Energy balance dimension model

The model mass and energy balance equations will have consistent units, throughout, i.e., kg/s, kmol/s or kj/s, and corresponding dimensions of mass/ time or energy/time. The major system variables, normally concentration or... 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

It is very well possible that the axial heat conduction in the wall has an influence on the temperature in the region close to the wall. A detailed description of the temperature in the bed as a function of the radial and axial positions can be obtained by extending the model to two dimensions. However, it is also possible to get more insights into the behaviour by adjusting the one-dimensional model. Instead of accounting for the heat capacity of the wall in the accumulation term, an extra energy balance for the wall is included in the model, in which heat is exchanged with the packed bed ... 

If the nonuniformities in the temperature and species concentrations among the catalyst channels need to be considered, a 2D or 3D modeling approach is required. In this case, the transient energy balance equation of the monolith should be extended to two or three dimensions. The heat conduction equation of the solid phase is formulated in polar coordinates for 2D simulations (13.18) and in Cartesian coordinates for 3D simulations (13.19). 

Given these general equations, one approach to mathematical modeling is to apply appropriately and simphly these equations for the particular problem. This involves a clear statement of assumptions and an estimation of the relative importance of various terms (e.g. using dimensional analysis see Chapter 4). In Example 3.3, the energy balance equation has been simplifed from its generic form to an equation describing the heat conduction in one dimension (Example 3.1). 

So what do the and terms in the quadratic phenomenological model mean A simple adaptation of the quadratic eq. 8 converts it into a form (eq. 9) that has units of energy, but some caution must be exercised. Work is the integral of the force times the distance travelled in the same direction. The Fand t/(force and dimension) vectors are not parallel, but an analysis in Reference 23 shows that the vertically applied force, F, can be transformed into a lateral wedging force that acts in the same direction as d Similar arguments were invoked by numerous authors with the hardness energy balance discussed above where vertical indentation distances can be correlated with the lateral indentation diagonal sizes. 

The box-like cell model of a PE star can be considered as a generalization of a classical mean-field Flory approach, which was first suggested to describe the swelling of a polymer chain in a good solvent [90], The Flory approach estimates the equilibrium dimensions of a macromolecule, as a function of its parameters, by balancing the free energy of intramolecular (repulsive) interactions with the conformational entropy loss of a swollen chain. Within the box-like approximation, the star is characterized by the radius of its corona, R (end-to-end distanee of the arms), or by the average intramolecular concentration of its monomers ... 

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