TRANSFER: BASICS OF MASS AND HEAT TRANSFER - Biomass as a Sustainable Energy Source for the Future - page 126

and time. Assume that the initial value inside the layer is X ( x ,0)= X 0 and outside
the layer is X ( x ,0)= X e . The normalized moisture concentration is defined
as

= X

X e
X 0 −

−

ϑ

X e .

a. Write an evolution equation for ϑ ( x , t) containing a transient term and a dif-

s law.
b. Assuming that the temperature remains constant during drying, solve the
equation by the method of separation of variables to obtain the solution

fusion term according to Fick '

! cos

π = ∞

n =1

n 2

2 D eff

ðÞ − 1 ð 2 n +1 Þ π

n + 1
2

x
L

ϑ ðÞ x ,t exp − t

ð 2 n +1 Þ π

4L 2

Hint: The problem is mathematically similar to the problem of diffusion of
heat from a slab considered, e.g., in Mills (1999). The factor ( D eff t/L 2 ) appear-
ing in the exponential function is the analogue of the Fourier number Fo
appearing in Table 4.1 and is the dimensionless time of the process.
c. Determine the surface moisture flux at x = L.
d. Identify accurate approximate solutions for the case of very long drying time
and very short drying time.
e. Describe a procedure to determine the pre-exponential factor and the activa-
tion energy from experiments.

4.3 The radiative transfer equation Equation (4.12) contains terms representing the
effects of scattering. For the simplest case of isotropic scattering, the scattering
phase function is constant Φ λ ( s ∗ , s ) 1. For this case, show, by integrating the

RTE over all directions, that scattering does not influence the value of the total
incident radiation G.

≡

4.4 There is currently substantial interest in utilizing eukaryotic algae for the renew-
able production of several bioenergy carriers, including starches for alcohols,
lipids for diesel fuel surrogates, and hydrogen for fuel cells. Algae can convert
solar energy into fuels at high photosynthetic efficiencies and can thrive in salt-
water systems. Part of an energy balance analysis of such a process is the com-
putation of the penetration of the radiative heat flux in a pool of water.

a. Considering the surface of the sun as a black surface at a temperature of 5777
K, determine the total (i.e., integrated over all wavelengths) solar heat flux
incident on the top of the Earth s atmosphere. The radius of the sun is

6.96 × 10 8 m, and a representative value of the sun-to-Earth distance is
1.496 × 10 11 m. Compare your result to the generally accepted annual mean
value of 1366 W

'

m −2 and discuss possible reasons for difference.
b. Assuming that at a certain location and time, when the sun is in the zenith, the
total heat flux arriving on the Earth surface is 1100 W m −2 , calculate the