Adiabatic conditions, steady-state

The temperature rise due to this exothermic reaction then approaches the adiabatic temperature rise. The final steady state is always characterized by conditions T = T, and c = 0. A batch reactor, in which a zero order reaction is carried out, always has a unique and stable mode of operation. This is also true for any batch and semibatch reactor with any order or combination of reactions. 

The final equality follows from the normalization of the conditional stochastic transition probability. This is the required result, which shows the stationarity of the steady-state probability under the present transition probability. This result invokes the preservation of the steady-state probability during adiabatic evolution over intermediate time scales. 

Enhancement of a rate by temperature can counteract the effect of falling concentration. Exothermic reaction rates in pores, as a consequence, can be much greater than at the surface condition. Another peculiarity that can arise with adiabatic reactions is multiple steady states. 

Fig. 10. Adiabatic and nonadiabatic steady-state Profiles, type I conditions.

Another type of stability problem arises in reactors containing reactive solid or catalyst particles. During chemical reaction the particles themselves pass through various states of thermal equilibrium, and regions of instability will exist along the reactor bed. Consider, for example, a first-order catalytic reaction in an adiabatic tubular reactor and further suppose that the reactor operates in a region where there is no diffusion limitation within the particles. The steady state condition for reaction in the particle may then be expressed by equating the rate of chemical reaction to the rate of mass transfer. The rate of chemical reaction per unit reactor volume will be (1 - e)kCAi since the effectiveness factor rj is considered to be unity. From equation 3.66 the rate of mass transfer per unit volume is (1 - e) (Sx/Vp)hD(CAG CAl) so the steady state condition is ... 

In the first case (Figure 8a), the side walls are adiabatic, and the reactor height (2 cm) is low enough to make natural convection unimportant. The fluid-particle trajectories are not perturbed, except for the gas expansion at the beginning of the reactor that is caused by the thermal expansion of the cold gas upon approaching the hot susceptor. On the basis of the mean temperature, the effective Rayleigh number, Rat, is 596, which is less than the Rayleigh number of 1844 necessary for the existence of a two-dimensional, stable, steady-state solution with flow in the transverse direction that was computed for equivalent Boussinesq conditions (188). 

Was the column cross-sectional area sufficient to effect essentially adiabatic behavior of the sorption fronts, and, if not, how much would the steady-state MTZ lengths change between operation at fully adia- batic conditions and operation at fully isothermal conditions ... 

For a resolution of question (3), either MASC or the simpler SSHTZ program was run under both isothermal and adiabatic conditions, with effective mass transfer coefficients chosen to simulate the stable portion of the sorption fronts. Fortunately, in most cases described below, the programs predicted that the steady-state MTZ lengths did not change by more than 10Z or so between the two extremes. Thus, an extensive analysis of the wall effects in the various columns was not required for proper interpretation of MTZ data. 

Write the steady-state mass and heat balance equations for this system, assuming constant physical properties and constant heat of reaction. (Note Concentrate your modeling effort on the adiabatic nonisothermal reactor, and for the rest of the units, carry through a simple mass and heat balance in order to define the feed conditions for the reactor.)... 

Tubular reactors often have high-temperature limitations because of the occurrence of undesirable reactions, catalyst degradation, or materials of construction. This means that the maximum temperature anywhere in the reactor cannot exceed this limit. An exothermic reaction in an adiabatic reactor produces a maximum temperature at the exit under steady-state conditions. An exothermic reaction in a cooled reactor can... 

There are five fundamental differences between CSTRs and tubular reactors. The first is the variation in properties with axial position down the length of the reactor. For example, in an adiabatic reactor with an exothermic irreversible reaction, the maximum temperature occurs at the exit of the reactor under steady-state conditions. However, in a cooled tubular reactor, the peak temperature usually occurs at an intermediate axial position in the reactor. To control this peak temperature, we must be able to measure a number of temperatures along the reactor length. 

The ordinary differential equations describing a steady-state adiabatic PFR can be written with axial length z as the independent variable. Alternatively the weight of catalyst w can be used as the independent variable. There are three equations a component balance on the product C, an energy balance, and a pressure drop equation based on the Ergun equation. These equations describe how the molar flowrate of component C, temperature T, and the pressure P change down the length of the reactor. Under steady-state conditions, the temperature of the gas and the solid catalyst are equal. This may or may not be true dynamically ... 

A maximum reactor temperature of 500 K is used in this study. This maximum temperature occurs at the exit of the adiabatic reactor under steady-state conditions. Plug flow is assumed with no radial gradients in concentrations or temperatures and no axial diffusion or conduction. 

If the number of reactors is reduced, the interaction is less and the system can be made closedloop-stable. This is illustrated by considering a system with three adiabatic cold-shot reactors. The optimum steady-state design of the three-stage system is studied (steady-state conditions are given in Fig. 6.3). 

Consider an exothermic irreversible reaction with first order kinetics in an adiabatic continuous flow stirred tank reactor. It is possible to determine the stable operating temperatures and conversions by combining both the mass and energy balance equations. For the mass balance equation at constant density and steady state condition,... 

The choice of the saturation gas is critical. When Ar and Kr were sparged in water irradiated at 513 kHz, an enhancement in the production of OH radicals of between 10% and 20%, respectively, was observed, compared with 02-saturated solutions [22]. The higher temperatures achieved with the noble gases upon bubble collapse under quasi-adiabatic conditions account for the observed difference. Because the rate of sonochemical degradation is directly linked to the steady state concentration of OH radicals, the acceleration of those reactions is expected in the presence of such background gases. The use of ozone as saturation gas (in mixtures with 02) provided new reaction pathways in the gas phase inside the bubbles, which also increase the measured reaction rates (see Sect. IV.G.l). 

Linearized or asymptotic stability analysis examines the stability of a steady state to small perturbations from that state. For example, when heat generation is greater than heat removal (as at points A— and B+ in Fig. 19-4), the temperature will rise until the next stable steady-state temperature is reached (for A— it is A, for B+ it is C). In contrast, when heat generation is less than heat removal (as at points A+ and B— in Fig. 19-4), the temperature will fall to the next-lower stable steady-state temperature (for A+ and B— it is A). A similar analysis can be done around steady-state C, and the result indicates that A and C are stable steady states since small perturbations from the vicinity of these return the system to the corresponding stable points. Point B is an unstable steady state, since a small perturbation moves the system away to either A or C, depending on the direction of the perturbation. Similarly, at conditions where a unique steady state exists, this steady state is always stable for the adiabatic CSTR. Hence, for the adiabatic CSTR considered in Fig. 19-4, the slope condition dQH/dT > dQG/dT is a necessary and sufficient condition for asymptotic stability of a steady state. In general (e.g., for an externally cooled CSTR), however, the slope condition is a necessary but not a sufficient condition for stability i.e., violation of this condition leads to asymptotic instability, but its satisfaction does not ensure asymptotic stability. For example, in select reactor systems even... 

In adiabatically operated industrial hydrogenation reactors temperature hot spots have been observed under steady-state conditions. They are attributed to the formation of areas with different fluid residence time due to obstructions in the packed bed. It is shown that in addition to these steady-state effects dynamic instabilities may arise which lead to the temporary formation of excess temperatures well above the steady-state limit if a sudden local reduction of the flow rate occurs. An example of such a runaway in an industrial hydrogenation reactor is presented together with model calculations which reveal details of the onset and course of the reaction runaway. 

As a consequence of this explanation the reaction runaway to total methanation is not a necessary condition for the observed phenomenon. Any simple exothermic two phase reaction in an adiabatic reactor ought to show the same behaviour provided that one phase with a high throughput is used to carry the heat out of the reactor and the flow is suddenly reduced. This will be shown in the following simulation results. Due to problems with the numerical stability of the solution (see Apendix) only a moderate reaction rate will be considered. Reaction parameters are chosen in such a way that in steady state the liquid concentration Cf drops from 4.42 to 3.11 kmol/m3 but the temperature rise is only 3°C (hydrogen in great excess). At t = 0 the uniform flow profile... 

Therefore, the total entropy produced within the system must be discharged across the boundary at stationary state. For a system at stationary state, boundary conditions do not change with time. Consequently, a nonequilibrium stationary state is not possible for an isolated system for which deS/dt = 0. Also, a steady state cannot be maintained in an adiabatic system in which irreversible processes are occurring, since the entropy produced cannot be discharged, as an adiabatic system cannot exchange heat with its surroundings. In equilibrium, all the terms in Eq. (3.48) vanish because of the absence of both entropy flow across the system boundaries and entropy production due to irreversible processes, and we have dJS/dt = d dt = dS/dt = 0. 

Steam expands in a nozzle from inlet conditions of 500°F, 250 psia, and a velocity of 260ft/s to discharge conditions of 95 psia and a velocity 1500 ft/s. If the flow is at lOlb/s and the process is at steady state and adiabatic, determine ... 

The primary reason for choosing a particular reactor type is the influence of mixing on the reaction rates. Since the rates affect conversion, yield, and selectivity we can select a reactor that optimizes the steady-state economics of the process. For example, the plug-flow reactor has a smaller volume than the CSTR for the same production rate under isothermal conditions and kinetics dominated by the reactant concentrations. The opposite may be true for adiabatic operation or autocata-lytic reactions. For those situations, the CSTR would have the smaller volume since it could operate at the exit conditions of a plug-flow reactor and thus achieve a higher overall rate of reaction. 

The preceding discussion points up a matter of tacit understanding. It is presumed that the reader possesses an intuitive grasp of the concept of mass and of mechanical variables, such as volume V or pressure P. By contrast, items relating to the transfer of energy must be very carefully explained through the various laws of thermodynamics before they are utilized. We shall also be able at a later point to supply better definitions for adiabatic and diathermic processes, and for steady-state conditions. 

---