Equations for mass and energy balance

The differential rate equations for the concentration of A and for the temperature within the reactor (the mass- and heat-balance equations) can be written in the form 

Here cp is the heat capacity (JK 1kg 1) and a the total density (kg m 3). If we divide the heat-balance equation throughout by cpa the Newtonian cooling time rN = cpa V/xS emerges naturally in the denominator of the last term, as does the group Q/cva which is related to the adiabatic temperature rise appropriate to the system, ATad = Qa0/cp j, in the second term. 

The most obvious choice for dimensionless concentration is that used in the previous chapter, i.e. 

For temperature, we can again recognize that we are more interested in the temperature rise rather than in T itself. We can use the same dimensionless temperature excess 9 as that introduced in chapters 4 and 5, with a slight modification. Instead of basing 9 on the ambient temperature Ta we will base 

In a similar way, the maximum temperature rise possible in the system (the adiabatic temperature rise ATad) has a dimensionless measure 0ad given by 

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In order to evaluate P2 we need to consider how the governing equations for mass and energy balance themselves vary with changes in the variables. In the case of the present model this means evaluating various partial derivatives of (5.1) and (5.2) with respect to a and 0. Before proceeding, however, we should take a look at the elements of the Jacobian matrix evaluated for Hopf bifurcation conditions ... 

The formal similarity allows us to carry over the equations for mass and energy balances in the tubular reactor, Eqs. (3.4.11)-(3.4.14). The momentum equation has no meaning. Care must be taken however to distinguish between a batch reactor working at constant volume and one that works at constant pressure. The latter has the Eqs. (3.4.12) or (3.4.14) which were derived from an enthalpy balance. In the former case the heat added would be equated to the internal energy change. Thus in this case c should replace Cp and the internal energy of reaction replace the heat of reaction. These... 

In order to search for the optimum, the relations between variables must be given to an optimization algorithm. This is here provided by the model equations developed by Mariano et al. [3]. The model calculates all mass and heat transfers, besides the hydrogenation reaction rate. Since there are three phases (the catalysts is solid, the hydrogen is a gas and the o-cresol is liquid), both reactants must come to the solid pores, where the reaction takes place, and the unreacted reactants and the reaction product must then leave the catalyst particle. All these phenomena are accounted for by partial differential equations for mass and energy balances for each component in each phase. 

For the first-order reaction, the steady-state equations for mass and energy balance in a CSTR can be combined into a single equation represented as... 

The mass transfer rate equations force values of the mass transfer coefficients such that the rates of each phase, N V and A L are equal. The above equations for material and energy balance across the interface are used to form these rate equations and are excluded from the final set of equations. Since the vapor and liquid are only at equilibrium at the interface, one equilibrium equation per component is drawn there ... 

The dimensionless steady-state equations of mass and energy balance for a CSTR are reproduced here ... 

Moving up into the reactor level, effects of convection, dispersion and generation are described in the conservation equations for mass and energy. The momentum balance describes the behavior of pressure. The interface between the reactor and the catalyst level is described by the external mass transfer conditions, most often represented in a Fickian format, i.e., a linear dependence of the rate of mass transfer on the concentration gradient. In cases where an explicit description of mixing and hydrodynamic patterns is required, the simultaneous integration of the Navier-Stokes equations is also conducted at this level. I f the reaction proceeds thermally, the conversion of mass and the temperature effect as a result of it are described here as well. 

Equations describing yioo and Tq can be obtained from the equations of mass and energy balance for a unit volume of the gas-liquid mixture, which, when written in the dimensionless variables introduced above, take the form ... 

The essential differences between sequential-modular and equation-oriented simulators are ia the stmcture of the computer programs (5) and ia the computer time that is required ia getting the solution to a problem. In sequential-modular simulators, at the top level, the executive program accepts iaput data, determines the dow-sheet topology, and derives and controls the calculation sequence for the unit operations ia the dow sheet. The executive then passes control to the unit operations level for the execution of each module. Here, specialized procedures for the unit operations Hbrary calculate mass and energy balances for a particular unit. FiaaHy, the executive and the unit operations level make frequent calls to the physical properties Hbrary level for the routine tasks, enthalpy calculations, and calculations of phase equiHbria and other stream properties. The bottom layer is usually transparent to the user, although it may take 60 to 80% of the calculation efforts. 

Microscopic Balance Equations Partial differential balance equations express the conservation principles at a point in space. Equations for mass, momentum, totaf energy, and mechanical energy may be found in Whitaker (ibid.). Bird, Stewart, and Lightfoot (Transport Phenomena, Wiley, New York, 1960), and Slattery (Momentum, Heat and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981), for example. These references also present the equations in other useful coordinate systems besides the cartesian system. The coordinate systems are fixed in inertial reference frames. The two most used equations, for mass and momentum, are presented here. 

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] ... 

The mass and energy balance equations for ideally mixed components where zero-order reaction proceeds are ... 

As for the mass and energy balance equations, steady-state conditions are obtained when the rate of change of momentum in the system is zero and... 

In principle, any type of process model can be used to predict future values of the controlled outputs. For example, one can use a physical model based on first principles (e.g., mass and energy balances), a linear model (e.g., transfer function, step response model, or state space-model), or a nonlinear model (e.g., neural nets). Because most industrial applications of MPC have relied on linear dynamic models, later on we derive the MPC equations for a single-input/single-output (SISO) model. The SISO model, however, can be easily generalized to the MIMO models that are used in industrial applications (Lee et al., 1994). One model that can be used in MPC is called the step response model, which relates a single controlled variable y with a single manipulated variable u (based on previous changes in u) as follows ... 

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). 

Component mass and energy balances and normalization equations are first rewritten using the method proposed by Crowe (1986) for the bilinear terms. Streams are divided into three categories depending on the combination of total flowrates (f), concentration (M), and temperature (t) measurements as shown in Table 1. 

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... 

For a single reaction in the CSTR, the transient mass- and energy-balance equations are... 

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... 

Mass- and energy-balance equations must be written and solved for each species in each phase. 

Module INPUT takes user-specified input and constructs proper initial conditions for the detailed mass and energy balance equations. 

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