Problem 57 - Master Resource Book in JEE Main Physics

Question

In an industrial process \(10 \mathrm{~kg}\) of water per hour is to be heated from \(20^{\circ} \mathrm{C}\) to \(80^{\circ} \mathrm{C}\). To do this steam at \(150^{\circ} \mathrm{C}\) is passed from a boiler into a copper coil immersed in water. The steam condenses in the coil and is returned to the boiler as water at \(90^{\circ} \mathrm{C}\). How many kg of steam is required per hour? (Specific heat of steam \(=1\) calorie per \(\mathrm{g}^{\circ} \mathrm{C}\), Latent heat of vaporisation \(=540 \mathrm{cal} / \mathrm{g}\) ) (a) \(1 \mathrm{~g}\) (b) \(1 \mathrm{~kg}\) (c) \(10 \mathrm{~g}\) (d) \(10 \mathrm{~kg}\)

Step-by-Step Solution

Answer

The steam required is 1 kg per hour.

Step 1: Calculate Heat Required to Heat Water

The water is heated from \(20^{\circ} \mathrm{C}\) to \(80^{\circ} \mathrm{C}\). The specific heat of water is \(1 \mathrm{cal/g}^{\circ} \mathrm{C}\), and the mass of water is \(10\,\mathrm{kg}\), which is \(10000\,\mathrm{g}\). \(Q = m \cdot c \cdot \Delta T\), where \(m\) is the mass, \(c\) is the specific heat, and \(\Delta T\) is the temperature change. Thus, \(Q = 10000 \times 1 \times (80 - 20) = 600,000 \mathrm{cal}\).

Step 2: Determine Heat Released by Steam

The steam first loses heat by condensation. The steam changes from vapor to liquid at \(150^{\circ} \mathrm{C}\). The heat released during this phase is \(m \times 540\), where \(m\) is the mass of the steam in grams. Then, the condensed steam cools from \(150^{\circ} \mathrm{C}\) to \(90^{\circ} \mathrm{C}\), releasing additional heat: \(m \times 1 \times (150 - 90) = m \times 60\). Total heat released is \(m \times (540 + 60)\).

Step 3: Equate Heat Required and Heat Released

Set the heat required by the water equal to the total heat released by the steam: \(600,000 = m \times (540 + 60)\). Simplify the equation: \(600,000 = m \times 600\). Therefore, \(m = \frac{600,000}{600} = 1000\,\mathrm{g}\).

Step 4: Convert Mass of Steam to Kilograms

The mass \(m\) of steam required is initially calculated in grams. Since \(1\,\mathrm{kg} = 1000\,\mathrm{g}\), \(m = 1000\,\mathrm{g} = 1\,\mathrm{kg}\).

Key Concepts

Heat Transfer

Heat transfer is the process of thermal energy moving from one place to another due to a temperature difference. This is a crucial concept in thermodynamics and is vital to understand, especially in industrial and engineering applications.
When heat is added to a substance, the temperature increases, and when heat is extracted, the temperature decreases. In our example of heating water in an industrial process, heat is transferred from steam to water.
The heat will move from the hotter steam to the cooler water until thermal equilibrium is reached. It's the temperature difference between the steam and the water that drives this heat flow.

• Heat flows naturally from a higher temperature to a lower temperature.
• Conduction, convection, and radiation are the three main methods of heat transfer.
• In this case, the process mostly involves conduction, with the steam condensing in the copper coil, transferring heat to the water.

Phase Change

A phase change refers to the process where a substance changes from one state of matter to another. Each phase change involves an exchange of energy. In the context of an industrial heating process, understanding phase changes is crucial.
In this scenario, steam condenses inside the copper coil. This phase change from gas (steam) to liquid (water) releases energy. This is known as latent heat, the energy released or absorbed during a phase change without changing temperature.

• Condensation is a phase change where a gas turns into a liquid.
• Latent heat during this process can transfer significant amounts of energy, facilitating the heating of other substances.
• The energy released during the phase change is critical in heating water efficiently in an industrial setup.

Specific Heat

Specific heat is a property that defines how much energy is required to raise the temperature of a substance by a specific amount. It's an essential concept in understanding energy transfer in thermodynamics.
During the heating of water in our example, we use the specific heat of water to calculate the total energy required. It's denoted by the symbol 'c' and generally measured in calories or Joules per gram per °C.

• The higher the specific heat, the more energy is needed to change the temperature.
• Water has a relatively high specific heat, meaning it takes more energy to increase its temperature compared to many other substances.
• In the exercise, specific heat helps calculate the amount of steam required to heat a specific amount of water over a temperature range.

Industrial Heating Process

Industrial heating processes involve the application of heat to materials or systems to facilitate change, such as melting, hardening, or chemical reactions. Understanding these processes is vital for efficiency and safety in industrial operations.
In our example, industrial heating is applied to convert thermal energy from steam into the heat needed to raise the temperature of water. This is a common setup in industries for processes like pasteurization, sterilization, or in heating systems.

• Often involves large-scale systems requiring careful calculation of energy input and transfer.
• Efficiency of heat transfer and phase change directly affects energy consumption and cost.
• Materials, like the copper coil used here, are chosen for their conductive properties, aiding effective heat transfer.