HEAT TRANSFER & THERMAL POWER PLANT, Mechanical Engg 3rd Semester Notes, ME3001

 

2. HEAT TRANSFER & THERMAL POWER PLANT

Heat transfer is the process by which thermal energy moves from one body or system to another. Understanding heat transfer is essential for designing systems such as thermal power plants, boilers, and heat exchangers.

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2.1 Modes of Heat Transfer

There are three primary modes of heat transfer:

1. Conduction: Heat transfer through a material without the movement of the material itself. It occurs when molecules collide, transferring energy from the hot region to the cold region. Conduction typically occurs in solids.

   Equation for conduction: Fourier's Law

   $$Q = -k A \frac{dT}{dx}$$

   Where:

   • $Q$ = heat transfer per unit time (W)
   • $k$ = thermal conductivity (W/m·K)
   • $A$ = cross-sectional area through which heat flows (m²)
   • $\frac{dT}{dx}$ = temperature gradient (K/m)

2. Convection: Heat transfer through the motion of a fluid (liquid or gas). When fluid particles move, they carry heat from one place to another. Convection is further divided into:

   • Forced convection: Fluid movement is caused by external means like fans or pumps.
   • Natural convection: Fluid movement occurs due to temperature-induced density differences.

3. Radiation: Heat transfer through electromagnetic waves without the need for a medium. All bodies emit radiation, and this process is governed by the Stefan-Boltzmann law:

   $$Q = \sigma A \epsilon (T^4 - T_0^4)$$

   Where:

   • $\sigma$ = Stefan-Boltzmann constant ($5.67 \times 10^{-8}$ W/m²·K⁴)
   • $\epsilon$ = emissivity (0 ≤ $\epsilon$ ≤ 1)
   • $T$ = temperature of the body (K)
   • $T_0$ = temperature of the surrounding (K)

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2.2 Conduction

Conduction is the process by which heat flows from the hot region to the cold region within a solid or between solids in direct contact. The rate of heat transfer is influenced by the material’s thermal conductivity, cross-sectional area, and temperature difference.

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2.2.1 Composite Walls and Cylinders

In practical applications, heat conduction often occurs through composite materials with different layers or through cylindrical objects.

Composite Walls:

When heat passes through a wall made of different materials, the total heat transfer rate depends on the resistance of each material. The thermal resistance of each material is the reciprocal of its thermal conductivity. The total heat transfer can be calculated as:

$$\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots$$

Where $R$ is the thermal resistance of each material, which is given by:

$$R = \frac{L}{kA}$$

Where:

• $L$ = thickness of the material (m)
• $k$ = thermal conductivity (W/m·K)
• $A$ = cross-sectional area (m²)

The rate of heat transfer $Q$ can be calculated using the temperature difference across the composite wall:

$$Q = \frac{T_1 - T_2}{R_{\text{total}}}$$

Composite Cylinders:

For a cylindrical shell, the heat transfer is more complex and involves the radial direction. The rate of heat transfer can be expressed as:

$$Q = \frac{2 \pi k (T_1 - T_2)}{\ln(r_2 / r_1)} L$$

Where:

• $r_1$ and $r_2$ are the inner and outer radii of the cylinder.
• $L$ is the length of the cylinder.
• $T_1$ and $T_2$ are the temperatures at the inner and outer surfaces.

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2.3 Simple Numerical Problems

Here are a couple of examples to illustrate how heat transfer problems are solved:

Example 1: Composite Wall Heat Transfer

Given:

• Wall consists of two layers of materials A and B.
• Material A: $k_A = 100 \, \text{W/m·K}$, Thickness $L_A = 0.2 \, \text{m}$
• Material B: $k_B = 200 \, \text{W/m·K}$, Thickness $L_B = 0.3 \, \text{m}$
• Temperature difference across the wall: $T_1 - T_2 = 60 \, \text{K}$

Find the rate of heat transfer through the composite wall.

Solution:

1. Calculate the thermal resistance of each material:

   $$R_A = \frac{L_A}{k_A A}, \quad R_B = \frac{L_B}{k_B A}$$

   Total resistance $R_{\text{total}} = R_A + R_B$

2. Calculate the heat transfer:

   $$Q = \frac{T_1 - T_2}{R_{\text{total}}}$$

Example 2: Heat Transfer in a Cylindrical Shell

Given:

• Inner radius $r_1 = 0.1 \, \text{m}$, Outer radius $r_2 = 0.2 \, \text{m}$
• Thermal conductivity $k = 50 \, \text{W/m·K}$
• Length $L = 5 \, \text{m}$
• Temperature difference $T_1 - T_2 = 100 \, \text{K}$

Find the rate of heat transfer.

Solution: Use the formula for heat transfer through a cylinder:

$$Q = \frac{2 \pi k (T_1 - T_2)}{\ln(r_2 / r_1)} L$$

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2.4 Thermal Power Plant Layout

A Thermal Power Plant uses heat energy to produce electricity by converting thermal energy into mechanical energy. The layout typically includes the following major components:

1. Boiler: Converts water into steam by heating it with the combustion of fuel.
2. Turbine: Steam expands in the turbine, causing it to spin and convert thermal energy into mechanical energy.
3. Generator: The turbine is connected to a generator that converts mechanical energy into electrical energy.
4. Condenser: Cools the steam after it has passed through the turbine, condensing it back into water.
5. Cooling Tower: Disposes of the waste heat from the condenser by cooling the water.
6. Feedwater Pump: Pumps water back into the boiler to start the cycle again.

Layout Diagram:

      +------------+     +-------------+     +-----------+
Fuel →|  Boiler   | →   |  Turbine    | →   | Generator |
      +------------+     +-------------+     +-----------+
             ↑                  ↓                  |
             |               +-------------+        |
             +--------------|  Condenser  | <-------+
                            +-------------+            
                                      ↓
                           +-------------------+
                           |  Cooling Tower    |
                           +-------------------+

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2.5 Fire Tube and Water Tube Boilers (Only Working Principle and Types)

Boilers are heat exchangers used to convert water into steam. They can be classified into two types based on the path of the hot gases and water.

Fire Tube Boilers:

• Working Principle: In fire tube boilers, the hot gases produced by combustion pass through tubes that are surrounded by water. The heat from the gases is transferred to the water, which turns into steam.

• Types:

  1. Scotch Marine Boiler: A common type, consisting of a large cylindrical shell and one or more fire tubes.
  2. Lancashire Boiler: Similar to the Scotch boiler, but with two large horizontal flues.

  Diagram:

  +----------------------+
  | Water                  |
  |                       |
  |  +-----+    +------>   |
  |  | Tube|    | Flue     |
  |  +-----+    | Gas      |
  |  +-----+    |          |
  |  | Tube|    |          |
  +----------------------+
  

Water Tube Boilers:

• Working Principle: In water tube boilers, water circulates inside the tubes, and hot gases from combustion pass around the tubes. Heat is transferred to the water, turning it into steam.

• Types:

  1. Bent Tube Boiler: The tubes are bent to form a serpentine shape, which allows for greater flexibility.
  2. Straight Tube Boiler: The tubes are arranged in a straight line.

  Diagram:

  +----------------------+
  | Fire Gases            |
  |                       |
  | +-------------------+ |
  | | Water Tubes       | |
  | +-------------------+ |
  |                       |
  +----------------------+