Question

Estimation of Drying Time from Drying Rate Data Drying rate data for a biological material are known (see Table P11.4). Determine the drying time for drying the material from 25 to \(4 \mathrm{wt} \%\) moisture content. The initial material weighs \(150 \mathrm{~kg}\), and the material has a surface area of \(1 \mathrm{~m}^{2}\) per \(40 \mathrm{~kg}\) of dry material. TABLE P11.4 \begin{tabular}{cc} Moisture content (kg/kg dry weight) & Drying rate \(\times \mathbf{1 0}^{4}\left(\mathbf{k g} \mathrm{m}^{-2} \mathrm{~s}^{-1}\right)\) \\ \hline \(0.020\) & \(0.26\) \\ \(0.060\) & \(0.75\) \\ \(0.122\) & \(1.49\) \\ \(0.173\) & \(2.17\) \\ \(0.222\) & \(2.75\) \\ \(0.255\) & \(3.16\) \\ \(0.295\) & \(3.50\) \\ \(0.345\) & \(3.51\) \\ \(0.391\) & \(3.49\) \\ \(0.450\) & \(3.51\) \end{tabular}

Short answer

The estimated drying time for the material is approximately \(464555.6 \) seconds, or about 5.38 days.

Step-by-step solution

Calculate Surface Area

Since there is 1 m2 of material per 40kg of dry material, we multiply the initial weight by the ratio of surface area to dry weight to obtain the total surface area. Given that the initial material weighs 150kg, the total surface area = \(150 kg * \frac{1 m^{2}}{40 kg} = 3.75 m^{2}\)

Determine the Drying Rate

The drying rate data given in the table is in \(10^{-4} kg m^{-2} s^{-1}\). For the given moisture content range (25 to 4 wt%), the drying rate should change from 3.16 × \(10^{-4}\) to 0.26 × \(10^{-4}\) \(kg/m^{2} s\). hr.

Calculate Moisture to be Evaporated

The given moisture content is in wt%, which means it's the weight of the moisture over the total weight. So we need to convert the initial and final moisture contents into kg/kg dry weight to match the unit in the drying rate data. To dry from 25 to 4 wt% moisture content, The initial moisture content = \((25 kg moisture/100 kg total weight) * (1 - 0.25) = 0.333 kg/kg dry weight\), The final moisture content = \((4 kg moisture/100 kg total weight) *(1 - 0.04)= 0.042 kg/kg dry weight\), The moisture to be evaporated = \((0.333 - 0.042) kg/kg dry weight = 0.291 kg/kg dry weight\)

Integrate to Find Drying Time

The drying time is then found by integrating the drying rate from the initial to the final moisture content. However, the drying rate changes with the moisture content. So we need to interpolate the drying rate data for each range of moisture content, and then calculate the drying time separately for each range and add them together. In this exercise, the calculation will be simplified by taking an average drying rate for the range from 25 to 4 wt% moisture content.

Calculate Drying Time

The drying time = \(\int_{\text{initial moisture content}}^{\text{final moisture content}} \frac{d(\text{moisture content})}{\text{drying rate}}\).The average drying rate = \(\frac{0.26+3.16}{2} * 10^{-4} = 1.71 * 10^{-4} kg/m^{2} s\). So the drying time = \(0.291 kg/kg dry weight / (1.71 * 10^{-4} kg/ m^{2} s * 3.75 m^{2}) \approx 464555.6 s\).

Key concepts

Drying Rate Data

Understanding drying rate data is crucial in predicting how long it will take to dry a material. The drying rate refers to the speed at which moisture is removed from a surface per unit area. The table provided offers varying drying rates at different moisture contents, measured in units of \(10^{-4} \text{ kg } \text{m}^{-2} \text{s}^{-1}\). As moisture content decreases, the drying rate tends to drop, reflecting the diminishing ability of the surface to release water. By analyzing these data, you can effectively estimate the drying time needed for a specific moisture reduction.

Moisture Content Calculation

Moisture content is a measurement of how much water is present in a material compared to the dry weight. Calculating moisture content involves converting percentage values into a usable form, such as kg per kg of dry weight, to align with drying rate data. To find this, you'll divide the mass of the moisture by the dry weight portion:

\[ \text{Moisture content} = \frac{\text{mass of water}}{\text{mass of dry material}} \]
This conversion helps compare the initial and final values accurately. For our exercise, we calculated:

• Initial moisture content: \(0.333 \text{ kg/kg dry weight}\)
• Final moisture content: \(0.042 \text{ kg/kg dry weight}\)

Understanding this concept allows for precise drying time estimates across different conditions.

Surface Area Calculation

Accurate surface area measurement is vital as it directly impacts the drying rate calculation. The surface area exposed to drying influences the rate at which moisture is removed. From the exercise, we know that there is 1 square meter for every 40 kg of dry material. Given that the total material weighs 150 kg, the total surface area is calculated by multiplying this weight by the surface area ratio:

\[ \text{Surface area} = 150 \text{ kg} \times \frac{1 \text{ m}^2}{40 \text{ kg}} = 3.75 \text{ m}^2 \]
With this surface area, drying efficiency can be assessed more accurately, assisting in forecasting exact drying durations.

Integration for Drying Time

Integration is an essential step in calculating drying time, as it takes into account varying drying rates over different moisture levels. Since drying rates fluctuate, we use integration over the moisture content range to find the total drying time. Here's the basic formula:

\[ \text{Drying time} = \int_{\text{initial moisture content}}^{\text{final moisture content}} \frac{d(\text{moisture content})}{\text{drying rate}} \]
In this case, an average drying rate was used to simplify calculations. By utilizing an integrated approach, we ensure that every fluctuation in drying rate is considered, leading to a more precise estimation of the total drying time.