Conduction

The Physics Hypertextbook™
© 1998-2008 by Glenn Elert -- A Work in Progress
All Rights Reserved -- Fair Use Encouraged

prev | up | next

Discussion

Heat conduction (as opposed to electrical conduction) is the flow of internal energy from a region of higher temperature to one of lower temperature by the interaction of the adjacent particles (atoms, molecules, ions, electrons, etc.) in the intervening space.

Factors affecting the rate of heat transfer by conduction.

  1. temperature difference
  2. length
  3. cross-sectional area
  4. material
P =  ΔQ  =  kAΔT
Δt ℓ

Fourier's law (compare to Ohm's law)

Φ =  P  =  ΔQ  = − k ∇T
A A Δt

Note: it's the rate at which heat is transferred, not the amount of heat transferred.

Conductivities vary for material being greatest for metallic solids, lower for nonmetallic solids, very low for liquids, and extremely low for gases. The best ordinary metallic conductors are (in decreasing order) silver, copper, gold, aluminum, beryllium, and tungsten. Diamond beats them all, and graphite beats diamond only if the heat can be forced to conduct in a direction parallel to the crystal layers. The material with the greatest thermal conductivity is a superfluid form of liquid helium called helium II, which only exists at temperatures below 2.17 K. Since it is highly unlikely you will encounter this substance, it is really not worth thinking about except in the fact that it is an exceptional material.

Thermal Conductivity for Selected Materials (~300 K except where otherwise indicated)
material k (W/m·K)   material k (W/m·K)
air, sea level 0.025   neoprene 0.15–0.45
air, 10,000 m 0.020   nickel 90.7
aluminum 237   particle board 0.15
asbestos 0.05–0.15   paper 0.04–0.09
asphalt 0.15–0.52   plaster 0.15–0.27
brass (273 K) 120   platinum 71.6
brick 0.18   plutonium 6.74
bronze (273 K) 110   plywood 0.11
carbon, diamond 895   polyester 0.05
carbon, graphite (∥) 1950   polystyrene foam 0.03–0.05
carbon, graphite (⊥) 5.7   polyurethane foam 0.02–0.03
carpet 0.03–0.08   sand 0.27
chromium 93.7   silica aerogel 0.026
concrete 0.05–1.50   silver 429
copper 401   soap powder 0.11
cotton 0.04   snow (< 273 K) 0.16
feathers 0.034   steel, plain (273 K) 45–65
fiberglas 0.035   steel, stainless  (273 K) 14
freon 12, liquid 0.0743   straw 0.05
freon 12, vapor 0.00958   teflon 0.25
felt 0.06   tin 66.6
glass 1.1–1.2   titanium 21.9
gold 317   tungsten 174
granite 2.2   uranium 27.6
helium gas 0.152   vacuum 0
helium I (< 4.2 K) 0.0307   water, ice (223 K) 2.8
helium II  (< 2.2 K) ~100,000?   water, ice (273 K) 2.2
ice cream powder 0.05   water, liquid (273 K) 0.561
iron 80.2   water, liquid (373 K) 0.679
lead 35.3   water, vapor (273 K) 0.016
limestone 1   water, vapor (373 K) 0.025
marble 1.75   wood 0.09–0.14
mercury 8.34   wool 0.03–0.04
mica 0.26   zinc 116
mylar 0.0001?   zirconia 0.056?

Thoughts on conductivity …

Related quantities: r value.

ΔT = R  Δq ⇒ R =  ℓ
Δt kA

The clo. studies of clothing have lead to the definition of the unit of clothing, which corresponds to the insulating value of clothing needed to maintain a subject in comfort sitting at rest in a room at 21 °C (70 °F) with air movement of 0.1 m/s and humidity less than 50%. One clo of insulation is equivalent to a lightweight business suit.

Newton's law of cooling Q/t ∝ ΔT. Heat leaks faster from a cool house than a warm house. Thus, it's more cost effective to turn your air conditioner off when you're away, than to leave it on hoping to keep your house cool.

Summary

Problems

practice

  1. The cook pot shown in the photograph to the right is quite unique. Pots of this sort are usually manufactured by stamping a single sheet of heavy gauge stainless steel or aluminum in a powerful metal press. This pot is different in that it was stamped from a "sandwich" of steel-copper-steel (note the copper core between the stainless steel bottom and sides). What advantage would such a complicated pot have over similar single metal pots when used in the kitchen? Why have a copper core? Why use stainless steel for the sides and bottom? What's the deal with this pot?

    Solution …

    Copper is one of the best conductors of heat available (only silver has a higher thermal conductivity), while stainless steel is a relatively mediocre conductor (mercury is one of the few metals with a lower thermal conductivity). Using copper in the base would increase the rate at which heat was transferred from the flames or heating element to the food, while using stainless steel on the sides would reduce the rate at which heat was lost from the food to the environment. Such an arrangement results in an effective cook pot that responds rapidly to changes in burner output. Capping the copper base with stainless steel seems to work against this efficiency, however.

    For many applications a base that was made only of copper would probably be too effective at transferring heat. Heat applied to a small region would be transferred so rapidly that it wouldn't have time to diffuse across the pot's bottom. This would result in uneven cooking and possibly even local areas of scorching. Capping the base with stainless steel slows the immediate rate of heat transfer from the burner or heating element, but once this heat enters the copper core its high conductivity would spread the heat rapidly and evenly to all parts of the base.

    There are also some esthetic issues behind this design. Copper is notoriously hard to clean, while stainless steel is, well, stainless. Cooks who insisted that their cookware sparkle at all times (even on the surfaces that are never in contact with food) would probably appreciate the use of low maintenance materials for most of the exterior. The thin, exposed band of copper near the bottom is no doubt there to catch the eye of wandering customers.
  2. Write something else.
    • Answer it.
  3. olive-oil.txt
    A liter of warm olive oil was placed in a freezer and allowed to solidify. Temperature measurements were taken every six seconds for one hour (3600 s). Determine its final equilibrium temperature.

    Solution …

    The temperature data appear to follow an exponential decay function. An automatic curve fit returns a limit value of −11.2 °C. The fit looks good early on, but near the end of the data collection period there is a noticeable deviation, which leads me to believe that the extrapolation to the limiting equilibrium value is unreliable.
     
    [magnify]
     
    Taking the logarithm of a quantity that varies exponentially will transform it into a linear relationship. This graph of log(temperature) vs. time does not fit a straight line, however, and is not technically exponential. The data needs to be massaged a bit to get it to fit.
     
    [magnify]
     
    Recall that the limit of an exponential decay process is zero. But the olive oil in our freezer will obviously end up below the freezing point of water (that is, below zero celsius). In order to get our temperature data to approach zero we will add a constant number to every value. The constant that results in the best linear fit will give the final temperature of the oil.

    Taking the logarithm of a quantity that varies exponentially will transform it into a linear relationship. This graph of log(temperature) vs. time does not fit a straight line, however, and is thus not strictly exponential. The data needs to be massaged a bit to get it to fit.
     
    [magnify]
     
    Of all the adjustments tried, the graph of log(T+09) best fits a straight line. Assuming the cooling curve follows an exponential decay function, the final temperature of olive oil in the freezer is −9 °C. I consider this a more reliable answer than −11.2 °C.
  4. Calculus problem. Show that Newton's law of cooling produces a temperature curve that decays exponentially.
    • Answer it.

numerical

  1. Some sort of blubber problem would be nice

    The ultrasound scans shoed a relatively uniform layer of blubber running virtually the entire length of the body, ranging from 1.6 to 2.4 inches thick, in adults that weigh between 900 and 1,100 pounds. Even in one-month-old pups, which are the size of mature Saint Bernards, the blubber is between 1.2 and 1.6 inches thick. (Sunbathing Seals of the Antarctic. Terrie M. Williams. Natural History. October 2003.)

Resources

prev | up | next

Another quality webpage by

Glenn Elert		 eglobe logo home | contact

bent | chaos | eworld | facts | physics