Donald Dean Adrian

A simplified solution technique for carbon adsorption model

Abstract A new method of solving the homogeneous surface diffusion model for activated carbon absorption makes use of Laplace transforms on the equations developed by applying orthogonal collocation. The simultaneous equations which are developed are non-linear so an interative method is introduced to make them solvable when one wishes to calculate the surface diffusion coefficient, Df, and the film transfer coefficient, Kf, from batch adsorption data. The proposed model has the advantage of being continuous in time in contrast to earlier models which relied on finite difference numerical methods to solve the system of equations comprising the homogeneous surface diffusion model. The model is applied to one data set from the literature. The proposed method gave values of Kf and D1 nearly identical to values found by other investigators. The model was applied to three new data sets obtained in our laboratories: an agricultural waste, a dye waste and an aqueous waste stream containing a herbicide. The industrial wastes were composed of a mixture of species, whereas the herbicide waste contained a single organic compound. The model produced a good match between experiment and prediction for all waste streams. The final form of the equations is presented in a form readily useable by the person interested in applications to carbon adsorption bed practice.

Oxygen sag equation for second-order BOD decay

Abstract A dissolved oxygen (DO) equation for a stream is developed in which the biochemical oxygen demand (BOD) deoxygenation rate is described as a second-order reaction. Rate constants for a second-order BOD equation may be evaluated by graphical or numerical methods. The DO sag equation incorporates exponential integral functions. These functions are available in tabular form or they may be calculated by exact or approximate series. The time at which the minimum DO concentration occurs is calculated numerically. The DO sag equation using a second-order BOD model fills a gap in the literature, as a number of investigators had explored describing BOD decay as a second-order reaction but the corresponding DO sag equation was not available to predict the impact of such wastes on a river.

Water quality modeling for a sinusoidally varying waste discharge concentration

Abstract Biochemical oxygen demand modeling in a river involves derivation and solution of the governing partial differential equation which describes concentration change with time and space brought on by convective, dispersive, and decay processes and the loading function. In this study, a sinusoidal variation in waste discharge concentration is considered. The governing partial differential equation is solved analytically by a transform method and by assuming that the solution varies periodically in time. The concepts of memory length and memory time are used to indicate when the solution becomes quasi-steady (periodic). The analytical solution is compared with two other solutions. The three solutions produce comparable results. However, the analytical solution is much easier to apply. The analytical solution extends the number of boundary conditions which a modeler can apply to describe real engineering problems.

Analytical water quality model for a sinusoidally varying BOD discharge concentration

Biochemical oxygen demand modeling in a river involves derivation and solution of the governing partial differential equation which describes concentration change with time and space brought on by convective, dispersive, and decay processes and the loading function. The boundary condition applied in this study describes a sinusoidal variation in waste discharge concentration. The governing partial differential equation is solved analytically by introducing complex variables, using a transform method and by applying the Laplace transform. The resulting exact analytical solution is compared with three other solutions. The analytical solution produces results that are exact for any location at any time. The analytical solution extends the number of boundary conditions which a modeler can apply to describe real engineering problems.

Water Quality Model Incorporates Unconventional Bod Reduction

Biochemical Oxygen Demand (BOD) reduction through the joint action of BOD decay expressed as an unconventional second order reaction and sedimentation is incorporated into a dissolved oxygen (DO) sag model for a river. A term named the Phelps-Thomas index, which is a function of the reaeration and sedimentation rate constants, is introduced and presented as a composite measure of river and wastewater rejuvenation characteristics. Calculations using ranges of reaeration and sedimentation rates show the Phelps-Thomas index has values between –2 and 10 or larger, with large values associated with rapid recovery of DO. The time at which the minimum DO occurs is calculated numerically. Examples apply the DO sag model to logging debris in a stream. INTRODUCTION For decades, the pioneering work of Streeter and Phelps served as a basis for water quality modeling in a river [1]. Their classic dissolved oxygen sag model *Financial support was provided by the Louisiana Transportation Research Center, Louisiana Water Resources Institute, and a sabbatical leave for Adrian. 303 2006, Baywood Publishing Co., Inc. related stabilization of an organic waste measured by the biochemical oxygen demand (BOD) and the dissolved oxygen (DO) resources of the river. Thomas accounted for settleable BOD in the DO sag equation [2]. The effect of dispersion on BOD and DO in small rivers has been found to be negligible [3]. The BOD reaction model is an empirical expression chosen on the basis of convenience to describe the complex transformations that take place as BOD stabilizes in a water body. A first order BOD decay equation has been widely applied to describe the deoxygenation rate of most municipal wastewaters, although an unconventional second order rather than a first order model was introduced by Thomas [4] prior to being applied by many investigators [5-11]. Conventional practice treats the BOD reaction as being first order; however, in this investigation the term unconventional reaction order designates a BOD reaction that is other than first order. Streams are subject to many sources of BOD loadings in addition to municipal wastewaters; this study draws upon the BOD characteristics of logging debris as measured by Ponce [12]. Rigorous statistical bases for deciding whether a first order or an unconventional second order BOD model provided a better fit to data are available [7, 13]. An analytical DO sag equation that incorporated second order reaction kinetics to describe BOD decay was not available until one was developed in 1998 [10], simplified in 2003 [11], and generalized in 2004 [14]. A notable deficiency of these BOD and DO models was that they did not include loss of BOD by sedimentation. As a result, an analytical DO sag model is not currently available that incorporates BOD decay as a second order reaction while also including loss of BOD by sedimentation. The primary objective of this study was to develop a methodology for including BOD loss through sedimentation in a DO sag model which included BOD decay expressed as a second order reaction. In addition, an expression was to be developed with which to calculate the time to reach the minimum dissolved oxygen concentration. A secondary objective was to show through examples how to determine the BOD rate constant and the ultimate BOD from logging debris data. Additional examples were to demonstrate the effect of sedimentation rates on the DO behavior of streams in which BOD decays by a second order reaction. SECOND ORDER BOD MODEL The second order BOD decay model applied to a moving plug flow reactor or a moving Lagrangian control volume as proposed in [4] is L t L k L t ( ) 0 2 0 1 (1) in which L(t) = BOD remaining at time t, g/m, L0 = BOD at time zero, t = elapsed time or flow time, day, and k2 = BOD deoxygenation reaction rate 304 / ROIDER ET AL.

SIMPLIFIED DEVELOPMENT OF OXYGEN SAG MODEL

A dissolved oxygen sag equation is developed by use of the Laplace transform and the convolution integral for a stream in which the biochemical oxygen demand (BOD) deoxygenation rate is described as a second-order reaction. The Laplace transform method simplifies the mathematical solution of the model equation by avoiding difficult-to-evaluate integrals. The dissolved oxygen sag equation incorporates exponential integral functions which are calculated by exact or approximate series. The time at which the minimum dissolved oxygen concentration occurs is calculated numerically. The dissolved oxygen sag model is applied using BOD data collected from Douglas Fir needles in stream water. The Douglas Fir needles had a small reaction rate constant which results in the stream being able to carry a BOD load without exhausting its dissolved oxygen supply. The model is useful in calculating Total Maximum Daily Loads (TMDL) of streams. INTRODUCTION Water quality modeling in a river has developed from the pioneering work of Streeter and Phelps [1], who developed a balance between the dissolved oxygen *Financial support was provided by a Louisiana State University Board of Regents fellowship to the senior author.

A river water quality model for time varying BOD discharge concentration

We consider a model for biochemical oxygen demand (BOD) in a semi-infinite river where the BOD is prescribed by a time varying function at the left endpoint. That is, we study the problem with a time varying boundary loading. We obtain the well-posedness for the model when the boundary loading is smooth in time. We also obtain various qualitative results such as ordering, positivity, and boundedness. Of greatest interest, we show that a periodic loading function admits a unique asymptotically attracting periodic solution. For non-smooth loading functions, we obtain weak solutions. Finally, for certain special cases, we show how to obtain explicit solutions in the form of infinite series.

Laplace transform application to a nontraditional dissolved oxygen model

Abstract A three-halves order biochemical oxygen demand (BOD) deoxygenation reaction is incorporated into a dissolved oxygen (DO) model to describe river water quality. The user is provided an alternative approach to the practice of using a first order BOD equation in river water quality models for DO. The new approach uses published BOD data sets of analyses of samples from rivers which are described best in the least squares sense by a three-halves order BOD reaction. The DO differential equation is solved by using the Laplace transform method. The time when the minimum DO concentration occurs is calculated numerically. The DO sag model is applied to examples to show the sensitivity of the model to changes in the three-halves order deoxygenation reaction rate constant and to changes in the BOD load.

Dilutions of Glucose and Glutamic Acid Analyzed as Multi-Order BOD Reactions

The Biochemical Oxygen Demand (BOD) of a mixture of glucose and glutamic acid is a standard test solution which provides a reasonably repeatable value of the 5-day BOD. The objective of this study was to evaluate the reaction order from respirometer data of BOD of glucose and glutamic acid mixtures. The mixtures ranged in increments of 10% from 10% strength (90% dilution) to 100% strength (no dilution). There were 10 replications of each strength of sample, so that the BOD of 100 samples measured at daily intervals for 5 days were available. The data were tested for goodness-of-fit to three BOD reaction models: a first-order model, a half-order model, and an order-n model. The root mean squared error measured the goodness-of-fit. Twenty-six percent of the samples fit the first-order model best, 63% fit the half-order model best, and 11% fit the order-n model best.

A LABORATORY MESOCOSM AS A TOOL TO STUDY PAH DEGRADATION IN A COASTAL MARSH WETLAND

ABSTRACT Polycyclic aromatic hydrocarbons (PAHs) are one of the contaminants of concern in coastal marsh environments which are subject to crude oil spills. A laboratory scale mesocosm can be used to complement field study of PAH degradation in coastal marshes. Coastal marsh wetland features, such as its soil, tidal cycles, and flushing, that may play roles in PAH degradation can be simulated in a laboratory mesocosm. The laboratory mesocosm tank is made of acrylic as the main construction material with an air chamber inside the tank which functions as a pneumatic system and tidal water storage compartment. Two trays filled with contaminated marsh wetland soil are situated at two different levels: the lower one is constantly submerged while the higher one is intermittently drained. When the air pressure inside the air chamber is high, the water will flow out from the air chamber to the tank to create high tide. When the air pressure inside the air chamber is low, the water will flow back from the tank to ...