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MODELING THE REACTION MECHANISM: THE USE OF EUCLIDIAN AND FRACTAL GEOMETRY
276 Chapier 10
10. MODELING THE REACTION MECHANISM: THE USE OF EUCLIDIAN AND FRACTAL GEOMETRY 10.1. Constitutive equations applied in chemical kinetics Chemical kinetics is based on the experimentally verified assumption that the rate of change, dx/dt, in the state of a system (characterized by) x is a function, f of the state alone, i.e., dx/dt = x^ = f(x). Using this traditional postulation, the appropriate constitutional approach to inaugurate the desired constitutive equation can be written in the principal form of the dependence of the reaction rate, expressed as the time development of the degree(s) of change (transformation) on the quantities that characterize the instantaneous state of the reacting system (chosen for its definition). In a simplified case, when Aeq=l (i.e., A=a), and under the unvarying experimental conditions (maintaining all intensive parameters around the sample constant, e.g., P = 0, P'= 0, JT = 0, etc.), we can write the set of constitutive equations with respect to all partial degrees of conversion, aj , a2 .. a^ , thus including all variables in our spotlight. In relation to our earlier description we can summarize it as [3,402] a' = fa(oc,p,T) {possibly including other ai' = d (a, P, T)} and T' = fr (a, P, T), where the apostrophe represents the time derivative. Such a set would be apparently difficult to solve and, moreover, experimental evidence usually does not account for the particularity of all fractional degrees of conversion, so that we can often simplify our account just for two basic variables, i.e., cir and T, only. Their change is traditionally depicted in the form of a set of two basic equations: oc' = fa (oc, T) = k(T) f(a) and r
= fr (a, T) =To
+f(t)outer
+f(t),
The analytical determination of the temperature function, (T, becomes the subject of both thermal effects so that let us see it firsts: (1) The external temperature, whose handling is given by programmed experimental conditions, i.e., thermostat (furnace) control when fx' (T) = To +f(TT)outer, and (2) The internal temperature, whose production and sink is the interior make up of the process investigated, tjfoc) = T'o ^f(To) inner • In the total effect these quantities govern the heat flows outwards and inwards the reaction interface. Their interconnection specifies the intimate reaction
10. MODELING THE REACTION MECHANISM: THE USE OF EUCLIDIAN AND FRACTAL GEOMETRY 10.1. Constitutive equations applied in chemical kinetics Chemical kinetics is based on the experimentally verified assumption that the rate of change, dx/dt, in the state of a system (characterized by) x is a function, f of the state alone, i.e., dx/dt = x^ = f(x). Using this traditional postulation, the appropriate constitutional approach to inaugurate the desired constitutive equation can be written in the principal form of the dependence of the reaction rate, expressed as the time development of the degree(s) of change (transformation) on the quantities that characterize the instantaneous state of the reacting system (chosen for its definition). In a simplified case, when Aeq=l (i.e., A=a), and under the unvarying experimental conditions (maintaining all intensive parameters around the sample constant, e.g., P = 0, P'= 0, JT = 0, etc.), we can write the set of constitutive equations with respect to all partial degrees of conversion, aj , a2 .. a^ , thus including all variables in our spotlight. In relation to our earlier description we can summarize it as [3,402] a' = fa(oc,p,T) {possibly including other ai' = d (a, P, T)} and T' = fr (a, P, T), where the apostrophe represents the time derivative. Such a set would be apparently difficult to solve and, moreover, experimental evidence usually does not account for the particularity of all fractional degrees of conversion, so that we can often simplify our account just for two basic variables, i.e., cir and T, only. Their change is traditionally depicted in the form of a set of two basic equations: oc' = fa (oc, T) = k(T) f(a) and r
= fr (a, T) =To
+f(t)outer
+f(t),
The analytical determination of the temperature function, (T, becomes the subject of both thermal effects so that let us see it firsts: (1) The external temperature, whose handling is given by programmed experimental conditions, i.e., thermostat (furnace) control when fx' (T) = To +f(TT)outer, and (2) The internal temperature, whose production and sink is the interior make up of the process investigated, tjfoc) = T'o ^f(To) inner • In the total effect these quantities govern the heat flows outwards and inwards the reaction interface. Their interconnection specifies the intimate reaction