Page 515 - Modelling in Transport Phenomena A Conceptual Approach

A.4. LEIBNITZS RULE FOR DIFFERENTLATION OF LNTEGRALS 495

The slope of a straight line, m, on a log - log graph paper is

(A.3-2)

On the other hand, the slope of a straight line, m, on a semi-log graph paper
(y-axis is logarithmic) is

m= (A.3-3)
22 - 51

A.4 LEIBNITZ’S RULE FOR
DIFFERENTIATION OF INTEGRALS

Let f(z, t) be continuous and have a continuous derivative af/& in a domain of the
zt plane which includes the rectangle a 5 z 5 b, tl 5 t 5 tz. Then for tl 5 t 5 tz

(A.41)

In other words, differentiation and integration can be interchanged if the limits of
the integration are fixed.
On the other hand, if the limits of the integral in Eq. (A.41) are dependent on
time, then

If f = f(z) only, then Eq. (A.42) reduces to

(A.43)

A.5 NUNPEMCAL DIFFEFUZNTIATION OF
EXPEMMENTAL DATA

The determination of a rate requires the differentiation of the original experimental
data. As explained by De Nevers (1966), given a table of x - y data, the value of
dy/dx can be calculated by:
1. Plotting the data on graph paper, drawing a smooth curve through the points
with the help of a Fkench curve, and then drawing a tangent to this curve.
2. Fitting the entire set of data with an empirical equation, such as a polynomial,
and then differentiating the empirical equation.