Typical scenario: • Volume of tank: $V$ units$^3$ • Amount of substance in tank at $t$: $x$

Question

Typical scenario: • Volume of tank: $V$ units$^3$ • Amount of substance in tank at $t$: $x$ mass • The rate of substance entering the tank: $m$ mass/time • Constant rate of water in and out of tank: $a$ units$^{3}$/time Assumption: Concentration of substance in liquid is well- mixed and uniform throughout the tank. To find: Rate of salt leaving tank $=\frac{\mathrm{d}x}{\mathrm{d}V}\times \frac{\mathrm{d}V}{\mathrm{d}t}=\frac{x}{V}\times a$ Total rate of substance, $\frac{\mathrm{d}x}{\mathrm{d}t}$ $=$ the rate of substance entering $–$ the rate of substance leaving $=$ $m-\frac{x}{V}\times a=m-\frac{ax}{V}$

Accepted Answer

$\text{Outflow mass rate}=\frac{ax}{V}$ $\frac{\mathrm{d}x}{\mathrm{d}t}=m-\frac{ax}{V}$

Symbol	Meaning	Units
\(V\)	constant liquid volume	volume
\(x(t)\)	amount of solute at time \(t\)	mass
\(a\)	equal volumetric inflow and outflow rate	volume per unit time
\(m\)	solute inflow rate	mass per unit time

Quantity	Expression	Units
Tank concentration	\(\displaystyle\frac{x(t)}{V}\)	mass per volume
Solute inflow	\(m\)	mass per unit time
Solute outflow	\(\displaystyle\frac{x(t)}{V}a\)	mass per unit time

Mixing-Tank Models

Mixing-Tank Models

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