Question

The human body transports heat from the interior tissues, at the temperature \(37.0^{\circ} \mathrm{C},\) to the skin surface, at the temperature \(27.0^{\circ} \mathrm{C},\) at a rate of \(100 .\) W. If the skin area is \(1.50 \mathrm{~m}^{2}\) and its thickness is \(3.00 \mathrm{~mm}\), what is the effective thermal conductivity, \(k,\) of skin?

Short answer

Answer: The effective thermal conductivity of skin is 0.02 Wm/°C.

Step-by-step solution

Write down Fourier's law of heat conduction

Fourier's law of heat conduction states that the rate of heat transfer (Q) through a material is proportional to the temperature difference (∆T) and the surface area (A) and inversely proportional to the material's thickness (d). Mathematically, this can be expressed as: \(Q = k \frac{A\Delta T}{d}\), where \(k\) is the thermal conductivity of the material.

List the given values

We are given the rate of heat transport \(Q = 100 \text{ W}\), temperature difference \(\Delta T = 37.0^{\circ} \mathrm{C} - 27.0^{\circ} \mathrm{C} = 10.0^{\circ}\mathrm{C}\), skin area \(A = 1.50\text{ m}^2\), and skin thickness \(d = 3.00\text{ mm}\). Let's convert skin thickness to meters: \(d = 3.00\text{ mm} \times \frac{1\text{ m}}{1000\text{ mm}} = 0.003\text{ m}\).

Rearrange Fourier's law to solve for thermal conductivity, \(k\)

We need to find the thermal conductivity \(k\). To do this, we can rearrange Fourier's law as follows: \(k = \frac{Qd}{A\Delta T}\)

Substitute the given values and calculate \(k\)

Now, we can plug in the given values into the equation for \(k\) and solve: \(k = \frac{(100\text{ W})(0.003\text{ m})}{(1.50\text{ m}^2)(10.0^{\circ}\mathrm{C})} = \frac{0.3\text{ Wm}}{15\text{ m}^{2}\cdot^{\circ}\mathrm{C}} = 0.02\frac{\text{Wm}}{^{\circ}\mathrm{C}}\) Hence, the effective thermal conductivity of skin is \(k = 0.02 \frac{\text{Wm}}{^{\circ}\mathrm{C}}\).

Key concepts

Fourier's law of heat conduction

Imagine putting your hand near a hot stove. You feel the warmth without touching it, right? Heat is traveling through the air to your skin. But what if you touched the stove? That's where Fourier's law of heat conduction comes into play. Heat conduction is the transfer of heat within a solid or between touching solids. Much like a group of people passing along information, atoms or molecules transfer heat energy to their neighbors.

Fourier's law gives us a formula to predict how quickly this heat will travel. It says the rate of heat transfer through a material is directly related to the temperature difference and the area through which the heat is being transferred, and inversely related to the material's thickness.

In simpler words, more heat will flow if there's a bigger temperature difference, if there's more area for the heat to move through, and if the material is thinner. This law can be used to calculate material properties, like in our exercise with the human skin, or in designing better insulation for buildings.

Heat transfer

When you wrap your hands around a warm mug of tea, the cozy feeling that spreads into your palms is all thanks to heat transfer. This is the movement of thermal energy from one thing to another thing of different temperature. There are three main types of heat transfer: conduction (heat moving through a solid), convection (heat moving through liquids or gases), and radiation (heat traveling as electromagnetic waves).

Our bodies, engines in cars, ovens, and even the Earth's climate system all rely on heat transfer. Understanding this concept allows scientists and engineers to create more efficient cooling systems, like air conditioners, and better safety measures, like firefighters' gear. It's also useful for everyday tasks, like cooking the perfect egg or keeping your home warm without a huge energy bill.

Rate of heat transport

Let's say you're sitting in a chilly room and start a space heater. The room gradually warms up, right? That's because there's a certain rate of heat transport - think of it as the speed at which heat energy is moving.

The rate at which heat transfers can be measured in watts (W), just like the power of a lightbulb. In our skin exercise, we mentioned the body transported heat at a rate of 100 W. Here's why that matters: if the rate is high, the room warms up quickly. If it's slow, you might need to wear a sweater for a bit longer. This concept is crucial in designing thermal systems. For instance, a cooling system needs to transport heat away from the engine fast enough to prevent overheating. By calculating this rate, we ensure devices operate safely and efficiently.