State the conservation of mass
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Conservation of Mass: Fundamental Principles and Applications
Traditional Conservation of Mass
The principle of conservation of mass states that mass cannot be created or destroyed in a closed system through ordinary physical or chemical processes. This fundamental law is derived using a first-order Taylor series to represent flux changes in a control volume, which is valid for linear changes in flux through the control volume1. This principle is crucial in various scientific and engineering disciplines, ensuring that the total mass remains constant over time.
Fractional Conservation of Mass
Recent advancements have introduced the concept of fractional conservation of mass, which uses higher-order Taylor series approximations for mass flux. This approach can exactly represent non-linear flux in a control volume with only two terms, similar to a first-order traditional Taylor series. The fractional-order conservation of mass equation is exact when the fractional order of differentiation matches the flux power-law, removing the restriction that flux must be linear or piece-wise linear1. This method introduces a fractional divergence term and a scaling term that may eliminate scale effects in parameters like hydraulic conductivity.
Mass Conservation in Numerical Models
Mass conservation is a critical aspect of numerical models, especially in geodynamics and climate modeling. However, many models fail to respect this principle, often conserving volume rather than mass. This discrepancy can lead to significant errors, particularly during phase transitions such as metamorphism or melting3. Ensuring mass conservation in these models is not optional but a necessity for accurate simulations.
Mass Conservation in Climate Models
In climate diagnostics, the conservation of mass is examined using data from the European Centre for Medium Range Weather Forecasts (ECMWF). Analyses have shown spurious residuals in the equation of continuity, particularly in the tropics, indicating significant errors. Improvements are noted when using higher resolution data, but challenges remain in achieving perfect mass conservation4.
Mass Conservation in Renal Research
The principle of conservation of mass is also fundamental in medical research, particularly in the treatment of end-stage renal disease (ESRD). Dialysis treatment relies on controlling solute concentrations and managing fluid volume, with mass balance being a key concept. Mathematical models incorporating mass conservation are essential for quantitatively guided studies and improving dialysis outcomes9.
Conservation of Mass in Physics and Chemistry
In classical physics and chemistry, conservation of mass is equivalent to conservation of matter, ensuring that the total mass of atoms remains constant during chemical reactions. This principle is a good approximation in these fields, as the variation in mass is typically undetectable by weighing. However, in nuclear and particle physics, conservation of mass is not valid for expressing conservation of matter, as mass is a form of energy and can vary significantly10.
Conclusion
The conservation of mass is a fundamental principle across various scientific disciplines, ensuring that mass remains constant in a closed system. While traditional methods rely on linear approximations, fractional conservation of mass offers a more accurate representation for non-linear fluxes. Ensuring mass conservation in numerical models and medical research is crucial for accurate simulations and effective treatments. Despite its limitations in nuclear physics, the principle remains a cornerstone of classical physics and chemistry.
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Fractional conservation of mass
Fractional-order Taylor series approximation accurately represents non-linear flux in a control volume, eliminating restrictions on linearity and infinitesimal control volumes.
Highly CitedConservation of mass and instability in a dynamic economy-environment system
The conservation of mass in a dynamic economy-environment system leads to continuous disequilibrating change in material transformations, uncontrollable through the price system.
Climate Diagnostics from Global Analyses: Conservation of Mass in ECMWF Analyses
ECMWF analyses show up to 100% spurious residuals in pressure coordinate equations, with largest errors in the tropics, and modest improvements occur when 14 versus 7 levels are used.
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