The Reynolds Transport Theorem

June 8, 2023

Keeping track of water is done through a scientific accounting methodology known as the Reynolds transport theorem, which is used to track the flows, fluxes, and storage of water in a physical system.

\( \frac{{dB}}{{dt}} = \iiint\limits_{cv} {\beta \rho d\forall } + \iint\limits_{cs} {\beta \rho v} \cdot dA \)

A simpler way to express this bookkeeping is as follows:

Wfinal = Winitial + Wadditions – Wwithdrawals

That is, total water in a system (Wfinal) is equal to the initial water (Winitial), plus added water (Wadditions), minus withdrawn water (Wwithdrawals). The paradigm resembles that of financial bookkeeping: the amount of money we have in the bank tomorrow is the sum of what we had yesterday, plus today’s deposits, minus withdrawals. A surplus depends on whether we withdraw more than we deposit. If society withdraws water from an aquifer faster than water returns from rainfall, then the aquifer will go dry.

The connection between water bookkeeping and financial bookkeeping is a way of life in some societies. For example, the unit of currency in Botswana is the pula, which literally translates as rain, (brought to the attention of the author by Dr. Ashlynn Stillwell). Terms, such as petrodollars and black gold, suggest that energy and money are also synonymous in some contexts.

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