# Thermal Conductivity – Resistance: Surface Area

## Abstract

As described by both part on and part two, Thermal Conductivity is derived from Thermal Resistance under steady state conditions. The final variable found in Thermal resistance which has yet to be investigated is Surface area. In this experiment, a simple set-up will be used to investigate the effect of surface area variation has on temperature difference between a given length of material. This concept plays an important role in a variety of industries including heat management and insulation.

## Introduction

The goal of this experiment is to demonstrate the effect which varying surface area plays on the thermal resistance of a given material component. This will ultimately provide insight into the final variable affecting thermal resistance and heat flow.

## Background Information and Equations

### Background Information

Laboratory testing of thermal conductivity involves first calculating thermal resistance, however; under those conditions, length and surface area are both known. Should the thermal conductivity be known, it is in fact possible to determine the determine both the length of surface area should one more variable be know.

Heat Flow Equation Q = ΔT / RΘ

• Q = Heat flow in Watts
• ΔT = Temperature difference in Degrees Celsius
• RΘ = Thermal Resistance (l / k ⋅ A)
• l = Length of a material in Meters
• k = Thermal conductivity constant in W/m-K
• A = Surface area in meters squared

This experiment will vary the bolded constant via different samples.

The design of this experiment is to observe the differences in temperature between components with varying length thus surface area, temperature difference and thermal conductivity must remain the same. It may be completed with non-metal samples, however; once again one may run the risk of a lengthy test time. In light of this metals are recommended and a table of various suitable metals can be found below.

Material Type Thermal Conductivity Price Location of Purchase
d = 2’’, A = 1’’ d = 2’’, A = 1.5’’ d = 2’’, A = 2’’
Copper 397.48 8.54$16.80$ 29.42$SpeedyMetals Brass 117.15 5.16$ 12.36$21.24$ SpeedyMetals
Stainless Steel, 316 13.531 4.84$11.82$ 18.92$SpeedyMetals Cast Iron 71.128 1.46$ 2.62$3.9$ SpeedyMetals

Above is the dimensions of the cylinder with A being the width and d being the height.

## Materials

Material Price Location of Purchase
Hot Plate 20$Amazon Three metal cylinders (Copper) 8.54 + 16.80 + 29.42 = 54.76$ SpeedyMetals
Thermometer (Infrared or Simple) 4-20$Amazon Ice Total 78.76$-94.76\$
• Heat up the hot plate to a stable temperature (steady state)
• Wrap the first cylinder in some insulating material (cloth, felt, cotton…)
• Place the wrapped cylinder with ice on top onto the hot plate (optional)
• Allow the material to heat up for 5-10 minutes or until ice is completely melted
• Record the degree to which the ice melted (photo) or the time passed until the ice completely melted OR Record the temperature of the top of the cylinder
• Carefully remove the cylinder (ice and insulation included) and remove any water present on the hot plate
• Repeat steps 1-6 for each cylinder to be tested

## Observation

Upon completing this experiment, one should obtain results similar to the results seen in Part One of this series. As the surface area increases, the temperature difference between the top and the bottom of the cylinder should also increase. Another way one can observe the difference in heat transfer rates without using a thermometer involves the ice. The degree to which the ice on top of the cylinder melted provides indication to the amount of heat transferred. More melted ice means more heat transferred, less melted ice means less heat transferred.

## Comparison

With data obtained several plots and tables can be made to compare the rates of heat transfer (time required to reach a certain temperature, temperature reached after a certain period of time…). These plots will provide an idea as to the changes in heat transfer rates across varying surface areas. Should more data be required to obtain a better understanding, testing more cylinders with varying surface areas can be done. However, there exists free thermal simulation software which can perform these further tests should one not want to conduct further laborious tests. Below is a video demonstrating the rate of heat transfer through varying surface areas using said software. A link to the download of this software can be found on the Thermtest Resource page or on the ‘For further information’ section of this experiment.