A MODEL OF INTENSIVE OIL BURNOUT FROM GLASS FABRIC

This paper continues the previous investigations for process of intensive oil burnout in glass fabric


INTRODUCTION
Several years ago the mathematical simulation group of Institute of Mathematics of Latvian Academy of Sciences and University of Latvia, in cooperation with the Valmiera Glass Plant, made a start to the modelling of the process of burning oil out from the glass fabric.
The glass fabric is produced from extremely small bers forming 'threads' from which the fabric is to be woven. During the initial stages of the technological process -manufacturing the threads and weaving the fabric -the glass is covered with an oil lm. Once the weaving is complete, the oil must be burned away, so that the fabric can serve f o r insulation, re-protective a n d other purposes. The burnout is performed in a special furnace through which the fabric is pulled. At the top and bottom of the furnace, in parallel with the fabric sheet plane there are placed steel plates heated by diesel/residual-oil fuel or gas. Therefore the fabric after it has entered the furnace is heated extremely fast the oil ignites and burns out from the fabric, which results in whitening the latter. The whole process in the furnace last a few seconds (typically, 3-4 s). In turn, having left the furnace, the fabric cools down rapidly, which may result in impermissible degradation of its mechanical strength. Our challenge is therefore to de ne, by means of mathematical modelling, the in uence of di erent factors (such as the pull-through speed of the fabric, the properties of the metallic plates, the fabric thickness, etc.) on the fabric. In the above mentioned works 1]-3] it was estimated that, to describe adequately the technological process by means of a mathematical model based on Stefan-Boltzmann's law of the radiant heat transfer, we should take i n to account the re ected radiation between the fabric surface and the heaters. This means that any point on the fabric surface receives heat from all the points on the heater plate (as well as re ects it to these points). In other words, the boundary conditions of Stefan-Boltzmann's type should be written in the integral form. It might be noted that in the mentioned publications a process was modelled in which the heater placed above the fabric had a lower temperature this means that the e ect of re ected radiation between the top of the fabric and the top heater could be discounted, with the classic Stefan-Boltzmann boundary condition employed instead.
As concerns thicker fabrics, to burn away all the oil it was necessary to rise the temperature of the top heater. In this case, the e ect of re ected radiation between the fabric and both heaters should be accounted for.
In this work a mathematical model is given that describes this more intensive t e c hnological process.

DESCRIPTION OF THE TECHNOLOGICAL PROCESS AND ITS MATHEMATICAL MODEL
The sizes of the active zone situated in the furnace in which the fabric is heated are as follows: length L, w i d t h D 0 and height (the distance between heating plates) H. The fabric of width D and thickness is pulled through the furnace with velocity v. Since in a real technological process the length of the fabric (L D) and each of the heaters possesses its own (though constant along the length and width of the furnace) temperature, we m a y assume (at least in the rst stage of the modelling) that the temperature changes only along the fabric movement a n d perpendicularly to the fabric plane, we m a y present the di erential equation for the temperature eld T(x y t) in the fabrics (see 1]) c p ( @T @ t + v @T @ x ) = @ @ x (k @T @ x ) + @ @ y (k @T @ y ) + R 1 (2.1) 0 < x < L 0 < y < t > 0 where , c p and k are respectively the density, speci c heat and thermal conductivity of the glass fabric, but x-axis is oriented along the fabric's movement and y-axis is perpendicular to the fabric. In turn, the termR 1 refers to the process of oil burnout (this term will be speci ed further).
The fabric thickness is relatively small as compared with other geometrical parameters we therefore will de ne the temperature averaged over the fabric thickness T(x t) = 1 Z 0T (x y t)dy: We shall now describe the heat transfer proceeding between the fabric surface and the surroundings. Here we will consider two mechanisms: 1) radiative heat exchange with the heaters according to Stefan-Bortzmann's law 2) convective heat exchange with the gas whose temperature in the furnace is T g .
In the literature, various techniques are proposed for de ning the heat transfer coe cient (T ). We used, after having performed a series of numerical experiments, a relatively simple expression from 4], because it became evident that the main role here was played by the radiative heat transfer: (T ) = N u k g L N u = 0 :044Re 0:77 T T g Re = LU g g where k g , g and U g are relatively the thermal conductivity, kinematic viscosity and velocity of the furnace gas. We assume that the gas velocity coincides with the speed of fabric movement: U g = v. Then the boundary condition at the top surface of fabric y = , w i l l b e : k @T @ y = " f T 4 ht ;T 4 ] + (T )(T g ;T) (2.4) and at the bottom one, y = 0, correspondingly: ;k @T @ y = " f T 4 hb ;T 4 ] + (T)(T g ;T): We are coming now to the description of the burnout process. Assuming that the oil burns via a simple one-step Arrhenius reaction, we obtain: where H is the heat of reaction of the oil combustion, A -pre-exponential Arrhenius constant, E a -activation energy of oil, R -gas constant. In turn, c(x y) is the non-burnt oil concentration varying as described by the di erential equation @ c @ t + v @ c @ x = ;cA exp(; E a RT ): For a real technological process in non-stop production the dependence on time t is lost. Further, taking into consideration that the fabric thickness is small, we assume that the temperature variations along the fabric thickness can be discounted. Then from relationship (2.2) we derive for a non-stationary processT

A MATHEMATICAL MODEL FOR THE INTENSIVE OIL BURNOUT
Glas fabric are used for various purposes, as decorative clothes, electric or thermal insulation material (the latter may be used, for example, for reprotective clothes). Accordingly, t h e t h i c knesses of such fabrics may be di erent. For a thicker fabric it is necessary, with the aim of attaining the complete oil burnout, to rise the temperature also on the top heating plate. For this case, one should take i n to account the heat re ection between the top surface of fabric and the top heater. For this, the set of equations (2.9)-(2.13) is to be substituted for a more generalized one. For a general non-stationary process this will read as follows: c p ( @ T @ t + v @ T @ x ) = " f (1 ; " f ) (J f t + J f b ; 2 T 4 ) + 2 (T ) (T g ; T) + c HA exp(; E a RT )

THE SOLUTION OF THE MATHEMATICAL MODEL
For a real industrial process constant conditions are needed. This means that we have to describe a stationary process: @ T @ t = 0, @ c @ t = 0. It should be noted that in the case of a non-stationary process the algorithm is essentially the same. Further we will consider a generalized process of intensive treatment (3.1) { (3.8). The calculations performed earlier ( 1]-3])show that at some local points (within the burning zone) the temperature varies very fast. Therefore for our numerical computations a piecewise-uniform grid (with a smaller step in the burning zone) was used. We have exploited an iterative algorithm, considering a process with boundary conditions (2.4), (2.5) as a rst approximation this means that at the rst step we nd the temperature distribution for the process with no re ection. Secondly, i n e a c h subinterval x n x n+1 ] w e consider uxes (3.3) { (3.6) and similarly to publication 1] we can derive (for example, from (3.3)) the following: x j+1 ; x n p (x j+1 ; x n ) 2 + a 2 t ; x j ; x n p (x j ; x n ) 2 + a 2 t # :  If for ux J f t (or for J f t and J f b ) formula (3.9) is employed, it is obvious that this computation stage becomes simpler. Comparative computations have been performed for the following variants of the stationary process: re ected radiation is not taken in to account, that is, ux J is computed by formula (3.9) the top heater is cooler, so the model (2.9) { (2.13) can be exploited there is an intensive burnout, therefore we u s e the "full" system (3.1) { (3.8).
Temperature distribution in the fabric is represented in the following table 1, wherẽ T 1 (x) = T(x)=T hb with T hb =1123 K, T ht =973 K and without the re ection T 2 (x) = T(x)=T hb with T hb =1123 K, T ht =973 K and with the re ection T 3 (x) = T(x)=T hb with T hb = T ht =1123 K and without the re ection T 4 (x) = T(x)=T hb with T hb = T ht =1123 K and with the re ection.
One can see that in the case of intensive burnout the maximum temperature in the burning zone increases and the burning process itself starts essentially nearer to the entrance of the furnace.