Chemical Processing - February 2008 - (Page 37)

WELCOME AGAIN TO DR. GOODDATA COUNTRY. It’s nice to have you back. I hope you enjoy the ride. Hold on, it’ll be a little bumpy. Last time (www.ChemicalProcessing.com/aricles/2007/ 157.html), we nished comparing uncertainty analysis results obtained by both the US/ASME method and the ISO method. As you undoubtedly recall, the engineering method groups errors and uncertainties by their effect on test results — that is, either random or systematic. The ISO method groups errors and uncertainties by the origin of the data available to estimate those uncertainties — that is, Type A if there are data to calculate a standard deviation for an uncertainty and Type B if not. A simple description of the two most widely accepted models for uncertainty analysis in the world — direct from Dr. Gooddata to you! Now let’s get complicated. Up to now, we’ve looked at uncertainty analyses where all the uncertainties are in the same units — that is, all pressure, temperature or length, etc. That’s ne for the simplest cases. However, the most important cases often require determining the uncertainty of a calculated result. Consider ow measurement with an ori ce. We calculate the ow based on upstream and downstream pressure measurements (psi), uid temperature (°F), ori ce diameter (in.) and, maybe, time (sec.). Obtaining the uncertainty in each of these measurements doesn’t result in knowing the uncertainty in the calculated result, ow. Oh, what shall we do? We must evaluate the effect on ow of errors in pressure, temperature, diameter and time. Those errors must be converted into the four corresponding uncertainties in ow units so they can be combined by root-sum-square. The evaluation of those effects on the rewww.chemicalprocessing.com sult is called “uncertainty propagation.” The propagation process uses the equation for ow and the four parameters in that calculation. It’s complicated, so we’ll rst consider something simpler. For this first case, we’ll calculate a result via a simple equation in which all parameters are in the same units. The example we’ll use is the sum of two weights. How does uncertainty propagation figure into that situation? Both measurements, the weights, are in the same units and yet uncertainty propagation is required because an equation is used to compute the test result. Let’s explore. First, we need to write the equation for the result. (Sometimes we don’t have to but we’ll cover that in a later installment.) It is: WT = W1 + W2 (1) where WT is the total of the two weights; W1 is the rst test weight; and W2 is the second test weight. Let’s now say that we have the uncertainties for each of the test weights — bW , bW , sW and sW , which represent 1 2 1 2 the systematic and random standard uncertainties of W1 and W2, respectively. How do we combine them? Let’s rst consider the case where all errors and uncertainties are independent. The basic equation for any result is: R = ƒ(X1, X2, ) In that case, the fundamental equation for combining independent systematic standard uncertainties is: bR = [∑( ) ] (∂XR) (∂Xi) 2 1/2 (bi)2 (2) February 2008 • 37 http://www.ChemicalProcessing.com/articles/2007/157.html http://www.ChemicalProcessing.com/articles/2007/157.html http://www.chemicalprocessing.com

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Chemical Processing - February 2008 Contents From the Editor ChemicalProcessing.com Field Notes In Process Energy Saver Compliance Advisor Nanoparticle Safety Raises Questions Take the Pressure Off Vacuum Systems Achieve Optimum Centrifugal Pump Performance Rethink Batch-Manufacturing Alarm Systems Dr. Gooddata Orlando Plant Pioneers HMI Migration Strategy Process Puzzler Plant InSites Equipment & Services Ad Index Product Spotlight/Classifieds End Point

Chemical Processing - February 2008

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