A carpenter builds an exterior house wall with a layer of wood 3.03.0 cm thick on the outside and a layer of Styrofoam insulation 2.22.2 cm thick on the inside wall surface. The wood has k=0.080 W/m⋅Kk=0.080\,W/m\cdot K , and the Styrofoam has k=0.027 W/m⋅Kk=0.027\,W/m\cdot K. The interior surface temperature is 19.019.0°C, and the exterior surface temperature is −10.0-10.0°C. What is the temperature at the plane where the wood meets the Styrofoam?

Ch 17: Temperature and Heat

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 17: Temperature and Heat

Problem 59a

Chapter 17, Problem 59a

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?

Verified step by step guidance

First, understand that this problem involves heat transfer through a composite wall made of two materials: wood and Styrofoam. We need to find the temperature at the interface between these two materials.

Use the concept of thermal resistance in series. The thermal resistance (R) for a material is given by the formula:

R = \frac{d}{k}

, where

d

is the thickness of the material and

k

is the thermal conductivity.

Calculate the thermal resistance for the wood layer:

R = \frac{0.03}{0.080}

Calculate the thermal resistance for the Styrofoam layer:

R = \frac{0.022}{0.027}

Apply the formula for temperature drop across a layer:

ΔT = R \cdot (T

_ext-T_int)R_total, where

T

_ext and

T

_int are the exterior and interior temperatures, respectively, and

R

_total is the sum of the thermal resistances. Calculate the temperature at the interface using this formula.

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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 Kelvin (W/m K). Materials with high thermal conductivity transfer heat more efficiently, while those with low thermal conductivity are better insulators. In this problem, wood and Styrofoam have different thermal conductivities, affecting the temperature gradient across the wall.

Heat Transfer Through Composite Walls

Heat transfer through composite walls involves calculating the temperature distribution across layers of different materials. Each layer has its own thermal resistance, which depends on its thickness and thermal conductivity. The temperature at the interface between two materials can be found by considering the heat flow continuity and the thermal resistances of each layer.

Steady-State Heat Transfer

Steady-state heat transfer assumes that temperatures and heat flow rates are constant over time. In this scenario, the temperature difference between the interior and exterior surfaces drives a constant heat flow through the wall. The temperature at any point within the wall can be determined by analyzing the balance of heat flow across the materials, ensuring that the heat entering one side equals the heat leaving the other.

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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 $-10.0$ °C. What is the rate of heat flow per square meter through this wall?

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