A carpenter builds an exterior house wall with a layer of wood 3.03.03.0 cm thick on the outside and a layer of Styrofoam insulation 2.22.22.2 cm thick on the inside wall surface. The wood has k=0.080 W/m⋅Kk=0.080\,W/m\cdot Kk=0.080W/m⋅K , and the Styrofoam has k=0.027 W/m⋅Kk=0.027\,W/m\cdot Kk=0.027W/m⋅K. The interior surface temperature is 19.019.019.0°C, and the exterior surface temperature is −10.0-10.0−10.0°C. What is the rate of heat flow per square meter through this wall?

Ch 17: Temperature and Heat

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 17: Temperature and Heat

Problem 57b

Chapter 17, Problem 57b

A carpenter builds an exterior house wall with a layer of wood $3.0$ cm thick on the outside and a layer of Styrofoam insulation $2.2$ cm thick on the inside wall surface. The wood has $k=0.080\,W/m$\cdot$ K$ , and the Styrofoam has $k=0.027\,W/m$\cdot$ K$ . The interior surface temperature is $19.0$ °C, and the exterior surface temperature is $-10.0$ °C. What is the rate of heat flow per square meter through this wall?

Verified step by step guidance

First, understand that the problem involves calculating the rate of heat flow through a composite wall made of two materials: wood and Styrofoam. This requires using the concept of thermal resistance and the formula for heat transfer through a layered wall.

The formula for the rate of heat flow (Q) through a wall is given by:

\frac{ΔT}{R_{total}}

, where

ΔT

is the temperature difference across the wall, and

R_{total}

is the total thermal resistance of the wall.

Calculate the temperature difference

ΔT

between the interior and exterior surfaces:

ΔT = 19.0 - - 10.0 = 29.0 °C

Calculate the thermal resistance for each layer using the formula:

R_{i} = \frac{d_{i}}{k_{i}}

, where

d_{i}

is the thickness of the layer and

k_{i}

is the thermal conductivity. Calculate

R_{wood}

and

R_{styrofoam}

Sum the thermal resistances to find the total resistance:

R_{total} = R_{wood} + R_{styrofoam}

. Finally, use the formula for heat flow to find the rate of heat flow per square meter:

\frac{ΔT}{R_{total}}

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Thermal Conductivity

Thermal conductivity (k) is a material property that indicates how well a material conducts heat. It is measured in watts per meter per degree Kelvin (W/m K). Materials with high thermal conductivity transfer heat quickly, while those with low thermal conductivity are better insulators. In this problem, wood and Styrofoam have different thermal conductivities, affecting the rate of heat flow through the wall.

Heat Transfer Through Composite Walls

Heat transfer through composite walls involves calculating the rate of heat flow through multiple layers of different materials. Each layer has its own thickness and thermal conductivity, which together determine the overall resistance to heat flow. The heat flow rate can be calculated using the formula for thermal resistance, considering the temperature difference across the wall and the properties of each layer.

Temperature Gradient

The temperature gradient is the change in temperature across a material, which drives heat flow. In this scenario, the temperature difference between the interior (19.0°C) and exterior (-10.0°C) surfaces creates a gradient that causes heat to flow from the warmer interior to the cooler exterior. Understanding this gradient is crucial for calculating the rate of heat transfer through the wall.

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Textbook Question

A carpenter builds an exterior house wall with a layer of wood $3.0$ cm thick on the outside and a layer of Styrofoam insulation $2.2$ cm thick on the inside wall surface. The wood has $k = 0.080 W / m \cdot K$ , and the Styrofoam has $k = 0.027 W / m \cdot K$ . The interior surface temperature is $19.0$ °C, and the exterior surface temperature is $negative 10.0$ °C. What is the temperature at the plane where the wood meets the Styrofoam?

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