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A sheet of cellulose acetate film containing 5.00 wt\% liquid acetone enters an adiabatic dryer where \(90 \%\) of the acetone evaporates into a stream of dry air flowing over the film. The film enters the dryer at \(T_{\mathrm{f} 1}=35^{\circ} \mathrm{C}\) and leaves at \(T_{\mathrm{f} 2}\left(^{\circ} \mathrm{C}\right) .\) The air enters the dryer at \(T_{\mathrm{al}}\left(^{\circ} \mathrm{C}\right)\) and 1.01 atm and exits the dryer at \(T_{\mathrm{a} 2}=49^{\circ} \mathrm{C}\) and 1 atm with a relative saturation of \(40 \% . C_{p}\) may be taken to be \(1.33 \mathrm{kJ} /\left(\mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) for dry film and \(0.129 \mathrm{kJ} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\) for liquid acetone. Make a reasonable assumption regarding the heat capacity of dry air. The heat of vaporization of acetone may be considered independent of temperature. Take a basis of \(100 \mathrm{kg}\) film fed to the dryer for the requested calculations. (a) Estimate the feed ratio [liters dry air (STP)/kg dry film]. (b) Derive an expression for \(T_{\mathrm{al}}\) in terms of the film temperature change, \(\left(T_{\mathrm{f} 2}-35\right),\) and use it to answer Parts (c) and (d). (c) Calculate the film temperature change if the inlet air temperature is \(120^{\circ} \mathrm{C}\). (d) Calculate the required value of \(T_{\mathrm{al}}\) if the film temperature falls to \(34^{\circ} \mathrm{C},\) and the value if it rises to \(36^{\circ} \mathrm{C}.\) (e) If you solved Parts (c) and (d) correctly, you found that even though the air temperature is consistently higher than the film temperature in the dryer, so that heat is always transferred from the air to the film, the film temperature can drop from the inlet to the outlet. How is this possible?

Step by step solution

01

Calculate the mass

As 90% of the acetone evaporates, only 10% which is equivalent to 0.5 kg remains in the 100 kg film. The mass of the dry film can therefore be calculated as \(100 - 0.5 = 99.5\) kg .

02

Calculate the mass of evaporated acetone

The 90% of the 5 wt% acetone in 100 kg film becomes 4.5 kg.

03

Formulate heat balance for the film

The heat gained by the film is equal to the heat supplied by the air and the heat of vaporization of acetone. Mathematically, this can be written as: \[99.5 \times 1.33 \times (T_{f2}-35) = 4.5 \times \Delta H_v + 99.5 \times 1.33 \times (T_{f2}-35) \] where \( \Delta H_v \) is the heat of vaporization for acetone.

04

Derive the expression for \(T_{al}\)

Using the fact that the heat lost by the air is the heat gained by the film, we can state that:\[V_{air} \times 1.01 \times 10^3 \times C_{p_{air}} \times (T_{al}-49) = 99.5 \times 1.33 \times (T_{f2} - 35)\]Solving for \(T_{al}\) in terms of \(T_{f2}\), we get: \[T_{al} = \frac{99.5 \times 1.33 \times (T_{f2} - 35)}{V_{air} \times 1.01 \times 10^3 \times C_{p_{air}}} + 49\]

05

Calculate the film temperature change

For an inlet air temperature of \(120^{\circ} C\), we will plug into the above equation to solve for \(T_{f2}\) which will give us the change in temperature of the film.

06

Derive the value for \(T_{al}\) for different film temperatures

We will replace \(T_{f2}\) by the given \(34^{\circ} C\) or \(36^{\circ} C\) in the equation from step 4 and calculate the necessary \(T_{al}\)

07

Understand the heat transfer phenomenon

The film temperature can drop despite the air temperature being consistently higher because of the latent heat consumed in the evaporation of acetone which effectively cools the film.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Adiabatic Dryer

An adiabatic dryer is a device where materials are dried using a stream of air or gas, and no external heat supply is provided during the process. The drying involves the removal of moisture through evaporation, which is accompanied by a drop in temperature due to the latent heat of vaporization. As this process occurs, the air becomes increasingly humid and leaves the dryer at a higher temperature and humidity level than when it entered.
This process is significant in chemical process engineering because it allows for drying without an external heat source, which can be both energy-efficient and cost-effective. In the case of our exercise, the acetone in the cellulose acetate film evaporates, cooling the film as it does so, even though the air itself might be warmer than the film.

Heat Capacity

Heat capacity is a measure of the amount of heat energy required to raise the temperature of a substance by a given amount; this property is crucial for understanding energy changes in chemical processes.
In the context of the adiabatic drying exercise, the heat capacity of the film, air, and acetone play a significant role in calculating the energy balance. The specific heat capacities provided allow us to estimate the amount of heat necessary to bring about the observed temperature changes, whether it be the drying film or the air stream. The heat capacity of dry air, which we assumed in the calculation, typically depends on temperature and pressure but is often approximated for engineering calculations like this one.

Heat of Vaporization

The heat of vaporization is the amount of heat energy required to convert a certain amount of a substance from its liquid state to its vapor state at a constant temperature. It's also known as enthalpy of vaporization and is a form of latent heat.
In the drying process, heat of vaporization is a critical factor because it represents the energy needed to evaporate the solvent—in this case, acetone—from the film. Although it is usually temperature-dependent, in this textbook problem, it is assumed to be constant for the sake of simplification. This assumption greatly streamlines the calculations but is often a close approximation in practical scenarios.

Vapor-Liquid Equilibrium

Vapor-liquid equilibrium (VLE) is the state where a liquid and its vapor are in equilibrium with each other at a given temperature and pressure, meaning that the rate of evaporation equals the rate of condensation.
Understanding VLE helps us to predict how the solvent will behave within the adiabatic dryer and at what temperature and pressure it will evaporate, which is essential for optimizing the drying process. In industrial settings, VLE data are foundational for designing distillation columns, evaporators, and other separation process units.

Mass and Energy Balances

Mass and energy balances are the cornerstone of chemical process calculations. They ensure that all material and energy entering a process equal all material and energy leaving, accounting for changes within the process.
In the example problem, we used mass balances to determine the amount of acetone evaporating and the mass of the dry film. For energy balances, we calculated the energy change of the film due to temperature change and the energy absorbed during acetone evaporation.
Understanding these balances is fundamental for engineers to design, analyze, and optimize chemical processes, ensuring sustainability and cost-efficiency of operations.