The thermal conductivity of the fur (including the skin) of...

The thermal conductivity of the fur (including the skin) of a male Husky dog is $0.026 \mathrm{W} /(\mathrm{m} \cdot \mathrm{K}) .$ The dog's heat output is measured to be $51 \mathrm{W}$, its internal temperature is $38^{\circ} \mathrm{C},$ its surface area is $1.31 \mathrm{m}^{2},$ and the thickness of the fur is $5.0 \mathrm{cm} .$ How cold can the outside temperature be before the dog must increase its heat output?

Key Concepts

Temperature Gradient

The temperature gradient is the change in temperature per unit distance across a material. It drives heat flow from warmer regions to cooler ones. In scenarios involving insulation, such as an animal's fur, the internal body temperature and the external environmental temperature create a gradient that directly affects the rate at which heat is conducted through the insulating layer.

Steady-State Heat Transfer

Steady-state heat transfer occurs when the temperature distribution in a system no longer changes with time, meaning that the heat entering one side of a material equals the heat leaving the other side. In problems of heat conduction through an insulating medium, assuming steady state allows us to set up a balance between internal heat generation (or metabolic heat output) and the conductive heat loss, which is used to determine the permissible environmental conditions.

Thermal Insulation

Thermal insulation refers to materials or systems designed to reduce the rate of heat transfer between objects or environments at different temperatures. In this context, an insulating layer, such as an animal's fur, helps to slow the loss of body heat to a colder environment by providing resistance to conductive heat flow. The effectiveness of insulation is directly related to the thermal conductivity and thickness of the insulating material.

Fourier's Law of Heat Conduction

Fourier's Law states that the rate of heat transfer through a material is proportional to the negative temperature gradient and the area through which the heat flows, and inversely proportional to the distance over which the temperature difference is applied. This principle is key in calculating how much heat is lost from a body through its insulating covering, using parameters such as thermal conductivity, surface area, temperature difference, and thickness.

Thermal Conductivity

Thermal conductivity is a property of a material that indicates its ability to conduct heat. It quantifies the rate at which heat passes through a material per unit area and per unit temperature gradient. Materials with low thermal conductivity, like insulating fur, restrict heat flow, which is essential in understanding how creatures maintain their body temperature in cold environments.

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