91Ó°ÊÓ

(a) Calculate the \(R\) value for a wall consisting of the following layers: concrete block \((R=1.93)\) \(1.0\) inch of insulating board \((R=4.3)\), and \(0.50\) inch of drywall \((R=0.45)\), all in U.S. Customary Units. ( \(b\) ) If the wall has an area of \(15 \mathrm{~m}^{2}\), find the heat flow per hour through it when the temperature just outside is \(20^{\circ} \mathrm{C}\) lower than inside. (a) \(R=R_{1}+R_{2}+\ldots+R_{N}=1.93+4.3+0.45=6.7\) in U.S. Customary Units. Using the fact that 1 U.S. Customary Unit of \(R=0.176 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\), we get \(R=1.18 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\) (b) \(\Delta Q=\frac{A \Delta T}{R}(\Delta t)=\frac{\left(15 \mathrm{~m}^{2}\right)\left(20^{\circ} \mathrm{C}\right)}{1.18 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}}(3600 \mathrm{~s})=0.915 \mathrm{MJ}=2.2 \times 10^{2} \mathrm{kcal}\)

Step by step solution

01

Calculate Total R-Value for the Wall

The total thermal resistance, or R-value, of a wall is the sum of the R-values of each layer. Add the R-values given for each material: \( R = R_{concrete} + R_{insulating\;board} + R_{drywall} = 1.93 + 4.3 + 0.45 = 6.7 \). This is in U.S. Customary Units.

02

Convert R-Value to SI Units

To convert the R-value from U.S. Customary Units to SI units, we use the conversion factor: 1 U.S. Customary Unit of \( R = 0.176 \, \mathrm{m^2 \cdot K / W} \). Thus, \( R_{SI} = 6.7 \times 0.176 = 1.18 \, \mathrm{m^2 \cdot K / W} \).

03

Use the Heat Flow Formula

The formula for heat flow through an object is \( \Delta Q = \frac{A \Delta T}{R} \Delta t \), where \( A \) is the area, \( \Delta T \) is the temperature difference, and \( R \) is the thermal resistance.

04

Substitute Values into the Formula

Use the information given: area \( A = 15 \, \mathrm{m^2} \), temperature difference \( \Delta T = 20^{\circ} \mathrm{C} \), \( R = 1.18 \, \mathrm{m^2 \cdot K / W} \), and \( \Delta t = 3600 \, \mathrm{s} \). Substitute these into the heat flow formula: \[ \Delta Q = \frac{15 \times 20}{1.18} \times 3600 \].

05

Calculate the Heat Flow

Perform the calculations: \[ \Delta Q = \frac{300}{1.18} \times 3600 = 915000 \, \mathrm{J} \] or \( 0.915 \, \mathrm{MJ} \). Convert this to kcal using the conversion 1 MJ = 239 kcal, resulting in \( \approx 2.2 \times 10^2 \mathrm{kcal} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat Flow Calculations

Heat flow calculations are essential in understanding how energy moves through materials. It helps us assess how well a building or object retains or loses heat. The primary equation to determine heat flow, or \(\Delta Q\), is \(\Delta Q = \frac{A \Delta T}{R} \Delta t\), where:

• \(A\) is the surface area through which heat is transferred.
• \(\Delta T\) is the temperature difference between the two sides of the material.
• \(R\) is the thermal resistance or R-value, measuring how well the material resists heat flow.
• \(\Delta t\) is the time duration over which heat transfer is calculated.

This formula helps engineers and architects design buildings to minimize energy consumption and maintain optimal indoor temperatures. By substituting the known values such as area, temperature difference, and time into the formula, you can calculate the amount of heat flow.

R-Value Conversion

The R-value indicates how well a material resists heat flow, and is given in either U.S. Customary Units or SI Units. To convert between these units, a conversion factor is applied. The U.S. Customary R-value can be converted to SI units (\(\mathrm{m^2 \cdot K / W}\)) using the factor of 0.176.

For example, a wall with an R-value of 6.7 in U.S. Customary Units converts to:

• \(R_{SI} = 6.7 \times 0.176 = 1.18 \, \mathrm{m^2 \cdot K / W}\)

This conversion is critical when comparing material insulation effectiveness globally, as the SI metric system is widely used in scientific contexts.

Thermal Conductivity

Thermal conductivity is a measure that indicates a material's ability to conduct heat. Unlike thermal resistance or R-value, which tells us how resistant a material is to heat flow, thermal conductivity tells us the opposite. A higher thermal conductivity means heat passes through a material more easily.

• Materials with low thermal conductivity make great insulators, such as fiberglass or insulating boards.
• Conversely, metals, which have high thermal conductivity, are not ideal for insulation.

Understanding thermal conductivity helps in choosing materials for thermal management, whether it's keeping a house warm, or electronic devices cool. It plays a vital role in any design where thermal control is necessary.

Physics Education

Physics education plays a crucial role in understanding the fundamental principles behind these thermal concepts. It emphasizes how various aspects of physics, such as heat transfer and thermodynamics, apply to real-world problems. This foundational knowledge is particularly useful for students interested in engineering or environmental sciences.

Educational topics like heat flow, R-value, and thermal conductivity are not only relevant for energy efficiency in homes but are also essential in industrial applications. By exploring these concepts, students can appreciate how physics contributes to technological advances and solves daily life challenges.
An understanding of physics is empowering, allowing students to analyze problems from multiple angles and develop innovative solutions.