Question 6.5.3: Production of a Hydrated Salt An aqueous solution of magnesi......
Production of a Hydrated Salt
An aqueous solution of magnesium sulfate at 104°C containing 30.1 wt% MgSO_{4} is fed to a cooling crystallizer that operates at 10°C. The stream leaving the crystallizer is a slurry of solid magnesium sulfate heptahydrate particles [MgSO_{4} · 7H_{2}O(s)] suspended in a liquid solution. Solubility data for magnesium sulfate [Perry’s Chemical Engineers’ Handbook (see Footnote 1), p. 18-40] show that a saturated solution at 10°C contains 23.2 wt% MgSO_{4}. Determine the rate at which solution must be fed to the crystallizer to produce 1 metric ton (1 tonne, 1000 kg) of magnesium sulfate heptahydrate per hour.
Basis: 1 tonne MgSO_{4} · 7H_{2}O(s) Produced/h
We assume that the solution leaving the crystallizer is in equilibrium with the solid crystals and is therefore saturated with MgSO_{4}. A flowchart of the crystallizer follows:
There are two unknowns on the chart (\dot{m}_{1} and \dot{m}_{2} ) and two independent molecular species on which balances may be written (MgSO_{4} \text{ and } H_{2}O), so that the problem can be solved. The atomic weights on the inside back cover may be used to show that the molecular weight of anhydrous magnesium sulfate is 120.4 and that of the heptahydrate salt is 246.4. The balances follow.
Total Mass Balance: \dot{m}_{1} =1 tonne/h + \dot{m}_{2}
MgSO_{4} Balance:
\begin{matrix}0.301 \dot{m}_{1}\left(\frac{1 tonne MgSO_{4}}{h} \right) = \begin{array}{c|c}tonne MgSO_{4} \cdot 7H_{2}O& 120.4 tonne MgSO_{4} \\ \hline h &246.4 tonne MgSO_{4} \cdot 7H_{2}O\end{array} \\ + \begin{array}{c|c} \dot{m}_{2} (tonne solution/h) & 0.232 tonne MgSO_{4}\\ \hline & tonne solution \end{array} \end{matrix}
Solving these two equations simultaneously yields \boxed{\dot{m}_{1}= 3.71 tonne/h } and \dot{m}_{2} = 2.71 tonne/h.
