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# Flow through a porous medium is widely encountered in various industries and scientific applications. In these

cases, due to the porosity of the catalytic bed, the superficial velocity along the radial direction of the bed is often assumed to be uniform, i.e. ideal plug flow. Relations of pressure drop and flow rate would then depend on the porosity and permeability of the bed, e.g. the Darcy's equation. In this assignment, governing equations for flow through a porous bed of spherical catalyst particles will be developed. Particularly, we are interested in the relationship between the temperature of the packed bed as a function of the heat of reaction and the effective thermal conductivity.

Task 1 (General description of transport phenomena and governing equations) Consider a tubular, ﬁxed bed ﬂow reactor packed with catalytic spherical particles. A reaction (“C02-
methanation”) is carried out within the bed, c02 +4h’2 &lt;—&gt; CH4 +H20. The catalytic conversion of carbon dioxide into methane is exothermic, generating heat at a rate of
Sc. For simplicity, we can assume that Sc varies linearly with temperature, i.e.S,_. =Sco(T — T0), with 5:20 = —66000 W/m3 and To = 298 K. It may also be assumed that the reactor wall is well
insulated (i.e. there is no temperature variation in the radial direction), and that the ﬂow through
the reactor behaves as an ideal plug ﬂow. The inner radius of the reactor is 25 mm, and the length of the catalytic bed is 0.5 m. It is filled with
spherical catalyst particle (diameter = 3 mm, bed porosity = 0.4, pellet thermal conductivity = 5
W/m.K). Gas ﬂows through the inlet at a rate of 1 kg/mzs, and it contains 20 mol% C02 (balance H2).
You may make assumptions for all other parameters. (a) Describe various transport phenomena occurring in the problem described above;
(5 marks) (b) Derive the governing equations for the temperature distribution along the catalytic bed.
Prescribe all boundary conditions. Clearly state your assumptions. (10 marks)

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