A feed mixture of two components is separated in a distillation column, as shown in Fig. 10.8 where F, D, and B are molar flow rates and z, y, and x are mole fractions of the more volatile component. Assume that the three flow rates are constant: F=4 mol/min, D=B=2 mol/min. During normal operation

Question

A feed mixture of two components is separated in a distillation column, as shown in Figure below where F, D, and B are molar flow rates and z, y, and x are mole fractions of the more volatile component. Assume that the three flow rates are constant: F=4 mol/min, D=B=2 mol/min. During normal operation, each composition measurement contains random errors. Based on previous experience, it can be assumed that the population means μ and population standard deviations σ (see Appendix F for definitions) have the following values :

x	y	z
0.200	0.800	0.500	μ
0.005	0.020	0.010	σ

At steady state, the measured compositions were
$\overline{z} = 0.485,\quad \overline{y}=0.825, \quad \overline{x} = 0.205$
Is there statistically significant evidence to support the assertion that one or more of the composition sensors are not working properly?

x	y	z
0.200	0.800	0.500	μ
0.005	0.020	0.010	σ

At steady state, the measured compositions were
[latex] \overline{z} = 0.485,\quad \overline{y}=0.825, \quad \overline{x} = 0.205 [/latex]
Is there statistically significant evidence to support the assertion that one or more of the composition sensors are not working properly?

Accepted Answer

The error of closure for the steady-state component balance is
[latex] E_{c} = Fz−Dy−Bx [/latex]
Substituting the measured values gives
[latex] E_{c} = 4(0.485)−2(0.825)−2(0.205)=− 0.120\space mol∕min [/latex]
As shown in Appendix F, the standard deviation of [latex] E_{c} [/latex] is
[latex] σ_{E_{c}} = \sqrt{F^{2}\sigma ^{2}_{z}+D^{2}\sigma ^{2}_{y}+B^{2}\sigma ^{2}_{x}} [/latex]
[latex] σ_{E_{c}}= \sqrt{(4)^{2}(0.01)^{2}+(2)^{2}(0.02)^{2}+(2)^{2}(0.005)^{2}} = 0.0574 \space mol/min [/latex]
Because the absolute value of the calculated value of [latex] E_{c} [/latex] is more than two standard deviations away from the expected value of zero, there is strong statistical evidence that an abnormal event has occurred. For example, if [latex] E_{c} [/latex] is normally distributed, with a mean of [latex] μ_{E_{c}}= 0 [/latex] and a standard deviation of [latex] σ_{E_{c}} = 0.0574,[/latex] the corresponding Z value (see Appendix F) is
[latex] Z = \frac{E_{c}-\mu _{E_{c}}}{\sigma_{ E_{c}}}=\frac{-0.120-0}{0.0574} = -2.09 [/latex]
The probability that an abnormal event has occurred is the probability that [latex] \left|Z\right| > 2.09. [/latex] From the table for the standard normal probability distribution (Montgomery and Runger, 2013), this probability is 0.963. Thus, there is statistically significant evidence to include that an abnormal event has occurred.
This error analysis can be extended to the situation where the flow measurements also contain random errors.

Question 10.3: A feed mixture of two components is separated in a distillat...

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