Question

The \(R\) factor for housing insulation gives the thermal resistance in units of \(\mathrm{ft}^{2}{ }^{\circ} \mathrm{F}\) h/BTU. A good wall for harsh climates, corresponding to about 10.0 in of fiberglass, has \(R=40.0 \mathrm{ft}^{2}{ }^{\circ} \mathrm{F} \mathrm{h} / \mathrm{BTU}\) a) Determine the thermal resistance in SI units. b) Find the heat flow per square meter through a wall that has insulation with an \(R\) factor of \(40.0,\) when the outside temperature is \(-22.0^{\circ} \mathrm{C}\) and the inside temperature is \(23.0^{\circ} \mathrm{C}\)

Short answer

Answer: The heat flow per square meter through the wall is approximately 6.39 W/m².

Step-by-step solution

Convert R factor to SI units

To convert \(R\) from \(\mathrm{ft}^{2}{ }^{\circ} \mathrm{F}\) h/BTU to SI units (m² K/W), we need to use the following conversion factors: 1 ft = 0.3048 m 1 °F = 5/9 K 1 h = 3600 s 1 BTU = 1055.06 J So, the conversion factor for \(\mathrm{ft}^{2}{ }^{\circ} \mathrm{F}\) h/BTU to m² K/W is: \(\frac{0.3048^2 \cdot 5/9 \cdot 3600}{1055.06} = 0.1761\) Now, multiply this conversion factor by the given \(R\) value (40.0 ft² °F h/BTU): \(R_{SI} = 40 \cdot 0.1761 = 7.044\) m² K/W

Calculate the heat flow per square meter

To calculate the heat flow per square meter, we use the following formula: \(Q = \frac{\Delta T}{R_{SI}}\) Where \(Q\) is the heat flow (in W/m²), \(\Delta T\) is the temperature difference, and \(R_{SI}\) is the \(R\) factor in SI units. The temperature difference is given by the difference between the inside and outside temperatures: \(\Delta T = 23.0^{\circ}\mathrm{C} - (-22.0^{\circ}\mathrm{C}) = 45.0\) K Now, we can calculate the heat flow: \(Q = \frac{45.0}{7.044} = 6.389\) W/m² So, the heat flow per square meter through the wall is approximately \(6.39\) W/m².

Key concepts

R Factor Insulation

Understanding the R factor in insulation is crucial for maintaining energy efficiency in buildings. The R factor represents thermal resistance, which is the measure of how well a material resists heat flow. The higher the R value, the better the insulating capability of the material. Therefore, a wall with an R factor of 40, as mentioned in the textbook exercise, is considered to offer excellent insulation for harsh climates.

In simple terms, thermal resistance works as a barrier that slows down the transfer of heat from a warmer area to a cooler one. Insulation materials with higher R values will reduce the amount of heat escaping during winter and entering during summer, thus maintaining a comfortable temperature indoors and decreasing the energy required for heating and cooling systems. It's important for students to recognize that insulation is not just about thickness; the type of material and its density also affect the R value.

SI Units Conversion

In physics, consistency in units is crucial for accurate calculations and comparisons. The International System of Units (SI) is the standard set of units used globally. In our exercise, the R factor native units are in \(\mathrm{ft}^{2}{ }^{\circ} \mathrm{F}\) h/BTU. However, to align with SI units, we convert them to m² K/W. The conversion process involves knowledge of basic conversion factors and their appropriate usage.

To help students contextualize the conversion, here are the steps broken down: first, translate the length from feet to meters, then convert Fahrenheit degrees to Kelvins, and finally, switch from BTUs (British Thermal Units) per hour to joules per second, which is watts. By understanding how each unit relates to the SI units, students can tackle a variety of problems where a conversion is necessary, not limited to thermal resistance.

Heat Flow Calculation

Heat flow calculation is a fundamental concept that helps us understand how heat energy transfers through materials. The formula used in the exercise, \(Q = \frac{\Delta T}{R_{SI}}\), illustrates the inverse relationship between heat flow and thermal resistance; as thermal resistance increases, the heat flow decreases. This is why materials with a high R value are preferred for insulation purposes.

The temperature difference (\(\Delta T\)) in the equation is the driving force for heat flow. It is the variance between the temperature inside and outside, in this case, represented in Celsius. When students calculate heat flow, the result expresses the amount of heat energy that passes through a specific area of material (per square meter) in a given time, hence the unit watts per square meter (W/m²). Understanding the dynamics of heat flow helps in designing more energy-efficient homes and can also be applied to various engineering problems regarding thermal management.