Isothermal Laminar Flow

Reactors in isothermal laminar flow are exactly scaleable using geometric similarity if diffusion is negligible in the pilot reactor. Converting Equation (8.2) to 

The temperature counterpart of 2At/R is art/R2 , and if art/R/2 is low enough, then the reactor will be adiabatic. Since ar 9 a, the situation of an adiabatic, laminar flow reactor is rare. Should it occur, then T(k, j ) will be the same in the small and large reactors, and blind scaleup is possible. More commonly, art/R2 will 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 typical monomer diffusivity in a polymerizing mixture is 1.0 x 10 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 symbolic manipulations, but do substantiate your results. 

It may not be feasible to have an adequately low value for Q AtlP and still scale using geometric similarity. Recall that reactor scaleups are done at constant t. The problem is that the pilot reactor would require too high a flow rate and consume too much material when is small enough (i.e., R is 

It may not be feasible to have an adequately low value for and still scale 

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Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian hquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic flmd described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length AP/L, the flow rate is given by... 

Kirelic/lranspon for Isothermal Laminar-Flow Reactor with no Axial Dispersion [See Shinohara and Christiansen (I974J for ilie non-isoihermul... 

Show that the volumetric flowrate of this fluid in a horizontal pipe of radius a under isothermal laminar flow conditions with a pressure gradient —AP/l per unit length is ... 

This chapter assumes isothermal operation. The scaleup methods presented here treat relatively simple issues such as pressure drop and in-process inventory. The methods of this chapter are usually adequate if the heat of reaction is negligible or if the pilot unit operates adiabatically. Although included in the examples that follow, laminar flow, even isothermal laminar flow, presents special scaleup problems that are treated in more detail in Chapter 8. The problem of controlling a reaction exotherm upon scaleup is discussed in Chapter 5... 

The same result is obtained when the fluid is compressible, as may be seen by substituting Sr = Si = S into Equations (3.40) and (3.41). Thus, using geometric similarity to scale isothermal, laminar flows gives constant pressure drop provided the flow remains laminar upon scaleup. The large and small reactors will have the same inlet pressure if they are operated at the same outlet pressure. The inventory and volume both scale as S. 

ISOTHERMAL LAMINAR FLOW WITH NEGLIGIBLE DIFFUSION... 

Consider isothermal laminar flow of a Newtonian fluid in a circular tube of radius R, length L, and average fluid velocity u. When the viscosity is constant, the axial velocity profile is... 

Equation (8.9) can be applied to any reaction, even a complex reaction where ctbatch(t) must be determined by the simultaneous solution of many ODEs. The restrictions on Equation (8.9) are isothermal laminar flow in a circular tube with a parabolic velocity profile and negligible diffusion. 

Consider an isothermal, laminar flow reactor with a parabolic velocity profile. Suppose an elementary, second-order reaction of the form A -h B P with rate SR- = kab is occurring with kui 1=2. Assume aj = bi . Find Uoutlam for the following cases ... 

The pilot reactor is a tube in isothermal, laminar flow, and molecular diffusion is negligible. The larger reactor wiU have the same value for t and will remain in laminar flow. The residence time distribution will be unchanged by the scaleup. If diffusion in the small reactor did have an influence, it wiU lessen upon scaleup, and the residence time distribution will approach that for the diffusion-free case. This wiU hurt yield and selectivity. 

Incompressible Flow and the Finite Element Method, Volume 1 Advection-Diffusion and Isothermal Laminar Flow,... 

Eigure 2.4a shows the velocity distribution in a steady isothermal laminar flow of an incompressible Newtonian fluid through a straight, round tube. The velocity distribution in laminar flow is parabolic and can be represented by... 

Cylinders Consider a Power Law model fluid placed between two long concentric cylinders of radii R, and R0. At a certain time the inner cylinder is set in motion at constant angular velocity O rads/s. Assuming steady isothermal laminar flow without slip at the walls, neglecting gravitational and centrifugal forces, the velocity profile is... 

Purging a Tubular Die A red polymer is pumped through a tubular die. At time f, the inlet stream is switched over to a white polymer for purging the die. Assuming Newtonian fluids, identical viscosities and densities, and fully developed isothermal laminar flow, calculate the volume fraction of red polymer left in the die at the time the first traces of white polymer appear at the exit. 

Strain Distribution Function in Poiseuille Flow (a) Derive the SDF F(y) for fully developed isothermal laminar flow of a Newtonian fluid in a tube, (b) Calculate the mean strain, (c) If the length of the tube is 1 m and its radius 0.01 m, what fraction of the exiting stream experiences a total strain of less than 100 ... 

Kinetic/transport for Isothermal Laminar-Flow Reactor with Axial Dispersion under Transient Open-Loop Operation... 

Gresho, P. M., Sani, R. L., Fugleman, M. S., Incompressible Flow and the Finite Element Method. Vol. 2 Isothermal Laminar Flow. Wiley New York, 1998. 

To begin, we restrict our attention to isothermal, laminar flow of an incompressible Newtonian fluid, in which all boundaries are solid surfaces, so that we can use the equations of continuity and motion in the forms (2 20) and (2 91), respectively, that is,... 

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