Continuity and Energy Equations

The continuity equation for the key reacting component A and the energy equation can now be written as follows, for a single reaction and steady state  

Note that the term accounting for effective transport in the axial direction has been neglected in this model, for the reasons already given in Section 11.6. This system of nonlinear second-order partial differential equations was integrated by Froment [1961, 1967] using a Crank-Nichokon procedure, to simulate a multitubular fixed bed reactor for a reaction involving yield problems. 

Mihail and lordache [145] compared the performance of some numerical techniques for integrating the system (11.7.b-l) Liu s average explicit scheme with 

---

The following example illustrates the derivation of the continuity and energy equations for model 1.2.2 in Figure 21.5, a pseudohomogeneous, two-dimensional model... 

Derive the continuity and energy equations for an FBCR model based on pseudohomogeneous, two-dimensional, DPF considerations. State any assumptions made, and include the boundary conditions for the equations. 

The starting points for the continuity and energy equations are again 21.5-1 and 21.5-6 (adiabatic operation), respectively, but the rate quantity7 (—rA) must be properly interpreted. In 21.5-1 and 21.5-6, the implication is that the rate is the intrinsic surface reaction rate, ( rA)int. For a heterogeneous model, we interpret it as an overall observed rate, (—rA)obs, incorporating the transport effects responsible for the gradients in concentration and temperature. As developed in Section 8.5, these effects are lumped into a particle effectiveness factor, 77, or an overall effectiveness factor, r]0. Thus, equations 21.5-1 and 21.5-6 are rewritten as... 

The fundamental physical laws governing motion of and transfer to particles immersed in fluids are Newton s second law, the principle of conservation of mass, and the first law of thermodynamics. Application of these laws to an infinitesimal element of material or to an infinitesimal control volume leads to the Navier-Stokes, continuity, and energy equations. Exact analytical solutions to these equations have been derived only under restricted conditions. More usually, it is necessary to solve the equations numerically or to resort to approximate techniques where certain terms are omitted or modified in favor of those which are known to be more important. In other cases, the governing equations can do no more than suggest relevant dimensionless groups with which to correlate experimental data. Boundary conditions must also be specified carefully to solve the equations and these conditions are discussed below together with the equations themselves. 

Since all properties have been assumed constant in Eqs. (1-1), (1-38), and (1-47), and the solute concentration has been assumed small, the Navier-Stokes equation may be solved independently of the species continuity and energy equations. We treat only one exception where the velocity field is considered to be affected by heat or mass transfer. This exception, natural convection, is covered in Chapter 10. 

The relationship (7.4) can also be derived, if the equation of motion (Navier-Stokes differential equations) are drawn up and dimensionlessly formulated under given boundary conditions (here the continuity and energy equations). W. Nusselt followed this path (1909/1915). The thus derived pi-numbers were later named by... 

Derive the steady-state continuity and energy equations and appropriate boundary conditions for the tubular reactor with turbulent flow, corresponding to the various situations represented in the following diagram (from Himmelblau and Bischoff [3]). 

For simple irreversible reactions a (semi) analytical solution of the continuity and energy equations is possible. Douglas and Eagleton [9] published solutions for zero-, first-, and second-order reactions, both with a constant and varying... 

The space velocity, often used in the technical literature, is the total volumetric feed rate under normal conditions, F o(Nm /hr) per unit catalyst volume (m X that is, PbF o/W. It is related to the inverse of the space time W/F g used in this text (with W in kg cat. and F q in kmol A/hr). It is seen that, for the nominal space velocity of 13,800 (m /m cat. hr) and inlet temperatures between 224 and 274 C, two top temperatures correspond to one inlet temperature. Below 224 C no autothermal operation is possible. This is the blowout temperature. By the same reasoning used in relation with Fig. 11.5.e-2 it can be seen that points on the left branch of the curve correspond to the unstable, those on the right branch to the upper stable steady state. The optimum top temperature (425°C), leading to a maximum conversion for the given amount of catalyst, is marked with a cross. The difference between the optimum operating top temperature and the blowout temperature is only 5°C, so that severe control of perturbations is required. Baddour et al. also studied the dynamic behavior, starting from the transient continuity and energy equations [26]. The dynamic behavior was shown to be linear for perturbations in the inlet temperature smaller than 5°C, around the conditions of maximum production. Use of approximate transfer functions was very successful in the description of the dynamic behavior. 

The continuity and energy equations for a number of models listed in Table 12.1 are included in Table 12.2 along with the methods of solution to be used. All of... 

Considerable, but not complete, information on the stability of a steady state can be derived from the combined steady-state continuity and energy equations. Therefore, (10.2.2-1) is considered simultaneously with the steady-state form of (10.2.1-3) ... 

Solve the set of species continuity and energy equations using a fourth-order Runge-Kutta routine. 

The PRC s are not strongly temperature-dependent. They can be calculated off-line for every 20 C in the interval encountered in the reactor and stored. This saves considerable computer time in the integration of the continuity and energy equations. 

---