- Email: [email protected]
PULP REFINING AT RESONANCE
Mechanical Systems and Signal Processing (1999) 13(4), 643}665 Article No. mssp.1999.1227, available online at http://www.idealibrary.com on
PULP REFINING AT RESONANCE R. WHALLEY AND D. MITCHELL Department of Mechanical and Manufacturing Engineering, University of Bradford, West Yorkshire, BD7 1DP, U.K. (Received 25 August 1998, accepted 4 March 1999) A disc re"ner model for stock preparation for paper and board manufacture, is derived. The torsional response of the machine under operating conditions is considered. Resonance excitation via drive and #ow pulsation variations is achieved enabling optimum torsional oscillatory behaviour to be attained. The e!ect of this action on the reciprocal shredding and shearing performance of the machine is commented upon and simulated response characteristics are presented. 1999 Academic Press
1. INTRODUCTION
Paper and board manufacturing comprises three basic processes all of which require considerable labour, energy, chemical and raw material commitments. Generally, these are stock preparation, sheet forming and drying followed by calendering and "nishing. Of these the initial processing of bleached pulp and recovered, recycled materials for stock preparation is one of the more expensive operations. The energy consumption and process plant employed, as described in [1, 2], are substantial and the operating costs in terms of energy consumption and attendant maintenance charges, as discussed in [3], form a considerable proportion of the manufacturing costs of the "nal product. Preparation of the basic raw material (stock) for the continuous Fourdrinier and Inverform sheet generating machines is usually a batch operation using processing machinery to disperse, dilute and shred mechanical and chemical wood pulp and/or waste paper in aqueous solution. Following treatment and the separation of contaminants large holding tanks for the reduced, diluted pulp/waste paper stock are "lled and continuously agitated in readiness for discharge to the forming machine #ow box. The basic preparation plant for each continuous paper making process comprises multiple pulping, beating and re"ning units which are used to simultaneously shred, delaminate and "brillate "bres to a given consistency commensurate with the product speci"cation. Moreover, setting the processing plant to achieve the required aqueous "bre solution is of paramount importance since the physical properties of the "nal product are largely determined by this conditioning. Usually, the diluted pulp or stock is sampled, in order to exercise quality control before embarking upon volume production. However, this is time-consuming and usually involves operating the re"ning, beating machines and separators for longer than that which is absolutely necessary in order to obtain the required degree of "brillation while manually starting-up, shutting down and resetting the re"ning units. The principal objectives achieved by the re"ning process are to increase the "bre matting and bonding properties, control "bre length and minimise the production of fractional length (cut) "bres. During the re"ning process, the "bres are mechanically treated to 0888}3270/99/040643#23 $30.00/0
1999 Academic Press
644
R. WHALLEY AND D. MITCHELL
increase #exibility, reduce lamination, maximise "brillation and increase their speci"c volume by water retention. Two types of pulp re"ners are widely employed in paper and board manufacturing: (a) the disc or Sutherland re"ner and (b) the conical plug and shell or Jordan re"ner. In (a) a powered grooved, rotating disc runs in close proximity to a grooved stationary disc with stock #owing under pressure between the two. Whilst in (b), stock is pumped through the annulus between the rotating conical plug and surrounding stationary shaft both of which contain bronze or stainless-steel bars which shred and "brillate the stock "bres. Owing to the superior performance of the disc re"ner, for chemical and mechanical pulps, the larger plug re"ners are steadily being replaced though the action of the two machines, in terms of their shearing e!ect on the stock, appears to be very similar. Both processes rely intrinsically on mechanical conditioning in that the re"ning action depends on the rotating plug and stationary shell separation, or alternatively, on the powered disc and idling disc clearance for equivalent running speeds and plup properties such as density, consistency, chemical content, etc. The principle of the disc processor is not new of course and its acceptance owes as much to the evolution of this machine from its #our milling and seed processing counterparts as from any technical or theoretical innovation. However, the longevity of the device is certainly impressive dating back to the mechanisation of wind power in the Middle Ages. Modern disc re"ners though have slight resemblance to their historical derivatives, run at high speed, deploying two or more grooved stainless-steel discs, in very close proximity, to achieve speci"c delaminating and shearing actions. Generally, the pulp or stock solution is cycled through twin stator disc arrangements, in series or parallel until the desired composition of the re"ned homogeneous mixture is achieved. Inevitably, there is more to the process than "rst appearances indicate and, as it transpires, the intimate reaction between the elements of the disc machine is the key to its operational e!ectiveness. Initially, the mechanics of the problem appear to be uncomplicated with simple analysis yielding no unexpected results. In this paper however, conventional analytical procedures are extended and enhanced with computer simulation studies arising from the dynamics of the non-linear and linear models employed, to explore the operational characteristics of the process in greater detail. Ultimately, the essential di!erences between the re"ning action of the two machines are exposed thereby and optimum operational conditions are then investigated.
2. MACHINE CONFIGURATION AND DEPLOYMENT
The disc re"ner is shown in Fig. 1(a) and that for the plug re"ner is shown in Fig. 1(b) in outline form. In the latter case, the stock #ows under pressure through the gap between the conical plug and shell which have embedded, in zig-zag patterns, bronze or stainless-steel rotor and stator shearing bars, respectively. Similarly in Fig. 1(a), stock is pumped through the input pipe to the serrated discs of the disc re"ner where the driven disc provides the power to shear the "bres apart ejecting the stock thereafter by pressurised, centrifugal action. Under the conditions of investigation the speed of the idling disc increases during this operation and the di!erential disc speed can be adjusted, using the brake dynamometer, while running the powered disc at a constant rate, if so desired. The stock consistency, type of stock and the distance between the discs a!ects the disc friction shredding capacity which provides the re"ning action. Additionally, pressure variations arising from the centrifugal pump feeding the re"ner, as described by Simpson et al. [4], generates a cyclic e!ect on the
PULP REFINING AT RESONANCE
645
Figure 1(a). Disc re"ner arrangement.
frictional force, arising from the powered disc thrust bearing reaction, while the journal bearing frictional component remains constant, under steady-state conditions. Modern re"ners often employ rotor discs which are grooved on both faces. A stator disc at each side of the rotor is then used with the stock #ow to the rear face being achieved through ori"ce ducts machined in the rotor hub, as shown in Fig. 2. Essentially, the analysis procedure for both disc con"gurations are identical though the total grooved surface in the latter case (Fig. 2) is twice that of Fig. 1(a), for equal disc
Figure 1(b). Conical re"ner.
646
R. WHALLEY AND D. MITCHELL
diameters. Moreover, production machines are not "tted with brake dynamometers, whereas with the experimental machine analysed here the e!ect of di!erent di!erential disc speeds can easily be investigated while retaining a constant speed, induction motor drive. A typical production re"ner is shown in Fig. 3 where the hinged housing has been opened for inspection purposes. In this photograph, the grooved stator and rotor discs and ori"ce ducts are clearly visible. Low-intensity re"ning which is characterised by high stock dilution, "ne disc grooving and least separation between the discs has been shown, to produce superior "brillation with minimum "bre length reduction. Ultimately, this results in high sheet strength compared to that obtainable from plug re"ners when set to run in a similar optimum mode, as con"rmed by Borsch et al. [5]. The fundamental question returns to the issue of why this should be so when employing machines which appear to have the same mechanical action. Moreover, this is maintained when treating hardwood or softwood chemical/mechanical pulps including sulphide, kraft, soda, groundwood, etc., and with non-wood "bres such as straw and re-cycled, de-inked waste. Much has been stated regarding the disc grooving pattern and depth, and the analogous problem of the Jordan plug and shell bar con"gurations, as discussed in [7] and doubtlessly re"ning action is in#uenced by these factors. However, as this analysis will show there are
Figure 2. Disc re"ner.
PULP REFINING AT RESONANCE
647
Figure 3. Disc re"ner.
reciprocating shear actions present in the disc machine to aid "brillation which do not occur in the plug re"ner and therein lies the advantage.
3. SYSTEM MODEL
Under processing conditions, the interaction of the twin disc arrangement shown in Fig. 1(a), which is analytically equivalent to Fig. 2, is instrumental in achieving a high quality, repeatable stock production preparation, in practice. Under experimental conditions, the brake mechanism, can be used to run the discs at variable di!erential speeds with this con"guration. The di!erential speed between the discs ensures that high shear forces shred and separate the "bres, whilst simultaneously regulating their length, in accordance with the disc separation and grooving pattern. From Fig. 1(a) the equations of motion are ¹ "J Dh #k(h !h ) k(h !h )"J Dh #(Dh !Dh )R (t)#RDh *
(1) (2)
and (Dh !Dh )R (t)"J Dh #R Dh (3) where D is Heavyside's operator and R (t) is the disc friction function. In equation (1), the damping coe$cient associated with h has been omitted, being relatively small in compari son with the inter-disc friction R and the disc bearing friction R . Equations (1)}(3) can be written in the matrix form as ¹ (t) J D#k !k 0 0 " !k J D#(R#R (t))D#k !DR (t) * 0 0 !DR (t) J D#(R (t)#R )D
h (t) h (t) . h (t) (4)
648
R. WHALLEY AND D. MITCHELL
Writing equation (4) as (A D#A D#A )h(t)"!AR (t)Dh(t)#¹ (t) where A (D)"A D#A D#A with J 0 0 0 0 A " 0 J 0 , A " 0 R * 0 0 J 0 0
k
0 0 R
!k 0
, A " !k 0
k
0
0
0
h"(h (t), h (t) h (t)), ¹ (t)"(¹ (t), 0, 0) and 0
0
0
1 !1 R (t)"R (1#r sin pt), and A" 0 0 !1 1
(5)
then Laplace transforming equation (5), with zero initial conditions, we get (A s#A s#A )h(s)"¹ (s)!¸+AR (t)Dh(t), (6) where the Laplace transformation for the non-linear term in equation (6) can be written as ¸+R (t)Dh(t),"R sh(s)#¸+R r sin ptDh(t),. More compactly, equation (6) which implies that the output response comprises the sum of the responses from the input torque and the pump-induced frictional force, becomes A (s)h(s)"¹ (s)!AsR h(s)!A¸+R r sin ptDh(t), where in the matrix quadratic form, A (s)"A s#A s#A Inverting and rearranging equation (6) gives
(7)
h(s)"(I #A (s)\u'(vR s)\A (s)\(¹ (s)!A¸+R r sin ptDh,) where in equation (8),
(8)
A (s)\"Adj A (s)/(det A (s)) u"(0, 1,!1), v"(0, 1,!1) and I "Diag(1, 1, 1). It is easy to obtain the inverses in equation (8), in that Adj A (s) ( J s#Rs#k)( J s#R s) k( J s#R s) 0 * " k( J s#R s) ( J s#R s)( J s#k) 0 0 0 ( J s#k)( J s#Rs#k)!k * (9)
PULP REFINING AT RESONANCE
649
det(A (s))"s( J s#R )[ J J s#J Rs#k( J #J )s#kR]. * *
(10)
and Evaluating (I #A (s)\u'(vsR ) R ( J s#R )k (0 1!1) 1 " det A (s)I #s R (J s#k)(J s#R ) det A (s) !R (J J s#J Rs#(J #J )ks#kR) * * (11) and 1 A (s)\¹ (s)" det (A (s))
(J s#Rs#k)(J s#R s) * ¹ . k(J s#R s) s 0
(12)
Since as stated earlier the transformed input vector for a step change of input torque ¹ (t) is ¹ ¹ (s)" , 0, 0 . s
If the inverse of the matrix in equation (11) is de"ned by b (s) b (s) b (s) 1 b (s) b (s) b (s) B(s)" (13) D(s) b (s) b (s) b (s) and from equation (11), b "b "0. Completing the manoeuvre yields b (s)"(det A (s)#R (J s#k)(J s#R )s#sR (J J s#J Rs)#(J #J )ks#kR) * * b (s)"R k(J s#R )s b (s)"R s(J s#k)(J s#R ) b (s)"!kR s(J s#R ) b (s)"(det A (s)#R s(J J s#J Rs#(J #J )ks#kR)) * * and b (s)"R s(J J s#J Rs#(J #J )ks#kR) * * b (s)"det A (s)#R (J s#k)(J s#R )s Upon multiplying equations (12) and (13) the linear part of the response owing to a driving torque change is h(s)"B(s)A (s)\(¹ (s)) where in equation (14), B(s)A (s)\"G(s)
(14)
650
R. WHALLEY AND D. MITCHELL
so that when r "0 the simple linear solution, corresponding to equation (8), is h (s) h (s) " h (s) J J s#(J (R#R )#(R #R ) J )s#((R #R )R#kJ #R R )s#k(R #R )s)/D(s) * * ¹ (s) k(J s#sR #sR )/D(s) kR s/D(s)
(15)
where in equation (13) and (15), D(s)"(det A (s)#s1v Adj A (s)R u2) "(s(J s#R )(J J s#J Rs#(J #J )ks#kR) * * #sR (J (J #J )s#J (R#R )s#(J #J #J )ks#(R#R )k). * *
(16)
4. DISC FRICTION FUNCTION
In production disc re"ning machines, the disc diameter may exceed 2m. Consequently, the pressure variation arising from the centrifugal stock #ow pump is &li"ed' by the disc area so that a relatively large cyclic separation force occurs between the discs at the frequency of the pressure pulsatance. Reducing the pressure pulsatance e!ect by feeding stock directly to both sides of a central driven disc which has stator discs on each side is sometimes preferred. However, this will be shown to be quite undesirable and detrimental to the shredding}"brillation action which is vital to e$cient re"ning. In practice, disc separation is maintained by the action of the powered disc thrust bearing and adjusting mechanism. However, since the parameters of the friction force R and r remain constant, the cyclic loading caused by the frictional force F(t) varies with the same frequency as the pulsating pump pressure. Normally the viscous frictional force is given by R Dh. In this case the frictional force F(t) is de"ned by the vector
dh F(t)"R r sin pt A (t) dt
(17)
where r sin pt is the pump pulsatance pressure variation of amplitude r and frequency p. Following the Laplace transformation of equation (17), a close linear approximation, as shown in Appendix A is
dh ¹R r p ¸ R r sin pt A (t) "AG(0) dt (s#p) where the transfer function matrix in equation (18) from equation (14) is: G(s)"B(s)A (s)\.
(18)
651
PULP REFINING AT RESONANCE
Then the complete output response, where dh/dt when Laplace transformed becomes u(s), can be obtained, for a step input change in torque, from u(s)"G(s)
¹ !AR r G(0)¹ p/(s#p) s
resulting in u 1 u " D (s) u
f (s)
s k(J s#s(R #R )) ¹ ! (RR #RR #R R )D (s) kR s
k(J s#R ) r p ; (J s#k)(J s#R ) R R ¹ (s#p) !(J J s#J Rs#(J #J )ks#kR) * *
(19)
where f (s)"(J J s#(J (R#R )#J (R #R ))s * * #((R #R )R#kJ #R R )s#k(R #R )s and D (s)"s(J J J s#J (J (R #R )#J (R#R ))s#(J (kJ #RR #RR #R R ) * * #kJ J )s#k((R #R )(J #J )#J (R#R ))s#k(RR #RR #R R )). * 5. APPLICATION STUDY
The speci"c machine parameters have not been stated in this section for reasons of con"dentiality. However, the relative performance of the system remains relevant in the following illustrative study. In this example, if in Fig. 1(a); J "J "J "750 kg m; k"10 N m/rad; R"80, * R "430, R "21000 N m s/rad, and ¹ "25 000 N m, then the steady-state operating conditions, with R constant, are: motor shaft speed, u "u "49.86 rad/s (476 rev/min); idling disc speed, u "1 rad/s (9.6 rev/min); no load power, 199 kW; full load power, 1246 kW.
6. ROOTS OF THE CHARACTERISTIC EQUATION
The parameter values given in Section 5 enable a study of the changes in the singularities of equation (19) to be undertaken. Of particular interest in this exercise are the variations in the roots of the denominator polynomial since these singularities determine the decay rates and frequency of oscillation of u (t), u (t), and u (t). In this respect, the purpose of arranging the denominator determinant in the form given in equation (16) now becomes apparent since the disc friction coe$cient R multiplies the bracketed factor in this expression leaving the remaining terms independent. Consequently, the denominator function which is equated to zero, to provide the characteristic equation, can be written, following the cancellation of the common s term in the numerator and denominator, as D /s"D(s)/s"0.
(20)
652
R. WHALLEY AND D. MITCHELL
Equation (20) can be arranged, using equation (16) as R (s(J #J )J #s(R#R )J #s(J #J #J )k#(R#R )k) * * . !1" (21) (J s#R )(J J s#J Rs#(J #J )ks#kR) * * This quotient is rendered monic by the extraction of the highest degree coe$cients so that equation (21) becomes
R#R (J #J #J ) k(R#R ) s# * ks# K s# (J #J ) J (J #J ) (J #J )J * * * !1" R R (J #J ) kR * ks# s# s# s# J J JJ JJ * * * where in equation (22) with the parameters from Section 5:
(22)
K"R (J #J )/J J "R /375.0 * * R#R "14.05 J #J * J #J #J * "2;10 k J (J #J ) * k(R#R ) "18.74;10 J (J #J ) * J R#J R * "28.106 JJ * R "0.106 J * (J #J ) * k"2.66 ;10 JJ * kR "142.2 . JJ * The poles and zeros of equation (22) for very small values of K are
!0.053, !0.0267$51.64i and !28.0 (poles) and !9.57, !2.24$44.183i (zeros). Increasing K, by changing R , causes the poles to move towards the zeros, one of which is located at !R. The loci of pole migration is shown in Fig. 4 where three values of K"K , K and K are shown in the calibration. From the root locus diagram given in Fig. 4 it is clear that the least oscillatory response is encountered initially by increasing K to K "100 where the damping ratio reaches its maximum. Moreover, it is evident that increasing K and hence R beyond this value causes increasing power to be absorbed by the re"ning action without achieving any improvement in the oscillatory motion of the powered disc over that available for values of K(100 N m s/rad.
PULP REFINING AT RESONANCE
653
Figure 4. Root locus corresponding to equation (22), calibrated in terms of changes in K"disc friction coe$cient/375.0.
Alternatively, for lower stock consistencies resulting in 0)K)30 N m S/rad, power dissipation is reduced and the damping ratio decreases generating higher frequency, maximum amplitude, oscillatory re"ning conditions thereby. Whereas high-energy re"ning determines the quantity of the processing performed, the quality of the re"ning action is determined by operations consuming the least power, owing to high dilution, ensuring that maximum disc}"bre contact is maintained. With low-intensity oscillatory re"ning, "bre treatment at the leading and at the trailing edge of the disc grooves occurs. Whereas with K'100 N m S/rad there are increasing penalties in terms of both higher energy costs and decreasing oscillatory amplitudes, at the driven disc, as R also becomes proportionally larger with higher stock consistencies and oscillatory motion migrates from the driven disc to the motor armature.
7. COMPARISON OF LINEAR AND NON-LINEAR MODELS
To validate the linear model, comparative studies aimed at establishing the parameter variations which could be accommodated are required. The principal non-linearity arises from the multiplication of R r sin pt Dh(t)
(23)
in equation (6). In equation (17), R is constant for a given consistency so tht r , p and u(t) form the set of coe$cients which are variable. However, since Dh(t) is the output speed vector only the pump frequency p and the amplitude of the pulsatance pressure r which
654
R. WHALLEY AND D. MITCHELL
Figure 5. Non-linear system analogue.
gives rise to the retarding torque induced by the pulsating thrust on the powered disc, are considered to be manipulatable variables here. Figures 5 and 6 show, for the speci"c application parameters of Section 5, the non-linear and the linearised models given in equations (4) and (19), respectively, in block diagram form. Following step changes the simulated outputs for the non-linear and linear responses
Figure 6. Linearised model block diagram.
655
PULP REFINING AT RESONANCE
Figure 7(a). Non-linear model driven disc response u (t), following an input torque change of 25 000 N m with pump speed p"30.00 rad/s, r "0.05.
for u (t) is shown in Fig. 7(a), with r "0.05, p"30 rads/s showing the response at arbitrarily selected operating conditions.
8. OSCILLATORY WORKING
If in this analysis the idling disc of the re"ner is set virtually to zero speed, by increasing the brake dynamometer resistance R to a high value rendering the con"guration identical to most production machines. Under steady-state running conditions the oscillatory behaviour of the powered disc is, from equation (19),
R R r ¹ (J s#k)(J s#R )s u (s)" u (s) N (RR #RR #R R ) D (s) where u (s)"p/(s#p). N
(24)
Factoring D (s) for the coe$cient values in Section 5 results in D (s)"sJ J J (s#0.33)(s#28.58)(s#0.34s#51.64) * J (s#k/J )"J (s#1.3.10) and
J (s#R /J )"750(s#28.0).
656
R. WHALLEY AND D. MITCHELL
Figure 7(b). Non-linear model driven disc response u (t), following an input torque change of 25 000 N m with pump speed p"51.64 rad/s, r "0.05.
Consequently, cancelling the pole and zero at s"!28.58 and s"!28.0, respectively, leads to u (s) (J s#k) R R r ¹ " u (s) J J (RR #RR #R R ) (s#2lu s#u )(s#a) N * L L
(25)
where 2lu "0.34, u "51.64 and a"0.33. L L The frequency response equation from equation (25) becomes (J s#k)p u (s)"C (26) (s#2lu s#u )(s#a)(s#p) L L where in equation (26), C"R R r ¹ /(J J (RR #RR #R R )) so that using equation * (26), the steady-state amplitude ratio and phase shift can be found from
C(J s#k) s#c s#c s#c QGN where c "2lu #a, c "2lu a#u and c "au . L L L L The modulus of equation (27) is given by M"C(k!J p)/((c !c p)#p(c !p)).
(27)
(28)
657
PULP REFINING AT RESONANCE
Figure 7(c). Linear model driven disc response u (t), following an input torque change of 25 000 N m with pump speed p"51.64 rad/s, r "0.05.
Extremum values of equation (28) for varying pump frequencies can be found by di!erentiating and equating this equation to zero. Consequently putting d (M)"0 dp we get (J p!k)[!4J p[c#(c!2c c )p#(c!2c )p#p] #(J p!k)[2(c!2c c )p#4(c!2c )p#6p]]"0. (29) It is evident that the common multiplier in equation (29) of (J p!k) will minimise the amplitude ratio of equation (28) whereas the remaining bracketed term which simpli"es to !p[4J c#2k(c!2c c )#(2J (c!2c c )#4k(c!2c ))p#6kp!2J p]"0 (30) will yield the maximum value. If now q"p then the equation to be solved from equation (30) for a non-trivial solution is
3k 2k k q! q!((c!2c c )# (c!2c ))q! 2c# (c!2c c ) "0 J J J
(31)
658
R. WHALLEY AND D. MITCHELL
Figure 8. Modulus ratio u (ip)/u (ip) db vs log frequency. N
where equation (31) upon inserting the parameter values becomes q!4.10q#7.11.10q!9.48.10"0.
(32)
The roots of equation (32) are q"2.667;10, (0.6667$1.764i);10
Figure 9(a). Non-linear model driven disc response u (t), following an input torque change of (25 000#500 sin 51.64 t) N m with pump speed p"51.64 rad/s, r "0.05.
659
PULP REFINING AT RESONANCE
Figure 9(b). Linear model driven disc response u (t), following (25 000#500 sin 51.64 t) N m with pump speed p"51.64 rad/s, r "0.05.
an
input
torque
change
of
so that the resonant frequency is given by the only real root yielding p"(2.667;10) "51.64 rads/s. This condition can be con"rmed from the frequency response plot shown in Fig. 8 where the amplitude of the output u exceeds that of the input amplitude by approximately 10 dB at 51.64 rads/s with the setting here, for demonstration purposes, of r "0.05. The corresponding time domain responses following an input step change of 25 000 N m are shown for the non-linear and linear models in Figs 7(b) and 7(c), respectively, with the pump speed again at p"51.64 rads/s and r "0.05. To improve the oscillatory working of the machine a sinusoidal perturbation is added to the input torque which becomes (25 000#500 sin 51.64t) N m while the pump speed is maintained at 51.64 rads/s and r "0.05. The improved oscillatory response to this excitation is shown in Figs 9(a) and 9(b) for the non-linear and linear models, respectively. Figures 10(a) and (b) show the magni"ed responses in the interval 5}6 s. Once again there is good correlation between the non-linear and linear realisations. Moreover, the linear model is quite tolerant to large changes in r and p, con"rming robustness thereby. 9. BATCH SAMPLING
During investigation trials it is necessary to operate the re"ner for relatively short periods of time in order that stock dilution, disc separation and running speed can be adjusted. This
660
R. WHALLEY AND D. MITCHELL
Figure 10(a). Magni"ed non-linear model driven disc response u (t), following an input torque change of (25 000#500 sin 51.64 t) N m with pump speed p"51.64 rad/s, r "0.05.
process is iterative and continues until the required degree of "brillation and freeness have been achieved for the particular stock preparation, commensurate with the desired "nal product. To facilitate this type of pilot plant deployment an interrupt facility can be incorporated into the control scheme as shown in Fig. 11. Essentially, the digital delay system shown there enables a &run-pause' programme to be automatically enabled, allowing thereby on-line adjustment of the machine without manually starting and stopping the driving motor and pumps, etc. The &run-pause' interval can be easily changed to provide the necessary batch sampling required, by adjusting the "nite time interval of &a' seconds. This is demonstrated in Fig. 12 where the cycle has been reduced to 60 s to illustrate e!ectiveness and the disc speed response u (t) has been computed under resonant conditions following the programme initiation.
10. CONCLUSION
The analysis of the non-linear and linearised models presented in this paper of a disc re"ner enables a general approach to this machine's performance to be investigated. Importantly, the out-of-balance pressure pulsatance arising from the stock #ow from the centrifugal pump to the disc face is shown, though small, to be instrumental in exciting torsional oscillatory action which may be maximised by invoking resonance conditions.
PULP REFINING AT RESONANCE
661
Figure 10(b). Magni"ed linear model driven disc response u (t), following an input torque change of (25 000#500 sin 51.64 t) N m with pump speed p"51.64 rad/s, r "0.05.
The presence of torsional oscillations in the disc re"ner distinguishes its re"ning action from that of the plug (Jordan) re"ner accounting thereby for the superior performance of this machine. Physically, the plug machine is rendered extremely rigid in comparison and any oscillatory motion would be commensurately small. This mode of operation may not be evident when running the stock feed pump at speeds which di!er from the resonant condition. However, by synchronising the drive and pump pulsatance frequency, a signi"cant increase in the amplitude of the oscillatory response at resonance has been demonstrated. These conditions, as shown by the frequency}response diagrams, are easily attainable with a well-de"ned maximum which could be adjusted by varying the sti!ness of the coupling and motor shaft. This results in an increase in e$ciency, by employing in e!ect, both the leading and trailing edges of the disc re"ner grooves in oscillatory shredding action. This action may be less pronounced at non-resonant conditions; however, it distinguishes this machine from its plug and shell (Jordan) counterpart, demonstrating that torsional oscillatory re"ning is an important aspect of disc machine behaviour. This method of pulp re"ning is the subject of a patent application which, when granted, would allow experimental results to be made available. The study also serves to emphasise the statements made by Borsch et al. [5] regarding low-intensity re"ning and the bene"ts arising thereby. From the dynamic analysis viewpoint this emphasises that low disc frictional levels which are necessary to achieve torsional oscillatory conditions are preferable to high frictional, non-oscillatory re"ning which is maximised by adjusting the disc friction level to give K "100 N m s rad. This may be
662
R. WHALLEY AND D. MITCHELL
Figure 11. Interrupt programmer.
achieved by increasing the stock density, for example. It is equally evident that further increases in K then result in improving the oscillatory performance. However, this is because the disc oscillations are in fact decreasing owing to K increasing, resulting in the oscillatory behaviour of the machine being transferred from the disc to the armature. The presence of the disc resonance reinforces the comments made on low-intensity re"ning hightening the "brillation e$ciency and e!ectiveness while minimising power consumption. All that has been promulgated on "bre}disc contact, see for example [3, 6], remains valid though the detailed view of the mechanics of re"ning presented here enables both design, analysis and tuning to be exercised to provide optimum conditions.
Figure 12. Non-linear model driven disc response u (t) to &run-pause' cycle.
PULP REFINING AT RESONANCE
663
While it is true that disc resonance can be obtained by tuning the pump frequency to achieve maximum oscillatory conditions the opposite is also available. As the frequency response diagram of Fig. 8 for u (ip) shows, running the pump at 36.5 rads/s minimises the oscillatory condition, as predicted by the analysis in Section 8, enforcing quiescence conditions and a loss of oscillatory action. Equally, the use of #exible or long piping runs to the re"ner may reduce or eliminate the pump pulsation e!ect. However, simple re-design of the piping connections or even using an active valve arrangement to induce a sinusoidally pulsed #ow could be used to achieve the desired resonant condition and the oscillatory re"ning action thereby.
ACKNOWLEDGEMENT
The authors wish to acknowledge the cooperation of Beloit-Walmsley Limited in supplying details of their DD4000 Double Disc Re"ner and supporting information.
REFERENCES 1. G. A. DUMONT 1986 Automatica 22, 143}153. Application of advanced control methods in the pulp and paper industry. 2. K. J. ASTROG M 1970 Introduction to Stochastic Control ¹heory. New York; Academic Press. 3. B. W. SMITH 1970 In ¹rans. Symposium of British Paper and Board Makers Association, Oxford. F. Bolan (ed.). Designing for control in papermaking systems. 4. H. C. SIMPSON, R. MACASKILL and T. A. CLARK 1966}67 Proceedings IMechE, 181 (pt 3A). 84}108. Generation of hydraulic noise in centrifugal pumps. 5. B. H. BORSCH, R. J. DEFOE and J. RIHS 1986 PIRA International Re,ning Conference, Birmingham, December. Searching to improve "bre quality by means of low consistency re"ning. 6. J. RIHS 1991 Report CPPA ¹ech. 91, Beloit Corporation, Fibre Systems Division, Dalton, MA. Fundamentals of paper mill re"ning for chemical, mechanical and secondary "bres. 7. H. H. ROSENBROCK 1974 Computer-Aided Control System Design. London: Academic Press Ltd.
APPENDIX A
The governing equation in terms of output speed corresponding with the terms in equation (6) is u(t)"u (t)!u (t) (A1) 2 N where u(t)"(u (t), u (t), u (t)) is the output response, u (t)"(u (t), u (t), u (t)) is 2 2 2 2 the output response due to torque inputs and u (t)"(u (t), u (t), u (t)) is the output N N N N response due to pump input in equation (A1). If
and
u (t)"G(D) ¹ (t) 2
(A2)
u (t)"G(D) AR r sin ptu(t) (A3) N where G(D) is a transfer matrix of dimensions 3;3 in equations (A2) and (A3) D"d/dt, A"u'(v with u"(0,!1, 1) and v"(0,!1, 1), R and r are scalars and p is the pump frequency. Consequently, u(t)"G(D) (¹ (t)!AR r sin pt u(t)). (A4)
664
R. WHALLEY AND D. MITCHELL
Laplace transforming equation (A4) yields u(s)"G(s) (¹ (s)!¸+AR r sin pt u(t),). Since r is relatively small in equation (A5) then $ u(s)"u (s) $ 2 and ¹ u (s)"G(s) 2 s
(A5)
(A6)
for a step input change of ¹ . The Laplace transform required therefore is AR r ¸+sin ptu (t),"AR r (u (s!ip)!u (s#ip))/2i. 2 2 2 From equation (A6) the expansion
(A7)
K ¹ K K L ¹ (A8) u (s)"G(s) " # #2# 2 s s#j s#j s L is easily obtained. This implies that equation (A7) can be written, using equation (A8) as
L L K /(s#j !ip! K /(s#j #ip) ¹ (A9) H H H H H H where in equation (A9) j "0 and all other terms containing j , 1)j)n, will decay in the H time domain if G(s) contains stable polynomials. Hence, AR r ¸ +sin ptu (t),"AR r 2
2iK p ¹ AR r ¸ +sin ptu (t),"AR r 2 s#p 2i p "AR r G(0)¹ . s#p
(A10)
Since K "G(s)" Q returning to equation (A5) the transformed output to input relationships using equation (A10) is, for a step change in torque, u(s)"G(s)
¹ p !AR r G(0)¹ . s s#p
(A11)
A good approximation to the time domain response is therefore
¹ u(t)"¸\ G(s) !R r "G(ip)"AG(0)¹ sin (pt#s) s where in equation (A12), tan s"arg(G(ip)A) where tan s"(tan s , tan s , tan s ).
(A12)
PULP REFINING AT RESONANCE
APPENDIX B: NOMENCLATURE J k ¹ J * R R R m u u u ¹ h(t) ¹ (t) A (s) A A A A D r p s I K D(s) A (s)\ G(D) G(s) G(0) F(t) c ,c ,c l u L !a M C u(t) u (t) 2 u (t) N !j H 1x, y2 x'(y e\?Q
motor armature and shaft inertia shaft and coupling sti!ness motor torque load inertia bearing and disc friction dynamometer friction bearing friction number of outputs motor speed"hQ load speed"hQ idling speed"hQ motor torque angular displacement vector"(h (t), h (t), h (t)) (m;1) torque vector (m;1) impedance matrix (m;m) sti!ness matrix (m;m) damping matrix (m;m) inertia matrix (m;m) disc friction matrix (m;m) Heavyside's operator d/dt amplitude of viscous variation force due to pressure drop frequency of viscous force variation due to pressure drop Laplace variable identity matrix (m;m) det A (s) B(s)"inverse of A (s)"Adj(A (s))/det(A (s)) (m;m) transfer matrix operator (m;m) transfer function matrix"B(s) A (s)\ (m;m) steady-state value of G(s) (m;m) frictional force function vector (m;1) polynomial coe$cients damping ratio natural frequency real root location modulus constant multiplier output vector (m;1) output vector component due to torque (m;1) output vector component due to pump (m;1) poles of G(s), 1)j)m inner product of vectors x and y outer product of vector x and y transformed "nite time delay (a seconds)
665
PULP REFINING AT RESONANCE R. WHALLEY AND D. MITCHELL Department of Mechanical and Manufacturing Engineering, University of Bradford, West Yorkshire, BD7 1DP, U.K. (Received 25 August 1998, accepted 4 March 1999) A disc re"ner model for stock preparation for paper and board manufacture, is derived. The torsional response of the machine under operating conditions is considered. Resonance excitation via drive and #ow pulsation variations is achieved enabling optimum torsional oscillatory behaviour to be attained. The e!ect of this action on the reciprocal shredding and shearing performance of the machine is commented upon and simulated response characteristics are presented. 1999 Academic Press
1. INTRODUCTION
Paper and board manufacturing comprises three basic processes all of which require considerable labour, energy, chemical and raw material commitments. Generally, these are stock preparation, sheet forming and drying followed by calendering and "nishing. Of these the initial processing of bleached pulp and recovered, recycled materials for stock preparation is one of the more expensive operations. The energy consumption and process plant employed, as described in [1, 2], are substantial and the operating costs in terms of energy consumption and attendant maintenance charges, as discussed in [3], form a considerable proportion of the manufacturing costs of the "nal product. Preparation of the basic raw material (stock) for the continuous Fourdrinier and Inverform sheet generating machines is usually a batch operation using processing machinery to disperse, dilute and shred mechanical and chemical wood pulp and/or waste paper in aqueous solution. Following treatment and the separation of contaminants large holding tanks for the reduced, diluted pulp/waste paper stock are "lled and continuously agitated in readiness for discharge to the forming machine #ow box. The basic preparation plant for each continuous paper making process comprises multiple pulping, beating and re"ning units which are used to simultaneously shred, delaminate and "brillate "bres to a given consistency commensurate with the product speci"cation. Moreover, setting the processing plant to achieve the required aqueous "bre solution is of paramount importance since the physical properties of the "nal product are largely determined by this conditioning. Usually, the diluted pulp or stock is sampled, in order to exercise quality control before embarking upon volume production. However, this is time-consuming and usually involves operating the re"ning, beating machines and separators for longer than that which is absolutely necessary in order to obtain the required degree of "brillation while manually starting-up, shutting down and resetting the re"ning units. The principal objectives achieved by the re"ning process are to increase the "bre matting and bonding properties, control "bre length and minimise the production of fractional length (cut) "bres. During the re"ning process, the "bres are mechanically treated to 0888}3270/99/040643#23 $30.00/0
1999 Academic Press
644
R. WHALLEY AND D. MITCHELL
increase #exibility, reduce lamination, maximise "brillation and increase their speci"c volume by water retention. Two types of pulp re"ners are widely employed in paper and board manufacturing: (a) the disc or Sutherland re"ner and (b) the conical plug and shell or Jordan re"ner. In (a) a powered grooved, rotating disc runs in close proximity to a grooved stationary disc with stock #owing under pressure between the two. Whilst in (b), stock is pumped through the annulus between the rotating conical plug and surrounding stationary shaft both of which contain bronze or stainless-steel bars which shred and "brillate the stock "bres. Owing to the superior performance of the disc re"ner, for chemical and mechanical pulps, the larger plug re"ners are steadily being replaced though the action of the two machines, in terms of their shearing e!ect on the stock, appears to be very similar. Both processes rely intrinsically on mechanical conditioning in that the re"ning action depends on the rotating plug and stationary shell separation, or alternatively, on the powered disc and idling disc clearance for equivalent running speeds and plup properties such as density, consistency, chemical content, etc. The principle of the disc processor is not new of course and its acceptance owes as much to the evolution of this machine from its #our milling and seed processing counterparts as from any technical or theoretical innovation. However, the longevity of the device is certainly impressive dating back to the mechanisation of wind power in the Middle Ages. Modern disc re"ners though have slight resemblance to their historical derivatives, run at high speed, deploying two or more grooved stainless-steel discs, in very close proximity, to achieve speci"c delaminating and shearing actions. Generally, the pulp or stock solution is cycled through twin stator disc arrangements, in series or parallel until the desired composition of the re"ned homogeneous mixture is achieved. Inevitably, there is more to the process than "rst appearances indicate and, as it transpires, the intimate reaction between the elements of the disc machine is the key to its operational e!ectiveness. Initially, the mechanics of the problem appear to be uncomplicated with simple analysis yielding no unexpected results. In this paper however, conventional analytical procedures are extended and enhanced with computer simulation studies arising from the dynamics of the non-linear and linear models employed, to explore the operational characteristics of the process in greater detail. Ultimately, the essential di!erences between the re"ning action of the two machines are exposed thereby and optimum operational conditions are then investigated.
2. MACHINE CONFIGURATION AND DEPLOYMENT
The disc re"ner is shown in Fig. 1(a) and that for the plug re"ner is shown in Fig. 1(b) in outline form. In the latter case, the stock #ows under pressure through the gap between the conical plug and shell which have embedded, in zig-zag patterns, bronze or stainless-steel rotor and stator shearing bars, respectively. Similarly in Fig. 1(a), stock is pumped through the input pipe to the serrated discs of the disc re"ner where the driven disc provides the power to shear the "bres apart ejecting the stock thereafter by pressurised, centrifugal action. Under the conditions of investigation the speed of the idling disc increases during this operation and the di!erential disc speed can be adjusted, using the brake dynamometer, while running the powered disc at a constant rate, if so desired. The stock consistency, type of stock and the distance between the discs a!ects the disc friction shredding capacity which provides the re"ning action. Additionally, pressure variations arising from the centrifugal pump feeding the re"ner, as described by Simpson et al. [4], generates a cyclic e!ect on the
PULP REFINING AT RESONANCE
645
Figure 1(a). Disc re"ner arrangement.
frictional force, arising from the powered disc thrust bearing reaction, while the journal bearing frictional component remains constant, under steady-state conditions. Modern re"ners often employ rotor discs which are grooved on both faces. A stator disc at each side of the rotor is then used with the stock #ow to the rear face being achieved through ori"ce ducts machined in the rotor hub, as shown in Fig. 2. Essentially, the analysis procedure for both disc con"gurations are identical though the total grooved surface in the latter case (Fig. 2) is twice that of Fig. 1(a), for equal disc
Figure 1(b). Conical re"ner.
646
R. WHALLEY AND D. MITCHELL
diameters. Moreover, production machines are not "tted with brake dynamometers, whereas with the experimental machine analysed here the e!ect of di!erent di!erential disc speeds can easily be investigated while retaining a constant speed, induction motor drive. A typical production re"ner is shown in Fig. 3 where the hinged housing has been opened for inspection purposes. In this photograph, the grooved stator and rotor discs and ori"ce ducts are clearly visible. Low-intensity re"ning which is characterised by high stock dilution, "ne disc grooving and least separation between the discs has been shown, to produce superior "brillation with minimum "bre length reduction. Ultimately, this results in high sheet strength compared to that obtainable from plug re"ners when set to run in a similar optimum mode, as con"rmed by Borsch et al. [5]. The fundamental question returns to the issue of why this should be so when employing machines which appear to have the same mechanical action. Moreover, this is maintained when treating hardwood or softwood chemical/mechanical pulps including sulphide, kraft, soda, groundwood, etc., and with non-wood "bres such as straw and re-cycled, de-inked waste. Much has been stated regarding the disc grooving pattern and depth, and the analogous problem of the Jordan plug and shell bar con"gurations, as discussed in [7] and doubtlessly re"ning action is in#uenced by these factors. However, as this analysis will show there are
Figure 2. Disc re"ner.
PULP REFINING AT RESONANCE
647
Figure 3. Disc re"ner.
reciprocating shear actions present in the disc machine to aid "brillation which do not occur in the plug re"ner and therein lies the advantage.
3. SYSTEM MODEL
Under processing conditions, the interaction of the twin disc arrangement shown in Fig. 1(a), which is analytically equivalent to Fig. 2, is instrumental in achieving a high quality, repeatable stock production preparation, in practice. Under experimental conditions, the brake mechanism, can be used to run the discs at variable di!erential speeds with this con"guration. The di!erential speed between the discs ensures that high shear forces shred and separate the "bres, whilst simultaneously regulating their length, in accordance with the disc separation and grooving pattern. From Fig. 1(a) the equations of motion are ¹ "J Dh #k(h !h ) k(h !h )"J Dh #(Dh !Dh )R (t)#RDh *
(1) (2)
and (Dh !Dh )R (t)"J Dh #R Dh (3) where D is Heavyside's operator and R (t) is the disc friction function. In equation (1), the damping coe$cient associated with h has been omitted, being relatively small in compari son with the inter-disc friction R and the disc bearing friction R . Equations (1)}(3) can be written in the matrix form as ¹ (t) J D#k !k 0 0 " !k J D#(R#R (t))D#k !DR (t) * 0 0 !DR (t) J D#(R (t)#R )D
h (t) h (t) . h (t) (4)
648
R. WHALLEY AND D. MITCHELL
Writing equation (4) as (A D#A D#A )h(t)"!AR (t)Dh(t)#¹ (t) where A (D)"A D#A D#A with J 0 0 0 0 A " 0 J 0 , A " 0 R * 0 0 J 0 0
k
0 0 R
!k 0
, A " !k 0
k
0
0
0
h"(h (t), h (t) h (t)), ¹ (t)"(¹ (t), 0, 0) and 0
0
0
1 !1 R (t)"R (1#r sin pt), and A" 0 0 !1 1
(5)
then Laplace transforming equation (5), with zero initial conditions, we get (A s#A s#A )h(s)"¹ (s)!¸+AR (t)Dh(t), (6) where the Laplace transformation for the non-linear term in equation (6) can be written as ¸+R (t)Dh(t),"R sh(s)#¸+R r sin ptDh(t),. More compactly, equation (6) which implies that the output response comprises the sum of the responses from the input torque and the pump-induced frictional force, becomes A (s)h(s)"¹ (s)!AsR h(s)!A¸+R r sin ptDh(t), where in the matrix quadratic form, A (s)"A s#A s#A Inverting and rearranging equation (6) gives
(7)
h(s)"(I #A (s)\u'(vR s)\A (s)\(¹ (s)!A¸+R r sin ptDh,) where in equation (8),
(8)
A (s)\"Adj A (s)/(det A (s)) u"(0, 1,!1), v"(0, 1,!1) and I "Diag(1, 1, 1). It is easy to obtain the inverses in equation (8), in that Adj A (s) ( J s#Rs#k)( J s#R s) k( J s#R s) 0 * " k( J s#R s) ( J s#R s)( J s#k) 0 0 0 ( J s#k)( J s#Rs#k)!k * (9)
PULP REFINING AT RESONANCE
649
det(A (s))"s( J s#R )[ J J s#J Rs#k( J #J )s#kR]. * *
(10)
and Evaluating (I #A (s)\u'(vsR ) R ( J s#R )k (0 1!1) 1 " det A (s)I #s R (J s#k)(J s#R ) det A (s) !R (J J s#J Rs#(J #J )ks#kR) * * (11) and 1 A (s)\¹ (s)" det (A (s))
(J s#Rs#k)(J s#R s) * ¹ . k(J s#R s) s 0
(12)
Since as stated earlier the transformed input vector for a step change of input torque ¹ (t) is ¹ ¹ (s)" , 0, 0 . s
If the inverse of the matrix in equation (11) is de"ned by b (s) b (s) b (s) 1 b (s) b (s) b (s) B(s)" (13) D(s) b (s) b (s) b (s) and from equation (11), b "b "0. Completing the manoeuvre yields b (s)"(det A (s)#R (J s#k)(J s#R )s#sR (J J s#J Rs)#(J #J )ks#kR) * * b (s)"R k(J s#R )s b (s)"R s(J s#k)(J s#R ) b (s)"!kR s(J s#R ) b (s)"(det A (s)#R s(J J s#J Rs#(J #J )ks#kR)) * * and b (s)"R s(J J s#J Rs#(J #J )ks#kR) * * b (s)"det A (s)#R (J s#k)(J s#R )s Upon multiplying equations (12) and (13) the linear part of the response owing to a driving torque change is h(s)"B(s)A (s)\(¹ (s)) where in equation (14), B(s)A (s)\"G(s)
(14)
650
R. WHALLEY AND D. MITCHELL
so that when r "0 the simple linear solution, corresponding to equation (8), is h (s) h (s) " h (s) J J s#(J (R#R )#(R #R ) J )s#((R #R )R#kJ #R R )s#k(R #R )s)/D(s) * * ¹ (s) k(J s#sR #sR )/D(s) kR s/D(s)
(15)
where in equation (13) and (15), D(s)"(det A (s)#s1v Adj A (s)R u2) "(s(J s#R )(J J s#J Rs#(J #J )ks#kR) * * #sR (J (J #J )s#J (R#R )s#(J #J #J )ks#(R#R )k). * *
(16)
4. DISC FRICTION FUNCTION
In production disc re"ning machines, the disc diameter may exceed 2m. Consequently, the pressure variation arising from the centrifugal stock #ow pump is &li"ed' by the disc area so that a relatively large cyclic separation force occurs between the discs at the frequency of the pressure pulsatance. Reducing the pressure pulsatance e!ect by feeding stock directly to both sides of a central driven disc which has stator discs on each side is sometimes preferred. However, this will be shown to be quite undesirable and detrimental to the shredding}"brillation action which is vital to e$cient re"ning. In practice, disc separation is maintained by the action of the powered disc thrust bearing and adjusting mechanism. However, since the parameters of the friction force R and r remain constant, the cyclic loading caused by the frictional force F(t) varies with the same frequency as the pulsating pump pressure. Normally the viscous frictional force is given by R Dh. In this case the frictional force F(t) is de"ned by the vector
dh F(t)"R r sin pt A (t) dt
(17)
where r sin pt is the pump pulsatance pressure variation of amplitude r and frequency p. Following the Laplace transformation of equation (17), a close linear approximation, as shown in Appendix A is
dh ¹R r p ¸ R r sin pt A (t) "AG(0) dt (s#p) where the transfer function matrix in equation (18) from equation (14) is: G(s)"B(s)A (s)\.
(18)
651
PULP REFINING AT RESONANCE
Then the complete output response, where dh/dt when Laplace transformed becomes u(s), can be obtained, for a step input change in torque, from u(s)"G(s)
¹ !AR r G(0)¹ p/(s#p) s
resulting in u 1 u " D (s) u
f (s)
s k(J s#s(R #R )) ¹ ! (RR #RR #R R )D (s) kR s
k(J s#R ) r p ; (J s#k)(J s#R ) R R ¹ (s#p) !(J J s#J Rs#(J #J )ks#kR) * *
(19)
where f (s)"(J J s#(J (R#R )#J (R #R ))s * * #((R #R )R#kJ #R R )s#k(R #R )s and D (s)"s(J J J s#J (J (R #R )#J (R#R ))s#(J (kJ #RR #RR #R R ) * * #kJ J )s#k((R #R )(J #J )#J (R#R ))s#k(RR #RR #R R )). * 5. APPLICATION STUDY
The speci"c machine parameters have not been stated in this section for reasons of con"dentiality. However, the relative performance of the system remains relevant in the following illustrative study. In this example, if in Fig. 1(a); J "J "J "750 kg m; k"10 N m/rad; R"80, * R "430, R "21000 N m s/rad, and ¹ "25 000 N m, then the steady-state operating conditions, with R constant, are: motor shaft speed, u "u "49.86 rad/s (476 rev/min); idling disc speed, u "1 rad/s (9.6 rev/min); no load power, 199 kW; full load power, 1246 kW.
6. ROOTS OF THE CHARACTERISTIC EQUATION
The parameter values given in Section 5 enable a study of the changes in the singularities of equation (19) to be undertaken. Of particular interest in this exercise are the variations in the roots of the denominator polynomial since these singularities determine the decay rates and frequency of oscillation of u (t), u (t), and u (t). In this respect, the purpose of arranging the denominator determinant in the form given in equation (16) now becomes apparent since the disc friction coe$cient R multiplies the bracketed factor in this expression leaving the remaining terms independent. Consequently, the denominator function which is equated to zero, to provide the characteristic equation, can be written, following the cancellation of the common s term in the numerator and denominator, as D /s"D(s)/s"0.
(20)
652
R. WHALLEY AND D. MITCHELL
Equation (20) can be arranged, using equation (16) as R (s(J #J )J #s(R#R )J #s(J #J #J )k#(R#R )k) * * . !1" (21) (J s#R )(J J s#J Rs#(J #J )ks#kR) * * This quotient is rendered monic by the extraction of the highest degree coe$cients so that equation (21) becomes
R#R (J #J #J ) k(R#R ) s# * ks# K s# (J #J ) J (J #J ) (J #J )J * * * !1" R R (J #J ) kR * ks# s# s# s# J J JJ JJ * * * where in equation (22) with the parameters from Section 5:
(22)
K"R (J #J )/J J "R /375.0 * * R#R "14.05 J #J * J #J #J * "2;10 k J (J #J ) * k(R#R ) "18.74;10 J (J #J ) * J R#J R * "28.106 JJ * R "0.106 J * (J #J ) * k"2.66 ;10 JJ * kR "142.2 . JJ * The poles and zeros of equation (22) for very small values of K are
!0.053, !0.0267$51.64i and !28.0 (poles) and !9.57, !2.24$44.183i (zeros). Increasing K, by changing R , causes the poles to move towards the zeros, one of which is located at !R. The loci of pole migration is shown in Fig. 4 where three values of K"K , K and K are shown in the calibration. From the root locus diagram given in Fig. 4 it is clear that the least oscillatory response is encountered initially by increasing K to K "100 where the damping ratio reaches its maximum. Moreover, it is evident that increasing K and hence R beyond this value causes increasing power to be absorbed by the re"ning action without achieving any improvement in the oscillatory motion of the powered disc over that available for values of K(100 N m s/rad.
PULP REFINING AT RESONANCE
653
Figure 4. Root locus corresponding to equation (22), calibrated in terms of changes in K"disc friction coe$cient/375.0.
Alternatively, for lower stock consistencies resulting in 0)K)30 N m S/rad, power dissipation is reduced and the damping ratio decreases generating higher frequency, maximum amplitude, oscillatory re"ning conditions thereby. Whereas high-energy re"ning determines the quantity of the processing performed, the quality of the re"ning action is determined by operations consuming the least power, owing to high dilution, ensuring that maximum disc}"bre contact is maintained. With low-intensity oscillatory re"ning, "bre treatment at the leading and at the trailing edge of the disc grooves occurs. Whereas with K'100 N m S/rad there are increasing penalties in terms of both higher energy costs and decreasing oscillatory amplitudes, at the driven disc, as R also becomes proportionally larger with higher stock consistencies and oscillatory motion migrates from the driven disc to the motor armature.
7. COMPARISON OF LINEAR AND NON-LINEAR MODELS
To validate the linear model, comparative studies aimed at establishing the parameter variations which could be accommodated are required. The principal non-linearity arises from the multiplication of R r sin pt Dh(t)
(23)
in equation (6). In equation (17), R is constant for a given consistency so tht r , p and u(t) form the set of coe$cients which are variable. However, since Dh(t) is the output speed vector only the pump frequency p and the amplitude of the pulsatance pressure r which
654
R. WHALLEY AND D. MITCHELL
Figure 5. Non-linear system analogue.
gives rise to the retarding torque induced by the pulsating thrust on the powered disc, are considered to be manipulatable variables here. Figures 5 and 6 show, for the speci"c application parameters of Section 5, the non-linear and the linearised models given in equations (4) and (19), respectively, in block diagram form. Following step changes the simulated outputs for the non-linear and linear responses
Figure 6. Linearised model block diagram.
655
PULP REFINING AT RESONANCE
Figure 7(a). Non-linear model driven disc response u (t), following an input torque change of 25 000 N m with pump speed p"30.00 rad/s, r "0.05.
for u (t) is shown in Fig. 7(a), with r "0.05, p"30 rads/s showing the response at arbitrarily selected operating conditions.
8. OSCILLATORY WORKING
If in this analysis the idling disc of the re"ner is set virtually to zero speed, by increasing the brake dynamometer resistance R to a high value rendering the con"guration identical to most production machines. Under steady-state running conditions the oscillatory behaviour of the powered disc is, from equation (19),
R R r ¹ (J s#k)(J s#R )s u (s)" u (s) N (RR #RR #R R ) D (s) where u (s)"p/(s#p). N
(24)
Factoring D (s) for the coe$cient values in Section 5 results in D (s)"sJ J J (s#0.33)(s#28.58)(s#0.34s#51.64) * J (s#k/J )"J (s#1.3.10) and
J (s#R /J )"750(s#28.0).
656
R. WHALLEY AND D. MITCHELL
Figure 7(b). Non-linear model driven disc response u (t), following an input torque change of 25 000 N m with pump speed p"51.64 rad/s, r "0.05.
Consequently, cancelling the pole and zero at s"!28.58 and s"!28.0, respectively, leads to u (s) (J s#k) R R r ¹ " u (s) J J (RR #RR #R R ) (s#2lu s#u )(s#a) N * L L
(25)
where 2lu "0.34, u "51.64 and a"0.33. L L The frequency response equation from equation (25) becomes (J s#k)p u (s)"C (26) (s#2lu s#u )(s#a)(s#p) L L where in equation (26), C"R R r ¹ /(J J (RR #RR #R R )) so that using equation * (26), the steady-state amplitude ratio and phase shift can be found from
C(J s#k) s#c s#c s#c QGN where c "2lu #a, c "2lu a#u and c "au . L L L L The modulus of equation (27) is given by M"C(k!J p)/((c !c p)#p(c !p)).
(27)
(28)
657
PULP REFINING AT RESONANCE
Figure 7(c). Linear model driven disc response u (t), following an input torque change of 25 000 N m with pump speed p"51.64 rad/s, r "0.05.
Extremum values of equation (28) for varying pump frequencies can be found by di!erentiating and equating this equation to zero. Consequently putting d (M)"0 dp we get (J p!k)[!4J p[c#(c!2c c )p#(c!2c )p#p] #(J p!k)[2(c!2c c )p#4(c!2c )p#6p]]"0. (29) It is evident that the common multiplier in equation (29) of (J p!k) will minimise the amplitude ratio of equation (28) whereas the remaining bracketed term which simpli"es to !p[4J c#2k(c!2c c )#(2J (c!2c c )#4k(c!2c ))p#6kp!2J p]"0 (30) will yield the maximum value. If now q"p then the equation to be solved from equation (30) for a non-trivial solution is
3k 2k k q! q!((c!2c c )# (c!2c ))q! 2c# (c!2c c ) "0 J J J
(31)
658
R. WHALLEY AND D. MITCHELL
Figure 8. Modulus ratio u (ip)/u (ip) db vs log frequency. N
where equation (31) upon inserting the parameter values becomes q!4.10q#7.11.10q!9.48.10"0.
(32)
The roots of equation (32) are q"2.667;10, (0.6667$1.764i);10
Figure 9(a). Non-linear model driven disc response u (t), following an input torque change of (25 000#500 sin 51.64 t) N m with pump speed p"51.64 rad/s, r "0.05.
659
PULP REFINING AT RESONANCE
Figure 9(b). Linear model driven disc response u (t), following (25 000#500 sin 51.64 t) N m with pump speed p"51.64 rad/s, r "0.05.
an
input
torque
change
of
so that the resonant frequency is given by the only real root yielding p"(2.667;10) "51.64 rads/s. This condition can be con"rmed from the frequency response plot shown in Fig. 8 where the amplitude of the output u exceeds that of the input amplitude by approximately 10 dB at 51.64 rads/s with the setting here, for demonstration purposes, of r "0.05. The corresponding time domain responses following an input step change of 25 000 N m are shown for the non-linear and linear models in Figs 7(b) and 7(c), respectively, with the pump speed again at p"51.64 rads/s and r "0.05. To improve the oscillatory working of the machine a sinusoidal perturbation is added to the input torque which becomes (25 000#500 sin 51.64t) N m while the pump speed is maintained at 51.64 rads/s and r "0.05. The improved oscillatory response to this excitation is shown in Figs 9(a) and 9(b) for the non-linear and linear models, respectively. Figures 10(a) and (b) show the magni"ed responses in the interval 5}6 s. Once again there is good correlation between the non-linear and linear realisations. Moreover, the linear model is quite tolerant to large changes in r and p, con"rming robustness thereby. 9. BATCH SAMPLING
During investigation trials it is necessary to operate the re"ner for relatively short periods of time in order that stock dilution, disc separation and running speed can be adjusted. This
660
R. WHALLEY AND D. MITCHELL
Figure 10(a). Magni"ed non-linear model driven disc response u (t), following an input torque change of (25 000#500 sin 51.64 t) N m with pump speed p"51.64 rad/s, r "0.05.
process is iterative and continues until the required degree of "brillation and freeness have been achieved for the particular stock preparation, commensurate with the desired "nal product. To facilitate this type of pilot plant deployment an interrupt facility can be incorporated into the control scheme as shown in Fig. 11. Essentially, the digital delay system shown there enables a &run-pause' programme to be automatically enabled, allowing thereby on-line adjustment of the machine without manually starting and stopping the driving motor and pumps, etc. The &run-pause' interval can be easily changed to provide the necessary batch sampling required, by adjusting the "nite time interval of &a' seconds. This is demonstrated in Fig. 12 where the cycle has been reduced to 60 s to illustrate e!ectiveness and the disc speed response u (t) has been computed under resonant conditions following the programme initiation.
10. CONCLUSION
The analysis of the non-linear and linearised models presented in this paper of a disc re"ner enables a general approach to this machine's performance to be investigated. Importantly, the out-of-balance pressure pulsatance arising from the stock #ow from the centrifugal pump to the disc face is shown, though small, to be instrumental in exciting torsional oscillatory action which may be maximised by invoking resonance conditions.
PULP REFINING AT RESONANCE
661
Figure 10(b). Magni"ed linear model driven disc response u (t), following an input torque change of (25 000#500 sin 51.64 t) N m with pump speed p"51.64 rad/s, r "0.05.
The presence of torsional oscillations in the disc re"ner distinguishes its re"ning action from that of the plug (Jordan) re"ner accounting thereby for the superior performance of this machine. Physically, the plug machine is rendered extremely rigid in comparison and any oscillatory motion would be commensurately small. This mode of operation may not be evident when running the stock feed pump at speeds which di!er from the resonant condition. However, by synchronising the drive and pump pulsatance frequency, a signi"cant increase in the amplitude of the oscillatory response at resonance has been demonstrated. These conditions, as shown by the frequency}response diagrams, are easily attainable with a well-de"ned maximum which could be adjusted by varying the sti!ness of the coupling and motor shaft. This results in an increase in e$ciency, by employing in e!ect, both the leading and trailing edges of the disc re"ner grooves in oscillatory shredding action. This action may be less pronounced at non-resonant conditions; however, it distinguishes this machine from its plug and shell (Jordan) counterpart, demonstrating that torsional oscillatory re"ning is an important aspect of disc machine behaviour. This method of pulp re"ning is the subject of a patent application which, when granted, would allow experimental results to be made available. The study also serves to emphasise the statements made by Borsch et al. [5] regarding low-intensity re"ning and the bene"ts arising thereby. From the dynamic analysis viewpoint this emphasises that low disc frictional levels which are necessary to achieve torsional oscillatory conditions are preferable to high frictional, non-oscillatory re"ning which is maximised by adjusting the disc friction level to give K "100 N m s rad. This may be
662
R. WHALLEY AND D. MITCHELL
Figure 11. Interrupt programmer.
achieved by increasing the stock density, for example. It is equally evident that further increases in K then result in improving the oscillatory performance. However, this is because the disc oscillations are in fact decreasing owing to K increasing, resulting in the oscillatory behaviour of the machine being transferred from the disc to the armature. The presence of the disc resonance reinforces the comments made on low-intensity re"ning hightening the "brillation e$ciency and e!ectiveness while minimising power consumption. All that has been promulgated on "bre}disc contact, see for example [3, 6], remains valid though the detailed view of the mechanics of re"ning presented here enables both design, analysis and tuning to be exercised to provide optimum conditions.
Figure 12. Non-linear model driven disc response u (t) to &run-pause' cycle.
PULP REFINING AT RESONANCE
663
While it is true that disc resonance can be obtained by tuning the pump frequency to achieve maximum oscillatory conditions the opposite is also available. As the frequency response diagram of Fig. 8 for u (ip) shows, running the pump at 36.5 rads/s minimises the oscillatory condition, as predicted by the analysis in Section 8, enforcing quiescence conditions and a loss of oscillatory action. Equally, the use of #exible or long piping runs to the re"ner may reduce or eliminate the pump pulsation e!ect. However, simple re-design of the piping connections or even using an active valve arrangement to induce a sinusoidally pulsed #ow could be used to achieve the desired resonant condition and the oscillatory re"ning action thereby.
ACKNOWLEDGEMENT
The authors wish to acknowledge the cooperation of Beloit-Walmsley Limited in supplying details of their DD4000 Double Disc Re"ner and supporting information.
REFERENCES 1. G. A. DUMONT 1986 Automatica 22, 143}153. Application of advanced control methods in the pulp and paper industry. 2. K. J. ASTROG M 1970 Introduction to Stochastic Control ¹heory. New York; Academic Press. 3. B. W. SMITH 1970 In ¹rans. Symposium of British Paper and Board Makers Association, Oxford. F. Bolan (ed.). Designing for control in papermaking systems. 4. H. C. SIMPSON, R. MACASKILL and T. A. CLARK 1966}67 Proceedings IMechE, 181 (pt 3A). 84}108. Generation of hydraulic noise in centrifugal pumps. 5. B. H. BORSCH, R. J. DEFOE and J. RIHS 1986 PIRA International Re,ning Conference, Birmingham, December. Searching to improve "bre quality by means of low consistency re"ning. 6. J. RIHS 1991 Report CPPA ¹ech. 91, Beloit Corporation, Fibre Systems Division, Dalton, MA. Fundamentals of paper mill re"ning for chemical, mechanical and secondary "bres. 7. H. H. ROSENBROCK 1974 Computer-Aided Control System Design. London: Academic Press Ltd.
APPENDIX A
The governing equation in terms of output speed corresponding with the terms in equation (6) is u(t)"u (t)!u (t) (A1) 2 N where u(t)"(u (t), u (t), u (t)) is the output response, u (t)"(u (t), u (t), u (t)) is 2 2 2 2 the output response due to torque inputs and u (t)"(u (t), u (t), u (t)) is the output N N N N response due to pump input in equation (A1). If
and
u (t)"G(D) ¹ (t) 2
(A2)
u (t)"G(D) AR r sin ptu(t) (A3) N where G(D) is a transfer matrix of dimensions 3;3 in equations (A2) and (A3) D"d/dt, A"u'(v with u"(0,!1, 1) and v"(0,!1, 1), R and r are scalars and p is the pump frequency. Consequently, u(t)"G(D) (¹ (t)!AR r sin pt u(t)). (A4)
664
R. WHALLEY AND D. MITCHELL
Laplace transforming equation (A4) yields u(s)"G(s) (¹ (s)!¸+AR r sin pt u(t),). Since r is relatively small in equation (A5) then $ u(s)"u (s) $ 2 and ¹ u (s)"G(s) 2 s
(A5)
(A6)
for a step input change of ¹ . The Laplace transform required therefore is AR r ¸+sin ptu (t),"AR r (u (s!ip)!u (s#ip))/2i. 2 2 2 From equation (A6) the expansion
(A7)
K ¹ K K L ¹ (A8) u (s)"G(s) " # #2# 2 s s#j s#j s L is easily obtained. This implies that equation (A7) can be written, using equation (A8) as
L L K /(s#j !ip! K /(s#j #ip) ¹ (A9) H H H H H H where in equation (A9) j "0 and all other terms containing j , 1)j)n, will decay in the H time domain if G(s) contains stable polynomials. Hence, AR r ¸ +sin ptu (t),"AR r 2
2iK p ¹ AR r ¸ +sin ptu (t),"AR r 2 s#p 2i p "AR r G(0)¹ . s#p
(A10)
Since K "G(s)" Q returning to equation (A5) the transformed output to input relationships using equation (A10) is, for a step change in torque, u(s)"G(s)
¹ p !AR r G(0)¹ . s s#p
(A11)
A good approximation to the time domain response is therefore
¹ u(t)"¸\ G(s) !R r "G(ip)"AG(0)¹ sin (pt#s) s where in equation (A12), tan s"arg(G(ip)A) where tan s"(tan s , tan s , tan s ).
(A12)
PULP REFINING AT RESONANCE
APPENDIX B: NOMENCLATURE J k ¹ J * R R R m u u u ¹ h(t) ¹ (t) A (s) A A A A D r p s I K D(s) A (s)\ G(D) G(s) G(0) F(t) c ,c ,c l u L !a M C u(t) u (t) 2 u (t) N !j H 1x, y2 x'(y e\?Q
motor armature and shaft inertia shaft and coupling sti!ness motor torque load inertia bearing and disc friction dynamometer friction bearing friction number of outputs motor speed"hQ load speed"hQ idling speed"hQ motor torque angular displacement vector"(h (t), h (t), h (t)) (m;1) torque vector (m;1) impedance matrix (m;m) sti!ness matrix (m;m) damping matrix (m;m) inertia matrix (m;m) disc friction matrix (m;m) Heavyside's operator d/dt amplitude of viscous variation force due to pressure drop frequency of viscous force variation due to pressure drop Laplace variable identity matrix (m;m) det A (s) B(s)"inverse of A (s)"Adj(A (s))/det(A (s)) (m;m) transfer matrix operator (m;m) transfer function matrix"B(s) A (s)\ (m;m) steady-state value of G(s) (m;m) frictional force function vector (m;1) polynomial coe$cients damping ratio natural frequency real root location modulus constant multiplier output vector (m;1) output vector component due to torque (m;1) output vector component due to pump (m;1) poles of G(s), 1)j)m inner product of vectors x and y outer product of vector x and y transformed "nite time delay (a seconds)
665