Heat Transfer Conduction

Heat Transfer ME 419

LAB #1

Transient Heat Conduction Lab

Victor Li (U91093053)

Nelson Chin

JingJiang Liu

Yaojun Shan

Section A1

2015 September 24 1:00 PM

Professor Duan

Erin Arai

1. Introduction

The purpose of this lab is to measure and model the transient thermal response of two

metal spheres of different thermal conductivities and same convection coefficients using either

the lumped capacitance method or the series solution. The lumped capacitance system assumes

that the solid possesses negligible spatial thermal gradients, where the solid exhibits the same

temperature everywhere. When spatial thermal gradients exist within a solid body, the lumped

capacitance method is invalid. Instead, a series solution approach, which incorporates more

complex mathematical formulations, must be taken to obtain the results. To validate the two

models, measurements are taken using two individual spheres consisting of copper and stainless

steel and two thermocouples located at the surface and center of each sphere. The copper sphere

has a Biot number that is much less than one, satisfying the lumped system method criteria. The

steel sphere, however, has a Biot number greater than one, signifying use of the series solution.

The varying parameters of the experiment are the temperatures of the two water baths that the

steel and copper spheres are submerged into. Therefore, two heating and two cooling curves will

ultimately be generated for each sphere by simultaneously transferring one sphere from the hot to

cold bath and the other from the cold to hot bath. The experiment involves four trials, with each

trial concluding once steady state is reached. Such an exposure of a solid body to a sudden

change in temperature allows for the examination of its thermal properties through the

manipulation of obtained temperature data.

2. Theory

Two methods of evaluating different cases of transient conduction, namely the lumped

system and the series solution, are utilized in this experiment. The Biot number, as shown below

in equation (1), is calculated to determine the appropriate method used to study the temperature

distribution within a solid body.

Bi = hrk (1)

The Biot number depends on the heat transfer coefficient,h, characteristic length or radius of a

sphere, r, and the thermal conductivity, k. This dimensionless number physically represents the

ratio of internal conduction thermal resistance to the external convection thermal resistance.

Thermal resistance is a heat property and measurement of temperature difference by which an

object resists heat flow. Conduction thermal resistance, displayed in equation (2), measures the

difficulty of heat to flow within a solid, while convection thermal resistance, as seen in equation

(3), measures the difficulty of thermal flow through a liquid where the solid is located.

Rth = L

kA (2)

Rth = 1

h A s (3)

The conduction resistance depends on the thickness, L, thermal conductivity, k, and cross

sectional area, A. Similarly, the convection resistance depends on the heat transfer coefficient, h,

and surface area, A s. The smaller the thermal resistance, the easier it is for heat to flow through

the solid or liquid. When the solid is a good conductor of heat, the internal thermal resistance is

small compared to the external thermal resistance, and a system, like the copper sphere, can be

modeled as a lumped system. This also means the Biot number is significantly less than 1. For

such a case, the temperature exponentially decays as a function of time, as Equation (4)

illustrates.

θ =T−T ∞

T i−T ∞=¿ e−t / τ (4)

Here, τ represents a positive quantity known as the time constant whose unit is seconds and is

equal to that shown in equation (5).

τ=ρV c p

h A s(5)

The time constant is dependent on the density,ρ , volume, V , specific heat,c p , heat transfer

coefficient, h, and surface area, A s.

In the case where the solid does not conduct heat relatively well, the temperature will vary based

on radial position and time, creating a spatial thermal gradient within the body. The Biot number

will consequently be greater than 1 because the internal conduction thermal resistance is large

compared to the external convection thermal resistance. This means that steel is not an efficient

conductor of heat compared to copper. Therefore, the lumped system solution cannot be used

here. Instead, one must apply the series solution shown below.

θ=T−T ∞

T i−T ∞=∑

n=1

∞

An e−λ n2 Fo f n (6)

The terms in the summation quickly decline due to the exponential decay function, especially

when the dimensionless Fourier number, Fo, is large. The Fourier number, Fo, represents the

ratio of conduction rate to the rate of thermal energy storage, as represented in equation (7).

τ=αtL2 (7)

The Fourier number depends on the thermal diffusivity, Fn, characteristic time, t , and length

through which conduction occurs, L.

If the Fourier number is greater than 0.2, the first-term approximation method is valid and,

equation (6) reduces to equation (8) for a sphere. Furthermore, Fn = 1 at the center of an object as

derived in equation (10).

θ=T (r , t)−T ∞

T i−T ∞=A1 e−λ1

2 Fo f n(8)

Fn for a sphere =sin( λ1 r

r0)

λ1r

r 0

(9)

limx→ o

sin (x)x

= 1 (10)

The Biot number is used to find the values of λ1 and A1 from Table 5.2 Using the one-term

approximation to find temperature at the center, equation (8) reduces to equation (11) for a

sphere.

θ=T 0−T ∞

T i−T ∞ = A1 e−λ1

2 Fo (11)

The two thermocouples, one in the middle and one on the surface of the sphere, will ultimately

reveal if the assumption that the copper sphere and stainless steel sphere temperature

distributions can be modelled using lumped capacitance and series solution method is true. If the

copper sphere is truly a lumped system, then the two temperatures measured will be nearly the

same. Similarly, if the steel sphere is not a lumped system, there will be a noticeable change

between surface and center temperatures with a temperature gradient in between.

3. Experimental Methods

Equipment used:

A copper sphere

A steel sphere Four thermocouples, two for each sphere National Instruments LabView Software Cold and hot water bath Liquid-in-gas thermometer

To initiate the experiment, the temperature of the cold and hot bath must be verified to be at 30 and

60 degrees Celsius with a liquid-in-gas thermometer. The LabView software, which converts voltages

from the thermocouples of both spheres to temperatures and records values until steady state of both

spheres is reached, is started. The aluminum and steel spheres are maintained at steady state in the cold

and hot bath respectively prior to the first trial of measurements. One individual is assigned the task of

starting and stopping the software, while two individuals are responsible for switching the two spheres

from the water baths. Once the software is started by pressing ‘record’, the copper sphere is quickly

transported from the cold to hot bath, and the steel sphere, at the same time, is taken out of the cold bath

and placed in the hot bath. The time that the spheres are in the air must be minimized to maintain the

same temperature as each respective water bath source. The software is stopped once the temperatures of

the two spheres remain constant, which takes several minutes. This procedure is iterated four times and,

as a result, two heating and two cooling data sets are generated for each sphere. The four sets of data

include the temperature at the surface and center of each sphere as a function of time.

4. Results

Property Copper Sphere

(diameter= 4.775 cm)

Steel Sphere

(diameter= 4.775cm)

k (W/m K) 401 13.4

ρ (kg /m3) 8933 8238

c p(J/kg K) 385 468

Table 1: Properties of copper and steel spheres used in calculating the heat transfer coefficients.

The two heating and cooling curves of copper and steel sphere centers and surfaces are plotted

below, where temperature is a function of time in figures 1 through 4. Next, the natural log of the

dimensionless temperatures are calculated from the cooling curve raw data, as illustrated in

figures 5 and 6. From figures 5 and 6, the linear regime of each copper and steel sphere

dimensionless temperature cooling curve plots are extracted, shown in figures 7 through 14, to

calculate the heat transfer coefficients of the centers and surfaces of each sphere. The method of

accomplishing this will further be elaborated in detail below.

Figure 1: Heating curve generated from raw temperature data of trial 1 copper and trial 2 steel sphere surfaces and centers.

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

70

Heating Curve of Copper and Steel Spheres

Copper Surface

Copper Center

Steel Surface

Steel Center

Time (s)

Tem

pera

ture

(C)

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

70Cooling Curve of Copper and Steel Spheres

Steel Surface

Steel Center

Copper Surface

Copper Center

Time (s)

Tem

pera

ture

(C)

Figure 2: Cooling curve generated from raw temperature data of trial 2 copper and trial 1 steel sphere surfaces and centers.

Figure 3: Heating curve generated from raw temperature data of trial 3 copper and trial 4 steel sphere surfaces and centers.

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

70Heating Curve of Copper and Steel Spheres

Copper Surface

Copper Center

Steel Surface

Steel Center

Time (s)

Tem

pera

utre

(C)

Figure 4: Cooling curve generated from raw temperature data of trial 4 copper and trial 3 steel sphere surfaces and centers.

Figure 5: Dimensionless cooling curve generated from calculating the natural logarithm of the dimensionless temperature, ϴ, and plotting it as a function of time for trial 2 copper and trial 1 steel sphere surfaces and centers.

0 50 100 150 200 250 300 350 400-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Dimensionless Temperature Cooling Curve of Copper and Steel Spheres

Steel Surface Trial 1

Steel Center Trial 1

Copper Surface Trial 2

Copper Center Trial 2

Time (s)

ln (ϴ

)

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

70

Cooling Curve of Copper and Steel Spheres

Steel Surface

Steel Center

Copper Surface

Copper Center

Time (s)

Tem

pera

ture

(C)

Figure 6: Dimensionless cooling curve generated from calculating the natural logarithm of the dimensionless temperature, ϴ, and plotting it as a function of time for trial 4 copper and trial 3 steel sphere surfaces and centers.

Figure 7: Linear regime of the natural log of dimensionless temperature cooling curve for the trial 2 copper sphere surface shown in figure 5 above. A trend line fitted to this linear region and its equation are shown.

0 50 100 150 200 250 300 350 400-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Dimensionless Temperature Cooling Curve of Copper and Steel Spheres





Time (s)

ln(θ

)

50 55 60 65 70-4

-3

-2

-1

0

f(x) = − 0.0296478837844917 x − 1.18042162526703

Linear Regime of Copper Sphere Dimensionless Temperature Cooling Curve


Linear (Copper Surface Trial 2)

Time (s)

ln(θ

)

Figure 8: Linear regime of the natural log of dimensionless temperature cooling curve for the trial 2 copper sphere center shown in figure 5 above. A trend line fitted to this linear region and its equation are shown.

Figure 9: Linear regime of the natural log of dimensionless temperature cooling curve for the trial 4 copper sphere surface shown in figure 6 above. A trend line fitted to this linear region and its equation are shown.

0 10 20 30 40 50 60 70 80-2.5

-2

-1.5

-1

-0.5

0

f(x) = − 0.0230474652352809 x − 0.185751868146651



Linear (Copper Center Trial 2)

Time (s)

ln(θ

)

45 55 65 75-4

-3.5

-3

-2.5

-2

-1.5

-1

f(x) = − 0.0311010789450932 x − 1.25561326518251




Time (s)

ln(θ

)

45 55 65 75-4

-3.5

-3

-2.5

-2

-1.5

-1

f(x) = − 0.0311107873986279 x − 1.24148236913552



Linear (Copper Surface Trial 4)

Time (s)

ln(θ

)

Figure 10: Linear regime of the natural log of dimensionless temperature cooling curve for the trial 4 copper sphere center shown in figure 6 above. A trend line fitted to this linear region and its equation are shown.

From figures 7 through 10, the slopes of the trend lines can be retrieved and used in the

calculation of the heat transfer coefficient for the copper sphere surfaces and centers in trials 2

and 4. As can be seen, the slopes of the trend lines for the centers and surfaces are, for the most

part, similar. Table 2 below displays the slopes of each respective trial.

Trial Slope, m

Copper Surface Trial 2 -0.0296

Copper Center Trial 2 -0.023

Copper Surface Trial 4 -0.0311

Copper Center Trial 4 -0.0311

Table 2: The slope of copper sphere surface and center trend lines for cooling trials used to calculate the heat transfer coefficient.

45 55 65 75-4

-3.5

-3

-2.5

-2

-1.5

-1

f(x) = − 0.0311010789450932 x − 1.25561326518251




Time (s)

ln(θ

)

Equation (4), also known as the lumped capacitance method, can be manipulated and rewritten

into a linear equation, where the slope can be equated to the values above in table 2. The

following steps detail the approach to obtain the heat transfer coefficient for the surface and

center cases of each trial.

ln (θ) ¿ln(e−t / τ)= −t /τ

Therefore, the slope is equal to −1/τ and τ=ρV c p

h A s. The slopes can now be equated to−1/τ .

Taking the slope of copper surface trial 2, the heat transfer coefficient can be found.

−1τ

=−0.0296

τ= 10.0296

ρV c p

h A s=

ρ( 4 π r3

3)c p

h(4 π r 2)=

ρ( r3)cp

h= 1

0.0296

h=0.0296 ρ( r3 )c p=0.0296∗8933∗( 0.04775

3 )(385 )=¿ 1620 W/m2*K

The other heat transfer coefficients listed in table 3 below are obtained in the same manner as

above.

Trial Heat Transfer Coefficient, h (W/m2*K)

Copper Surface Trial 2 1620

Copper Center Trial 2 1259

Copper Surface Trial 4 1702

Copper Center Trial 4 1702

Table 3: The heat transfer coefficients calculated from cooling curves of copper sphere surfaces and centers in trials 2 and 4.

30 35 40 45 50 55 60 65 70 75 80-2

-1.5

-1

-0.5

0

f(x) = − 0.0283436062544533 x + 0.208746453560516

Linear Regime of Steel Sphere Dimensionless Temperature Cooling Curve

Steel Surface Trial 1Linear (Steel Surface Trial 1)

Time (s)

ln(θ

)

Figure 11: Linear regime of the natural log of dimensionless temperature cooling curve for the trial 1 steel sphere surface shown in figure 5 above. A trend line fitted to this linear region and its equation are shown.

30 40 50 60 70 80 90 100 110-3

-2.5

-2

-1.5

-1

-0.5

0

f(x) = − 0.0284217056494812 x + 0.303315211297358


Steel Center Trial 1Linear (Steel Center Trial 1)

Time (s)

ln(θ

)

30 35 40 45 50 55 60 65 70 75 80-2

-1.5

-1

-0.5

0

f(x) = − 0.0283436062544533 x + 0.208746453560516



Time (s)

ln(θ

)

Figure 12: Linear regime of the natural log of dimensionless temperature cooling curve for the trial 1 steel sphere center shown in figure 5 above. A trend line fitted to this linear region and its equation are shown.

Figure 13: Linear regime of the natural log of dimensionless temperature cooling curve for the trial 3 steel sphere surface shown in figure 6 above. A trend line fitted to this linear region and its equation are shown.

30 35 40 45 50 55 60 65 70 75 80-2.5

-2

-1.5

-1

-0.5

0

f(x) = − 0.0287993370953905 x + 0.332930589349517


Steel Center Trial 3Linear (Steel Center Trial 3)

Time (s)

ln(θ

)

30 40 50 60 70 80 90 100 110 120-3

-2.5

-2

-1.5

-1

-0.5

0

f(x) = − 0.0259933744791697 x + 0.0841194147290059



Time (s)

ln(θ

)

Figure 14: Linear regime of the natural log of dimensionless temperature cooling curve for the trial 3 steel sphere center shown in figure 6 above. A trend line fitted to this linear region and its equation are shown.

From figures 11 through 14, the slopes of the trend lines can be retrieved and used in the

calculation of the heat transfer coefficient for the steel sphere surfaces and centers in trials 1 and

3. As can be seen, the slopes of the trend lines for the centers and surfaces are quite similar..

Because steel is not a decent conductor compared to copper, the lumped capacitance method

cannot be used to model the sphere. Instead, the first term approximation will be used to

calculate the heat transfer coefficients associated with the center and surface of the steel sphere.

For the first term approximation to be accurate, the Fouriers number, as seen in equation (7),

should be greater than 0.2. Thus, similar to the approach applied to the copper sphere, a trend

line must be fitted to the linear regime of the natural log plots where the time yields a Fouriers

number greater than 0.2. The calculation below details the minimum time needed to satisfy this

condition.

Fo= αtL2 =

αtr 2

α= kρ∗c p

= 13.48238∗468

=3.4756558975∗10−6

t= Fo∗r2

α=

0.2∗( 0.047752

)2

3.4756558975∗10−6 =32.8 s

Therefore, the linear regime should be chosen so that the minimum time approximately exceeds

32.8s, giving a Fouriers number greater than 0.2. Table 4 below displays the slopes of each

respective trial.

Trial Slope

Steel Surface Trial 1 -0.0283

Steel Center Trial 1 -0.0284

Steel Surface Trial 3 -0.026

Steel Center Trial 3 -0.0288

Table 4: The slope of steel sphere surface and center trend lines for cooling trials used to calculate the heat transfer coefficient.

Employing equations 8, 11, and 1 in the theory section, values in table 4, table 4-2 in the textbook, the heat transfer coefficient can be calculated following this method detailed below.

θ=T 0−T ∞

T i−T ∞ = A1 e−λ1

2 Fo (11) (center of sphere)

θ=T (r , t)−T ∞

T i−T ∞=A1 e−λ1

2 Fo

sin ( λ1rr 0

)λ1 r

r0

(8) (one term approximation for

sphere)

From (11) for the center of a sphere, taking the natural log of both sides:

ln (θ )=ln ( A1 )−λ12 Fo

ln (θ )=ln ( A1 )−λ12 αt

r2 =ln ( A1 )−λ1

2 ktρc p r2

Recalling that the slope intercept form of a line is y=mx+b, where m is the slope, the slope of the

equation can be found.

m=− λ1

2 kρ c pr 2

Solving for λ1 and substituting known values for steel and from trial 1,

λ1=√−mρ c pr2

k=√−(−0.0284)(8238)(468)( 0.04775

4

2

)

13.4

λ1=2.158161652

From table 4-2 values for a sphere, interpolate to obtain corresponding Biot number.

2.2889−2.02882.158161652−2.0288

= 3−2Bi−2

Bi=2.497353018=hrk

=h( 0.04775

2 )13.4

h=1402W /m2∗K

Following the same approach, the heat transfer coefficient for the surface of a steel sphere can be

calculated using equation (8) by taking the natural log of both sides.

ln (θ)=ln (A1

sin( λ1rr0 )

λ1r

r0)− λ1

2 Fo=−λ12 Fo+constant

As one can see, the slopes for the surface and center of the steel sphere have the same

expression.

−λ12 Fo=

−λ12kt

ρ c pr2

m=− λ1

2 kρ c pr 2

Solving for λ1 and substituting known values for steel and from trial 1.

λ1=√−mρ c pr2

k=√−(−0.0283)(8238)(468)( 0.04775

4

2

)

13.4

λ1=2.154358721

From table 4-2 values for a sphere, interpolate to obtain corresponding Biot number.

2.2889−2.02882.154358721−2.0288

= 3−2Bi−2

Bi=2.4827324913=hrk

=h( 0.04775

2 )13.4

h=1393W /m2∗K

The Biot number, as expected, is greater than 1 in both cases, verifying that the lumped

capacitance method cannot be used. The values for the trial 3 center and surface heat coefficient

values are obtained following the above method. Table 5 displays the values for each trial.

Trial Heat Transfer Coefficient, h (W/m2*K)

Steel Surface Trial 1 1393

Steel Center Trial 1 1402

Steel Surface Trial 3 1201

Steel Center Trial 3 1434

Table 5: The heat transfer coefficients calculated from cooling curves of copper sphere surfaces and centers in trials 1 and 3.

From the λ1 values calculated above for the steel sphere surface and center in trial 1, a

corresponding A1 value can be obtained from interpolation using table 4-2 in the textbook.

Consequently, the first term approximation is used to model a best fit curve for the original

cooling curve of the steel sphere surface and center from Figure 2. The calculation is detailed

below.

For the center of the steel sphere in Trial 1:

θ=T 0−T ∞

T i−T ∞ = A1 e−λ 1

2 Fo

T 0=T ∞+(T i−T ∞)( A1 e−λ12 Fo)

T 0=T ∞+(T i−T ∞)(A1 e−λ 1

2∗k∗tρ∗cp∗r 2

)

2.2889−2.02882.158161652−2.0288

=1.6227−1.4793A1−1.4793

λ1=2.158161652; A1=1.550620496

T 0=30+(63.921135−30)(1.550620496 e−2.1581616522∗13.4∗t8238∗468∗0.04775 2/4 )

T 0=30+52.59880718∗e−0.0284 t

For the surface of the steel sphere in Trial 1:

T (r , t)−T ∞

T i−T∞=A1 e−λ1

2 Fo

sin ( λ1rr 0

)λ1 r

r0

T (r , t )=T ∞+(T i−T ∞)( sin( λ1 rr0 )

λ1r

r 0)(A1e

− λ12∗k∗t

ρ∗cp∗r2

)

λ1=2.154358721; A1=2.154358721;r=r0

T (r , t )=30+(63.927486−30)( sin (2.154358721 )2.154358721 )(2.154358721 e

−2.1543587212∗13.4∗t8238∗468∗0.047752 /4 )

T (r , t )=30+20.35075545∗e−0.0283t

Figures 15 and 16 below show the best fit models for the steel sphere center and surface

respectively for trial 1. Trial 3 best fit models are not shown, as the method is the same as that

outlined above and results are quite similar.

Figure 15: Best fit model for the steel sphere center of trial 1 using the first term approximation.

Figure 16: Best fit model for the steel sphere surface of trial 1 using the first term approximation.

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

70

80

90

Center First Term Approximation Best Fit


Best Fit Center

Time (s)

Tem

pera

ture

(C)

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

70

Surface First Term Approximation Best Fit


Best Fit Surface

Time (s)

Tem

pera

ture

(C)

From the heat transfer coefficients in tables 3 and 5, the Biot number can be calculated using

equation 1. Table 6 displays these figures.

Trial Biot NumberCopper Surface Trial 2 0.096

Copper Center Trial 2 0.075

Copper Surface Trial 4 0.164

Copper Center Trial 4 0.164

Steel Surface Trial 1 2.482

Steel Center Trial 1 2.498

Steel Surface Trial 3 2.140

Steel Center Trial 3 4.147

Table 6: The calculated Biot number of all copper and steel spheres used in the cooling trials.

5. Discussions

The results obtained from the measured raw temperature data of the two spheres in each

cooling trial reveal the importance of selecting the correct model used in representing the

temperature distribution within a solid body. For the copper sphere, the lumped capacitance

method is valid because, as seen from table 6, the Biot number is much less than 1. This means

that copper possesses little conduction internal resistance compared to external convective

resistance and can be modeled as an object with uniform temperature throughout, even though

there may be small differences. The heat transfer coefficients of the copper sphere in table 3

confirm the validity of applying the lumped capacitance method here. For the second trial, the

heat transfer coefficients of the surface and center of the copper sphere are 1620 and 1259 W/m2

*K respectively, exhibiting some discrepancy and a percent difference of approximately 25%.

This can be attributed to several possible sources error. During the experiment, the criteria that

the time of the spheres in the air must be minimized may not have been satisfied, changing the

surface temperature of the ball slightly. Additionally, the method in which the spheres were

mounted in the bath was unsteady and caused movement of the body during several of the trials.

This may have caused a shift in position and caused such a difference in the heat transfer

coefficients. However, the largest factor affecting results potentially could have been the manner

in which the linear regime of the dimensionless temperature cooling curve was selected. Since a

lot of freedom is given here, selecting fewer and more data points affects the slope of the trend

line fitted to the regime. Because the slope is directly used to solve for the heat transfer

coefficients, a large number of values were possible. The heat transfer coefficients for trial 4 of

the surface and center of the copper sphere, on the other hand, are identical and demonstrate the

high conductivity of the material.

The steel spheres cannot be modelled using the lumped capacitance method due to its

high conduction thermal resistance. Therefore, a temperature gradient will develop between the

surface and center of the sample, where temperature varies with time and position. To further

confirm such a claim, the Biot number for the steel sphere surfaces and centers in trials 1 and 3

shown in table 6 exceed 1. In terms of the heat transfer coefficients, trials 1 and 3 exhibit nearly

identical values, which is surprising due to the relatively high conductive thermal resistance of

steel compared to that of copper. This may be attributed to the aforementioned sources of error.

The largest potential factor affecting the values, again, is the way in which the linear trend is

fitted to the linear regime, where picking different number of points within the region affects the

value of the slope. Furthermore, the trend fit can only be applied to a linear region where the

time range yields a Fouriers number of 0.2 for the one term approximation to be accurate. This

step is critical and will give inaccurate results if done incorrectly. Thus, a larger number of terms

can be used in the calculation and model if this lab were to be performed again for more accurate

results. The purity of materials could also be considered as a source of error. The fabrication of a sphere

containing trace elements of other materials could give different thermal property values. Thus, the

impurity of the materials could also influence the heat transfer coefficient values that calculated from

each trial. Because the surface temperature should always be higher than that of the center, the best fit

models in figures 15 and 16 demonstrate the contrary, where the initial starting temperatures of the center

and surface are 81.1℃ and 49.6℃ based on the first term approximation model. This is due to the fact

that the lambda values found for trial 1 surface and center were found to be 2.154358721 and

2.158161652, which will yield a very similar Biot number and, in turn, a very similar h value.

Very similar lambda values will cause the temperature at the center to be higher than that of the

surface due to the extra sine function in equation 8. In order to produce accurate results, the

experiment needs to be repeated many times to reduce as many possible sources of error as

possible and yield more sets of data to manipulate.

6. Conclusion

In this experiment, a lesson on making the necessary assumptions and deciding which model to

employ to two different materials was taught. The heat transfer coefficient values and, consequently, the

Biot numbers obtained from the analysis verified that our assumptions were correct. The lumped

capacitance method gave a decent approximation for the copper sphere due to the small differences in the

cooling curve temperatures at the center and the surface. Similarly, the series solution can be used to

model the temperature distribution within the steel sphere, as the Biot number exceeds 1. The importance

of meeting the criteria of each of these two methods was also taught in this experiment, where the Biot

number reveals which method to employ. If the series solution must be used, one must ensure the

accuracy of the first term approximation by confirming that the Fouriers number is greater than 0.2.

Otherwise, more terms or the full series solution must be used. Although several discrepancies existed

between the heat transfer coefficients of copper and steel, possible sources of error were identified and

can be minimized in future experiments. Furthermore, a more methodical method of choosing a linear

regime in the dimensionless cooling curves of copper and steel can be developed for even more accurate

results. This experiment ultimately revealed that different materials possessing various geometries and

thermal properties will give rise to different temperature distributions and heat transfer coefficients. The

lessons taught in this experiment are exceptionally applicable to an engineer because the skill to choose

an appropriate model and make accurate assumptions to precisely describe the temperature distribution

and obtain key thermal values of a specific geometric object consisting of any material is critical in the

design and optimization of an efficient thermal system.

References

Cengel, Yunus A., and Afshin J. Ghajar. Heat and Mass Transfer: Fundamentals & Applications. New York: McGraw Hill, 2015. Print.

Prof. D. Wroblewski, Prof. M. Isaacson, and G. Dewar, 1999, Lab 1: Transient Heat Conduction handout, Experiment,; Revised by D. Kelly, M. Raymond, M. Dvilyanskiy, 2002; B. Vattiat, J. Estano, D. Campbell, 2005; D.Karabacak, 2007; B. Nagpure and Prof. M. Isaacson, 2009; Prof. A. Schmidt, J. Estano, D.Campbell, 2012.

Heat Transfer Conduction

Documents

Transcript of Heat Transfer Conduction