Statistics for Environmental Engineers

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How Measurement Errors Are Transmitted into Calculated Values

KEY WORDS error transmission, experimental planning, linear approximation, propagation of errors, propagation of uncertainty, reactor kinetics, sensitivity coefficients, titration errors, variance, uncertainty analysis.

When observations that contain errors are used to calculate other values, the errors are transmitted, to a greater or lesser extent, into the calculated values. It is useful to know whether the errors in a particular measured variable will be magnified or suppressed. This could help in selecting measurement methods and planning the number of replications that will be required to control error propagation in the variables that most strongly transmit their errors. Here are three such situations.

1.    A process material balance is described by m equations that involve n variables, say m = 8 and n = 14. If n m = 14 — 8 = 6 variables are measured, the remaining eight variables can be calculated. Errors in the measured variables will be reflected in the calculated values.

2.    An air pollution discharge standard might be based on the weight of NO* emitted per dry standard cubic foot of gas corrected to a specified CO2 content. To calculate this quantity, one needs measurements of NO* concentration, velocity, temperature, pressure, moisture content, and CO2 concentration. The precision and bias of each measurement will affect the accuracy of the calculated value. Which variables need to be measured most precisely?

3.    An experiment to estimate the reaction rate coefficient к involves measuring reactor solids content X and effluent organic concentration S over a series of runs in which detention time t = V/Q is controlled at various levels. The influent organic concentration S0 can be prepared precisely at the desired value. The calculation of к from the known values is к = (S0S)Q/SXV. Both X and S are difficult to measure precisely. Replicate samples must be analyzed to improve precision. Questions during the experimental planning stage include: “How many replicate samples of X and S are needed if к is to be estimated within plus or minus 3%? Must both X and S be measured precisely, or is one more critical than the other? Measuring S is expensive; can money be saved if errors in S are not strongly transmitted into the calculated value of к?”

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