Time-conversion relationships, reaction equation

Definition of the maxima of intermediates Effectiveness of this method has already been demonstrated for the Type III system and arises from the fact that the net rate of reaction of intermediate is zero at the maximum, allowing one to work with the rate equation directly, rather than the integral time-conversion relationship. This is addressed in a number of the exercises given later. 

Since the various mole numbers can be expressed in terms of the extent of reaction, equation 10.2.9 expresses the relationship that must exist between the extent of reaction at time t and the temperature at that time. In terms of the fraction conversion where the fraction conversion at zero time is taken as zero,... 

Develop a batch-reactor design equation from the mass balance. To find the required holding time, a relationship between reaction time and the rate of conversion of acetylene must be developed. This may be developed from a mass balance on the batch reactor. Since the molar density of the reacting mixture is not constant (there is a net change in the number of moles due to reaction), the pressure of the reactor will have to change accordingly. 

Whereas the calculation of the time to gelation is relatively simple, the calculation of the time to vitrification (tyu) is not so elementary. The critical point is to obtain a relationship between T, and the extent of conversion at T, (Pvu)- Once the conversion at Tg is known, then the time to vitrification can be calculated from the kinetics of the reaction. Two approaches have been examined one calculates tyu based on a relationship between T, and Pyj, in conjunction with experimental values of Pvit the other approach formulates the Tg vs. pyj, relationship from equations in the literature relating Tg to molecular weight and molecular weight to extent of reaction... 

In the case of an isothermal mode of operation for a measurement and provided that the reference temperature is identical with the isothermal test temperature, this equivalent reaction time automatically becomes identical with the true reaction time. In the case of a none-isothermal mode of operation or the choice of a reference temperature which does not correspond to an isothermal test temperature, all thermal conversion data X(t) are plotted over h(t). In order to obtain the different h(t) data, a first estimation of the activation temperature has to be made. If the activation temperature was correctly chosen and if, at the same time, the process can be described with a single gross reaction equation with sufficient accuracy, then all the different data sets plotted over h(t) must take an identical course. The first estimate on the activation temperature can be obtained from the slope of the linearized functional relationship... 

Separating variables and integrating Equation fB.8.51 yield the conversion as a function of batch reaction time. This relationship is shown in Figure B.8.2. 

The relationship between the conversion of heavy oil and the time factor, WIF, at different reaction temperatures is shown in Figure 6.12. Conversion was defined as the mass fraction of heavy oil (components above C12) converted to gasoline, gas, and coke. This value was calculated from the following equation ... 

Equation (3-11) relates the temperature to the reaction variables and, with Eq. (3-9), establishes the relationships of conversion, temperature, and time for nonisothermal operation. 

In the next section various reaction models are considered. Then global rate equations are developed in Sec. 14-3 for one model (shrinking core). In Sec. 14-4 integrated conversion-vs-time relationships (for single particles) are presented. Such relationships are suitable for use in design of reactors in which the fluid phase is completely mixed. In Secs. 14-5 and 14-6 all these results will be applied to reactor design. 

The experimental conversion-time relationship was found to agree with Equation (CS9.3). Furthermore, the experimental value of t was found to be proportional to R. From these two observations, and also from the value of activation energy, it was concluded that the oxydesulfurization process of coal is chemical reaction controlled. For coal of 100 pm size, the time required for complete... 

For liquid-phase reactions and for gas-phase reactions with no pressure drop (F = Pq), one can combine the information in levels and , to express the rate of reaction as a function of conversion and arrive at level . It is now possible to detemiine either the time or reactor volume necessary to achieve the desired conversion by substituting the relationship linking conversion and rate of reaction into the appropriate design equation (level ). 

The experimental measurements produced concentration-time plots of ethylene oxide and ethylene glycol in the liquid phase, as shown in Figure 8.18. The physical picture of this reaction/reactor system is most closely approximated by the plug-flow gas phase, well-mixed batch liquid phase. The appropriate relationships to model this system are given in equations (8-176) to (8-178), (8-183), and (8-188). The bubble volume is variable, and the nature of the variation changes with the extent of conversion (i.e., concentration of glycol in the liquid phase), however, the pure oxide gas phase allows yg = l. The modified equations specific to this reactor are then... 

The rate laws we have examined so far enable us to calculate the rate of a reaction from the rate constant and reactant concentrations. These rate laws can also be converted into equations that show the relationship between concentrations of reactants or products and time. The mathematics required to accomplish this conversion involves calculus. We do not expect you to be able to perform the calculus operations, but you should be able to use the resulting equations. We will apply this conversion to three of the simplest rate laws those that are first order overall, those that are second order overall, and those that are zero order overall. 

The mean residence time is thus of crucial importance. Whether equation (7.5) or (7.6) is used there is clearly a simple relationship between size of reactor, reaction rate, extent of reaction and initial concentration. Thus knowing any three allows the fourth to be found directly. In design, the feed conditions and the required extent of reaction will generally be fixed and thus the equation will be used to obtain the required size. (The economic conversion will be dependent upon reactor cost, product sales price and a number of other factors. For fixed feed conditions the equation may be used to investigate how reactor costs, and ultimately profitability, are likely to vary with the extent of reaction.)... 

An equation relating temperature T and conversion Xa is required to design the non-isothermal reactors. This relationship between temperature T and conversion is obtained by setting up a heat balance equation around the reactor (Section 3.1.5.3). In certain cases, reactor temperature T is deliberately varied with conversion by regulating the heat supply to the reactor or heat removal from the reactor. One such case is the non-isothermal reactor in which a reversible exothermic reaction is carried out. In the case of a reversible exothermic reaction, there is an optimum temperature T for every value of conversion x at which the rate is maximum. A specified conversion Xaj will be achieved in a CSTR or a PFR with the smallest volume or in a batch reactor in the shortest reaction time if the temperature in the reaction vessel is maintained at the optimum level. This optimal temperature policy in which temperature is varied as a function of conversion x,i is known as the optimal progression of temperature presented in the following section. 

Radioactive decay of nuclei is a first-order reaction decay rate (activity A) is therefore dependent on the concentration (content) of the radionuclide and is the product of this concentration (more precisely, the number of atoms of radionuclide N) and the decay constant X (in s" ) A =-(dN/dt) = X.N. The basic unit of activity, according to the System International (SI system) is the Bq (becquerel). One Bq (in s ) is defined as the activity of a quantity of radioactive material in which one nucleus decays per second. Previously, the frequently used unit was the Ci (curie) defined as 3.7 x 10 decays per second. For conversion, the following relationship can be used 1 Ci = 3.7 x lO Bq. Number of radionuclide atoms transformed in time t is f T=NQ.e", where Nq is the initial number of atoms of the radionuclide at the time t=0. During conversion, the number of radioactive atoms of the radioactive nuclide is continuously decreasing. Combining both equations we get the relation expressing the dependence of activity on time A = -(dN/dt) The... 

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