Steady-State Mass and Heat Balance Equations

1 Steady-State Mass and Heat Balance Equations 

The formulation of the model equations is based on the transport of the reactants between the gas-liquid-soUd phases that takes place in trickle-bed reactors (Alvarez and Anchey ta, 2008). Hydrogen, being the main gaseous reactant, is first transferred from the gas phase to the liquid bulk. The reactants in the liquid phase (chemical lumps and dissolved H2) travel to the catalyst particle in order to react. Products such as H2S and CH4 are released to the gas phase passing through the liquid phase, whereas hydrocarbon products return to the liquid. 

The change in the molar flow of gaseous compounds along the reactor is equal to the gas-liquid transport rate  

The change in the concentration of gaseous compounds in the liquid phase is attributed to the gas-liquid transport and mass transfer to the solid phase  

The chemical lumps are transferred from the liquid bulk to the catalyst surface  

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Write the steady-state mass and heat balance equations for this system, assuming constant physical properties and constant heat of reaction. (Note Concentrate your modeling effort on the adiabatic nonisothermal reactor, and for the rest of the units, carry through a simple mass and heat balance in order to define the feed conditions for the reactor.)... 

Steady state mass- and heat-balance equations ... 

We have seen that there can be two stable steady states, one at a low temperature corresponding to quench conditions and the other at a high temperature corresponding to ignition conditions. An important consideration is the approach to the steady state, for that would determine the fate of a reaction. Thus, if we started at any initial condition, will the reaction approach one of the steady states, and if so how quickly Alternatively, if we started the reaction at a condition close to a stable steady state, will it approach that steady state To answer these questions we must modify the steady-state mass and heat balance equations to include a time-dependent component. The resulting transient equations are... 

Fixed-beds packed with inert pellets are usually used for the determination of transport properties and transfer coefficients. The solution of a steady-state mass or heat balance equation is compared with concentration or temperature measurements for the determination. For thermal properties, for instance, radial temperature... 

Effective axial transport properties can be determined using an adiabatic reactor. Steady state mass and heat balances result in second-order ordinary differential equations when the axial dispersion is taken into consideration, solutions of which can readily be obtained. Based on these solutions and temperature or concentration measurements, the effective transport properties can be calculated in a manner similar to the procedures used for the radial transport properties. As indicated earlier, a transient experiment can also be used for the determination. Here, experimental and analytical procedures are illustrated for the determination of the effective axial transport property for mass. An unsteady state mass balance for an adiabatic reactor can be written as ... 

Mass and heat balance equations for typical gas-liquid reactors in heterogeneous systems at steady state... 

The mass- and heat-balance equations for the steady-state model are the equations (7.25) to (7.47). In the following, we describe a simple procedure to compute the model parameters. 

Some of the tex ms, for example, in the mass and heat balance equations can be neglected without serious errors, depending on the condition. The terms relating to the b ilk flow are not imr-portant except during pyrolysis or hydropyrolysis. The accmu-lation term for gaseous species in the mass b8j.ance equation can usually be ignored, and a pseudo steady state assiimption can be applied without serious errors. 

Chapter 8 is dedicated to the modeling of heavy oil upgrading via hydroprocessing. Experimental studies for generation of kinetic data, catalyst deactivation, and long-term stability test are explained. Mass and heat balance equations are provided for the reactor modeling for steady-state and dynamic behavior. Simulations of bench-scale reactor and commercial reactor for different situations are also reported. 

The fractional conversions in terms of both the mass balance and heat balance equations were calculated at effluent temperatures of 300, 325, 350, 375, 400, 425, 450, and 475 K, respectively. A Microsoft Excel Spreadsheet (Example6-ll.xls) was used to calculate the fractional conversions at varying temperature. Table 6-7 gives the results of the spreadsheet calculation and Eigure 6-24 shows profiles of the conversions at varying effluent temperature. The figure shows that die steady state values are (X, T) = (0.02,300), (0.5,362), and (0.95,410). The middle point is unstable and die last point is die most desirable because of die high conversion. 

The reactor models considering complete mixing may be subdivided into batch and continuous types. In the continuous stirred tank reactor (CSTR) models, an entering fluid is assumed to be instantaneously mixed with the existing contents of the reactor so that it loses its identity. This type of reactor operates at uniform concentration and temperature levels. For this reason the species mass balances and the temperature equation may be written for the entire reactor volume, not only over a differential volume element. Under steady-state conditions, the species mass and heat balances reduce to algebraic equations. 

In particular cases simplified reactor models can be obtained neglecting the insignificant terms in the governing microscopic equations (without averaging in space) [9]. For axisymmetrical tubular reactors, the species mass and heat balances are written in cylindrical coordinates. Himelblau and Bischoff [9] give a list of simplified models that might be used to describe tubular reactors with steady-state turbulent flow. A representative model, with radially variable velocity profile, and axial- and radial dispersion coefficients, is given below ... 

Salnikov specifically reported multiple singular points and a limit cycle establishing the existence of oscillations in chemical reactions. Bilous and Amundson (1955) referred to Salnikov s (1948) paper as the first work where periodic phenomenon in reaction systems was discussed. They also indicated that a reaction A -> B in CSTR is irreversible, exothermic, and kinetically first order. Considering mass balance and heat balance equations it is known that at the steady states, the heat consumption... 

The steady-state concentration and temperature profiles cU e governed by the mass and energy balance equations. These involve reaction and diffusion and reaction and heat conduction, respectively. In terms of the wave-fixed coordinates introduced previously, these can be written as... 

In designing a double-pipe heat exchanger, mass balance, heat balance, and heat-transfer equations are used. The steady-state heat balance equation is... 

Equation 10.4.a-7 is a necessary but not sufficient condition for stability. In Other words, if the criterion is satisfied, the reactor may be stable if it is violated, the reactor will be unstable. (Aris [1] prefers to use the reverse inequality as a sufficient condition for instability.) The reason is that in deriving Eq. 10.4.a-7, it was implicitly assumed that only the special perturbations in conversion and temperature related by the steady-state heat generation curve were allowed. To be a general criterion giving both necessary and sufficient conditions, arbitrary perturbations in both conversion and temperature must be considered. Van Heerden s reasoning actually implied a sense of time ( tends to move... ), and so the proper criteria can only be clarified and deduced by considering the complete transient mass and energy balances. 

Now, consider a one-dimensional parallel flow of two phases either in co- or countercurrent flow, exchanging mass and heat with each other. Neglecting diffusional (or dispersion) terms, in steady state the balance equations become... 

The objective of this chapter is to give a comprehensive overview of both basics and peculiarities of RD modeling. A detailed description covers balance equations, mass and heat transfer, reaction kinetics including reaction-mass-transfer coupling, as well as steady-state and dynamic modeling issues. The achievements in the theoretical description are illustrated with several case studies supported by laboratory- and pilot-scale experimental investigations. 

So far, we have admitted only measurement errors. Another source of discrepancy (imbalance) can be a fault in the model itself. We do not consider the cases when some well-defined item (say, mass flowrate of some stream) has been omitted. Still, some unforeseen material or heat (energy) loss can occur perhaps also a significant change of accumulation in some node can have been neglected by the steady-state hypothesis, and the like. Then the m-th equation (balance) reads correctly... 

The steady state model is based on the mass and energy balances which were carried out over the furnace, heat exchanger area, economising section, dolezal, drum, generating section and superheaters. A set of nonlinear algebraic equations were used to accurately predict changes in the physical state of each system. Fig. 1 shows a simplified PFD for recovery boiler steam generation cycle. 

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