specific heat of ceramic
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A proposed experiment to determine the specific heat of a ceramic material.
ME 321-03
Team 4: Joe Kaltenthaler
Joey Arthur
Andrew Niemann
Zach Lehman
An experiment using an insulated vessel with a mass of liquid and solid material at different temperatures is
designed to find a specific heat, πΆπ , of the solid.
Insulated Dewar flask
π1,πΏ
π1,π
Heated in water for π1,π then submerged in liquid at π1,πΏ
π2,ππ ππΏ
ππ
The design problem is to select values for ππΏ, ππ, πΆπΏ, πππ π2,ππ to ultimately determine:
the specific heat, πΆπ, within a value range of 0.7 to 1.0 π½
πβπΎ
an uncertainty for πΆπ of Β± 10%
the expense and practicality of the design
a simple experimental procedure
The experimental procedure is outlined in the schematic below.
ππ
ππΏ π1,πΏ
π1,π
π2,ππ
Β°C
The specific heat of a solid is determined by the heat transferred between the solid and fluid.
This ratio yields the DRE:
πΆπ =ππΏπΆπΏβππΏ
ππβππ
Where βππΏ = (π1,πΏ β π2,ππ) and βππ = (π2,ππ β π1,π)
Relative uncertainty targets were determined by developing the uncertainty magnification factor form of
the DRE.
π€πΆπ
πΆπ
2
= (ππΆπ
πππΏ
ππΏ
πΆπ )2(
π€ππΏ
ππΏ)2+(
ππΆπ
πππ
ππ
πΆπ )2(
π€ππ
ππ )2+(
ππΆπ
πβππΏ
βππΏ
πΆπ )2(
π€βππΏ
βππΏ)2+(
ππΆπ
πβππ
βππ
πΆπ )2(
π€βππ
βππ )2+(
ππΆπ
ππΆπΏ
πΆπΏ
πΆπ )2(
π€πΆπΏ
πΆπΏ)2+(
π€ππππ
πΆπ )2
All UMFs are 1. All relative uncertainties have the same bounds. π€πΆπ
πΆπ
2
= 6(π€ππ
ππ)2
0.1 2 = 6(π€ππ
ππ)2
π€ππ
ππ= 4.082%
This percentage is a target value for random uncertainty and each measurandβs relative uncertainty
UMF πΆπΏ UMF ππ UMF ππΏ UMF ππ UMF ππΏ
The mass of the liquid and the solid were limited by the size of the Dewar flask and scale uncertainty.
π€π2 = π€π,πππ
2 + π€π,ππππ2
π€π = 0.0141 π
π€π
πβ€ 3.5%
π β₯ 0.45 π
Acculab vic-612 Scale Values (g)
Rated Input 610
Accuracy 0.005
Resolution 0.01
Readability 0.005
The mass of the liquid and the solid were limited by the size of the Dewar flask and scale uncertainty.
Max Dewar Volume = 350 mL
Assuming ππ ππππ = 2.5π
ππΏ and ππ€ππ‘ππ = 1.0
π
ππΏ
π = πβ
ππ ππππ
ππ ππππ+
ππ€ππ‘ππ
ππ€ππ‘ππβ€ 350 ππΏ
The mass of the liquid and the solid were limited by the size of the Dewar flask and scale uncertainty.
A useful value for this experiment is the ratio of liquid and solid masses
ππΏ
ππ β€
350 π
ππ β 0.4
π
π
The lower limit of this ratio is dependent upon the amount of water
(liquid) needed to completely submerge the solid object.
The minimum mass ratio to ensure that the solid is submerged is dependent on the solidβs geometry.
The size of the Dewar requires small solid objects to ensure they are fully submerged.
Assuming the pieces are small (ππ β€ 14 π and smaller than Dewar diameter) then the minimum mass ratio is 4*.
Sample Sample Mass (g) Liquid Mass to Submerge (g)
Mass Ratio (l/s)
1 10.45 40 3.83
2* 14.87 169 11.37
3 16.84 55 3.27
Average 14.05 88 6.26
The minimum and maximum masses are determined by the limits of the mass ratios and the uncertainty in mass.
Since the mass ratio is a minimum of 4, the solid mass is the minimum determined by uncertainty in measurement but the minimum liquid is not.
ππ β₯ 0.45 π πππ ππΏ
ππβ₯ 4
ππΏ β₯ 1.8 π
The geometric constraint for the solid is based on the diameter of the Dewar. This coupled with the maximum volume of the Dewar determines the maximum solid volume.
ππ β€ 14π πππ ππΏ
ππβ€
350
ππβ 0.4 β€
350
14β 0.4 β€ 24.6
Therefore,
ππΏ β€ 344.4 π
π. ππ π β€ ππΊ β€ ππ π π. ππ β€ ππ³ β€ πππ. π π π β€ππ³
ππΊβ€ ππ. π
The liquid temperature difference is limited by the uncertainty and the range of the thermometer.
π€βππΏ
βππΏβ€ 3.5%
π€βππΏ
2 = (πβππΏ
π1πΏ)2(π€π1πΏ,πππ
2 +π€π1πΏ,ππππ2 ) + (
πβππΏ
π2)2(π€π2,πππ
2 +π€π2,ππππ2 )
π€βππΏ= 0.1 β
Omega ASTM 3964C Thermometer Values (Β°C)
Range 25 to 55
Accuracy 0.05
Resolution 0.1
Readability 0.05
The liquid temperature difference is limited by the uncertainty and the range of the thermometer.
βππΏβ₯ 2.86β
From the Thermometer Range:
ππππ = 25β and ππππ₯ = 55β
Therefore,
2.86β β€ βππΏβ€ 30.0β
This is true for both the solid and liquid if the accurate, Omega ASTM
thermometer is utilized.
The solid temperature difference is limited by the uncertainty and the range of the thermometer.
π€βππ
βππ β€ 3.5%
π€βππ
2 = (πβππ
π1π )2(π€π1π ,πππ
2 +π€π1π ,ππππ2 ) + (
πβππ
π2)2(π€π2,πππ
2 +π€π2,ππππ2 )
π€βππ = 0.711 β
Enviro-Safe Thermometer Values (Β°C)
Range -20 to 110
Accuracy 0.5
Resolution 1
Readability 0.5
The thermometerβs lower limit and the practical limit of the boiling point of water determine the range of solid
temperatures.
βππβ₯ 20.31β
ππΏ,πππ = 25β and ππΏ,πππ₯ = 100β
Therefore,
20.31β β€ βππβ€ 75.0β
This is true for the solid if the accurate, Omega ASTM thermometer is utilized for
the equilibrium temperature (π2,ππ) and the Enviro-Safe thermometer measures
the elevated temperature (π1,π)
A design space was developed based on sensor and relative uncertainty constraints.
Design Point
(72.5, 6)
ΞTL=2.5Β°C
0
2
4
6
8
10
12
0 20 40 60 80
mL/m
S (
-)
ΞTS (Β°C)
ΞTL=2Β°C at Max CS
ΞTL,max=3.14Β°C at Min CS
Using 5% relative uncertainty in ΞTL
Our experimental design can effectively determine πΆπ within a 10% uncertainty.
Parameter Representative
Value Systematic Uncertainty
Relative Uncertainty
(%) UMF
RSSC (%)
UPC (%)
βππΏ (Β°πΆ) 2.5 Β± 0.1 4 1 4 54.7
πΆπππππππ
π½
π πΎ - - 3.5 1 3.5 41.9
βππ (Β°πΆ) 72.5 Β± 0.711 0.980 1 0.980 3.29
ππ (π) 10 Β± 0.0141 0.141 1 0.141 0.0684
ππΏ (π) 60 Β± 0.0141 0.0235 1 0.0235 0.00189
πΆπΏ π½
π πΎ 4.179 πππππππππππ - - - -
πΆπ
π½
π πΎ Expected value: 0.87 - -
π€πΆπ
πΆπ: 5.41 100
The design utilizes simple, practical components.
Omega ASTM 64C Thermometer
Enviro-Safe (-20β to 100β) Thermometer
Acculab vic-612 Scale
Pope 8600/0099 350mL Dewar
Simple Hot Plate
Large Beaker
Stir Rod
In conclusion, the proposed experimental design successfully:
Measures the specific heat of a solid piece of ceramic between 0.7 and 1.0 π½
πβπΎ .
Measures the specific heat of a solid piece of ceramic with a relative uncertainty below 10%.
Utilizes practical measurement devices at a relatively low cost to the experimental team.
Follows a simple experimental procedure.
Backup Slide Menu
Alternate Temperature
Sensor UMF Derivation
CL Temperature Model
CL Uncertainty
βππΏ,πππ₯ Limit Derivation
DRE Derivation CDewar
Discussion Detailed
Procedure
An infrared heat gun introduces large uncertainty when measuring solid temperatures in the applicable
temperature range. Extech Model 42560 Values
Range β50β π‘π 1050β
Resolution 0.1β
Readability 0.05β
Accuracy Β±1.5% Γ πππππππ + 2β
At βππ,πππ₯= 75β where the readings are π1,π = 100β and π2,ππ = 25β
π€βππ
2 = 0.015 β 100 + 2 2
+ 0.05 2 + 0.015 β 25 + 2 2
+ 0.05 2
π€βππ= 4.23β
π€βππ
βππ =
4.23β
75β= 5.64%
UMF derivations
UMF derivations
A model exists to calculate specific heat of water at varying temperatures.
πΆπΏ (15Β°πΆ) = 4.1855π½
π Β°πΆ
β
*International Committee for Weights and Measures (Paris 1950)
πΆπΏ = 0.996185 + 0.0002874ππΏ + 100
100
5.26
+ 0.011160 Γ 10β0.036ππΏ πΆπΏ (15Β°πΆ)
Source: CODATA Key Values for Thermodynamics, Cox, Wagman, and Medvedev
The uncertainty for liquid specific heat is negligible.
4.16
4.17
4.18
4.19
4.2
4.21
20 25 30 35 40 45 50
CL
(J/
(gΒ°C
)
Temperature (Β°C)
For a large range, CL changes little with changing temperature
UPC=0.001%
Our experimental procedure is easy to follow and utilizes accurate measuring devices.
1) Obtain the mass of a ceramic chip using the Acculab vic-612 Scale
2) Zero a beaker on the scale and add deionized water until the mass measurement is equal to the mass of the solid multiplied by the mass ratio.
3) Heat the beaker of water on hot plate or cool in ice bath until the temperature on the OMEGA thermometer reads 25Β°C. Stir well with stir rod.
4) Place water in dry Dewar flask.
5) Heat large beaker of water to boil and add solid ceramic piece. Let sit 10 minutes so ceramic reaches equilibrium with water. Record π1,π with the Enviro-safe thermometer. Stir well throughout.
6) Carefully, add solid to Dewar. After considerable time (5~10 minutes) and constant stirring record final temperature π2,ππ
The DRE is a simple variation of the exchange of energy between two substances.
Energy contained in a material:
πΈ = πβ = ππΆβπ
From the conservation of energy:
πΈππ β πΈππ’π‘ = ππππ‘ + ππππ‘ + (βπΈπ ππππ) + πΈππππ’ππ
There is no work, heat transfer, or net energy change of the system so:
πΈππππ’ππ = πΈπ ππππ
ππΏπΆπΏβππΏ= πππΆπβππ
Solving for the specific heat of the solid
πΆπ =ππΏπΆπΏβππΏ
ππβππ
The specific heat of the Dewar could be a contributing factor in the experiment
Like the liquid, the Dewar can absorb energy as well:
πΈπ ππππ = πΈπΏπππ’ππ + πΈπ·ππ€ππ
πππΆπβππ= ππΏπΆπΏβππΏ + ππ·πΆπ·βππ·
The new DRE becomes:
πΆπ =ππΏπΆπΏβππΏ
ππβππ+
ππ·πΆπ·βππ·
ππβππ
Since specific heats are material properties, the Dewar itself could impact the
temperature changes and the determination of πΆπ .
The specific heat of the Dewar could be a contributing factor in the experiment
Since the Dewar is glass:
πΆπ· = 0.84 π½
π β πΎ
This is in the same range as the specific heat of the ceramic solid in question.
However, since the heat cannot propagate throughout the whole Dewar, the mass contacting the
water is small.
Therefore,
ππ·πΆπ·βππ·
ππβππ
Is negligible as both ππ· and βππ· are small.
The DRE was utilized to identify the βππΏ,πππ₯value corresponding to our Design Space.
Given that our mass ratio was bounded by 4 and a βππ,πππ₯of 75 Β°C we can solve for the
minimum allowable βππΏ,πππ. Using the minimum value of 0.7 π½
πβπΎ
βππΏ,πππ₯ =πΆπ,πππππβππ
ππΏπΆπΏ(π1,πΏ=25Β°πΆ)
Also given the Cox, Wagman, and Medvedev relationship at T1,L=25 Β°C
βππΏ,πππ₯ =0.7
1
4(75)
(4.1793)= 3.14Β°C
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