Simultaneous reactions, mass balance

Note that for multiple-reaction systems we can simply substitute 52 v,jr/ for vjr in the mass-balance expressions for a single-reaction mass-balance equation. The difference with multiple reactions is that now we must solve R simultaneous mass-balance equations rather than the single equation we had with a single reaction. 

The sweetener aspartame exists as a hemihydrate (C14H18N2O5 O.5H2O ASH), under ambient conditions. When heated, ASH converted to aspartame anhydrate (ASA), which on further heating decomposed to form a diketopiperazine (DKP) derivative. The XRD patterns of ASH, ASA, and DKP showed pronounced differences (Fig. 12). XRD was used to simultaneously quantify the (i) disappearance of ASH and appearance of ASA in the first reaction and (ii) disappearance of ASA and appearance of DKP in the second reaction. For studying the kinetics of the first reaction, the peaks unique to ASH at 15.9 and 16.4° 20 and the 17.1° 20 peak of ASA were used. For the second reaction, the sum of the integrated intensities of the peaks at 10.2, 11.0, and 11.8° 20 of ASA and the 13.0° 20 peak of DKP were used. While the dehydration of ASH appeared to follow first-order kinetics, the cyclization of ASA was a nucleation-controlled process. Figs. 13 and 14 contain the dehydration and cyclization data at 118 and 180°C respectively. Since the concentrations of the crystalline reactant as well as the product were simultaneously monitored, mass balance calculations of the crystalline phases were possible at each time point. The reaction... 

How can we express the equilibrium state of such a system A direct approach would be to write each reaction that could occur among the system s species, minerals, and gases. To solve for the equilibrium state, we would determine a set of concentrations that simultaneously satisfy the mass action equation corresponding to each possible reaction. The concentrations would also have to add up, together with the mole numbers of any minerals in the system, to give the system s bulk composition. In other words, the concentrations would also need to satisfy a set of mass balance equations. 

Recommendation (Blue Grass) EH-1. Tetrytol should not be processed simultaneously with any lead-containing energetic material until such time as it can be confirmed that lead picrate will not precipitate from the hydrolysate. The solubility of lead picrate in the reaction medium should be determined as a function of temperature, and the lead in the reaction should be accounted for in a mass balance. 

These are two simultaneous differential equations with two initial conditions for a single reaction. For R simultaneous reactions we have to solve R + 1 simultaneous differential equations with R + 1 initial equations because there are R independent mass-balance equations and one temperature equatiorr... 

For R simultaneous reactions we have R different Xj values and R simultaneous maSS-balance equations to solve along with the energy-balance equation. In the energy-balance equation the flow and heat removal terms are identical, but the energy generation has terms for each of the R reactions,... 

These equations apply to the total mass or mass density of the system, while we use moles when describing chemical reaction. Therefore, whenever we need to solve these equations simultaneously, we must transform our species mass balances into weight fraction when including momentum and total mass-balance equations. 

In a single-phase reactor we have to solve a mass-balance equation for each reaction. In a multiphase reactor we have to solve one mass balance equation for each reaction in each phase so for R reactions in P phases we would have to solve R x P simultaneous mass-balance equations. 

In Ref. 139 a purely numerical approach to the solution of the considered complex RA problem was suggested. The liquid film is treated as an additional balance region, in which reaction and mass transfer occur simultaneously. Therefore, the reactions are considered both in the liquid-bulk-phase mass balances, Eq. (Al), and in the differential balances for the liquid film, Eq. (A10). 

One of the simplest models used to describe the performance of tubular reactors is the well-known isothermal one-dimensional plug flow tubular reactor (PFTR) model. The mass balance of this model for steady-state conditions, the simultaneous occurrence of M reactions and a constant volumetric flow rate V is ... 

When a fast reaction is highly exothermic or endothermic and, additionally, the effective thermal conductivity of the catalyst is poor, then significant temperature gradients across the pellet are likely to occur. In this case the mass balance (eq 32) and the enthalpy balance (eq 33) must be simultaneously solved using the corresponding boundary conditions (eqs 34-37), to obtain the concentration profile of the reactant and the temperature profile inside the catalyst pellet. The exponential dependence of the reaction rate on the temperature thereby imposes a nonlinear character on the differential equations which rules out an exact analytical treatment. Approximate analytical solutions [83, 99] as well as numerical solutions [13, 100, 110] of eqs 32-37 have been reported by various authors. 

Kosinski and Bostian (12) reported the extraction of lanthanum by HDEHP from aqueous nitrate solutions over a wide concentration range. Using mass balance data and IR analysis of the organic phase, they proposed three extraction reactions which occur simultaneously ... 

Thermodynamic data, whether determined through calorimetry or solubility studies, are subject to refinement as more exact values for the components in the reaction scheme, or more complete description of the solution phases, become available. Many of the solubility studies on clays were done before digital-computer chemical equilibrium programs were available. One such program, SOLMNEQ, written by one of the authors ( ) solves the mass-action and mass-balance equations for over 200 species simultaneously. SOLMNEQ was employed in this investigation to convert the chemical analytical data into the activities of appropriate ions, ion pairs, and complexes. 

To solve the mass balance, it must be accompanied by the simultaneous solution of the energy balance (i.e., the solution of Equation (9.2.9)). To do this, Equation (9.2.9) can be written in more convenient forms. Consider that the enthalpy contains both sensible heat and heat of reaction effects. That is to say that Equation (9.2.10) can be written as ... 

Given the transport fluxes for all species inside the catalyst particle, as modeled in Section 3.4.3. a steady-state mass balance considering the simultaneous transport and chemical reaction gives... 

In discussing the general algorithm for a cold shot reactor with m simultaneous reactions we will return to the extent variables c and actual temperature T. Let Co and To denote the state of the cold unreacted feed and g, be the flow rate of reactants through the bed r of volume F,. The bypass that is mixed with the process stream between beds (r - - 1) and r is (g, — g, i) so that mass and heat balances give... 

In the next sections, the design equations for the three ideal reactor t)q>es will be derived for isothermal conditions. In practice, the heat effects associated with chemical reactions usually result in non-isothermal conditions. The application of the law of conservation of energy leads to the so-called energy balance equation. This derivation is analogous to the derivation of the mass balance equations, and will not be treated here (see for instance, [4,5]). However, it should be noted that under non-isothermal conditions, the energy balance equation should always be solved simultaneously with the corresponding mass balance equation, since the reaction rate depends not only on composition but also on temperature. 

The structure of the chapter should be clear from Fig. 7.1. The design must be based on the proper mass and energy balances for the reactor. When there is only one reaction, conversion and yield are equivalent concepts, but with simultaneous reactions, the primary concern of the design... 

This may be illustrated in an example which, though very simple, is often useful as a first approximation to a more complex situation. If the two irreversible reactions A —> B — C take place simultaneously and are first order with rate constants ki and ki, we have (for a feed of pure A) the mass balance equations... 

Solution Since there will be heat transfer from the reaction gases to the glycol jacket, the temperature in the reactor will depend on the length z, and the design calculations have to be carried out by the stepwise integration of the rate and energy equations. The conversion resulting from each of the simultaneous reactions must be obtained separately. It is convenient in this problem to define the conversion as moles of chlorine reacted per mole of total feed rather than per mole of chlorine in the feed. If this conversion is (x,)i for reaction 1 and (x,)2 for 2, the mass-balance equation, Eq. (3-12), are... 

We next consider the mass balances for a mixture. In this case while total mass is conserved, there will be a change af mass of some or all species if one or more chemical reactions occur. For this case, it is more convenient to develop the mass balance for mixtures on a-molar basis, as chemical reaction stoichiomentry is much easier to write on a molar basis than on a mass basis. In this chapter we will consider only the ca.se of a single chemical reaction in later chapters the more general case of several chemical reactions occuring simultaneously will be considered. 

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