air chamber length, 161; the packing gland screw number, i062; total length, h7, of suction air chamber; the valve ring number, n9; the rib height, h~9, and number, m~9, for multi-ring valve; the valve spring wire diameter, d~l, coil mean diameter, DH, and coil number, nil; the screw full diameter, dz12, for valve pilot bar (thread root diameter, dri2 = combine variable); and the connecting-rod small end length, 113, thickness, g13, arid inside diameter, D013; 1-section width, B, height, H, flange thickness, gp, and web thickness, gs, for connecting-rod shank; the connecting-rod big end inside diameter, Dt; the crank-pin diameter, dk; the connecting-rod big end length, lsk; the connecting-rod screw full diameter, dz14 (thread root diameter dr14=combine variable); the crankweb thickness, gr, and height, hr; the crankshaft main bearing journal diameter, d91, and length, 191. The subsystem parameter vector/~1 for the first subsystem is ffi = (flu, Ds ^ Dks, ls7, D* ^ Okt, It6, n, r/v, eg A bg ^ Sg, 3g, hmax, C~9); u = 1, 21, 22, 3, 41, 421, 5, 6, 7, 81, 82, 83, 841, 91, 101, 111, (6) 1211, 1221, 1231 and for the second subsystem
~2 = (Pv);
'/3 = 131, 132, 133, 141, 142, 1431, 15
(7)
where Po, = mass density of the wth design element material D~ = suction pipeline diameter (Dk~= suction pipe flange; combine parameter)
42
Jerzy Kowalski, Konrad Pylak
suction pipeline length for air chamber D* = f o r c e pipeline diameter ( D k t = f o r c e pipe flange; combine parameter) /t6 = force pipeline length for pump cylinder n = crank rotation speed r/v = pump volumetric efficiency eg = ring mean spacing for valve (bg= ring thickness; Sg = gland ring thickness; combine parameters) /3g = rib inclination angle for valve hmax = maximum valve lift a~9 = ratio of mean valve diameter to real delivery of single-cylinder pump. The following operation is the constraint set decomposition. Based on the recommendations for separating the constraint set in optimum mechanical system design 5,6 and the second author's idea, the set of 66 inequality constraints formulated for the total system has been separated into three subsystems: ls7 =
(a)
Constraints exclusively concerning the coordinative variable space W °'t >- 0; W 0"12~" 0;
(b)
(yl variables) (37 variables)
Constraints concerning the variables of the subsystem 1 W~, -> 0; W~ '1 >- 0; W~,'~2 >- 0;
(c)
i = 1 , . . . , 12 j = 1,... , 4
k = 1. . . . 4 (£1 variables) l = 1 . . . . 12 (£~ and )71 variables) n = 1, 2 (£1 and )7 variables)
Constraints concerning the variables of the subsystem 2
w~,--- o; W~q,' -> o; W 2,12 > ft. W 2,2 s >- O;
p = 1, 2, 3 (£z variables) q = 1 , . . . 13 (£z and )7~ variables) r = 1, . . . 15 (£2 and f variables) s = 1 (£2 and )72 variables).
The particular permissible regions (sets of permissible vectors) are determined as follows: (a)
for the subsystem level I Xlff)
= {.1~1:W l ( . ~ 1) ~ 0 ("] W:'l()~ 1, y a ) ~ 0 ('~ Wln'12(£ 1, ; ) ~ O}
X 2 f f ) = (..1~2"W 2 ( £ 2) ~ 0 ("l Wq'21(~32, y l ) ~ 0 ~ ('~ w~2,2( ~2 ~ , y~) - o}
x f f ) = x'(y) u x~(y)
~2 ;)--w~2,12(.~,
0
(8)
Minimum mass design of a three-throw plunger pump
43
(b) for the coordinative level II y = {37:Wi°,1071 ) _> 0 n .
W ~'~ > " ' j° ' 1 2 ",9'/ -- 0 n
1,12 *1 * n W n (Xopt(y) , y) ~on
~1 w , 1,1(~op,(Y), Yl)---0 -
2,1 *2 "~ W q (XoptQ~) , ~ 1 ) ~ 0
n w22(~op,(y), ~) - 0 n wv(~op, fy), D - 0}
(9)
The constraint list is as follows: 0,1 V¢ 1-3
= design conditions determining the plunger basic dimensions; = condition eliminating the possibility of vibration resonance of liquid column on the pump delivery side, 7 (ensuring that the work is within subcritical region); wo,1 = condition eliminating the possibility of vibration resonance of liquid column on the pump suction side; WO,1 = condition limiting the pump noise intensity level; 8 WO,1 = stress condition for plunger wall; W% = stress conditions for slider-pin-guide design pair; 0,1 W l o , ll = design conditions determining the pin dimensions; wo~ = design condition determining the slider ~guide diameter; WO,12 = stress conditions for slider-pin-guide design pair; 1,2 WO,12 = design conditions limiting the connecting-rod length factor; 3,4 = stress condition for cylinder wall; w] = design condition determining the height of suction air w~ chamber; = design condition determining the height of valve body rib; w~ = design condition determining the screw full diameter for w~ valve pilot bar; W~,1 = design condition determining the basic dimensions of force air chamber; = stress condition for bending the valve body rib; w~. 1 = design condition determining the basic dimensions of suction air chamber; w~,' = stress condition determining the screw number for packing gland; w~:~ = stress and stiffness conditions for valve spring; w~, ~ = design condition determining the flange diameter for packing gland; 1,1 Ws...lO = design conditions determining the valve body dimensions; Wh1 = stress condition for screw of the valve pilot bar; w L 1 = design condition limiting the conic part length of pump main part; W1,12 = stress conditions of slider tube including joint; 1,2 WO,1
44
Jerzy Kowalski, Konrad Pylak W~ W~,3 WE'1 W~:~ W42:~ W2'~ 2,1 Ws-lO 2,1 Wl1,12
W2~1 W 2 14 ,12 W72'~2 W2,12
8,9
W2,12 10-12 W2,12
13,~4
W2~~2
W 2,2
= design condition determining the spacing of connecting-rod screws; = design conditions determining the dimensions of connectingrod small end; =design condition determining the sleeve thickness for connecting-rod small end; = design conditions determining the dimensions of connectingrod small end; = connecting-rod stress conditions; =stress and heating conditions for slide bearing of the connecting-rod in top dead centre; p = crankshaft stress conditions for $15, E~5 and $15 sections in top dead centre; = stress and heating conditions for B15 crankshaft slide bearing in top dead centre; = design condition determining the crankweb height; = stress conditions for connecting-rod including cover and screws; = condition eliminating the collision between the slider tube and crankshaft; =stress and heating conditions for slide bearing of the connecting-rod in crank position corresponding to maximum crank pin effort; = crankshaft stress conditions in crank position corresponding to maximum crank pin effort; = stress and heating conditions for B15 crankshaft slide bearing in crank position corresponding to maximum crank pin effort; = condition eliminating the possibility of resonance of the crankshaft torsional vibration by Geiger method 9 (ensuring that the work is within subcritical region); = condition eliminating the possibility of resonance of the connecting-rod transverse vibration.
Figure 6 shows the solution chart for the pump decomposed optimization model. 3 TWO-LEVEL HIERARCHIC OPTIMIZATION MODELING SYSTEM F O R T H E PUMP To increase the efficiency of pump mathematical modelling, the idea of a two-level hierarchic optimization modelling system has been used. 6,1°
Minimum mass design of a three-throw plunger pump System level (II)
..................... ] rain
m (-~)
45
mmin -~opt ~'opt +
-~¢y T Iteration
of 7 se,ectio.
/ Sets of
/ Xop,<7); / mu, ¢x% *(7)
LYL_°P' Subsystem level (1) -
Fig. 6,
-
rain m u (-~u,-~u) -'-Pu u "-" J "x eX ( y )
u=1-2
Solution chart for pump decomposed optimization model.
The modelling system is based on the orderly sequence of the optimization models in which the quantity model is a minute detail of the analytic-structural model. The analytic-structural model of the design object is an effective means to determine structural relations, correlations amongst the material parameters, and the sets of nonmaterial parameters occurring in the constraints. The formulation of the analytic-structural model also enables the creation of the object quantity model, by which this model is directly systematized. The structural relations are given by the optimization criterion and constraint matrices. The objective function matrix (Table 1) determines the occurrence of coupling amongst particular components of the objective function and the variables, and their repeatability in the component set. It is sparse. Because of additive type of the optimization criterion, the matrix must have the block-diagonal form. The exact subsystem segregation so that the objective function is separable regarding variables, appears to be impossible. The approximation has been used based on the overestimation of mr31 sleeve mass for the connecting-rod small end. It is exact when the W~'1 design condition is equally satisfied. The possible inaccuracy of this component has very little effect on the total objective function. On the other hand, the constraint matrix is given in Table 2. It determines the structural relations amongst the constraints and particular variables. The constraint matrix is complex and sparse. The correlations amongst the material parameters are presented in the model by means of the network diagram and the matrix of outer material parameters. Let us limit to representing the vector of material
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Minimum mass design of a three-throw plunger pump
51
parameters occurring in the constraints. It can be expressed as: ffM = (krcu, krlc132, krjp,
P~y-e, P,~15b, k~oe, k~t, k'~i132, kcj132, ksi111,
~x~r132, /~yoe132, E132, G,~, A132, B132, C15_133, C15_b)
(10)
where = allowable stress for alternate tension and compres-
krcac
sion; a~ = 1, 41, 6 (kr'c132 for the connecting-rod small end); krjp = allowable pulsatory tensile stress; fl = 421, 841, 1221, 1431; allowable unit pressure; y-t~ = 2 1 - 3 , 22-41, 3-131, P ~v-6 = 15-133 (P~ls-b) for crankshaft and sleeve of crankshaft main bearing journal); allowable oscillatory bending stress; e =3, 15 (to kgoe compensate the effect of flywheel mass and belt tension, kgo15 value must be reduced at 20%); allowable pulsatory bending stress; ~ = 91, 141 (k~132 for the connecting-rod big end); kcj132 = allowable pulsatory compression stress for the connecting-rod shank; ksjll I = allowable pulsatory torsional stress for the valve spring; nxa¢132, nyor132 --'~ safety factors for buckling the connecting-rod shank; E132 -- connecting-rod elastic modulus; GZ = shear modulus; r/= 111, 15, A132, B132 = factors occurring in Tetmayer buckling formula; C15-133, C15-b = heat dissipation factors for crankshaft slide bearings. The sets of the nonmaterial parameters occurring in the constraints may be presented with the aid of nonmaterial parameter vector of the first kind (characterizing the significant parameters) and vector of the second kind (enclosing the parameters of less significance). The parameter vector of the first kind is given by :1 = [a, ~
A p, L*,/-/~, Ls*, n, r/v, (S/O)min, (S/O)max, LI~]
where Q = H~t = p = L* = =
pump calculation delivery static force lift positive gauge pressure in the pump cylinder force pipeline length static suction lift
(11)
Jerzy Kowalski, Konrad Pylak
52
L* = suction pipeline length (S/D)min , (S/D)max minimum and maximum values of the relation of plunger stroke to diameter; LI~ = admissible pump noise intensity level. =
On the other hand, the parameter vector of the second kind can be expressed as: 132 = [Kp, 6p,, 6ps, eg A bg A Sg, fig, a~9, (P)homin, (P)homax, ~9, ~'min, ~'max,ap, ap, e*]
(12)
Here Kp = overload factor; 6pt, 6ps=factors of pressure fluctuation for force and suction chambers; (P)homi.,(P)h . . . . = m i n i m u m and maximum loads for multi-ring valve; ~9 = calculation factor for valve weight including hydrostatic lift; Zmi,, Amax= minimum and maximum values of the connecting-rod length factor; ap, ap = casing allowances increasing theoretical wall thickness of the plunger and cylinder; e* = relation of delivery dynamic component to average delivery. Based on principles for the model transformation, 6'1° the quantity model for the pump has been formulated. In this model, one quantifies all kinds of relations as the mathematical formulas, i.e. equalities and inequalities. It contains the nonlinear objective function of 43 variables and 66 inequality constraints including 10 linear ones. Let us look at one of the typical elements of the objective function, i.e. the cylinder mass which strongly influences the mass, m 1, of the pump main part (subsystem 1). It may be presented as: f:r D2k6 - D62)gk6 + ffl( ~-[(D6 + 2g6) 2 - D 2] (0.5.16ol - gk6 + D* m6 = p6/~( + 16o2+ 16o3+ 16o4) + 2 . 4 [(D6 + 2g6) 2 - d~]h6 q- 4 [(D6 -t- 2g6)2-d21](h6-1 - 5) q-4 (D261-d21)gk61 + ~" [(d61 + 2g6) 2 - d21]1611- ~~ Dt,2g6 - ~ D23g6
53
Minimum mass design of a three-throw plunger pump 7f 2 7f -I- ~" (D2t - D* )g6 + ~- [(D* + 2g6) 2 - D*2](lt6-g6)
-t- ]'~/63[(D64 + 2g6) 2 4" (D64 + 2g6)(D63 + 2g6) + (063 -4- 2g6) 2]
~2 1 6 3 ( 0 2 + 0 6 4 0 6 3 "~ 0 2 3 ) -4- -~- (064 -q- 2g6)2166 flit
"~
2
"~ 2
•
ff~'
+ ~ (D65 + 2g6)2(/67 -t- 168) + ~ Dk4g4 -- ~- d~g4t04 +~- D 2 • 2 7f
Jt
2
"" - T 4f d 2 062/56 '~ i 062 - - 4 D64164-~D25165~4 "[- "4" D 2w62~6
J
(13)
where 16o2= Hnl2 -~ 5 -J¢-1121 + Hll + 0.6//9 - 0-5Dt*
(14)
with nut height for valve pilot bar Hn12 = 0-8dz12
(15)
1121 = hm~, + 2dll
(16)
H9 = 1.2bg + h~9 + 5
(17)
Hll = 4 x 1-2(1.2bg - 10) + n~(d~l + 1)
(18)
D64 = D + 2 V ~
(19)
063---- 1"6/94
(20)
163 = S + 0"50 - 2 - / 9 6 - 2g6
(21)
06 = d6 + 15
(22)
D6s = D + 4 V ~
(23)
/65 = 5 + Is2 + g6 = 5 + D + g6
(24)
/64 =/66 + 167 - 5 = 0 - 5
(25)
loaded spring height
multi-ring valve height
and valve plate height
Consequently
Let us examine one of the simple constraints which appeared, however, to be active in numerical calculations, i.e. the stress condition
JerzyKowaiski,KonradPylak
54
for cylinder wall. It is formed as:
x/kr~6+_0.4p W~=g6- [~ ( ¥kr~6__ l.3p
1)+ a~]-0
(26)
Other typical objective function components and constraints are presented in a previous paper. 6
4 NUMERICAL EXAMPLE Let us consider the existing pump with the following basic parameters: 4 Q = 0.05
m3/s,
/~t = 125 m,
Lt* = 23 0a,
n = 170 rpm
The design materials are assumed as follows: •
•
•
• •
• • • •
plungers, sliders, slider tubes, force air chambers, cylinders, suction air chambers are grey cast iron Z120 (equivalent GG-20 DIN); slider pins, screws of the slider tube-and-cylinder joints, packing gland screws, connecting-rod screws, crankshaft are constructional carbon steel St5 (St50 DIN); valve plates and pilot bars, nuts and nut washers of valve pilot bars, connecting-rods are constructional carbon steel St3 (St37 DIN); packing glands, covers of connecting-rod big ends are constructional Carbon steel St4 (St42 DIN); sleeve packings, sleeves of connecting-rod big ends, sleeves of crankshaft main bearing journal are tin-phosphor bronze CuSnl0P (GBz-10 DIN); bearing bushes of slider bases are Babbit metal SnSb11Cu6 (WM80 DIN); multi-ring valves are grey cast iron Z135 (GG-35 DIN); valve springs are silicon steel 50S2 (50Si2 DIN); plunger packings are packing cord.
By analysis, it was stated that the existing design fulfills all constraints. It enables us to determine the starting point for numerical computations. To solve the two-level optimization problem, the method of systematic searching is used based on optimization of a variable and constraint arrangement and progressive reduction of the searching step and variation block. Because of the large number of variables and their discrete distribution, as well as the large number of constraints including complex types of functional relations, it is recommended that
Minimum mass design of a three-throw plunger pump
55
TABLE 3
The Values of the Objective Function and Its Components for Existing and Optimum Pumps Objective function component
Existing pump
Optimum pump
m (kg) m I (kg) m2 (kg)
5 296-826 4 972.576 324.250
4 081-044 3 860.903 220.141
this m e t h o d be used. For that reason, the suitable systematizing of the constraint set is very important. A PC/AT-386 computer has been used. C P U time was equal to about 3 h for o p t i m u m and 26 suboptimum solutions, for the preliminary reduced variable block. Table 3 gives the values of the objective function and its components for existing and o p t i m u m pumps. The p u m p mass reduction at 22.95% (1215.782 kg) has been achieved compared with the existing design. It corresponds to a 23.25% material cost reduction (that means about $120). For lot production of pumps, it is possible to receive significant material cost reduction. On the other hand, the values of coordinative variables for existing and o p t i m u m pumps are presented in Table 4. The values of variables of both subsystems are given in Tables 5 and 6, respectively. The optimum p u m p is characterized by a smaller average piston speed at 4.29% (0-102 m/s) and a decrease in the noise intensity level at 0.44% TABLE 4
The Values of Coordinative Variables for Existing and Optimum Pumps Variable
Existing design
Optimum design
D (mm) S (mm) gl (mm) L2 (mm) B2 (mm) D20(mm) 12o(mm) D~2 (mm) d3 (mm) 13(mm) do~2(mm) lk (ram)
140 420 12 310 210 200 76 380 50 126 16 1 150
136 402 11 290 186 180 80 360 42 120 14 1 080
Jerzy Kowalski, Konrad Pylak
56
TABLE 5
The Values of Variables of Subsystem 1 for Existing and Optimum Pumps
Variable g4 (mm) io4 do4 (mm) d6 (mm) g6 (mm) 1~ol(mm) 161(mm) i6~ (ram) h7 (mm) n9 h~9 (mm)
Existingdesign
Optimum design
20 16 20 415 22 594 400 8 660 3 105
16 8 16 380 20 570 370 8 640 3 85
mz9
6
6
dll (mm) Dll (ram) nit d~lz (ram)
7 70 8 14
7 80 6 10
TABLE 6
The Values of Variables of Subsystem 2 for Existing and Optimum Pumps
Variable 113(mm) g13 (mm) Do~3(mm) B (ram) H (mm) gp (mm) gs (ram) Dt (mm) dk (mm) lsk (mm) dz14 (mm) g~ (mm) h r (mm)
d91 (mm) 194 (mm)
Existingdesign
Optimum design
72 35 58 65 120 10 10 200 160 90 20 35 400 150 80
60 30 54 54 105 8 8 180 130 80 16 28 380 135 65
Minimum mass design of a three-throw plunger pump
57
TABLE 7 The Absolute and Per Cent Margins of Active Constraints for Optimum Pump
Most active constraints
Optimum design Absolute margin
Per cent margin
W19J W~1
0.641 30 12-000 0 0"002 25 0.044 12 3 066 248 36 496"0 0"13685 5"901 5 15"649 4
3-31 1-80 4"50 1"47 2"86 0"04 1"99 4"16 0"21
w~
o.ooo o
o.oo
W32
1.000 0
0.88
W~ W~ W °'1 W °J W~"1 W~'1
W~'1
I ,38,
,;,~
i
--
Ji, / / ~ ~ . /~"""'""!/"~"/"~//.~./ /~
E
E 13E - - ~ 7 " ~
d 135 I'.~ - / ~
'"t .... 1331
390
I
/ / / V / / / ~ ' / - . r ~ . ~'-% " /h I
f
1~
- ~''
/
i :392
i
394
I
396
398
,1"1
400
402
404
406
408
S, m m
Fig. 7. D - S plane of coordinative variable space: - - , constraint lines only depending on D.S. variables; . . . . , constraint lines also depending on variables of subsystem 1; [~77~, suboptimum solution region; (~), ith solution (the first is the optimum one.
58
Jerzy Kowalski, Konrad Pylak
(0.89 dB). Based on Ref. 11, the flywheel has been selected. Its mass is reduced at 11.94% (11-673 kg). Table 7 gives the absolute and percentage margins of active constraints for the optimum pump. Let us note that the other constraints, e.g. W~4, W 2'1, W 2'12 have limited the permissible region for the suboptimum solution area. Amongst 39 constraints depending on material parameters, 33 conditions have been improved in the direction of better utilization of material properties; W~i1, W216~, w2,~2 became rr 2,6 worse, but W2~12 and W~'2, practically, do not show any change. One of the more interesting sections of the variable space is the plane of D - S coordinative variable space (Fig. 7). Full lines show constraint lines only depending on S and D variables. On the other hand, the broken lines represent constraint lines also depending on the variables of the subsystem 1. The lining area shows the suboptimum solution region; i is the ith solution ( i - - 1 , 2 , . . . 27). The values of m ~ component for particular solutions are equal to 3860.903, 3861.205, 3861.506, 3861.400, 3861-416, 3861.731,... 3865.238kg, respectively. The solutions 1 , 2 , . . . 2 3 are given for h T = 6 4 0 m m . On the other hand, the solutions 24, 25, 26, 27 located out of W~,~ constraint are for h7=645 mm. The remaining variables have the same values. The component m E = 220.141 kg is the same for all solutions. Figure 7 points at flatness of the objective function in the neighbourhood of the minimum, at least in the D - S section.
5 CONCLUSIONS The presented optimization method can be easily adapted, e.g. to high pressure plunger pumps.
REFERENCES 1. Reddy, Y. R. & Kar, S., Optimum vane number and angle of centrifugal pumps with logarithmic vanes. J. Basic Engng. 93 (1971). 2. Sav~enko, V. S., Introduction into Optimum Machine Design. Nauka i Technika Publishers, Minsk, 1974 (in Russian). 3. Kowalski, J., Method of mathematical modeling the horizontal centrifugal pump for optimum design. La Ventana de la UAZ, No. 159, 4-5 (1989) (in Spanish). 4. Demeter, T., Collection of Design Problems. WSIP Publishers, Warsaw, 1974 (in Polish).
Minimum mass design of a three-throw plunger pump
59
5. Kowalski, J., Large object mathematical modelling in optimal design of machine construction. Elements of methodology of creating the optimization models. Strojarstvo, 24 (1982) 125-31. 6. Kowalski, J,, Modeling of Design Objects for the Optimum Design. W N T Publishers, Warsaw, 1983 (in Polish). 7. Brill, K. F., Berechnungen von Druckstoi3schwingungen in hydraulischen PreBanlagen", Konstruktion, 12 (1960) 31-5. 8. Harris, C., Handbook of Noise Control. McGraw-Hill, New York, 1957. 9. Rytel, Z., Piston machines. Mechanics of crankshaft assembly. In: Mechanical Engineer Handbook, vol. 2. WNT Publishers, Warsaw, 1970 (in Polish). 10. Kowalski, J., An appropriate strategy for mathematical modeling in optimum design of machine construction. J. Mech. Transmiss. Autom. Des., 107 (1985) 463-76. 11. Bouch6, Ch., Die kolbenpumpen. In: Dubbels Taschenbuch fiir den Maschinenbau, Bd. II. Springer-Verlag, Berlin/G6ttingen/Heidelberg, 1963.