Dead-zone Model Research Articles

Distribution of water quality parameters in two cross‐sections of the river Elbe measured with high local, temporal, and analytic resolution

AbstractThe river Elbe is a large eutrophic lowland river with high primary production and high phytoplankton biomass in the growing season. The objective of the study was to gain basic knowledge of the role of dead zones for phytoplankton distribution and the oxygen balance of the river Elbe. At two sampling stations water temperature, oxygen concentration, conductivity, pH value, turbidity, and chlorophyll fluorescence were measured with high precision to learn about the distribution of these parameters across the cross‐sections. The structures indicate differences in the intensity of physical and biological processes between the shallow waters near the shores, which are characterized by groyne fields, and the deep bulk flow. The conductivity clearly shows the high transverse mixing intensity in the groyne fields in contrast to the bulk flow. Groyne fields and the first, the groyne head near margin lamella of the bulk flow can be regarded as a unit. Groyne field results indicate higher primary production of phytoplankton, oxygen release, and higher pH. The turbidity of water entering the groyne fields decreases rapidly because of sedimentation within a short time. The groyne head lamella of the bulk flow sometimes shows higher chlorophyll fluorescence than the central lamella of the bulk flow and the water of the groyne field. The processes which contribute to the observed distribution are discussed, but could not yet be quantified. A modification of the dead zone model is proposed.

Discharge‐dependent pollutant dispersion in rivers: Estimation of aggregated dead zone parameters with surrogate data

Much has been done to mitigate the effects of intermittent discharges of pollutants; however, pollution incidents still occur, and the downstream transport and dispersion of pollutants must be predicted. The application of transient storage models based on the advection dispersion equation is often limited by the strong dependence of the parameters on changes in discharge. In this paper a methodology is outlined for estimating the parameters of the simple aggregated dead zone model using surrogate data derived from continuous water quality measurements such as conductivity, including a full treatment of the errors and prediction uncertainties within a Bayesian framework. This methodology is demonstrated in the prediction of a tracer experiment on a reach of the river Elbe in the Czech Republic.

Mountain Streams—Modeling Hydraulics and Substance Transport

One-dimensional models are useful tools for predicting hydraulics, substance transport, temperature, and water quality in rivers for flow distances that are long compared to the lateral mixing distance. However, the application of such models to mountain streams is difficult because, in addition to small scale bed roughness, the effects of large-scale roughness, irregularity in bed geometry, and pool and riffle sequences have to be taken into account at a spatially averaged scale. Representation of pools, stagnant zones due to bed irregularity, and interstitial zones in the gravel bed with a dead zone model leads to a description of substance transport that is strongly simplified but correct in the representation of basic mechanisms. However, this approach leads to more severe problems regarding parameter identifiability compared to inclusion of these effects in an enlarged friction coefficient. To overcome these problems, we applied a friction estimator developed for river sections without pools and were then able to identify the dead zone volume and exchange parameters. In a case study in the southern Swiss Alps, the dead zone volume fraction and the residence time in the dead zone did not strongly depend on stream discharge, but these parameters varied strongly from one stream section to another. This indicates that the evaluation of a tracer test for one single flow condition significantly decreases prediction uncertainty of substance transport for a specific stream reach.

Tracer experiments in the Rhine Basin: evaluation of the skewness of observed concentration distributions

Field studies reporting on the propagation of a pollution wave travelling down a river mostly show persistence of the temporal skewness. As a result, in the Rhine Alarm-Model a constant skewness coefficient (equal to 1) has been applied. The appropriateness of this assumption has been proven by tracer experiments. This finding seems to be in conflict with the solution of the transient storage equations of the one-dimensional Fickian-type diffusion equations, the so-called dead-zone model, showing a continuous decrease of the skewness with the distance. On the other hand, based on these equations as an initial-boundary value problem for the transport of a spill in a river with dead zones Schmid [Schmid, B.H., 2002. Persistence of skewness in longitudinal dispersion data: can the dead zone model explain it after all?. Journal of Hydraulic Engineering 128 (9), 848–854, September 1, ASCE], showed that the skewness can locally increase, if there are river reaches with different values of the mass-transfer coefficient between the main stream and the dead zone, or due to changing topography. This paper shows that by applying Schmid's [Schmid, B.H., 2002. Persistence of skewness in longitudinal dispersion data: can the dead zone model explain it after all?. Journal of Hydraulic Engineering 128 (9), 848–854, September 1, ASCE] approach to the River Rhine and its tributaries Mosel (Germany) and Aare (Switzerland), the observed persistence of the skewness can be reproduced, taking into account the changes in the river topography. Moreover, it is demonstrated that irregularities of the riverbed and banks, and vegetation along the river borders, resulting in ‘natural dead zones’, contribute to the persistence of the skewness. In addition, the physical processes behind the observed mass-transfer coefficient have been analysed.

Simplification in Longitudinal Transport Modeling: Case of Instantaneous Slug Releases

Tracer experiments are widely used tools in field studies on solute transport in streams. In addition, numerous injections are carried out each year as part of the so-called salt dilution technique of stream gauging, producing a large body of data (in principle) available for analysis, and most of these latter experiments are instantaneous slug releases. Interpretation of the observed breakthrough curves is often done with the aid of either the transient storage (TS) or the advection-dispersion (AD) model, the latter being a subset or simplification of the former. The work reported here aimed at the development of a criterion to determine when the use of the TS model is required to explain a given breakthrough curve and in what circumstances the AD model can be expected to produce comparable results. The transient storage (TS) or dead zone model was solved for 216 parameter combinations, believed to cover a large portion of the situations possibly encountered in practice. The advection-dispersion (AD) model was then calibrated to reproduce the TS breakthrough curves as closely as possible. The outcome was judged to determine whether a particular breakthrough curve was unique to the TS model or could have been produced by the AD model as well. Criteria in terms of Damköhler index, DaI, and skewness coefficient, Gt, have been derived, and an application to a real world case is given. Use of an artificial neural network to improve the “diagnostics” is demonstrated.

Robust adaptive control of a class of nonlinear systems with unknown dead-zone

This paper deals with the adaptive control of a class of continuous-time nonlinear dynamic systems preceded by an unknown dead-zone. By using a new description of a dead-zone and by exploring the properties of this dead-zone model intuitively and mathematically, a robust adaptive control scheme is developed without constructing the dead-zone inverse. The new control scheme ensures global stability of the adaptive system and achieves desired tracking precision. Simulations performed on a typical nonlinear system illustrate and clarify the validity of this approach.

Temporal moments routing in streams and rivers with transient storage

Closed-form expressions for the temporal moments of concentration–time distributions in streams and rivers with transient storage (TS) have been derived. In addition to the mathematical relationships known already for the case of an instantaneous slug release, further and more general results were obtained for stream reaches with arbitrary inflow pollutographs. Thus, the temporal moments of the concentration-versus-time distribution at the lower end of a reach can be linked to the respective moments at the upper end and the transport and mixing parameters of the so-called TS or Dead Zone Model. Therefore, starting from the site of a release (or some arbitrary cross-section within the zone of validity of the TS model), the first to fourth temporal moments of concentration–time distributions can be routed in flow direction along a cascade of stream reaches. The moment-routing equations presented are expected to be useful in the interpretation of measured field data (including estimation of the transport and mixing parameters) as well as for simulation purposes and as an easily used tool to check the accuracy of numerical codes.

Changes in water quality parameters due to in-sewer processes

Combined sewer systems contain a large number of organic and inorganic pollutants from both domestic and industrial sources. These pollutants are often retained within the combined sewer system for significant lengths of time before entering sewage treatment works, or being spilt to a watercourse via a combined sewer overflow (CSO) during storm conditions. Currently little knowledge exists concerning the effects of in sewer processes on pollutants. Understanding of in-sewer processes is important for the effective and efficient design of treatment works and CSO chambers and for impact assessments on receiving waters. A series of studies covering storm and dry weather flow conditions were undertaken with the aim of investigating the nature of in-sewer processes. These studies consisted of marking a body of water with a fluorescent tracer. The tracer was then monitored at a series of downstream sites, and discrete samples collected from the body of water as it progressed through the sewer. The samples were analysed for water quality parameters and these results investigated in tandem with the detailed hydraulic information gained through the tracer studies. The results highlight the hydraulic differences between storm and dry weather conditions such as increased travel times and mixing under storm conditions. The Advection Dispersion Equation (ADE) and Aggregated Dead Zone (ADZ) model parameters have been quantified for the tracer data. The ADE mixing coefficient is shown to increase by an order of magnitude for storm conditions. The ADZ dispersive fraction parameter is shown to be approximately constant with flow. Chemical reactions and decay within the sewer system were found to be consistent with oxygen limitation.

Adaptive fuzzy PIMD controller for systems with uncertain deadzones

In this paper, a new adaptive fuzzy Proportional-Integral (of a modified error function)-Derivative (PIMD) controller is designed for systems with uncertain deadzones. Instead of using the summation of the system output error to be one of the input variables, the fuzzy mechanism in PIMD controller takes the summation of a proposed error function as one essential part of the output fuzzy singleton. Together, with the linearly combined error and difference of the error as the only input variables, the complexity reduced fuzzy mechanism of the fuzzy PIMD controller is constructed. The adaptation processes are provided to determine the parameters of the PIMD controller to reduce the overshoot and to accelerate the system with deadzone to the desired output. The fuzzy PIMD controller is indicated to be flexible to the variations of deadzone parameters. Also, the proposed fuzzy PIMD controller is flexible to the change of deadzone model to contain jump discontinuity points. Moreover, the fuzzy PIMD controller can perform well for the system with time-varying deadzone model. Simulation results are included to indicate the effectiveness of the adaptive fuzzy PIMD controller.

Persistence of Skewness in Longitudinal Dispersion Data: Can the Dead Zone Model Explain It After All?

Field studies reporting coefficients of temporal skewness that do not decrease in the main flow direction have cast doubt on the transient storage (TS) or dead zone model of longitudinal dispersion in rivers and streams. In this study, the conditions under which the TS model predicts persistent or growing skewness coefficients are investigated. The findings clearly show that, though not outright impossible, and instantaneous slug release into a uniform channel reach is, indeed, extremely unlikely to result in persistent or growing skewness coefficients. In contrast, the passage of a tracer or pollutant along a sequence of (hydraulically) different subreaches may easily give rise to nondecaying skewness coefficients, the occurrence of which is governed by the parameter sets of the subreaches concerned. Thus, the TS model does show a certain potential to explain the persistence of skewness. The findings reported here are expected to be useful in guiding future field studies on the subject. An application of the newly derived criterion to stream tracer data has been successful.

Longitudinal dispersion in natural channels: 2. The roles of shear flow dispersion and dead zones in the River Severn, U.K.

Abstract. The classical one-dimensional advection-diffusion equation (ADE) gives an inadequate description of tracer cloud evolution in the River Severn, U.K. A solute transport model incorporating the effects of tracer storage in dead zones is presented in which the channel is conceived as being divided into two parallel regions. The bulk flow region occurs in the central part. Its longitudinal dispersive properties are described by the ADE. Adjacent to this, an additional cross-sectional area is defined in which tracer can be stored temporarily in regions of slowly moving water called dead zones. Exchange between the two regions follows a first order rate equation. Applying the model to the River Severn shows that a dispersing cloud’s evolution occurs in two distinct stages with a rapid transitional phase. Initially, shear-dispersion is dominant while the tracer particles mix fully over the bulk flow. Once this has occurred, dead zone storage accounts well for the non-Fickian evolution of the cloud. After the transitional phase the dead zone storage mechanism clearly dominates over shear-dispersion. Overall, the combined shear flow dispersion – dead zone model (D-DZM) provides a much better, physically consistent description of the tracer cloud’s evolution than the simple classical ADE approach can do alone. Keywords: Channels; dispersion; dead zones; tracers; River Severn

Longitudinal dispersion in natural channels: 3. An aggregated dead zone model applied to the River Severn, U.K.

Abstract. An Aggregated Dead Zone (ADZ) model is presented for longitudinal dispersion of tracer in river channels, in which the channel cross-section is divided into two parallel regions: the bulk flow and dead zone storage. Tracer particles in the bulk flow are assumed to obey plug-flow advection at the discharge velocity U without any mixing effects. The dispersive properties of the model are completely embodied in the residence time for tracer storage in the dead zone. The model provides an excellent description and prediction of empirical concentration-time distributions, for times t < x/U. Its physical realism is demonstrated by using it to describe the evolution of a tracer cloud in the River Severn, U.K., and by comparing it with a more complex model which incorporates the additional effects of shear flow dispersion within the bulk flow. The ADZ model is a potentially useful tool for practical prediction of dispersion in natural channels. Keywords: Channels; dispersion; dead zones; tracers; River Severn

A model of indoor releases with recirculating ventilation

AbstractModeling indoor dispersion is an important problem for risk analysis used to justify design improvements that increase safety. Of particular interest is a method to estimate what fraction of the volume of a room could have flammable concentrations when the source rate is too weak to fill the entire room. An analytic model is described here that allows a time‐varying source rate such as typically occurs with an evaporating liquid pool. The model allows automatic increases and decreases in the ventilation rate and/or fresh air makeup ratio such as could be activated by a gas sensor system. This provides a design tool for such gas sensor systems. An approximate approach is developed to estimate the effects of imperfect mixing, such as can occur with density stratification or dead zones. Two options are described, a dead zone model and a “bypass” model.

Dispersion processes in reservoirs and natural streams

The application of dispersion models in estimating the concentration levels of a solute, or a pollutant constituent, resulting from discharging waste into reservoirs and natural streams is presented. For the reservoir case, the mass balance principle of the aggregated dead zone (ADZ) model is employed, from which a general formulation for the dispersion of a pollutant constituent is derived. A hypothetical case is presented where an accidental spillage of a chemical discharge into a reservoir is considered. The prediction of the transient change in the concentration levels is analytically derived for the general case of reservoirs of arbitrary volumes and of various influx rates and duration. For the case of natural streams, the classic version of the Fickian diffusion model is modified, from which an expression for quantifying the coefficient of dispersion is proposed. The proposed dispersion model has been applied to a case study carried out on Waikato River, in New Zealand, where a dye is released in the river for the purpose of providing information about flushing rates at low steady flows. It has been demonstrated that the proposed dispersion model gives a more realistic estimation of the concentration profiles than those obtained from other existing models. Reasonably good comparisons were obtained with the measured concentration levels in the river at both the near-field and the far-field locations from the injection site.

On the relationship of transient storage and aggregated dead zone models of longitudinal solute transport in streams

The relationship between the distributed transient storage (TS) and lumped aggregate dead zone (ADZ) models of longitudinal solute transport in rivers and streams is examined by a parallel application to tracer data and through an investigation of parameter relationships. Both models accurately describe observed solute transport in a stream where the effects of storage or dead zones significantly affect longitudinal dispersion. A moment matching technique, based on theoretical temporal moments, is used to develop parameter relationships. Tests using the previously calibrated parameters, in addition to simulation experiments, show that the moment matching procedure allows ADZ model parameters to be reliably derived from TS model parameters and vice versa. An investigation of these parameter relationships reveals an important difference between the effective solute transport velocity and the average reach flow velocity in streams subject to transient storage or dead zone processes. A number of practical uses for the derived relationships are suggested, including the ability to utilize powerful methods of system identification in the estimation of TS model parameters.

Hydroecology of river plankton: the role of variability in channel flow

The mechanisms by which entrained planktonic organisms survive in river systems, despite an inexorable, unidirectional downstream transport, are revisited. The importance of channel retentivity to downstream population recruitment is emphasized. The aggregated dead-zone (ADZ) model is shown to be adequate to explain downstream recruitment of a growing population. The ADZ behaviour is more prevalent in sinuous, low-gradient reaches than in other parts of the river. Plankton selection and dynamics relate conspicuously to flow at higher discharges but other environmental features are important at low flows. Discharge variability is pivotal to the opportunities for potamoplankton to thrive. Copyright © 2000 John Wiley & Sons, Ltd.

Dispersion Model for Mountain Streams

Major differences have often been observed between experimental measurements and solutions of the one-dimensional dispersion equation for the spreading of tracers in rivers. This paper compares calculated values of peak concentration decay rates, variances, and concentration distributions for three different one-dimensional models with experimental results obtained by T. J. Day. The calculations show that the Fickian and dead-zone models have similar behaviors for peak decay rates and variances, but that these behaviors do not agree with Day's measurements. The third model, in which the dispersion coefficient increases linearly with distance downstream, gives results that agree closely with Day's measurements. It is suggested that the increase in the dispersion coefficient with distance downstream may be the result of a relatively large amount of dispersion from velocity shear near the leading and trailing edges of the tracer cloud.

Experimental and numerical validation of the dead-zone model for longitudinal dispersion in rivers

Experimental results of longitudinal dispersion in rivers are discussed in the light of new findings with respect to the “dead-zone” theory. Both the experiments made by authors as well as some literature results are taken into consideration. The dead-zone model parameters, namely the relative dead-zone volume, penetration time of the tracer into dead-zones, constant mean flow velocity and dispersion coefficient, are obtained with the use of the nonlinear least square technique applied to the image functions of concentration time-distributions. The statistical moments of the concentration time distributions as functions of distance are also analyzed. Expressions for statistical moments, recently obtained by authors, are found to agree well with both experimental and computational results.

Properties of the dead-zone model of longitudinal dispersion in rivers

The dead zone equations were solved with the use of the Laplace transform technics. The solution was a base to derive the three moments of the pollution concentration distribution in a river. They differ from the moments published so far, because they were calculated from the solution of the pure boundary problem. This approach is easier to apply in calculations of the pollution concentration distributions and it covers a wider range of cases when we deal with natural field data. Main features of the model equations were analysed from the point of view of the theory of dynamic systems. The transfer function was derived and analysed as well as the frequency-response function. The dispersion relation for the dead zone equations was also obtained and analysed for different parameters of the model.

New models for backlash and gear play

Backlash is the most important non-linearity that limits the performance of speed control in industrial drives and is an important impediment to position control as well. A new simple model based on phase plane analysis is derived for an elastic shaft with internal damping connected to a backlash The model is extended to include the case of a so-called flexible coupling, where some of the backlash gap is filled with rubber. Extensive measurements on an industrial drive with a flexible coupling are compared with simulations and fit very well. With internal damping the classical dead-zone model gives an unphysical behaviour of the dynamics. However, the new model converges to the classical dead-zone model when the damping tends to zero. © 1997 by John Wiley & Sons, Ltd.