4 Introduction to pr is and selecti The constituents of concern found in wastewater are removed by physical, chemical, and biological methods. The individual methods usually are classified as physical unit operations, chemical unit processes, and biological unit processes. Treatment methods in which the application of physical forces predominate are known as physical unit operations. Examples of physical unit operations include transfer, filtration, and adsor hich the al or conversion of constituents is brought about reactions are known as chemical unit processes. Examples of chemical unit processes include disinfection. oxidation. and precipitation Treatment methods in which the removal of constituents is brought about by biological activity are known as biological unit processes. Biological treatment is used primarily to remove the biodegradable organic constituents and nutrients in wastewater. Examples of biological activated-sludge and trickling-filter processes. Unit operations and ombinations in treatment flow diagrams ontact basin Gravity thickeners ons and conversions Dechlorination Gri Liquid biosolids completion and eftluent pumping storage lagoons generally a function of the temperature. and the type which the reactions take place). Hence. both the effects of mperature and the ty important in the selection of treatment process physical constraints(约束 must be considered in Fig 4-1 Overview of a biological nutrient Anaerobic Dew Influent pump Septage station an removal(BNR) wastewater-treatment Harford County, MD. The capacity is 76,000 m /d) The fundamental basis for the analysis of the physical, chemical, and biological unit operations and processes used for wastewater treatment is the materials mass balance principle in which an accounting of made before and after Therefore, the purpose of this chapter is to introduce and discuss(1)the types of reactors used for wastewater treatment;(2)the preparation of mass balances to determine process performance;(3) modeling ideal flow in reactors; (4)the analysis of reactor hydraulics using tracers: (5)modeling nonideal flow in reactors;(6) reactions, reaction rates, and reaction rate coefficients;(7)modeling treatment kinetics, which involves the coupling of reactors and reaction rates;(8) treatment processes involving mass transfer; and(9)important factors involved in process analysis and selection The information in this chapter is intended to serve as an introduction to the subject of process analysis, and to provide a basis for the analysis of the unit operations and processes that will be presented in ubsequent chapters. 4-1 Reactors used for the treatment of wastewater 4-1
4-1 4 Introduction to Process Analysis and Selection The constituents of concern found in wastewater are removed by physical, chemical, and biological methods. The individual methods usually are classified as physical unit operations, chemical unit processes, and biological unit processes. Treatment methods in which the application of physical forces predominate are known as physical unit operations. Examples of physical unit operations include screening, mixing, sedimentation, gas transfer, filtration, and adsorption. Treatment methods in which the removal or conversion of constituents is brought about by the addition of chemicals or by other chemical reactions are known as chemical unit processes. Examples of chemical unit processes include disinfection, oxidation, and precipitation. Treatment methods in which the removal of constituents is brought about by biological activity are known as biological unit processes. Biological treatment is used primarily to remove the biodegradable organic constituents and nutrients in wastewater. Examples of biological treatment processes include the activated-sludge and trickling-filter processes. Unit operations and processes occur in a variety of combinations in treatment flow diagrams. The rate at which reactions and conversions occur, and the degree of their completion, is generally a function of the constituents involved, the temperature, and the type of reactor (i.e., container or tank in which the reactions take place). Hence, both the effects of temperature and the type of reactor employed are important in the selection of treatment processes. In addition, a variety of environmental and other physical constraints(约束) must be considered in process selection. Fig. 4-1 Overview of a biological nutrient removal(BNR) wastewater-treatment plant (Harford County, MD. The capacity is 76,000 m3 /d) The fundamental basis for the analysis of the physical, chemical, and biological unit operations and processes used for wastewater treatment is the materials mass balance principle in which an accounting of mass is made before and after reactions and conversions have taken place. Therefore, the purpose of this chapter is to introduce and discuss (1) the types of reactors used for wastewater treatment; (2) the preparation of mass balances to determine process performance; (3) modeling ideal flow in reactors; (4) the analysis of reactor hydraulics using tracers: (5) modeling nonideal flow in reactors; (6) reactions, reaction rates, and reaction rate coefficients; (7) modeling treatment kinetics, which involves the coupling of reactors and reaction rates; (8) treatment processes involving mass transfer; and (9) important factors involved in process analysis and selection. The information in this chapter is intended to serve as an introduction to the subject of process analysis, and to provide a basis for the analysis of the unit operations and processes that will be presented in subsequent chapters. 4-1 Reactors used for the Treatment of Wastewater
Wastewater treatment involving physical unit operations and chemical and biological unit processes is carried out in vessels or tanks commonly known as"reactors Types of Reactors The principal types of reactors used for the are(1) the batch reactor(2) the complete-mix reactor(also known as the continuous-flow stirred-tank reactor(CFSTR) reactor(also known as a tubular-flow reacto (4) complete-mix reactors in series. (5) the acked-bed reactor and(6)the fluidized-bed Batch Reactor. In the batch reactor. flow is flow enters. is treated. and then is discharged the reactor are mixed completely. For example the bod test is carried out in a batch reactor although it should be noted that the contents are not mixed completely during the Uflow incubation period. Batch r ften chemicals Packing matera Fig. 4-2 Definition sketch for Outfiow ofreactors used for wastewater treatment (a) batch reactor; (b)complete-mix reactor (c]plug-flow open reactor, (dplug-flow closed reactor(tubular reactor); (e)complete-mix reactor in series. o) packed-bed reactor ( g)paked-bed upflow reactor; (h)expanded-bed upflow reactor Complete-Mix Reactor. In the complete-mix reactor, it is assumed that complete mixing occurs instantaneously and uniformly throughout the reactor as fluid particles enter the reactor. Fl leave the reactor in proportion to their statistical population. Complete mixi round or square reactors if the contents of the reactor are uniformly and continuously redistributed. The actual time required to achieve completely mixed conditions will depend on the reactor geometry and the Plug-Flow Reactor Fluid particles pass through the reactor with little or no longitudinal mixing and ch they entered The particles retain their identity and remain in the reactor for a time equal to the theoretical detention time. This type of flow is approximated in long open tanks with a high length-to-width ratio in which longitudinal dispersion is minimal or absent or closed tubular reactors(e.g, pipelines Complete -Mix Reactors in Series. The series of complete- mix reactors is used to model the flow regime that exists between the ideal hydraulic flow patterns corresponding to the complete-mix and lug-flow reactors. If the series is composed of one reactor. the complete-mix regime prevails. If the series consists of an infinite number of reactors in series. the plug-flow regime prevails. Packed-Bed Reactors. The packed-bed reactor is filled with some type of packing material, such as rock, slag, ceramic, or, now more commonly, plastic. With respect to flow, the packed-bed reactor can be operated in either the downflow or upflow mode, Dosing can be continuous or intermittent(e. g. trickling filter). The packing material in packed-bed reactors can be continuous or arranged in multiple stages. with Fluidized-Bed Reactor. The fluidized-bed reactor is similar to the packed-bed reactor in many respects but the packing material is expanded by the upward movement of fluid (air or water) through the bed. The expanded porosity of the fluidized-bed packing material can be varied by controlling the flowrate of the fuid Application of re The principal applications of reactor types used for wastewater treatment are reported in Table 4-1
4-2 Wastewater treatment involving physical unit operations and chemical and biological unit processes is carried out in vessels or tanks commonly known as "reactors." Types of Reactors The principal types of reactors used for the treatment of wastewater, illustrated on Fig. 4-2, are (1) the batch reactor, (2) the complete-mix reactor (also known as the continuous-flow stirred-tank reactor (CFSTR) in the chemical engineering literature), (3) the plug-flow reactor (also known as a tubular-flow reactor), (4) complete-mix reactors in series, (5) the packed-bed reactor, and (6) the fluidized-bed reactor. Batch Reactor. In the batch reactor, flow is neither entering nor leaving the reactor (i.e, flow enters, is treated, and then is discharged, and the cycle repeats). The liquid contents of the reactor are mixed completely. For example, the BOD test is carried out in a batch reactor, although it should be noted that the contents are not mixed completely during the incubation period. Batch reactors are often used to blend chemicals or to dilute concentrated chemicals. Fig. 4-2 Definition sketch for various types of reactors used for wastewater treatment (a) batch reactor;(b)complete-mix reactor;(c)plug-flow open reactor; (d)plug-flow closed reactor(tubular reactor);(e)complete-mix reactor in series; (f)packed-bed reactor;(g)paked-bed upflow reactor;(h)expanded-bed upflow reactor Complete-Mix Reactor. In the complete-mix reactor, it is assumed that complete mixing occurs instantaneously and uniformly throughout the reactor as fluid particles enter the reactor. Fluid particles leave the reactor in proportion to their statistical population. Complete mixing can be accomplished in round or square reactors if the contents of the reactor are uniformly and continuously redistributed. The actual time required to achieve completely mixed conditions will depend on the reactor geometry and the power input. Plug-Flow Reactor. Fluid particles pass through the reactor with little or no longitudinal mixing and exit from the reactor in the same sequence in which they entered. The particles retain their identity and remain in the reactor for a time equal to the theoretical detention time. This type of flow is approximated in long open tanks with a high length-to-width ratio in which longitudinal dispersion is minimal or absent or closed tubular reactors (e.g., pipelines). Complete-Mix Reactors in Series. The series of complete-mix reactors is used to model the flow regime that exists between the ideal hydraulic flow patterns corresponding to the complete-mix and plug-flow reactors. If the series is composed of one reactor, the complete-mix regime prevails. If the series consists of an infinite number of reactors in series, the plug-flow regime prevails. Packed-Bed Reactors. The packed-bed reactor is filled with some type of packing material, such as rock, slag, ceramic, or, now more commonly, plastic. With respect to flow, the packed-bed reactor can be operated in either the downflow or upflow mode. Dosing can be continuous or intermittent (e.g., trickling filter). The packing material in packed-bed reactors can be continuous or arranged in multiple stages, with flow from one stage to another. Fluidized-Bed Reactor. The fluidized-bed reactor is similar to the packed-bed reactor in many respects, but the packing material is expanded by the upward movement of fluid (air or water) through the bed. The expanded porosity of the fluidized-bed packing material can be varied by controlling the flowrate of the fluid. Application of Reactors The principal applications of reactor types used for wastewater treatment are reported in Table 4-1
Tab. 4- Principal applications ofreactor types used for wastewater treatment ted-sludge biological treatment in a sequence batch reactor, mixing of orking solution ated lagoons, aerobic sludge digestions C Activated-sludge biological treatment g-flow reactors Packed-bed submerged and submerged trickling-filter biolog ical treatment units, depth atural treatment systems, air stripping Fluid ized-bed luidized-bed reactors for aerobic and anaerob ic biological treatment, upflow ludge blanket reactors, air stri Operational factors that must be considered in the selection of the type of reactor or reactors to be used in the treatment process include(1) the nature of the wastewater to be treated. (2)the nature of the reaction ocess.( 4) the rocess performance requirements. and(5) local environmental conditions. In practice, the construction costs and operation and maintenance costs also affect reactor selection. Because the relative importance of these factors varies with each factor should be considered separately when the type of reactor is to be selected Hydraulic Characteristics of Reactors Complete-mix and plug-flow reactors are the two reactor types used most commonly in the field of wastewater treatment. The hydraulic flow characteristics of complete-mix and plug-flow reactors can be described as varying from ideal and nonideal, depending on the relationship of the incoming flow to outgoing flow Ideal Flow in Complete-Mix and Plug-Flow Reactors. The ideal hydraulic flow characteristics of complete-mix and plug-flow reactors are illustrated on Fig. 4-3 in which dve tracer response craves are presented for pulse (slug-dose) and step inputs(continuous iniection). On Fig. 4-3, t is the actual time and t is equal to the theoretical hydraulic detention time defined as follows where t=hydraulic detention time, T V=volume of the reactor. L: Q=volumetric flowrate, LT-I If a pulse(slug) input of a conservative (i.e, nonreactive) tracer is injected and dispersed instantaneously in an ideal-flow complete-mix reactor, with a continuous inflow of clear water, the output tracer concentration would appear as shown on Fig. 4-3 (a-1)If a continuous step input of a conservative tracer at concentration Co is injected into the inlet of an ideal complete-mix reactor, initially filled with clear water, the appearance of the tracer at the outlet would occur as shown on Fig 4-3(a-2 In the case of an ideal plug-flow reactor, the reactor is initially filled with clear water before being deal,ebr subjected to a pulse or a step input of tracer. If an the appearance of the tracer in the effluent for a pulse input, distributed uniformly across the reactor cross section, would occur as shown on Fig. 4-3(b-1). If a continuous step input of a tracer were injected into such a reactor at an initial concentration Co. the tracer would appear in the effluent as shown on Fig 4-3(b-2) Fig. subject to pulse and step inputs ofatrmcer 4-3
4-3 Tab. 4-1 Principal applications of reactor types used for wastewater treatment Type of reactor Application in wastewater treatment Batch Activated-sludge biological treatment in a sequence batch reactor, mixing of concentrated solutions into working solutions Complete-mix Aerated lagoons, aerobic sludge digestions Complete-mix with recycle Activated-sludge biological treatment Plug-flow Chlorine contact basin, natural treatment systems Plug-flow with recycle Activated-sludge biological treatment, aquatic treatment systems Complete-mix reactors in series Lagoon treatment systems, used to simulate nonideal flow in plug-flow reactors Packed-bed Nonsubmerged and submerged trickling-filter biological treatment units, depth filtration, natural treatment systems, air stripping Fluidized-bed Fluidized-bed reactors for aerobic and anaerobic biological treatment, upflow sludge blanket reactors, air stripping Operational factors that must be considered in the selection of the type of reactor or reactors to be used in the treatment process include (1) the nature of the wastewater to be treated, (2) the nature of the reaction (i.e., homogeneous or heterogeneous), (3) the reaction kinetics governing the treatment process, (4) the process performance requirements, and (5) local environmental conditions. In practice, the construction costs and operation and maintenance costs also affect reactor selection. Because the relative importance of these factors varies with each factor should be considered separately when the type of reactor is to be selected. Hydraulic Characteristics of Reactors Complete-mix and plug-flow reactors are the two reactor types used most commonly in the field of wastewater treatment. The hydraulic flow characteristics of complete-mix and plug-flow reactors can be described as varying from ideal and nonideal, depending on the relationship of the incoming flow to outgoing flow. Ideal Flow in Complete-Mix and Plug-Flow Reactors. The ideal hydraulic flow characteristics of complete-mix and plug-flow reactors are illustrated on Fig. 4-3 in which dye tracer response craves are presented for pulse (slug-dose) and step inputs (continuous injection). On Fig. 4-3, t is the actual time and τ is equal to the theoretical hydraulic detention time defined as follows: τ= V/Q where τ = hydraulic detention time, T V = volume of the reactor, L3 Q = volumetric flowrate, L3T -1 If a pulse (slug) input of a conservative (i.e., nonreactive) tracer is injected and dispersed instantaneously in an ideal-flow complete-mix reactor, with a continuous inflow of clear water, the output tracer concentration would appear as shown on Fig. 4-3 (a-1) If a continuous step input of a conservative tracer at concentration Co is injected into the inlet of an ideal complete-mix reactor, initially filled with clear water, the appearance of the tracer at the outlet would occur as shown on Fig. 4-3(a-2). In the case of an ideal plug-flow reactor, the reactor is initially filled with clear water before being subjected to a pulse or a step input of tracer. If an observer were positioned at the outlet of the reactor, the appearance of the tracer in the effluent for a pulse input, distributed uniformly across the reactor cross section, would occur as shown on Fig. 4-3(b-1). If a continuous step input of a tracer were injected into such a reactor at an initial concentration Co, the tracer would appear in the effluent as shown on Fig. 4-3(b-2). Fig. 4-3 Output tracer response curves from reactors subject to pulse and step inputs of a tracer (a)complete-mix reactor; (2)plug-flow reactor
Nonideal Flow in Complete-Mix and Plug-Flow Reactors. In practice the flow in complete-mix and olug- flow reactors is seldom ideal. For example, when a reactor is designed, how is the flow to be introduced to satisfy the theoretical requirement of instantaneous and complete dispersion? In practice, there is al ways some deviation from ideal conditions, and it is the precautions taken to minimize these effects that are important Nonideal flow 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. Nonideal flow is illustrated on Fig 4-3a and 4-3b. The important issue with nonideal flow is that a portion of the flow will not remain in the reactor as long as may be required for a biological or chemical reaction to go to completion 4-2 Mass-balance Analysis The fundamental approach used to study the hydraulic flow characteristics of reactors and to delineate the changes that take place when a reaction is occurring in a reactor(eg, a container), or in some definable portion of a body of liquid, is the mass-balance analysis Inflow Outflow Q C ystem boundary amass balance Fig 4-4 Definition sketch for the balanc jsis for nix reactor with o are mixed completely The ph pical commplefe-mix actinated shudge reactor used for the biological treatment of wastewater The Mass-Balance Principle form of the mass can be altered (e. g. liquid to a gas). The mass-balance analysis affords a convenient way of defining what occurs within treatment reactors as a function of time. To illustrate the basic concepts involved in the preparation of a mass-balance analysis, consider the reactor shown on Fig 4-4. The system boundary is drawn to identify all of the liquid and constituent flows into and out of the system. The control volume is used to identify the actual volume in which change is occurring In most cases, the system and control volume boundaries will coincide. For a given reactant. the general mass-balance analysis is given by 1. General word statement Rate of accumulation Rate of flow of of reactant within reactant out of the tant within the e system boundary ystem boundary ystem boundary 2. The corresponding simplified word statement is Accumulation= inflow- outflow t generation (4) The mass balance is made up of the four terms cited above. Depending on the flow regime or treatment process, one or more of the terms can be equal to zero. For example, in a batch reactor in which there is no inflow or outflow the second and third terms will be equal to zero. A positive sign is used for the ecause the necessary (e.g. re=-kC for a decrease in the reactant or r.=+ kc for all increase in the reactant). Preparation of Mass Balances In preparing mass balances it is helpful if the following steps are followed, especially as the techniques involved are being mastered
4-4 Nonideal Flow in Complete-Mix and Plug-Flow Reactors. In practice the flow in complete-mix and plug-flow reactors is seldom ideal. For example, when a reactor is designed, how is the flow to be introduced to satisfy the theoretical requirement of instantaneous and complete dispersion? In practice, there is always some deviation from ideal conditions, and it is the precautions taken to minimize these effects that are important. Nonideal flow 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. Nonideal flow is illustrated on Fig.4-3a and 4-3b. The important issue with nonideal flow is that a portion of the flow will not remain in the reactor as long as may be required for a biological or chemical reaction to go to completion. 4-2 Mass-balance Analysis The fundamental approach used to study the hydraulic flow characteristics of reactors and to delineate the changes that take place when a reaction is occurring in a reactor (e.g., a container), or in some definable portion of a body of liquid, is the mass-balance analysis. Fig. 4-4 Definition sketch for the application of materials mass-balance analysis for a complete-mix reactor with inflow and outflow. The presence of a mixer is used to represent symbolically the fact the contents of the reactor are mixed completely. The photo is of a typical complete-mix activated sludge reactor used for the biological treatment of wastewater. The Mass-Balance Principle The mass-balance analysis is based on the principle that mass is neither created nor destroyed, but the form of the mass can be altered (e.g., liquid to a gas). The mass-balance analysis affords a convenient way of defining what occurs within treatment reactors as a function of time. To illustrate the basic concepts involved in the preparation of a mass-balance analysis, consider the reactor shown on Fig. 4-4. The system boundary is drawn to identify all of the liquid and constituent flows into and out of the system. The control volume is used to identify the actual volume in which change is occurring. In most cases, the system and control volume boundaries will coincide. For a given reactant, the general mass-balance analysis is given by 1. General word statement: ＝ － ＋ 2. The corresponding simplified word statement is Accumulation ＝ inflow － outflow ＋ generation (1) (2) (3) (4) The mass balance is made up of the four terms cited above. Depending on the flow regime or treatment process, one or more of the terms can be equal to zero. For example, in a batch reactor in which there is no inflow or outflow the second and third terms will be equal to zero. A positive sign is used for the rate-of-generation term because the necessary sign for the operative process is past of the rate expression (e.g., rc = -kC for a decrease in the reactant or rc = + kC for all increase in the reactant). Preparation of Mass Balances In preparing mass balances it is helpful if the following steps are followed, especially as the techniques involved are being mastered. Rate of accumulation of reactant within the system boundary (1) Rate of flow of reactant into the system boundary (2) Rate of flow of reactant out of the system boundary (3) Rate of generation of reactant within the system boundary (4)
1. Prepare a simplified schematic or flow diagram of the system or process for which the mass balance is to be prepared 2. Draw a system or control volume boundary to de its over which mass balance is to be extremely import e mass List all of the pertinent data and assumptions that will be used in the preparation of the materials balance on the schematic or flow diagram 4. List all of the rate expressions for the biological or chemical reactions that occur within the control volume 5. Select a convenient basis on which the numerical calculations will be based It is recommended that the above steps be fol slowed routinely. to avoid the errors that are often made in the preparation of mass-balance analyses Application of the Mass-Balance Analysis To illustrate the application of the mass-balance analysis, consider the complete-mix reactor shown on Fig 4-4. First, the control volume boundary must be established so that all the flows of mass into and out of the system can be identified. On Fig 4-4a, the control volume boundary is shown by the inner dashed line To apply a mass-balance analysis to the liquid contents of the reactor shown on Fig 4-4. it will be volumetric flowrate into and out of the control volume is constant 2. The liquid within the control volume is not subiect to evaporation(constant volume The liquid within the control volume is mixed completely. 4.A chemical reaction i a reactant a is occurring within the reacte The rate of change in the concentration of the reactant a that is occurring within the control volume is overned by a first-order reaction(rc=-KC). Using the above assumptions, the mass balance can be formulated as follows 1. Simplified word statement Accumulation=[nflow-5utflow+ generation 2. Symbolic representation(refer to Fig. 4-4) y=C -kC+ry Substituting-kC for r'eyields =C-4C+(-kC where dC/dt=rate of change of reactant concentration within the control volume, MLT V= volume contained within control volume L3 0=volumetric flowrate into and out of control volume L'T-l Co=concentration of mactunt entering the control volume ML-3 C=concentration of reactant leaving the control volume ML-3 rc= first-order reaction, (-kC). ML-ST k=first-order reaction rate coefficient, T-I Before attempting to solve any mass-balance expression, a unit check should always be made to assure that units of the individual quantities are consistent. If the following units are substituted into the above. dC/dt=g/' / Co, C=g/ V =QCo -QC +(-kC)V (g/n.)m'=m /s(g/m)-m'(g/m)+(-1/s)(8/m)m
4-5 1. Prepare a simplified schematic or flow diagram of the system or process for which the mass balance is to be prepared. 2. Draw a system or control volume boundary to define the limits over which mass balance is to be applied. Proper selection of the system or control volume boundary is extremely important because, in many situations, it may be possible to simplify the mass-balance computations. 3. List all of the pertinent data and assumptions that will be used in the preparation of the materials balance on the schematic or flow diagram. 4. List all of the rate expressions for the biological or chemical reactions that occur within the control volume. 5. Select a convenient basis on which the numerical calculations will be based. It is recommended that the above steps be followed routinely, to avoid the errors that are often made in the preparation of mass-balance analyses. Application of the Mass-Balance Analysis To illustrate the application of the mass-balance analysis, consider the complete-mix reactor shown on Fig. 4-4. First, the control volume boundary must be established so that all the flows of mass into and out of the system can be identified. On Fig. 4-4a, the control volume boundary is shown by the inner dashed line. To apply a mass-balance analysis to the liquid contents of the reactor shown on Fig. 4-4, it will be assumed that: 1. The volumetric flowrate into and out of the control volume is constant. 2. The liquid within the control volume is not subject to evaporation (constant volume). 3. The liquid within the control volume is mixed completely. 4. A chemical reaction involving a reactant A is occurring within the reactor. 5. The rate of change in the concentration of the reactant A that is occurring within the control volume is governed by a first-order reaction (rc = -kC ). Using the above assumptions, the mass balance can be formulated as follows: 1. Simplified word statement: Accumulation ＝ inflow － outflow ＋ generation 2. Symbolic representation (refer to Fig. 4-4): c dC V QC QC rV dt = − + 0 Substituting -kC for rc yields 0 ( ) dC V QC QC kC V dt = − + − where dC/dt = rate of change of reactant concentration within the control volume, ML-3T -1 V = volume contained within control volume, L3 Q = volumetric flowrate into and out of control volume, L3T -1 Co = concentration of mactunt entering the control volume, ML-3 C = concentration of reactant leaving the control volume. ML-3 rc = first-order reaction, (-kC). ML-3T -1 k = first-order reaction rate coefficient, T-1 Before attempting to solve any mass-balance expression, a unit check should always be made to assure that units of the individual quantities are consistent. If the following units are substituted into the above equations: V=m3 dC/dt = g/m3·s Q = m3 /s Co, C = g/m3 k = 1/s the resulting unit check yields 0 ( ) dC V QC QC kC V dt = − + − (g/m3·s)m3=m3 /s(g/m3 )-m 3 (g/m3 )+(-1/s)(g/m3 )m3