A solid food was dried from 50 to 10% moisture content in 2.5 h in a batch drier with constant air conditions. The drying rate remained constant down to a moisture content of 18%. If the equilibrium moisture content is 2%, calculate the total time required to dry from 50 to 4% moisture content. All moisture contents are given on a dry basis.

To solve this problem, we'll break it down into two stages: the constant rate drying period and the falling rate drying period. Given that the moisture content (MC) in the food is initially high, the drying rate remains constant until it reaches a certain critical moisture content, after which it starts to slow down (falling rate period). 

### Given:
- Initial moisture content \( MC_1 = 50\% \) dry basis.
- Final moisture content \( MC_3 = 4\% \) dry basis.
- Critical moisture content \( MC_c = 18\% \) dry basis (end of constant rate period).
- Equilibrium moisture content \( MC_e = 2\% \) dry basis.
- Constant drying time from \( 50\% \) to \( 18\% \) is 2.5 hours.
- Drying rate remains constant from \( 50\% \) to \( 18\% \).

### Step 1: Constant Rate Period (from \( 50\% \) to \( 18\% \))
The constant drying rate can be calculated based on the drying time provided. The moisture loss during this period is \( 50\% - 18\% = 32\% \).

Thus, the drying rate \( R_c \) (moisture removed per unit time) in this period is:
\[
R_c = \frac{50\% - 18\%}{2.5 \text{ hours}} = \frac{32\%}{2.5} = 12.8\%/\text{hour}.
\]
This rate will be used for the constant drying period.

### Step 2: Falling Rate Period (from \( 18\% \) to \( 4\% \))
In the falling rate period, the moisture content decreases from \( 18\% \) to \( 4\% \), with the drying rate no longer constant. The process now depends on the difference between the current moisture content and the equilibrium moisture content (\( MC_e = 2\% \)).

In the falling rate period, the drying is typically modeled by a linear approximation, which can be written as:

\[
\frac{dMC}{dt} \propto (MC - MC_e)
\]

Thus, the drying time \( t_f \) for the falling rate period can be estimated using the relationship:

\[
t_f = \frac{MC_2 - MC_3}{R_f},
\]
where \( R_f \) is the average drying rate in the falling rate period, calculated as:

\[
R_f = R_c \cdot \frac{MC_2 - MC_e}{MC_1 - MC_e} = 12.8 \cdot \frac{18\% - 2\%}{50\% - 2\%} = 12.8 \cdot \frac{16\%}{48\%} = 12.8 \cdot \frac{1}{3} = 4.27\%/\text{hour}.
\]

Now, the drying time \( t_f \) for the falling rate period (from 18% to 4%) is:

\[
t_f = \frac{18\% - 4\%}{4.27\%/\text{hour}} = \frac{14\%}{4.27} = 3.28 \text{ hours}.
\]

### Step 3: Total Drying Time
The total drying time is the sum of the constant rate period and the falling rate period:

\[
t_{total} = t_c + t_f = 2.5 \text{ hours} + 3.28 \text{ hours} = 5.78 \text{ hours}.
\]

### Final Answer:
The total time required to dry the solid food from 50% to 4% moisture content is approximately **5.78 hours**.

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Would you like further details or have any questions?

Here are five related questions for further practice:
1. How would the drying time change if the equilibrium moisture content were 5% instead of 2%?
2. What is the significance of the equilibrium moisture content in drying processes?
3. How is the drying rate affected if air conditions (temperature or humidity) change during the drying process?
4. If the drying rate was not constant initially, how would you approach calculating the total drying time?
5. How could you experimentally determine the critical moisture content in a drying process?

**Tip:** The constant rate period typically corresponds to when the surface of the material is saturated with moisture, while the falling rate period occurs as moisture needs to diffuse from inside the material to the surface.

This problem involves calculating the total drying time for a solid food product, drying from 50% to 4% moisture content. It includes both constant and falling rate periods, using linear drying models. Key concepts include drying rates, moisture content, and the application of related formulas. Suitable for undergraduate students studying food processing or chemical engineering.

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