Temperature Profile and Concentration Profiles

As mentioned above, two similar transport phenomena may take place on a heated electrode, namely  

Heat transfer— the transport of thermal energy from a body with higher temperature (the electrode) to another body with lower temperature (the electrolyte solution) 

Diffusion of particles from regions with higher concentration to regions with lower concentration 

Transport of heat by conduction, and of matter by diffusion, follows analogous mathematical principles, namely Fourier s law of heat conduction and Fick s first law of diffusion. Both are differential equations. If we simplify the problem and start with the one-dimensional form, we consider two ordinary differential equations. 

In the above equations, Q means thermal energy, n the number of moles transferred, t the time, jc the distance between electrode surface and a point inside electrolyte, T the temperature and c the concentration of particles taking part in the electrochemical reaction considered. The proportionality constants at the right side of equations are A the thermal conductivity (common unit W cm and D the diffusion coefficient (common unit cm s ). D and X are considered here to be constants. 

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In a recent article, Revel et al. [224] demonstrated a systematic method for the production of a global mechanism fitted to a detailed mechanism for NO generation in methane flames. The global reactions were deduced on the basis of element flux calculations and pathway analyses. The global mechanism of 10 species and 6 reactions is able to reproduce accurately the ignition delays, temperature profiles, and concentration profiles of major species and NO over a wide range of fuel-to-air ratios and initial temperatures. 

For the analysis of the chemical structure of flames, laser methods will typically provide temperature measurement and concentration profiles of some readily detectable radicals. The following two examples compare selected LIF and CRDS results. Figure 2.1 presents the temperature profile in a fuel-rich (C/O = 0.6) propene-oxygen-argon flame at 50 mbar [42]. For the LIF measurements, 1% NO was added. OH-LIF thermometry would also be possible, but regarding the rather low OH concentrations in fuel-rich flames, especially at low temperatures, this approach does not capture the temperature rise in the flame front [43]. The sensitivity of the CRDS technique, however, is superior, and the OH mole fraction is sufficient to follow the entire temperature profile. Both measurements are in excellent agreement. For all flames studied here, the temperature profile has been measured by LIF and/or CRDS. 

Equations 1 through 6 can be solved numerically for temperature, pressure and concentration profiles along the length of the reactor, provided that the appropriate initial or boundary conditions are given. Solution methods are now discussed for both the initial and boundary value problems. However, emphasis will be placed on the latter since most industrial applications fall into this category. 

We have reviewed the literature (3-20) and found that many concurrent and consecutive chemical reactions can occur as a result of the interaction between nitrite and sulfite ions. We will summarize the kinetic results of these reactions and present the results of model calculations that give the concentration profile of species produced in this system as a function of reaction time. The effects of temperature, pH, and concentrations of reactant will be demonstrated. 

Goh and Gollahalli [20] measured temperature profiles in piloted and nonpiloted propane and propylene flames in crossflow at R = 32-97. The profiles are generally characterized by off-axis single peak structure for all flames. The nonpiloted flames produced higher peak temperatures signifying increased oxidation of O2 under the same flow conditions. Tsue, Kadota, and Kono [71] measured the structure of propane diffusion flame in crossflow, including the flame temperature, velocity and concentration fields. Further, Tsue, Kadota, and... 

Figure 4.6 Temperature (T) and concentration ([/I]) profiles (a) over time for the batch reactor and (b) along reactor length for the PFR.

The profiles of temperature, conversion, and concentration of the reactants in nonisothermal reactions taking place in batch, tank, or tubular reactors are shown in Figure 14.13. 

FIGURE 3.11 Temperature (left) and concentration (right) profiles in a homogeneous tube reactor. Curve 1—adiabatic reactor curve 2— small heat transfer coefficient curves 3 and 4—intermediate heat transfer coefficient and curve 5— very high heat transfer coefficient. Saponification of ethyl adipate with NaOH [3] A + B R + E,A = diethyl adipate, B = NaOH, E = ethanol R-FB- S-FE, R = primary hydrolysis product, and S = secondary hydrolysis product (Na salt of adipic acid). 

Figure 9-10 Profiles of temperature (T) and concentration of reactant A (Ca) through the boimdaiy layer surroimding a catalyst particle in which an exothermic reaction is taking place at steady state. The width of the boimdary layer is exaggerated relative to the size of the catalyst particle. The profiles of Ca and T are shown to be linear because the actual thickness of the boimdary layer is small relative to the radius of the particle.

Laser Raman diagnostic teclmiques offer remote, nonintnisive, nonperturbing measurements with high spatial and temporal resolution [158], This is particularly advantageous in the area of combustion chemistry. Physical probes for temperature and concentration measurements can be debatable in many combustion systems, such as furnaces, internal combustors etc., since they may disturb the medium or, even worse, not withstand the hostile enviromnents [159]. Laser Raman techniques are employed since two of the dominant molecules associated with air-fed combustion are O2 and N2. Flomonuclear diatomic molecules unable to have a nuclear coordinate-dependent dipole moment caimot be diagnosed by infrared spectroscopy. Other combustion species include CFl, CO2, FI2O and FI2 [160]. These molecules are probed by Raman spectroscopy to detenuine the temperature profile and species concentration m various combustion processes. 

Nonisothermal Gas Absorption. The computation of nonisothermal gas absorption processes is difficult because of all the interactions involved as described for packed columns. A computer is normally required for the enormous number of plate calculations necessary to estabUsh the correct concentration and temperature profiles through the tower. Suitable algorithms have been developed (46,105) and nonisothermal gas absorption in plate columns has been studied experimentally and the measured profiles compared to the calculated results (47,106). Figure 27 shows a typical Hquid temperature profile observed in an adiabatic bubble plate absorber (107). The close agreement between the calculated and observed profiles was obtained without adjusting parameters. The plate efficiencies required for the calculations were measured independendy on a single exact copy of the bubble cap plates installed in the five-tray absorber. 

The equiHbrium approach should not be used for species that are highly sensitive to variations in residence time, oxidant concentration, or temperature, or for species which clearly do not reach equiHbrium. There are at least three classes of compounds that cannot be estimated weU by assuming equiHbrium CO, products of incomplete combustion (PlCs), and NO. Under most incineration conditions, chemical equiHbrium results in virtually no CO or PlCs, as required by regulations. Thus success depends on achieving a nearly complete approach to equiHbrium. Calculations depend on detailed knowledge of the reaction network, its kinetics, the mixing patterns, and the temperature, oxidant, and velocity profiles. 

Fig. 7. Constitutional supercooling, (a) impurity concentration profile during solidification (b) actual temperature T and equilibrium freezing temperature T...

To obtain the concentration and temperature profiles, the two transitions are first assumed to Le gradual. Equation (16-135) is written in the form... 

Plug Flow Reactor (PFR) A plug flow reactor is a tubular reactor where the feed is continuously introduced at one end and the products continuously removed from the other end. The concentration/temperature profile in the reactor varies with position. 

Ethylene oxidation was studied on 8 mm diameter catalyst pellets. The adiabatic temperature rise was limited to 667 K by the oxygen concentration of the feed. With the inlet temperature at 521 K in SS and the feed at po2, o=T238 atm, the discharge temperature was 559 K, and exit Po =1.187 atm. The observed temperature profiles are shown on Figure 7.4.4 at various time intervals. The 61 cm long section was filled with catalyst. 

Methanol is frequently used to inhibit hydrate formation in natural gas so we have included information on the effects of methanol on liquid phase equilibria. Shariat, Moshfeghian, and Erbar have used a relatively new equation of state and extensive caleulations to produce interesting results on the effeet of methanol. Their starting assumptions are the gas composition in Table 2, the pipeline pressure/temperature profile in Table 3 and methanol concentrations sufficient to produce a 24°F hydrate-formation-temperature depression. Resulting phase concentrations are shown in Tables 4, 5, and 6. Methanol effects on CO2 and hydrocarbon solubility in liquid water are shown in Figures 3 and 4. 

Fig ure 6-3. Profiles of outlet temperature and concentration with time in a CFSTR. 

There are also formulas for calculation of temperature and concentration distribution along and across an air jet. These are based on the similarity profile of the jet. ... 

FIGURE 4-1. Concentration and temperature profiles through a premixed flame. 

In all tests, the temperature in the first- and second-stage reactors was kept within the necessary temperature limits of 288°-482°C. Because the carbon monoxide concentration was low in many of the tests, the second stage was not used to full capacity as is indicated by the temperature rise in runs 23, 24, and 27. The temperature profile shows the characteristic rise to a steady value. With the space velocities used (<5000 ft3/ft3 hr), the temperature profile is fully developed in the first stage within 30.0 in. of the top of the catalyst bed. A characteristic dip in temperature was observed over the first 8-10 in. of the catalyst bed in all runs. This temperature profile may indicate the presence of deactivated catalyst in this region, but, until the catalyst can be removed for examination, the cause of the temperature drop cannot be determined. There is no evidence that this low temperature zone is becoming progressively deeper. It is possible that an unrecorded brief upset in the purification system may have poisoned some of the top catalyst layers. 

A schematic representation of temperature and concentration profiles in a temperature-jump experiment. All scales are arbitrary, and the matter to be emphasized is that the temperature jump occurs rapidly compared with the re-equilibration reaction. 

In fact, this phenomenon has been used as the basis of a very sensitive detecting system. An example of the temperature profile of an adsorbent as a peak passes over it is shown in figure 2. Unfortunately, it was found almost impossible to produce a true simulation of the concentration profile of the peak from the temperature profile and interest in the detector declined. 

Continuous Polymerizations As previously mentioned, fifteen continuous polymerizations in the tubular reactor were performed at different flow rates (i.e. (Nj g) ) with twelve runs using identical formulations and three runs having different emulsifier and initiator concentrations. A summary of the experimental runs is presented in Table IV and the styrene conversion vs reaction time data are presented graphically in Figures 7 to 9. It is important to note that the measurements of pressure and temperature profiles, flow rate and the latex properties indicated that steady state operation was reached after a period corresponding to twice the residence time in the tubular reactor. This agrees with Ghosh s results ). 

The ability to manipulate reactor temperature profile in the polymerization tubular reactor is very important since it directly relates to conversion and resin product properties. This is often done by using different initiators at various concentrations and at different reactor jacket temperature. The reactor temperature response in terms of the difference between the jacket temperature and the peak temperature (0=Tp-Tj) is plotted in Figure 2 as a function of the jacket temperature for various inlet initiator concentrations. The temperature response not only depends on the jacket temperature but also, for certain combinations of the variables, it is very sensitive to the jacket temperature. 

The boundary layers for these three variables (gas velocity, temperature, and concentration) may sometimes coincide, although in slow reactions, the profiles of velocity and temperature may be fully developed at an early stage while the deposition reaction is spread far downstream the tube. 

We turn now to the numerical solution of Equations (9.1) and (9.3). The solutions are necessarily simultaneous. Equation (9.1) is not needed for an isothermal reactor since, with a flat velocity profile and in the absence of a temperature profile, radial gradients in concentration do not arise and the model is equivalent to piston flow. Unmixed feed streams are an exception to this statement. By writing versions of Equation (9.1) for each component, we can model reactors with unmixed feed provided radial symmetry is preserved. Problem 9.1 describes a situation where this is possible. 

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