Cube3D 立方体数理Physics · Middle school / High school
Why can two rods reach the same temperature profile at different speeds?
One end of each rod is held at 80 °C and the other at 20 °C. The rods are the same size and need the same amount of energy per unit volume to warm by one degree. Only their thermal conductivities differ. In the experiment, place the thermometer at 10 cm and observe how the temperature difference between the rods changes over time.
Compare the rods before they settle
At time zero the inside of both rods is at 20 °C. The left end has just been set to 80 °C. Heat passes from the warmer region near that end to cooler parts of the rod. With the probe at 10 cm and time set to 120 seconds, the model with conductivity 20 W/(m·K) reads about 22.5 °C. The model with conductivity 80 W/(m·K) reads about 38.3 °C.
Match the two conductivities and the curves overlap. With the heat capacity per volume held fixed, the higher conductivity gives a higher thermal diffusivity. The same distance then warms sooner. Now move the probe to the two ends. Both rods show the set temperatures there at any time because the ends are held fixed.
Steady temperature can coexist with heat flow
Set the time to 1200 seconds with the starting conductivities. The midpoint readings are about 48.0 °C and 50.0 °C, approaching the same dashed line. At steady state there is no heat building up anywhere, and the sides are insulated. The same amount of heat per second crosses every section of a rod. With uniform conductivity, this requires the same temperature drop over each equal length. Halfway between 80 °C and 20 °C, the steady temperature is 50 °C.
At steady state the power is P = kAΔT/L, where k is conductivity, A is cross-sectional area, and L is length. For 0.0001 m², a temperature difference of 60 K and a 0.20 m rod, the two powers are 0.60 W and 2.40 W. The model labels these as steady predictions; it does not use them as the current heat flow during warm-up.
Check your understanding
Set the left end to 60 °C and keep the right end at 20 °C. Predict the steady temperature at 5 cm and the ratio of steady powers when B has four times A’s conductivity.
Show the explanation
Five centimetres is one quarter of the rod. The steady temperature is 60 − 40/4 = 50 °C. B conducts four times as much power, because conductivity is the only changed factor in the power formula.
What the model assumes
This one-dimensional approximation uses constant thermal properties, insulated sides, a common volumetric heat capacity of 2,000,000 J/(m³·K), and two fixed temperatures. The plotted curves use discrete spatial and time steps. Conductivities are adjustable model values; real rods can differ in heat capacity, lose heat from their sides, or have imperfect thermal contacts. The interactive experiment requires Family Plus or Teacher access.