Tracer response curve linear

An RTD, however, does not represent the mixing behavior in a vessel uniquely, because several arrangements of the internals of a vessel may give the same tracer response, for example, any series arrangements of reactor elements such as plug flow or complete mixing. This is a consequence of the fact that tracer behavior is represented by linear differential equations. The lack of uniqueness limits direct application of tracer studies to first-order reactions with constant specific rates. For other reactions, the tracer curve may determine the upper and lower limits of reactor performance. When this range is not too broad, the result can be useful. Tracer data also may be taken at several representative positions in the vessel in order to develop a realistic model of the reactor. 

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. 

Equation (7) has been used traditionally in medicine as the Stewart-Hamilton indicator-dilution technique (23-25) to measure the unknown flow rate through isolated organs. In this method a known bolus, mr, of tracer is injected into an artery and the tracer dilution curve, Cp(t), is measured at the outflow in a major vein. In chemical engineering systems eq. (7) should be used as a basic mass balance consistency check on the accuracy of the tracer experiment. Failure to satisfy eq. (7) indicates errors and nonidealities, and requires repeats of the experiment at different mij, s to establish linearity. On occasion, only a response R(t), which is linearly proportional to tracer concentration, is measured and the mass injected m,j, is unknown. Then the experiment should be repeated to establish linearity of R(t) with respect to HLp. The obtained age density function R(t)// R(t)dt should be invariant to the mass injected. Although the above simple rules are rather evident, they are all too often neglected by practitioners. Unfortunately, not all tracers at concentrations used behave ideally, and erroneous conclusions result when the nonideal tracer residence time distribution is accepted as the F curve of the system. The tracer tests needed to determine E(t), F(t) and W(t) functions are schematically represented in Figure 1. 

A further important conclusion is that for a given C-curve or residence time distribution obtained from tracer studies, a unique value of the conversion in a chemical reaction is not necessarily obtainable unless the reaction is first order. Tracer measurements can certainly tell us about departures from good macromixing. However, tracer measurements cannot give any further information about the extent of micromixing because the tracer stimulus-response is a first-order (linear) process as is a first-order reaction. 

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