Question

What is a conduction shape factor? How is it related to the thermal resistance?

Short answer

Answer: The conduction shape factor (S) is a dimensionless quantity used in heat conduction problems to characterize the geometry of a system and compare its efficiency of heat conduction to a simple, reference geometry. It is related to thermal resistance (R) through the equation R = 1/(kS), where k is the thermal conductivity of the material. The lower the thermal resistance, the more quickly heat flows through the material.

Step-by-step solution

Definition of Conduction Shape Factor

The conduction shape factor, commonly denoted as S, is a dimensionless quantity used in heat conduction problems, specifically in systems where a three-dimensional heat transfer is reduced to a one-dimensional problem. It's a way to compare the actual complex geometry of a system to a simple, reference geometry and gives an indication of the efficiency of heat conduction for a given system.

Relation to Thermal Resistance

The thermal resistance (R) for conductive heat transfer in a system is the inverse of the conduction shape factor multiplied by the conductivity (k) of the material. Mathematically, this is represented as: R = \frac{1}{kS} Where: - R is the thermal resistance (K/W or °C/W) - k is the thermal conductivity of the material (W/mK) - S is the conduction shape factor The lower the thermal resistance, the more quickly heat flows through the material, which is generally desired to improve heat transfer efficiency.

Example Calculation

Let's consider a simple example to understand these concepts better. Suppose we have a wall made of brick with uniform thickness, and we want to compute its conduction shape factor and thermal resistance. Given: - Brick Thermal Conductivity k = 0.72 W/mK - Dimensions of the wall: 10 m x 10 m (height x width) - Thickness of the wall: 0.2 m Step 1: Compute the area (A) of the wall: A = height x width = 10 m x 10 m = 100 m^2 Step 2: Compute the conduction shape factor (S): For a simple plane wall of uniform thickness, the conduction shape factor is calculated using the equation: S = \frac{A}{L} Where L is the thickness of the wall. S = \frac{100 m^2}{0.2 m} = 500 Step 3: Compute the thermal resistance (R): Now, using the previously mentioned formula (R = \frac{1}{kS}), we calculate the thermal resistance: R = \frac{1}{(0.72 W/mK)(500)} = \frac{1}{360} K/W ≈ 0.00278 K/W Therefore, the conduction shape factor for the brick wall is 500 and the thermal resistance is approximately 0.00278 K/W. This information can be used to analyze the heat transfer properties of the wall and to design energy-efficient building systems.