Page 73 - Introduction to Computational Fluid Dynamics

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52
1D HEAT CONDUCTION
22. It is proposed to remove NO from exhaust gases of an internal combustion
engine by passing them over a catalyst surface. It is assumed that chemical
reactions involving NO are very slow so that NO is neither generated nor
destroyed in the gas phase. At the catalyst surface, however, NO is absorbed

at the rate of ˙ m = Kρ m ω 0 , where the rate constant K = 0.075 m/s and ω 0
is the mass fraction of NO at the catalyst surface. In the exhaust gases (T =
◦
500 C, p = 1 bar, M = 30) the mole fraction of NO is X NO = 0.002. Now,
it is assumed that NO diffuses to the catalyst surface over a stagnant layer of
2
1 mm with effective diffusivity = 3 × D, where D = 10 −4 m /s. Determine
2
the steady-state absorption rate (kg/m -s) of NO and its mass fraction at the
surface.
23. The mass fraction of carbon in a low-carbon steel rod (2 cm diameter) is 0.002.
◦
To case-harden the rod it is preheated to 900 C and packed in a carburising
mixture at 900 C. The mass fraction of carbon at the rod surface is now 0.014
◦
and is maintained at this value. Calculate the time required for the carbon mass
fraction to reach 0.008 at a depth of 1 mm from the rod surface. Assume radial
diffusion only. In this case, cross-sectional area A = 2πr. However, since the
penetration depth is only 10% of the rod radius, one may take A = 2π R =
constant (i.e., assume plane diffusion). Compare the time required in the two
2
cases. Take the diffusivity of carbon in steel to be D = 5.8 × 10 −10 m /s.
◦
24. Gaseous H 2 at 10 bar and 27 C is stored in a 10-cm inside diameter spher-
ical tank having a 2-mm-thick wall. If diffusivity of H 2 in steel is D =
2
3
0.3 × 10 −12 m /s and solubility S = 9 × 10 −3 kmol/m -bar, estimate the time
required for the tank pressure to reduce to 9.9 bar. Also, plot the time variation
and the instantaneous hydrogen loss rate. Take ρ steel =
of tank pressure p H 2
3
8,000 kg/m . The density of hydrogen at the inner surface of the tank is given
. Is an exact solution possible for this problem?
by ρ H 2 ,i = Sp H 2 M H 2
25. Consider steady-state heat transfer through the composite slab shown in
Figure 2.14. Assume k 1 = 0.05(1 + 0.008 T ), k 2 = 0.05(1 + 0.0075 T ), and
k 3 = 2 W/m-K, where T is in degrees centigrade. Calculate the rate of heat
transfer and the temperatures of the two interfaces. Ignore radiation.

26. Repeat Exercise 25 including the effect of radiation. The emissivities at x = 0
and x = 17 cm are 0.1 and 0.8, respectively. In this problem, one must use the
concept of effective heat transfer coefficient h eff = h + h rad . Thus, at x = 0,
for example,

2 2
h eff = 50 + 0.1σ (T ∞ + T x=0 ) T + T ,
∞ x=0
2
4
where the Stefan–Boltzmann constant σ = 5.67 × 10 −8 W/m −K , and T ∞
T x=0 are in Kelvin.
27. Consider fully developed turbulent heat transfer in a circular tube under con-
stant wall heat flux conditions. Equations 2.80 and 2.81 are again applicable