Transport balance modelling

Transport balance or box models have been used by many workers in the past in efforts to understand the trace element and isotope characteristics of the Earth s major silicate reservoirs, i.e. continental crust, and upper and lower mantle (e.g. Jacobsen Wasserburg 1979 Zartman Haines 1988). Although simple mass balance calculations can be applied to present-day trace element concentrations and Pb, Nd and Hf isotope compositions of major reservoirs, e.g. continental crust and depleted mantle, to test the hypothesis that these reservoirs are complementary, transport balance models are needed to test ideas on their evolution in time. The reason is that the isotope ratio variations are the result of time-integrated trace element variations in the reservoir, modified by fluxes between them. Below, recent transport balance models in which the evolution of the continental crust is examined using Th-U-Pb (Kramers T olstikhin 1997) and... 

Fig. 1. Principle of forward transport balance modelling used.

Principles and methods of transport balance models previous work... 

Although global transport balance modelling thus provides some perspective on the rate at which CO2 could have been removed from the atmosphere in early Earth history, the resolution of the method is inherently poor and it can provide only broad outlines. The large uncertainties and the problems encountered highlight the need to find independent constraints on weathering and erosion rates in Archaean time to understand the early evolution of the Earth s atmosphere, climate and life. 

A number of models have been proposed which link crustal evolution with the mantle Nd-isotope evolution curve, the most realistic of which is probably the transport-balance model of Nagler and Kramers (1998). This model is based upon their empirically derived Nd-isotope mantle evolution curve and assumes that the upper mantle melts to form basaltic oceanic crust, which is then reprocessed to form continental crust. An important aspect of the model is that it also includes crustal recycling, such that as the volume of continental crust grows with time, proportionately some crust is recycled back into the mantle through erosion and subduction. 

FIGURE 4.8 Crustal growth curves based upon transport-balance modeling (Nagler Si Kramers, 1998), U-Pb zircon ages from juvenile crust (Condie, 2000) and Nb/U ratios for the depleted mantle (Collerson Si Kamber, 1999). The Nagler and Kramers model assumes either 0% crust at 4.4 Ga (A) or 10% crust at 4.4 Ga (B), and their calculations are consistent with Pb isotope modeling and Hf-isotope studies. 

The transport balance model of Kramers and Tolstikhin (1997) for the U-Th-Pb isotope system is particularly sensitive to the extent to which crustal material can be recycled into the mantle and offers some constraints on the mechanism of crustal recycling. They concluded that the rate of recycling of crustal material back into the Earth s mantle has increased with time, particularly since 2.0 Ga. We know from Pb-isotope studies that some sediment recycling took place during the Archaean (Halla, 2005), but it would seem as though the proportion was small. 

The basic assumption for a mass transport limited model is that diffusion of water vapor thorugh air provides the major resistance to moisture sorption on hygroscopic materials. The boundary conditions for the mass transport limited sorption model are that at the surface of the condensed film the partial pressure of water is given by the vapor pressure above a saturated solution of the salt (Ps) and at the edge of the diffusion boundary layer the vapor pressure is experimentally fixed to be Pc. The problem involves setting up a mass balance and solving the differential equation according to the boundary conditions (see Fig. 10). 

The transport-dispersive model consists of one differential mass balance equation for each component, i, in the mobile phase ... 

Other Applications of the Multiple-Core Approach. The bulk of this chapter has dealt with the specific application of multiple-core methodology to questions of atmospheric Hg deposition. Whole-basin Hg accumulation rates for seven lakes, calculated from multiple sediment cores, were used in a simple mass-balance model to estimate atmospheric fluxes and Hg transport from catchment soils. This approach can be used to answer other limnological questions, and the model is not restricted to Hg or atmospheric deposition. 

If whole-basin accumulation rates for a substance are produced for multiple lakes in a geographic region, it is possible to use a simple mass-balance model to estimate both the atmospheric deposition rate and transport from the terrestrial catchment. The model was applied to both modern and preindustrial Hg accumulation in seven undisturbed lakes in the upper midwest... 

Atmospheric transport of toxaphene from the southern USA continues [38, 45]. MacLeod et al. [19] attributed 70% of the atmospheric inputs to the Great Lakes to long-range atmospheric transport and deposition from the southern US and northern Mexico. The dynamic mass balance model [65] indicates that the net exchange direction of toxaphene in Lake Superior was... 

The discussion above provides a brief qualitative introduction to the transport and fate of chemicals in the environment. The goal of most fate chemists and engineers is to translate this qualitative picture into a conceptual model and ultimately into a quantitative description that can be used to predict or reconstruct the fate of a chemical in the environment (Figure 27.1). This quantitative description usually takes the form of a mass balance model. The idea is to compartmentalize the environment into defined units (control volumes) and to write a mathematical expression for the mass balance within the compartment. As with pharmacokinetic models, transfer between compartments can be included as the complexity of the model increases. There is a great deal of subjectivity to assembling a mass balance model. However, each decision to include or exclude a process or compartment is based on one or more assumptions—most of which can be tested at some level. Over time the applicability of various assumptions for particular chemicals and environmental conditions become known and model standardization becomes possible. 

The construction of a mass balance model follows the general outline of this chapter. First, one defines the spatial and temporal scales to be considered and establishes the environmental compartments or control volumes. Second, the source emissions are identified and quantified. Third, the mathematical expressions for advective and diffusive transport processes are written. And last, chemical transformation processes are quantified. This model-building process is illustrated in Figure 27.4. In this example we simply equate the change in chemical inventory (total mass in the system) with the difference between chemical inputs and outputs to the system. The inputs could include numerous point and nonpoint sources or could be a single estimate of total chemical load to the system. The outputs include all of the loss mechanisms transport... 

Dimensionless groups for a process model can be easily obtained by inspection from Table 13-2. Each of the three transport balances is shown (in vector/tensor notation) term-by-term under the description of the physical meanings of the respective terms. The table shows how various well-known dimensionless groups are derived and gives the physical interpretation of the various groups. Table 13-3 gives the symbols of the dimensions of the terms in Table 13-2. 

Himmelblau [32] and Himmelblau and Bischoff [33] have considered three types of model which are useful in process analysis, i.e. empirical models, population balance models and transport phenomena models. Empirical models involve mathematical relationships between dependent and independent variables, which are postulated either entirely a priori, or by considering the nature of the experimental data, or by analogies, etc. On the other hand, transport phenomena models are based on the laws of... 

Mass budgets represent snapshots in time, constructed mostly with measurements. However, the processes themselves can be described quantitatively, and combined to describe contaminant fate and transport in mass balance models. The resource managers of the Great Lakes have promoted the use of mass balance models to be used as tools to help guide decisions and strategies for lake-wide management... 

The multimedia urban model (MUM) is a fugacity-based mass balance model that treats the movement of POPs in an urban environment and links emissions to ambient chemical concentrations, and thus outdoor exposure (Diamond et al., 2001). MUM considers longterm, average conditions of chemical transport and transformation among six environmental compartments in urban areas (air, soil, surface water, sediment, vegetation and surface film see Figure 6.1) shows a concepmal version of the model). The model does not estimate event-specihc processes as do meteorological-based air or stormwater models. 

Available reaction-transport models describe the second regime (reactant transport), which only requires material balances for CO and H2. Recently, we reported preliminary results on a transport-reaction model of hydrocarbon synthesis selectivity that describes intraparticle (diffusion) and interparticle (convection) transport processes (4, 5). The model clearly demonstrates how diffusive and convective restrictions dramatically affect the rate of primary and secondary reactions during Fischer-Tropsch synthesis. Here, we use an extended version of this model to illustrate its use in the design of catalyst pellets for the synthesis of various desired products and for the tailoring of product functionality and molecular weight distribution. 

In the following the most relevant models for liquid chromatography are derived in a bottom-up procedure related to Fig. 6.2. To illustrate the difference between these models their specific assumptions are discussed and the level of accuracy and their field of application are pointed out. The mass balances are completed by their boundary conditions (Section 6.2.7). For the favored transport dispersive model a dimensionless representation will also be presented. 

As mentioned in Section 6.2.2, the mass transfer term in Eq. 6.3 is defined by the linear driving force approach. Therefore, the transport dispersive model consists of the balance equations in the mobile phase (Eq. 6.71) written with the pore concentration... 

Introducing Eqs. 6.101-6.105 into the equations of the transport dispersive model leads to the following dimensionless mass balances ... 

Because of the analogy between simulated and true counter-current flow, TMB models are also used to design SMB processes. As an example, the transport dispersive model for batch columns can be extended to a TM B model by adding an adsorbent volume flow Vad (Fig. 6.38), which results in a convection term in the mass balance with the velocity uads. Dispersion in the adsorbent phase is neglected because the goal here is to describe a fictitious process and transfer the results to SMB operation. For the same reason, the mass transfer coefficient feeff as well as the fluid dispersion Dax are set equal to values that are valid for fixed beds. 

A common modeling approach for chromatographic batch reactors is the equilibrium-transport dispersive model (Chapter 6.2.4.1). Therefore, only the equations for this approach are discussed here. The differential mass balance (Fig. 8.6) takes into account axial dispersion as well as mass transfer between fluid and both solid phases. 

Coupling CFD with one of the much simpler zone models is potentially particularly valuable. It is difficult and time consuming to add the additional complexity of mixing, heat and mass transfer, and dynamic population balance modeling to the CFD model, plus it makes the CFD model very slow. Sufficient accuracy may be achievable for many applications from applying the population balance modeling to the simpler zone models, although accuracy will be limited because the effects of the particle concentration, distribution and PSD will not be fed back to the transport models. Alternately, comprehensive CFD models can be used to understand the flow and its variation for a limited number of conditions, but simpler zone models may be used for application of the model where speed and convenience are important and detailed accuracy is not, e.g., process control. 

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