What is the Weir equation for REE? Clarifying a Scientific Misconception

November 19, 2025 •

3 min read

While the phrase "Weir equation for REE" is often searched, it typically arises from a common misunderstanding between two different scientific fields. The Weir equation is a formula for calculating Resting Energy Expenditure, a physiological metric, completely unrelated to the chemical extraction or separation of rare earth elements.

Quick Summary

The Weir equation calculates metabolic rate or volumetric discharge, depending on the scientific field, and is not used for rare earth element processing. This guide clarifies the equation's actual applications in calorimetry and hydraulics and explains the distinct methods used in rare earth element metallurgy.

Key Points

Misconception: The phrase "Weir equation for REE" stems from confusing two different scientific fields that use the same acronym, REE.
Physiological Application: In medicine, the Weir equation is used in indirect calorimetry to measure Resting Energy Expenditure (REE) from oxygen consumption ($VO_2$) and carbon dioxide production ($VCO_2$).
Hydraulics Application: In engineering, a different Weir equation calculates the volumetric discharge of fluid over a weir based on the head of water.
REE Processing Methods: The separation of Rare Earth Elements (REEs) involves complex hydrometallurgical processes like solvent extraction, ion exchange, and precipitation, not a simple empirical equation like the Weir formula.
Solvent Extraction Principles: REE separation relies on chemical equilibrium and the distribution ratio, which quantifies the separation efficiency between aqueous and organic phases.

Understanding the Weir Equation in Different Contexts

The notion of a single "Weir equation for REE" is a source of confusion because the acronym REE and the term 'Weir equation' apply to entirely separate scientific disciplines. The Weir equation is not a universal formula and has no application in the field of rare earth element processing. Instead, it is prominently known in two unrelated areas: indirect calorimetry (where REE stands for Resting Energy Expenditure) and hydraulics (where it calculates discharge over a weir).

This article aims to clarify this scientific misconception by detailing the true purpose of the Weir equation in its proper contexts and outlining the distinct and complex methods used for rare earth element metallurgy.

The Weir Equation for Resting Energy Expenditure (REE)

In medicine and nutritional science, the Weir equation is a formula used in indirect calorimetry to measure a person's metabolic rate. In this context, REE stands for Resting Energy Expenditure and the equation calculates calories burned at rest by quantifying oxygen consumption ($VO_2$) and carbon dioxide production ($VCO_2$).

The most cited version of the equation is: $REE (kcal/day) = (3.941 \times VO_2) + (1.106 \times VCO_2) \times 1440$. A simplified version for clinical use is also common: $REE = (3.94 \times VO_2) + (1.1 \times VCO_2)$, where $VO_2$ and $VCO_2$ are in liters per minute. This equation is a fundamental tool for assessing metabolic health.

The Weir Equation in Hydraulics and Fluid Dynamics

In civil and environmental engineering, a different formula is also called the Weir equation. This formula calculates the volumetric discharge (flow rate) of water over a weir, a structure like a dam used to measure flow. The standard formula for a rectangular weir is:

$Q = C \times L \times H^{3/2}$

Here, Q is the volumetric discharge, C is a discharge coefficient, L is the weir crest length, and H is the water height above the crest. This equation is important for designing and managing water systems and is based on fluid mechanics principles.

Why Rare Earth Element (REE) Processing Does Not Use a Weir Equation

There is no Weir equation for rare earth elements because REE processing falls under hydrometallurgy and chemical engineering. It uses chemical and physical principles distinct from those of metabolic rate or fluid flow. Rare earth elements (REEs) are a group of seventeen elements with very similar chemical properties, making separation challenging.

Instead of a simple equation, REE separation involves complex multistage chemical processes that exploit small differences in properties like ionic size. Key industrial methods include:

Common REE Processing Techniques

Leaching: Extracting REEs from crushed ore using acids or alkalis to create a solution.
Solvent Extraction (SX): A primary method for separating individual REEs from the solution. This involves mixing the REE solution with an organic solvent containing an extractant. REE ions move into the organic phase based on their affinity for the extractant. Separation efficiency is measured by the distribution ratio ($D$). The process is repeated in many stages for high purity.
Ion Exchange: Another method where REEs attach to resins and are then separated using a chelating agent.
Precipitation: Using reagents to precipitate REEs from the solution.

These methods rely on complex chemical equilibrium, unlike the gas-exchange-based Weir equation for metabolic rate.

Comparison of Weir Equation Contexts


Feature	Indirect Calorimetry	Hydraulics	REE Processing
Abbreviation REE Meaning	Resting Energy Expenditure	Not applicable to Weir equation	Rare Earth Elements
Purpose of Equation	Calculate metabolic rate from gas exchange	Determine flow rate over a weir or dam	Not Applicable (No single Weir Equation)
Associated Field	Medicine, Nutrition, Physiology	Civil Engineering, Water Management	Hydrometallurgy, Chemical Engineering
Governing Principles	Thermodynamic, Biological Gas Exchange	Fluid Mechanics, Hydrodynamics	Chemical Equilibrium, Solvent Extraction

Conclusion

The phrase "Weir equation for REE" results from confusing the Weir equation used in indirect calorimetry (for Resting Energy Expenditure) and hydraulics (for fluid discharge) with the separate field of rare earth element (REE) processing. The metallurgical separation of REEs uses complex chemical methods like solvent extraction and ion exchange, not the Weir equation. Understanding the distinct applications of the Weir equation and the methods for REE processing clarifies this common scientific misconception.

Frequently Asked Questions

In medicine, the Weir equation is a formula used in indirect calorimetry to calculate Resting Energy Expenditure (REE). It uses measurements of oxygen consumption ($VO_2$) and carbon dioxide production ($VCO_2$) to determine a person's metabolic rate.

When associated with the Weir equation in medicine, REE stands for Resting Energy Expenditure. This is a measure of the energy expended by the body at rest.

Rare earth elements (REEs) are processed using sophisticated hydrometallurgical methods. Key techniques include acid or alkali leaching, followed by multistage solvent extraction or ion exchange to separate the individual elements based on their chemical properties.

No, the Weir equation is not used for mineral processing. That field employs complex chemical and physical separation techniques, such as leaching and solvent extraction, governed by chemical equilibrium and thermodynamic principles.

In engineering, the Weir equation is a hydraulics formula used to calculate the volumetric flow rate of water over a weir or dam. It depends on factors like the length of the weir and the height of the water above it.

Separating rare earth elements is difficult because they have very similar chemical properties, particularly their valence states. Modern hydrometallurgical techniques exploit the subtle differences in their ionic size and stability to achieve separation.

The Weir equation is a single formula for calculating metabolic rate or fluid flow. Solvent extraction is a complex, multi-stage chemical process used to separate different rare earth elements from each other based on differences in their distribution coefficients between two liquid phases. There is no connection between the two.