91Ó°ÊÓ

Following the hot vacuum forming of a paper-pulp mixture, the product, an egg carton, is transported on a conveyor for 18 s toward the entrance of a gas- fired oven where it is dried to a desired final water content. It is observed that very little water evaporates during the travel time. So, to increase the productivity of the line, it is proposed that a bank of infrared radiation heaters. which provide a uniform radiant flux of \(5000 \mathrm{~W} \mathrm{ma}^{2}\). be installed over the conveyor. The carton has an exposed area of \(0.0625 \mathrm{~m}^{2}\) and a mass of \(0.220 \mathrm{~kg}, 75 \%\) of which is water after the forming process. The chicf engincer of your plant will approve the purchase of the heaters if they can reduce the water content by \(10 \%\) of the total mass. Would you recemmend the purchase? Axsume the heat of vaporization of water is \(h_{e}=2400 \mathrm{k} / \mathrm{kg}\).

Step by step solution

01

Calculate Initial Water Content

First, determine the initial mass of water in the carton. Since 75% of the mass is water, the initial water content is: \[ m_{water, initial} = 0.75 \times 0.220 \text{ kg} = 0.165 \text{ kg} \]

02

Determine Desired Reduction in Water Content

Calculate the reduction in water content needed to achieve a 10% total mass reduction. This is: \[ \text{Reduction} = 0.10 \times 0.220 \text{ kg} = 0.022 \text{ kg} \] This means the water content should be reduced by 0.022 kg.

03

Calculate Energy Required for Evaporation

The energy needed to evaporate 0.022 kg of water is calculated with the heat of vaporization:\[ Q_{required} = 0.022 \text{ kg} \times 2400 \text{ kJ/kg} = 52.8 \text{ kJ} \]

04

Determine Energy Supplied by Heaters

Calculate the energy supplied by the infrared heaters over the 18-second period. Convert the given flux into total energy:\[ Q_{supplied} = \text{radiant flux} \times \text{area} \times \text{time} = 5000 \text{ W/m}^2 \times 0.0625 \text{ m}^2 \times 18 \text{ s} \]Convert energy from watts to kilojoules: \[ Q_{supplied} = 5000 \times 0.0625 \times 18 \times 0.001 = 5.625 \text{ kJ} \]

05

Compare Energy Required and Supplied

Compare the energy required (52.8 kJ) to evaporate the water with the energy supplied by the heaters (5.625 kJ). Since 52.8 kJ is substantially more than 5.625 kJ, the heaters do not provide enough energy to achieve the desired reduction.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Evaporation

Evaporation is an essential process in drying, especially in manufacturing. It involves the phase transition of water from liquid to vapor. When drying objects like egg cartons, evaporation is crucial to reduce moisture content. In this problem, the goal was to reduce water by 10% of the total mass.
To calculate evaporation needs, we must determine how much water should be removed. Here, this is 0.022 kg. Knowing how much water needs to evaporate helps in determining energy requirements, but it’s just the start.
Remember, evaporation rates can be modified by factors such as temperature, surface area, and air movement. In this scenario, infrared heaters aim to speed up evaporation along the conveyor by providing additional energy.

Heat Transfer

Heat transfer is the movement of thermal energy from a hotter object to a cooler one. In drying processes like this, heat transfer is vital for increasing the temperature of the carton, facilitating water evaporation.
Infrared heaters were proposed to supplement heat transfer since regular conveyor belts didn't suffice. These heaters emit thermal radiation directly absorbed by the carton, increasing its core temperature.
Heat transfer can occur in different modes:

• Conduction: Transfer through solid materials.
• Convection: Heat moved by fluid motion.
• Radiation: Energy transfer by electromagnetic waves.

Infrared heaters utilize radiation, making them suitable for drying applications where direct heating is required without physical contact.

Radiant Flux

Radiant flux is the measure of total power of electromagnetic radiation, essentially how much energy the heaters deliver per unit area. It's pivotal in deciding if the heaters supply enough energy.
For the given scenario, each heater provides a radiant flux of 5000 W/m². To find total energy provided over time, multiply radiant flux by area and time duration.
Infrared heaters aim to directly radiate energy onto the carton, absorbing it to increase its temperature and reduce water content.
Understanding radiant flux is fundamental because insufficient flux results in inadequate heating, as demonstrated by the 5.625 kJ delivered here, which wasn't enough.

Energy Calculation

Calculating energy needs is core to evaluating heater efficiency. Start with how much energy is required to achieve the intended drying, using the evaporation process and the water's heat of vaporization.
For instance, to evaporate 0.022 kg of water, multiply the mass by the heat of vaporization (2400 kJ/kg), concluding an energy requirement of 52.8 kJ.
Then, calculate energy supplied:

• Use heater’s radiant flux, area, and available time.
• Convert results to the same units for comparison.

In this example, heaters provide only 5.625 kJ, a gap from required 52.8 kJ suggests these heaters are inadequate to meet drying needs within time constraints. Energy calculation is a decisive step in approving processes like purchasing new heaters.