Heat Generation and Removal

A reactor will be isothermal at the feed inlet temperature Tq if (1) reactions do not generate or absorb significant heat or (2) the reactor is thermostatted by contact with a temperature bath at coolant temperature Tq. For any other situation we will have to solve the energy-balance equation long with the mass balance to find the temperature in the reactor. We therefore must set up these equations for our mixed and unmixed reactors. 

Before we develop the energy balances for our reactors, it is worthwhile to define some quantities from thermocfynamics because the energy balances we need are thermal energy balances. We begin with the First Law of Thermodynamics, 

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

In thermodynamics these quantities are usually expressed in energy per mole (quantities in bold), while we are interested in the rate of energy change or energy flow in energy per time. For these we replace the molar enthalpy H (cal/mole) of the fluid by the rate of enthalpy flow in a flow system 

For the energy balance in a flow system we therefore assume that we can make an enthalpy balance on the contents of the reactor. We can write the rate of enthalpy generation Aff in any flowing or closed system as 

Preliminary Guide for Selecting Size Reduction Equipment 

Equipment Max. Feed Size (mm) Min. Prod. Size (mm) Capacity (ton/day) Hardness Limit (Mohs) Applications Examples 

Roller mill 30 1 1 - 103 7.5 Cereals, vegetables, calcite, kaolin 

Hammer mill 40 0.01 1 - 103 4 Phosphates, pigments, dried fruits 

Disc attrition mill 12 0.07 1 - 103 3 Cellulose, asbestos, rubber 

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Setting Equation (5.32) equal to Equation (5.33) gives the general heat balance for a steady-state system. Figure 5.6(c) shows the superposition of the heat generation and removal curves. The intersection points are steady states. There are three in the illustrated case, but Figure 5.6(d) illustrates cases that have only one steady state. 

Oxidations of ammonia display ignition/extinction characteristics and auto-thermal reaction behavior. At low heat supply, only low conversion is observed and temperature remains nearly constant. With increasing heat supply and approaching a certain temperature, the reaction heat generated can no longer be transferred completely totally to the reactor construction material. At this stage, the reaction starts up . Suddenly, the temperature is raised by increased heat production until heat generation and removal are in balance. The reaction can now be carried out without a need for external heat supply, namely in autothermal mode. 

Recall from Chapter 6 that the steady-state energy-balance equations in a CSTR can be reduced to a single equation, which we wrote by considering the rates of heat generation and removal. We wrote these as... 

An example of an incredibly complex multiphase chemical reactor is iron ore refining in a blast furnace. As sketched in Figure 12-22, it involves gas, liquid, and solid phases in countercurrent flows with complex temperature profiles and heat generation and removal processes. 

The steady-state equations can be manipulated to take the form of heat generation and heat removal functions, i.e., a modified Van Heerden diagram. This manipulation can be carried out in different ways, all leading to the same results and here we choose to obtain the heat generation and removal functions of the regenerator as a function of the reactor... 

Figure 8-25 Heat generation and removal functions for feed mixture of 0.8 M Na7S203 and 1.2M H2OJ at 0°C. By S. A. Vejtasa and R. A. Schmitz, AlChE J., 16(3), 415, (1970). (Reproduced by permission of the American Institute of Chemical Engineers. Copynght 1970AIChE All nght reserved,) See Problem P8C-4,...

The rate of heat removal from the washcoat is proportional to the heat transfer coefficient and to the temperature difference between the gas and the washcoat, while the rate of heat generation is the product of the rate of mass transfer times the heat of reaction (-AH), as indicated by Eqs. (3) and (4) below, where kr is a first-order reaction rate constant. At the steady state the rates of heat generation and removal are equal. 

Fig. 3 Heat generated and removed at the inlet of a monolith combustor vs. temperature, calculated from Eqs. (3) and (4) for the conditions presented in Table 1. The straight lines represent the heat transfer curves in the absence of radiation losses. When the inlet gas temperature is 280 C, Eq. (5) is satisfied for three values of 297 C, 371 C, and 1326 C. As the temperature of the inlet gas is increased, the two lower intersection points approach each other and eventually both points merge at = 335 C when the inlet gas temperature is 292°C. This is referred to as the catalytic ignition or light-off temperature. A further increase in the inlet gas temperature results in a situation where there is only one intersection point. (View this art in color at www.dekker.com.)...

Still be very sensitive to a particular variable. On the other hand, an unstable condition is such that the least perturbation will lead to a finite change and such a condition may be regarded as infinitely sensitive to any operating variable. Sensitivity can be fully explored in terms of steady state solutions. A complete discussion of stability really requires the study of the transient equations. For the stirred tank this was possible since we had only to deal with ordinary differential equations for the tubular reactor the full treatment of the partial differential equations is beyond our scope here. Nevertheless, just as much could be learned about the stability of a stirred tank from the heat generation and removal diagram, so here something may be learned about stability from features of the steady state solution. 

Figure ff-22 Heat generation and removal Figure 8-23 Multiple steady states, functions for feed mixture of 0.8 M Na2StOj and l.2AfH20 atO C.

The stability problem for an exothermic reaction in a catalyst particle is similar to that for a reaction in a CSTR, in that multiple solutions of the heat and mass balance equations are possible. A typical plot of heat generation and removal rates is shown in Figure 5.11. The values of Qg and Qr are in cal/sec, g, and a is the external area in cm /g. The plot differs from the one for a CSTR (Fig. 5.2) in that the highest possible value for Qg is a mass transfer limit corresponding to Cj = 0 and not to complete conversion. The mass transfer limit increases with temperature because of the increase in diffusivity, and the limit also increases with gas velocity. The heat removal... 

Figure I Heat generation and removal functions for feed mixture of 0.8M NaiS O and I.2M H2O2 at 0°C (from Vejtassa and Schmitz [29]).

Experimental measurements were made of the several relevant variables so that an evaluation of the above criteria could be made. First, the steady-state heat generation and removal rates were determined as shown in Fig. 1. 

Heat generation and removal functions versus reactor temperature for a feed mixture of 0.8-MNa2S2O3 and I.2-MH2O2 at 0°C. From Vejtassa and Schmitz [1970]. 

Heat Generation and Removal in Nuclear Reactors which gives... 

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