- Email: [email protected]
A spatially dependent model for washing wool
Available online at www.sciencedirect.com
Applied Mathematical Modelling 32 (2008) 389–404 www.elsevier.com/locate/apm
A spatially dependent model for washing wool J.F. Caunce, S.I. Barry *, G.N. Mercer School of Physical, Environmental and Mathematical Sciences, UNSW@ADFA, Canberra ACT, Australia Received 1 March 2006; received in revised form 1 November 2006; accepted 12 December 2006 Available online 22 December 2006
Abstract We analytically model the transport of dirt in the industrial washing of wool using the advection–diffusion equation in two dimensions. Separation of variables leads to a Sturm–Liouville problem where the analytic solution reveals how contamination is distributed both along and down the wool and indicates the operating parameter regimes that optimise the cleaning efficiency. 2006 Elsevier Inc. All rights reserved. Keywords: Sturm–Liouville; Eigenfunction; Wool scouring
1. Introduction The industrial process of washing shorn wool is called wool scouring. In an aqueous wool scour machine, wool is fed as a thin layer along the top of bowls of water. Sets of harrows located evenly and almost continuously along the bowls drive the wool forward and agitate it, freeing caught dirt particles. Some dirt is bound to the wool by grease and is removed at the squeeze rollers at the end of the bowl rather than in the bowl itself, but most is loosely bound to the wool and removed relatively easily. Once released, dirt particles settle away from the wool, eventually collecting in tanks at the bottom of the bowl. A typical scour bowl consists of four tanks. The tanks are periodically drained to remove the dirt, while more water is added, creating a cross-flow along the top of the bowl. A schematic of the conventional scour bowl is shown in Fig. 1. There have been previous mathematical models of wool scouring that looked at the contaminant movement through the entire scour machine. The simplest model [1] balanced the amount of contamination entering and leaving a scouring machine through a single differential equation to determine the contamination in the entire machine at steady state. Later models [2–6] extended this idea to find the contamination in each of the bowls within a machine and introduced more physical factors. A study group report [7] considered the contaminant concentration within a compartmentalised scour bowl and how it changed with time. This was improved [8] by discretising the scour bowl more finely and using finite differences to get more detailed results. An analytical *
Corresponding author. Tel.: +61 2 6268 8881; fax: +61 2 6268 8786. E-mail address: [email protected] (S.I. Barry).
0307-904X/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2006.12.010
390
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
Fig. 1. A schematic of the conventional scour bowl. The harrows move the wool across the top of the water in the bowl and contamination settles out of the wool into the settling tanks. The harrows are shown schematically to indicate their movement, but in reality there are more of them located almost continuously along the bowl.
model of the washing action around the wool at the top of a scour bowl was introduced [9] and some of this work is briefly reproduced in this paper, for clarity and completeness. The model presented in this paper looks at the concentration of dirt in the wool and water at the top of a single scour bowl. It calculates the steady-state distribution of contamination at the top of the bowl in a two dimensional scenario. In particular, the mixing action of the harrows through and around the wool is taken into consideration. Our aim is to gain a better understanding of how contamination in the wool behaves and see how this model compares with the numerical model of a full scour bowl from [8] and the analytical model of [9]. This model is also a nice example of an application of Sturm–Liouville theory to an industrial problem. 2. Model This model looks at the top of the scour bowl, where the wool is gently mixed by the harrows. This ‘mixing zone’ is divided into two layers, the wool and the washing zone, as shown in Fig. 2. The washing zone is where water is mixed with the wool by the harrows. Underneath is the settling zone where dirt moves to the bottom of the tanks under gravity. The movement of contamination in the scour bowl is modelled by the advection– diffusion equation oc o2 c o2 c oc oc ¼ Dx 2 þ Dy 2 v w ; ot ox oy ox oy
ð1Þ
where x is the coordinate down the tank in the direction of settling, y is the coordinate along the tank in the direction of cross-flow, t is time, c is the concentration of contamination at position ðx; yÞ, D is a diffusion coefficient which represents the mixing action of the harrows, v is the vertical velocity and w is the horizontal velocity, representing both the wool velocity, w1 and the cross-flow velocity, w2.
Fig. 2. The mixing zones (wool and washing zone) at the top of the scour bowl with the governing equation and boundary conditions of the model shown.
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
391
The movement of contamination is modelled as a continuum rather than as discrete particles because the dirt particles in wool are very small, with radii of around 1.3 · 105 m and concentrations are low, around 0.4 kg/m3. Chemicals inducing flocculation, where dirt particles clump together, are used in wool scouring, however these clumps are still small enough for the continuum assumption to be valid and simply increases the settling velocities, v in our model. Typically, there are approximately 2 · 1010 particles per m3. A particle tracking model was tried (simulating far less particles), with a result shown in [8] and while this model produced similar results to our numerical solution [8] it was too computationally intensive for practical use. There are four further assumptions made about the system. The first is that the amount of mixing, represented by the diffusion coefficient, D ¼ ðDx ; Dy Þ, is constant in each direction throughout the mixing zones. Diffusion is assumed independent of contaminant concentration since dirt concentrations are always small and hence particle–particle interactions are minimal. In addition constant diffusion is required to generate an exact solution. Second, it is assumed that the velocity of the wool and the cross-flow are both w ¼ w1 ¼ w2 , which is required to generate an analytical solution, that is so Eqs. (12) and (14) have constant coefficients. The third assumption is that the height of the wool, x0/2, is the same as the washing zone beneath it, as they are similar heights in reality. Finally, the system is assumed to be at steady state, so oc ¼ 0. ot The boundary condition at the top of the bowl is of no flux, oc Dx vc ¼ 0; ð2Þ ox x¼0 as no contamination can leave through the top of the wool. At the bottom of the washing zone, there are two boundary conditions ensuring continuity of flux with the settling zone below, ocþ oc vþ cþ ¼ D v c ; x ox ox and continuity of concentration Dþ x
ð3Þ
cþ ¼ c :
ð4Þ
As there is assumed to be negligible diffusion in the settling zone the two boundary conditions combine to give oc ¼ 0: ð5Þ ox x¼x0 At the inlet side of the bowl, the boundary condition, cðx; 0Þ ¼ hðxÞ;
ð6Þ
specifies the contamination brought in on the wool and the cross-flow. Typically, it is assumed that the concentration is constant throughout the incoming wool, cwin , and cross-flow, cin , so that 8 h x 0 w > > c ; x 2 0; ; < in 2 ð7Þ hðxÞ ¼ x > > cin ; x 2 0 ; x : : 2 0 We also know that eventually all the contamination will leave the wool, so cðx; y ! 1Þ ! 0:
ð8Þ
This boundary condition is required to find a solution as there is no natural boundary condition at a finite length. Eq. (1) is non-dimensionalised by the length of a single tank, y0, and the height of the mixing zones, x0, to give 0 ¼ Dx
2 o2 c oc oc o c þ D v w ; y ox2 oy 2 ox oy
ð9Þ
392
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
where Dx ¼
Dx ; x20
Dy ¼
Dy ; y 20
v ¼
v ; x0
w ¼
w ; y0
x ¼
x ; x0
y ¼
y : y0
ð10Þ
This reduces the domain to [0, 1] in the x-direction and [0, 4] in the y-direction, in a typical bowl with four tanks. Applying separation of variables, cðx ; y Þ ¼ f ðx Þgðy Þ;
ð11Þ
Eq. (9) can be split into Dx
o2 f of v lf ¼ 0; 2 ox ox
where l is the eigenvalue in Sturm–Liouville theory, with the boundary conditions of of Dx v f ¼ 0; ¼ 0; ox ox x ¼0 x ¼1
ð12Þ
ð13Þ
and Dy
o2 g og w lg ¼ 0; 2 ox ox
ð14Þ
with the boundary condition from Eq. (8), gðy ! 1Þ ! 0:
ð15Þ
Eq. (12) is a constant coefficient ordinary differential equation, solved using a standard transformation, f ¼ eFx , so that
ðDx F 2 v F lÞeFx ¼ 0;
ð16Þ
which implies F ¼
v
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v 2 þ 4Dx l : 2Dx
ð17Þ
Therefore, there are three different kinds of solutions for f ðx Þ ¼ expðFx Þ. If the discriminant, v2 þ 4Dx l, is zero or positive, then the application of the boundary conditions (13) lead to a trivial solution. If the discriminant is negative, Eq. (12) can be solved in terms of eigenfunctions b fn ðx Þ ¼ ebx cos zn x þ sin zn x ; ð18Þ zn
v where b ¼ 2D and x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 þ 4Dx ln zn ¼ 2Dx
ð19Þ
are convenient shorthands. Note that there are now multiple solutions, hence l is replaced by ln. With these definitions, zn is determined by the implicit equation 2bzn ¼ tan zn : b2
z2n
ð20Þ
Fig. 3 shows the left and right hand sides of Eq. (20), where b = 0.6, with four of the zn values indicated at the intersection points. Note that the singularity of the left hand side at zn = b must be taken into account: a value of b > p=2 would lead to zn less than b. Our numerical method, which is dependent on the fzero command in MATLAB [10], looks for the first two values in the regions ½b p=2; bÞ and ðb; b þ p=2, then works outwards
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
393
Fig. 3. The values of zn associated with the eigenfunctions shown in Fig. 4 with b = 0.6. The curves represent the right and left hand sides of Eq. (20), with the zn occurring where they intersect.
Fig. 4. An example of the eigenfunctions, f1 ; f2 ; f3 and f4 from Eq. (4) with b = 0.6.
with values occurring approximately every p in each direction. The fzero command uses a combination of bisection, secant and inverse quadratic interpolation methods to find the roots of a function. An example of the first four eigenfunctions from Eq. (18) are plotted in Fig. 4. The other separated equation, Eq. (14), after applying the boundary condition (15), has the specific solution
gn ðy Þ ¼ c1 ekn y ;
ð21Þ
394
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
where w kn ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w2 4Dy ln 2Dy
:
ð22Þ
Eq. (19) relates kn to zn . Diffusion in the y-direction, Dy , usually has a very small effect on the system, so often it is helpful to assume that it is negligible, especially when comparing results with the averaged model presented in Section 4. Eq. (22) can be rearranged as Dy k2n w kn þ ln ¼ 0; so if Dy is assumed to be negligible, then, l kn n : w Combining (18) and (21) gives the general solution 1 X b cðx ; y Þ ¼ cn ekn y ebx cos zn x þ sin zn x ; zn n¼1
ð23Þ
ð24Þ
ð25Þ
where cn are constants yet to be determined. To apply the final boundary condition (6), the appropriate inner product is found from Sturm–Liouville theory as Z 1 hf ðx Þ; gðx Þi ¼ e2bx f ðx Þgðx Þdx ð26Þ 0
for all
f(x*)
and
g(x*).
Thus, each of the constants are found through orthogonality as
cn ¼
hhðx Þ; fn i : hfn ; fn i
ð27Þ
The norm of the eigenfunctions is evaluated analytically as hfn ; fn i ¼
z2n sinð2zn Þ þ 2z3n 2bzn cosð2zn Þ þ 2bzn b2 sinð2zn Þ þ 2b2 zn : 4z3n
ð28Þ
The eigenfunctions and eigenvalues were checked by numerically solving the inner product to verify that hfn ; fm i ¼ 0, n 6¼ m. Physically, the eigenvalues, kn, determine the rate at which the concentration decays along the bowl, the expðkn y Þ term in Eq. (25), with the first eigenvalues being the most significant. The eigenfunctions have little physical significance except to give the natural modes for the system. 3. Results Typical results from the model are shown in Figs. 5–7. The physically realistic ‘typical’ scenario used the parameters: Dx = 0.005 m2/min, Dy = 0.0005 m2/min, v = 0.06 m/min, w = 6 m/min, each tank being 2.15 m long and the mixing zones are each 0.1 m high. The inlet side boundary condition (7) becomes hðx Þ ¼ 0:4ð1 H ðx 1=2ÞÞ when non-dimensionalised, where H denotes the Heaviside function, meaning that the contamination of the incoming wool is 0.4 kg/m3 and the water entering the cross-flow area is clean. Eq. (25) was approximated using the first 60 terms of the summation. Fig. 5 shows the concentration along three horizontal cross-sections through the wool region. Initially, dirty wool and clean water enter the bowl, the wool gets cleaner and the water dirtier as contamination settles out of the wool and the two zones are mixed together. Eventually, the concentration in the washing zone surpasses that of the wool and both layers become cleaner as the exponential, expðkny Þ, in Eq. (25) decays, simulating contamination moving into the settling zone. At the inlet side, the bottom of the wool is cleaned quickly compared to the middle because it is adjacent to clean water.
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
395
Fig. 5. A typical result from the model, showing a cross-section of concentration along the middle and bottom of the wool and in the middle of the washing zone. The four tanks of the bowl are indicated by the dotted lines.
Fig. 6. A typical vertical cross-section of contamination through the wool and washing zones, at 8:6 103 m from the inlet side of the bowl (beginning), a third the way along (middle) and the outlet side (end).
Fig. 6 shows three vertical cross-sections through the mixing zones; at the beginning of the bowl (y ¼ 8:6 103 m), a third the way along (y = 2.86 m) and the end of the bowl (y = 8.6 m). Contamination at the beginning of the wool is determined by the boundary condition, with wool contaminated and the
396
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
Fig. 7. Contour plot of contamination in the mixing zones, combining the data shown in Figs. 5 and 6. Contamination settles out of the wool, in the top half of the graph, and diffuses out along the bowl.
cross-flow clean. At y = 0, the discontinuity in h(x) causes Gibbs phenomena oscillation in Eq. (25), with the number of oscillation determined by the finite (60) number of terms used. Beyond y = 0, diffusion removes the discontinuity and smooth the approximation, so the oscillation does not cause any more physically unrealistic effects, as seen at y ¼ 8:6 103 m. Further along the tank contamination has mixed, spreading it more evenly, meaning contamination has moved down from the top of the wool. By the end of the bowl the washing zone is dirtier than the wool and contamination has settled into the lower parts of the bowl. Fig. 7 shows a contour plot of the concentration in the mixing zones, which is a summing up of Figs. 5 and 6. Greater contamination is shown as a darker shade. The dirty wool enters above the clean cross-flow on the left side of the plot, and the contamination noticeably settles and diffuses.
4. Averaged model A weakness of the previous model is that both the wool and the cross-flow in the washing zone must be moving with the same velocity, which in reality is often not the case, to make an analytical solution possible. The next model avoids this by averaging contamination over the two layers to simplify the equations. The disadvantage of this simplification is the solutions are not fully two dimensional. A more complete discussion of this model is contained in [9]. In this model, similar to the previous one, the top of the bowl is divided into two layers with a settling zone beneath and the movement of contamination is modelled by the advection–diffusion Eq. (1). The same assumptions and boundary conditions are used as previously. For convenience, Eq. (1) is non-dimensionalised by the length of a single tank, y0, and the height of the wool, x0/2, rather than the height of both mixing zones, in keeping with the notation used in [9]. For each of the layers, this gives oc1 ^ o2 c1 oc1 ¼ D 2 ^v ; o^y o^x o^x 2 oc2 ^ o c2 a^v oc2 ; ¼ aD o^y o^x2 o^x
^x 2 ½0; 1; ^x 2 ½1; 2;
ð29Þ ð30Þ
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
where the parameters are ^ ¼ 4Dy 0 ; ^v ¼ 2vy 0 ; D w1 x0 w1 x20
w1 ; w2
a¼
^x ¼
x ; x0
^y ¼
y : y0
397
ð31Þ
The concentration of the wool and washing zones are averaged over their height, which has been non-dimensionalised to unity so that, Z 1 Z 2 c1 ¼ c1 d^x; c2 ¼ c2 d^x: ð32Þ 0
1
Averaging Eqs. (29) and (30) in the same way leads to 1 oc1 ^ oc1 1 ¼D ^v½c1 0 ; o^y o^x 0 2 oc2 ^ oc2 a^v½c2 2 : ¼ aD 1 o^y o^x 1
ð33Þ ð34Þ
The boundary condition (2) at the top of the wool eliminates the terms at x = 0, while the continuity boundary condition (5) between the washing zone and the settling zone removes the oco^x2 ^x¼2 term. The remaining derivative term is replaced by the central difference oc1 ¼ c2 ð^x ¼ 3=2Þ c1 ð^x ¼ 1=2Þ: ð35Þ o^x Finally, the concentrations are replaced by the average concentration for each region, so c1 ð^x ¼ 1=2Þ c1 ð^x ¼ 1Þ c1 and c2 ð^x ¼ 3=2Þ c2 ð^x ¼ 2Þ c2 . Hence, oc1 ^ þ ^vÞc1 þ Dc ^ 2; ¼ ðD ð36Þ o^y oc2 ^ þ ^vÞc1 aðD ^ þ ^vÞc2 : ¼ aðD ð37Þ o^y These coupled equations have the solution, c1 ð^y Þ ¼ k 1 v1 ek1 ^y þ k 2 v2 ek2 ^y ; c2 ð^y Þ where 1 ^ þ ^vÞ ð1 aÞ k1 ; k2 ¼ ð D 2 and
" v1 ¼
# ^ D ; ^ þ ^v k1 þ D
" v2 ¼
ð38Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4a^v 2 ð1 þ aÞ ; ^ D þ ^v
^ D ^ þ ^v k2 þ D
are the eigenvalues and eigenvectors of " # ^ þ ^vÞ ^ ðD D ; ^ þ ^vÞ aðD ^ þ ^vÞ aðD
ð39Þ
# ð40Þ
ð41Þ
and ^ in þ cw ðk2 þ D ^ þ ^vÞ Dc in ; ^ 2 k1 Þ Dðk ^ in cw ðk1 þ D ^ þ ^vÞ Dc in k2 ¼ ^ Dðk2 k1 Þ k1 ¼
ð42Þ ð43Þ
398
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
satisfy the inlet side boundary condition (6), which in the averaged model becomes
w cin ; ^x 2 ½0; 1Þ; cð^x; 0Þ ¼ cin ; ^x 2 ½1; 2;
ð44Þ
where cwin is the contamination of incoming wool and cin is the incoming cross-flow contamination. This model displays the same ek^y exponential dependence along the bowl as the x-dependent model; numerically it has been shown that the leading eigenvalue of both models are comparable. When a = 1, as in the x-dependent model, the dimensionalised form of the eigenvalues of the averaged model are vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 u 2v u y 0 4D 2v @ t1 x 0 A ; k1 ; k2 ¼ þ ð45Þ 1 4D x0 w x20 þ 2v x0 x2 0
which are inversely proportional to the horizontal velocity, w, as are the x-dependent model eigenvalues in Eq. (24). 5. Comparisons Comparisons can be made between the x-dependent model, the averaged model and the numerical discretisation model presented in [8]. The time dependent discretisation model simulates the whole scour bowl until steady state is reached. The concentration at the single discretised points in each tank’s wool region is used in these comparisons. Although the discretisation model has several advantages over the analytical models, it is complimented by them because the analytical models take far less time to run, they give continuous solutions and give greater insight into the behaviour of the bowl. Fig. 8 shows the results from each of the three models for concentration in the middle of the wool along the bowl. The same ‘typical’ parameters were used in this simulation as were used in Section 3, except Dx = 0.01 m2/min, Dy = 0.001 m2/min. These parameters were chosen to allow better comparison with the discretisation model which has a more restricted parameter range because the numerical approximation becomes unstable at some parameter values. All three models compare well along the bowl, with a slight discrepancy
Fig. 8. The concentration of contamination along the scour bowl from the x-dependent model, compared to the average model and the numerical discretisation model.
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
399
Fig. 9. The effect of horizontal velocity, w, on the concentration of contamination in the wool, with results from both of our models and data from a discretisation simulation.
between the discretisation model and the other two in the first tank. The dominant eigenvalues in the x-dependent model and the averaged model are comparable. Fig. 9 shows how the three models compare as the horizontal velocity is changed, that is the speed of the wool and cross-flow. In all three models, as w ¼ w1 ¼ w2 ! 0 the concentration in the wool approaches zero because the slower the wool moves the longer there is for contamination to be removed from it. This is true regardless of the contamination of the incoming water because as w2 ! 0 less contamination is brought into the bowl. However, if only w1 ! 0 and cin > 0, then the wool would not be completely cleaned. As w ! 1 contamination approaches the input contamination of the wool, in this case 0.4 kg/m3, as no contamination is cleaned from the wool. As would be expected, given the similarity of the models, the x-dependent and averaged models closely agree, whereas the discretisation model, while showing the same trends, give slightly different results for this parameter set. Each of the models have advantages and disadvantages, but compliment each other. The x-dependent model gives continuous solutions in both the x- and y-directions, but has the disadvantage that the horizontal velocities must be equal to make an analytical solution possible. The averaged model does not have this limitation, allowing a more realistic relationship between w1 and w2, but only gives continuous results in the ydirection with the x-direction averaged. The discretisation model gives time dependent results throughout the scour bowl, but only at discrete points within it. Results from each of these models allow a more complete picture of the scour bowl to be gained.
6. Producing ‘clean wool’ One purpose of these models is to determine which parameter values will produce ‘clean wool’ at the end of the bowl. That is, when c is less than or equal to some contamination threshold ce at the end of the bowl, y = ye. For the simulations discussed in this paper, ce is arbitrarily defined as 0.08 kg/m3 and the bowls have four tanks, so ye = 4. Three parameters that an operator of a wool scour production line has control over are the contamination of water put into the bowls, cin, the horizontal velocity of the wool, w1, and the horizontal cross-flow velocity, w2. The main input parameter is the incoming contamination in the wool, cwin . If the
400
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
horizontal wool velocity is too fast or the input cross-flow is too contaminated, the wool will still be dirty by the end of the bowl. Using the x-dependent model, wool is clean if c 6 ce at the end of the bowl, y ¼ y e , and half way up the wool, x ¼ 1=4. Therefore, the boundary between clean and dirty wool is defined, using Eq. (25), by 1 X b cn ekn y e ebx cos zn x þ sin zn x ; ð46Þ ce ¼ zn n¼1 where the constants, cn, are defined by Eq. (27). The non-dimensionalised form of the inlet side boundary condition (7) is
w cin ; x 2 ½0; 1=2Þ; hðx Þ ¼ ð47Þ cin ; x 2 ½1=2; 1; so cn ¼
cwin
1 hfn ; fn i
Z
1=2
e
bx
0
Z 1 b 1 b bx cos zn x þ sin zn x dx þ cin e cos zn x þ sin zn x dx ; zn hfn ; fn i 1=2 zn
ð48Þ which is rewritten as cn ¼ cwin c1n þ cin c2n ;
ð49Þ
where b cos zn x þ sin zn x dx ; zn 0 Z 1 1 b ebx cos zn x þ sin zn x dx : c2n ¼ hfn ; fn i 12 zn
c1n ¼
1 hfn ; fn i
Z
1 2
ebx
ð50Þ ð51Þ
For simplicity and better comparison with the averaged model, it is assumed that diffusion in the y-direction is negligible, so kn wln as in Eq. (24). Substituting Eq. (49) into Eq. (46) and rearranging gives, P1 ce cin n¼1 c2n jn ðw Þ w P1 1 ; ð52Þ cin ¼ n¼1 cn jn ðw Þ where
jn ðw Þ ¼ e
ln y w
e
bx
b cos zn x þ sin zn x ; zn
ð53Þ
the incoming wool contamination, cwin , required to produce clean wool given the incoming cross-flow contamination, cin, and the horizontal velocities, w ¼ w1 ¼ w2 . Using the averaged model, the boundary between clean and dirty wool at the end of the tank is defined by Eq. (38). Expanded and rearranged this gives a relationship for the incoming wool contamination required for clean wool similar to Eq. (52), cwin ¼
^ k2 ^y e ek1 ^y e Þ ce ðk2 k1 Þ cin Dðe ; ^ þ ^vÞ ek2 ^y e ðk1 þ D ^ þ ^vÞ ek1 ^y e ðk2 þ D
ð54Þ
^ and ^v, and contained within k1 and where the horizontal velocities, w1 and w2, are non-dimensionalised into D k2. The relationships required for clean wool determined using the x-dependent model in Eq. (52) and the averaged model in Eq. (54) are shown in Figs. 10–12 by varying some of the parameters and fixing the others. Unless otherwise specified the parameters are fixed at the physically reasonable values: cwin ¼ 0:2 kg=m3 , cin = 0 kg/m3, w1 = 6 m/min and w2 = 0.7752 m/min.
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
401
Fig. 10. The relationship between horizontal velocity and incoming wool contamination required to produce clean wool. The velocity axis represents either the wool velocity, w1, the cross-flow velocity, w2, or both w ¼ w1 ¼ w2 depending on the context. Stationary wool will always be cleaned regardless of its input contamination.
Fig. 11. The relationship of the horizontal velocities to input contamination of the cross-flow required to clean the wool. The velocity axis represents either the wool velocity, w1, the cross-flow velocity, w2, or both w ¼ w1 ¼ w2 depending on the context.
Fig. 10 illustrates the relationships between the incoming wool contamination, cwin , and the horizontal velocities required to produce clean wool. The first two curves are generated using the averaged model, the first curve varying only the wool velocity, w1 and the second curve varying both the wool and cross-flow velocities together, w ¼ w1 ¼ w2 . In both cases, as wool velocity becomes smaller the required input wool concentration
402
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
Fig. 12. The relationship of the cross-flow velocity to input cross-flow contamination required to clean the wool from two initial wool contaminations as defined by Eq. (54).
approaches infinity. This is because if the wool is stationary eventually all contamination will settle out of it. As the velocities increase, the required input wool contamination approaches ce as infinitely fast moving wool is not cleaned at all. The final curve in the figure is from the x-dependent model, where both velocities are varied together. It compares well to the corresponding curve from the averaged model. Any wool velocity and contamination combination below the curve will result in clean wool, but those on the curve give the optimal relationship because there the wool velocity, and therefore the output of the bowl, is maximised. So, this relationship can be used to determine the optimal wool velocity given an input wool contamination and fixed input cross-flow contamination. For example, if the contamination of the incoming wool is 2.7 kg/m3, then the optimal processing strategy is to run the scour at the maximum wool velocity of 0.5 m/min. There are, however, some physical constraints on the wool velocity because the wool needs to travel fast enough to keep it from sinking down the bowl and slow enough to prevent excessive felting. Fig. 11 illustrates the relationships between incoming cross-flow contamination, cin, and the horizontal velocities required to produce clean wool. The first two curves shown are from the averaged model, the first varying wool velocity, w1, and the second varying both horizontal velocities together, w ¼ w1 ¼ w2 . Both display the same singularity when velocity is zero as those in Fig. 10. One difference between these relationships and those in the Fig. 10 is that as the wool velocity increases the input cross-flow contamination required to clean the wool decreases and becomes negative and therefore unphysical. This decrease is quicker when only wool velocity is varied than when both velocities are increased because faster cross-flow carries contamination away, partially compensating for the faster wool. The final curve in the figure is from the x-dependent model, which compares well with the corresponding curve from the averaged model. This relationship allows wool velocity to be maximised given fixed incoming wool contamination and an input cross-flow contamination. Generally, a wool scour operator has control over only one parameter. The solid curve in Fig. 11 hence allows the operator to optimise the throughput while still producing clean wool. For example, if the input cross-flow contamination is 2.5 kg/m3 then the maximum wool velocity would be 1.6 m/min as shown in Fig. 11. Fig. 12 shows the input cross-flow contamination, cin, required for clean wool as the cross-flow velocity, w2, is varied for two incoming wool contaminations. There is a singularity when cross-flow velocity is zero because when there is no incoming cross-flow it does not matter what concentration the incoming cross-flow has, other parameters decide if the wool will be cleaned. When the wool concentration is 0.1 kg/m3, the wool will be
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
403
cleaned, so the the required cross-flow concentration tends to infinity, when wool concentration is 0.2 kg/m3 the wool will not be cleaned, so required cross-flow concentration tends to negative infinity. To highlight the behaviour of the averaged model, an approximation of Eq. (54) can be made by assuming that y^e is large enough to make the non-dominant exponential term, expðk1 y^e Þ, negligible, giving ^ ce ðk2 k1 Þek2 ^y e cin D cwin ¼ : ð55Þ ^ þ ^v k1 þ D When the horizontal velocities are equal, that is w ¼ w1 ¼ w2 , as in the x-dependent model, the eigenvalues as defined in Eq. (39) reduce to those defined in Eq. (45), where the effect of w can be easily seen. Substituting this into the redimensionalised Eq. (55) and using vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 u 2v u 4D 2v @ x c1 ; c2 ¼ þ ð56Þ 1 t1 4D 0 2vA 2 x0 x0 þ x0 x2 0
as a convenient shorthand gives c2
cwin
¼
cin 4D ce ðc2 c1 Þe w y e x2 0
c1 þ 4D þ 2v x0 x2
;
ð57Þ
0
which shows clearly the exponential relationship between cwin and w and the linear relationship between cwin and cin. This approximation matches the exact solution extremely well. A complete optimisation of a scour bowl, in terms of minimising cost and maximising profit, is a difficult undertaking. This is not just because of the variability in cost estimates, but also in the interaction of numerous parameters – each with physical constraints. For example, increased wool velocity increases productivity, but wool velocity is constrained by mechanical limitation on the harrows, felting of the wool mat and bottlenecks at the squeeze rollers and dryers. Similarly, decreasing incoming cross-flow contamination requires use of either costly fresh water or extra centrifuging and filtration capacity. The goal of optimising scour performance, using the models outlined here, as well as models of roller and dryer efficiency is the subject of our current research. 7. Conclusion The washing of wool at the top of a wool scour bowl can be modelled continuously in two dimensions using an analytic Sturm–Liouville method. This model is limited because the wool and cross-flow must have the same velocity. Averaging contamination over the height of the two layers overcomes this problem, but removes the continuity in the x-direction. Comparisons of the x-dependent model, averaged model and a previous numerical model show a good agreement for the parameters chosen. The models show that, given initially dirty wool and clean cross-flow, the wool becomes cleaner along the bowl. Wool closer to the bottom is initially cleaned much faster, but towards the end of the bowl, contamination increases down the wool. The models reveal the effect of varying some of the operating parameters of the bowl and also give a relationship between the horizontal velocities and incoming contamination required to produce clean wool at the end of the bowl. Acknowledgements This work was funded by Michell Pty. Ltd. and the Australian Research Council. The authors would like to thank the reviewers for their helpful comments. References [1] G.F. Wood, Aqueous jet scouring of raw wool. Part IV. Continuous scouring – The approach to equilibrium, J. Textile Inst. 58 (1967) 49–53. [2] J.R. McCracken, A. Samson, M. Chaikin, The systematic optimization of the aqueous compression jet wool scour. Part I: Pre-optimization procedures, J. Textile Inst. 63 (1) (1972) 1–23.
404
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
[3] D.B. Early, A theoretical analysis of equilibrium conditions in the continuous scouring of raw wool, J. Textile Inst. 68 (2/3) (1978) 60– 67. [4] D.B. Early, The effects of operating variables in raw wool scouring, J. Textile Inst. 69 (12) (1979) 518–524. [5] J.J. Warner, A theoretical analysis of the wool-scouring process at very low water consumption, J. Textile Inst. 71 (5) (1981) 216–227. [6] B.R. Benjamin, J. Staniforth, Use of detergent in wool scouring, in: N.G. Barton (Ed.), Proceedings of the 1991 Mathematics-inIndustry Study Group, 1992, pp. 41–55. [7] S.I. Barry, T.R. Marchant, G.N. Mercer, Grease recovery and dirt removal in wool scouring, in: J. Hewitt, K. White (Eds.), Proceedings of the 2002 Mathematics-in-Industry Study Group, 2003, pp. 106–124. [8] J.F. Caunce, S.I. Barry, G.N. Mercer, T.R. Marchant, Numerical simulation of the contaminant flow in a wool scour. Mathematical and Computer Modelling, in press, doi:10.1016/j.mcm.2006.11.032. [9] J.F. Caunce, S.I. Barry, G.N. Mercer, A simple mathematical model of wool scouring, in: Andrew Stacey, Bill Blyth, John Shepherd, A.J. Roberts (Eds.), Proceedings of the 7th Biennial Engineering Mathematics and Applications Conference, EMAC-2005, ANZIAM J., vol. 47, 2006, pp. C34–C47. [10] MATLAB..
Applied Mathematical Modelling 32 (2008) 389–404 www.elsevier.com/locate/apm
A spatially dependent model for washing wool J.F. Caunce, S.I. Barry *, G.N. Mercer School of Physical, Environmental and Mathematical Sciences, UNSW@ADFA, Canberra ACT, Australia Received 1 March 2006; received in revised form 1 November 2006; accepted 12 December 2006 Available online 22 December 2006
Abstract We analytically model the transport of dirt in the industrial washing of wool using the advection–diffusion equation in two dimensions. Separation of variables leads to a Sturm–Liouville problem where the analytic solution reveals how contamination is distributed both along and down the wool and indicates the operating parameter regimes that optimise the cleaning efficiency. 2006 Elsevier Inc. All rights reserved. Keywords: Sturm–Liouville; Eigenfunction; Wool scouring
1. Introduction The industrial process of washing shorn wool is called wool scouring. In an aqueous wool scour machine, wool is fed as a thin layer along the top of bowls of water. Sets of harrows located evenly and almost continuously along the bowls drive the wool forward and agitate it, freeing caught dirt particles. Some dirt is bound to the wool by grease and is removed at the squeeze rollers at the end of the bowl rather than in the bowl itself, but most is loosely bound to the wool and removed relatively easily. Once released, dirt particles settle away from the wool, eventually collecting in tanks at the bottom of the bowl. A typical scour bowl consists of four tanks. The tanks are periodically drained to remove the dirt, while more water is added, creating a cross-flow along the top of the bowl. A schematic of the conventional scour bowl is shown in Fig. 1. There have been previous mathematical models of wool scouring that looked at the contaminant movement through the entire scour machine. The simplest model [1] balanced the amount of contamination entering and leaving a scouring machine through a single differential equation to determine the contamination in the entire machine at steady state. Later models [2–6] extended this idea to find the contamination in each of the bowls within a machine and introduced more physical factors. A study group report [7] considered the contaminant concentration within a compartmentalised scour bowl and how it changed with time. This was improved [8] by discretising the scour bowl more finely and using finite differences to get more detailed results. An analytical *
Corresponding author. Tel.: +61 2 6268 8881; fax: +61 2 6268 8786. E-mail address: [email protected] (S.I. Barry).
0307-904X/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2006.12.010
390
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
Fig. 1. A schematic of the conventional scour bowl. The harrows move the wool across the top of the water in the bowl and contamination settles out of the wool into the settling tanks. The harrows are shown schematically to indicate their movement, but in reality there are more of them located almost continuously along the bowl.
model of the washing action around the wool at the top of a scour bowl was introduced [9] and some of this work is briefly reproduced in this paper, for clarity and completeness. The model presented in this paper looks at the concentration of dirt in the wool and water at the top of a single scour bowl. It calculates the steady-state distribution of contamination at the top of the bowl in a two dimensional scenario. In particular, the mixing action of the harrows through and around the wool is taken into consideration. Our aim is to gain a better understanding of how contamination in the wool behaves and see how this model compares with the numerical model of a full scour bowl from [8] and the analytical model of [9]. This model is also a nice example of an application of Sturm–Liouville theory to an industrial problem. 2. Model This model looks at the top of the scour bowl, where the wool is gently mixed by the harrows. This ‘mixing zone’ is divided into two layers, the wool and the washing zone, as shown in Fig. 2. The washing zone is where water is mixed with the wool by the harrows. Underneath is the settling zone where dirt moves to the bottom of the tanks under gravity. The movement of contamination in the scour bowl is modelled by the advection– diffusion equation oc o2 c o2 c oc oc ¼ Dx 2 þ Dy 2 v w ; ot ox oy ox oy
ð1Þ
where x is the coordinate down the tank in the direction of settling, y is the coordinate along the tank in the direction of cross-flow, t is time, c is the concentration of contamination at position ðx; yÞ, D is a diffusion coefficient which represents the mixing action of the harrows, v is the vertical velocity and w is the horizontal velocity, representing both the wool velocity, w1 and the cross-flow velocity, w2.
Fig. 2. The mixing zones (wool and washing zone) at the top of the scour bowl with the governing equation and boundary conditions of the model shown.
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
391
The movement of contamination is modelled as a continuum rather than as discrete particles because the dirt particles in wool are very small, with radii of around 1.3 · 105 m and concentrations are low, around 0.4 kg/m3. Chemicals inducing flocculation, where dirt particles clump together, are used in wool scouring, however these clumps are still small enough for the continuum assumption to be valid and simply increases the settling velocities, v in our model. Typically, there are approximately 2 · 1010 particles per m3. A particle tracking model was tried (simulating far less particles), with a result shown in [8] and while this model produced similar results to our numerical solution [8] it was too computationally intensive for practical use. There are four further assumptions made about the system. The first is that the amount of mixing, represented by the diffusion coefficient, D ¼ ðDx ; Dy Þ, is constant in each direction throughout the mixing zones. Diffusion is assumed independent of contaminant concentration since dirt concentrations are always small and hence particle–particle interactions are minimal. In addition constant diffusion is required to generate an exact solution. Second, it is assumed that the velocity of the wool and the cross-flow are both w ¼ w1 ¼ w2 , which is required to generate an analytical solution, that is so Eqs. (12) and (14) have constant coefficients. The third assumption is that the height of the wool, x0/2, is the same as the washing zone beneath it, as they are similar heights in reality. Finally, the system is assumed to be at steady state, so oc ¼ 0. ot The boundary condition at the top of the bowl is of no flux, oc Dx vc ¼ 0; ð2Þ ox x¼0 as no contamination can leave through the top of the wool. At the bottom of the washing zone, there are two boundary conditions ensuring continuity of flux with the settling zone below, ocþ oc vþ cþ ¼ D v c ; x ox ox and continuity of concentration Dþ x
ð3Þ
cþ ¼ c :
ð4Þ
As there is assumed to be negligible diffusion in the settling zone the two boundary conditions combine to give oc ¼ 0: ð5Þ ox x¼x0 At the inlet side of the bowl, the boundary condition, cðx; 0Þ ¼ hðxÞ;
ð6Þ
specifies the contamination brought in on the wool and the cross-flow. Typically, it is assumed that the concentration is constant throughout the incoming wool, cwin , and cross-flow, cin , so that 8 h x 0 w > > c ; x 2 0; ; < in 2 ð7Þ hðxÞ ¼ x > > cin ; x 2 0 ; x : : 2 0 We also know that eventually all the contamination will leave the wool, so cðx; y ! 1Þ ! 0:
ð8Þ
This boundary condition is required to find a solution as there is no natural boundary condition at a finite length. Eq. (1) is non-dimensionalised by the length of a single tank, y0, and the height of the mixing zones, x0, to give 0 ¼ Dx
2 o2 c oc oc o c þ D v w ; y ox2 oy 2 ox oy
ð9Þ
392
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
where Dx ¼
Dx ; x20
Dy ¼
Dy ; y 20
v ¼
v ; x0
w ¼
w ; y0
x ¼
x ; x0
y ¼
y : y0
ð10Þ
This reduces the domain to [0, 1] in the x-direction and [0, 4] in the y-direction, in a typical bowl with four tanks. Applying separation of variables, cðx ; y Þ ¼ f ðx Þgðy Þ;
ð11Þ
Eq. (9) can be split into Dx
o2 f of v lf ¼ 0; 2 ox ox
where l is the eigenvalue in Sturm–Liouville theory, with the boundary conditions of of Dx v f ¼ 0; ¼ 0; ox ox x ¼0 x ¼1
ð12Þ
ð13Þ
and Dy
o2 g og w lg ¼ 0; 2 ox ox
ð14Þ
with the boundary condition from Eq. (8), gðy ! 1Þ ! 0:
ð15Þ
Eq. (12) is a constant coefficient ordinary differential equation, solved using a standard transformation, f ¼ eFx , so that
ðDx F 2 v F lÞeFx ¼ 0;
ð16Þ
which implies F ¼
v
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v 2 þ 4Dx l : 2Dx
ð17Þ
Therefore, there are three different kinds of solutions for f ðx Þ ¼ expðFx Þ. If the discriminant, v2 þ 4Dx l, is zero or positive, then the application of the boundary conditions (13) lead to a trivial solution. If the discriminant is negative, Eq. (12) can be solved in terms of eigenfunctions b fn ðx Þ ¼ ebx cos zn x þ sin zn x ; ð18Þ zn
v where b ¼ 2D and x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 þ 4Dx ln zn ¼ 2Dx
ð19Þ
are convenient shorthands. Note that there are now multiple solutions, hence l is replaced by ln. With these definitions, zn is determined by the implicit equation 2bzn ¼ tan zn : b2
z2n
ð20Þ
Fig. 3 shows the left and right hand sides of Eq. (20), where b = 0.6, with four of the zn values indicated at the intersection points. Note that the singularity of the left hand side at zn = b must be taken into account: a value of b > p=2 would lead to zn less than b. Our numerical method, which is dependent on the fzero command in MATLAB [10], looks for the first two values in the regions ½b p=2; bÞ and ðb; b þ p=2, then works outwards
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
393
Fig. 3. The values of zn associated with the eigenfunctions shown in Fig. 4 with b = 0.6. The curves represent the right and left hand sides of Eq. (20), with the zn occurring where they intersect.
Fig. 4. An example of the eigenfunctions, f1 ; f2 ; f3 and f4 from Eq. (4) with b = 0.6.
with values occurring approximately every p in each direction. The fzero command uses a combination of bisection, secant and inverse quadratic interpolation methods to find the roots of a function. An example of the first four eigenfunctions from Eq. (18) are plotted in Fig. 4. The other separated equation, Eq. (14), after applying the boundary condition (15), has the specific solution
gn ðy Þ ¼ c1 ekn y ;
ð21Þ
394
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
where w kn ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w2 4Dy ln 2Dy
:
ð22Þ
Eq. (19) relates kn to zn . Diffusion in the y-direction, Dy , usually has a very small effect on the system, so often it is helpful to assume that it is negligible, especially when comparing results with the averaged model presented in Section 4. Eq. (22) can be rearranged as Dy k2n w kn þ ln ¼ 0; so if Dy is assumed to be negligible, then, l kn n : w Combining (18) and (21) gives the general solution 1 X b cðx ; y Þ ¼ cn ekn y ebx cos zn x þ sin zn x ; zn n¼1
ð23Þ
ð24Þ
ð25Þ
where cn are constants yet to be determined. To apply the final boundary condition (6), the appropriate inner product is found from Sturm–Liouville theory as Z 1 hf ðx Þ; gðx Þi ¼ e2bx f ðx Þgðx Þdx ð26Þ 0
for all
f(x*)
and
g(x*).
Thus, each of the constants are found through orthogonality as
cn ¼
hhðx Þ; fn i : hfn ; fn i
ð27Þ
The norm of the eigenfunctions is evaluated analytically as hfn ; fn i ¼
z2n sinð2zn Þ þ 2z3n 2bzn cosð2zn Þ þ 2bzn b2 sinð2zn Þ þ 2b2 zn : 4z3n
ð28Þ
The eigenfunctions and eigenvalues were checked by numerically solving the inner product to verify that hfn ; fm i ¼ 0, n 6¼ m. Physically, the eigenvalues, kn, determine the rate at which the concentration decays along the bowl, the expðkn y Þ term in Eq. (25), with the first eigenvalues being the most significant. The eigenfunctions have little physical significance except to give the natural modes for the system. 3. Results Typical results from the model are shown in Figs. 5–7. The physically realistic ‘typical’ scenario used the parameters: Dx = 0.005 m2/min, Dy = 0.0005 m2/min, v = 0.06 m/min, w = 6 m/min, each tank being 2.15 m long and the mixing zones are each 0.1 m high. The inlet side boundary condition (7) becomes hðx Þ ¼ 0:4ð1 H ðx 1=2ÞÞ when non-dimensionalised, where H denotes the Heaviside function, meaning that the contamination of the incoming wool is 0.4 kg/m3 and the water entering the cross-flow area is clean. Eq. (25) was approximated using the first 60 terms of the summation. Fig. 5 shows the concentration along three horizontal cross-sections through the wool region. Initially, dirty wool and clean water enter the bowl, the wool gets cleaner and the water dirtier as contamination settles out of the wool and the two zones are mixed together. Eventually, the concentration in the washing zone surpasses that of the wool and both layers become cleaner as the exponential, expðkny Þ, in Eq. (25) decays, simulating contamination moving into the settling zone. At the inlet side, the bottom of the wool is cleaned quickly compared to the middle because it is adjacent to clean water.
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
395
Fig. 5. A typical result from the model, showing a cross-section of concentration along the middle and bottom of the wool and in the middle of the washing zone. The four tanks of the bowl are indicated by the dotted lines.
Fig. 6. A typical vertical cross-section of contamination through the wool and washing zones, at 8:6 103 m from the inlet side of the bowl (beginning), a third the way along (middle) and the outlet side (end).
Fig. 6 shows three vertical cross-sections through the mixing zones; at the beginning of the bowl (y ¼ 8:6 103 m), a third the way along (y = 2.86 m) and the end of the bowl (y = 8.6 m). Contamination at the beginning of the wool is determined by the boundary condition, with wool contaminated and the
396
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
Fig. 7. Contour plot of contamination in the mixing zones, combining the data shown in Figs. 5 and 6. Contamination settles out of the wool, in the top half of the graph, and diffuses out along the bowl.
cross-flow clean. At y = 0, the discontinuity in h(x) causes Gibbs phenomena oscillation in Eq. (25), with the number of oscillation determined by the finite (60) number of terms used. Beyond y = 0, diffusion removes the discontinuity and smooth the approximation, so the oscillation does not cause any more physically unrealistic effects, as seen at y ¼ 8:6 103 m. Further along the tank contamination has mixed, spreading it more evenly, meaning contamination has moved down from the top of the wool. By the end of the bowl the washing zone is dirtier than the wool and contamination has settled into the lower parts of the bowl. Fig. 7 shows a contour plot of the concentration in the mixing zones, which is a summing up of Figs. 5 and 6. Greater contamination is shown as a darker shade. The dirty wool enters above the clean cross-flow on the left side of the plot, and the contamination noticeably settles and diffuses.
4. Averaged model A weakness of the previous model is that both the wool and the cross-flow in the washing zone must be moving with the same velocity, which in reality is often not the case, to make an analytical solution possible. The next model avoids this by averaging contamination over the two layers to simplify the equations. The disadvantage of this simplification is the solutions are not fully two dimensional. A more complete discussion of this model is contained in [9]. In this model, similar to the previous one, the top of the bowl is divided into two layers with a settling zone beneath and the movement of contamination is modelled by the advection–diffusion Eq. (1). The same assumptions and boundary conditions are used as previously. For convenience, Eq. (1) is non-dimensionalised by the length of a single tank, y0, and the height of the wool, x0/2, rather than the height of both mixing zones, in keeping with the notation used in [9]. For each of the layers, this gives oc1 ^ o2 c1 oc1 ¼ D 2 ^v ; o^y o^x o^x 2 oc2 ^ o c2 a^v oc2 ; ¼ aD o^y o^x2 o^x
^x 2 ½0; 1; ^x 2 ½1; 2;
ð29Þ ð30Þ
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
where the parameters are ^ ¼ 4Dy 0 ; ^v ¼ 2vy 0 ; D w1 x0 w1 x20
w1 ; w2
a¼
^x ¼
x ; x0
^y ¼
y : y0
397
ð31Þ
The concentration of the wool and washing zones are averaged over their height, which has been non-dimensionalised to unity so that, Z 1 Z 2 c1 ¼ c1 d^x; c2 ¼ c2 d^x: ð32Þ 0
1
Averaging Eqs. (29) and (30) in the same way leads to 1 oc1 ^ oc1 1 ¼D ^v½c1 0 ; o^y o^x 0 2 oc2 ^ oc2 a^v½c2 2 : ¼ aD 1 o^y o^x 1
ð33Þ ð34Þ
The boundary condition (2) at the top of the wool eliminates the terms at x = 0, while the continuity boundary condition (5) between the washing zone and the settling zone removes the oco^x2 ^x¼2 term. The remaining derivative term is replaced by the central difference oc1 ¼ c2 ð^x ¼ 3=2Þ c1 ð^x ¼ 1=2Þ: ð35Þ o^x Finally, the concentrations are replaced by the average concentration for each region, so c1 ð^x ¼ 1=2Þ c1 ð^x ¼ 1Þ c1 and c2 ð^x ¼ 3=2Þ c2 ð^x ¼ 2Þ c2 . Hence, oc1 ^ þ ^vÞc1 þ Dc ^ 2; ¼ ðD ð36Þ o^y oc2 ^ þ ^vÞc1 aðD ^ þ ^vÞc2 : ¼ aðD ð37Þ o^y These coupled equations have the solution, c1 ð^y Þ ¼ k 1 v1 ek1 ^y þ k 2 v2 ek2 ^y ; c2 ð^y Þ where 1 ^ þ ^vÞ ð1 aÞ k1 ; k2 ¼ ð D 2 and
" v1 ¼
# ^ D ; ^ þ ^v k1 þ D
" v2 ¼
ð38Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4a^v 2 ð1 þ aÞ ; ^ D þ ^v
^ D ^ þ ^v k2 þ D
are the eigenvalues and eigenvectors of " # ^ þ ^vÞ ^ ðD D ; ^ þ ^vÞ aðD ^ þ ^vÞ aðD
ð39Þ
# ð40Þ
ð41Þ
and ^ in þ cw ðk2 þ D ^ þ ^vÞ Dc in ; ^ 2 k1 Þ Dðk ^ in cw ðk1 þ D ^ þ ^vÞ Dc in k2 ¼ ^ Dðk2 k1 Þ k1 ¼
ð42Þ ð43Þ
398
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
satisfy the inlet side boundary condition (6), which in the averaged model becomes
w cin ; ^x 2 ½0; 1Þ; cð^x; 0Þ ¼ cin ; ^x 2 ½1; 2;
ð44Þ
where cwin is the contamination of incoming wool and cin is the incoming cross-flow contamination. This model displays the same ek^y exponential dependence along the bowl as the x-dependent model; numerically it has been shown that the leading eigenvalue of both models are comparable. When a = 1, as in the x-dependent model, the dimensionalised form of the eigenvalues of the averaged model are vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 u 2v u y 0 4D 2v @ t1 x 0 A ; k1 ; k2 ¼ þ ð45Þ 1 4D x0 w x20 þ 2v x0 x2 0
which are inversely proportional to the horizontal velocity, w, as are the x-dependent model eigenvalues in Eq. (24). 5. Comparisons Comparisons can be made between the x-dependent model, the averaged model and the numerical discretisation model presented in [8]. The time dependent discretisation model simulates the whole scour bowl until steady state is reached. The concentration at the single discretised points in each tank’s wool region is used in these comparisons. Although the discretisation model has several advantages over the analytical models, it is complimented by them because the analytical models take far less time to run, they give continuous solutions and give greater insight into the behaviour of the bowl. Fig. 8 shows the results from each of the three models for concentration in the middle of the wool along the bowl. The same ‘typical’ parameters were used in this simulation as were used in Section 3, except Dx = 0.01 m2/min, Dy = 0.001 m2/min. These parameters were chosen to allow better comparison with the discretisation model which has a more restricted parameter range because the numerical approximation becomes unstable at some parameter values. All three models compare well along the bowl, with a slight discrepancy
Fig. 8. The concentration of contamination along the scour bowl from the x-dependent model, compared to the average model and the numerical discretisation model.
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
399
Fig. 9. The effect of horizontal velocity, w, on the concentration of contamination in the wool, with results from both of our models and data from a discretisation simulation.
between the discretisation model and the other two in the first tank. The dominant eigenvalues in the x-dependent model and the averaged model are comparable. Fig. 9 shows how the three models compare as the horizontal velocity is changed, that is the speed of the wool and cross-flow. In all three models, as w ¼ w1 ¼ w2 ! 0 the concentration in the wool approaches zero because the slower the wool moves the longer there is for contamination to be removed from it. This is true regardless of the contamination of the incoming water because as w2 ! 0 less contamination is brought into the bowl. However, if only w1 ! 0 and cin > 0, then the wool would not be completely cleaned. As w ! 1 contamination approaches the input contamination of the wool, in this case 0.4 kg/m3, as no contamination is cleaned from the wool. As would be expected, given the similarity of the models, the x-dependent and averaged models closely agree, whereas the discretisation model, while showing the same trends, give slightly different results for this parameter set. Each of the models have advantages and disadvantages, but compliment each other. The x-dependent model gives continuous solutions in both the x- and y-directions, but has the disadvantage that the horizontal velocities must be equal to make an analytical solution possible. The averaged model does not have this limitation, allowing a more realistic relationship between w1 and w2, but only gives continuous results in the ydirection with the x-direction averaged. The discretisation model gives time dependent results throughout the scour bowl, but only at discrete points within it. Results from each of these models allow a more complete picture of the scour bowl to be gained.
6. Producing ‘clean wool’ One purpose of these models is to determine which parameter values will produce ‘clean wool’ at the end of the bowl. That is, when c is less than or equal to some contamination threshold ce at the end of the bowl, y = ye. For the simulations discussed in this paper, ce is arbitrarily defined as 0.08 kg/m3 and the bowls have four tanks, so ye = 4. Three parameters that an operator of a wool scour production line has control over are the contamination of water put into the bowls, cin, the horizontal velocity of the wool, w1, and the horizontal cross-flow velocity, w2. The main input parameter is the incoming contamination in the wool, cwin . If the
400
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
horizontal wool velocity is too fast or the input cross-flow is too contaminated, the wool will still be dirty by the end of the bowl. Using the x-dependent model, wool is clean if c 6 ce at the end of the bowl, y ¼ y e , and half way up the wool, x ¼ 1=4. Therefore, the boundary between clean and dirty wool is defined, using Eq. (25), by 1 X b cn ekn y e ebx cos zn x þ sin zn x ; ð46Þ ce ¼ zn n¼1 where the constants, cn, are defined by Eq. (27). The non-dimensionalised form of the inlet side boundary condition (7) is
w cin ; x 2 ½0; 1=2Þ; hðx Þ ¼ ð47Þ cin ; x 2 ½1=2; 1; so cn ¼
cwin
1 hfn ; fn i
Z
1=2
e
bx
0
Z 1 b 1 b bx cos zn x þ sin zn x dx þ cin e cos zn x þ sin zn x dx ; zn hfn ; fn i 1=2 zn
ð48Þ which is rewritten as cn ¼ cwin c1n þ cin c2n ;
ð49Þ
where b cos zn x þ sin zn x dx ; zn 0 Z 1 1 b ebx cos zn x þ sin zn x dx : c2n ¼ hfn ; fn i 12 zn
c1n ¼
1 hfn ; fn i
Z
1 2
ebx
ð50Þ ð51Þ
For simplicity and better comparison with the averaged model, it is assumed that diffusion in the y-direction is negligible, so kn wln as in Eq. (24). Substituting Eq. (49) into Eq. (46) and rearranging gives, P1 ce cin n¼1 c2n jn ðw Þ w P1 1 ; ð52Þ cin ¼ n¼1 cn jn ðw Þ where
jn ðw Þ ¼ e
ln y w
e
bx
b cos zn x þ sin zn x ; zn
ð53Þ
the incoming wool contamination, cwin , required to produce clean wool given the incoming cross-flow contamination, cin, and the horizontal velocities, w ¼ w1 ¼ w2 . Using the averaged model, the boundary between clean and dirty wool at the end of the tank is defined by Eq. (38). Expanded and rearranged this gives a relationship for the incoming wool contamination required for clean wool similar to Eq. (52), cwin ¼
^ k2 ^y e ek1 ^y e Þ ce ðk2 k1 Þ cin Dðe ; ^ þ ^vÞ ek2 ^y e ðk1 þ D ^ þ ^vÞ ek1 ^y e ðk2 þ D
ð54Þ
^ and ^v, and contained within k1 and where the horizontal velocities, w1 and w2, are non-dimensionalised into D k2. The relationships required for clean wool determined using the x-dependent model in Eq. (52) and the averaged model in Eq. (54) are shown in Figs. 10–12 by varying some of the parameters and fixing the others. Unless otherwise specified the parameters are fixed at the physically reasonable values: cwin ¼ 0:2 kg=m3 , cin = 0 kg/m3, w1 = 6 m/min and w2 = 0.7752 m/min.
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
401
Fig. 10. The relationship between horizontal velocity and incoming wool contamination required to produce clean wool. The velocity axis represents either the wool velocity, w1, the cross-flow velocity, w2, or both w ¼ w1 ¼ w2 depending on the context. Stationary wool will always be cleaned regardless of its input contamination.
Fig. 11. The relationship of the horizontal velocities to input contamination of the cross-flow required to clean the wool. The velocity axis represents either the wool velocity, w1, the cross-flow velocity, w2, or both w ¼ w1 ¼ w2 depending on the context.
Fig. 10 illustrates the relationships between the incoming wool contamination, cwin , and the horizontal velocities required to produce clean wool. The first two curves are generated using the averaged model, the first curve varying only the wool velocity, w1 and the second curve varying both the wool and cross-flow velocities together, w ¼ w1 ¼ w2 . In both cases, as wool velocity becomes smaller the required input wool concentration
402
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
Fig. 12. The relationship of the cross-flow velocity to input cross-flow contamination required to clean the wool from two initial wool contaminations as defined by Eq. (54).
approaches infinity. This is because if the wool is stationary eventually all contamination will settle out of it. As the velocities increase, the required input wool contamination approaches ce as infinitely fast moving wool is not cleaned at all. The final curve in the figure is from the x-dependent model, where both velocities are varied together. It compares well to the corresponding curve from the averaged model. Any wool velocity and contamination combination below the curve will result in clean wool, but those on the curve give the optimal relationship because there the wool velocity, and therefore the output of the bowl, is maximised. So, this relationship can be used to determine the optimal wool velocity given an input wool contamination and fixed input cross-flow contamination. For example, if the contamination of the incoming wool is 2.7 kg/m3, then the optimal processing strategy is to run the scour at the maximum wool velocity of 0.5 m/min. There are, however, some physical constraints on the wool velocity because the wool needs to travel fast enough to keep it from sinking down the bowl and slow enough to prevent excessive felting. Fig. 11 illustrates the relationships between incoming cross-flow contamination, cin, and the horizontal velocities required to produce clean wool. The first two curves shown are from the averaged model, the first varying wool velocity, w1, and the second varying both horizontal velocities together, w ¼ w1 ¼ w2 . Both display the same singularity when velocity is zero as those in Fig. 10. One difference between these relationships and those in the Fig. 10 is that as the wool velocity increases the input cross-flow contamination required to clean the wool decreases and becomes negative and therefore unphysical. This decrease is quicker when only wool velocity is varied than when both velocities are increased because faster cross-flow carries contamination away, partially compensating for the faster wool. The final curve in the figure is from the x-dependent model, which compares well with the corresponding curve from the averaged model. This relationship allows wool velocity to be maximised given fixed incoming wool contamination and an input cross-flow contamination. Generally, a wool scour operator has control over only one parameter. The solid curve in Fig. 11 hence allows the operator to optimise the throughput while still producing clean wool. For example, if the input cross-flow contamination is 2.5 kg/m3 then the maximum wool velocity would be 1.6 m/min as shown in Fig. 11. Fig. 12 shows the input cross-flow contamination, cin, required for clean wool as the cross-flow velocity, w2, is varied for two incoming wool contaminations. There is a singularity when cross-flow velocity is zero because when there is no incoming cross-flow it does not matter what concentration the incoming cross-flow has, other parameters decide if the wool will be cleaned. When the wool concentration is 0.1 kg/m3, the wool will be
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
403
cleaned, so the the required cross-flow concentration tends to infinity, when wool concentration is 0.2 kg/m3 the wool will not be cleaned, so required cross-flow concentration tends to negative infinity. To highlight the behaviour of the averaged model, an approximation of Eq. (54) can be made by assuming that y^e is large enough to make the non-dominant exponential term, expðk1 y^e Þ, negligible, giving ^ ce ðk2 k1 Þek2 ^y e cin D cwin ¼ : ð55Þ ^ þ ^v k1 þ D When the horizontal velocities are equal, that is w ¼ w1 ¼ w2 , as in the x-dependent model, the eigenvalues as defined in Eq. (39) reduce to those defined in Eq. (45), where the effect of w can be easily seen. Substituting this into the redimensionalised Eq. (55) and using vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 u 2v u 4D 2v @ x c1 ; c2 ¼ þ ð56Þ 1 t1 4D 0 2vA 2 x0 x0 þ x0 x2 0
as a convenient shorthand gives c2
cwin
¼
cin 4D ce ðc2 c1 Þe w y e x2 0
c1 þ 4D þ 2v x0 x2
;
ð57Þ
0
which shows clearly the exponential relationship between cwin and w and the linear relationship between cwin and cin. This approximation matches the exact solution extremely well. A complete optimisation of a scour bowl, in terms of minimising cost and maximising profit, is a difficult undertaking. This is not just because of the variability in cost estimates, but also in the interaction of numerous parameters – each with physical constraints. For example, increased wool velocity increases productivity, but wool velocity is constrained by mechanical limitation on the harrows, felting of the wool mat and bottlenecks at the squeeze rollers and dryers. Similarly, decreasing incoming cross-flow contamination requires use of either costly fresh water or extra centrifuging and filtration capacity. The goal of optimising scour performance, using the models outlined here, as well as models of roller and dryer efficiency is the subject of our current research. 7. Conclusion The washing of wool at the top of a wool scour bowl can be modelled continuously in two dimensions using an analytic Sturm–Liouville method. This model is limited because the wool and cross-flow must have the same velocity. Averaging contamination over the height of the two layers overcomes this problem, but removes the continuity in the x-direction. Comparisons of the x-dependent model, averaged model and a previous numerical model show a good agreement for the parameters chosen. The models show that, given initially dirty wool and clean cross-flow, the wool becomes cleaner along the bowl. Wool closer to the bottom is initially cleaned much faster, but towards the end of the bowl, contamination increases down the wool. The models reveal the effect of varying some of the operating parameters of the bowl and also give a relationship between the horizontal velocities and incoming contamination required to produce clean wool at the end of the bowl. Acknowledgements This work was funded by Michell Pty. Ltd. and the Australian Research Council. The authors would like to thank the reviewers for their helpful comments. References [1] G.F. Wood, Aqueous jet scouring of raw wool. Part IV. Continuous scouring – The approach to equilibrium, J. Textile Inst. 58 (1967) 49–53. [2] J.R. McCracken, A. Samson, M. Chaikin, The systematic optimization of the aqueous compression jet wool scour. Part I: Pre-optimization procedures, J. Textile Inst. 63 (1) (1972) 1–23.
404
J.F. Caunce et al. / Applied Mathematical Modelling 32 (2008) 389–404
[3] D.B. Early, A theoretical analysis of equilibrium conditions in the continuous scouring of raw wool, J. Textile Inst. 68 (2/3) (1978) 60– 67. [4] D.B. Early, The effects of operating variables in raw wool scouring, J. Textile Inst. 69 (12) (1979) 518–524. [5] J.J. Warner, A theoretical analysis of the wool-scouring process at very low water consumption, J. Textile Inst. 71 (5) (1981) 216–227. [6] B.R. Benjamin, J. Staniforth, Use of detergent in wool scouring, in: N.G. Barton (Ed.), Proceedings of the 1991 Mathematics-inIndustry Study Group, 1992, pp. 41–55. [7] S.I. Barry, T.R. Marchant, G.N. Mercer, Grease recovery and dirt removal in wool scouring, in: J. Hewitt, K. White (Eds.), Proceedings of the 2002 Mathematics-in-Industry Study Group, 2003, pp. 106–124. [8] J.F. Caunce, S.I. Barry, G.N. Mercer, T.R. Marchant, Numerical simulation of the contaminant flow in a wool scour. Mathematical and Computer Modelling, in press, doi:10.1016/j.mcm.2006.11.032. [9] J.F. Caunce, S.I. Barry, G.N. Mercer, A simple mathematical model of wool scouring, in: Andrew Stacey, Bill Blyth, John Shepherd, A.J. Roberts (Eds.), Proceedings of the 7th Biennial Engineering Mathematics and Applications Conference, EMAC-2005, ANZIAM J., vol. 47, 2006, pp. C34–C47. [10] MATLAB.