A Mixing Process A mixing tank is shown in Figure 2.1.4. Pure water flows into the tank of volume V = 600 m³ at the constant volume rate of 5 m³/s. A solution with a salt concentration of si kg/m³ flows into the tank at a constant volume rate of 2 m³/s. Assume that the solution in the tank is well

Question

A Mixing Process

A mixing tank is shown in Figure 2.1.4. Pure water flows into the tank of volume V = 600 m³ at the constant volume rate of 5 m³/s. A solution with a salt concentration of [latex]s_i[/latex] kg/m³ flows into the tank at a constant volume rate of 2 m³/s. Assume that the solution in the tank is well mixed so that the salt concentration in the tank is uniform. Assume also that the salt dissolves completely so that the volume of the mixture remains the same. The salt concentration [latex]s_o[/latex] kg/m³ in the outflow is the same as the concentration in the tank. The input is the concentration [latex]s_i(t)[/latex], whose value may change during the process, thus changing the value of [latex]s_o[/latex]. Obtain a dynamic model of the concentration [latex]s_o.[/latex]

Accepted Answer

Two mass species are conserved here: water mass and salt mass. The tank is always full, so the mass of water [latex]m_w[/latex] in the tank is constant, and thus conservation of water mass gives

[latex]\frac{dm_w}{dt}=5\rho_w+2\rho_w-\rho_w q_{vo}=0[/latex]

where [latex]\rho_w[/latex] is the mass density of fresh water, and [latex]q_{vo}[/latex] is the volume outflow rate of the mixed solution. This equation gives [latex]q_{vo}=5+2=7 \ m^3/s[/latex] The salt mass in the tank is [latex]s_oV[/latex], and conservation of salt mass gives

[latex]\frac{d}{dt}(s_oV)=9(5)+2 s_i-s_oq_{vo}=2s_i-7s_o[/latex]

or, with V = 600,

[latex]600\frac{s_o}{dt}=2s_i-7s_o[/latex]

This has the same form as (2.1.11) where a = 7/600 and b = [latex]s_i/300[/latex]. If [latex]s_i[/latex] is constant, we can solve for [latex]s_o(t)[/latex] using (2.1.12).

[latex]\frac{dx}{dt}+ax=b \quad (2.1.11)[/latex]

[latex]x(t)=\frac{b}{a}+[x(0)-\frac{b}{a}] e^{-at} \quad (2.1.12)[/latex]

[latex]s_o(t)=\frac{2s_i}{7}+[s_o(0)-\frac{2s_i}{7}]e^{-7t/600}[/latex]

Question 2.1.2: A Mixing Process A mixing tank is shown in Figure 2.1.4. Pur...

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