Tubular energy balancing

Much of the basic theory of reaction kinetics presented in Sec. 7 of this Handbook deals with homogeneous reaclions in batch and continuous equipment, and that material will not be repeated here. Material and energy balances and sizing procedures are developed for batch operations in ideal stirred tanks—during startup, continuation, and shutdown—and for continuous operation in ideal stirred tank batteries and plug flow tubulars and towers. 

Figure 4.7. Energy balancing for the tubular plug-flow reactor. 

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. 

The coupling of the component and energy balance equations in the modelling of non-isothermal tubular reactors can often lead to numerical difficulties, especially in solutions of steady-state behaviour. In these cases, a dynamic digital simulation approach can often be advantageous as a method of determining the steady-state variations in concentration and temperature, with respect to reactor length. The full form of the dynamic model equations are used in this approach, and these are solved up to the final steady-state condition, at which condition... 

There are a variety of limiting forms of equation 8.0.3 that are appropriate for use with different types of reactors and different modes of operation. For stirred tanks the reactor contents are uniform in temperature and composition throughout, and it is possible to write the energy balance over the entire reactor. In the case of a batch reactor, only the first two terms need be retained. For continuous flow systems operating at steady state, the accumulation term disappears. For adiabatic operation in the absence of shaft work effects the energy transfer term is omitted. For the case of semibatch operation it may be necessary to retain all four terms. For tubular flow reactors neither the composition nor the temperature need be independent of position, and the energy balance must be written on a differential element of reactor volume. The resultant differential equation must then be solved in conjunction with the differential equation describing the material balance on the differential element. 

The material and energy balances of a tubular vessel are based on the conservation law, Eq 2.42, applied to a differential ring between r and r+dr and z and z+dz. A material balance is derived, for example, in problem P5.08.01, and is quoted in Table 2.6 along with the heat balance. The result is a pair of second order partial differential equations, usually nonlinear, that must be solved numerically. Table 2.6 indicates one possible procedure, but computer software is plentiful. 

These are the fundamental thermodynamic equations from which we can develop our energy balances in batch, stirred, and tubular reactors. 

The combined stream is preheated to 122°C in a FEHE. A heater (HX3) is installed after the FEHE so that inlet temperature of the coolant stream in REACT2 can be adjusted to satisfy the energy balance when the exit temperature of the coolant stream is specified in this countercurrent tubular reactor. This temperature is 150°C, and the heat load in HX3 is 9.34 x 106 kcal/h. The stream is further preheated to 265°C in the tube side of reactor REACT2 by the heat transfer from the reactions that are occurring in the hot shell side of this vessel. There is no catalyst on the cold tube side, so the feed stream does not react but its temperature is increased. The stream is then fed to reactor REACT 1, which contains 48,000 kg of catalyst. This reactor is cooled by generating steam. The coolant temperature is 265°C (51 bar steam). This vessel contains 3750 tubes, 0.0375 m in diameter, and 12.2 m in length. The overall heat transfer coefficient between the process gas and the steam is 244 kcal h-1 m-2 °C 1. The heat transfer rate is 42 x 106 kcal/h. 

The energy balance for a tubular reactor is PuCpdT-u CAO( )dXA = U7tdt(TE-T)dl... 

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 ... 

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... 

Consider a PFR operating at nonisothermal conditions (refer to Figure 9.4.1). To describe the reactor performance, the material balance. Equation (9.1.1), must be solved simultaneously with the energy balance. Equation (9.2.7). Assuming that the PFR is a tubular reactor of constant cross-sectional area and that T and C, do not vary over the radial direction of the tube, the heat transfer rate Q can be written for a differential section of reactor volume as (see Figure 9.4.1) ... 

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) ... 

Consider a tubular fixed-bed reactor accomplishing a highly exothermic gas-phase reaction. Assuming that axial dispersion can be neglected, the mass and energy balances can be written as follows and allow for radial gradients ... 

In this section we apply die general energy balance [Equation (8-22)] to the CSTR and to the tubular reactor operated at steady state. We then present example problems showing how the mole and energy balances are combined to size reactors operating adiabadcally. 

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. 

Consider an adiabatic tubular reactor (Davis, 1984)[15] with the following data length L = 2 m, radius Rp = 0.1 m, inlet reactant concentration cO = 30 moles/m3, inlet temperature TO = 700K, enthalpy AH = -10000 J/mole, specific heat capacity Cp = 1000 J/kg/K, activation energy Ea = 100 J/mole, p = 1200 kg/m3, velocity uO = 3 m/s, and rate constant kO = 5 s-1. Dimensionless concentration (y) and dimensionless temperature (9) are governed by material and energy balances as ... 

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