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Propagation of Error Ch En 475 Unit Operations

Propagation of Error Ch En 475 Unit Operations. Quantifying variables (i.e. answering a question with a number). Directly measure the variable. - referred to as “measured” variable ex. Temperature measured with thermocouple

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Propagation of Error Ch En 475 Unit Operations

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  1. Propagation of Error Ch En 475 Unit Operations

  2. Quantifying variables (i.e. answering a question with a number) • Directly measure the variable. - referred to as “measured” variableex. Temperature measured with thermocouple • Calculate variable from “measured” or “tabulated” variables - referred to as “calculated” variable ex. Flow rate m = r A v (measured or tabulated) Each has some error or uncertainty

  3. Uncertainty of Calculated Variable Calculate variable from multiple input (measured, tabulated, …) variables (i.e. m = rAv) What is the uncertainty of your “calculated” value? Each input variable has its own error Example: You take measurements of r, A, v to determine m = rAv. What is the range of m and its associated uncertainty? Details provided in Applied Engineering Statistics, Chapters 8 and 14, R.M. Bethea and R.R. Rhinehart, 1991).

  4. To obtain uncertainty of “calculated” variable • DO NOTjust calculate variable for each set of data and then average and take standard deviation • DO calculate uncertainty using error from input variables: use uncertainty for “calculated” variables and error for input variables Plan: Obtain max error (d) for each input variable then obtain uncertainty of calculated variable Method 1: Propagation of max error - brute forceMethod 2: Propagation of max error - analytical Method 3: Propagation of variance - analytical Method 4: Propagation of variance - brute force – Monte Carlo simulation

  5. Value and Uncertainty • Value used to make decisions - need to know • uncertainty of value • Potential ethical and societal impact • How do you determine the uncertainty of the value? • Sources of uncertainty (from Rhinehart, Applied Engineering Statistics, 1991): • Estimation - we guess! • Discrimination - device accuracy (single data point) • Calibration - may not be exact (error of curve fit) • Technique - i.e. measure ID rather than OD • Constants and data - not always exact! • Noise - which reading do we take? • Model and equations - i.e. ideal gas law vs. real gas • Humans - transposing, …

  6. Estimates of Error (d) for input variables (d’s are propagated tofind uncertainty) • Measured: measure multiple times; obtain s; d≈ 2.5s Reason: 99% of data is within ± 2.5sExample: s = 2.3 ºC for thermocouple, d = 5.8 ºC • Tabulated :d ≈ 2.5 times last reported significant digit (with 1) Reason: Assumes last digit is ± 2.5 (± 0 assumes perfect, ± 5 assumes next left digit is fuzzy) Example: r = 1.3 g/ml at 0º C, d = 0.25 g/ml Example: People = 127,000 d = 2500 people

  7. Estimates of Error (d) for input variables • Manufacturer spec or calibration accuracy: use given spec or accuracy dataExample: Pump spec is ± 1 ml/min, d = 1 ml/min • Variable from regression (i.e. calibration curve):d≈ 2.5*standard error (std error is stdev of residual) Example: Velocity is slope with std error = 2 m/s • Judgment for a variable: use judgment for d Example: Read pressure to ± 1 psi, d = 1 psi

  8. Estimate of Error for Calculated Variables (i.e., Propagation of Error) • Max error: • propagating error with brute force • propagating error analytically • Probable error: • propagating variance analytically • propagating variance with brute force (i.e., Monte Carlo)

  9. Example Problem Data from a computer show that the flow rate is 562 ml/min ± 3 ml/min (stdev of computer noise). Your calibration shows 510 ml/min ± 8 ml/min (stdev). What flow rate do you use and what is d? In the following propagation methods, it’s assumed that there is no bias in the values used - let’s assume this for all lab projects.

  10. Method 1: Propagation of max error- brute force • Brute force method: obtain upper and lower limits of all input variables (from maximum errors); plug into equation to get uncertainty of calculated variable (y). • Uncertainty of y is between ymin and ymax. • This method works for both symmetry and • asymmetry in errors (i.e. 10 psi + 3 psi or - 2 psi)

  11. Example: Propagation of max error- brute force = r A v Brute force method: r = 2.0 g/cm3 (table) A = 3.4 cm2 (measured avg) v = 2 cm/s (slope of graph) Additional information: sA = 0.03 cm2 std. error (v) = 0.05 cm/s All combinations What is d for each input variable? min < < max

  12. Example: Propagation of max error- brute force = r A v Compute  (look at cheat sheet) r = 2.0 g/cm3 (table) 2.5  lowest digit = 0.25 A = 3.4 cm2 (measured avg) 2.5  sA = 0.075 v = 2 cm/s (slope of graph) 2.5  std. error (v) = 0.125 Additional information: sA = 0.03 cm2 std. error (v) = 0.05 cm/s What is d for each input variable?

  13. Example: Propagation of max error- brute force = r A v Brute force method: r = 2.0 g/cm3 (table) A = 3.4 cm2 (measured avg) v = 2 cm/s (slope of graph) Additional information: sA = 0.03 cm2 std. error (v) = 0.05 cm/s All combinations What is d for each input variable? min < < max 10.9 16.6 13.6  2.69 3.01

  14. Method 2: Propagation of max error- analytical • Propagation of error: Utilizes maximum error of input variable (d) to estimate uncertainty range of calculated variable (y) • Uncertainty of y: y = yavg± dy • Assumptions: • input errors are symmetric • input errors are independent of each other • equation is linear (works o.k. for non-linear equations if input errors are relatively small) * Remember to take the absolute value!!

  15. Example: Propagation of max error- analytical = r A v y x1 x2 x3 Avrv rA 3.42 2 2 2 3.4 r = 2.0 g/cm3 (table) A = 3.4 cm2 (measured avg) v = 2 cm/s (slope of graph) m = mavg ± dm = rAv ± dm = 13.6 ± 2.85 g/s Remember from above: For r, s = 0.1 g/cm3d= 0.25 sA = 0.03 cm2, dA= 0.075 std. error (v) = 0.05 cm/s dv= 0.125 = (3.4)(2)(0.25) = 0.60 (2.85) ferror,r= (fractional error)

  16. Propagation of max error • If linear equation, symmetric errors, and input errors are independent  brute force and analytical are same • If non-linear equation, symmetric errors, and input errors are independent  brute force and analytical are close if errors are small. If large errors (i.e. >10% or more than order of magnitude), brute force is more accurate. • Must use brute force if errors are dependent on each other and/or asymmetric. • Analytical method is easier to assess if lots of inputs. Also gives info on % contribution from each error.

  17. Method 3: Propagation of variance- analytical • Maximum error can be calculated from max errors of input variables as shown previously: • Brute force • Analytical • Probable error is more realistic • Errors are independent (some may be “+” and some “-”). Not all will be in same direction. • Errors are not always at their largest value. • Thus, propagate variance rather than max error • You need variance (s2) of each input to propagate variance. If s (stdev) is unknown, estimate s = d/2.5

  18. Method 3: Propagation of variance- analytical y = yavg ± 1.96 SQRT(s2y) 95% y = yavg ± 2.57 SQRT(s2y) 99% • gives propagated variance of y or (stdev)2 • gives probable error of y and associated confidence • error should be <10% (linear approximation) • use propagation of max error if not much data, use propagation of variance if lots of data

  19. Example: Propagation of variance Calculate r and its 95% probable error All independent variables were measured multiple times (Rule 1); averages and s are given M = 5.0 kg s = 0.05 kg L = 0.75 m s = 0.01 m D = 0.14 m s = 0.005 m

  20. Propagation of Variance

  21. Method 4: Monte Carlo Simulation (propagation of variance – brute force) • Choose N (N is very large, e.g. 100,000) random ±δi from a normal distribution of standard deviation σi for each variable and add to the mean to obtain N values with errors: • rnorm(N,μ,σ) in Mathcad generates N random numbers from a normal distribution with mean μ and std dev σ • Find N values of the calculated variable using the generated x’ivalues. • Determine mean and standard deviation of the N calculated variables.

  22. Monte CarloExample 1

  23. Monte Carlo Simulation Example 2 • Estimate the uncertainty in the critical compressibility factor of a fluid if Tc = 514 ± 2 K, Pc = 61.37 ± 0.6 bar, and Vc = 0.168 ± 0.002 m3/kmol?

  24. Overall Summary • Measured variables: (use all) • Average • Stddev (data range) • Confidence interval from student t-test (mean range and mean comparison) • Calculated variable: determine uncertainty (pick one) • Max error: propagating error with brute force • Max error: propagating error analytically • Probable error: propagating variance analytically • Probable error: propagating variance with brute force (i.e., Monte Carlo)

  25. Data and Statistical Expectations (for UO Lab reports) • Summary of raw data (table format) • Sample calculations– including statistical calculations • Summary of all calculations- table format is helpful • If measured variable: average and standard deviation for all, confidence of mean for at least one variable • If calculated variable: 1 of the 4 methods. Pleasestate in report. If messy equation, you may show 1 of 4 methods for small part and then just average (with std dev.) the value (although not the best method).

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