Activation energy dimensionless

The Tempered activation energy , is the activation energy divided by R, the gas constant, and is dimensionless. It will be shown here with a superscript T, e.g. 10 000. ... 

A dimensionless number equal to the activation energy of a reaction (E ) divided by the product of the universal gas constant (R) and the absolute temperature, T thus, EJ(RT). 

The temperature dependence of the rate constant for the step A -> B leads to the term /(0) in the dimensionless mass- and heat-balance eqns (4.24) and (4.25). The exact representation of an Arrhenius rate law is f(9) — exp[0/(l + y0)], where y is a dimensionless measure of the activation energy RTa/E. As mentioned before, y will typically be a small quantity, perhaps about 0.02. Provided the dimensionless temperature rise 9 remains of order unity (9 < 10, say) then the term y9 may be neglected in the denominator of the exponent as a first simplification. 

Thus rres is the dimensionless residence time, as in the previous chapter, and tn is the dimensionless Newtonian cooling time. High values of rN correspond to slow heat transfer across the reactor walls, indicating well-insulated vessels which approach adiabatc operation as tn tends to infinity. Small values of tn correspond to systems which have fast heat transfer and hence which would be expected not to have great departures from isothermal operation. Finally, we need a measure of the activation energy... 

The dimensionless temperature excess and activation energy have the same forms as those used in chapter 4 6 = (T — TJE/RT2 and y = RTJE, where Ta is the ambient (reservoir) temperature. The dimensionless concentration X is simply c/c , where c0 is the reservoir concentration of the gaseous reactant. The group 3 is given by... 

The equations studied by Uppal, Ray, and Poore were written in terms of a dimensionless conversion, x, and a dimensionless temperature, x2, with origin at feed conditions and scaled by the dimensionless activation energy, y. This left three parameters Da, the Damkohler number, the ratio of reaction rate to flow rate B, a dimensionless heat of reaction /3, a dimensionless heat transfer coefficient and x2c, a dimensionless coolant temperature. The equations were... 

The forcing variable is the coolant temperature jc2c as in Sincic and Bailey (1977) and more recently in Mankin and Hudson (1984). In eq. (10) jti is a dimensionless reactant concentration while x2 is a dimensionless reactor temperature. These equations hold at the limit of infinite reaction activation energy. All models were thus chosen so that extensive simulation results existed in the literature, and they cover a wide range of lumped reactor types. 

Here t, x are the dimensionless time and coordinate along the propagation of the front 6 = (T — Tc)E/RTq is the dimensionless temperature counted down from the combustion temperature Tc and measured in the characteristic intervals RTq/E E is the activation energy f3 = RTC/E and A is the scale coefficient. In the numerical calculations we took (3 = 0. 

For the thermicity factor (3 = 1.2, the dimensionless activation energy 7 = 18, and with the constant dimensionless cooling jacket temperature yc = 0.85, compute the following ... 

The apparent kinetic data as discussed in the previous section is given in terms of the dimensionless frequency factors Ai and the activation energy ), measured in kJ/kmol with... 

Two additional dimensionless values connected with the activation energies are also used ... 

Differential Eqs (9.10) and (9.11), with initial and boundary conditions (9.12) and (9.13), may be numerically solved for different sets of values of the four dimensionless parameters, W3-W4 (Williams et al., 1985). To illustrate the evolution of temperature and conversion profiles during the cure, values of W2-W4 will be kept constant and Wi will be varied to simulate the influence of the part thickness. The particular case of W2 = 40, W3 = 1.5, and W4 =0.125 will be analyzed. This represents a process characterized by high values of both the activation energy and the adiabatic temperature rise. 

Next, following Semenov (40) we define another dimensionless temperature 0, which is the product of the dimensionless temperature define in Eq. 11.2-3 with the dimensionless activation energy... 

The dimensionless heat generation term re = 0/(1 — 0) is plotted as a function of the dimensionless temperature Q = T/e for various values of the inverse dimensionless activation energy, e, in Fig. 11.6. 

We note two important features on this graph. First, the transition from stable to potentially runway conditions increases dramatically with decreasing e, that is, increasing the reaction constant activation energy in the limit at e — 0 we have explosive conditions. Second, the transition from stable to potentially unstable reactions occurs when the dimensionless at 0 = 7 / 1. Furthermore, for 0 = 10-1 the reaction is stable with 0/(1 + 0) = 1 and for >10 there are is a significant increase (of the order of 104 to 1010) in the dimensionless heat-generation term, denoting the potential of unstable, runway reactions. 

Fig. 11.6 Dimensionless heat generation rates for various values of the dimensionless activation energy. [Reprinted by permission from D. H. Sebastian, Non-Isothermal Effects in Polymer Reaction Engineering, in Temperature Control Principles for Process Engineers, E. P. Dougherty, Ed., Hanser, Munich, 1993.]...

A->B- C has also been considered by Jorgensen (65). This last is a system of three equations for u and v, the concentrations of A and B, and w, the temperature, with seven parameters a, the Damkohler number for A->B 3, its dimensionless heat of reaction y, its Arrhenius number k, a dimensionless heat transfer coefficient v, the ratio of activation energies of the two reactions p, the ratio of the heats of reaction a, the ratio of the Damkohler numbers. They contain a characteristic non-linearity... 

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