Error Analysis Example W10

Error Analysis Example W10 - Department of Chemical...

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Department of Chemical Engineering University of California, Santa Barbara Ch.E. 132C Winter, 2010 Error Analysis for Nonlinear Equations Consider a general nonlinear equation 12 () = p Y f X ,X ,. ..,X (1) In order to derive analytical expressions for E( Y ) and V( Y ), Eqn. (1) is linearized by performing a Taylor Series Expansion at ii XX = , and truncating after the first-order terms: 1 ⎛⎞ =≈+− ⎜⎟ ⎝⎠ = pp i I i i X p f Y f X ,X ,. f X ,X ,. X X X i (2) Note that Eqn. (2) is a linear with respect to the { X i } because function f and its derivatives become constants when they are evaluated at specific numerical values, the sample means. The following important results can be derived 1-3 : , ,..., p E Yf X X X (3) 2 2 1 = ⎡⎤ ⎢⎥ ⎣⎦ p i i i X f V(Y ) s X (4) 2 1 where denote the sample mean of and denotes its sample variance. s Example: Error Analysis for a Chemical Reaction Rate Expression Consider a reaction rate expression for a nonisothermal, irreversible reaction, A B: / 0 (, ) ≈= ER T AA A rf C Tk e C
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This note was uploaded on 12/29/2011 for the course CHE 132C taught by Professor Peters during the Fall '11 term at UCSB.

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Error Analysis Example W10 - Department of Chemical...

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