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, 𝐶𝑠, of the solid.

Insulated Dewar flask

𝑇1,𝐿

𝑇1,𝑆

Heated in water for 𝑇1,𝑆 then submerged in liquid at 𝑇1,𝐿

𝑇2,𝑒𝑞 𝑚𝐿

𝑚𝑆

The design problem is to select values for 𝑚𝐿, 𝑚𝑆, 𝐶𝐿, 𝑎𝑛𝑑 𝑇2,𝑒𝑞 to ultimately determine:

the specific heat, 𝐶𝑆, within a value range of 0.7 to 1.0 𝐽

𝑔∙𝐾

an uncertainty for 𝐶𝑆 of ± 10%

the expense and practicality of the design

a simple experimental procedure

The experimental procedure is outlined in the schematic below.

𝑚𝑆

𝑚𝐿 𝑇1,𝐿

𝑇1,𝑆

𝑇2,𝑒𝑞

°C

The specific heat of a solid is determined by the heat transferred between the solid and fluid.

This ratio yields the DRE:

𝐶𝑆 =𝑚𝐿𝐶𝐿∆𝑇𝐿

𝑚𝑆∆𝑇𝑆

Where ∆𝑇𝐿 = (𝑇1,𝐿 − 𝑇2,𝑒𝑞) and ∆𝑇𝑆 = (𝑇2,𝑒𝑞 − 𝑇1,𝑆)

Relative uncertainty targets were determined by developing the uncertainty magnification factor form of

the DRE.

𝑤𝐶𝑠

𝐶𝑠

2

= (𝜕𝐶𝑠

𝜕𝑚𝐿

𝑚𝐿

𝐶𝑠)2(

𝑤𝑚𝐿

𝑚𝐿)2+(

𝜕𝐶𝑠

𝜕𝑚𝑠

𝑚𝑠

𝐶𝑠)2(

𝑤𝑚𝑠

𝑚𝑠)2+(

𝜕𝐶𝑠

𝜕∆𝑇𝐿

∆𝑇𝐿

𝐶𝑠)2(

𝑤∆𝑇𝐿

∆𝑇𝐿)2+(

𝜕𝐶𝑠

𝜕∆𝑇𝑠

∆𝑇𝑠

𝐶𝑠)2(

𝑤∆𝑇𝑠

∆𝑇𝑠)2+(

𝜕𝐶𝑠

𝜕𝐶𝐿

𝐶𝐿

𝐶𝑠)2(

𝑤𝐶𝐿

𝐶𝐿)2+(

𝑤𝑟𝑎𝑛𝑑

𝐶𝑠)2

All UMFs are 1. All relative uncertainties have the same bounds. 𝑤𝐶𝑠

𝐶𝑠

2

= 6(𝑤𝑋𝑖

𝑋𝑖)2

0.1 2 = 6(𝑤𝑋𝑖

𝑋𝑖)2

𝑤𝑋𝑖

𝑋𝑖= 4.082%

This percentage is a target value for random uncertainty and each measurand’s relative uncertainty

UMF 𝐶𝐿 UMF 𝑇𝑠 UMF 𝑇𝐿 UMF 𝑚𝑠 UMF 𝑚𝐿

The mass of the liquid and the solid were limited by the size of the Dewar flask and scale uncertainty.

𝑤𝑚2 = 𝑤𝑚,𝑎𝑐𝑐

2 + 𝑤𝑚,𝑟𝑒𝑎𝑑2

𝑤𝑚 = 0.0141 𝑔

𝑤𝑚

𝑚≤ 3.5%

𝑚 ≥ 0.45 𝑔

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 𝜌𝑠𝑜𝑙𝑖𝑑 = 2.5𝑔

𝑚𝐿 and 𝜌𝑤𝑎𝑡𝑒𝑟 = 1.0

𝑔

𝑚𝐿

𝑚 = 𝜌∀

𝑚𝑠𝑜𝑙𝑖𝑑

𝜌𝑠𝑜𝑙𝑖𝑑+

𝑚𝑤𝑎𝑡𝑒𝑟

𝜌𝑤𝑎𝑡𝑒𝑟≤ 350 𝑚𝐿

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

𝑚𝐿

𝑚𝑠≤

350 𝑔

𝑚𝑠− 0.4

𝑔

𝑔

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 solid’s geometry.

The size of the Dewar requires small solid objects to ensure they are fully submerged.

Assuming the pieces are small (𝑚𝑆 ≤ 14 𝑔 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.

𝑚𝑆 ≥ 0.45 𝑔 𝑎𝑛𝑑 𝑚𝐿

𝑚𝑆≥ 4

𝑚𝐿 ≥ 1.8 𝑔

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.

𝑚𝑆 ≤ 14𝑔 𝑎𝑛𝑑 𝑚𝐿

𝑚𝑆≤

350

𝑚𝑆− 0.4 ≤

350

14− 0.4 ≤ 24.6

Therefore,

𝑚𝐿 ≤ 344.4 𝑔

𝟎. 𝟒𝟓 𝒈 ≤ 𝒎𝑺 ≤ 𝟏𝟒 𝒈 𝟏. 𝟖𝒈 ≤ 𝒎𝑳 ≤ 𝟑𝟒𝟒. 𝟒 𝒈 𝟒 ≤𝒎𝑳

𝒎𝑺≤ 𝟐𝟒. 𝟔

The liquid temperature difference is limited by the uncertainty and the range of the thermometer.

𝑤∆𝑇𝐿

∆𝑇𝐿≤ 3.5%

𝑤∆𝑇𝐿

2 = (𝜕∆𝑇𝐿

𝑇1𝐿)2(𝑤𝑇1𝐿,𝑎𝑐𝑐

2 +𝑤𝑇1𝐿,𝑟𝑒𝑎𝑑2 ) + (

𝜕∆𝑇𝐿

𝑇2)2(𝑤𝑇2,𝑎𝑐𝑐

2 +𝑤𝑇2,𝑟𝑒𝑎𝑑2 )

𝑤∆𝑇𝐿= 0.1 ℃

Omega ASTM 3964C Thermometer Values (°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.

∆𝑇𝐿≥ 2.86℃

From the Thermometer Range:

𝑇𝑚𝑖𝑛 = 25℃ and 𝑇𝑚𝑎𝑥 = 55℃

Therefore,

2.86℃ ≤ ∆𝑇𝐿≤ 30.0℃

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.

𝑤∆𝑇𝑠

∆𝑇𝑠≤ 3.5%

𝑤∆𝑇𝑠

2 = (𝜕∆𝑇𝑠

𝑇1𝑠)2(𝑤𝑇1𝑠,𝑎𝑐𝑐

2 +𝑤𝑇1𝑠,𝑟𝑒𝑎𝑑2 ) + (

𝜕∆𝑇𝑠

𝑇2)2(𝑤𝑇2,𝑎𝑐𝑐

2 +𝑤𝑇2,𝑟𝑒𝑎𝑑2 )

𝑤∆𝑇𝑠= 0.711 ℃

Enviro-Safe Thermometer Values (°C)

Range -20 to 110

Accuracy 0.5

Resolution 1

Readability 0.5

The thermometer’s lower limit and the practical limit of the boiling point of water determine the range of solid

temperatures.

∆𝑇𝑆≥ 20.31℃

𝑇𝐿,𝑚𝑖𝑛 = 25℃ and 𝑇𝐿,𝑚𝑎𝑥 = 100℃

Therefore,

20.31℃ ≤ ∆𝑇𝑆≤ 75.0℃

This is true for the solid if the accurate, Omega ASTM thermometer is utilized for

the equilibrium temperature (𝑇2,𝑒𝑞) and the Enviro-Safe thermometer measures

the elevated temperature (𝑇1,𝑆)

A design space was developed based on sensor and relative uncertainty constraints.

Design Point

(72.5, 6)

ΔTL=2.5°C

0

2

4

6

8

10

12

0 20 40 60 80

mL/m

S (

-)

ΔTS (°C)

ΔTL=2°C at Max CS

ΔTL,max=3.14°C at Min CS

Using 5% relative uncertainty in ΔTL

Our experimental design can effectively determine 𝐶𝑆 within a 10% uncertainty.

Parameter Representative

Value Systematic Uncertainty

Relative Uncertainty

(%) UMF

RSSC (%)

UPC (%)

∆𝑇𝐿 (°𝐶) 2.5 ± 0.1 4 1 4 54.7

𝐶𝑆𝑟𝑎𝑛𝑑𝑜𝑚

𝐽

𝑔 𝐾 - - 3.5 1 3.5 41.9

∆𝑇𝑆 (°𝐶) 72.5 ± 0.711 0.980 1 0.980 3.29

𝑚𝑆 (𝑔) 10 ± 0.0141 0.141 1 0.141 0.0684

𝑚𝐿 (𝑔) 60 ± 0.0141 0.0235 1 0.0235 0.00189

𝐶𝐿 𝐽

𝑔 𝐾 4.179 𝑛𝑒𝑔𝑖𝑙𝑖𝑔𝑖𝑏𝑙𝑒 - - - -

𝐶𝑆

𝐽

𝑔 𝐾 Expected value: 0.87 - -

𝑤𝐶𝑆

𝐶𝑆: 5.41 100

The design utilizes simple, practical components.

Omega ASTM 64C Thermometer

Enviro-Safe (-20℃ to 100℃) 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 𝐽

𝑔∙𝐾 .

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

∆𝑇𝐿,𝑚𝑎𝑥 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 −50℃ 𝑡𝑜 1050℃

Resolution 0.1℃

Readability 0.05℃

Accuracy ±1.5% × 𝑟𝑒𝑎𝑑𝑖𝑛𝑔 + 2℃

At ∆𝑇𝑆,𝑚𝑎𝑥= 75℃ where the readings are 𝑇1,𝑆 = 100℃ and 𝑇2,𝑒𝑞 = 25℃

𝑤∆𝑇𝑠

2 = 0.015 ∗ 100 + 2 2

+ 0.05 2 + 0.015 ∗ 25 + 2 2

+ 0.05 2

𝑤∆𝑇𝑆= 4.23℃

𝑤∆𝑇𝑆

∆𝑇𝑠=

4.23℃

75℃= 5.64%

UMF derivations

UMF derivations

A model exists to calculate specific heat of water at varying temperatures.

𝐶𝐿 (15°𝐶) = 4.1855𝐽

𝑔 °𝐶

∗

*International Committee for Weights and Measures (Paris 1950)

𝐶𝐿 = 0.996185 + 0.0002874𝑇𝐿 + 100

100

5.26

+ 0.011160 × 10−0.036𝑇𝐿 𝐶𝐿 (15°𝐶)

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/

(g°C

)

Temperature (°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 25°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 𝑇1,𝑆 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 𝑇2,𝑒𝑞

The DRE is a simple variation of the exchange of energy between two substances.

Energy contained in a material:

𝐸 = 𝑚ℎ = 𝑚𝐶∆𝑇

From the conservation of energy:

𝐸𝑖𝑛 − 𝐸𝑜𝑢𝑡 = 𝑄𝑛𝑒𝑡 + 𝑊𝑛𝑒𝑡 + (−𝐸𝑠𝑜𝑙𝑖𝑑) + 𝐸𝑙𝑖𝑞𝑢𝑖𝑑

There is no work, heat transfer, or net energy change of the system so:

𝐸𝑙𝑖𝑞𝑢𝑖𝑑 = 𝐸𝑠𝑜𝑙𝑖𝑑

𝑚𝐿𝐶𝐿∆𝑇𝐿= 𝑚𝑆𝐶𝑆∆𝑇𝑆

Solving for the specific heat of the solid

𝐶𝑆 =𝑚𝐿𝐶𝐿∆𝑇𝐿

𝑚𝑆∆𝑇𝑆

The specific heat of the Dewar could be a contributing factor in the experiment

Like the liquid, the Dewar can absorb energy as well:

𝐸𝑠𝑜𝑙𝑖𝑑 = 𝐸𝐿𝑖𝑞𝑢𝑖𝑑 + 𝐸𝐷𝑒𝑤𝑎𝑟

𝑚𝑆𝐶𝑆∆𝑇𝑆= 𝑚𝐿𝐶𝐿∆𝑇𝐿 + 𝑚𝐷𝐶𝐷∆𝑇𝐷

The new DRE becomes:

𝐶𝑆 =𝑚𝐿𝐶𝐿∆𝑇𝐿

𝑚𝑆∆𝑇𝑆+

𝑚𝐷𝐶𝐷∆𝑇𝐷

𝑚𝑆∆𝑇𝑆

Since specific heats are material properties, the Dewar itself could impact the

temperature changes and the determination of 𝐶𝑠.

The specific heat of the Dewar could be a contributing factor in the experiment

Since the Dewar is glass:

𝐶𝐷 = 0.84 𝐽

𝑔 ∙ 𝐾

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,

𝑚𝐷𝐶𝐷∆𝑇𝐷

𝑚𝑆∆𝑇𝑆

Is negligible as both 𝑚𝐷 and ∆𝑇𝐷 are small.

The DRE was utilized to identify the ∆𝑇𝐿,𝑚𝑎𝑥value corresponding to our Design Space.

Given that our mass ratio was bounded by 4 and a ∆𝑇𝑆,𝑚𝑎𝑥of 75 °C we can solve for the

minimum allowable ∆𝑇𝐿,𝑚𝑖𝑛. Using the minimum value of 0.7 𝐽

𝑔∙𝐾

∆𝑇𝐿,𝑚𝑎𝑥 =𝐶𝑆,𝑚𝑖𝑛𝑚𝑆∆𝑇𝑆

𝑚𝐿𝐶𝐿(𝑇1,𝐿=25°𝐶)

Also given the Cox, Wagman, and Medvedev relationship at T1,L=25 °C

∆𝑇𝐿,𝑚𝑎𝑥 =0.7

1

4(75)

(4.1793)= 3.14°C