Reactor Mass and Energy Balances

Finally, since engineers must be concerned with making money in a chemical process, the economic balance equations must also be solved along with the above equations. Managers fiequently beheve that the preceding equations are not important at all, and that one should only worry about profit and loss. The problem with this idea is that these equations underhe any economic considerations. 

There should be some sort of a money continuity equation for a chemical process, which might look like 

We can easily recover the mass- and energy-balance equations used in previous chapters from these equations. First we assume variation in only one spatial dimension, the direction of flow z, to obtain 

The heat removal term Q has to be inserted in the one-dimensional energy balance by implicitly solving the ener balance in the radial direction. The heat flux in the radial direction is = —kf dT/d R, and we replace the radial flux to the wall by assuming that the fluid at any position j is at a constant temperature T while the wall is at temperature to obtain 

Note also that these equations can be simplified to obtain the batch reactor mass- and energy-balance equations by setting Dj = 0 and w = 0 to give 

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Notice how the catalyst particle balances are coupled to the reactor mass and energy balances. Thus, the catalyst particle balances [Equation (10.2.7)] must be solved at each position along the axial reactor dimension when computing the bulk mass and energy balances [Equation (10.2.6)]. Obviously, these solutions are lengthy. 

Vasco de Toledo et al. (2001) developed dynamic models for catalytic slurry reactors. Mass and energy balances, as well as an equation for a coolant fluid were proposed for the hydrogenation of o-cresol on Ni/Si02 to produce 2-methylcyclohexanol. 

A model can be defined as a set of relationships between the variables of interest in the system being investigated. A set of relationships may be in the form of equations the variables depend on the use to which the model is applied. Therefore, mathematical equations based on mass and energy balances, transport phenomena, essential metabolic pathway, and physiology of the culture are employed to describe the reaction processes taking place in a bioreactor. These equations form a model that enables reactor outputs to be related to geometrical aspects and operating conditions of the system. 

Figure 1.19. Information flow diagram for modelling a non-isothermal, chemical reactor, with simultaneous mass and energy balances.

In gas-phase reactors, the volume and volumetric flow rate frequently vary, owing to the molar changes caused by reaction and the effects of temperature and pressure on gas phase volume. These influences must be taken into account when formulating the mass and energy balance equations. 

Population Balance Approach. The use of mass and energy balances alone to model polymer reactors is inadequate to describe many cases of interest. Examples are suspension and emulsion polymerizations where drop size or particle distribution may be of interest. In such cases, an accounting for the change in number of droplets or particles of a given size range is often required. This is an example of a population balance. 

Fermentation systems obey the same fundamental mass and energy balance relationships as do chemical reaction systems, but special difficulties arise in biological reactor modelling, owing to uncertainties in the kinetic rate expression and the reaction stoichiometry. In what follows, material balance equations are derived for the total mass, the mass of substrate and the cell mass for the case of the stirred tank bioreactor system (Dunn et ah, 2003). 

The second case study corresponds to an existing pyrolysis reactor also located at the Orica Botany Site in Sydney, Australia. This example demonstrates the usefulness of simplified mass and energy balances in data reconciliation. Both linear and nonlinear reconciliation techniques are used, as well as the strategy for joint parameter estimation and data reconciliation. Furthermore, the use of sequential processing of information for identifying inconsistencies in the operation of the furnace is discussed. 

Data reconciliation of the pyrolysis reactor was performed on the basis of the following mass and energy balance equations (Weiss et al., 1996). 

In most jacketed reactors or steam-heated reboilers the volume occupied by the steam is quite small compared to the volumetric flow rate of the steam vapor. Therefore the dymamic response of the jacket is usually very fast, and simple algebraic mass and energy balances can often be used. Steam flow rate is set equal to condensate flow rate, which is calculated by iteratively solving the heat-transfer relationship (Q = UA AT) and the valve flow equation for the pressure in the jacket and the condensate flow rate. 

For the nonisothermal reactors we need to solve the mass- and energy-balance equations... 

The reactor configuration might look as shown in Figure 6-15 for the jacketed reactor and with an internal cooling cod. ff is not constant, we have to solve an energy balance on the coolant along with the mass and energy balance on the reactor, where now the coolant has inlet temperature T o and outlet temperature and the coolant flows with volumetric flow rate ii(. and has contact area A, with the reactor. 

Any fluid flow situation is described completely by momentum, mass, and energy balances. We have thus far looked at only simplified forms of the relevant balance equations for our simple models, as is done implicitly in aU engineering courses. It is interesting to go back to the basic equations and see how these simple approximations arise. We need to examine the full equations to determine the errors we are making in describing real reactors with... 

Leung s method is given in 6.3.2 below. The method.is an approximate solution to the differential mass and energy balances for the reactor during relief and takes account of both emptying via the relief system and the tempering effect of vapour production due to relief. The method makes use of adiabatic experimental data for the rate of heat release from the runaway reaction (see Annex 2). Nomenclature is given in Annex 10. 

Non-isothermal and non-adiabatic conditions. A useful approach to the preliminary design of a non-isothermal fixed bed reactor is to assume that all the resistance to heat transfer is in a thin layer near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the approximate design of reactors. Neglecting diffusion and conduction in the direction of flow, the mass and energy balances for a single component of the reacting mixture are ... 

A more quantitative analysis of the batch reactor is obtained by means of mathematical modeling. The mathematical model of the ideal batch reactor consists of mass and energy balances, which provide a set of ordinary differential equations that, in most cases, have to be solved numerically. Analytical integration is, however, still possible in isothermal systems and with reference to simple reaction schemes and rate expressions, so that some general assessments of the reactor behavior can be formulated when basic kinetic schemes are considered. This is the case of the discussion in the coming Sect. 2.3.1, whereas nonisothermal operations and energy balances are addressed in Sect. 2.3.2. 

The mathematical model of the reactor consists of the mass and energy balances written for all the compartments and an energy balance written for the jacket. The mass balance written for the reactant and for a first-order reaction in a generic compartment on the central level holds ... 

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