Adiabatic temperature increase equation

Chapter 2 The adiabatic temperature increase equation 2.1 Introduction... 

Eq. (44) is obtained directly from Eq. (43), i.e., an equation ealled the adiabatic temperature increase equation. 

An alternative method to derive the adiabatic temperature increase equation... 

The derivation method of the adiabatic temperature increase equation, which is introduced in the preceding section, is very difficult to understand. First of all, wc cannot understand very well the reason why dimensionless numbers, such as 0, 6 and T, are needed to derive the equation. Fortunately, however, we have an alternative method to derive, though qualitatively, the equation in a far simpler manner. 

In Chapter 2, is derived the adiabatic temperature increase equation, which holds between the rate of heat generation per unit volume per unit time in the early stages of the self-heating process of a small-scale chemical of the TD type, including every small-scale gas-permeable oxidatively-heating substance, having the spatially uniform distribution of internal temperature, subjected to either of the two kinds of adiabatic tests, and, the rate of increase in temperature of the chemical, assuming the effect of the concentration of the chemical on the... 

The adiabatic temperature increase for an ideal gas is computed from the thermodynamic adiabatic compression equation ... 

The preceding equation assumes the reaction is completely quenched immediately after the relief point is reached. This behavior is closely approximated if the reaction stops in the quench pool and the reactor empties quickly and thoroughly. If the reaction continues in the quench pool, the temperature Tr should be increased to the maximum adiabatic exotherm temperature. An equation is presented by CCPS (AIChE-CCPS, 1997) that includes the heat of reaction. In some cases, an experiment is necessary to confirm that the reaction indeed stops in the quench pool. 

The line retains the same slope as that given above, but its intercept moves up the 0ad axis as y increases, tending to infinity as y approaches The equation for the cusp is slightly more complex and is again most easily expressed parametrically. The appropriate values for the adiabatic temperature excess must be obtained from a quadratic equation before it can be used to determine tn. Thus, for any given y [Pg.196]

Analysis of the non-isothermal polymerization of E-caprolactam is based on the equations for isothermal polymerization discussed above. At the same time, it is also important to estimate the effect of non-isothermal phenomena on polymerization, because in any real situation, it is impossible to avoid exothermal effects. First of all, let us estimate what temperature increase can be expected and how it influences the kinetics of reaction. It is reasonable to assume that the reaction proceeds under adiabatic conditions as is true for many large articles produced by chemical processing. The total energy produced in transforming e-caprolactam into polyamide-6 is well known. According to the experimental data of many authors, it is close to 125 -130 J/cm3. If the reaction takes place under adiabatic conditions, the result is an increase in temperature of up to 50 - 52°C this is the maximum possible temperature increase Tmax- In order to estimate the kinetic effect of this increase... 

Mrf represents the mass of the reaction mixture at the end of the feed, MrW the instantaneous mass of reactant present in the reactor, and Xal the fraction of accumulated reactant The ratio of both masses accounts for the correction of the specific energy, since the adiabatic temperature rise is usually calculated using the final reaction mass, that is, the complete batch. In Equation 2.5, the concentration corresponds to the final reaction mass this is also the case for the specific heat of reaction obtained from calorimetric experiments, which is also expressed for the total sample size. Since in the semi-batch reaction, the reaction mass varies as a function of the feed, the heat capacity of the reaction mass increases as a function of time and the adiabatic temperature rise must be corrected accordingly. 

Since, in the case of an adiabatic reaction the temperature increases linearly with the achieved conversion A.v according to the equation... 

This provides a pair of coupled, non-linear (through the Arrhenius temperature dependence) ordinary differential equation for the two variables a and T. If the temperature increases, the reaction rate increases through the increase in k. The consequent increase in T will lead to increases in the heat transfer rates and also to a decrease in the concentration of A, which in turn tends to decrease the reaction rate term ka. To quantify this effect, we can examine the adiabatic case a = 0- In this situation, the temperature rise above the inflow is uniquely linked to the extent of reaction = ao a)/ao through the condition... 

The effect of compression of biomaterials can be illustrated by first considering the effects in liquids. Water and hexane are of particular interest because they represent t)q)ical examples of polar and non-polar materials, respectively. If one performs the compression under adiabatic conditions, then the following equation gives the temperature increase for a given pressure increase ... 

Thermal effects are often the key concern in reactor scaleup. The generation of heat is proportional to the volume of the reactor. Note the factor of V in Equation 5.31. For a scaleup that maintains geomedic similarity, the surface area increases only as Sooner or later, temperature can no longer be controlled by external heat transfer, and the reactor will approach adiabatic operation. There are relatively few reactions where the full adiabatic temperature change can be tolerated. Endothermic reactions will have poor yields. Exothermic reactions will have thermal runaways giving undesired byproducts. It is the reactor designer s job to avoid limitations of scale or at least to understand them so that a desired product will result. There are many options. The best process and the best equipment at the laboratory scale are rarely the best for scaleup. Put another way, a process that is less than perfect at a small scale may be better for scaleup precisely because it is scaleable. 

Additionally, the upper integration limit is also known, as the temperature increases in a steady and monotone way in this special case. On the other hand, the dimensionless temperature increase d may not become greater than 1 by definition. This way, for the special case of a zero order reaction, the equation for the adiabatic induction time is reduced to ... 

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