Gas Phase Energy Balance

We first start with the general thermal energy balance for the fluid phase within the reactor, 

We need to add the local sources and sinks of thermal energy to this equation  

Qc = heat transfer between the catalyst and fluid phases Qj. = heat of reaction 

This gives for the general thermal energy balance 

Equation (6.14.4) becomes upon expanding the substantial derivative, 

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Gas phase energy balance is represented as follows. Gas temperature is increased by heat transfer between gas and catalyst surface. 

A simple dynamic model for describing a nylon 6,6 SSP reactor is shown in Table 7.6. This partial differential equation (PDE) model is a set of material balances (on water, amine groups, carboxyl groups and amide links in the particles and on water in the gas phase). Energy balances are also required to predict temperatures within the reactor [30, 31, 40]. Concentrations (and temperature) vary radially within the polymer particles, and with vertical position and time as the particles move downward through the reactor. As a result, the independent variables that appear in the model are the radial direction within the particles, the vertical direction z within the bed, and time, t. 

We can now integrate the gas phase energy balance (6.14.32) over the radial cross-sectional area for flow,... 

A completely analogous derivation to the gas-phase energy balance yields... 

The gas-phase energy balance (Equation 9.5i) can in turn be formally integrated to yield the familiar HTU-NTU relation ... 

The basic scheme for the numerical solution is the same as that used for the 1 -D model, except that in this case the solid temperature field used to solve the DAE system for each monolith channel must be calculated from the three-dimensional solid-phase energy balance equation. The three-dimensional energy balance equation can be solved by a nonlinear finite element solver (such as ABAQUS) for the solid-phase temperature field while a nonlinear finite difference solver for the DAE system calculates the gas-phase temperature and... 

Heat release or consumption by surface reactions contributes to the energy balance at a gas-surface interface. Diffusive and convective fluxes in the gas phase are balanced by thermal radiative and chemical heat release at the surface. This balance is stated as... 

Solid Energy Balance. The solid-phase energy balance is composed of terms related to thermal communication with the walls of neighboring unit cells, with the two gas-phase compartments of unit cell (k, ), and of the electrical work produced. For clarity, we have elected to show these terms in detail. 

The purpose of the developed model is to describe the 3-way catalyst warm-up behaviour by predicting time dependent NO, THC and CO conversions and temperature profiles. The model has been presented in an earlier paper (Kangas et al., 2002). The converter is assumed to be adiabatic with uniform radial temperature and flow rate distribution. Thus, the profiles of the whole converter are obtained by modeling one channel of the monolith. The channel model equations consist of gas phase mass and energy balances, solid phase energy balance and heat transfer model... 

Material balances, often an energy balance, and occasionally a momentum balance are needed to describe an adsorption process. These are written in various forms depending on the specific application and desire for simplicity or rigor. Reasonably general material balances for various processes are given below. An energy balance is developed for a fixea bed for gas-phase application and simphfied for liquid-phase application. Momentum balances for pressure drop in packed beds are given in Sec. 6. 

When a gas comes in contact with a solid surface, under suitable conditions of temperature and pressure, the concentration of the gas (the adsorbate) is always found to be greater near the surface (the adsorbent) than in the bulk of the gas phase. This process is known as adsorption. In all solids, the surface atoms are influenced by unbalanced attractive forces normal to the surface plane adsorption of gas molecules at the interface partially restores the balance of forces. Adsorption is spontaneous and is accompanied by a decrease in the free energy of the system. In the gas phase the adsorbate has three degrees of freedom in the adsorbed phase it has only two. This decrease in entropy means that the adsorption process is always exothermic. Adsorption may be either physical or chemical in nature. In the former, the process is dominated by molecular interaction forces, e.g., van der Waals and dispersion forces. The formation of the physically adsorbed layer is analogous to the condensation of a vapor into a liquid in fret, the heat of adsorption for this process is similar to that of liquefoction. 

Because of this heat generation, when adsorption takes place in a fixed bed with a gas phase flowing through the bed, the adsorption becomes a non-isothermal, non-adiabatic, non-equilibrium time and position dependent process. The following set of equations defines the mass and energy balances for this dynamic adsorption system [30,31] ... 

Thus we see that the equilibrium solubility of a gas again involves a balance between randomness and energy as it does for a solid, but the effects are opposite. For a gas, the tendency toward maximum randomness favors the gas phase, opposing dissolving. The tendency toward minimum energy favors the liquid state, hence favors dissolving. 

Equation (22) was obtained, essentially, with examination of the energy balance equation with respect to flows of gas-containing polymer melts. The key moment of this analysis is, in our view, comprehension of the fact that the energy of gas dissolved in the polymer is transformed into the energy of movement of the two-phase medium. 

As the gas-liquid mixture travels down the vent line, the phases will slip past each other and the fluids will accelerate. This contribution to the energy balance can be most significant for high pressure blowdown. Pressure increments are calculated and when the pressure gradient becomes infinite the flow is choked. If this occurs at the end of the pipe the assumed flowrate is the converged choked flow solution. If choked flow does not occur and the end of the line is reached at the reservoir pressure, the non-choked flow solution is obtained. 

A model based on energy balance was developed by Garcia-calvo et al. [5]. Energy input during gas expansion is dissipated in the flow field and in the phase interfaces, therefore ... 

The balance of average kinetic energy (red line and intermolecular forces of attraction favors the gas phase... 

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