The adiabatic temperature increase equation

Chapter 2 The adiabatic temperature increase equation 2.1 Introduction 

As a matter of fact, it is very difficult to calculate the for a chemical of the TD type, including every gas-permeable oxidatively-heating substance, having an arbitrary shape and an arbitrary size, placed in the atmosphere under isothermal conditions, by applying the Semenov or the F-K equation only, because, as stated in Preface, the chemical quantities, E, Ao and AH, which specify as a whole the rate of heat generation per unit volume per unit time in the early stages of the self-heating process of the chemical and are included in both the Semenov and the F-K equation, are very difficult to measure precisely by means of conventional experimental techniques. 

however, becomes very easy to calculate the if both the Semenov and the F-K equation are reduced to the simpler forms by substituting the two coefficients, a and b, of an empirical formula, nAt = atT + b, i.e., Eq. (44), which is derived in Section 2.4, into each equation, respectively. 

(44) holds between a T, and the time, A t, required for the temperature of a definite quantity of a chemical of the TD type, including every gas-permeable oxidatively-heating substance, charged, or confined, in some one of the open-cup, the draft or the closed cell, in accordance with the self-heating property of the chemical, and subjected to the adiabatic self-heating test, or to the adiabatic oxidatively-heating test, which is started from the T to increase by a definite temperature difference, A T, from the T,. 

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

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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. 

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

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]

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. 

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 ... 

In many processes exothermic reactions are carried out in tubular reactors in a practically adiabatic manner. This means that the reaction temperature increases with the length of the reactor, and the reaction rate increases at first exponentially. The conversion can he calculated from mass and heat balances, that are written in the form of two coupled differential equations. For simple first order processes the mass balance was formulated in eq. (7.18). However, the reaction rate constant varies with temperature, therefore eq. (8.1) has to be substituted in eq. (7.18). The boundary conditions were given by eqs. (7.19a) and (7.19b). The heat balance is analogous to the mass balance ... 

Adiabatic-Saturation Temperature, or Constant-Enthalpy Lines If a stream of air is intimately mixed with a quantity of water at a temperature t, in an adiabatic system, the temperature of the air will drop and its humidity will increase. If t, is such that the air leaving the system is in equihbrium with the water, t, will be the adiabatic-saturation temperature, and the line relating the temperature and humidity of the air is the adiabatic-saturation line. The equation for the adiabatic-saturation line is... 

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. 

Hetsroni et al. (2005) evaluated the effect of inlet temperature, channel size and fluid properties on energy dissipation in the flow of a viscous fluid. For fully developed laminar flow in circular micro-channels, they obtained an equation for the adiabatic increase of the fluid temperature due to viscous dissipation ... 

Consider the case of pure methanol for which the values of Cp and a are known. Using a specific volume of 0.00127 m3/kg, a temperature of 25°C (298 K), and a compression pressure of 1000 bar, Equation 13.2 predicts the eluent temperature will increase approximately 15°C assuming adiabatic conditions. In actual practice, the increase in eluent temperature entering the column will be lower than this upper limit due to thermal losses in the pump, connecting tubing, and injection system, as well as entropic changes (AS A 0). 

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... 

As an example, let us analyze mold filling with a model polyurethane formulation. Let the kinetics of curing be described by an equation with the self-deceleration term (as was discussed above). The following values of the parameters were used U = 49.1 kJ/mol ko = 3.8xl06 = 1.1 ATmax = 25.8°C where ATmax is the maximum expected increase in temperature for adiabatic curing. 

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