PURPOSE

You will investigate the heat transfer involved in a cooling process and determine the thermal time constant of a thermometer.

EQUIPMENT  Styrofoam cup, mercury thermometer (0°C to 100°C), 2 Vernier temperature probes, plastic sleeve, Bunsen burner, teakettle, stand w/ clamp.

RELEVANT EQUATIONS

DISCUSSION

All dynamic instruments used to make physical measurements require a period of time to respond to the change in conditions which they measure. Some instruments respond very quickly and their response times go unnoticed. However, the times required for some instruments to give "correct" readings are readily observable and measurable. An example of this is a thermometer, which requires a noticeable time for its temperature to become equal to the surrounding temperature. Because of this delay, you must wait 2 to 3 minutes to get a correct reading when measuring your body temperature, for example.

Newton's law of cooling is an empirical principle which states that the rate of change of temperature of a body is directly proportional to the difference in temperature of the body and its surroundings, provided that the temperature difference is small.

Mathematically Newton's law of cooling may be expressed as:

(1)

where ΔT/Δt is the time rate of change in the temperature T of the object; T_r is the temperature of the surroundings (room temperature in this experiment); and K is a measure of the thermal conduction for heat flow from the hot object to its surroundings. The minus sign indicates that temperature decreases with time when T is greater than T_r.

The temperature T of an object can be determined at any time t by the equation:

(2)

where the 'e' factor is the exponential function (the base of natural logarithms, e = 2.718...), and T_o is the temperature of the object at time zero. The above equation for T is often written

(3)

where τ = 1 / K is called the time constant of the process and has the same units as time.

When t = τ (one time constant), then e^{- t / τ} = e^-1 = 0.368. Thus the temperature difference of the object from its surroundings will fall to about 37% of its initial value in one time constant. In the next time constant it will fall another 37%, or 37% x 37% = 14% of the initial value, and so on. A plot of how the temperature of the cooling object varies with time is shown in Fig. 15-1.

The temperature vs. time plot shown above is referred to as an exponential decay curve. The temperature of the object decays or decreases exponentially with time because of the negative exponent. The shorter the time constant, the greater the negative exponent and the faster the decay or cooling.