Page 57 - Modelling in Transport Phenomena A Conceptual Approach

38 CHAPTER 2. MOLECULAR AND CONVECTIVE TRANSPORT

(ii) The temperature gradient at a = 0 is equal to the geothermal gradient of the
earth, i.e., 25 "C/ km.
(iii) The thermal conductivity, the density and the heat capacity of the earth are
3 W/ m. K, 5500 kg/ m3 and 2000 J/ kg. K, respectively.
Estimation of the age of the earth, based on the above model, is first used by
Lord Kelvin (1864). However, he knew nothing about radioactivity and heating
of the earth's crust by radioactive decay at that time. As a result, his estimates,
ranging from 20 to 200 million years, were completely wrong. Today, the geologists
generally accept the age of the earth as 4.55 billion years.
(Answer: 85.3 x lo6 year)
2.9 A slab is initially at a uniform temperature To and occupies the space from
z = 0 to z = 03. At time t = 0, the temperature of the surface at z = 0 is suddenly
changed to TI (TI > To) and maintained at that temperature for t > 0. Under
these conditions the temperature distribution is given by
- (i>
Ti -T
Tl -To = erf 2 6
If the surface area of the slab is A, determine the amount of heat transferred into
the slab as a function of time.
Answer: Q=

2.10 Air at 20 "C and 1 atm pressure flows over a porous plate that is soaked in
ethanol. The molar concentration of ethanol in the air, CA, is given by

CA = 4 e- 1.5'

where CA is in kmol/m3 and z is the distance measured from the surface of the
plate in meters. Calculate the molar flux of ethanol from the plate.
(Answer: 0.283 kmol/ m2. h)

- .=(E)
2.11 The formal definition of the partial molar volume is given by

TJ',nj#i
Substitute
n
V=-
C
into Eq. (1) and show that the volume fraction is equal to the mole fraction for
constant total molar concentration, c, i.e.,