Tracer perfect step

Note that when k] = k.2 = k3 = 0, Cao = 1> Cbo = 0- the program solves the case of a step input of tracer solution, which can be used to generate the typical F-diagram for a single perfectly mixed tank. Compare this result with the analytical solution. 

Z+1 designates the number of states, i.e. Z perfectly mixed reactors in the flow system as well as the tracer collector designated by As shown later, the probabilities Si(0) may be replaced by the initial concentration of the fluid elements in each state, i.e. Cj(0) and S(0) will contain all initial concentrations of the fluid elements. The one-step transition probability matrix is given by Eqs.(2-16) and (2-20) whereas pjk represent the probability that a fluid element at Cj will change into Ck in one step, pjj represent the probability that a fluid element will remain unchanged in concentration within one step. 

The inlet concentration most often takes the form of either a perfect pitlse input (Dirac delta function), imperfect pulse injection (see Figure 13-4), or a step input-Just as the RTD function (/) can be determined directly from a pulse input, the cumulative distribution Fit) can be determined directly from a step input. We will now analyze a step input in the tracer concentration for a system with a constant volumetric flow rate. Consider a constant rate of tracer addition to a feed that is initiated at time t = 0. Before this time no tracer was added to the feed, Stated symbolically, we have... 

Figure 3.5 Response to a step input of a tracer from a perfect (a) PFR and (b) MFR.

F(t) is a probability distribution which can be obtained directly from measurements of the system s response in the outflow to a step-up tracer input in the inflow. Consider that at time t = 0 we start introducing a red dye at the entrance of the vessel into a steady flow rate Q of white carrier fluid. The concentration of the red dye in the inlet flow is C. At the outlet we monitor the concentration of the red dye, C(t . If our system is closed, i.e. if every molecule of dye can have only one entry and exit from the system (which is equivalent to asserting that input and output occur by convection only), then QC(t)/QCQ is the residence time distribution of the dye. This is evident since all molecules of the dye appearing at the exit at time t must have entered into the system between time 0 and time t and hence have residence times less than t. Only if our red dye is a perfect tracer, i.e.. if it behaves identically to the white carrier fluid, then we have also obtained the residence time distribution for the carrier fluid and F(t) = C(t)/C. To prove that the tracer behaves ideally and that the F curve is obtained, the experiment should be repeated at different levels of C. The ratio C(t)/C at a given time should be invariant to C, i.e. the tracer response and tracer input must be linearly related. If this is not the case, then C(t)/CQ is only the step response for the tracer, which includes some nonlinear effects of tracer interactions in the system, and which does not represent the true residence time distribution for the system. 

If. eq. (4) is not satisfied for step-up and step-down tracer tests, then the proper distribution functions have not been obtained by the tracer test. The cause might be either that the system was not at steady state, or that the tracer did not behave perfectly and underwent a nonlinear process within the system. Another requirement for establishing the F(t) curve correctly from tracer step-up or step-down experiments, which is particularly important for systems with laminar flow, is that tracer must be introduced proportionally to flow (flow tagging) and that its mixing cup concentration must be monitored at,the outflow (18-21). 

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