Nonisothermal Laminar Flow

The temperature counterpart of 3JjR is atjR, and if atjR is low enough, then the reactor will be adiabatic. Since a the situation of an adiabatic laminar flow 

The temperature counterpart of Q aVR ccj-F/R and if ccj-F/R is low enough, then the reactor will be adiabatic. Since aj 3 a, the situation of an adiabatic, laminar flow reactor is rare. Should it occur, then T i, will be the same in the small and large reactors, and blind scaleup is possible. More commonly, ari/R wiU be so large that radial diffusion of heat will be significant in the small reactor. The extent of radial diffusion will lessen upon scaleup, leading to the possibility of thermal runaway. If model-based scaleup predicts a reasonable outcome, go for it. Otherwise, consider scaling in series or parallel. 

Polymerizations often give such high viscosities that laminar flow is inevitable. A t5rpical monomer diffusivity in a polymerizing mixture is 1.0 X 10 ° m/s (the diffusivity of the polymer will be much lower). A pilot-scale reactor might have a radius of 1 cm. What is the maximum value for the mean residence time before molecular diffusion becomes important What about a production-scale reactor with R= 10 cm  

The velocity profile for isothermal, laminar, non-Newtonian flow in a pipe can sometimes be approximated as 

Repeat Example 8.1 and obtain an analytical solution for the case of first-order reaction and pressure-driven flow between flat plates. Feel free to use software for the S5anbolic manipulations, but do substantiate your results. 

Equation (8.4) defines the average concentration, Ugut, of material flowing from the reactor. Omit the V ir) term inside the integral and normalize by the cross-sectional area, Ac = ttR, rather than the volumetric flow rate, Q. The result is the spatial average concentration a patiai, and is what you would measure if the contents of the tube were frozen and a small disk of the material was cut out and analyzed. In-line devices for measuring concentration may measure a panai rather than Uout- Is the difference important  

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Temperature Profile n (1) In extrusion or injection molding, the sequence of barrel temperatmes from feed opening to head, sometimes presented as a plot of temperatme versus longitudinal position, hence profile. (2) The sequence of metal temperatmes across a sheet or film die, or around a large blown-fihn die. (3) The sequence of temperatmes across the width of a slab of newly extruded or cast plastic foam, is indicated by temperatme sensors placed laterally at the same depth in the foam. (4) In analysis of Nonisothermal, laminar flow of very viscous liquids (e.g., polymer melts) within tubes and dies, the sequence of temperatmes from the one sidewall through the center to the opposite sidewall at any point along the axis of flow. Such profiles have also been measured experimentally with traversing thermocouples. 

All pipe-line work to date has dealt with fluids which are not thixotropic and rheopectic. To an extent this may be justified because the limiting conditions (at startup—for thixotropic materials, and after long times of shear for rheopectic fluids) in pipe flow and some mixing problems are of primary importance. Design for these conditions would be similar to the techniques discussed herein for other fluids. This is not true of problems in heat transfer, however, and inception of work on the laminar flow of thixotropic fluids in round pipes would appear to be in order as a prerequisite to an understanding of such more complex nonisothermal problems. 

The nonisothermal polymerization process in a tubular reactor at laminar flow is investigated in141 Experimental data on the polymerization of styrene under these conditions are presented in142). 

What models should be used either for scaleup or to correlate pilot plant data Section 9.1 gives the preferred models for nonisothermal reactions in packed beds. These models have a reasonable experimental basis even though they use empirical parameters D, hr, and Kr to account for the packing and the complexity of the flow field. For laminar flow in open tubes, use the methods in Chapter 8. For highly turbulent flows in open tubes (with reasonably large L/dt ratios) use the axial dispersion model in both the isothermal and nonisothermal cases. The assumption D = E will usually be safe, but do calculate how a PFR would perform. If there is a substantial difference between the PFR model and the axial dispersion model, understand the reason. For transitional flows, it is usually conservative to use the methods of Chapter 8 to calculate yields and selectivities but to assume turbulence for pressure drop calculations. 

For nonisothermal systems a general differential equation of conservation of energy will be considered in Chapter 5. Also in Chapter 7 a general differential equation of continuity for a binary mixture will be derived. The differential-momentum-balance equation to be derived is based on Newton s second law and allows us to determine the way velocity varies with position and time and the pressure drop in laminar flow. The equation of momentum balance can be used for turbulent flow with certain modifications. 

We should note that the Navier-Stokes equation holds only for Newtonian fluids and incompressible flows. Yet this equation, together with the equation of continuity and with proper initial and boundary conditions, provides all the equations needed to solve (analytically or numerically) any laminar, isothermal flow problem. Solution of these equations yields the pressure and velocity fields that, in turn, give the stress and rate of strain fields and the flow rate. If the flow is nonisothermal, then simultaneously with the foregoing equations, we must solve the thermal energy equation, which is discussed later in this chapter. In this case, if the temperature differences are significant, we must also account for the temperature dependence of the viscosity, density, and thermal conductivity. 

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