Overall material balances
$$ [1] \quad m_{feed} = m_{2} + m_{4} + m_{5} $$
$$ [2] \quad z_{w} \, m_{feed} = m_{2} + x_{w,5} \, m_{5} $$
$$ [3] \quad z_{k} \, m_{feed} = m_{4} + x_{k,5} \, m_{5} $$
\( m_{feed} = \) mass flow rate of fresh feed (kg/h)
\( m_{2} = \) mass flow rate of water exiting evaporator (kg/h)
\( m_{4} = \) mass flow rate of solid KCl crystals exiting the crystallizer and filter (kg/h)
\( m_{5} = \) mass flow rate of slurry exiting the crystalizer and filter (kg/h)
\( z_{k} = \) mass fraction of KCl in fresh feed
\( z_{w} = \) mass fraction of water in fresh feed
Material balance around crystallizer and filter
$$ [4] \quad m_{3} = m_{4} + m_{5} + m_{R} $$
$$ [5] \quad x_{k,3} \, m_{3} = m_{4} + x_{k,5} \, m_{5} + x_{k,R} \, m_{R} $$
\( m_{3} = \) mass flow rate of solution exiting evaporator (kg/h)
\( m_{R} = \) mass flow rate of recycle (kg/h)
\( x_{k,3} = \) mass fraction of KCl in stream exiting evaporator
\( x_{k,5} = \) mass fraction of KCl in solution exiting crystallizer and filter
\( x_{w,5} = \) mass fraction of water in solution exiting crystallizer and filter (this is equivalent to \( x_{k,R} \) as the exit stream from the crystallizer is split).
\( x_{k,R} = \) mass flow rate of solution exiting evaporator (kg/h)
Material balances around mixing point
$$ [6] \quad m_{feed} + m_{R} = m_{1} $$
$$ [7] \quad z_{k} \, m_{feed} + x_{k,R} \, m_{R} = x_{k,1} \, m_{1} $$
\( m_{1} = \) mass flow rate entering evaporator (kg/h)
\( x_{k,1} = \) mass fraction of KCl in stream entering evaporator
Material balance around evaporator
$$ [8] \quad m_{1} = m_{2} + m_{3} $$
Dependence of solubility on temperature
$$ [9] \quad s_{3} = 0.2881 \, T_{evap} + 28.123 $$
$$ [10] \quad x_{k,3} = \frac{s_{3}}{s_{3} + 100 \; \mathrm{kg}} $$
$$ [11] \quad s_{5} = 0.2881 \, T_{crys} + 28.123 $$
$$ [12] \quad x_{k,3} = \frac{s_{5}}{s_{5} + 100 \; \mathrm{kg}} $$
\( s_{3} = \) solubility of KCl in 100 kg H2O
\( T_{evap} = \) evaporator operating temperature (°C)
\( s_{5} = \) solubility of KCl in 100 kg H2O
\( T_{crys} = \) crystallizer and filter operating temperature (°C)
Note that \( z_{k} + z_{w} = 1 \) and \( x_{k,i} + x_{w,i} = 1 \) where \( i \) is the stream. Recovery rate \( R = m_{4} / ( z_{k} \, m_{feed} ) \).
View
this screencast for more practice with material balances on a crystallizer.