Rough rice were dried at 50 °C and 60% relative humidity. The moisture content at time 3 and 5 hrs are 12.0 and 10.0%, respectively. If the drying rate constant be 0.49 per hour, c... Rough rice were dried at 50 °C and 60% relative humidity. The moisture content at time 3 and 5 hrs are 12.0 and 10.0%, respectively. If the drying rate constant be 0.49 per hour, calculate the equilibrium moisture content of the sample.
Understand the Problem
The question is asking to calculate the equilibrium moisture content of rough rice given specific drying conditions and moisture content values at different times. The calculation involves using the drying rate constant and the moisture content measurements provided.
Answer
The equilibrium moisture content of the sample is approximately $10.0\%$.
Answer for screen readers
The equilibrium moisture content of the sample is approximately $10.0%$.
Steps to Solve
- Identify given values
The problem provides the following values:
- Initial moisture content ( M_1 = 12.0% ) at ( t_1 = 3 ) hours
- Final moisture content ( M_2 = 10.0% ) at ( t_2 = 5 ) hours
- Drying rate constant ( k = 0.49 ) per hour
- Calculate the moisture content difference
Determine the change in moisture content between the two time points:
$$ \Delta M = M_1 - M_2 $$ $$ \Delta M = 12.0% - 10.0% = 2.0% $$
- Determine the time interval
Calculate the time interval ( \Delta t ) between the points:
$$ \Delta t = t_2 - t_1 $$ $$ \Delta t = 5 \text{ hours} - 3 \text{ hours} = 2 \text{ hours} $$
- Calculate the average drying rate
The average drying rate ( R ) can be calculated as follows:
$$ R = \frac{\Delta M}{\Delta t} $$ $$ R = \frac{2.0%}{2 \text{ hours}} = 1.0% \text{ per hour} $$
- Apply the drying rate expression
In drying processes, we use the equation:
$$ R = k(M - M_e) $$
Where ( M_e ) is the equilibrium moisture content that we want to find. Rearranging gives:
$$ M - M_e = \frac{R}{k} $$
- Calculate equilibrium moisture content
Substituting the known values from the first time point ( M_1 ):
$$ M_1 - M_e = \frac{1.0%}{0.49} $$ $$ M_e = M_1 - \frac{1.0%}{0.49} $$
Calculating the right side:
$$ M_e = 12.0% - \frac{1.0}{0.49} \approx 12.0% - 2.04% \approx 9.96% $$
Thus, we conclude that:
$$ M_e \approx 10.0% $$
The equilibrium moisture content of the sample is approximately $10.0%$.
More Information
The equilibrium moisture content (EMC) is the moisture level at which the rice neither gains nor loses moisture in the given conditions. This calculation is essential in food processing to ensure quality and to prevent spoilage.
Tips
- Confusing the initial and final moisture content values.
- Miscalculating the time difference which can lead to incorrect drying rate.
- Forgetting to convert percentage values correctly when using them in calculations.
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