This is a more general version of Equation (1.24). For a first-order reaction, the number of molecules of the reactive component decreases exponentially with time. This is true whether or not the density is constant. If the density happens to be constant, the concentration of the reactive component also decreases exponentially as in Equation (1.24). [Pg.59]

This gives = 0.8047 h. The molecular weight of the monomer, Ma, is not actually used in the calculation. Extrapolation of the first-order kinetics to a 4-h batch predicts that there will be 900exp(-3.22) = 36kg or 4% by weight of monomer left unreacted. Note that the fraction unreacted, Ya, must be defined as a ratio of moles rather than concentrations because the density varies during the reaction. [Pg.59]

Clearly, we must determine F or p as a function of composition. The integration will be easier if is treated as the composition variable rather than a since this avoids expansion of the derivative as a product d Va) = Vda- -adV. The numerical methods in subsequent chapters treat such products as composite variables to avoid expansion into individual derivatives. Here in Chapter 2, the composite variable, Na = Va, has a natural interpretation as the number of moles in the batch system. To integrate Equation (2.32), F or p must be determined as a function of Na- Both liquid- and gas-phase reactors are considered in the next few examples. [Pg.60]

Example 2.9 Repeat Example 2.8 assuming that the polymerization is second order in monomer concentration. This assumption is appropriate for a binary polycondensation with good initial stoichiometry, while the pseudo-first-order assumption of Example 2.8 is typical of an addition polymerization. [Pg.60]

Example 2.10 Suppose 2A B in the liquid phase and that the density changes from po to Poo = Po + Ap upon complete conversion. Find an analytical solution to the batch design equation and compare the results with a hypothetical batch reactor in which the density is constant. [Pg.60]

The density change in this example increases the reaction rate since the volume goes down and the concentration of the remaining A is higher than it would be if there were no density change. The elfect is not large and would be negligible for many applications. When the real, variable-density reactor has a conversion of 50%, the h5rpothetical, constant-density reactor would have a conversion of 47.4% (T = 0.526). [Pg.62]

The effect of the density change is larger than in the previous example, but is still not major. Note that most gaseous systems will have substantial amounts of inerts (e.g. nitrogen) that will mitigate volume changes at constant pressure. [Pg.63]

We have considered volume changes resulting from density changes in liquid and gaseous systems. These volume changes were thermodynamically determined using an equation of state for the fluid that specifies volume or density as a function of composition, pressure, temperature, and any other state variable that may be important. This is the usual case in chemical engineering problems. In Example 2.10, the density depended only on the composition. In Example 2.11, the density depended on composition and pressure, but the pressure was specified. [Pg.63]

Volume changes also can be mechanically determined, as in the combustion cycle of a piston engine. If V=V(i) is an explicit function of time. Equations like (2.32) are then variable-separable and are relatively easy to integrate, either alone or simultaneously with other component balances. Note, however, that reaction rates can become dependent on pressure under extreme conditions. See Problem 5.4. Also, the results will not really apply to car engines since mixing of air and fuel is relatively slow, flame propagation is important, and the spatial distribution of the reaction must be considered. The cylinder head is not perfectly mixed. [Pg.63]

It is possible that the volume is determined by a combination of thermodynamics and mechanics. An example is reaction in an elastic balloon. See Problem 2.20. [Pg.63]

Mass transfer in the continuous phase is less of a problem for liquid-liquid systems unless the drops are very small or the velocity difference between the phases is small. In gas-liquid systems, the resistance is always on the liquid side, unless the reaction is very fast and occurs at the interface. The Sherwood number for mass transfer in a system with dispersed bubbles tends to be almost constant and mass transfer is mainly a function of diffusivity, bubble size, and local gas holdup. [Pg.347]

For a batch system, with no inflow and no outflow, the total mass of the system remains constant. The solution to this problem, thus involves a liquid-phase, component mass balance for the soluble material, combined with an expression for the rate of mass transfer of the solid into the liquid. [Pg.34]

You should be able to describe a system at equilibrium both qualitatively and quantitatively. Rigorous solutions to equilibrium problems can be developed by combining equilibrium constant expressions with appropriate mass balance and charge balance equations. Using this systematic approach, you can solve some quite complicated equilibrium problems. When a less rigorous an- [Pg.176]

S] Low mass-flux with constant property system. Use with arithmetic concentration difference. [Pg.605]

Equations (1.1) to (1.3) are diflerent ways of expressing the overall mass balance for a flow system with variable inventory. In steady-state flow, the derivatives vanish, the total mass in the system is constant, and the overall mass balance simply states that input equals output. In batch systems, the flow terms are zero, the time derivative is zero, and the total mass in the system remains constant. We will return to the general form of Equation (1.3) when unsteady reactors are treated in Chapter 14. Until then, the overall mass balance merely serves as a consistency check on more detailed component balances that apply to individual substances. [Pg.2]

For some systems qiiasiperiodic (or nearly qiiasiperiodic) motion exists above the unimoleciilar tlireshold, and intrinsic non-RRKM lifetime distributions result. This type of behaviour has been found for Hamiltonians with low uninioleciilar tliresholds, widely separated frequencies and/or disparate masses [12,, ]. Thus, classical trajectory simulations perfomied for realistic Hamiltonians predict that, for some molecules, the uninioleciilar rate constant may be strongly sensitive to the modes excited in the molecule, in agreement with the Slater theory. This property is called mode specificity and is discussed in the next section. [Pg.1027]

The model consists of a two dimensional harmonic oscillator with mass 1 and force constants of 1 and 25. In Fig. 1 we show trajectories of the two oscillators computed with two time steps. When the time step is sufficiently small compared to the period of the fast oscillator an essentially exact result is obtained. If the time step is large then only the slow vibration persists, and is quite accurate. The filtering effect is consistent (of course) with our analytical analysis. Similar effects were demonstrated for more complex systems [7]. [Pg.278]

T] Low M.T. rates. Low mass-flux, constant property systems. Ns, % L local k. Use with arithmetic difference in concentration. Coefficient 0.323 is Blasius approximate solution. [Pg.605]

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