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Chapter 17: Problem 108

A wood ceiling with thermal resistance \(R_{1}\) is covered with a layer of insulation with thermal resistance \(R_{2} .\) Prove that the effective thermal resistance of the combination is \(R=R_{1}+R_{2}\) .

Short Answer

Expert verified
The effective thermal resistance is \(R=R_1+R_2\).

Step by step solution

01

Understanding Thermal Resistance

Thermal resistance measures a material's ability to resist the flow of heat. When two layers with different thermal resistances are put together, the overall or effective thermal resistance can affect the total heat transfer across these layers.
02

Setup the System Together

Consider two layers: the first layer (the wood ceiling) with thermal resistance \(R_1\) and the second layer (the insulating material) with thermal resistance \(R_2\). These layers are stacked together, creating a composite wall system.
03

Applying the Series Resistance Formula

When thermal resistances are in series (occurring one after the other), their resistances add up. Thus, the effective thermal resistance for these two layers, connected in series, is given by the sum of their individual resistances: \(R = R_1 + R_2\).
04

Proof by Definition

By the definition of series resistances, the effective resistance \(R\) can be calculated by directly summing up the thermal resistances of each layer: \(R = R_1 + R_2\). This rule stems from the same principles used in electrical resistance calculations, following the path where heat in our system must sequentially pass through each of the two resistances.

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Key Concepts

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

Heat Transfer
Heat transfer is the process by which thermal energy moves from a higher temperature area to a lower temperature area. This process can occur through three main methods: conduction, convection, and radiation. In the context of our problem, we are primarily concerned with conduction, which involves heat moving through materials without the bulk movement of the substance itself.
Conduction in materials, especially in composite systems like a wood ceiling with insulation, essentially depends on the temperature gradient, cross-sectional area, material properties, and thickness of the material. In simple terms, heat will flow faster if the difference in temperature is bigger, or if the material is a good conductor.
A greater understanding of conduction allows us to predict and control heat transfer in various applications, such as heating systems, building insulation, or electronic devices.
Composite Materials
Composite materials are formed by combining two or more materials with different properties to create a new, distinct material with enhanced performance. In our exercise, we see an example of a composite ceiling system formed by wood and insulation. Each component contributes to the overall thermal behavior.
Wood, typically a poor conductor of heat, provides a basic layer of thermal resistance. The additional insulation can be made from various materials, each selected for its specific thermal resistance properties. When stacked together, wood and insulation form a composite material that benefits from the properties of both components.
This kind of systematic approach allows for greater control over properties like thermal resistance, offering an advantage in applications where energy efficiency, temperature control, or material strength is critical.
Series Resistance Formula
The series resistance formula is a simple yet powerful tool used to calculate the total resistance in systems where resistances are arranged in series. This concept is not limited to electrical circuits; it applies equally well to thermal resistance in composite systems.
For thermal resistance, when layers are arranged one after another, the total resistance to heat flow is simply the sum of each layer鈥檚 resistance. This is mathematically represented as:
  • For two resistances in series: \[R = R_1 + R_2\]
  • If more layers are added, continue adding their resistances: \[R = R_1 + R_2 + R_3 + \ldots\]
This additive property arises because heat must pass through each layer in sequence, just like electrons through a series of resistors. By applying this formula, it's easy to estimate the overall insulating effect of composite materials used in construction, packaging, and even clothing.

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Most popular questions from this chapter

Bicycling on a Warm Day. If the air temperature is the same as the temperature of your skin (about \(30^{\circ} \mathrm{C} ),\) your body cannot get rid of heat by transferring it to the air. In that case, it gets rid of the heat by evaporating water (sweat). During bicycling, a typical 70 -kg person's body produces energy at a rate of about 500 \(\mathrm{W}\) due to metabolism, 80\(\%\) of which is converted to heat. (a) How many kilograms of water must the person's body evaporate in an hour to get rid of this heat? The heat of vaporization of water at body temperature is \(2.42 \times 10^{6} \mathrm{J} / \mathrm{kg}\) . (b) The evaporated water must, of course, be replenished, or the person will dehydrate. How many 750 -mL bottles of water must the bicyclist drink per hour to replenish the lost water? (Recall that the mass of a liter of water is 1.0 \(\mathrm{kg.}\)

A technician measures the specific heat of an unidentified liquid by immersing an electrical resistor in it. Electrical energy is converted to heat transferred to the liquid for 120 s at a constant rate of 65.0 W. The mass of the liquid is \(0.780 \mathrm{kg},\) and its temperature increases from \(18.55^{\circ} \mathrm{C}\) to \(22.54^{\circ} \mathrm{C}\) . (a) Find the average specific heat of the liquid in this temperature range. Assume that negligible heat is transferred to the container that holds the liquid and that no heat is lost to the surroundings. (b) Suppose that in this experiment heat transfer from the liquid to the container or surroundings cannot be ignored. Is the result calculated in part (a) an overestimate or an underestimate of the average specific heat? Explain.

Like the Kelvin scale, the Rankine scale is an absolute temperature scale: Absolute zero is zero degrees Rankine \(\left(0^{\circ} \mathrm{R}\right)\) However, the units of this scale are the same size as those of the Fahrenheit scale rather than the Celsius scale. What is the numerical value of the triple-point temperature of water on the Rankine scale?

What is the rate of energy radiation per unit area of a blackbody at a temperature of \((\) a ) 273 \(\mathrm{K}\) and (b) 2730 \(\mathrm{K} ?\)

Why Do the Seasons Lag? In the northern hemisphere, June 21 (the summer solstice) is both the longest day of the year and the day on which the sun's rays strike the earth most vertically, hence delivering the greatest amount of heat to the surface. Yet the hottest summer weather usually occurs about a month or so later. Let us see why this is the case. Because of the large specific heat of water, the oceans are slower to warm up than the land (and also slower to cool off in winter). In addition to perusing pertinent information in the tables included in this book, it is useful to know that approximately two-thirds of the earth's surface is ocean composed of salt water having a specific heat of 3890 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) and that the oceans, on the average, are 4000 m deep. Typically, an average of 1050 \(\mathrm{W} / \mathrm{m}^{2}\) of solar energy falls on the earth's surface, and the oceans absorb essentially all of the light that strikes them. However, most of that light is absorbed in the upper 100 \(\mathrm{m}\) of the surface. Depths below that do not change temperature seasonally. Assume that the sunlight falls on the surface for only 12 hours per day and that the ocean retains all the heat it absorbs. What will be the rise in temperature of the upper 100 \(\mathrm{m}\) of the oceans during the month following the summer solstice? Does this seem to be large enough to be perceptible?
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