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Chapter 4: Problem 54

A tank of fixed volume contains brine with initial density, \(\rho_{l},\) greater than water. Pure water enters the tank steadily and mixes thoroughly with the brine in the tank. The liquid level in the tank remains constant. Derive expressions for (a) the rate of change of density of the liquid mixture in the tank and (b) the time required for the density to reach the value \(\rho_{f}\) where \(\rho>q>\rho_{\mathrm{H}_{2} \mathrm{O}}\)

Short Answer

Expert verified
The rate of change of density of the liquid mixture in a tank is given by \(\frac{d\rho}{dt} = -\frac{q}{V}\rho\). The time required for the density to reach the value \(\rho_{f}\) is given by \( t = -\frac{V}{q}\ln\left(\frac{\rho_{f}}{\rho_{l}}\right)\).

Step by step solution

01

Understanding the Situation

The tank contains a mixture of brine and pure water is entering the tank at a constant rate. The level of liquid in the tank remains constant and the brine is being diluted by the incoming water. The density of the overall mixture decreases over time because the water is less dense than the brine. Express the total mass of brine in the tank at time \(t\) as \(M(t) = V \rho(t)\), where \(V\) is the volume of the tank and \(\rho(t)\) is the density of the mixture at time \(t\).
02

Derive the rate of change of density

Consider the change in the mass of the brine in the tank. This is given by the inflow minus the outflow. The inflow is zero as pure water is entering the tank. The outflow is the amount of brine leaving the tank which is given by the product of the density, \(\rho(t)\), and the rate of outflow, \(q\). By conservation of mass, this gives the differential equation: \(\frac{dM}{dt} = -q \rho\). Substituting \(M(t) = V\rho(t)\) into the differential equation gives the equation: \(V\frac{d\rho}{dt} = -q\rho\). This simplifies to give: \(\frac{d\rho}{dt} = -\frac{q}{V}\rho\).
03

Derive the time required for density to reach a given value

The differential equation \(\frac{d\rho}{dt} = -\frac{q}{V}\rho\) is a first order linear differential equation that can be solved analytically. The general solution to this equation is \(\rho(t) = \rho_{l}e^{-qt/V}\), where \( \rho_{l}\) is the initial density of the brine. To find the time for the density to reach a specified value \( \rho_{f}\), equate \(\rho(t) = \rho_{f}\) giving \( \rho_{f} = \rho_{l}e^{-qt/V}\) and solving for \( t\), gives \( t = -\frac{V}{q}\ln\left(\frac{\rho_{f}}{\rho_{l}}\right)\).

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