Modelling and feedforward control of a glassheat-conditioning process via least-squares

Patterson collocationA.H.Whitfield, B.Sc, Ph.D.

Indexing terms: Control equipment and applications, Modelling, Mathematical techniques

Abstract: The approximate solution of systems governed by linear differential operators in two space dimen-sions is achieved by a new least-squares collocation technique using Patterson quadrature points and weights.Boundary conditions which are identically imposed on the approximate solution may be optimised, and thescheme may be used to rapidly predict feedforward control parameters. Boundary conditions which cannotbe imposed on the approximation are also considered. The technique is applied to a short-forehearth heat-conditioning green glass, and its use in the online control of such a process is discussed.

List of symbols

Aa

Bb

CccDmax

D(X)D{x)

Dm{x)

d\

dr(X)db(X)/,(y)

Gfem)

Si(x,y)

In

IM{NQ))Jj(x,y)

Kb

Kg

Kr

Keff(X)

L

b

M

Mf

= matrix defined in eqn. 18= normalised linear depth par-

ameter in zone 3= matrix defined in eqn. 20= normalised linear depth par-

ameter in zone 3= general cost function, eqn. 54= specific heat of glass= vector defined in eqn. 18= maximum depth of glass in

forehearth= glass depth in forehearth= normalised glass depth of non-

dimensional system= normalised glass depth in

zonera= normalised glass depth in zones

1 and 2= thickness of refractory= thickness of brick= combination of boundary con-

dition parameters= element defined and used in

eqn.67= functions multiplying bound-

ary condition parameters= integral evaluated by quadra-

ture formula, eqn. 7= integral evaluation, eqn. 8= determinant of Jacobian

matrix in/th subdomain= conductivity of brick= conductivity of glass= conductivity of refractory= effective conductance of

refractory/brick at glass/refrac-tory interface

= length of forehearth= linear time-invariant interior

operator, eqn. 3= linear time-invariant boundary

operator, eqn. 11= upper limit of summation in

polynomial approximation= mass flow rate

N.

Paper 1852D, first received 4th August and in revised form 11thDecember 1981The author is with the Department of Engineering Mathematics,University of Technology, Loughborough LE11 3TU, England

IEEPROC, Vol. 129, Pt. D, No. 3, MA Y 1982

Pi

Pi

R(a,x,y)bR(«,x,y.

jR(x,y)

R(«,x,y)

mRo(x,y)

mRj(x,y)

bR0(x,y),bRj(x,y)

T(X, Y)TaTdTe

U(x,y)

= highest order of polynomialapproximant in one spacedimension

= number of combinations ofboundary condition parameters

= number of multiplicative andadditive boundary conditionparameters

= number of interior collocationpoints

= number of boundary collo-cation points

= /th-order normalised Legendrepolynomial on [0, 1]

= /th-order non-normalisedLegendre polynomial on [0, 1 ]

= interior residual= boundary residual

= interior residual in / th sub-domain

= boundary residual from /thnonimposed boundary con-dition

= weighted residual= contribution to interior residual

from imposed boundary con-ditions

= contribution to interior residualfrom basis function

= contribution to interior residualin subdomain m from imposedboundary conditions

= contribution to interior residualin subdomain m from basisfunction

= boundary residuals correspond-ing to latter

= boundary residuals= as above with residual defined

over normalised region, i.e.[0, 1] x [0, 1] for interior or[0, 1] for boundary

= /th subdomain of S= temperature distribution= constant reference temperature= desired outlet temperature= T(0,0)-T(0,D(0))= constant reference temperature= temperature of brick/air inter-

face= dimensionless variable, eqn. 26

0143-7054/82/030081 +15 $01.50/0 81

U(X, Y)W(x)

W(X)

wmax

Wg{X)

X, Y,Zx,y,z

x,y

aPi

r

71 (x)

72 (X)

7i

y

7o(x,y)

7io,7n,7i2

'ba

Mo,Mi

P

£i(.y)> • • • , I

Superscripts*

82

= dimensionless variable U{x, y)in zone /

= velocity profile= nondimensional width of fore-

. hearth= width of forehearth= maximum width of forehearth= quadrature weights, expr. 14= width of glass at bottom of

forehearth= width of refractory/brick inter-

face= width of brick/air interface= co-ordinate variables= dimensionless co-ordinate vari-

ables= normalised dimensionless

co-ordinate variables in 2DVBmodel

= normalised distance to end to/th zone

= coefficient vector, eqn. 2= 2DVB model parameter= domain of definition of non-

imposable boundary condition= domain of /th nonimposable

boundary condition= boundary temperature profile

in dimensionless system= boundary polynomial in

dimensionless system= boundary polynomial in

dimensional system= vector of boundary condition

parameters 7,, . . . , yN usedin eqn. 20

= expression of imposed bound-ary conditions, eqn. 2

= quadratic approximation to72(x) in zone/

= glass surface temperature coef-ficients, eqn. 51

= convergence criterion constantfor number of collocationpoints

= convergence criterion constantfor order of polynomial

= dimensionless temperature vari-able

= approximate solution= nondimensional minimum

allowable surface temperature= nondimensional brick/air-inter-

face temperature= inlet temperature profile= inlet temperature profile in

dimensionless system= parameters in linear inlet pro-

file, eqn.50= glass density= functions used in eqn. 67= general cost function weighting= optimisable boundary con-

dition parameter= basis function in general MWR

eqn. 2

= optimum

d_

dy

_ d_

~ dx

= vector transpose

1 Introduction

1.1 Solution and control of distributed-parameter systems

For many engineering systems, various states may have bothspatial and temporal variations which may not be neglected,and the equations governing the behaviour of the states arethen described by one or more partial differential equations(PDE) with their associated boundary conditions. The classicaltext by Carslaw and Jaeger [1] contains many examples ofsuch distributed-parameter systems (DPS) which have ananalytic solution. Unfortunately, many practical systems arenot amenable to such solution techniques, and methods ofapproximation must be considered. The wide variety of suit-able approximation techniques includes the methods ofcharacteristics [2], finite differences [3], finite elements [4],variational methods and a class of techniques known collec-tively as the method of weighted residuals (MWR) [5].Accuracy of solution is given by the methods of character-istics, finite differences and finite elements, but their placein online control applications is at present limited by com-puting time and memory storage. The general class of MWRtechniques probably demands more offline work than theselatter methods, but it may also offer solution accuracy with arelatively small computing requirement once the offline workis completed.

When the MWR is applied to partial differential equationswhose boundary conditions are identically imposable on anassumed form of approximate solution, then the methods aregenerally referred to as interior. Such applications are quitecommon [5—7]. The techniques are also applied to boundaryproblems in which the general solution of the partial differen-tial equation is known [8, 9] and less frequently to mixedproblems in which both the general solution is unknown andvarious boundary conditions cannot be identically imposedon an assumed approximate solution [10]. Within these broadtitles, the individual MWR techniques include Galerkin'smethod, the method of moments and least-squares approxi-mation. The majority of MWR applications require explicitevaluation of certain integrals and approximate the variationin one space dimension to leave a set of coupled ordinarydifferential equations in one other dimension, either temporalor spatial. However, many applications do not yield readilysolvable ordinary differential equations, and it is thereforenecessary to consider projection of two or more independentvariables. Such a projection usually implies that the previouslymentioned integrals must be calculated numerically, and, toavoid such explicit evaluations, the method of collocationwas forwarded. Villadsen and Stewart [11] point out thatthis method may actually provide identical results to certainMWR, although this is not the case in general.

When the collocation procedure does not concur with theintegration approach in MWR, a sufficient number of collo-cation points is required to at least give a suitably accurateapproximation. With orthogonal collocation as proposed byVilladsen and Stewart [11], Villadsen and Michelson [12],this is not feasible, and one purpose of this paper is to outlinea systematic and efficient procedure to overcome thisproblem.

IEE PROC, Vol. 129, Pt. D, No. 3, MA Y1982

In addition to the rapid solution of DPS, MWR has a majoradvantage over other solution techniques in that it may bepossible to embed boundary conditions within the assumedapproximate solution. Control of DPS is usually via theboundary, and thus, by use of MWR, the internal behaviourof the system may be determined explicitly in terms of thecontrol functions and may therefore be optimised, in somesense, with respect to such functions. This optimisation isdemonstrated for a general class of control functions and isillustrated by application in a glass heat-conditioning process.

1.2 Glass heat-conditioning process

In the manufacture of many types of glassware, one of themost critical phases is the heat conditioning of the glass afterit has been melted and prior to its forming. For many pro-ducts, such as containers, tubing, lenses, light bulbsand table-ware, this conditioning starts in a refining zone and is com-pleted in a forehearth. If such heat conditioning is not per-formed correctly, then the glass product may be defective inmany ways [13]. The avoidance of such defects, togetherwith consideration of mounting fuel costs and the desirefor increased throughput, indicate that the correct designand operation of the refiner and forehearth is of extremeimportance.

skimmer block back block needleburners /cooling wind \ burners

Fig. 1 Schematic diagram of a forehearth

A forehearth which is heat-conditioning glass flowingto a gob feeder is illustrated in Fig. 1. This 'short forehearth'is made up of two sections, each of which is fired by a numberof burners. The first section is known as the cooling zone andis additionally equipped with a wind cooling system. Theburners of the second section, known as the heating or con-ditioning zone, are operated in such a manner as to equalisethe temperature of the glass as it reaches the spout. Tempera-tures of the surface glass are measured at two points alongthe length of the forehearth, with control of the outlettemperature being achieved by controlling the burners of thecooling and conditioning zones so that the two measuredtemperatures agree with two prescribed temperatures. Thecontrol of the two temperatures is currently rapid and accu-rate, with two- and three-term control usually provingadequate. However, the two set points, which ultimatelydetermine the glass quality, are set and adjusted manually,and, in consequence, the overall control of such a forehearthcould be improved.

In an effort to approach this and related problems, severalmathematical models of the process of heat and mass transferwithin a forehearth have been developed. The flow of glass inan open channel was considered by Hearn and Booth [14],who show that the assumption that the flow of glass is laminaris not unreasonable. Henry [15] suggests that such laminarflow, which implies a lack of mixing, is one of the basicdifficulties in balancing forehearth temperatures. Duffin andJohnson [16] were early workers in the development ofsuitable heat transfer equations and produced relativelycomplicated models which, when solved by finite-difference

techniques, gave reasonable agreement with measured tempera-ture profiles. Catling [17] has produced further heat andmass transfer equations and, while giving good agreementwith measured temperatures, they are also extremely com-plicated. Hamilton [18] has used the simplest 2DL modelof Carling to predict approximately the optimum tempera-ture profile of the glass as it enters a particular short fore-hearth. Even this simpler 2DL model demands quite extensivecomputing facilities and can therefore only be used in selectivedesign studies, and the need for a model which may be rapidlyoptimised and hence used in an online control scheme is clear.

2 Least-squares Patterson collocation

2.1 Least-squares collocation

The basic PDE to be considered in this paper has the generalform

JS?[0J = 0 (1)

where Sf is a linear time-invariant differential operator definedover some finite two-dimensional domain S. A point within Sis denoted by (x, y)£S, and the PDE eqn. 1 is subject tocertain boundary conditions which make the problem wellposed.

A global approximate solution d(x,y) to eqn. 1 and itsassociated boundary conditions is to be expressed as

= yo(x,y)M

7 = 1(2)

It is desirable to impose as many boundary conditions ontothe approximate solution as possible, and yo(x,y), tyj(x,y),7 = 1 , 2 , . . . ,M, are chosen to this end. In particular, bound-ary conditions which are to be optimised should be containedexplicitly within the function Jo(x,y). The vector of con-stante a = [a , , . . . , aM]T is to be determined in such a waythat d(x,y) represents, in some sense, an approximate solutionof the equation

Sf[B] = 0 (3)

Application of the linear operator JZf to 6 yields a residualR(a,x,y). Thus

i.e.

= R(«,x,y)

= R0(x,y)+M

where

Rj(x,y) =In the least-squares context, the basic problem to be solvedis then

mint R2 (a jXty)dSa Js

(4)

The evaluation of terms within the integral expr. 4 may bedifficult, and numerical quadrature techniques would, ingeneral, be required. Explicitly using such a quadrature for-mula gives an approximately equivalent problem

min fwkR2(a,xk,yk) (5)

fe=i

Clearly, the least-squares solution of the Nq >M equations

= 0

IEE PROC, Vol. 129, Pt. D, No. 3, MA Y 1982 83

i.e. M

%otjRj(xk,yk) = - (6)

k = l,...,NQ

will also be a solution of expr. 5 and hence an approximatesolution of expr. 4.

Thus the original least-squares integral problem is nowreposed as a least-squares collocation problem, which willbe shown to be useful in determining values for the numberof collocation points Nq and the number of basis functionsM and will ultimately provide the approximate solution ofthePDEeqn. 1.

2.2 Least-squares collocation using Pattersonquadrature points and weights

Consider the basic approximate solution eqn. 2 in which theupper limit of summation M is fixed. An efficient quadraturescheme is required such that:

(a)Nq is small, yet the quadrature formula is accurate(b) the computation involved in the scheme is minimal.

It is well known that the Gaussian quadrature formulas satisfythe first requirement, and, within this class, the Gauss-Legendre formula is particularly easy to apply since, in onedimension, the collocation points xk are the zeros of Legendrepolynomials. Indeed, it is these points which often form thebasis of orthogonal collocation. Explicitly, this quadratureformula is expressed as

jj(x)dx * £ Wkf(xk) = In0 fe = i(7)

where xk(k = 1, 2, . . . ,n) are the zeros of the wth-orderLegendre polynomial Pn{x) on [0 ,1] , and where Wk(k =1,2, ... ,n) are known constants. If the nth-order approxi-mation /„ is not considered to be accurate, then a higher-orderapproximation InAnx > n) will have to be evaluated. How-ever, the zeros of the Legendre polynomials of orders n and«! are all different for n > 1, and hence f(x) has to be evalu-ated at «j new points. Patterson [19, 20] proposed a newquadrature formula in which all the quadrature points areretained and used in any higher-order approximation. Thisquadrature formula is only slightly less accruate than thecomparable Gauss-Legendre scheme, and it has obviouscomputational advantages. In practice, when determiningthe convergence of the quadrature formula, n is usuallychosen to follow the sequence {1, 3, 7, 15, 31, . . .}, i.e.

number of quadrature points at rth iteration =

Thus the standard Gauss-Legendre scheme requires 2 2

l~l

1"1

additional function evaluations at the rth iteration, whilePatterson quadrature requires only 2r~l similar evaluationsat the same iteration. For integrals in higher dimensions, thedifference in the number of additional function evaluations iseven more significant. In two dimensions, for example, overthe square domain [0,1] x [0 ,1] , the difference at therth (r > 1) iteration is equal to the number of points at the(r — l)th iteration, i.e.

r -1 |2

X2'-1'1 = 1

Patterson quadrature points and weights therefore satisfyboth requirements (a) and (b) and will now be used as thebasis of the collocation procedure. The points and weights in

84

two dimensions are readily generated by extending Patterson'sone-dimensional integration routine once the domain S hasbeen transformed to the square [0, 1] x [0, 1]. Assume thatthe domain S has been transformed to [0, 1] x [0, 1] andthat the residual defined over [0,1] x [0, 1] i s ^ ( o , x,y).This assumption is not restrictive, as will be shown later.With the order of approximate solution M and the collocationpoints and weights specified, the number of such collocationpoints is determined as follows. The parameters in the bound-ary conditions are initially assigned numerical values. Thenthe right-hand sides of eqn. 6 are known, and this set ofequations is readily solved by recursive least squares. Hence,for a given value of Nq=Nql, the values a,-, i = 1, . . . , M,are evaluated without the need of an explicit matrix inver-sion. The values of the residuals &j(xk,yk), / = 1,. . . ,M,&o(xk,yk) are stored during the process of the recursiveleast-squares algorithm, and the summation

(8)k=i

is evaluated on termination of the algorithm. NQl is thenincreased to NQ2 and the resulting a,-, i=l,...,M, arere-evaluated, again using recursive least squares. SincePatterson collocation points are employed, the residuals&j(Xk,y~k), j=l,...,M, and &0{xk,yk) need only to beevaluated at a few new points with the previously storedvalues being recalled within the algorithm. The relative errorbetween IM(NQI) and IM(NQ2) is then used as a convergencecriterion for NQ. Thus, if ec is a given constant and

< e. (9)

then Nq is set equal toNQ2, and the process is terminated. Thefinal value of Nq is dependent onM, a suitable value of whichwill now be determined. M is initially chosen to be rax, andthe corresponding Nq is evaluated by the scheme outlinedabove. M itself may be incremented to a new value m2, andthe whole process of determining a suitable Nq will be recom-menced. The incrementation of M is then terminated when

and the final value of M is chosen to be m2 . Thus there existsa rational method for calculating appropriate values of theconstants Nq andAf.

2.3 Treatment of nonimposed boundary conditions

For many models described by PDEs, it may be difficult, oreven impossible, to simultaneously impose all the given bound-ary conditions on to the approximate solution. The effect ofthe nonimposed boundary conditions must still be includedand should clearly influence the final approximate solution.Two situations are likely to lead to such a nonimposition:

(i) There are a large number of boundary conditions.(ii)One, or more, of the boundary conditions is defined

over a complicated domain.

Consider first that only one boundary condition cannot beexactly imposed on the approximate solution. Let this bound-ary condition be of the form

= f (10where jgf b is a time-invariant linear differential operatorand/ is some function, both defined over a given boundary V.The boundary residual bR(<*, x, y), (x, ^ ) £ r , may then bedefined as

bR(«,x,y) =Sfb[6]-f (12)

IEEPROC, Vol. 129, Pt. D, No. 3, MAY 1982

The vector a must be found such that both the interior andboundary residuals are simultaneously forced to zero. Oneobvious and convenient formulation which would achievethis is

min f R2(a,x,y)dS+ ( bR2(a,x,y)dr (13)

as JrOnce again, the equivalent collocation problem is sought,and therefore expr. 13 is reposed as

min X Wk^2(a,xk,yk) + I Wt b J ? 2 (a ,x , ) (14)

where the interior residual J% and the boundary residualb& are defined over [0, 1] x [0, 1] and [0, 1], respectively.Clearly, the least-squares solution of the overdeterminedsystem

= 0 k = \,...,NQ

(xk,yk)e[0tl] x [0,1]

(15)

/ = h-..,NQ

(16)

is also a solution of expr. 14The three constants M, Nq and NQl and the associated

collocation points and weights must now be determined. Asbefore, M is given an initial value mx and Nqim^ is evaluatedby the procedure described in Section 2.2. At the end of thisprocess, a will have a fixed value, say ax. The same procedureis then repeated to determine NQl(nii), the initial valueof a being taken as ax with the residual h*% being evaluatedat various collocation points on [0, 1]. When NQl is judgedto have converged, o will have a new value, say ot2> whichis a solution of expr. 14 for the given M. M is then incremented,and this whole process is repeated. The convergence criterionfor M is taken as the error between two successive evaluationsof the sum of the two total residuals, i.e. the function beingminimised in expr. 14.

Extension of this method to incorporate more than onenonimposed boundary condition is trivial, and a more generalsituation may also be solved. Consider a total of q nonimposedboundary conditions with residuals lR(a,x,y), where(x,y)£Tj,j = 1, . . . ,q. Consider also that:

(a) the domain S is not readily transformable to [0 ,1] x[0 ,1] but that subdomains Sx, . . . ,SPl are individuallytransformable

(b) the parameters which are involved in the basic PDE varywithin S but are constant within each of the subdomains

Then an approximate solution is provided by

minain I J iR2(a,x,y)dSi + I \ b

iR2(*,x,y)dTj

(17)

which is readily solved by extending the process outlined forone nonimposed boundary condition.

2.4 Optimisation of imposed boundary conditions

The method outlined above utilises a recursive least-squaresalgorithm, which in turn requires that the boundary conditionparameters have been assigned specific numerical values. When

a simulation of a PDE is required, then all parameters willhave values, and thus the above method will give an accuratesolution without the need to perform any direct matrixinversion. However, the real power of the general polynomialapproximation method is that boundary conditions may beexplicitly contained in the final expansion, and any sub-sequent optimisation of boundary condition parameters mayoften be performed analytically. The above discussion is stilluseful in this more general context, particularly if the unspeci-fied parameters enter the function yo(x,y) as multiplicativeand additive constants, as is often the case, e.g. polynomialboundary conditions. In such an instance, after assigningvalues to the optimisable parameters, application of theabove technique will generate collocation points and weightswhich are valid for any parameter value and will yield accurateresults.

After domain normalisation, the most general situationposed in expr. 17 is equivalent to

min >; = 1

Thus the problem solved by application of the recursiveleast-squares approach is of the form

A a = c

where

(18)

A =PA

U

u

c =

and

for collocation points (xh yt) G [0, 1] x [0, 1] and associatedweights Wh 1= I,..., Nq(i); m = 1, . . . , M; i = 1, . . . , p ,and

bCr = ~

for collocation points xr E [0, 1] and associated weightsWr, r = 1, . . . ,Nq(i); m = 1, .. . ,M; i = 1,._. . ,q. Through-out the recursive algorithm, the values of \^m{xh y{) andb&m(xi) have been calculated and stored, and hence thematrix A is available. Thus the pseudoinverse (A TA)~l AT

can be calculated, and the vector of unknowns a is thenexpressed as a combination of the boundary condition para-meters by

<*= (ATA)'1 ATc (19)

In particular, if the boundary condition parameters, y =

IEE PROC, Vol. 129, Pt. D, No. 3, MA Y1982 85

(Ti > • • • , 7NC) say, enter into the boundary conditions onlyas multiplicative and additive constants, then the residualsdue to the imposed boundary conditions may be expressedas

Nb

b

=1and hence

- (7T(ArAYlATBf (20)

where

B =

PB

iB

\B

/ = l,...,Nq(i)

The final approximate solution is thus determined as anexplicit function of the parameters in the boundary con-ditions. Hence any index of performance which involves thesolution of a PDE at any point or over any region of S can beaccurately represented by a function of the boundary con-dition parameters and analytic optimisation may then bepossible.

2.5 Choice of basis functionsGiven the basic approximate solution of the form of eqn. 2,the functions jo(x, y), ylfj(x, y), / = 1 2, . . . ,M, are tobe selected. As mentioned previously, it is desirable thatas many boundary conditions as possible are imposed on tothis approximate solution, and certainly those boundaryconditions which are to be optimised. Hence yo(x, y) andtyj(x, y), / = 1,2, . . . ,M, may contain multiplying factorswhich are chosen to force the approximate solution eqn. 2to satisfy such boundary conditions identically. The basisfunctions \fy(x, y) are required to be linearly independentand, for largely theoretical reasons, they must form a completeset. Apart from these requirements, the choice of Wj(x, y) isfree. Therefore 4^(x, y) are chosen tqinvolve functions whichsimplify the computation of the A matrix and c vector.Monomials of the form {1, x, y, x2, xy, y2, . . .} give particu-larly easy evaluations at collocation points, but Jhey aregenerally not favoured since the corresponding A matrixtends to be ill conditioned [21, 22]^ Orthogonal polynomials,however, yield a well conditioned A matrix and are thereforeused in the majority of applications of the MWR. The basis

{Po (x),Pl

where Pj(x) is the /th-order Legendre polynomial, orthogonalto all other such Legendre polynomials over the maximumspan of x, is suggested in this application.

The collocation process does not explicitly use the ortho-gonality of such polynomials, but it does exploit the simplerecurrence relationships [23] which relate a polynomial ofhigh degree and its derivative to those of lower degree.A general element of the A matrix is of the form, to withina multiplicative constant,

for the appropriate domain and operator J?7. Applicationof-S t̂o ^m(x> y) and use of the corresponding recurrencerelationships will then show that A lm largely consists ofterms which have already been computed to give A n , . . . ,A im.i. Thus the evaluation of the A matrix is particularlyrapid when orthogonal polynomials are used within tyj(x, y).

3 Equations governing the glass heat-conditioning process

3.1 Equation of internal heat transferThe construction of a typical short forehearth is illustratedin Fig. 1 and the process of steady-state heat transfer in glasswhich is being heat-conditioned in such a forehearth is to beconsidered. The variations of forehearth width and depth withlongitudinal direction X will be accounted for, and hencethe model developed within this paper will be referred to asthe two-dimensional variable boundary (2DVB) model.

The conduction of heat in the longitudinal and lateraldirections, X and Z, respectively, is to be neglected sincetemperature gradients in these directions are small whencompared with the temperature gradient in the vertical direc-tion. An 'effective conductivity', which includes both aconduction and a radiation component [24], is used and isconsidered to be temperature independent. Such an assump-tion is not unreasonable for opaque, i.e. coloured, glasses.

The flow of molten glass through open channels of rectan-gular cross-section has been previously studied [14], and theassumption of laminar flow is extended to this model in whichthe width and depth of the forehearth vary piecewise linearlywith distance along the forehearth. Therefore, extendingthe results for laminar flow in the rectangular channel tothe 2DVB model, the velocity profile is given as

U(X, Y) =3Mf f i-J

2pD(X)W(X)[ LP(X)(21)

L J

With the assumptions outlined above, steady-state heat trans-fer in the glass is described by the partial differential equation[25]

(22)

The temperature variable is made dimensionless by thesubstitution

6 {X, Y) =T(X, Y)-Tr

T —T1 a * r

(23)

where Ta and Tr are constant reference temperatures, anadvantageous choice of Tr being the desired outlet tempera-ture. Dimensionless length and depth, x and y, respectively,are introduced using

x = — and y = Dr

(24)

(y),P2 (*),/>! 00, • • Iwhere L is the length of the forehearth and Dmax is themaximum depth of glass in the forehearth. Eqn. 23 is then

86 IEE PROC, Vol. 129, Pt. D, No. 3, MA Y1982

reformulated in dimensionless form as

320 30

where

U(x,y) =3cMfDr

and

D(x) =

2KgL D(x) W{x) Wr

W(x) =

1 -D\x)

(25)

(26)

(27)

3.2 Boundary conditionsThe three boundary conditions which are required for thesolution of the heat transfer equation eqn. 25 are providedby:

(i) the inlet temperature profile

7\0, Y) = RY)or

(28)

(ii) the temperature profile along the glass surface

T(X,0) = 7i(*)

or

(iii) the heat loss through the bottom of the forehearth.

A realistic approximation of this boundary condition has beenshown [26] to be

-Kgjjr (X, D(X)) = Keff {X) [T(X, D(X)) - Tba (X)]

where

K(x)Keff(X) = - { ~

Kr Wg(X)-Wrb(X)

db(X) Wg(X)

Wrb(X)-Wba{X)

In

In

(30)

glasso

o o

v o

o o/ refractory

oo

oo

oo °0

oo o

. W r b ( X , ) .

Fig. 2 Forehearth section at X = Xl

IEEPROC, Vol. 129, Pt. D, No. 3, MA Y1982

is a measure of the effective conductance of the bottom ofthe forehearth; the terms Wg(X), Wrb{X), Wba(X) beingillustrated in Fig. 2.

In the nondimensional system, the third boundary con-dition is therefore

K' ljwhere

72 (x) = -Keff(x)Dmax

4 Application of least-squares Patterson collocation

(32)

4.1 General approachThe 2DVB model provides a useful illustration of the powerand application of the theory developed in Section 2. Thespecific forehearth to be considered is illustrated in Figs. 3 and4. The body of the glass is subdivided into regions Slt S2, S3

and its lower boundary into sections Tlf F2, F3, as showndiagramatically in Fig. 5. Both the PDE eqn. 25 and its asso-

inlet back block spout061H 1.22

Fig. 3 Plan of a specific short forehearth

inlet-3.05-

back block spout4- 1.22 H

T—glasslevel

0 22

° ° „" „ o ° °o ° refractory (sillimanite)

i—conditioning-)zone

Fig. 4 View of a specific short forehearth

Fig. 5 Domain definitions for the short forehearth

87

ciated third boundary condition eqn. 31 are defined overnonregulai domains. Moreover, the parameters involved inthe terms W(x) and D(x) vary over the spatial domain S,but are constant within each subdivision S{, S2, S3. There-fore least-squares Patterson collocation is immediately appli-cable to the 2DVB model, while other forms of MWR ororthogonal collocation would not be.

It is easy to impose the first two boundary conditions

H0,y) =

6(x,0) = 7l(x)

on to an approximate solution. Thus one convenient formof 6{x, y) which satisfies both of these equations simulta-neously is

M

(33)

As discussed previously, the choice of Legendre polynomials asbasis functions in one space dimension yields well conditionedequations, and therefore a natural choice for ^fe(x, y) is aproduct of Legendre polynomials. Thus, in the approximatesolution eqn. 33, the basis function ^ ( x , y) is taken as

(34)

where

k = 2+ 1 ) + 0 - + l ) for I , / = 0 , 1 , . . . , 7 V

i+j=n

n = 0,\,...,N

and Pi(x), Pj(y) are non-normalised Legendre polynomialswhich are orthogonal on [0, 1].

Therefore

M =

N being the highest degree of Legendre polynomial in eitherco-ordinate direction.

The interior residual is given by

b2dR(«,x,y) = 7 - 5 - U(x,y) —

dy01dx

(35)

where

U(x,y) =

and

Pi =

W(x)D(x)

3MfcDmax

2KgLWmax

1 -y

D\x)

The residual Ro (x, y) due to the imposed boundary conditionsis

(36)

where

d_

dyd_

dx

and

Ut(xty) = U(x,y)

in zone /, / = 1, 2, 3.The contribution to the interior residual from a basis

function xy^k (x,y)= xyPi(x)Pj(y) is

d - _—j [xyPfa)Pj(y)] -ay

3—ox

2xPi(x)P;(y)+xyPi(x)Pj"(y)

- Ufay) [yPt(x)Pj(y)+xy?t(pc)Pj(y)] (37)

Since the interior residual has different parameters in each ofthe three zones Slf S2, S3, it is necessary to transform theintegral of residual squared over each zone into the equivalentintegral over [0, 1] x [0, 1]. The most general integral is thatover S3, the integrals over Si and S2 being simplifications of it.Therefore the determinant J3(x, y), (x, y) G [0, 1] x [0, 1],of the Jacobian matrix is sought such that

R2(a,x,y)dydx =a + bx

(a,x,y)dydx

3^2(a,x,y)dydx (38)

Two simple transformations from S3 to [0, 1] x [0, 1] are

x-x2 _ yy = a + bxx3 -x2

ThereforeJ3(x,y) = (a + bx)(x3-x2) (39)

The corresponding transformations in zones Si and S2 have anidentical form to eqn. 39 with

X2

X2

b->0 in S2

in

(40)The third boundary condition, which is defined by

= 72 (x) [e-eba] y=DMy=D(x)

where the parameters in y2(x) vary over F(y = D(x)) but areconstant within each subsection Vx, T2, P3 , must now beaccounted for. The boundary residual due to the imposedboundary conditions is

(41)

while the boundary residual arising from the basis functionx,y) = xyPi(x)Pj(y) is

-Dmix) (42)

where my2(x) and Dm(x) denote the functions y2(x) and

IEEPROC, Vol. 129, Pt. D, No. 3, MA Y1982

D(x) in zone F m , m = 1, 2, 3. Each line integral is now rede-fined over [0, 1] by means of a simple linear transformation.Thus

"1

•J.

(43)

where

x = (x3 -x2)x +

Identical transformations with the substitutions of expr. 40also redefine the integrals over F2 and Fi .

The approximate solution of the 2DVB model is thereforegiven by

mma £ f1

1=1 JO

, x,y)dydx

(44)

where

N

,J=o<*k&k(x,y)-&o(x,y)\ (45)

N

/ = 1,2,3

»,;=o

m = 1 ,2,3 (46)

N ,

£ denotes that the index k within the summation sign isi, J=o

chosen by k = n(n + l ) /2 + (i + 1) for /, / = 0, l,...,N;i + j = n; n = 0, 1 , . . . ,N, and the Jacobians, residuals andtransformations x->x, y-+y have all been defined. Thusthe 2DVB model can now be solved by least-squares Pattersoncollocation.

4.2 ResultsBefore considering the optimisation of boundary-conditionparameters, the validity of the whole Patterson collocationtechnique must be demonstrated. This will be done byshowing that, when all the boundary condition and PDEparameters are specified:

(a) the solution converges for increasing M(b) the convergent solution is correct.

Since the 2DVB model does not possess an analytic solution,the accuracy of the convergent polynomial approximationcan only be demonstrated by a direct comparison with asolution provided by a conventional numerical technique.A suitable finite-difference scheme has been produced byWhitfield [26] for this comparison.

The dimensions of a specific forehearth, which is heat-conditioning green glass, are illustrated in Figs. 3 and 4, whileall other physical constants are listed below:

Kg = 6Wm~1 °C"1

c = 840.12Jkg-loC"1

p = 2350.0 kg m"3

Kr = 1.54Wm-1°C"1

Kb = 0.34 Wm

throughput = 23t/day

i.e.Mf = 0.2662 kgs"1

Eqns. 30 and 32, which express the effective conductanceof the refractory at the bottom of the forehearth as a functionof the conductivities Kr and Kb and the corresponding dimen-sions, are extremely cumbersome, and therefore, with allthe constants as supplied above, an approximation to Keff{x)was sought which could rapidly be evaluated.

Quadratic polynomials ly2 (X)weie derived for each zone/, / = 1, 2, 3, via quadratic interpolation as

ly2(X) = (-2A98)Dmax

272(X) = (-0.7399 +3 T 2 W = (-0.08059+

(47)

(48)

(49)

A further simplification of the third boundary condition wasachieved by assuming that the temperature at the brick/airinterface Tba was constant and equal to 130°C.

It has already been shown [7] that the inlet and surfacetemperature profiles, which describe the first two boundaryconditions, may be well approximated by a linear and aquadratic polynomial, respectively. Therefore fi(y) and Ji(x)are taken to be

= 7io

(50)

(51)

the back block and spout being convenient points at whichto compare both measured, finite-difference and Pattersoncollocation results.

For the inlet and surface boundary conditions as shown inFig. 7, the collocation procedure described in Section 2was followed with the convergence constant ec set equal to0.02, yielding a relatively accurate quadrature formula. Thenumber of collocation points was found to be

]\fQ = 49 Nq = 7 in each zone for N - 2

NQ = 225 NQi = 1 5 in each zone for 3 < TV < 7

Convergence of the Patterson collocation technique is clearlyshown in Fig. 6 by the monotonically decreasing sum ofsquared residuals. Figs. 7 and 8 show that the approximatepolynomial solution does in fact converge to the finite-difference solution and that N = 6 provides an accuratesolution of the 2DVB model.

20r

0 1 2 3 U 5 6 7maximum polynomial degree in one space dimension (N)

Fig. 6 Convergence of the sum of squared residuals to zero withincreasing order of approximation

IEEPROC, Vol. 129, Pt. D, No. 3, MA Y1982 89

With the value of N determined, a comparison of thetemperature profiles predicted by the 2DVB model withmeasured profiles is now possible. On the short forehearthunder consideration, there are two temperature sensors atdistances 2.76m and 4.04m along the forehearth. The quad-ratic polynomial approximation to the surface temperatureis determined by interpolation between the temperatures at

the two sensors and the projected surface temperature at theinlet. Fig. 9 shows the agreement between profiles measuredby Hamilton and the 2DVB model when solved by finitedifference and by Patterson collocation with N = 6. Clearly,the 2DVB model compares favourably with the measuredvalues and also with other analytic models, e.g. the model ofAbbott and Whitfield [7] or the 2DL model of Carling.

0.00

temperature ,°C1220 1260 1300

0.05

0.10

0.151

temperature, °C1080 1120 1160

0.001200

N= 3

temperature ,°C1060 1100 1140

0001 '

0.05

1180

finitedifference

a•a 0.10

back block0.15

020

spout

10500.00 1.00 2.00 3.00 A.00

distance along forehearth , m

Fig. 7 Comparison of a finite-difference solution of the 2D VB model with least-squares Patterson collocation solutions for N = 2, 3, 4

—X- -X— finite-difference

Polynomial approximations

temperature fC1220 1260 1300

0.00

temperature ,°C1080 1120 1160 1200

0.00

temperature,°C1080 1120 1160

0.00!1200

N=5

back block

0.05

" 0.10

a.<h

0.15

0.20

finitedifference

spout

10500.00 1.00 2.00 3.00 4.00

distance along forehearlh , m

Fig. 8 Comparison of a finite-difference solution of the 2D VB model with least-squares Patterson collocation solutions for N = 5, 6, 7

—X X— finite difference

90 IEEPROC, Vol. 129, Pt. D, No. 3, MA Y1982

5 Optimisation of steady-state forehearth operatingconditions

5. / Identification of control parametersThe purpose of the forehearth is to heat condition glasswhich has been melted down in a furnace so that the glassleaves the forehearth with a uniform desired temperatureand is then suitable for moulding. Thus the temperatureof the glass at the spout is required to be a given constantthroughout its depth. For a specific forehearth and a fixedthroughput of glass, only the inlet and surface temperatureprofiles may be varied to control the outlet temperature.Indeed, the inlet temperature may also be predetermined bythe operating conditions within the furnace, leaving only thesurface profile as variable, and this will itself be constrained bythe geometry of the cooling and conditioning zones. Under theassumption that the inlet and surface temperature profiles maybe adequately approximated by a linear and a quadraticpolynomial, respectively, then, in the most general operatingsituation, there are four parameters nx, 710, yn, j l 2 whichmay be available to control the outlet temperature. Thisfollows since the further assumption that the inlet and surfaceprofiles concur at (X, Y) = (0, 0) implies that ju0 =7io- Ifthe inlet temperature is controllable at all, it is only to theextent that its mean value may be varied while the shape of itsprofile is fixed. Thus the difference in the temperature of theglass between its surface and the bottom of the forehearth isassumed to be a prescribed constant Te, and this in turndetermines Hi:

0(O,d)- 0(0,0)

Therefore

d(Ta-Tr)

(52)

(53)

and, in general, only the three parameters 71 0, 7 U , yl2 willbe controllable.

5.2 Development of suitable performance criteriaIt is now required to consider performance criteria whichindicate the deviation of the predicted temperature distri-bution at the spout from the desired value. Since the poly-nomial approximation method yielded an expression forthe inlet temperature as an explicit combination of the controlparameters, a general analytic performance criterion is sought.Thus, in the nondimensional co-ordinate system, a suitable'cost function' is

C= dy (54)

and if the reference temperature in the nondimensionaltemperature variable is chosen as Tr = Td, then C is simplyexpressed as

(55)C= (Ta-Tr? \l cj(y)[0{\,y)]2dy

Two specific forms of the weighting function oj(y) will beapplied. The first is co(y) = 1, which is a Legendre weightingon [0, 1] and gives a strict least-squares cost function. Theminimisation of such a cost function is known to yield anoptimum profile with an uneven error distribution over thenondimensional depth y. In particular, the errors tend tobe larger at both ends of the interval [0, 1], and this couldlead to an undesirable outlet temperature distribution. There-fore a second cost function which weights both ends of theinterval [0, 1] more heavily than the remainder of [0, 1] isalso considered. This cost function has

1

-y)

which is a Chebyshev weighting on [0, 1] and is known toyield a smoother optimum error distribution. Thus the fol-

temperature,°C1220 1260 1300

0.00

1050

temperature , °C1080 1120 1160 1200

0.00

tempera tu re , °C1080 1120 1160

O.OOl1200

0.05

0.10

back block0.15

0.20

finitedifference

N=6

spout

0.00 1.00 2.00 3.00 4.00distance along forehearth, m

Fig. 9 Comparison of measured temperature distributions with 2D VB model solutions from least-squares Patterson collocation (N = 6) and afinite-difference scheme

—X X— measured• model boundary condi t ion

IEE PROC, Vol. 129, Pt. D, No. 3, MA Y1982 91

lowing two cost functions will be used:

cost function 1 = (Ta-Tr)2 f * 02 (1 ,y) dy

Jo

cost functionion 2 = (ja — Tr)2 \

e2(i,y)dy— i,M 1/2

(56)

(57)

and such a minimum is assumed to occur at the back block,nondimensional distance x2 along the forehearth, then eqn. 61simplifies to

5.3 Imposition of constraints on the surface temperatureprofile

A range of operating conditions were investigated, and itproved necessary to impose one hard constraint on the surfacetemperature profile. Specifically, if the temperature of theglass at its surface falls below 1080 °C, then a 'freezing' effect,which is to be avoided, ensues. The short forehearth underconsideration has a cooling zone which starts at the inlet andextends to the back block, nondimensional distance x2 alongthe forehearth, followed by a conditioning zone from thatpoint to the end of the forehearth. It is therefore reasonable toassume that the minimum temperature along the length of theglass surface will occur at the back block. In nondimensionalterms, the above constraint is then expressed as

Tii = ~2y12x2

and expr. 60 reduces to

7l0~ 7l2*2 > Qm

which is clearly linear in 710 and yl2 .

(62)

(63)

e(x, o) > 0,

where1080 ~Tr

(58)

(59)

5.4 Optimisation via polynomial approximationIt is shown in Appendix 9 that the boundary-condition controlparameters enter linearly into the approximate polynomialsolution d(x, y) and quadratically into the general cost func-tion eqn. 55. Hence, the unconstrained optimisation of thegeneral cost function eventually yields a set of linear equationswhich are solved to give optimum parameter values.

The restriction on the surface temperature given by theconstraint of Section 5.3 is applied by first performingunconstrained optimisation of the parameters 710, yn , yl2 ifthe inlet temperature T(0, 0) is not specified, or similaroptimisation of yn , yl2 if T(0, 0) is specified. If the resultingoptimum parameters violate the inequality constraint expr. 63,then the equality constraint

m T —T

and, under the further assumption of a quadratically distri-buted surface temperature, the constraint becomes

7io — 7i2*2 = 0 m (64)

must certainly be satisfied. Thus eqn. 64 yields the substitution

~T [7io -6m]x2

(65)

7io > (60)

Now, since the minimum of the quadratic polynomial

.27io +

occurs at7u (61)

and the problem becomes unconstrained optimisation of aquadratic cost function, and hence the optimum parameterssubject to the inequality constraint expr. 58 are also foundfrom the solution of a set of linear equations.

5.5 ResultsThe theory outlined above has been applied to the forehearthshown in Figs. 3 and 4 with green glass constants as previously

temperature, °C1160 1200 1240

0.00

temperature, °C1080 1120 1160

0.001200

temperature ,*C1060 1100 1U0

QOOl1180

back block

0.05

• 0.10

0.15

0.20

spout

1050L0.00 1.00 2.00 3.00 4.00

distance along forehearth , m

Fig. 10 Optimum temperature distributions with surface profile constrained by T(x2, 0) > 1080°C

—X X— cost function 1 —a o— cost function 2

92 IEE PROC, Vol. 129, Pt. D, No. 3, MA Y 1982

specified. Following Section 4, the highest order of Legendrepolynomial in the basis functions was chosen to be N = 6, thedifference between the inlet temperatures at the glass surfaceand the bottom of the forehearth fixed at Te = 80° C and therequired temperature at the spout was taken to be 7^ =1100°C.

For the standard throughput of 23 t/day, the unconstrainedsurface and inlet profiles predicted by the analytic formulaswere found to be physically unfeasible. Thus, for a throughputof 23 t/day, the inequality constraint expr. 58, which guaran-tees a realisable profile, must be imposed. The resultant opti-mum surface and inlet profiles, together with the predictedback-block and spout temperature distribution, are showngraphically in Fig. 10 for both cost functions. Both sets ofprofiles compare favourably with those Hamilton obtained bynumerical experimentation [18]. The optimum distributionsat the spout also illustrate the applicability of the Chebyshevweighting which defines cost function 2; the large error at thebottom of the forehearth with the Legendre weighting pro-ducing an undesirable deviation from the required temperatureat this point.

The most feasible range of inlet surface temperatures is1200-1300°C, and the effect of constraining T(Q, 0) to liewithin this range was investigated for throughputs rangingfrom 15 to 35 t/day. With the inequality constraint expr. 58

8400r

7200"

6000"

4800-

•2 3600-

2400-

1200-

1200 1220 1240 1260 1280temperature T (0 .0 ) , 'C

1300

Fig. 11 Variation of cost function 2 with inlet temperature T(0, 0)with surface profile constrained by T(x2, 0) > 1080°C

Throughput in tonnes per day:—X X— 35

30>— 25*— 2 0>—15

imposed on the optimisation of the surface profile, the varia-tions of cost function 2 with inlet temperature T(0, 0) areshown in Fig. 11. The graphs clearly demonstrate the impor-tance of maintaining T(0, 0) at the optimum value. This isparticularly true of the higher throughputs, for which the largecurvature of both cost functions indicates that a small stepaway from the optimum T(0, 0) will lead to a considerablypoorer outlet profile. The approximately constant levelof optimum cost-function values in Fig. 11 illustrates another,less obvious, fact, i.e. an increase of throughput from 15 to35 t/day will yield the same quality of glass output providedthat the respective T(0, 0) optima are maintained.

A numerical optimisation of the boundary conditionsbased on the finite-difference scheme has also been performedto verify the results produced by the polynomial approxi-mation and to illustrate typical computational requirements.This technique involved the optimisation of the cost functionswith respect to the boundary-condition parameters, suchparameters being adjusted by the Powell method [27]. Eachupdate of the parameters requires a re-evaluation of the costfunction by a new finite-difference simulation, and, as Table 1shows, computation time increases rapidly with the numberof optimisable parameters.Table 1: Comparison of CPU times for various simulations of the

2DVB model and optimisations of boundary temperature profiles

Simulation/optimisation

12345

6

Simulation/optimisation

1

Least-squaresPatterson collocation

s299.041

3.4151.1271.5281.5001.429

Description

Finite difference

s1.0331.0331.0331.630

21.15693.076

Simulation, with output, for specified forehearthstructure, glass constants, T^ and boundarytemperature parameters. Patterson collocationprocedure has N = 6

2 Assuming simulation 1 has been performed once,this is a simulation, with output, for a change inthe parameter 0,

3 Assuming simulation 1 has been performed once,this is a simulation, with output, for a change inany of the boundary temperature profile con-stants Mo- Mi .7 io '7n .7 i2

4 Optimisation with surface profile constrained,inlet temperature profile specified: a one-parameter optimisation

5 Optimisation with surface profile constrained, Te

specified: a two-parameter optimisation6 Optimisation with only Te specified: a three-

parameter optimisation

With reference to Table 1, all offline computation is per-formed in simulation 1, and, for a specified structure, such asimulation is only required once. Optimisations 4, 5 and 6are a little misleading in that they appear to indicate anincrease in computation time for a decreasing number ofoptimisable parameters when applying the polynomialapproximation; this is simply due to the functioning of thecomputer program [26], which requires more input/outputfor an increasing number of constraints. In several situations,e.g. the overall refiner/forehearth operation, many more thanthree parameters may need to be optimised, and the classicalapproach of coupling a finite difference with numericaloptimisation would be impracticable, while the polynomialapproximation obtained by least-squares Patterson collocationremains entirely feasible.

Table 2, showing storage requirements, provides a furtherillustration of the applicability of polynomial approximation

IEEPROC, Vol. 129, Pt. D, No. 3, MA Y 1982 93

Table 2: Computer storage requirements for least-squares Pattersoncollocation and finite-difference solution of the 2DVB model with

boundary profile optimisation

8 References

Least-squaresPatterson collocation

Finite difference

Overall simulationand optimisation

Simulation andoptimisation for aspecified forehearth

k bytes

68

k bytes

12

10

once a single least-squares Patterson collocation solution hasbeen performed.

6 Conclusions

A new technique has been developed to solve a specifiedtype of PDE by producing a closed-form approximationwith explicit boundary condition parameter dependence.The general application of this collocation technique is basedon the use of Patterson quadrature points and weights and istherefore more efficient than other collocation schemes,which would typically use Gauss-Legendre points and weights.This additional efficiency is produced by the retention andreuse of previously calculated collocation function values ifcollocation at a certain number of points is not consideredto be sufficiently accurate. A computational scheme forthe determination of an appropriate number of collocationpoints and the order of polynomial approximation has beenproposed.

With the determination of collocation points and approxi-mation order complete, the Patterson collocation techniqueprovides a closed-form approximate solution to a given partialdifferential equation. Therefore simulation of the systemgoverned by the PDE is particularly easy, requiring littlecomputer time or storage, and optimisation of imposedboundary-condition parameters may be performed analy-tically. Thus, even for complicated DPSs defined over non-regular domains, feedforward control parameters may bepredicted with a relatively small computing effort.

A practical application of the method to the heat con-ditioning of glass in a forehearth has been illustrated. The2DVB model of this process was not amenable to solutionby standard MVVR techniques. However, the least-squaresPatterson collocation approach was successful, the conver-gence to the correct solution with increasing order of approxi-mation being demonstrated by comparison with a finitedifference solution.

Several optimisation studies have been performed. Onesuch study confirmed the optimal profiles Hamilton [18]obtained by numerical experimentation, while another showedthat a greatly increased throughput was feasible withoutimpairing the quality of the glass. The computation timesand storage shown in Tables 1 and 2 indicate that the approxi-mate solution of the 2DVB model obtained by least-squaresPatterson collocation can be readily incorporated into anoverall forehearth control scheme as feedforward action.

7 Acknowledgments

The author gratefully acknowledges significant discussionswith K.M. Abbott of the Department of Electrical, Electronic& Control Engineering, Sunderland Polytechnic, and Dr.D.M. Burley of the Department of Applied Mathematics &Computing Science, Sheffield University. The experimentaldata were most kindly provided by Dr. B. Hamilton and thestaff at Rockware Glass Ltd., Knottingley.

1 CARSLAW, H.S., and JAEGER, J.C.: 'Conduction of heat in solids'(Clarendon Press, 1959), 2nd edn.

2 SAUERWEIN, J.: 'Numerical calculations of multidimensional andunsteady flows by the method of characteristics',/. Comput. Phys.,1967, 1, pp. 406-432

3 SMITH, G.D.: 'Numerical solution of partial differential equations'(Oxford University Press, 1969)

4 ZIENKIEWICZ, O.C.: 'The finite element method' (McGraw-Hill,1977), 3rd edn.

5 FINLAYSON, B.A., and SCRIVEN, L.E.: 'The method of weightedresiduals - a review' Appl. Mech. Rev., 1966, 19, pp. 735-748

6 LeCROY, R.C., and ERASLAN, A.H.: 'The solution of temperaturedevelopment in the entrance region of an MHD channel by theB.G. Galerkin method', Trans. ASME. J. Heat Transfer, 1969, 91,pp. 212-220

7 ABBOTT, K.M., and WHITFIELD, A.H.: 'Modelling of the glassheat-conditioning process by the application of orthogonal projec-tion in the method of weighted residuals', Proc. IEE, 1979, 126, (2),pp. 189-197

8 SPARROW, E.M., and HAJI-SHEIKH, A.: 'Transient and steadyheat conduction in arbitrary bodies with arbitrary boundary andinitial conditions', Trans. ASME. J. Heat Transfer, 1968, 90, pp.103-108

9 CHENG, K.C.: 'Dirichlet problems for laminar forced convectionwith heat sources and viscous dissipation in regular polygonalducts', Amer. I. Ch. E. J., 1967, 13, pp. 1175-1180

10 SHULESHKO, P.: 'A new method of solving boundary-value prob-lems of mathematical physics', Aust. J. Appl. Sci., 1959, 10, pp.1-7

11 VILLADSEN, J.V., and STEWART, W.E.: 'Solution of boundaryvalue problems by orthogonal collocation', Chem. Eng. Sci., 1967,22, pp. 1483-1501

12 VILLADSEN, J.V., and MICHELSEN, M.L.: 'Solution of differ-ential equation models by polynomial approximation' (Prentice-Hall,1978)

13 GREEN, J.R.: 'Automatic control of glass forehearths', Glass, 1955,32, pp. 277-280

14 HEARN, G.E., and BOOTH, F.: 'Flow of molten glass throughan open channel', Glass Technol., 1974, 15, pp. 81-86

15 HENRY, H.L.: 'A forehearth control system for a glass containerplant'. IEEE Ind. Appl. Soc. Annual Meeting, Atlanta, GA, Sept.28th - Oct. 2nd, 1975, 10th Conference Record, pp. 532-534

16 DUFFIN, J., and JOHNSON, K.: 'Glass container process: fore-hearth simulation'. IBM Systems Development Division, San Jose,USA, report 02-472-1, July 1965

17 CARLING, J.C.: 'Two- and three-dimensional models of flow andheat transfer in forehearths' Glastechn. Ber., 1976, 49, pp. 269-277

18 HAMILTON, B.: 'The control of forehearths and feeders forcoloured glasses', Glass Technol, 1976, 17, pp. 243-248

19 PATTERSON, T.N.L.: 'The optimum addition of points to quad-rature formulae', Math. Comput., 1968, 22, pp. 847-856

20 PATTERSON, T.N.L.: 'Algorithm 468. Algorithm for automaticnumerical integration over a finite interval', Commun. ACM, 1973,16, pp. 694-699

21 PRABHU, S.S., and McCAUSLAND, I.: 'Time-optimal control oflinear diffusion processes using Galerkin's method', Proc. IEE, 1970,117,(7), pp. 1398-1404

22 LYNN, L.L., and ZAHRADNIK, R.L.: 'The use of orthogonalpolynomials in the near-optimal control of distributed systems bytrajectory approximation', Int. J. Control, 1970, 12, pp. 1079-1087

23 ABRAMOWITZ, M., and STEGUN, I.A.: 'Handbook of mathe-matical functions' (US National Bureau of Standards, 1965)

24 SIEGEL, R., and HOWELL, J.R.: Thermal radiation heat transfer'(McGraw-Hill, 1972)

25 WELTY, J.R., WICKS, C.E., and WILSON, R.E.: 'Fundamentalsof momentum, heat and mass transfer' (Wiley, 1969)

26 WHITFIELD, A.H.: 'Optimal heating policies for glass conditioningforehearths'. Ph.D., CNAA, 1980

27 POWELL, M.J.D.: 'An efficient method of finding the minimumof a function of several variables without calculating derivatives',Comput. J., 1964, 7, pp. 155-162

9 Appendix

The basic approximate solution of the 2DVB model is givenby least-squares Patterson collocation as

i, J=0

94 IEE PROC, Vol. 129, Pt. D, No. 3, MA Y 1982

N,

where £ denotes that the index k within the summation

sign is chosen by

k =n(n+\)

for / , / = 0,l,...,N

i + j — n

n = 0,1,...,N

o is the solution of the linear equations

where Ax, A2, A3, AA, As are matrices generated by thecollocation procedure and

c = [0i7n,0i 7i2, l ,Mi ,7 io ,7n ,7 i2 ] T

Consider optimisation of the general cost function eqn. 55with respect to ILX , yl0, yn , y12 , i.e. both the inlet and surfaceprofiles are unconstrained. If ff denotes any one of theseparameters, then the optimum with respect to f,- is given bythe solution of

cj(y)O2(l,y)dy\ = 0

Thus, if Ta is chosen as a constant independent of nl} j l 0 ,7n 5 7i2 , then the required solution is that of

= 0

The following definitions simplify the notation:

(66)

(4)

(S)

i, ;=o

Gk + ^i ,(D,

bk + Pi t/fe

N,

i, J=0

where

M

M = (N+ \)(N+2)12

2m

and [/Ix +A2(li +^43J3?]^ denotes the element in the kthrow and the /th column of [Ay + -42p\ +^43/3i]~1

5 anda similar notation holds for [A4 +A5 j3j].

Then eqn. 66 has the general form

Z \ \° ° (67)

and optimisation with respect to any combination of the f,- isattained by solving the corresponding set of linear equationsprovided by eqn. 67. The Hessian matrix should also beproved positive-definite to guarantee a minimum of costfunction.

IEE PROC, Vol. 129, Pt. D, No. 3, MA Y 1982 95