Mole balances reaction rate

To obtain a plot of heat generated, G(T), as a function of temperature, we must solve for X as a function of T using the CSTR mole balance, the rate law, and stoichiometry. For example, for a first-order liquid-phase reaction, the CSTR mole balance becomes... 

For a packed-bed reactor, the approach is quite similar to that described for a CSTR. For a first-order reaction, the combined mole balance and rate law is... 

One can model the clotting process in a manner identical to the series reactions by writing a mole balance and rate law for each species such as... 

This method is the quickest method to use to determine the rate law if (he order turns out to zero, first, or second order. In the integral method, we guess the reaction order, a, in the combined batch reactor mole balance and rate law equation... 

We will now apply nonlinear regres.sion to reaction rate data to determine the rate law parameters. Here we make initial estimates of the parameter values (e.g.. reaction order, specific rate constant) in order to calculate the concentration for each data point. Cj,., obtained by. solving an integrated fonn of the combined mole balance and rate law. We then compare the measured concentration at that point. C(, . with the calculated value, for the parameter values chosen. We make this comparison by calculating the sum of the. squares of the differences at each point S(C, —We then continue to choose new parameter values and search for those values of the rate law that will minimize the sum of the squared differences of the measured concentrations. Cm,. and the calculated concentrations values, C,v.. That is, we want to find the rate law parameters for which the sum of all data points S(C, — C,) is a minimum. If we carried out N experiments, we would want to find the parameter values (e.g , activation energy, reaction orders) that minimize the quantity... 

In this chapter, we discuss reactor selection and general mole balances, net rates and relative rales for mulUpie reactions. 

Solution The obvious way to solve this problem is to choose a pressure, calculate Oq using the ideal gas law, and then conduct a batch reaction at constant T and P. Equation (7.38) gives the reaction rate. Any reasonable values for n and kfCm. be used. Since there is a change in the number of moles upon reaction, a variable-volume reactor is needed. A straightforward but messy approach uses the methodology of Section 2.6 and solves component balances in terms of the number of moles, Na, Nb, and Nc-... 

The reversible reaction A 2B is conducted at 540°F and 3 atma in a tubular-flow reactor. The feed contains 30 mole % A and the balance inert material, the total being at the rate of 75 lb moles/hr. The rate equation is... 

Finally, an algebraic model relationship is included in order to check on the total component material balance achieved in the simulation. The last lines specify the chemical reaction rate terms and calculate the total number of moles present at any time during the reaction. 

For isothermal, first-order chemical reactions, the mole balances form a system of linear equations. A non-ideal reactor can then be modeled as a collection of Lagrangian fluid elements moving independe n tly through the system. When parameterized by the amount of time it has spent in the system (i.e., its residence time), each fluid element behaves as abatch reactor. The species concentrations for such a system can be completely characterized by the inlet concentrations, the chemical rate constants, and the residence time distribution (RTD) of the reactor. The latter can be found from simple tracer experiments carried out under identical flow conditions. A brief overview of RTD theory is given below. 

Stoichacmetry and reaction equilibria. Homogeneous reactions kinetics. Mole balances batch, continuous-shn-ed tank and plug flow reactors. Collection and analysis of rate data. Catalytic reaction kinetics and isothermal catalytic radar desttpi. Diffusion effects. 

In catalytic reactors we assume that there is no reaction in the fluid phase, and all reaction occurs on the surface of the catalyst. The surface reaction rate has the units of moles per unit area of catalyst per unit time, which we will call r". We need a homogeneous rate r to insert in the mass balances, and we can write this as... 

Note that the reaction rate is in (moles/m3 solid) (1/s), and thus by multiplying with ss m3 solid/m3 reactor) we obtain the material balance in (moles/m3 reactor) (1/s). 

The measurement of a small concentration gradient requires more analytical work, and often gives less accurate kinetic data. For this reason, in the differential recycle reactor a fraction of the reaction mixture leaving a thin catalyst bed is recycled and added again to the feed (Fig. 3.3-4). This results in a larger difference of concentration, c0, or mole fraction, x , between the feed and c or x at the reactor outlet, which is used to determine the reaction rate from the material balance ... 

When quoting a reaction rate, it s important to specify the reactant or product on which the rate is based because the rates of product formation and reactant consumption may differ, depending on the coefficients in the balanced equation. For the decomposition of N205,4 mol of N02 form and 2 mol of N205 disappear for each mole of 02 that forms. Therefore, the rate of formation of 02 is one-fourth the rate of formation of N02 and one-half the rate of decomposition of N205 ... 

Kinetic models utilize a set of algebraic or differential equations based on the mole balances of the main species involved in the process (ozone in water and gas phases, compounds that react with ozone, presence of promoters, inhibitors of free radical reactions, etc). Solution of these equations provides theoretical concentration profiles with time of each species. Theoretical results can be compared with experimental results when these data are available. In some cases, kinetic modeling allows the determination of rate constants by trial and error procedures that find the best values to fit the... 

The governing equations - that is, mainly the component and the total mass balances in the anode channels - are provided here in dimensionless form. The five ordinary differential equations (ODE) with respect to the spatial coordinate describe the development of the five unknowns in one single anode channel, namely the mole fractions, with i = CH4, H2O, H2, CO2, as well as the molar flow density inside the anode channel, y. Here, the Damkohler numbers, Da/, are the dimensionless reaction rate constant of the reforming and the oxidation reaction, respectively, the rj are the corresponding dimensionless reaction rates, and the v, j are the stoichiometric coefficients ... 

A final simplification of the rate laws can be made by incorporating mole-balance conditions that follow from the stoichiometry of the reactions in Eq. 1.52 (cf. Eq. 1.29) ... 

The atomic processes that are occurring (under conditions of equilibrium or non equilibrium) may be described by statistical mechanics. Since we are assuming gaseous- or liquid-phase reactions, collision theory applies. In other words, the molecules must collide for a reaction to occur. Hence, the rate of a reaction is proportional to the number of collisions per second. This number, in turn, is proportional to the concentrations of the species combining. Normally, chemical equations, like the one given above, are stoichiometric statements. The coefficients in the equation give the number of moles of reactants and products. However, if (and only if) the chemical equation is also valid in terms of what the molecules are doing, the reaction is said to be an elementary reaction. In this case we can write the rate laws for the forward and reverse reactions as Vf = kf[A]"[B]6 and vr = kr[C]c, respectively, where kj and kr are rate constants and the exponents are equal to the coefficients in the balanced chemical equation. The net reaction rate, r, for an elementary reaction represented by Eq. 2.32 is thus... 

A differential characteristic which demands a lower degree of standardization is the reaction rate. The rate of a chemical reaction with respect to compound B at a given point is defined as the rate of formation of B in moles per unit time per unit volume. It cannot be measured directly and is determined from the rates of change of some observable quantities such as the amount of substance, concentration, partial pressure, which are subject to measurements. Reaction rates are obtained from observable quantities by use of the conservation equations resulting from the mass balance for the given reactor type. 

Eqs. 1 to 3 relate the rate of production Rj of the balanced reaction component y to the molar amounts or their derivatives with respect to the time variable (reaction time or space time, see above). From the algebraic eq. 2 for the CSTR reactor the rate of production, Rj, may be calculated very simply by introducing the molar flow rates at the inlet and outlet of the reactor these quantities are easily derived from the known flow rate and the analytically determined composition of the reaction mixture. With a plug-flow or with a batch reactor we either have to limit the changes of conversion X or mole amount n7 to very low values so that the derivatives or dAy/d( //y,0) or dn7/d/ could be approximated by differences AXj/ (Q/Fj,0) or An7/A, (differential mode of operation), or to measure experimentally the dependence of Xj or nj on the space or reaction time in a broader region this dependence is then differentiated graphically or numerically. 

Related Calculations. For a batch reactor, the material balance is Rate of accumulation of species A = rate of generation of species A, or dNA/dt = rA, where N is number of moles at time t and r is rate of reaction (which can be, for example, per unit of catalyst mass in the reactor, in which case it must be multiplied by the number of such units present). The rate at any given time can be found by plotting Na against residence time and measuring the slope, but this technique can lead to large errors. A better approach is to use the Taylor-series interpolation formula (see mathematics handbooks for details). 

Hie chemical reaction rate is usually dependent on the molar concentrations of the reactants and not on their mass fractions, because it depends on the chance of collision of molecules. However, here the definition of in terms of mass fractions is preferred, because it can readily be incorporated into mass balances. A definition in terms of moles or molar concentrations might invite the use of mole balances instead of mass balances. Since, contrary to conservation of mass, there is no such thing as conservation of moles (because one molecule might divide into several molecules, or several might condense into one), the use of mole balances is strongly dissuaded. More information concerning the definition of conversion can be found elsewhere [2]. 

If U = -1, then the reaction rate is equal to the number of moles of the limiting reactant fed to the reactor per unit time and per unit volume of the reacting fluid times the fractional conversion. For any product p not present in the feed stream, a material balance on p is easily obtained from Equation (3.3.1) with F = 0 to give ... 

The ethanol dehydrogenation reaction of Example 9.5-3 is carried out with the feed entering at 300°C. The feed contains 90.0 mole% ethanol and the balance acetaldehyde and enters the reactor at a rate of 150 mol/s. To keep the temperature from dropping too much and thereby decreasing the reaction rate to an unacceptably low level, heat is transferred to the reactor. When the heat addition rale is 2440 kW, the outlet temperature is 253 C. Calculate the fractional conversion of ethanol achieved in the reactor. 

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