Question

Lumped system analysis of transient heat conduction situations is valid when the Biot number is (a) very small (b) approximately one (c) very large (d) any real number (e) cannot say unless the Fourier number is also known.

Short answer

Question: For lumped system analysis to be valid, the Biot number should be: Answer: (a) very small

Step-by-step solution

Understand the Biot Number

The Biot number (Bi) is a dimensionless quantity that relates the rate of heat conduction within an object to the rate of heat convection at the surface of the object. It is defined as the ratio of the thermal resistance inside the object to the thermal resistance at the surface of the object. Mathematically, the Biot number is given by: Bi = \frac{h L_{c}}{k} Where, h = convective heat transfer coefficient (W/m²K) L_c = characteristic length of the object (m) k = thermal conductivity of the object (W/mK)

Analyzing the Biot Number values

Lumped system analysis assumes that the temperature within the object is uniform, which means the effects of heat conduction within the object can be neglected. This is only possible when the effects of convection at the surface of the object dominate heat conduction within the object. When Bi << 1 (the Biot number is very small), heat conduction within the object is much faster than surface heat convection. In this case, any temperature differences within the object are rapidly equalized, supporting the assumption of a uniform temperature within the object. Hence, lumped system analysis is valid when the Biot number is very small. Now let's see the given options and find the correct one based on our understanding: (a) very small: This is the correct answer, as explained above. (b) approximately one: Lumped system analysis is not valid when the Biot number is approximately one, as this implies comparable rates of heat conduction within the object and surface heat convection. (c) very large: When the Biot number is very large, heat conduction within the object becomes dominant, contradicting the assumption of a uniform temperature in lumped system analysis. (d) any real number: Lumped system analysis is not valid for any real number, as explained in options (b) and (c). (e) cannot say unless the Fourier number is also known: The Fourier number is related to the rate of heat conduction, but the validity of lumped system analysis primarily depends on the Biot number. So, the correct answer is (a) very small.

Key concepts

Transient Heat Conduction

Transient heat conduction is a fascinating process that occurs when temperature changes with time within an object. This happens because heat flows from areas of higher temperature to areas of lower temperature until equilibrium is reached. During transient heat conduction, the rate and direction of heat transfer can vary depending on several factors:

• Material properties: Different materials conduct heat differently. Some materials, like metals, conduct heat rapidly, while others, like wood, do so slowly.
• Initial temperature distribution: How heat is initially spread throughout an object greatly affects transient conduction.
• External conditions: Changes in the surrounding environment, like temperature and airflow, can influence heat flow.

Transient heat conduction is key in applications like cooling electronics, cooking, and even climate control systems.

Lumped System Analysis

Lumped system analysis is a simplified method to solve transient heat conduction problems. It assumes that the temperature within a solid object is uniform, greatly simplifying calculations. This assumption is valid when the Biot number is very small (Bi << 1). This means that heat conduction within the object is much faster than the heat transfer across its surface. With this, the internal resistance to heat flow is negligible compared to the external resistance. This analysis is particularly useful in:

• Small objects: Like sensors or small electronics, where variations in internal temperature are minimal compared to their surface interactions.
• Rapid heat changes: Situations requiring quick responses, where detailed internal temperature profiles aren't necessary.

In practice, lumped system analysis simplifies complex heat transfer by reducing spatial details to more manageable calculations.

Thermal Conductivity

Thermal conductivity is a material's ability to conduct heat. It tells us how quickly or slowly heat will flow through a material. The equation that defines thermal conductivity is:\[ q = -k \frac{dT}{dx} \]where:

• \( q \) is the heat flow per unit time (W/m²).
• \( k \) is the thermal conductivity (W/mK).
• \( \frac{dT}{dx} \) is the temperature gradient across the material.

Materials with high thermal conductivity, like metals, efficiently transfer heat, making them ideal for applications that require quick dissipation of heat, such as heat sinks and radiators. Conversely, materials with low thermal conductivity, like wood or foam, are excellent insulators, helpful in minimizing heat loss in homes.

Heat Convection

Heat convection is the transfer of heat through the movement of fluid, which can be a liquid or gas. This process occurs naturally or can be forced, like using a fan to cool down a computer. Convection depends on:

• Fluid properties: Different fluids have different capacities to carry heat, influenced by factors like density and viscosity.
• Velocity: Fast-moving fluids enhance heat transfer, essential in cooling systems.
• Surface area and temperature difference: Larger areas and higher temperature differences increase heat transfer rates.

Overall, understanding heat convection helps improve designs of systems like air conditioning, car radiators, and even efficient cooking methods.