Mass balance tubular reactor

Steady-state reactors with ideal flow pattern. In an ideal isothermal tubular pZi/g-yZovv reactor (PFR) there is no axial mixing and there are no radial concentration or velocity gradients (see also Section 5.4.3). The tubular PFR can be operated as an integral reactor or as a differential reactor. The terms integral and differential concern the observed conversions and yields. The differential mode of reactor operation can be achieved by using a shallow bed of catalyst particles. The mass-balance equation (see Table 5.4-3) can then be replaced with finite differences ... 

The component mass balance equation, combined with the reactor energy balance equation and the kinetic rate equation, provide the basic model for the ideal plug-flow tubular reactor. 

Figure 4.9. Mass balancing for a gas-phase, tubular reactor.

At steady-state conditions, the mass balance design equations for the ideal tubular reactor apply. These equations may be expressed as... 

In Section 11.1.3.1 we considered the longitudinal dispersion model for flow in tubular reactors and indicated how one may employ tracer measurements to determine the magnitude of the dispersion parameter used in the model. In this section we will consider the problem of determining the conversion that will be attained when the model reactor operates at steady state. We will proceed by writing a material balance on a reactant species A using a tubular reactor. The mass balance over a reactor element of length AZ becomes ... 

In this liquid phase reaction, it may be assumed that the mass density of the liquid is unaffected by the reaction, allowing the material balance for the tubular reactor to be applied on a volume basis (Section 1.7.1, Volume 3) with plug flow. 

Figure 2-9 Sketch showing correspondence between time in a batch reactor and position a in a plug-flow tubular reactor. The mass-balance equations describe both reactors for the constant-density situations.

We can therefore replace dt by dz/u in all of the preceding differential equations for the mass balance in the batch reactor and use these equations to describe reactions during flow through a pipe. This reactor is called the plug-flow tubular reactor, which is the most important continuous reactor encountered in the chemical industry. 

To illustrate the problem of thermal sensitivity we will analyse the simple one-dimensional model of the countercurrent cooled packed tubular reactor described earlier and illustrated in Fig. 3.25. We have already seen that the mass and heat balance equations for the system may be written ... 

To illustrate we consider a homogeneous tubular reactor. The simplest model is given by plug flow and the design equations are obtained from the mass-balance equations by taking the mass balance over an element of length Al. This is expressed in the formula... 

The reaction is also influenced by the heat of reaction developing during the conversion of the reactants, which is a problem in tubular screening reactors. In microstructures, the heat transport through the walls of the channels is facilitated by their small dimensions, which allows the development of isothermal reaction conditions. Thus, by decoupling the heat and mass balance, an analytical description of the flow in the screening reactor is achievable. 

In a tubular reactor, the reactants are fed in at one end and the products withdrawn from the other. If we consider the reactor operated at steady state, the composition of the fluid varies inside the reactor volume along the flow path. Therefore, the mass balance must be established for a differential element of volume dV. We assume the flow as ideal plug flow, that is, that there is no back mixing along the reactor axis. Hence, this type of reactor is often referred to as Plug Flow Reactor (PFR). 

Industrial design problems often occur in tubular reactors that involve the simultaneous solution of AP, energy, and mass balances. 

Consider a steady flow of reactant A to products at constant density through an element of radius r, width 8r, and height 81 in a tubular reactor at isothermal condition. Suppose that radial and axial mass transfer is expressed by Fick s law, with (D and (De)r as effective diffusivities. The rate at which A reacts is (-rA), mol/m3 sec. A material balance on a tubular element of radii r and r + 8r and height 81 is carried out from... 

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 ... 

Plug Flow Reactor A plug flow reactor (PFR) is an idealized tubular reactor in which each reactant molecule enters and travels through the reactor as a plug, i.e., each molecule enters the reactor at the same velocity and has exactly the same residence time. As a result, the concentration of every molecule at a given distance downstream of the inlet is the same. The mass and energy balance for a differential volume between position Vr and Vr + dVr from the inlet may be written as partial differential equations (PDEs) for a constant-density system ... 

Tubular Reactor with Dispersion An alternative approach to describe deviation from ideal plug flow due to backmixing is to include a term that allows for axial dispersion De in the plug flow reactor equations. The reactor mass balance equation now becomes... 

For the same type of catalyst we have observed in a recirculation laboratory reactor multiplicity, periodic and chaotic behavior. Unfortunately, so far we are not able to suggest such a reaction rate expression which would be capable of predicting all three regimes [8]. However, there is a number of complex kinetic expressions which can describe periodic activity. One can expect that such kinetic expressions combined with heat and mass balances of a tubular nonadiabatic reactor may give rise to oscillatory behavior. Detailed calculations of oscillatory behavior of singularly perturbed parabolic systems describing heat and mass transfer and exothermic reaction are apparently beyond, the capability of both standard current computers and mathematical software. 

Each zone of the tubular reactor is simulated as a sequence of N perfectly mixed elementary volumes, as shown in Figure 2. Each volume can receive a feed side-stream, and exchanges heat with a corresponding volume in the oil jacket. For volume i and chemical species j (either initiator, oxygen, radicals or monomer) the mass balance is written as ... 

Hydroprocessing studies were carried out in continuous fixed-bed tubular reactors containing 50-200 cm3 of catalyst. All reactors were operated in downflow mode. To facilitate mass balance data acquisition,... 

Numerous reactions are performed by feeding the reactants continuously to cylindrical tubes, either empty or packed with catalyst, with a length which is 10 to 1000 times larger than the diameter. The mixture of unconverted reactants and reaction products is continuously withdrawn at the reactor exit. Hence, constant concentration profiles of reactants and products, as well as a temperature profile are established between the inlet and the outlet of the tubular reactor, see Fig. 7.1. This requires, in contrast to the batch reactor, the application of the law of conservation of mass over an infinitesimal volume element, dV, of the reactor. In contrast to a batch reactor the existence of a temperature profile does not allow us to consider the mass balances for the reacting components and the energy balance separately. Such a separation can only be performed for isothermal tubular reactors. 

Example 9.11 Modeling of a nonisothermal plug flow reactor Tubular reactors are not homogeneous, and may involve multiphase flows. These systems are called diffusion convection reaction systems. Consider the chemical reaction A -> bB described by a first-order kinetics with respect to the reactant A. For a nonisothermal plug flow reactor, modeling equations are derived from mass and energy balances... 

These basic rate models were Incorporated Into a differential mass balance In a tubular, plug-flow reaction. This gives a set of coupled, non-llnear differential equations which, when Integrated, will provide a simulation model. This model corresponds to the Integral reactor data provided by experimentation. A material balance Is written for each of the four components In our system ... 

Consider first the tubular reactor. From the material balance (Table 3.5.1), it is clear that in order to solve the mass balance the functional form of the rate expression must be provided because the reactor outlet is the integral result of reaction over the volume of the reactor. However, if only initial reaction rate data were required, then a tubular reactor could be used by noticing that if the differentials are replaced by deltas, then ... 

An axially-dispersed, adiabatic tubular reactor can be described by the following mass and energy balances that are in dimensionless form (the reader should verify that these descriptions are correct) ... 

For the isothermal tubular plug-flow reactor (PFR) discussed previously, the mass balance for the G gaseous components is... 

Though in later applications we may return to the concentration unit of moles per unit volume, let us take the opportunity, in discussing the tubular reactor, to use the unit of moles per unit mass. In this we follow Amundson (1958), whose work has done so much to set chemical reactor design on a sound analytical basis. We shall still assume that the flow is uniform and that there is no longitudinal diffusion. Thus G, the flow rate in mass per unit area per unit time, is constant throughout the reactor under all circumstances. The linear velocity v and the density p may vary, but their product is constant, pv = G. Then a mass balance of A, over an element of length yields the differential equation... 

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 contrast to a batch reactor, the existence of a temperature profile does not allow us to consider the mass balances for the reacting components and the energy balance separately. Such a separation can only be performed for isothermal tubular reactors. 

In all these cases, the correct design must grow from the equations of mass, energy, and momentum balance to which we now turn in the next few sections. From these we proceed to the design problem (Sec. 9.5) and hence to elementary considerations of optimal design (Sec. 9.6). The stability and sensitivity of a tubular reactor is a vast and fascinating subject. Since the steady state equations are ordinary differential equations, the equations describing the transient behavior are partial differential equations. This... 

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