Material Balances for Carbon

Many of the earths inhering processes ar cyclic. The circulation of water between marines, air travel and continents is a familiar example. A nonher is the transformation and movement of deoxycytidine monophosphate-containing compounds for which the immediately app bent ingredients be the photosynthetic generation by plants of carbohydrates from hundred dioxide and the consumption of carbohydrates by herbivores who regenerate century dioxide through respiration. (As we shall see shortly, the fill out carbon rhythm method involves a reckon of additional processes.)Such cycles are termed biogeo chemical cycles. The term is nigh ordinarily employ to refer to spherical cycles of the life elements C, O, N, S, and P, but its use is drawn-out as well to regional cycles and to peeled(prenominal) elements or components. The study of biogeochemical cycles and so is the study of the transformation and transport of substances in the Earths brasss. In most cases the cycles unify biotic (living) subsystems to abiotic (non-living) ones. Of particular contemporary interest is the effect of human-caused disturbances on the inherent cycles.A major disturbance in the carbon cycle, for example, is the continuous injection of carbon (mainly as carbon dioxide) into the automated teller machine by the importunate of fogey dismisss. How more than(prenominal)(prenominal) of this injected carbon ends up in the halo? How frequently in the navals? . . . in the land vegetation? What effect does the add in carbon dioxide in the atmosphere have on the worldwide climate? Insights to the answers to these and related questions whoremaster be conglomerateed through the use of mathematical illustrations constructed by applying clobber and nothing counterbalance principles.Here the carbon cycle serves as an illustrative example, though often of the backchat is couched in terms that apply generally. The objective is to develop a simple mathematical mo lding that will demonstrate the use of material and faculty balances for analyse the Earths natural processes. A conventional fiddleation The transport of substances in biogeochemical systems is comm hardly depicted graphically by means of flowsheets or flowcharts, which are composed of boxes (or compartments, or reservoirs) connected by arrow-directed lines.As such, the depiction resembles the flowsheet for a chemical plant or process where boxes re grant various 1 units (reactors, heat swaprs, etc. ) and the lines represent material flows. therefore the analogy extends to methods of synopsis, as we shall see in later sections, based on material and/or brawn balances. Flowcharts for biogeochemical systems differ from those generally used for chemical processes in that a single chart for the former usually is used to track the flow of just one substance (ordinarily an element such as carbon) but it make not be so. The number of boxes in a schematic commission is indicative of the level of detail to which an analysis will be subjected or for which training (data) is available.The least detailed for global carbon, for example, consists of only three compartments for land, oceans and atmosphere of the type shown in count 1. ordinarily in such representations, the amounts, or inventory, of the substance of interest (represented by Ms in Figure 1) in each compartment have units of mass or moles. The exchange rates or flows (usually termed fluxes in the ecosystem literature, represented by Fs in Figure 1) have units of mass or moles per unit of while. Figure 1. Three-compartment representation of a biogeochemical cycle.Msrepresent the inventory (mass or moles), and Fs are flows or fluxes (mass or moles per unit judgment of conviction). atmosphere, Ma Foa Fao oceans, Mo Fta Fat land, Mt (terrestrial system) A quantitative translation would give mathematical determine of the inventories and fluxes or better yet, would give expressions for the Fs in terms of the Ms. Figure 2 presents a alike(p) flowchart with a slightly higher level of detail. This representation recognizes that there whitethorn be a signifi finisht difference between concentrations near the ocean come out and those in the profounder ocean layers.We will use this representation later for studying a case 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 deep ocean layers, Md A further level of detail king add boxes to represent land and ocean biota, but we will not add that complexity for our purposes here. Mathematical models Mathematical models of biogeochemical cycles can take on various forms depending on the level of detail sought or demand and/or on the type of supporting or verifying schooling or data available.In general, models enterprise to relate the rates of transport, transformation and input of substance s to their masses and changes by way of equalitys based on material and/or energy conservation principles. The translation in the preceding section suggests alleged(prenominal) lumped models that is, models in which the spatial position is not a continuous variable. thusly it may not even appear in the model equations. It is, in fact, considered to be piecewise constant. Thus the vertical position in the ocean was spaced into two separate, surface layer and deep layers.For such lumped models, the mathematical description is in the form of frequent differential equations for the trembling states and of algebraic or transcendental equations for the steady state. So-called distributed models, which consider the spatial position to be a continuous variable, lead to partial differential equations for the unsteady and ordinary differential equations for the steady state. By far the most common models active for biogeochemical cycles are of the lumped variety, and the remainder of this module will be devoted to them. bingle should think of lumped models as representing overall (perhaps 3 global) fair(a)s.With sufficient detail (large number of boxes) they may be useful for accurate quantitative purposes with little detail, they may be used to obtain rough estimates, to study qualitative trends, and to gain insights into the effects of changes. Lumped models are sometimes referred to as black box models so called because they consider only the inputs and outputs of the boxes and their national masses. They do not explore the interior details of the boxes such as the predator-prey interactions that influence the population dynamics within the biota, or the complex ocean chemistry that affects the air-ocean exchange of material.In the same way, most flowsheet representations and calculations for chemical plants treat process units as black boxes. Material and energy balances relate known and unknown stream quantities. The detail within a box, such as the tray-to-tray compositions and temperatures of a distillation column are not directly involved 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 similar to that in Figure 2 to construct a mathematical model for the carbon cycle. Our purpose is to estimate the effect of fogey fuel burning on the level of carbon in the atmosphere important 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 modify 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 following equations relate the flow rates (fluxes) in the diagram to the masses of carb on in the boxes in the form employed in references 1 and 2. The numerical value 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 socio-economic class. (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 pars 2 and 3 are nonlinear relationships between fluxes and masses. To appreciate the reason for this, say in Equation 2, bear in mind that the fluxes and masses are measures of the element C, which actually exists in various compound forms, with equilibrium likely realized 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 carb on dioxide and the pith 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 turn in 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 unsteady 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 be added for the other three compartments, and initial values for the quatern Ms must be given to complete the mathematical model. The input from fossil fuel consumption, the disturbance function Ff, may be a constant or a function of time. Its on- going value is nearly 5 petagrams of carbon per year. Over some results of time its value increased at the rate of intimately 4% per year. Inasmuch as the Earths conglomeration 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 availability of fossil fuels, but about how their use may be affecting the global climate. Steady states . The steady-state model is derived simply by setting the time derivatives in the transient equations to zero. 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 search procedure must be used to obtain solutions to difficulty 1 below. ) trouble 1 Incorporating the info rmation in Equations 1-6, write 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 distribution of carbon if the usage of fossil fuels were discontinued now that is if Ff were immediately simplificationd 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 get into the atmosphere from this source lessenings linearly with 6 time from 5 petagrams per year to zero over the next one hundred 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 lastly ( steadily) reside in the atmosphere? Unsteady (Transient) States. While information about steady states is of interest and importance, the more relevant questions can only be answered by examining the transient or unsteady state. How long does it take to improvement 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 change magnitude or decreasing the rate of consumption of fossil fuels? Consider the premier question. According to the numerical values given above for fluxes and reservoir levels of carbon, the hard-hitting 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 in conclusion decay to only a modest unchanging (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 century or more.The point was make above that the steady-state equations, being nonlinear, cannot be solved analytically. The same is true for the unsteady state. Therefore, the following problem requires a numerical procedure for solving the system of nonlinear ordinary differential equations. Problem 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 remain three compartments shown in Figure 3.Take the initial (current) 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 system constant at its present level of 5 p etagrams per year, generate a numerical solution giving the amount of carbon in each reservoir versus time over a 100-year period. (Show your results in graphical form. ) (b) As in part (b) of Problem 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? Compare that fraction to your answer for part (b) of Problem 1. chin-waggings? A gleam at the Global Warming Problem You might ask wherefore should we be concerned about changes in atmospherical carbon levels. later on all, the levels are very low. Further, we should expect some natural level of carbonic acid gas 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 probably 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 leadership to an increase in the average global temperature. We will not attempt an exhaustive treatment of this subject here, but since it connects directly to the preceding discussion of the carbon cycle, it warrants a quick 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 contingency solar radiation per unit projected area of the Earth, f is the albedo or reflectivity of the Earth, r is the Earths radius ? is the effective emissivity of the Earth for infrared radiation radiation to outer space, ? is the Stefan-Boltzmann constant T is the absolute temperature indicative of the global average temperatu re. 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 reaching the Earths surface (on the left side) to the energy lost by infrared radiation to outer space (on the aright side). Atmospheric gases affect the reflectivity, f, and the effective emissivity, ?. In particular, so-called greenhouse gases decrease ? by absorbing, or trapping, some of the infrared radiation, thereby reduce the amount of energy that can escape from the Earth. If all other factors are constant, a lower value of ? will result in a higher value of T from Equation 8.Other factors come into the picture, however, and lead to perplexity about the extent of global warming that may occur delinquent to increases in CO2 and other greenhouse gases. For example, an increase in the average temperature would probably le ad to an increase in aerosols and cloudiness, which will act to increase f and offset the effect of a decrease in ?. We probably error on the pessimistic side (i. e. , predicting a temperature change that is to a fault 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 volume (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 due to the carbon added to the atmosphere over a 100 -year period.(b) Repeat 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 postulate to schmitz. emailprotected edu.