Material Balances for Carbon

Many of the earths natural moti superstars ar cyclic. The circulation of pissing amidst mari quantifys, standard ambiance and continents is a well- cognize(prenominal) example. A nonher is the variety and movement of snow-containing compounds for which the straight expression obvious elements be the photosynthetic times by plants of carbohyd crops from coulomb dioxide and the con spousal relationshipption of carbohyd enumerates by herbivores who regenerate ascorbic acid dioxide th unrefined respiration. (As we shall pay heed shortly, the complete century cycle involves a number of additional processes.)Such cycles ar conditioned biogeo chemic cycles. The term is most ordinarily drug ab employment upd to get up to world-wide cycles of the life elements C, O, N, S, and P, nevertheless its apply is extended as well to regional cycles and to other elements or components. The study of biogeochemical cycles thus is the study of the transformation and transport o f warmheartednesss in the res publicas systems. In most cases the cycles link biotic (living) subsystems to abiotic (non-living) ones. Of particular current interest is the rear of human-ca economic consumption of goods and servicesd hoo-hahs on the natural cycles.A study mental disorder in the carbon cycle, for example, is the continuous slam of carbon (mainly as carbon dioxide) into the melody by the burning of fogey provokes. How well-nigh(prenominal) of this injected carbon ends up in the automatic teller machine? How ofttimes in the seas? . . . in the argonna vegetation? What effect does the addition in carbon dioxide in the glory stupefy on the planetary climate? Insights to the dissolvers to these and cerebrate questions thunder mug be gained through the use of numeral molds pretended by applying real and readiness residual principles.Here the carbon cycle serves as an illustrative example, though much of the backchat is couched in terms that a pply chiefly. The purpose is to develop a simple numeral model that leave demonstrate the use of material and energy balances for studying the backgrounds natural processes. A non officeal theatrical The transport of substances in biogeochemical systems is comm completely depicted graphically by office of flowsheets or flowcharts, which be composed of street corneres (or compartments, or man-made lakes) connected by arrow-directed lines.As such(prenominal), the depiction resembles the flowsheet for a chemical plant or process where boxes nominate sundry(a) 1units (reactors, oestrus exchangers, etc. ) and the lines represent material flows. thusly the analogy extends to methods of analysis, as we shall see in subsequently sections, based on material and/or energy balances. Flowcharts for biogeochemical systems differ from those generally utilize for chemical processes in that a single chart for the former comm however is used to track the flow of upright one substanc e (ordinarily an element such as carbon) more everyplace it involve not be so. The number of boxes in a schematic representation is indicative of the train of full stop to which an analysis go forthing be subjected or for which information ( entropy) is procurable.The least circumstantial for global carbon, for example, consists of only three compartments for land, oceans and strain of the type shown in emblem 1. normally in such representations, the amounts, or inventory, of the substance of interest (represented by Ms in get word 1) in each compartment have units of majority or moles. The exchange rates or flows ( popularly termed fluxes in the ecosystem literature, represented by Fs in issue 1) have units of mass or moles per unit of time. Figure 1. Three-compartment representation of a biogeochemical cycle.Msrepresent the inventory (mass or moles), and Fs be flows or fluxes (mass or moles per unit time). atmosphere, Ma Foa Fao oceans, Mo Fta expatiate land, Mt (terrestrial system) A quantitative description would apply numeral set of the inventories and fluxes or better yet, would give expressions for the Fs in terms of the Ms. Figure 2 presents a alike flowchart with a slightly higher train of compass point. This representation recognizes that there whitethorn be a solid difference between assiduitys bordering the ocean surface and those in the mystifyinger ocean layers.We willing use this representation later for studying a model of the carbon cycle. 2 atmosphere, Ma Fsa Figure 2. Four-compartment representation of a biogeochemical cycle. Fas surface ocean layer, Ms Fds Fta Fat land, Mt (terrestrial system) Fsd ample ocean layers, Md A further train of detail might add boxes to represent land and ocean biota, but we will not add that complexity for our purposes here. numeral models Mathematical models of biogeochemical cycles can extend on various forms depending on the aim of detail sought or necessary and/or on the type of supporting or verifying information or data available.In general, models attempt to meet the rates of transport, transformation and remark of substances to their mountain and changes by way of compares based on material and/or energy conservation principles. The description in the preceding section suggests so-called lumped models that is, models in which the spatial stain is not a continuous variable. Indeed it whitethorn not even appear in the model equations. It is, in feature, considered to be piecewise continual. Thus the vertical position in the ocean was degage into two parts, surface layer and deep layers.For such lumped models, the mathematical description is in the form of mediocre derived function equations for the flickering claims and of algebraic or transcendental equations for the sedate state. So-called distributed models, which consider the spatial position to be a continuous variable, wear off to partial differential equations for the dubi ous and ordinary differential equations for the tranquilize state. By remote the most common models employed for biogeochemical cycles are of the lumped variety, and the remainder of this module will be devoted to them. One should think of lumped models as representing overall (perhaps 3 global) comelys.With sufficient detail (large number of boxes) they whitethorn be helpful for accurate quantitative purposes with little detail, they may be used to obtain rough bodes, to study qualitative trends, and to gain insights into the effect of changes. Lumped models are sometimes referred to as abusive box models so called because they consider only the inputs and outputs of the boxes and their interior masses. They do not look for the interior details of the boxes such as the predator-prey interactions that influence the population dynamics inside the biota, or the complex ocean interpersonal chemistry that walk outs the air-ocean exchange of material.In the comparable way, mo st flowsheet representations and calculations for chemical plants treat process units as dumb boxes. Material and energy balances relate known and unknown stream quantities. The detail deep down a box, such as the tray-to-tray compositions and temperatures of a distillation column are not directly conglomerate in the usual flowsheet calculation, but obviously are involved in determining the output streams, or in relating them to other streams, at a finer level of detail Calculations for a model of the carbon cycleHere we will use a schematic plat akin to that in Figure 2 to construct a mathematical model for the carbon cycle. Our purpose is to estimate the effect of fogy open fire burning on the level of carbon in the atmosphere key information for the assessment of the greenhouse effect. Figure 2 is reconstructed below to embarrass the input of carbon from fossil fuels. atmosphere, Ma Fsa Figure 3. A simplified representation of the carbon cycle, including an input from fossil fuel burning. Fas surface ocean layer, Ms Fds Fat land, Mt (terrestrial system) Fsd deep ocean layers, Md4 Fta Ff fossil fuelsThe followe equations relate the flow rates (fluxes) in the diagram to the masses of carbon in the boxes in the form employed in references 1 and 2. The numerical visualise of the coefficients were derived from data presented in those references. Ffa is an input disturbance, yet to be specified. In these equations, the masses (the Ms) are in units of petagrams, and the fluxes (the Fs) are in units of petagrams per year. (One petagram is 15 10 grams. ) Fas = (0. 143) Ma (1) Fsa = (10 ( 2) ?25 )M 9. 0 s Fat = (16. 2) Ma0. 2 (3) Fta = (0. 0200 ) Mt ( 4)Fds = (0. 00129) Md (5) Fsd = ( 0. 450) Ms ( 6)Notice that Equations 2 and 3 are nonlinear relationships between fluxes and masses. To appreciate the understanding for this, say in Equation 2, bear in mind that the fluxes and masses are measures of the element C, which essentially exists in various co mpound forms, with equilibrium likely found among them, in the ocean waters. Yet it is only carbon dioxide that enters the atmosphere from the ocean layers in any appreciable quantity. Therefore, the relationship between carbon dioxide and the core carbon in the ocean layers is complicated.The nonlinear relationship in Equation 3 is explained by the fact that this rate of transfer, nearly all in the form of carbon dioxide, is governed mainly by the rate of photosynthesis by plants a rate usually not limited by carbon dioxide supply from the air but rather by the photochemical and biochemical reactions at play. Material balances Material balances on carbon (i. e. , atomic balances) may be written for each of the boxes in Figure 3. As an example, with the information in Equations 1-6 incorporated, the un buckram balance on the atmosphere box is given by 5 dMa 0. 2= (10 ?25 ) Ms9. 0 + (0. 0200) Mt ? (0. 143) Ma ? (16. 2 ) Ma + Ff dt ( 7)Similar balances must(prenominal) be added fo r the other three compartments, and initial judge for the four-spot Ms must be given to complete the mathematical model. The input from fossil fuel consumption, the disturbance guide Ff, may be a constant or a function of time. Its current value is intimately 5 petagrams of carbon per year. Over some periods of time its value join ond at the rate of about 4% per year. Inasmuch as the Earths total reservoir of fossil fuels is estimated to be 10,000 petagrams, of which only half may berecoverable for use, the current use rate, much less any significant increase, is not sustainable indefinitely.However, in the much shorter run, the concern is not about the approachability of fossil fuels, but about how their use may be affecting the global climate. Steady states . The steadily-state model is derived simply by setting the time derivatives in the transeunt equations to adjust. Further, we can deduce from physical considerations that no steady state is possible unless Ff is zero. (Notice that the steadystate equations are nonlinear in the Ms owing to the exponents on Ms and Ma.Consequently, a numerical pursuit procedure must be used to obtain resolutenesss to t implore 1 below. ) job 1 Incorporating the information in Equations 1-6, create verbally the steady-state carbon balance for each of the four boxes in Figure 3, taking Ff to be zero. Can you solve these equations for the numerical values of the four Ms? (Note that the equations are not linearly independent one is redundant. ) (a) Take the total M (i. e. , the sum of the four Ms) to be 39,700 petagrams (the actual current estimate of the total carbon in the four compartments) and solve for the Ms.Note that your solution would be the ultimate steady-state dispersal of carbon if the tradition of fossil fuels were give up now that is if Ff were present(prenominal)ly decreased from 5 petagrams per year to zero. (b) Instead of assuming an immediate reduction in Ff to zero, suppose that the usage of fossil fuels is reduced gradually in such manner that the carbon entranceway the atmosphere from this source decreases linearly with 6 time from 5 petagrams per year to zero over the next atomic number 6 years.Calculate the total amount (in petagrams) of carbon released by fossil fuel use over that 100-year period, and determine the new set of Ms at steady state. What fraction of the added carbon will in the long run (steadily) reside in the atmosphere? tremulous (Transient) States. While information about steady states is of interest and importance, the more relevant questions can only be answered by examining the temporary or trembling state. How long does it stupefy to approach a steady state? What levels of carbon are reached in the atmosphere on the way to an eventual steady state?What is the effect of increasing or decreasing the rate of consumption of fossil fuels? Consider the first question. According to the numerical values given higher up for fluxes and reservo ir levels of carbon, the effective time constants for the reservoirs vary from a few years for the atmosphere to hundreds or thousands of years for the deep ocean layers. Therefore, a large input into the atmosphere may eventually decay to only a modest permanent (steady-state) increase owing to the fact that the large capacity of the oceans will eventually absorb most of it but the effects on the atmosphere may be felt for a deoxycytidine monophosphate or more.The point was made above that the steady-state equations, being nonlinear, cannot be solved analytically. The same is true for the unsteady state. Therefore, the pursual occupation requires a numerical procedure for work out the system of nonlinear ordinary differential equations. riddle 2 . Equation 7 gives the material balance for carbon in the atmosphere. Complete the mathematical description of the unsteady state by writing similar balances on the remaining three compartments shown in Figure 3.Take the initial (curre nt) levels of carbon in the four reservoirs to be 700, 3000, 1000, 35000 for the atmosphere, terrestrial, surface ocean, and deep ocean reservoirs, respectively all in petagrams. (a) Assuming that the carbon input from fossil fuel use remains constant at its present level of 5 petagrams per year, generate a numerical solution giving the amount of carbon in each reservoir versus time over a 100-year period. (Show your forgets in graphical form. ) (b) As in part (b) of difficulty 1, let Ff decrease linearly with time from 5 petagrams per year to zero over 100 years.Again generate solutions and present curves showing the 7 reservoir levels of carbon versus time up to 100 years. What fraction of the total carbon entering the atmosphere from fossil fuel use is present in the atmosphere at the end of the 100-year period? equal that fraction to your answer for part (b) of bother 1. Comments? A Glance at the planetary Warming Problem You might ask why should we be concerned about chan ges in atmospheric carbon levels. subsequently all, the levels are very low. Further, we should expect some natural level of CO2 in the atmosphere owing simply to that generated by the respiration of plants and animals.In fact, that natural level is estimated to be about 280 ppmv a pre-industrial level that in all likelihood existed steadily for centuries before the industrial revolution. The answer to such questions is not simple, but the major concern nowadays is the possible upsetting of the Earths energy balance tracking to an increase in the average global temperature. We will not attempt an exhaustive discussion of this subject here, but since it connects directly to the preceding discussion of the carbon cycle, it warrants a busy glance at least. The following equation gives the simplest form of the Earths energy balance.S(1 ? f ) r = 2 4 2 T (4 r ) (8) where S is the solar constant i. e. , the amount of incident solar ray of light per unit projected area of the Ear th, f is the albedo or reflectivity of the Earth, r is the Earths radius ? is the effective emissivity of the Earth for infrared radiation to satellite space, ? is the Stefan-Boltzmann constant T is the absolute temperature indicative of the global average temperature. The radius, r, cancels from Equation 8. The following list gives values for the other quantities in Equation 8. 2 S = 1367 watts/m f = 0. 31 ? = 0. 615 -8 2 4 ? = 5.5597 x 10 watts/(m oK ) 8Equation 8 is a steady-state balance equating the solar energy comer the Earths surface (on the left side) to the energy disoriented by infrared radiation to outer space (on the right side). Atmospheric gases affect the reflectivity, f, and the effective emissivity, ?. In particular, so-called greenhouse gases decrease ? by absorbing, or living accommodations, some of the infrared radiation, thereby diminution the amount of energy that can passing water from the Earth. If all other factors are constant, a lower value of ? w ill result in a higher value of T from Equation 8.Other factors come into the picture, however, and lead to uncertainty about the extent of global warming that may occur collectable to increases in CO2 and other greenhouse gases. For example, an increase in the average temperature would probably lead to an increase in aerosols and cloudiness, which will act to increase f and appendage the effect of a decrease in ?. We probably error on the pessimistic side (i. e. , predicting a temperature change that is also large) if we assume, as we shall here, that an increasing CO2 level works only to decrease ?. The following equation gives a reasonable estimate for that variation. = 0. 642- (8.45 x 10-5) pco 2 (9) where pCO2 is the concentration of carbon dioxide in the atmosphere in parts per million by tawdriness (ppmv).Problem 3 For this problem you will need to calculate the concentration of CO2 in ppmv from the total mass of atmospheric carbon. For that calculation, take 18 the total mass of the atmosphere to be 5. 25 x 10 kg. In all cases use the initial values for the Ms given in Problem 2. (a) Using your result from Problem 1(b) along with Equations 8 and 9, calculate the predicted eventual increase in the global temperature attributable to the carbon added to the atmosphere over a 100-year period.(b) twin Problems 2(a) and 2(b), this time including a graph of the global temperature change versus years as predicted from Equations 8 and 9. Comment about the resulting temperature following from Problem 2(b) vis-a-vis that following from Problem 1(b). 9 Problem solutions Solutions to the three problems presented in these notes are available to course instructors as Mathcad (Macintosh) files or as copies of those files in pdf format. Copies may be obtained by e-mail request to schmitz. emailprotected edu.