Nonisothermal Piston Flow

FIGURE 5.3 Differential element in a nonisothermal piston flow reactor. 

Unlike a molar flow rate—e.g, aQ—the mass flow rate, pQ, is constant and can be brought outside the differential. Note that Q = uAc and that is the external surface area per unit length of tube. Equation (5.22) can be written as 

This equation is coupled to the component balances in Equation (3.9) and with an equation for the pressure e.g., one of Equations (3.14), (3.15), (3.17). There are A +2 equations and some auxiliary algebraic equations to be solved simultaneously. Numerical solution techniques are similar to those used in Section 3.1 for variable-density PFRs. The dependent variables are the component fluxes I , the enthalpy H, and the pressure P. A necessary auxiliary equation is the thermodynamic relationship that gives enthalpy as a function of temperature, pressure, and composition. Equation (5.16) with Tref=0 is the simplest example of this relationship and is usually adequate for preliminary calculations. 

With a constant, circular cross section, A = 2jiR (although the concept of piston flow is not restricted to circular tubes). If Cp is constant, 

This is the form of the energy balance that is usually used for preliminary calculations. Equation (5.24) does not require that u be constant. If it is constant, we can set dz = udt and 2IR = AextlAc to make Equation (5.24) identical to Equation (5.19). A constant-velocity, constant-properties PER behaves 

Steady state and differential quantities are now used. The pQH terms cancel and Az factors out to give 

This equation is coupled to the component balances of Equation 3.9 and with an equation for the pressure, for example, one of Equations 3.18, 3.19, or 3.21. There are iV + 2 equations and some auxiliary algebraic equations that need to be solved simultaneously. Numerical solution techniques are similar to those used in Section 

1 for variable-density PFRs. The dependent variables are the component fluxes the enthalpy H, and the pressure P. A necessary auxiliary equation is the thermodynamic relationship that gives enthalpy as a function of temperature, pressure, and composition. Equation 5.15 with = 0 is the simplest example of this relationship and is usually adequate for preliminary calculations. 

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Most kinetic experiments are run in batch reactors for the simple reason that they are the easiest reactor to operate on a small, laboratory scale. Piston flow reactors are essentially equivalent and are implicitly included in the present treatment. This treatment is confined to constant-density, isothermal reactions, with nonisothermal and other more complicated cases being treated in Section 7.1.4. The batch equation for component A is... 

Correlations for E are not widely available. The more accurate model given in Section 9.1 is preferred for nonisothermal reactions in packed-beds. However, as discussed previously, this model degenerates to piston flow for an adiabatic reaction. The nonisothermal axial dispersion model is a conservative design methodology available for adiabatic reactions in packed beds and for nonisothermal reactions in turbulent pipeline flows. The fact that E >D provides some basis for estimating E. Recognize that the axial dispersion model is a correction to what would otherwise be treated as piston flow. Thus, even setting E=D should improve the accuracy of the predictions. 

Example 9.6 Compare the nonisothermal axial dispersion model with piston flow for a first-order reaction in turbulent pipeline flow with Re= 10,000. Pick the reaction parameters so that the reactor is at or near a region of thermal runaway. 

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