## Physics Laboratory 3

### Objective:

In this lab you will measure the rate of heat loss of three objects and you will think about how the human body regulates its temperature.

Open a Microsoft Word document to keep a log of your experimental procedures and results.  Complete all the tasks (in blue).  Answer all questions.

### Newton's law of cooling

A CSI investigator is called to the scene of a crime where a dead body has just been found.  One of the first things the investigator does is to measure the temperature of the body and the temperature of the environment.  After the crime scene has been studied further, the temperature of the body is measured once more, approximately an hour later.  The two temperature readings and the assumption that the victim's body temperature was normal (98.6o F) prior to death can be used to estimate the  victim's time of death.

How fast does a warmer object cool to the temperature of its environment?  Let ΔT = TB - TE be the difference between the temperature of the warmer body  or object, TB,  and the temperature of its environment, TE.  In a small time interval dt, the rate at which this temperature difference changes is often found to be proportional to the temperature difference itself.

dΔT /dt = -σΔT

This is called Newton's law of cooling.  It is an empirical law and there are conditions under which it does not hold.  The rate constant σ depends on physical properties of the system, such as its mass, specific heat capacity, surface, etc.  Newton's law of cooling can be rewritten as dΔT /ΔT = -σdt and integrated to yield

∫ΔT(0)ΔT(t')dΔT/ΔT = -σ ∫0t'dt ,  or ln(ΔT(t')/ΔT(0)) = -σt',  or  ΔT(t') = ΔT(0)e-σt'.

TB(t')= TE + (TB(0) - TE)e-σt'.

### Experiment:

In this experiment you will explore if three objects with initial temperature TB(0), submerged in an environment with a large heat capacity and temperature TE, cool, as expected,  according to Newton's law of cooling.  You will will determine the rate constant σ for three calorimeter cans filled with hot water.

The surface of one can is unpainted, the surface of the second can is painted black, and the surface of the third can is painted white.  The cans are exposed to air in the laboratory environment.  The cans will transfer heat to the environment via conduction, convection and radiation.  The surfaces of the three cans have different emissivities, so we may expect different σ's for the three cans.  You will determine σ for each can and compare the measured rate constants for the different cans.  To determine σ, you will record each can's temperature as a function of time for approximately 1 hour.  Newton's law of cooling states that

ln(ΔT(t')/ΔT(0)) = -σt',  or  ΔT(t') = ΔT(0)e-σt'.

This equation is of the form y = ax + b, with y = ln(TB - TE)(t') x = t, a = -σ and b = ln(TB - TE)(0).  You will determine the slope a = -σ of a straight line fit of ln(TB - TE)(t) versus t and thus determine the rate constant.

Procedure:

With the temperature sensor in air, determine the temperature of the environment, i.e.  the room temperature TE.

Record the temperature TE in the spreadsheet.

Monitor the calorimeter cans filled with with hot water for approximately 1 hour.  The thermometer inserted into the water reads the temperature of the water.  For each can record the temperature as a function of time in your spreadsheet.  (Click on each of the thumbnails above to view an enlarged picture.)

Clock Can 1(unpainted) Can 2 (black) Can 3 (white) Elapsed Can 1 Can 2 Can 3
TE (oC) Time  Temp (oC) Temp (oC) Temp (oC) Time (s) ln(TB - TE) ln(TB - TE) ln(TB - TE)

Data Analysis:

• For each measurement find the elapsed time in seconds since the start of the experiment and record it in your spreadsheet.
(Link:  Convert time to seconds in Excel)
• In your spreadsheet, subtract TE from the measured temperature TB of each can and find the natural logarithm of this difference.  Use the regression function or the trendline to find the slope of a straight line fit of ln(TB - TE)(t) versus time t in seconds for each can.  Format the trendline label in scientific format with 2 decimal places.
Set the rate constant σ equal to the magnitude of the slope.

• In a few sentences summarize the experiment.
What are the rate constants you found for the different cans?
How do these rate constant compare?  Are the different surfaces of the calorimeter cans associated with different rate constants?
Did you expect a difference?  Explain!

Exploration:

Regulating the temperature of the human body

The human body has the ability to regulate its temperature so that it remains very close to 37oC.  If the body is overheating as a result of strenuous exercise, the surface blood vessels dilate to increase the blood flow to the surface areas.  Heat is carried by the blood to the surface where it causes the skin temperature to increase.  The body also begins to produce sweat.  The rate of sweating increases strongly with body temperature above 37oC.

Conduction, evaporation, convection, and radiation can now transfer heat from the skin to the environment.  The chart below shows the relative importance of these heat transfer mechanisms at different environmental temperatures.

Questions:

• For environmental temperature below 32 oC, convection and radiation are the dominant heat removal mechanisms.  What can a person do to increase the amount of heat carried away by these mechanisms?
• Why is conduction usually not a dominant heat removal mechanism?  Can you think of situations when it becomes important?
• Why do convection and radiation no longer remove heat from the body at environmental temperature above 37 oC?

Evaporation

The body secrets water on to the skin from sweat glands.  As this water evaporates, the latent heat of vaporization is removed from the body.  The rate of evaporation increases as the degree of saturation of the surrounding air decreases.  In warm humid weather, the rate may be so low that a layer of water accumulates on the skin.  In still air, the air near the skin will become saturated and evaporation will stop.  Evaporation will increase if this saturated air is continually removed by wind or an artificially produced air stream.

Questions:

• If there is a cool wind blowing, what would be the consequences, as far as heat loss from the body is concerned, of wearing wet clothes?  Explain your answer.
• Cloudy nights are usually warmer than cloudless nights.  Why?