Calculator

Fiber Dissolution Rate Calculator

Fill in the weight percent oxide composition of the fiber in the column "Given", then click on the "Show" button, and an estimate of the dissolution rate is displayed on the right. A more detailed description and the theory behind this calculation are outlined below.

Comments on this calculator may be sent to John Hoffman.

Description

This calculator computes the dissolution rate of a fiber from its composition. The composition in oxide weight percent is entered into the boxes in the column labeled "Given" in the usual way, by moving the pointer with the mouse to the desired box and clicking with the left mouse button. Then the value is typed in. One may move to another box by clicking it with the mouse or to the box below with the tab key. Striking the tab key with the Shift key pressed moves to the box directly above. Any box left blank will be considered to be zero percent. When all oxides have been entered, one moves the pointer to the "Show" button and presses the left mouse button again. The dissolution rate constant is then shown just below the buttons in the conventional units of nanograms of fiber material dissolved per square centimeter of fiber surface area per hour (ng/cm2/hr). One may change the values in the "Given" column by clicking in the box, backspacing over the digits, and retyping them. It is easy to change a value and click the "Show" button again to see the effect on dissolution rate of changes in the composition. In addition, the "Clear" button erases all values from the boxes so that one may start over. There are a couple of keys one can use in place of the buttons: The "Enter" key does the same as clicking on the "Show" button and pressing "C" is the same as the "Clear" button.

There is a lot more information available in this calculator. The column labeled "Calculated" gives the fiber composition normalized to 100% in all of the oxides used in the calculation of dissolution rate. These values are what the program is using to compute the dissolution rate. Any oxides for which the effect on dissolution rate is incompletely known are not included in the calculation and either do not appear on this calculator or their value is shown as 0.00 in the "Calculated" column. The bottom line "Sum" is useful here, as it gives the sum of all the oxides that have been entered. If the sum does not add up to about 100%, say between 95% and 105%, then it is an indication that the fiber may contain enough of some oxides whose effect on dissolution rate is unknown that the calculated dissolution rate is not accurate.

There are three different equations used in this calculator to compute the dissolution rate constant depending on the composition. If the composition falls into the range for which the equation for borosilicate glass fibers is estimated to be accurate, then that equation is used. The borosilicate equation does not have information about FeO, since there is so little of this oxide in most borosilicate glass fibers. If the composition is not a borosilicate, then either the low alumina or the high alumina rock wool equation is used. A notation is made just under the calculated dissolution rate as to which equation was used. If the composition is outside the ranges from which all three equations were developed, no estimate of accuracy is made, and a notation appears warning that the prediction may be far from accurate.

The three equations differ in their accuracy, which is noted under the equation in the calculator. The accuracy is expressed as a geometric standard deviation, which can be interpreted in the following way: If the calculated dissolution rate constant were 100, for example, and the accuracy were given as x / 1.5, then one would expect the true value of the dissolution rate constant to lie between 100 / 1.5 and 100 x 1.5, or between 67 and 150 about 70% of the time. The borosilicate equation is considerably more accurate than the other two, but the range of composition over which it has this accuracy is more limited. The high and low alumina rock wool equations have been found to be approximately correct for a wide variety of different compositions, from conventional rock wools, to high alumina rock wools, to various slag wools, to refractory ceramic fibers, and even E glass. But the accuracy of these two equations are somewhat uneven and thus the stated geometric standard deviation is large.

The borosilicate equation for estimating dissolution rate constant is based on in-vitro data, which were found to agree well with in-vivo biopersistence results. The high and low alumina equations, on the other hand, are based entirely on in-vivo data, to avoid any of the known problems with accurate measurement of the high alumina rock wool fiber compositions in vitro. Thus these equations and this calculator produces estimates of dissolution rate that happen in vivo. More information about the procedure for obtaining the in-vivo dissolution rate from biopersistence data is found in this paper.

How It Works

The theory behind these calculations of fiber dissolution rate from composition has been published in a series of papers and will be outlined here. It is based on the Arrhenius equation in which the logarithm of a reaction rate constant is proportional to the free energy required to initiate the process. The dissolution rate constant is a reaction rate constant for the dissolution process, and thus what is needed to use this equation is the free energy to initiate dissolution for a given fiber composition. The assumption made here is that the free energy for the dissolution of a fiber composition is the weighted sum of the free energies to dissolve each component, weighted by the amount of that component present in the fiber. It should be noted that the free energies used here are those required to create the transition state starting from the original glass in contact with water. In particular, it is not the free energy difference between products and reactants. This latter free energy is related to the equilibrium between dissolved glass components and solid glass, whereas we are interested here in the dissolution rate far from equilibrium. It is further assumed that the components are approximately the same as the oxides. Now all that is needed are the free energies in the sense just described to dissolve each component. A convenient way to obtain these is to fit the weighted sum equation to a set of data consisting of the measured dissolution rate for many fibers and their compositions.

Precisely this fitting procedure was done for a large set of borosilicate glass compositions with measured dissolution rate constants to obtain the parameters for borosilicate glass fibers. A previously published scientific paper describing this procedure is available here. It is clear that these same parameters cannot be used with any confidence to compute the dissolution rate for other types of fibers, such as rock wool fibers, since their glass structure is known to be different. Therefore, the same type of sum was fit to a set of low alumina fibers and to a set of high alumina fibers to obtain two more equations for these types of fibers. A paper describing the application to rock and slag wool and other fibers is available here.

The decision about which equation to use is made by the calculator based on the composition. If it falls within the range of the data over which the borosilicate equation was developed and was verified, then that equation is used. If not, then either the high alumina or the low alumina equation is used, depending on which applies.