## Brazilian Journal of Chemical Engineering

##
*Print version* ISSN 0104-6632*On-line version* ISSN 1678-4383

### Braz. J. Chem. Eng. vol.17 n.4-7 São Paulo Dec. 2000

#### http://dx.doi.org/10.1590/S0104-66322000000400059

**RECENT EXTENSIONS OF THE RESIDENCE TIME DISTRIBUTION CONCEPT: UNSTEADY STATE CONDITIONS AND HYDRODYNAMIC MODEL DEVELOPMENTS. **

**S.Claudel ^{1}, J.P.Leclerc^{1*}, L.Tétar^{ 1}, H.G.Lintz^{1} and A.Bernard^{2 }**

^{1}PROGEPI, Laboratoire des Sciences du Génie Chimique,-CNRS-ENSIC,

1 rue Grandville B.P. 451, 54001, Nancy Cedex France

^{2}Laboratoire dEnergétique et de Mécanique Théorique et Appliquée-CNRS-ENSEM

2 Avenue de la forêt de Haye, 54516 Vandoeuvre-les-Nancy Cedex France

*(Received: November 11, 1999 ; Accepted: May 30, 2000 )*

Abstract -Two recent extensions of the residence time distribution concept are developed. The first one concerns the use of this method under transient conditions, a concept theoretically treated but rarely confirm by relevant experiments. In the present work, two experimental set-ups have been used to verify some limits of the concept. The second extension is devoted to the development of hydrodynamic models. Up to now, the hydrodynamics of the process are either determined by simple models (mixing cells in series, plug flow reactor with axial dispersion) or by the complex calculation of the velocity profile obtained via the Navier-Stokes equations. An alternative is to develop a hydrodynamic model by use of a complex network of interconnected elementary reactors. Such models should be simple enough to be derived easily and sufficiently complex to give a good representation of the behavior of the process.

Keywords:residence time distribution, transient state, hydrodynamic model, industrial processes.

**INTRODUCTION**

**Background**

The Residence Time Distribution (RTD) is a chemical engineering concept introduced by Danckwerts in 1953. It has been described in a multitude of scientific papers and applied for various industrial processes. Every year around 50 articles dealing with the RTD method and its applications are published. However, most of them are restricted to new applications. The purpose of this paper is to present recent fundamental developments and extensions of this method.

The development of Computer Fluid Dynamics will improve the understanding and optimization of processes. This method has been successfully tested for simple flow pattern like sedimentation tanks in wastewater treatment [Matko, 1996]. However for complex systems, this approach remains difficult. Therefore, the extension of the RTD concept is a way to obtain hydrodynamic data and to permit the improvement of processes. The main limiting assumption is that the studied process runs under steady state during the tracer experiment. Unfortunately, this assumption is too restrictive for many processes such evaporator or wastewater treatment plant. The first publication relative to the use of the systemic approach in transient state is due to Nauman (1969) who developed this concept for a perfect mixing cell. Niemi (1977) has extended this work to different models. More recently, Niemi et al. (1998) and Fernandez-Sempere et al. (1995) have experimentally developed this concept. The discussion between the authors [Niemi, 1997 and Fernandez-Sempere, 1997] shows that this concept is still questionable.

The models derived from tracer experiments are often limited to simple elementary reactors such as perfect mixing cells in series [Lin, 1999], plug flow with axial dispersion [van Hasselt, 1999], mixing cells in series exchanging with a dead zone [Leclerc, 1996] or simple compartment models [Levenspiel 1999]. The obtained results are not sufficient for understanding complex processes. The creation of a complex network of interconnected elementary reactors makes possible to get better results more easily than with Computer Fluid Dynamics. However, this induces two problems: the first one is the necessity to simulate easily any complex network and the second one is to obtain a realistic model. Indeed, the complex model has so many parameters that two different models can give the same result or the same model can give same results with different sets of parameters.

**RTD UNDER TRANSIENT STATE CONDITIONS**

The residence time distribution for steady state condition can be directly derived from a pulse injection of tracer in the studied process. However, for many industrial processes, the inlet flow Q_{E}(t) and the outlet flow Q_{S}(t) are time dependent. V(t) is the volume and M the initial quantity of tracer injected in the reactor.

The initial concentration of tracer is and C(t) is the concentration of tracer at current time. Then, using mass balance, transfer functions could be easily found for different models and states (cf. Table 1), with

(E(t) / Q_{S}(t)) depend only of the integral of Q_{S}(t) (or the integral of Q_{S}(t) and the integral of V(t)) which remains a constant.

Niemi *et al.* (1997) carried out experiments with a continuous flow vessel with mecanical agitator. They used chemical and radioisotope tracers for variations (sinusoidal) of flow-rate (0.450 to 0.980l/min) and volume (0.800 to 1.200 liters). Their results for unsteady state conditions were close to those carried out in steady state. Fernandez-Sempere *et al.* (1995) investigated experiments with wastewater of their university, which is discharged in sewage system of Alicante. Flow-rate (outlet flow around 1E+3m^{3}/s) and volume variations are not periodic. Results are shown either in steady state or transient state but not together in order to see if a general RTD theory, derived from regular RTD for stationary conditions, is confirmed by experiments.

In the present work, different amplitudes and frequencies of the inlet flow-rate fluctuations (cf. Table 2) have been investigated with two experimental set-ups. The first one is the pilot of an activated sludge channel reactor used in a wastewater treatment plant (cf. Fig. 1). This pilot has been chosen because the inlet flow-rate in the wastewater treatment plant varies all the time. It is divided into several compartments by baffles. Its dimensions are 0.500m long, 0.300m large and 0.205m high. Aeration is done uniformly, over the entire length, by tubes of stainless steel pierced with small holes. The experiments are undertaken with water and the tracer used is sodium chloride, detectable by conductrimetry. Impulse injection of the salty solution is performed using a seringe. At steady state, the flow pattern can be modeled by perfect mixing cells in series: N=4.6 for 1 l/min; N=3.9 for 0.58 l/min and N=7.5 for 1.88 l/min.

The second set-up is a fixed bed column with a loop of recirculation (cf. Fig. 2). Its dimension is 1.100m high and 0.050m diameter. This pilot is furnished with beads of glass of 2mm diameter, the porosity of bed is 0.38. The experiments are undertaken with water and the tracer used is the nigrosine, detectable by absorption. A Dirac impulsion of tracer is introduced at the inlet of the reactor. The flow has been also modeled by perfect mixing cells in series; in this case, the flow pattern is well reproduced for a number of cells ranging between 80 and 180 in the range of studied flow-rates.

During the tests, the liquid volume was kept constant. Figures 3A and 3B represent RTD curves in the aeration sludge reactor at a mean flow-rate Q_{mean}=1 l/min. Figure 3A shows results for steady state at flow-rate Q=1l/min and for transient state with a change of flow-rate every 10 min period between two values of the amplitudes: ±25% around the mean flow-rate Q_{mean} (dotted line) and ±95% around the mean flow-rate Q_{mean} (thin solid line). Figure 3B shows RTD curves for the same parameters but with a change of flow-rate at a period of 20 min. Using the transfer function defined in table 1 and the number of cells corresponding to the average flow-rate (N=4.6), the obtained RTD curves are similar in transient and steady states, despite the important influence of the flow-rate on the number of cells. The observed fluctuations are mainly due to the difficulty to estimate accurately the flow-rate. Figures 3C and 3D represent results for the fixed bed column for a mean flow-rate Q_{mean}=3 l/min. Figure 3D shows results for steady state and for transient state with a change of flow-rate every 40s period at two different amplitudes: ±50% around the mean flow-rate Q_{mean} (dotted line) and ±75% around the mean flow-rate Q_{mean} (thin solid line). Transient responses are very similar to that of steady state flows for studied amplitudes. Figure 3C shows results for the same parameters but with a change of flow-rate at a period of 10 s. The form of transient responses looks like to some extent that of steady state flow, but that is less accurate than for a period of 40 s. Once again the fluctuations are mainly due to flow-rate measurement, especially for high frequencies of fluctuations. Further experiments will carry out to confirm these results with other models in particular model using recirculation loop of the fixed bed column. The influence of the period compare to the mean residence time should be also study for different configurations. Other experiments will be also conduct with a better control of flow-rate measurement to detect possible unexpected internal flow behaviour.

**HYDRODYNAMIC MODEL DEVELOPMENT**

The systemic approach allows to determine a hydrodynamic model by interconnection of elementary reactors (plug flow reactor, mixing cells in series ). The models developed are often limited to a single model with one elementary reactor (mixing cells in series, plug flow reactor with axial dispersion ). These models make it possible to determine an axial dispersion or a number of mixing cells but they give often a poor representation of the real hydrodynamic behavior. The use of the Computer Fluid Dynamics permits the determination of the velocities and pressures profiles along the reactors. An alternative is to develop through the systemic approach a proper network of interconnecting elementary reactors which is complex enough to represent the real hydrodynamic behavior of the reactor. This approach has been already successfully tested for complex geometries. However, these models involve a large number of parameters, which should be properly determined to avoid the representation of the experimental results with a model without physical meaning. Nevertheless, the analysis of the shape of the RTD curve and of the process itself coupled with colored tracing experiments (when it is possible) are sufficient to overcome the problem. The optimization of the parameters can be easily solved using the existing DTS software [Leclerc, 1995]. The problem remains complex for the optimization of the internal flow-rate of the network.

**Simulation of Complex Network, Optimization of The Parameters**

A software package has been developed to allow the creation of complex networks of interconnected elementary reactors. Figure 4 shows a typical example of such a model. The network is described in terms of so called branch and node. The software can simulate the response to an input of any complex network [Leclerc, 1995]. Eight different elementary reactors can be chosen (see Table 3). The transfer function and the characteristics of these reactors has been already described by many authors [e.g. Levenspiel, 1999 and Villermaux, 1995]. The parameters can be optimised by comparison of experimental data with the response of the model.

**Optimization of Flow-Rates**

The optimization of the parameters, which are only depending on the elementary reactors (see table 3) can be run easily. The optimization of the flow-rate is much more difficult for two reasons:

(a) First, when one flow-rate is changing they other ones should change also in order to respect the mass balance,

(b) second, when one flow-rate is changing, the corresponding "mean residence time" is changing too.

The preliminary network analysis enables us to determine the fixed flow-rates of the network (inlet and outlet flow-rate, flow-rate fixed by the user ). Since the problem has more non-fixed flow rate than equations, the program generates itself the possible sets of flow-rates, which can be, optimized. The remaining ones are determined through the mass balance equations. Consequently, each branch has to be connected to, at least, one node for which all the other flow-rates are determined. The "matrix of connection" (CONNEC) indicates the links between the different nodes and branches and the flow-rate in these branches, which can be optimized. The dimensions of this matrix are given by the total number of nodes equal to the number of lines and the total number of branches equal to the number of columns. When the flow-rate of the branch number "J" can be optimized and when this branch is connected to the node number "I" so CONNEC(I, J)= J elsewhere CONNEC(I, J) = 0.

To each set, one assigns the similar matrix. Values of each set are withdrawn from their matrix. If on a line of a matrix, which corresponds to a node, there is one non-null value, which corresponds to an unknown flow-rate, it means that this flow-rate, can be estimated through mass balance, etc... At the end, if a matrix is null, then the corresponding set allows us to determine all the flow-rates of the network. Once all the possible sets obtained, the optimization using the method of box complex can be run for each set. The solutions without any physical meaning are removed by the operator. The solution is considered to be the set with the lowest objective function, which is physically meaningful.

A simple example (see network Fig. 4) has been chosen to demonstrate the methodology. The same procedure has been successfully tested for complex networks involving a high number of branches and complex internal recirculation. In the present case, the flow-rates in the four branches (2, 3, 4 and 5) are not fixed. The program allows us to determine that two flow-rates are enough to describe the system. Thus, the matrix CONNEC has 5 lines and 6 columns as shown in table 4a and 4d. If the set chosen is (2; 3), matrix CONNEC is transformed as shown in table 4b. For the set (2; 5), CONNEC is changed as shown in table 4e. After calculation on lines, CONNEC on table 4c is null while that on table 4f is not, therefore the set (2; 3) is one of the solutions required whereas the (2; 5) is not. Solutions required are the five following sets (2; 3), (2; 4), (3; 4), (3; 5) and (4; 5).

The outlet of the model with different fixed flow-rates has been simulated using the DTS software [Leclerc, 1995]. Using the obtained result as a possible experimental curve, the new algorithm has been used to recover the possible sets to be optimized and to obtain the flow-rates with a good accuracy. Other examples have been tested to optimize complex network of flow-rate. The precision of the optimization decreases with the number of flow-rates to optimize.

**Automatic Generation of the Network**

The existing software performs the estimation of parameters for fixed model structures provided by the user. The next step is to perform the generation of the model structures. This concept has been first published by Laquerbe (1998). However, the methodology used is debatable. First, the model takes into account only two elementary reactors (plug flow reactor and perfect mixing cell). Secondary the example describing the methodology used is very atypical for the mere observation of the RTD curve allows to determine unequivocally the structure of the model by optimization. This case is not representative of typical curve, which can be described by different models. Moreover, the concept is based on a mathematical approach, which is in contradiction with the main problem of the methodology. Indeed a mathematical approach will choose the best transfer function based only on mathematical considerations, which may have no physical meaning. Physical aspects bring to the system a knowledge, which the mathematical aspect cannot bring from its nature.

As a consequence, the automatic generation of RTD models needs both physical information about the studied process and optimization procedure. Complex geometry induces a more complex flow behavior and requires an adapted corresponding model. Although a simple model can be sufficient for representing correctly the RTD, it is not complete enough to describe the real flow and bring physical information on this flow. A very good knowledge of the geometry of the reactor and operating conditions are thus necessary in order to establish a model as faithful as possible to the real flow, and thus to permit to draw some useful informations for future experiments (better operating conditions, realistic simulations, ). Typical examples of information about RTD models deduced from the characteristics of the process are given in table 5. A compilation of such relations has been created using literature and results from industry.

**CONCLUSION**

Up to now, the determination of hydrodynamic behavior is mainly limited to two extreme approaches. The first one is the RTD interpretation through simple models, which gives the dispersion of general parameters such as the number of mixing cells. The second one is the use of Computer Fluid Dynamics, which provides an accurate description of the flow pattern. The first method is poor whereas the second one is time consuming and sometime limited due to the complexity of the studied process. The extension of the first method through the creation of complex network of elementary reactors and use under transient state can overcome this problem. The theory of RTD under transient state has been experimentally confirmed for different operating conditions. The results show that even when the model is strongly depending on the flow-rate, the theory of RTD under transient state can be applied using the model of the average flow-rate at steady state. Further studies will be conduct to determine the limits of this concepts utilization for more complex internal flow structures. Random flow-rate fluctuations will be also tested to simulate more reliable industrial situations. Software has been previously developed to make easier the creation of complex hydrodynamic model by interconnection of elementary reactors. All the parameters of the structure can be optimized through comparison with experimental tracer experiments. New developments allow us to determine the optimum flow-rate distribution in complex hydrodynamic models. The creation of an expert system for deriving directly the structure of the model from experimental data and user experience is under development.

**ACKNOWLEDGEMENTS**

The authors are grateful to the workshop team of the LSGC who build the two experimental set-up and to Alain Chenu and Mathieu Weber who designed the data acquisition and control systems. The authors are indebted to Virginie Loiseau for her assistance in some experiments

**NOMECLATURE**

C | outlet concentration of tracer at time t. |

C_{0 } | initial concentration of tracer. |

E_{q} | dimensionless function of residence time distribution. |

_{Eq' } | dimensionless function of residence time distribution (Q_{0} variable). |

E_{q''} | dimensionless function of residence time distribution (Q_{0} et V variable). |

G | transfer function. |

M | quantity of tracer injected. |

N | number of reactors. |

Q | volumic flow-rate of fluid. |

Q_{E } | volumic inlet flow-rate. |

Q_{S } | volumic outlet flow-rate. |

*Greek Letters*

t | space time V/Q. | |

q | dimensionless time. | |

q' | dimensionless time, EV/Q_{0} (Q_{0} variable). | |

q'' | dimensionless time, EV/Q_{0} (Q_{0} et V variable). |

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*To whom correspondence should be addressed