Statistics for Environmental Engineers

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Case Study: Solution


The system shown in Figure 48.1 will be solved for the case where all seven flows have been measured. Each value is measured with the same precision, so the weights are wi = 1 for all values. The sum of squares equation is:


minimize G = (Qx 21 )2 + (Q2 35)2 + (Q3 10)2 + (Q4 62)2 + (Q5 — 16)2 + (Q6 20)2 + (Q7 98)2


subject to satisfying the two conservation of mass constraints:


Q1 + Q2 + Q3 — Q4 = 0    and    Q4 + Q5 + Q6 — Q7 = 0


Incorporating the constraints gives a function of nine unknowns (seven Q’s and two Я^):


minimize G = (Qx 21 )2 + (Q2 — 35)2 + (Q3 10)2 + (Q4 — 62)2 + (Q5 — 16)2 + (Q6 20)2 + (Qi — 98)2


+ Я1( Q1 + Q2 + Q3 — Q4) + Я2( Q4 + Q5 + Q6 — Q7)


The nine partial derivative equations are all linear:


2QX +Я! = 21( 2 ) = 42    2Q2 + Я! = 35 (2) = 70 2Q3 + Я! = 10( 2) = 20


2Q4 — Я1 + Я2 = 62 (2) = 124 2Q5 + Я2 = 16( 2) = 32    2Q6 + Я2 = 20 (2) = 40


2Q7 — Я2 = 98( 2)= 196    Q1 + Q2 + Q3 — Q4 = 0 Q4 + Q5 + Q6 — Q7 = 0

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