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Chapter 18: Problem 61

The human body transports heat from the interior tissues, at temperature \(37.0^{\circ} \mathrm{C},\) to the skin surface, at temperature \(27.0^{\circ} \mathrm{C},\) at a rate of \(100 . \mathrm{W}\). If the skin area is \(1.5 \mathrm{~m}^{2}\) and its thickness is \(3.0 \mathrm{~mm}\), what is the effective thermal conductivity, \(\kappa,\) of skin?

Short Answer

Expert verified
Answer: The effective thermal conductivity of the skin is 0.02 W/m·K.

Step by step solution

01

Identify the given values

We are given: - The temperature of the interior tissues, \(T_1 = 37.0 ^{\circ} \mathrm{C}\) - The temperature of the skin surface, \(T_2 = 27.0 ^{\circ} \mathrm{C}\) - The rate of heat transfer, \(Q = 100 \mathrm{W}\) - The skin area, \(A = 1.5 \mathrm{~m}^{2}\) - The thickness of the skin, \(d = 3.0 \mathrm{~mm}\)
02

Convert all values to SI units

To work with the SI units, the temperatures need to be in Kelvin, and the thickness of the skin needs to be in meters. - Convert \(T_1\) and \(T_2\) to Kelvin: \(T_1 = 37.0 ^{\circ} \mathrm{C} + 273.15 = 310.15 \mathrm{K}\) and \(T_2 = 27.0 ^{\circ} \mathrm{C} + 273.15 = 300.15 \mathrm{K}\) - Convert the thickness of the skin to meters: \(d = 3.0 \mathrm{~mm} \times \frac{1 \mathrm{~m}}{1000 \mathrm{~mm}} = 0.003 \mathrm{~m}\)
03

Calculate the temperature difference

Calculate the temperature difference between the interior tissues and the skin surface. \(\Delta T = T_1 - T_2 = 310.15 \mathrm{K} - 300.15 \mathrm{K} = 10 \mathrm{K}\)
04

Rearrange Fourier's Law to find the thermal conductivity

Rearrange Fourier's Law to solve for \(\kappa\): \(\kappa = \frac{-Q \times d}{A \times \Delta T}\)
05

Calculate the effective thermal conductivity

Substitute the given values into the formula to find the effective thermal conductivity of the skin: \(\kappa = \frac{-(-100 \mathrm{W}) \times 0.003 \mathrm{~m}}{1.5 \mathrm{~m}^{2} \times 10 \mathrm{K}} = \frac{0.3 \mathrm{W}}{15 \mathrm{m}^2 \mathrm{K}} = 0.02 \mathrm{W/m} \cdot \mathrm{K}\) The effective thermal conductivity of the skin is \(\kappa = 0.02 \mathrm{W/m} \cdot \mathrm{K}\).

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Most popular questions from this chapter

Suppose you mix 7.00 L of water at \(2.00 \cdot 10^{1}{ }^{\circ} \mathrm{C}\) with \(3.00 \mathrm{~L}\) of water at \(32.0^{\circ} \mathrm{C}\); the water is insulated so that no energy can flow into it or out of it. (You can achieve this, approximately, by mixing the two fluids in a foam cooler of the kind used to keep drinks cool for picnics.) The \(10.0 \mathrm{~L}\) of water will come to some final temperature. What is this final temperature?

A gas enclosed in a cylinder by means of a piston that can move without friction is warmed, and 1000 J of heat enters the gas. Assuming that the volume of the gas is constant, the change in the internal energy of the gas is a) 0 . b) 1000 J. c) -1000 J. d) none of the above.

The latent heat of vaporization of liquid nitrogen is about \(200 . \mathrm{kJ} / \mathrm{kg} .\) Suppose you have \(1.00 \mathrm{~kg}\) of liquid nitrogen boiling at \(77.0 \mathrm{~K}\). If you supply heat at a constant rate of \(10.0 \mathrm{~W}\) via an electric heater immersed in the liquid nitrogen, how long will it take to vaporize all of it? What is the time for \(1.00 \mathrm{~kg}\) of liquid helium, whose heat of vaporization is \(20.9 \mathrm{~kJ} / \mathrm{kg}\) ?

A \(2.0 \cdot 10^{2}\) g piece of copper at a temperature of \(450 \mathrm{~K}\) and a \(1.0 \cdot 10^{2} \mathrm{~g}\) piece of aluminum at a temperature of \(2.0 \cdot 10^{2} \mathrm{~K}\) are dropped into an insulated bucket containing \(5.0 \cdot 10^{2} \mathrm{~g}\) of water at \(280 \mathrm{~K}\). What is the equilibrium temperature of the mixture?

A cryogenic storage container holds liquid helium, which boils at \(4.2 \mathrm{~K}\). Suppose a student painted the outer shell of the container black, turning it into a pseudoblackbody, and that the shell has an effective area of \(0.50 \mathrm{~m}^{2}\) and is at \(3.0 \cdot 10^{2} \mathrm{~K}\). a) Determine the rate of heat loss due to radiation. b) What is the rate at which the volume of the liquid helium in the container decreases as a result of boiling off? The latent heat of vaporization of liquid helium is \(20.9 \mathrm{~kJ} / \mathrm{kg} .\) The density of liquid helium is \(0.125 \mathrm{~kg} / \mathrm{L}\).
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