Page 532 - Modelling in Transport Phenomena A Conceptual Approach

512 APPENDIX A. MATHEMATICAL PRELlMINARLES

PROBLEMS

A.3 Caffeine is extracted from coffee grains by means of a crossflow extractor.
The standard error of the exit concentration versus time curve was found as CT =
1.31, where standard deviation, cr2, is given as

2 2
(1
a2 = - + - -e-")
Pe Pe2
Solve this equation and determine the Peclet number, Pe, which is a measure of
axial dispersion in the extractor.
(Answer: 1.72)

A.4 The roof of a building absorbs energy at a rate of 225kW due to solar
radiation. The roof loses energy by radiation and convection. The loss of energy
flux as a result of convection from the roof to the surrounding air at 25°C is
expressed as
q=2.5(T-Tm) 1.25
where T and T' are the temperatures of roof and air in degrees Kelvin, respectively,
and q is in W/ m2. Calculate the steady-state temperature of the roof if it has the
dimensions of 10 m x 30 m and its emissivity is 0.9.
(Answer: 352 K)

A.8 METHODS OF INTEGRATION

Analytical evaluation of a definite integral

(A.8-1)

is possible only for limited cases. When analytical evaluation is impossible, then
the following techniques can be used to estimate the value of the integral.

A.8.1 Mean Value Theorem

As stated in Section A.2, if f (x) is continuous in the interval a 5 x 5 b, then the
value of I is

I = JD" f(x) cix = (f)(b - a) (A.8-2)
where (f) is the average value of f in the interval a 5 x 5 b.