Final Coefficient of Expanisitivity

7/31/2019 Final Coefficient of Expanisitivity

1/20Determining The Thermal Coefficient Of Expansion In Some Solids And Liquids. Page 1

: Dhywill Dzikunoo ( 5044710)

Dorcas Addo (5039510)

: EXPERIMENTAL PHYSICS 2

: 084

: One (1)

: 13th February 2012.



The experiment was performed to determine the linear expansion of two solidscopper and brass at various temperatures. The volume expansion of water wasalso observed. Also the coefficients of linear expansibility were also determinedfor copper and brass as a function of temperature. And the coefficient of volumeexpansion of water as a function of temperature also determined. Increasing thetemperature in steps of about 10from a room temperature of 25to a temperatureof about 90 was observed to cause an increase in the length which is an of the copper andbrass rods being observed on the dial gauge. The linear expansion was observed on the dialgauge. For that of the water the volume expansion was observed by increasing thetemperature of water in which the pycometer sits in a water bath and the temperatureincreasedtemperature in steps of about 10from a room temperature of 25to atemperature of about 90 was observed to cause an increase in the volume of the water inthe pycynometer. Brass was found to have a higher linear coefficient. The linear coefficient of expansitivity of copper and brass was found to be about 1.659 10-5 K 2.3 10-7 K and1.82 10-5 K 2.04 10-7 K respectively. Water was also found to have a coefficient of volume expansitivity of 2.066 10-5 K 2.3 10-7 K. It was observed that an increase intemperature causes a corresponding increase in the volumes and length of the liquid (water)and solids (brass and copper) respectively


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Expansion in solids

Materials generally change their size when subjected to a temperature change while thepressure is held constant. In the special case ofsolid materials, the pressure does notappreciably affect the size of an object, and so, for solids, it's usually not necessary to specifythat the pressure be held constant.Common engineering solids usually have coefficients of thermal expansion that do not varysignificantly over the range of temperatures where they are designed to be used, so whereextremely high accuracy is not required, practical calculations can be based on a constant,average, value of the coefficient of expansion.

Linear expansionThe linear thermal expansion coefficient relates the change in a material's linear dimensions toa change in temperature. It is the fractional change in length per degree of temperature change.Ignoring pressure, we may write:

where L is the linear dimension (e.g. length) and dL / dT is the rate of change of that lineardimension per unit change in temperature.

The change in the linear dimension can be estimated to be:

This equation works well as long as the linear expansion coefficient does not change muchover the change in temperatureT. If it does, the equation must be integrated.

Effects on strainFor solid materials with a significant length, like rods or cables, an estimate of the amount of thermal expansion can be described by the material strain, given by anddefined as:

where is the length before the change of temperature and is the length afterthe change of temperature.

Therefore thermal expansitivity can be defined as

=



For most solids, thermal expansion is proportional to the change in temperature:

Thus, the change in either the strain or temperature can be estimated by:

Where

is the difference of the temperature between the two recorded strains, measured indegrees Celsius or kelvins, and is the linear coefficient of thermal expansion in inversekelvins.Area expansionThe area thermal expansion coefficient relates the change in a material's area dimensions to achange in temperature. It is the fractional change in area per degree of temperature change.Ignoring pressure, we may write:

where A is some area of interest on the object, and dA / dT is the rate of change of that areaper unit change in temperature.The change in the linear dimension can be estimated as:

This equation works well as long as the linear expansion coefficient does not change muchover the change in temperatureT. If it does, the equation must be integrated.Volumetric expansionFor a solid, we can ignore the effects of pressure on the material, and the volumetric thermalexpansion coefficient can be written

where V is the volume of the material, and dV / dT is the rate of change of that volume withtemperature.This means that the volume of a material changes by some fixed fractional amount. Forexample, a steel block with a volume of 1 cubic meter might expand to 1.002 cubic meterswhen the temperature is raised by 50 C. This is an expansion of 0.2%. If we had a block of steel with a volume of 2 cubic meters, then under the same conditions, it would expand to2.004 cubic meters, again an expansion of 0.2%. The volumetric expansion coefficient wouldbe 0.2% for 50 C, or 0.004% per degree C.



If we already know the expansion coefficient, then we can calculate the change in volume

whereV / V is the fractional change in volume (e.g., 0.002) andT is the change intemperature (50 C).The above example assumes that the expansion coefficient did not change as the temperaturechanged. This is not always true, but for small changes in temperature, it is a goodapproximation. If the volumetric expansion coefficient does change appreciably withtemperature, then the above equation will have to be integrated:

where T0 is the starting temperature andV(T) is the volumetric expansion coefficient as afunction of temperature T.



Fig 1.

Set up for the thermal expansion of water and metal.

Apparatus;

Dilatometer with clock gauge, Copper tube, Brass tube, Immersion thermostat , Accessoryset, Bath for thermostat, Lab thermometer, -10+100C, Rubber tubing, Cannula,Measuring tube, Wash bottle, Flask, flat bottom, Glass beaker, tall, 100 ml

Thermostat

Dilatometer

Thermometer

Water bath



Raw experimental values.

Initial length of rod = 600m Initial volume of water 100ml

Initial temperature, room temperature = 25C

TABLE 1.

COPPER VALUES

FinalTemperature/

C

TemperatureChange/ C

Expansion/mm

25 0 030 5 0.0540 15 0.1450 25 0.2460 35 0.3670 45 0.4680 55 0.5690 65 0.66

.

TABLE 2

BRASS. VALUES.

Temperature/C


Expansion/mm

25 0 030 5 0.0540 15 0.1650 25 0.2760 35 0.3970 45 0.5080 55 0.6190 65 0.72



TABLE 3.

Water

Temperature/C


Linearexpansion/mm

Volumeexpansion/ mm3

Volumeexpansion/ ml 3

25 0 0 0 030 5 14.61 103.3 0.103340 15 43.88 310.2 0.310250 25 73.19 517.4 0.517460 35 102.45 724.3 0.7243



For the thermal expansibility Ethermal= Change in temperature Final temperature - Initial Temperature

25 degrees = 25 - 25 = 0 K

30 degrees = 30 25 = 5 K

40 degrees = 40 25 = 15 K

50 degrees= 50 25 = 25 K

60 degrees = 60 25 = 35 K

70 degrees = 70 25 = 45 K

80 degrees= 80 25 = 55 K

90 degrees = 90 25= 65 K

Calculations For copper

Expansibility at 30 degrees Celsius = = 8.33 10-5



Expansibility at 60 degrees Celsius = =6.00 10-4






The coefficient of thermal expansibility of solids implies

For copper at 25 degrees Celsius 0

For copper at 30 degrees Celsius = 1.67 10-5


For copper at 50 degrees Celsius = 1.60 10-5 For copper at 60 degrees Celsius = 1.71 10-5


For copper at 80 degrees Celsius 1.69 10-5


Average = (1.67 + 1.55 + 1.60 + 1.71 + 1.70 + 1.69 + 1.69) 10-5

7= 1.659 10-5 -K

-

FOR BRASS

Expansibility at 25 degrees Celsius = =0










For brass at 25 degrees Celsius = 0

For brass at 30 degrees Celsius = 1.70 10-5







Average = (1.70 + 1.78 + 1.80 + 1.85 + 1.86 + 1.87 + 1.85) 10-5

7

= 1.82 10-5 -K



Diameter of tube = 3mm

Radius of tube = 1.5mm

Volume of water rise = height of rise area of hole

Area of hole = R 2

= (1.50) 2

= 7.069 mm2

Volume of water rise = height of rise area of hole

At 30 degrees = volume = 7.069 mm2 14.61 mm

= 103.3 mm3


= 310.2 mm3


=517.4 mm3


=724.3 mm3



For the volumetric expansion the coefficient of expansion is as follows

For the expansion of water at 25 degrees = = 0For the expansion of water at 30 degrees = = 2.060 10-4 For the expansion of water at 40 degrees = = 2.067 10-4 For the expansion of water at 50 degrees = = 2.068 10

-4

For the expansion of water at 60 degrees = = 2.069 10-4

Average = = 2.066 10-4-K



COPPER

FinalTemperature/

C


Expansion/mm

expansibility Coefficientof thermal

conductivity/1 10 -5 -K

25 0 0 0 030 5 0.05 8.33 10-5 1.67

40 15 0.14 2.33 10-4

1.5550 25 0.24 4.00 10-4 1.6060 35 0.36 6.00 10-4 1.7170 45 0.46 7.67 10-4 1.7080 55 0.56 9.33 10-4 1.6990 65 0.66 1.10 10-3 1.69

.

BRASS.

Temperature/C


Expansion/mm

expansibility Coefficientof thermal

conductivity/1 10 -5 -K

25 0 0 0 030 5 0.05 8.33 10-5 1.6740 15 0.16 2.67 10-4 1.7850 25 0.27 4.50 10-4 1.8060 35 0.39 6.50 10-4 1.8570 45 0.50 8.33 10-4 1.8680 55 0.61 1.02 10-4 1.8790 65 0.72 1.02 10-3 1.85



Water

Temperature/C


Linearexpansion/mm

Volumeexpansion/ mm3

Volumeexpansion/ ml 3

Volumetricexpansion

Coefficientof expansibility/1 10 -5 -K

25 0 0 0 0 0 030 5 14.61 103.3 0.1033 1.03 10-3 2.06040 15 43.88 310.2 0.3102 3.10 10-3 2.06750 25 73.19 517.4 0.5174 5.17 10-3 2.068

60 35 102.45 724.3 0.7243 7.24 10-4 2.069



)

1.67 10-5 1.1 10-7 1.21 10-14 1.55 10-5 -1.09 10-6 1.1881 10-14 1.60 10-5 -5.9 10-7 3.44 10-14 1.71 10-5 5.1 10-7 2.601 10-14 1.70 10-5 4.1 10-7 1.681 10-14 1.69 10-5 3.1 10-7 9.61 10-14 1.69 10-5 3.1 10-7 9.61 10-14

= 2.1687

10-5

Mean = = 1.659 10-5

Standard deviation = = = 2.3 10-7 -K

For volumetric expansion

) 2.060 10-4 -6 10-7 3.6 10-13 2.067 104 1 10-7 1 10-14 2.068 10-4 2 10-7 4 10-14 2.069 10-4 3 10-7 9 10-14

= 5 10 -13

Mean = = 6.71 10-5

Standard deviation = = = 2.04 10-7 -K



DISCUSSIONFrom the experiment performed to determine the volume expansion of water and the linear

expansion of brass and copper as a function of temperature using the pycynometer and adilatometer respectively. The coefficient of expansion was seen to be directly proportional tothe expansion. That means the greater the expansion the greater the coefficient of expansionand the linear of volume expansion and vice versa. The coefficient of expansion was seen to bedirectly proportional to the expansion. It was observed from the graphs that were thetemperature increases there is a corresponding increase in length. From the graph it was seenthat brass which had the higher coefficient if linear expansitivity for the solids had a steeperslope than copper which expands a little less than brass. Also an increase in temperature causes

an increase in the volume of water. With the linear expansions brass was found to have thegreater linear expansion. That means brass expands more than copper.


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CONCLUSION.

The linear expansion of brass and copper was observed to

and respectively. Thevolume expansivity of water was .

It was observed that as temperature increases, there is a corresponding increase in the

volume of expansion of liquids (water) and the linear expansivity of solids (brass and copper).

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Final Coefficient of Expanisitivity