ELEM.PRIN.OF CHEMICAL...ABRIDGED (LL)
ELEM.PRIN.OF CHEMICAL...ABRIDGED (LL)
4th Edition
ISBN: 9781119540632
Author: FELDER
Publisher: WILEY
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Chapter 2, Problem 2.49P
Interpretation Introduction

(a)

Interpretation:

The given data for system temperature versus time is to be fit into a straight-line using the method of least squares. Also, the formula for T(t) is to be derived and converted to t(T).

Concept introduction:

Method of least squares:

If (xi,yi) are the points plotted for (i=1,2,,n), then Ri is the residual for each point corresponding to i.

To fir the data into the straight-line plot, the value of ϕ(a,b) should be minimized.

ϕ(a,b)=i=1nRi2=i=1n(yiaxib)2

Perform differentiation on equation (1) to calculate the values of a and b, thus values defined are:

Sx=1ni=1nxiSxx=1ni=1nxi2Sy=1ni=1nyiSxy=1ni=1nxiyi ......... (1)

Now, use these values to calculate a and b as:

a=SxySxSySxx(Sx)2 ......... (2)

And,

b=SxxSySxySxSxx(Sx)2 ......... (3)

The straight-line that fits into the given data is now,

y=ax+b

With a and b being substituted as calculated from equations (2) and (3).

Interpretation Introduction

(b)

Interpretation:

The given data is to be plotted on the rectangular axes and the expression for T(t) is to be determined. Also, the value of R2 is to be checked, if it is closer to 1.

Concept introduction:

Given an equation, y=ax+b, it represents a straight-line equation for the expression y(x).

Here, a is the slope of the line and b is the y- intercept.

According to the method of least square used to fit data into a straight-line, R2 is the square of the residual. For the best fit, R2 should be closer to 1.

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