Mechanistic, Mass Balance Models in Surface Water Quality Engineering1. The Basic Mass BalanceA common approach in surface water quality modeling is to develop a mass balance on a completely-mixed flow reactor, i.e.dMsources sinksdt= -.Given that,M C V= �,the equation may be written as,( )d C Vsources sinksdt�= -and for a constant volume,dCV sources sinksdt� = -Here, the source is loading and the sinks include outflow and, potentially, reaction and settling.[T] Chapra, Figure 3.12. LoadingMass may enter a lake from a variety of sources, including point and tributary discharges, direct runoff and from the atmosphere and sediments. In a modeling context, these inputs represent the loading (W, e.g. g·d-1) to the system, Loading W=Loading is described mathematically in two ways, depending on the nature of the source. For tributary and point sources, ,i in iW Q C= � where Qi is the flow (e.g. m3·d-1) and Cin,i the concentration of the material of interest (e.g., g·m-3)in tributary or point source i. For atmospheric and sediment inputs, an areal approach is taken, tributary or point sourceW J A= �where J is the flux from the atmosphere or sediment (e.g. g·m-2·d-1) and A is the area of the sediment or, for atmospheric inputs, the lake surface (e.g., m2).3. OutflowMass is removed from the system in the outflow stream,Outflow Q C= �where Q (e.g. m3·d-1) is the sum of the inflows and C (e.g., g·m3) is the concentration of the material of interest in the lake (note the impact of the assumption of complete mixing on the use of C versus Cin here). Reaction k V C or Reaction V k C= �� = ��[T] Excel – behavior of first order decay reactions.4. SettlingLosses due to settling may be formulated as a flux across the sediment-water interface,sSettling J A= �with J and As (sediment area) having units as described above. Because J is not an intrinsic property of the material of interest (i.e. A is a system characteristic), it is recognized that J may be represented as,J v C= �(note dimensionality here and that v is an intrinsic property)[T] Chapra, Figure 3.2and the settling relationship written as,sSettling v A C= � � 5. The Total Mass BalanceThese representations of the source and sink terms may then be combined to yield the mass balance for a well-mixed lake.sdCV W Q C V k C v A Cdt� = - � - �� - � �Note in this equation that C and t are the dependent and independent variables, respectively, because we seek to predict concentration as a function of time. The loading term if the forcing function, as it describes the manner in which external conditions influence or ‘force’ the system, and V, Q, k, v, and As are parameters or coefficients particular to the system or material of interest.6. The Steady State SolutionIf a system receives a constant loading for a sufficient time (and the sink terms remain unchanged), it will reach a dynamic equilibrium termed a steady state.sdCV W Q C V k C v A Cdt� = - � - �� - � �(discuss how the system approaches steady state)Here the rate of accumulation is zero, 0sdCV W Q C V k C v A Cdt� = = - � - �� - � �and the mass balance can be solved for C at steady state,sssWCQ V k v A=+ � + �Recalling that,1C Wa= �the assimilation factor may be defined as,sa Q V k v A= + �+ �Thus the steady state solution has served to define the assimilation factor in terms of measureable coefficient and parameters that reflect,( , , )C f biology chemistry physics=Finally, we can use W, and Css times the various components of the assimilation factor to look at the relative importance of the source-sink terms as they determine the fate of the material of interest.[T] Chapra, Figure 3.37. Transfer FunctionThe transfer function,  (dimensionless), specifies the manner in which an input to the system (Cin) is transformed or “transferred” to an output (C). Remembering that,inW Q C= �and substituting this to the steady state solution,sssWCQ V k v A=+ �+ �and dividing both sides by Cin,ssin sCQC Q V k v A=+ �+ �where,sQQ V k v Ab =+ �+ �Thus, if  << 1 then sink processes are acting to significantly reduce levels of the material of interest before it leaves the lake. Conversely, if   1, there is little transfer and the outflow concentration approaches the inflow concentration. Note in this equation that increases in V, k, v and As all act to reduce the outflow concentration. Flow, which appears in both the numerator and denominator of the equation acts to both increase (delivery) and decrease (flushing) concentration. 8. Residence TimeResidence time (, e.g. yr) is defined as the mean amount of time that a molecule or particle remains or ‘resides’ in a system. The residence time of water in a lake is given by, wVQt = and the residence time of a material of interest by,csVQ V k v At =+ �+ �[T] Excel - residence time example (how do Q, k, V and A act to influence the response time for remedial actions).9. Time Variable SolutionTo this point we have focused on steady state, an average water quality that will result if loadings and factors mediating assimilation are held constant for a sufficiently long period. Our consideration of the time-variable solution will proceed in two steps. First we will look at what happens when a pollutant is removed from a lake and then we will consider the response to time-variable loads.a. Temporal Aspects of Pollutant ReductionBeginning with the mass balance model developed previously (with W now = f(t)),( )sdCV W t Q C V k C v A Cdt� = - � - �� - � �dividing both sides by V,( )sAdC QW t C k C v Cdt V V= - � - � - � �noting that A/V = H,( )dC Q vW t C k C Cdt V H= - � - � - �and collecting terms,( )dC W tCdt Vl+ � =Where  is termed the eigenvalue and is equivalent to the sum of all the processes that purge the material of interest from the lake, Q VkV Hl = + +This equation consists of two parts, a general solution (cg) on the left there W(t)=0 and a particular solution (cp) on the right for specific forms of W(t),g pc c c= +The general solution may be developed by integrating the left side of the equation yielding,0tc c el- �= �Note that the units of  (d-1) and its component terms suggest that they may be conceptualized as first order losses yielding a characteristic shape to the lake response.[T] Excel – temporal response following loading termination for various values of Although the