Adiabatic temperature dimensionless

In order to derive specific numbers for the temperature rise, a first-order reaction was considered and Eqs. (10) and (11) were solved numerically for a constant-density fluid. In Figure 1.17 the results are presented in dimensionless form as a function of k/tjjg. The y-axis represents the temperature rise normalized by the adiabatic temperature rise, which is the increase in temperature that would have been observed without any heat transfer to the channel walls. The curves are differentiated by the activation temperature, defined as = EJR. As expected, the temperature rise approaches the adiabatic one for very small reaction time-scales. In the opposite case, the temperature rise approaches zero. For a non-zero activation temperature, the actual reaction time-scale is shorter than the one defined in Eq. (13), due to the temperature dependence of the exponential factor in Eq. (12). For this reason, a larger temperature rise is foimd when the activation temperature increases. 

In order to show how specific guidelines for the reactor layout can be derived, the maximum allowable micro-channel radius giving a temperature rise of less than 10 K was computed for different values of the adiabatic temperature rise and different reaction times. For this purpose, properties of nitrogen at 300 °C and 1 atm and a Nusselt number of 3.66 were assumed. The Nusselt number is a dimensionless heat transfer coefficient, defined as... 

Xu et al. [124] numerically computed the adiabatic temperature rise in a micro channel due to viscous heating and expressed their results by a correlation based on dimensionless groups. They introduced a dimensionless temperature rise AT = AT/Tjgf with a reference temperature of 1 K. The correlation they found is given by... 

In a similar way, the maximum temperature rise possible in the system (the adiabatic temperature rise ATad) has a dimensionless measure 0ad given by... 

Dividing eqn(7.18) throughout by the dimensionless adiabatic temperature rise 0ad and then adding these two conditions gives the stationary-state... 

Fig. 7.3. The existence and loss of multiplicity as the dimensionless adiabatic temperature 9ai decreases through 4 (a) flow diagram for 6ai = 7.2 (b) corresponding stationary-state locus (1 — ot ,)-r 9 (c) flow diagram for 0a<1 = 3.6 (d) corresponding stationary-state locus.

Equation (7.28) only has real roots, and hence ignition and extinction can only occur if the discriminant under the square root sign is positive. This then makes a requirement on the size of the dimensionless adiabatic temperature excess 0ad. In particular, tangency only occurs if the reaction is sufficiently exothermic such that... 

The condition on the size of the dimensionless adiabatic temperature rise is similar to the condition for multiple stationary states in terms of the dimen-... 

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. 

A more comprehensive approach consists of studying the variation of the Semenov criterion as a function of the reaction energy. Such an approach is presented in [12], where the reciprocal Semenov criterion is studied as a function of the dimensionless adiabatic temperature rise. This leads to a stability diagram similar to those presented in Figure 5.2 [11, 13]. The lines separating the area of parametric sensitivity, where runaway may occur, from the area of stability is not a sharp border line it depends on the models used by the different authors. For safe behavior, the ratio of cooling rate over heat release rate must be higher than the potential of the reaction, evaluated as the dimensionless adiabatic temperature rise. 

Figure 5.2 Stability diagram presenting the variation of the reciprocal Semenov criterion as a function of the dimensionless adiabatic temperature rise.

The four scenarios were originally discussed by Fortuin et al. [7] with reference to the effect of variation of the dimensionless adiabatic temperature raise, but their validity can be extended to other model parameters as far as their effect on the maximum reactor temperature is monotonic. 

Here we have denoted y conversion, 0 Frank-Kameneckii dimensionless temperature, Da Damkohler number, Pe Peclet number for axial mass dispersion, Pe Peclet number for Sxial heat dispersion, Y dimensionless activation energy, B dimensionless adiabatic temperature rise, 3 dimensionless cooling parameter, 6 temperature of the cooling medium, A mass capacity, AT heat capacity. 

The dimensionless variables and parameters are axial coordinate 0 < < 1, conversion temperature 0(1 ), recycle flow rale f and reactor-inlet concentration z3 plant Damkohler number Da, activation energy y, adiabatic temperature rise B, heat-transfer capacity p, coolant temperature 0 concentration of recycle and product streams z3, z4. For convenience, X = x(l) will stand for conversion at reactor outlet. 

S/m and fl/ x = Biot numbers for mass and heat transfer 4 and 4 x = Thiele modulus Le = Lewis number A0 i = dimensionless adiabatic temperature rise t) = effectiveness factor kg =mass transfer coefficient (ms-1) Rp = radius of catalyst pellet (m) Da = effective diffusion coefficient (ms-2) r =rate of reaction (molm-3s-1) C —concentration of reactant (molm-3) ... 

B Zeldovich number (or dimensionless adiabatic temperature rise) Bi Biot number... 

P dimensionless adiabatic temperature rise, (-AH)C0/Pg Cp T0 7s dimensionless activation energy (E/RT) e bed void fraction V effectiveness factor... 

This set of equations describes the behaviour of multiple, first order reactions in a tubular reactor using the relative conversion to desired product Xp and to undesired product Xx, the dimensionless temperature T and the dimensionless reactor length Z. The is characterized by the ratio of the reaction heats H in addition to kR, TR, y and p. The operating and design are determined by PC, the dimensionless cooling medium temperature Da, the dimensionless residence time in the reactor U, the dimensionless cooling capacity per unit of reactor volume and ATacp the dimensionless adiabatic temperature rise for the desired reaction, which, of course, depends on the initial concentration of the reactant A. 

A review of the physical significance of the dimensionless parameters is appropriate at this point. S is the maximum possible adiabatic temperature rise, scaled by the temperature coefficient of reaction rate. It is a measure of the sensitivity of the reaction to changes in operating conditions, and is sometimes designated by "B" in the literature. "D" is the residence time multiplied by the specific rate constant at the inlet, often called the "number of reaction units", x is the temperature rise, scaled by the temperature coefficient of reaction rate. The maximum possible value of x is S. 

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. 

Sinee in the DPF the gas flows through a solid bed, with a non-Arrhenius reaction rate given by k = koTQxp —E/RT), we redefine, for the DPF, the following parameters the charaeteristic reaction time tr d (such that Td = tjty d) the eharacteristic time for thermal convection tc,d (such that the cooling parameter 3 = tr /tc ), the dimensionless adiabatic temperature rise Bd, and the Lewis number. Led, s the ratio of the total heat capaeity of the soot bed to that of the substrate wall. Z = w/w is the dimensionless soot layer thickness, and 6 is the dimensionless temperature, defined earlier. 

The relevant dimensionless parameters are a modified Thiele modulus, the normalized adiabatic temperature rise 0), and the Arrhenius number — a IRT. Plots for a first-order reaction in a spherical particle are shown in Figure 9.4 (next page). For highly exothermic reactions and large values of (3 the rate can be multivalued at modified Thiele moduli around 0.5, with two stable and one unstable steady states. At which state the particle performs depends on its prior history. 

We start by recalling some results concerning the stand-alone adiabatic CSTR. Its steady state can be described by the following dimensionless equation, where the Damkohler number (Da ) and adiabatic temperature rise (B ) are calculated using the reactor-inlet flow rate (Fi) and concentration (ci) as reference values. 

This ratio of the two characteristic numbers has already been introduced in Section 4.3.1.1, Equ.(4-95). At that time it had already been referred to the safety technical significance of this ratio. This shall now be proven. Besides the adiabatic temperature increase this is the only other parameter which influences the maximum driving temperature differences of batch processes. It is important to recognize at this point that the reaction time does not have any impact on this safety relevant value. The ignition point criterion for isothermal batch processes may also be expressed in a dimensionless form ... 

Use if the order of the reaction is positive and > 95 % conversion is the target, and for consecutive reactions with an intermediate as the target product. Exchange heat generated if the product of the Arrhenius number and the dimensionless adiabatic temperature rise > 10. 

Figure 6.14 Dimensionless maximum temperature vs. Damkohler number for several values of dimensionless wall temperature, temperatures are made dimensionless by the adiabatic temperature rise ATad = 115K. This To = 55"C and ranges from 55 to 200 °C. =...

---