Unsteady Stirred Tanks

A well-mixed stirred tank (which we will continue to call a CSTR despite possibly discontinuous flow) has p = Pout- The unsteady-state balance for total mass is obtained just by including the accumulation term  

Liquid-phase systems with approximately constant density are common. Thus, the usual simplification of Equation (14.1) is 

The general case treats time-dependent volumes, flow rates, and inlet concentrations. The general case must be used to for most startup and shutdown transients, but some dynamic behavior can be effectively analyzed with the constant-volume, constant-flow rate version of Equation (14.2)  

The case of = a (t) will force unsteady output as will sufficiently complex kinetics. 

A still more general case is discussed in Problem 14.15. Typical simplifications are constant volume and flow rate, constant density, and replacement of enthalpy with Cp T — T ,f)- This gives 

Chemical Reactor Design, Optimization, and Scaleup, Second Edition. By E. B. Nauman Copyright 2008 John Wiley Sons, Inc. 

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An inherent property of the LES approach is that the simulated flow field is no longer steady, but exhibits a transient character due to the presence and motion of large-scale eddies. The LES methodology has proven to be a powerful tool for studying and visualizing stirred tank flows (Eggels, 1996 Derksen et al. 1999 Bakker et al., 2000 Derksen, 2001 Bakker and Oshinowo, 2004), as it inherently takes the unsteady and periodic behavior of the flow (around impeller and baffles) into account. 

Large eddy simulations explicitly resolves the inherently unsteady character of the turbulent flow in a stirred tank into account, including the periodic phenomena associated with the motion of the impeller and their interaction with... 

Batch-stirred tank reactor (BSTR) In this type of reactor, the reactants are fed into the container, they are well mixed by means of mechanical agitation, and left to react for a certain period of time. This is an unsteady-state operation, where composition changes with time. However, the composition at any instant is uniform throughout the reactor. 

Fig. 1.9. Continuous stirred-tank reactor showing steady state operation (a) and two modes of unsteady state operation (b) and (c)...

One of the simplest practical examples is the homogeneous nonisothermal and adiabatic continuous stirred tank reactor (CSTR), whose steady state is described by nonlinear transcendental equations and whose unsteady state is described by nonlinear ordinary differential equations. 

Many chemical and biological processes are multistage. Multistage processes include absorption towers, distillation columns, and batteries of continuous stirred tank reactors (CSTRs). These processes may be either cocurrent or countercurrent. The steady state of a multistage process is usually described by a set of linear equations that can be treated via matrices. On the other hand, the unsteady-state dynamic behavior of a multistage process is usually described by a set of ordinary differential equations that gives rise to a matrix differential equation. 

Reactor Tracer Responses Continuous Stirred Tank Reactor (CSTR) With magnitude Cf, the unsteady material balance of tracer a step input of... 

After specifying the energy form, the catalyst and the phases in contact, the next task is to decide whether to conduct the reaction in a batch or continuous mode. In the batch mode, the reactants are charged to a stirred-tank reactor (STR) and allowed to react for a specified time. After completing the reaction, the reactor is emptied to obtain the products. This operating mode is unsteady state. Other unsteady-state reactors are (1) continuous addition of one or more of the reactants with no product withdrawal, and (2) all the reactants added at the beginning with continuous withdrawal of product. At steady-state, reactants flow into and products flow out continuously without a change in concentration and temperature in the reactor. 

It is usual in laminar mixing simulations to represent the flow using tracer trajectories. The computation of such flow trajectories in a coaxial mixer is more complex than in traditional stirred tank modelling due to the intrinsic unsteady nature of the problem (evolving topology, flow field known at a discrete number of time steps in a Lagrangian frame of reference). Since the flow solution is periodic, a node-by-node interpolation using a fast Fourier transform of the velocity field has been used, which allowed a time continuous representation of the flow to be obtained. In other words, the velocity at node i was approximated... 

Finally the work of Versteeg et al. [564] should be mentioned, since it reported a different way of determining kr from those thus far described. Versteeg et al. pointed out that the effect of molecular diffusivity D and hence of Sc upon kt or Sh could only be established, if it was decoupled from the hydrodynamics (Re). Thus experiments were carried out at pressures of 1-10 bar. kc was determined by absorption measurements under steady-state conditions and some were carried out with reaction gases diluted with inert gas. On the other hand, kt was determined with measurements under unsteady-state conditions and with pure gases. In both cases a stirred tank was used with a known plane surface. 

These are systems where the state variables describing the system are lumped in space (invariant in all space dimensions). The simplest chemical reaction engineering example is thp perfectly mixed continuous stirred tank reactor. These systems are described at steady state by algebraic equations while in the unsteady state they are described by initial value ordinary differential equations where time is the independent variable. 

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