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Understand the Real World of Mixing

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Understand the Real World of Mixing Back to Basics Understand the Real World of Mixing Thomas Post Post Mixing Optimization and Most chemical engineering curricula do not Solutions adequately address mixing as it is commonly practiced in the chemical process industries. This article attempts to fi ll in some of the gaps by explaining fl ow patterns, mixing techniques, and the turbulent, transitional and laminar mixing regimes. or most engineers, college memories about mix- This is the extent of most engineers’ collegiate-level ing are limited to ideal reactors. Ideal reactors are preparation for real-world mixing applications. Since almost Fextreme cases, and are represented by the perfect- everything manufactured must be mixed, most new gradu- mixing model, the steady-state perfect-mixing model, or the ates are ill-prepared to optimize mixing processes. This steady-state plug-fl ow model (Figure 1). These models help article attempts to bridge the gap between the theory of the determine kinetics and reaction rates. ideal and the realities of actual practice. For an ideal batch reactor, the perfect-mixing model assumes that the composition is uniform throughout the Limitations of the perfect-mixing reactor at any instant, and gives the reaction rate as: and plug-fl ow models Classical chemical engineering focuses on commodity (–rA)V = NA0 × (dXA/dt) (1) chemicals produced in large quantities in continuous opera- tions, such as in the petrochemical, polymer, mining, fertil- The steady-state perfect-mixing model for a continuous process is similar, with the composition uniform throughout the reactor at any instant in time. However, it incorporates Feed the inlet molar fl owrate, FA0: Uniformly Mixed (–rA)V = FA0 × XA (2) The steady-state plug-fl ow model of a continuous process assumes that the composition is the same within a differential volume of the reactor in the direction of fl ow, typically within a pipe: Product (–r )V = F × dX (3) Uniformly A A0 A Mixed These equations can be integrated to determine the time required to achieve a conversion of XA. Many models are developed as combinations of the two continuous extremes to describe nonideal fl ow, such as in Feed Product recirculation loops, short-circuiting paths, and dead zones S Figure 1. The three types of ideal reactors are the perfect-mixing model (Figures 2 and 3) (1). These extremes are seldom achieved in (left), steady-state perfect-mixing model (right), and the steady-state plug- real life. fl ow model (bottom). CEP March 2010 www.aiche.org/cep 25 Back to Basics izer, and pulp and paper industries. Many of these industries viscosity non-Newtonian fl uids, all of which are common in are mature, and their plants have already been built and the chemical process industries (CPI). optimized. In contrast, industries such as specialty chemi- Reaction control. Engineers employ the perfect-mixing cals, pharmaceuticals, biochemicals, personal care products, and plug-fl ow models to identify reaction-rate constants; foods and beverages, and paints and coatings, as well as this implies a kinetically controlled reaction. Although the emerging fi eld of nanoscale materials, make products in the literature contains much research on kinetics, most of much smaller volumes, employing batch processes as the it provides no proof that the process discussed was truly main mode of mixing. kinetically controlled. Consequently, most students never Perfect-mixing and plug-fl ow models imply a single- learn how to determine whether a process is controlled by phase process. They do not address solids suspensions, kinetics or by mass transfer. liquid-liquid dispersions, gas-liquid dispersions, and high- To prove kinetic control, it must be demonstrated that increases in impeller speed have no effect on the rate of the reaction, and such tests must be conducted at speeds well above the minimum required to carry out the reaction. Most Inlet papers in the literature state only that the reactions were run at a certain speed (which may or may not have been the minimum), without elaborating on whether the impel- ler speed had any impact on the reaction rate. Some papers claim that kinetic reaction rate constants vary for different C impeller speeds. However, these conditions actually describe a mass-transfer-controlled reaction, not a kinetically con- A B trolled reaction. If a process is truly kinetically controlled, it does not Recirculation Loop Recirculation matter whether the impeller is a Rushton turbine, a hydro- C foil, or a rotor-stator. For kinetically controlled processes, B the speed of the impeller and its power input are also irrel- evant. These things should be taken into account only for mass-transfer-controlled processes. Pump The real mixing world The fl ow fi eld of a mixing tank is three-dimensional and Outlet very complex. To get an exact account of all the velocities and shear rates in a mixing tank, computational fl uid dynam- S Figure 2. This schematic shows tank fl ow patterns, including well- ics (CFD) is used to solve the partial differential equations mixed zones (A), the short-circuiting path (B), and dead zones (C). (PDEs) associated with the Navier-Stokes momentum, energy and continuity equations (2). Any solution method- Perfectly Mixed ology must address all three sets of PDEs simultaneously. CFSTR Plug Flow Analytical solutions derived directly from the PDEs are only useful for the simplest of cases. For more complicated Short-Circuited problems, the PDEs are solved in terms of gradients using X A approximate solution methods, such as fi nite difference, Recirculation Loop fi nite volume, and fi nite element techniques. The greatest diffi culty in solving these three equations is the nonlinear- ity of the momentum equation, which makes the resulting implementations extremely unstable. Therefore, approxima- Dead Zone tions to the solutions can only be as accurate as the accuracy of the discretized domain at representing the gradients of the t variables (e.g., velocity, pressure, temperature, etc.). Many CFD software programs include lines of code that introduce artifi cial diffusion to stabilize the non- S Figure 3. This compares the conversion rates of the perfectly mixed (blue) and plug-fl ow (red) models with real-life performance (green). linear solutions. Doing this, however, may be problematic because it violates one of the Navier-Stokes equations 26 www.aiche.org/cep March 2010 CEP Nomenclature C = constant in liquid-liquid correlation in Eq. 14 T = tank diameter, m D = impeller diameter, m V = liquid volume in the reactor, m3 (D/T)opt = the optimum ratio relating impeller diameter to vsg = superfi cial gas velocity, m/s 2 3 tank diameter, dimensionless We = Weber number = (ρcn D )/μ dp = particle size, m x = an exponent for impeller diameter, D, that informs E = a correlation factor that collectively describes the scale-up design effects of fl uids other than water, as well as tem- X = solids loading = 100wt.%/(100 – wt.%) perature, viscosity, and impeller design on the gas- XA = fraction of reactant A converted into product liquid mass-transfer coeffi cient Z = liquid level, m FA0 = initial molar fl owrate of substance A at time 0, mol/s Greek Letters Fr = Froude number = n2D/g αLL = exponent in Eq. 14 hP = process-fl uid-side heat-transfer coeffi cient, α = factor that describes the effect of fl uids other than W/m2-K water on gas-liquid mass transfer 2 g = gravitational constant, 9.81 m/s ξimp = term representing the effect of the impeller design KF = factor reducing impeller power in the presence of on gas-liquid mixing gas = Pgassed/Pungassed ξvis = term representing the effect of the viscoscity on –1 kLa = gas-liquid mass-transfer coeffi cient, s gas-liquid mixing n = impeller speed, rev/s ρ = density, kg/m3 3 NA0 = initial number of moles of reactant A at time 0, mol ρS = solid density, kg/m 3 3 Nae = aeration number = QG/(nD ) ρL = liquid density, kg/m 3 NaeF = aeration number at the fl ood point ρSlurry = slurry density, kg/m 3 nJS = impeller speed at which the last particle is just ρc = continuous-phase density, kg/m 3 suspended, rev/s ρdispersion = dispersed-phase density, kg/m nmin = minimum impeller speed to achieve a liquid-liquid μ = viscosity, Pa-s dispersion, rev/s μc = continuous-phase viscosity, Pa-s Nmix = dimensionless mixing time = nΘmix μd = dispersed-phase viscosity, Pa-s Np = power number = P/(ρn3D5) ν = kinematic viscosity, m2/s Pgassed = impeller power under gassed conditions, W σ = interfacial tension, N/m Pimp = impeller power, W Θ = temperature factor for gas-liquid mass transfer Pieg = power of the isothermal expansion of gas bubbles, W Θmix = mixing time, s PJS = impeller power at the just-suspended speed, W Φ = gas hold-up, unitless Pmin = minimum impeller power to achieve a liquid-liquid dispersion, W Subscripts Pungassed = impeller power under ungassed conditions, W A = reactant descriptor 3 QG = gas fl owrate, m /s An = anchor 3 QL = liquid fl owrate, m /s HR = helical ribbon 3 rA = rate of reaction based on liquid volume, mol/m /s T = turbulent Re = Reynolds number = (ρnD2)/μ TL = the boundary between the transitional and laminar S = Zwietering constant in solid-liquid correlation regimes SV = settling velocity, m/s TT = the boundary between the turbulent and transitional t = time, s regimes (usually the continuity equation). Correctly performed CFD analysis describes the non- For fi nite-element analysis, the Galerkin/least-squares ideal world of real fl ow situations. However, producing CFD fi nite element (G/L-SFE) method has a completely self- models often consumes large amounts of time and money, consistent and mathematically rigorous formulation that and analyzing the results can be diffi cult for even a mixing provides stability without sacrifi cing accuracy.
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