# Allowable Upper Bounds of Elemental Errors

In the design of PSP experiments, we need to give the allowable upper bounds of the elemental errors for the required pressure accuracy. This is an optimization problem subject to certain constraints. In matrix notations, Eq. (7.3) is expressed as ap = aTA a, where the notations are defined as ap = var(p)/p2, Atj = St Sjptj, and ai = [var( ) ]1//2 /£7 . For required pressure uncertainty ap, we look for a vector aup to maximize an objective function H = WT a, where W is the weighting vector. The vector aup gives the upper bounds of the elemental errors for a given pressure uncertainty aP. The use of the Lagrange

multiplier method requires H = WT a + X (ap – aT A a) to be maximal, where Я is the Lagrange multiplier. The solution to this optimization problem gives the upper bounds

= A – JW

( wta -1 W )1/2 ^P. (.)

For the uncorrelated variables with ptj = 0 (i Ф j), Eq. (7.6) reduces to ( X-1/2 -2W2

V k 7

When the weighting factors Wi equal the absolute values of the sensitivity coefficients I St |, the upper bounds can be expressed in a very simple form

(°i)Up /*p = NP1/2S] -1, (i = 1,2,■■■ ,NV ) (7.8)

where NV is the total number of the variables or the elemental error sources. The relation Eq. (7.8) clearly indicates that the allowable upper bounds of the elemental uncertainties are inversely proportional to the sensitivity coefficients and the square root of the total number of the elemental error sources. Figure 7.6 shows a distribution of the upper bounds of 15 variables for PSP Bath Ruth + silica-gel in GE RTV 118 at p/pref = 0.8 and T = 293 K. Clearly, the

allowable upper bound for temperature is much lower than others, and therefore the temperature effct of PSP must be tightly controlled to achieve the required pressure accuracy.