# A vat of volume 10001000 gallons initially contains 44 lbs of salt. For t > 0 pure water is pumped

A vat of volume 10001000 gallons initially contains 44 lbs of salt. For t > 0 pure water is pumped into the vat at the rate of 66 gallons per minute; the perfectly stirred mixture is pumped out at the same flow rate. Derive a formula for the concentration of salt in the tank at any time t.

## This Post Has 3 Comments

1. Expert says:

step-by-step explanation:

2. Expert says:

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3. miyapooh9447 says:

$c(t) = 0.044\cdot e ^{-0.066\cdot t}$

Step-by-step explanation:

The quantity of salt inside the tank is modelled after the Principle of Mass Conservation:

Salt

$\dot m_{in, salt} - \dot m_{out, salt} = \frac{dm_{tank,salt}}{dt}$

Water

$\dot m_{in,water} - \dot m_{out,water} = \frac{dm_{tank,water}}{dt}$

Given that water is an incompressible fluid, the expression can be simplified into the following expression:

$\dot V_{in, water} - \dot V_{out,water} = \frac{dV_{tank, water}}{dt}$

Both flows have the same rate and tank can be modelled as a steady state system.

$\dot V_{in, water} - \dot V_{out,water} = 0$

The expression for salt concentration in the tank is:

-$-\dot V_{tank}\cdot c = V_{tank} \cdot \frac{dc}{dt}$

After some handling, the following homogeneous first-order linear differential equation is found:

$\frac{V_{tank}}{\dot V_{tank}} \cdot \frac{dc}{dt} + c = 0$

Where $c (0) = 0.044\,\frac{lbm}{gal}$. The solution is obtained by using Laplace transforms:

$\frac{V_{tank}}{\dot V_{tank}} \cdot \left[s\cdot C(s) - c(0)\right] + C(s) = 0$

$\left(\frac{V_{tank}}{\dot V_{tank}}\cdot s + 1\right)\cdot C(s) = \frac{V_{tank}}{\dot V_{tank}}\cdot c(0)$

$C(s) = \frac{\frac{V_{tank}}{\dot V_{tank}}\cdot c(0) }{\left(\frac{V_{tank}}{\dot V_{tank}} \right)\cdot \left(s + \frac{\dot V_{tank}}{V_{tank}} \right)}$

$c(t) = c(0) \cdot e^{-\frac{\dot V_{tank}}{V_{tank}}\cdot t }$

Where $\frac{\dot V_{tank}}{V_{tank}} = 0.066\,min^{-1}$.

The formula for the concentration of salt in the tank is:

$c(t) = 0.044\cdot e ^{-0.066\cdot t}$