g/s=g/s-g/s-g/s(units are consistent) The analytical procedures that are adopted for the solution of mass-balance equations usually are governed by(1)the nature of the rate expression, (2)the type of reactor under consideration, (3)the mathematical form of the final materials-balance expression (i.e, ordinary or partial differential equation), and(4)the corresponding boundary conditions Steady-State Simplificati Fortunately in most applications in the field of wastewater treatment. the solution of mass-balance that the steady-statelie. long-term) concentration is of principal concern. If it is assumed that only the steady-state effluent concentration is desired, then above equation can be simplified by noting that, under steady-state conditions. the rate accumulation is zero(dC/di=0). Thus, the equatin can be written as When solved for rc, the equation yields the following expression Q The solution to the expression given by the equation will depend on the nature of the rate expression(e.g 4-3 Analysis of Nonideal Flow in Reactors Using Tracers The discussion of nonideal flow in this section will serve as an introduction to the modeling of nonideal flow considered in the following section. Attention is called to this subject because often it is neglected o not considered properly. Because of a lack of appreciation for the hydraulics of reactors. many of the. treatment plants that have been built do not perform hydraulically as designed. Factors Leading to nonideal flow in reactors As noted previously, nonideal flow is often defined as short circuiting that occurs when a portion of the flow that enters the reactor during a given time period arrives at the outlet before the bulk of the flow that entered the reactor during the same time period arrives. Factors leading to nonideal flow in reactors include ature differences. In flow reactors, nonideal flow( short rents due to temperature differenc colde or warmer than the water in the tank a portion of the water can travel to the outlet along the bottom of across the top of the reactor without mixing completely(see Fig 4-5a). 2. Wind-driven circulation patterns. In shallow reactors. wind-circulation patterns can be set ort a portion of the incoming water to the outlet in a ion of the actual detention time(see Fig 4-5 3. Inadequate mixing. Without sufficient energy input, portions of the reactor contents may not mix with the incoming water(see Fig. 4-5c). 4 Poor d the design of the inlet and outlet of the reactor relative to the reactor aspe ratio, dead zones may develop within the reactor that will not mix with the incoming water(see Fig. 4-5d 5. Axial dispersion in plug-flow reactors. In plug-flow reactors the forward movement of the tracer is due to advection and dispersion. Advection is the term used to describe the movement of dissolved or colloidal material with the current velocity. Dispersion is the term used to describe the axial and longitudinal transport of material brought about by velocity differences, turbulent eddies. and molecular diffusion. The distinction between molecular diffusion, turbulent diffusion, and dispersion is considered in the notion e subsequent discussion dealing with"Modeling Nonideal Flow In reactors. "In a tubular plug-flow reactor(.g, a pipeline), the early arrival of the tracer at the outlet can be reasoned partially by remembering that the velocity distribution in the pipeline will be parabolic 八人 Ultimately, the inefficient use of the reactor4-6 g/s=g/s-g/s-g/s(units are consistent) The analytical procedures that are adopted for the solution of mass-balance equations usually are governed by (1) the nature of the rate expression, (2) the type of reactor under consideration, (3) the mathematical form of the final materials-balance expression (i.e., ordinary or partial differential equation), and (4) the corresponding boundary conditions. Steady-State Simplification Fortunately, in most applications in the field of wastewater treatment, the solution of mass-balance equations, such as the one given by the equations, can be simplified by noting that the steady-state(i.e., long-term) concentration is of principal concern. If it is assumed that only the steady-state effluent concentration is desired, then above equation can be simplified by noting that, under steady-state conditions, the rate accumulation is zero (dC/dt = 0). Thus, the equatin can be written as c dC V QC QC rV dt = − + 0 When solved for rc, the equation yields the following expression: 0 ( ) c Q r C C V = − The solution to the expression given by the equation will depend on the nature of the rate expression (e.g., zero-, first-, or second-order). 4-3 Analysis of Nonideal Flow in Reactors Using Tracers The discussion of nonideal flow in this section will serve as an introduction to the modeling of nonideal flow considered in the following section. Attention is called to this subject because often it is neglected or not considered properly. Because of a lack of appreciation for the hydraulics of reactors, many of the treatment plants that have been built do not perform hydraulically as designed. Factors Leading to Nonideal Flow in Reactors As noted previously, nonideal flow is often defined as short circuiting that occurs when a portion of the flow that enters the reactor during a given time period arrives at the outlet before the bulk of the flow that entered the reactor during the same time period arrives. Factors leading to nonideal flow in reactors include: 1. Temperature differences. In complete-mix and plug-flow reactors, nonideal flow (short circuiting) can be caused by density currents due to temperature differences. When the water entering the reactor is colder or warmer than the water in the tank, a portion of the water can travel to the outlet along the bottom of or across the top of the reactor without mixing completely (see Fig. 4-5a). 2. Wind-driven circulation patterns. In shallow reactors, wind-circulation patterns can be set up that will transport a portion of the incoming water to the outlet in a fraction of the actual detention time (see Fig. 4-5b). 3. Inadequate mixing. Without sufficient energy input, portions of the reactor contents may not mix with the incoming water (see Fig. 4-5c). 4. Poor design. Depending on the design of the inlet and outlet of the reactor relative to the reactor aspect ratio, dead zones may develop within the reactor that will not mix with the incoming water (see Fig. 4-5d). 5. Axial dispersion in plug-flow reactors. In plug-flow reactors the forward movement of the tracer is due to advection and dispersion. Advection is the term used to describe the movement of dissolved or colloidal material with the current velocity. Dispersion is the term used to describe the axial and longitudinal transport of material brought about by velocity differences, turbulent eddies, and molecular diffusion. The distinction between molecular diffusion, turbulent diffusion, and dispersion is considered in the subsequent discussion dealing with"Modeling Nonideal Flow In Reactors." In a tubular plug-flow reactor (e.g., a pipeline), the early arrival of the tracer at the outlet can be reasoned partially by remembering that the velocity distribution in the pipeline will be parabolic. Ultimately, the inefficient use of the reactor