Paddy grain is dried in a thin layer dryer from an initial moisture content of $24\%$ (w.b.) to a final moisture content of $14\%$ (w.b.). The equilibrium moisture content under the drying air conditions is $10\%$ (d.b.). Assuming drying follows the thin layer Lewis exponential equation $\frac{M - M_e}{M_0 - M_e} = e^{-k t}$ with drying constant $k = 0.25\text{ h}^{-1}$, the drying time required is ________ hours (round off to two decimal places).
Answer Key:
4.94
Step-by-Step Solution & Explanation
1. Convert moisture contents to dry basis (d.b.):
$M_0 = \frac{0.24}{1 - 0.24} = \frac{0.24}{0.76} = 0.31579\text{ kg water/kg dry solid}$
$M = \frac{0.14}{1 - 0.14} = \frac{0.14}{0.86} = 0.16279\text{ kg water/kg dry solid}$
$M_e = 0.10\text{ kg water/kg dry solid}$
2. Compute Moisture Ratio ($MR$):
$MR = \frac{M - M_e}{M_0 - M_e} = \frac{0.16279 - 0.10}{0.31579 - 0.10} = \frac{0.06279}{0.21579} = 0.29098$
3. Calculate drying time $t$ using Lewis model:
$\ln(MR) = -k t \implies t = -\frac{\ln(0.29098)}{0.25} = -\frac{-1.23447}{0.25} = 4.9379\text{ hours} \approx 4.94\text{ hours}$