Mass Balance Calculations PDF Free: A Review of Mass Balances and Their Use in Environmental Analysi

The Forbes equation relating fat-free mass (FFM) to fat mass (FM) has been used to predict longitudinal changes in FFM during weight change but has important limitations when paired with a one dimensional energy balance differential equation. Direct use of the Forbes model within a one dimensional energy balance differential equation requires calibration of a translate parameter for the specific population under study. Comparison of translates to a representative sample of the US population indicate that this parameter is a reflection of age, height, race and gender effects.

Modeling body weight regulation, and thus energy balance, involves quantifying appropriate system inputs, outputs, and balances. Mathematical models based on the energy balance equation provide descriptions of the impact of physiological changes and quantitative predictions of body mass during weight change. The development of energy balance models can have two approaches: 1. descriptions of the impact of physiological changes 2. quantitative predictions of body mass during weight change applying minimal individual baseline information.

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Position of different Forbes curves on the NHANES band. A. Female FFM (values (grey) obtained from substituting NHANES FM values [14, 15] into NAA female polynomial in Table 3 overlaying NHANES actual fat free mass (black). B. Three different translates of the Forbes curve from Equation 2 overlaying NHANES actual FFM values for females.

In addition to providing a class of models that can be paired to a one dimensional energy balance differential equations, there are several interesting observations produced by the NHANES polynomials. One application of the NHANES generated models is determination of the width of the band for specific FM values. For example, we applied the NHANES models to determine BMI as a function of height for zero fat (Figure 5). Theoretically, these values estimate the lowest possible BMI and are supported by the close correlation to mean data collected during the 1992-93 Somalia famine in [13] also plotted in Figure 5. For the two data points, the mean age for males was 32 and the mean age for females was 35. Out of the 261 subjects in the famine study, 51 had severe edema which impacted their BMI. However, we can compare the subjects from [13] who had severe edema to the 24 week data from the Minnesota Starvation Experiment some of whom also experienced edema. The mean BMI for the famine subjects with edema was 15.4 at a height of 167 cm and the mean BMI of the Minnesota subjects at 24 weeks was 16.4 with a mean height of 179.5 cm [13, 23]. Although these results are confounded by edema, it provides experimental support for the conclusion that BMI at zero fat is an increasing function of height. The polynomial regression models have several limitations. The effect of physical activity was not incorporated as a covariate. However, as observed with the placement of the athletes within the NHANES band (Figure 2B), physical activity does affect the position of the FFM-FM value on the band. Moreover, increased free-living physical activity was observed during weight gain in [24, 25] and therefore can also be a factor for longitudinal body composition during weight change. It also appears that for the case of gastric bypass, changes in FFM for a cohort of subjects are less than estimated by the NHANES band and their resulting regression models (Figure 2A). This may be due to available active tissue within FFM or increased physical activity as a result of weight loss and remains to be investigated although we point out that this observation does not apply to energy balance equations that model weight change as a function of changes in energy intake and activity.

In conclusion, the relationship between FFM and FM is an integral component for differential equation energy balance models. By examining correlations of FFM with FM through the cross-sectional NHANES data and longitudinal data from calorie restriction, calorie restriction combined with exercise, bariatric surgery, gastric bypass surgery, and overfeeding studies, we established that the trend set by NHANES is traveled during weight change except in certain subjects with massive weight loss during surgery (Figure 2). Our central focus was to supply a FFM function of FM that preserves the predictive properties of the Forbes model for the change in FFM during weight loss, has a non-negative physically realistic intercept, and also accounts for individual variability due to age, height, race and gender. Since the development of the FFM formula, we have applied the new FFM formula within an energy balance equation with satisfactory results [26].

Abstract:Municipal waste management system modeling based on the mass balance of individual waste streams allows us to answer the question of how the system will react to organizational changes, e.g., to the expected reduction in the amount of plastics or the introduction of a deposit for glass and/or plastic packaging. Based on the data on Polish municipal solid waste and the forecast of changes in its quantity and composition, as well as demographic data, a balance model was prepared to assess the impact of introducing higher and higher levels of recycling, in accordance with the circular economy assumptions on the waste management system. It has been shown that, for the Polish composition of municipal waste, even if the assumed recycling levels of individual streams are achieved, achieving the general target level of 65% recycling in 2025/30 may not be feasible. The possibility of achieving a higher level of recycling will be possible due the introduction of selective ash collection from individual home furnaces, while the impact of reducing the amount of plastics or introducing a deposit on packaging is minimal. The calculations also showed that, to complete the waste management system in Poland, we need at least 3.5 million Mg/year of incineration processing capacity and the present state (approx. 1.3 million Mg/year) is insufficient.Keywords: MSW management; recycling; circular economy; mathematical modeling

where, Cu is the ultrafiltrate drug concentration and Cp is the plasma drug concentration after incubation. % Protein Binding, % Recovery and % Stability using the Centrifree device with plasma were calculated by mass balance as follows:

Correlation of mean %PPB in mouse plasma of 27 compounds determined by ultrafiltration versus rapid equilibrium dialysis. %PPB was determined by mass balance as described in equation (2) using ultrafiltration or as described in equation (1) using RED for twenty seven known and investigational compounds in mouse plasma. The values were plotted and a linear regression line plotted with the y intercept set to 0. Compounds deviating most significantly from the regression line are indicated by a lighter degree of shading and are discussed in the text.

Therefore, mass balances are used widely in engineering and environmental analyses. For example, mass balance theory is used to design chemical reactors, to analyse alternative processes to produce chemicals, as well as to model pollution dispersion and other processes of physical systems. Closely related and complementary analysis techniques include the population balance, energy balance and the somewhat more complex entropy balance. These techniques are required for thorough design and analysis of systems such as the refrigeration cycle.

In environmental monitoring, the term budget calculations is used to describe mass balance equations where they are used to evaluate the monitoring data (comparing input and output, etc.). In biology, the dynamic energy budget theory for metabolic organisation makes explicit use of mass and energy balance.

Suppose that the slurry inlet composition (by mass) is 50% solid and 50% water, with a mass flow of 100 kg/min. The tank is assumed to be operating at steady state, and as such accumulation is zero, so input and output must be equal for both the solids and water. If we know that the removal efficiency for the slurry tank is 60%, then the water outlet will contain 20 kg/min of solids (40% times 100 kg/min times 50% solids). If we measure the flow rate of the combined solids and water, and the water outlet is shown to be 65 kg/min, then the amount of water exiting via the conveyor belt must be 5 kg/min. This allows us to completely determine how the mass has been distributed in the system with only limited information and using the mass balance relations across the system boundaries. The mass balance for this system can be described in a tabular form:

Such systems are common in grinding circuits, where grain is crushed then sieved to only allow fine particles out of the circuit and the larger particles are returned to the roller mill (grinder). However, recycle flows are by no means restricted to solid mechanics operations; they are used in liquid and gas flows, as well. One such example is in cooling towers, where water is pumped through a tower many times, with only a small quantity of water drawn off at each pass (to prevent solids build up) until it has either evaporated or exited with the drawn off water. The mass balance for water is M = D + W + E.

A mass balance can also be taken differentially. The concept is the same as for a large mass balance, but it is performed in the context of a limiting system (for example, one can consider the limiting case in time or, more commonly, volume). A differential mass balance is used to generate differential equations that can provide an effective tool for modelling and understanding the target system.

The differential mass balance is usually solved in two steps: first, a set of governing differential equations must be obtained, and then these equations must be solved, either analytically or, for less tractable problems, numerically. 2ff7e9595c