THERMO/FLUIDS LAB ME 415

Design Project:Transient Heat Conduction

In a Semi-Infinite Plate

ME 415 LabInstructor: Dr. RossGroup Members: Ezcurra, Johnson, GibsonDate of Experiment: 5/6/02Date of Report: 5/20/02

Ezcurra: pp 9-21Johnson: pp 35-48 (all hand calculations), 11-18Gibson: pp 1-8, 11-15, 23-25

TABLE OF CONTENTS

PageAbstract 3

Theory 4

Published Results 7

Equipment 8

Procedure 9

Data for Calculations 10

Tabulated Results with Graphs 11

Discussion 18

Diagrams and Schematics 19

References 21

Equations Used 23

Data for Each Trial 24

Original Data Sheets 25

Original Hand Calculations 35

2

Abstract

In this lab, a stainless steel rod one foot in length was heated to 100C and then

cooled with ice water. The ice water was pumped through a special fitting attached at

one end of the rod that cooled only the very end of the bar. Temperatures at five nodes

were taken in 30-second intervals up to ten minutes. Actual temperatures were closer to

theoretically values closer to the end of the bar with the fitting. Temperatures digressed

more at further nodes. The maximum difference between theoretical and actual

temperatures was approximately 12K. More detailed results are provided in the

Tabulated Results section of this report.

3

Theory

Under certain circumstances a bar of steel only 12” can act theoretically as if it

were infinite in length. This rod can be considered semi-infinite in length because it

provides the practicality of simulating situations where a variable, such as temperature in

this case, would need to be considered over an infinite length. Despite our rod only being

12” long, it is theoretically similar to a rod of infinite length. This allows us to accurately

estimate the temperature distribution throughout the entire rod over a period of time.

Temperature distribution was calculated using the finite element method. The bar

was divided into ten nodes, each with identical lengths. Each node has a formula to

calculate its temperature depending on both the node after it and before it. In other

words, one cannot calculate the temperature at one point without analyzing temperatures

before and past the point. To prevent a perpetual loop of calculations, the last node is

counted twice so that one does not need to perform an infinite amount of calculations.

Although this method works well, it does have a disadvantage in that a stability

requirement must be met. Usually a very small or uneven time interval must be used in

order to use finite element analysis. The time interval for this experiment was only 30

seconds making recording temperature values a bit of a challenge.

Even though calculations were done with ten nodes, only the first five were

considered for analysis. A sixth thermocouple was added to the end opposite to the

fitting where cooling was taking place to ensure we had an infinite rod. The five

equations shown on the proceeding page were used to calculate the temperatures for the

first five nodes.

4

Fo represents the Fourier number and Bi represents the Biot numbers. They are

both dimensionless numbers that help to simplify the equations. The equations to solve

for both of them are provided below:

Fourier Number

Biot Number

Since water was used as a cooling agent, it was necessary to calculate the

convection h from the water. However, this could not be done directly. The velocity

was calculated first by using the cross sectional area and the known flow rate.

Velocity of water

From there a corresponding Reynold’s number was found. Care was taken to

ensure that a laminar flow was used for cooling.

5

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111

541

5

4531

4

3421

3

2311

2

FoTTTFoT

FoTTFoT

FoTTTFoT

FoTTTFoT

FoTTTFoT

pppp

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pppp

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pppp

pwa ter

pp TFoTBiFoFoFoBiTT 211

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Reynold’s Number

Once Reynold’s number was found, the Nusselt number could be calculated using

the known Prandtl number from the textbook.

Nusselt Number

Convection can then be determined knowing the Nusselt number by solving the

below equation for h.

Nusselt Number

Once the convection is known than calculations for estimating temperature can be

performed. Hand calculations are provided in the appendix of this report. Complete

spreadsheet calculations for each of the ten nodes are provided in the Tabulated Data

section

6

Published Results

The table below shows calculations for what should be obtained theoretically.

The exact formulae used are provided in the handwritten calculations near the end of this

report. There is also a separate section for formulae used in this experiment.

Calculations were done for ten nodes for a maximum time of ten minutes.

7

Theoretical Calculations Date of Experiment: 5/6/02Group Members: Ezcurra, Gibson, Johnson

diffusivity L=delta X Fo h density Cp k Bi Re Nu3.75944E-06 0.03 0.125315 133.7148 8042.75 476.25 14.4 0.278572 1724.80 3.66

Step P T (sec) To (K) T1 (K) T2 (K) T3 (K) T4 (K) T5 (K) T6 (K) T7 (K) T8 (K) T9 (K) T10 (K)0 0 273.15 369 370 374 376 378 379 380 381 382 3831 30 273.15 366 370 374 376 378 379 380 381 382 3832 60 273.15 363 370 374 376 378 379 380 381 382 3833 90 273.15 361 369 373 376 378 379 380 381 382 3824 120 273.15 359 368 373 376 378 379 380 381 382 3825 150 273.15 357 367 373 376 377 379 380 381 382 3826 180 273.15 355 367 372 375 377 379 380 381 382 3827 210 273.15 354 366 372 375 377 379 380 381 382 3828 240 273.15 353 365 371 375 377 379 380 381 382 3829 270 273.15 351 364 371 375 377 379 380 381 382 382

10 300 273.15 350 363 370 374 377 379 380 381 381 38211 330 273.15 349 362 370 374 377 379 380 381 381 38212 360 273.15 348 361 369 374 377 378 380 381 381 38213 390 273.15 347 361 369 374 376 378 380 381 381 38114 420 273.15 346 360 368 373 376 378 380 381 381 38115 450 273.15 345 359 368 373 376 378 380 381 381 38116 480 273.15 345 358 367 373 376 378 380 381 381 38117 510 273.15 344 358 367 372 376 378 379 380 381 38118 540 273.15 343 357 366 372 376 378 379 380 381 38119 570 273.15 342 356 366 372 375 378 379 380 381 38120 600 273.15 342 356 365 371 375 378 379 380 381 381

Equipment

1. 12” long stainless steel rod. Because of uncertainty over what kind of

stainless steel it was, all constants used in calculations were an average

between the various types found in the textbook.

2. Custom made Delrin fitting for water flow past rod

3. Six Type-K thermocouples

4. Digital thermocouple scanner

5. Foam insulation

6. 12V DC motor

7. Hosing and miscellaneous fittings

8. PVC bucket for ice water reservoir

9. Electrical tape, tie wraps and screws

8

Procedure

The following steps were followed:

1. Connect all thermocouples to the scanner.

2. Remove rod assembly from Delrin base (to facilitate even heating)

3. Turn the heating wire on (power supply set at 75)

4. Heat the bar until all thermocouples register 400K

5. Allow the bar to cool down until thermocouple #1 reads 373K (by this time

temperature in the bar should have stabilized)

6. Insert the bar in the Delrin base and fill the reservoir with ice water.

7. Turn pump on (power supply set at 5V)

8. Start recording all six temperatures every 30 seconds for 10 minutes.

9

Data For Calculations

The following data is the average of three runs (see appendix for each individual

data set).

Step P Time (s) Twater (K) T1(K) T2(K) T3(K) T4(K) T5(K) T6(K)0 0 273.15 369 370 374 376 378 3831 30 273.15 364 367 372 374 376 3842 60 273.15 361 365 371 373 376 3823 90 273.15 360 363 370 373 375 3814 120 273.15 359 362 369 372 375 3805 150 273.15 358 361 367 371 374 3796 180 273.15 357 360 367 370 373 3787 210 273.15 356 359 366 370 373 3778 240 273.15 355 358 366 369 372 3769 270 273.15 354 357 365 368 371 375

10 300 273.15 353 356 364 367 370 37411 330 273.15 352 355 363 366 370 37312 360 273.15 351 355 362 366 369 37313 390 273.15 351 354 361 365 368 37214 420 273.15 350 353 360 364 368 37115 450 273.15 349 352 360 363 367 37016 480 273.15 349 352 359 363 366 37017 510 273.15 348 351 358 362 365 36918 540 273.15 347 350 357 361 365 36819 570 273.15 347 350 356 360 364 36820 600 273.15 346 349 356 360 363 367

Time: +/- 1 sec.

Temperature: +/- 1K

10

Tabulated Results

This graph shows the temperature distribution of all six nodes, both theoretical

and experimental. For the sake of clarity we will now show each individual temperature

on the following pages.

11

Theoretical vs Actual Temperature Comparison

340

345

350

355

360

365

370

375

380

0 100 200 300 400 500 600

Time (sec)

Tem

p (K

)

T1 (Theoretical)

T1 (Actual)T2 (Theoretical)

T2 (Actual)

T3 (Theoretical)T3 (Actual)

T4 (Theoretical)

T4 (Actual)

T5 (Theoretical)T5 (Actual)

Figure 1: Comparison of all temperatures – 5/20/02

T1 THEORETICAL VS. ACTUAL

y = 2E-10x4 - 3E-07x3 + 0.0002x2 - 0.1092x + 368.91R2 = 1

y = 2E-14x6 - 3E-11x5 + 3E-08x4 - 1E-05x3 + 0.002x2 - 0.2142x + 368.93R2 = 0.9987

340

345

350

355

360

365

370

0 100 200 300 400 500 600

Time (sec)

Tem

p (K

) T1 (Theoretical)T1 (Average)Poly. (T1 (Theoretical))Poly. (T1 (Average))

Figure 2: Comparison of T1 - 5/20/02

T2 THEORETICAL VS. ACTUAL

y = 5E-13x5 - 9E-10x4 + 6E-07x3 - 0.0002x2 + 0.0027x + 370.03R2 = 1

y = 5E-15x6 - 1E-11x5 + 8E-09x4 - 3E-06x3 + 0.0007x2 - 0.1206x + 370.02R2 = 0.9986

345

350

355

360

365

370

375

0 100 200 300 400 500 600

Time (sec)

Tem

p (K

)

T2(Theoretical)T2 (Average)

Poly. (T2(Theoretical))Poly. (T2(Average))

Figure 3: Comparison of T2 - 5/20/02

12

T3 THEORETICAL VS. ACTUAL

y = 2E-08x3 - 3E-05x2 - 0.0058x + 373.97R2 = 1

y = -1E-16x6 - 2E-14x5 + 5E-10x4 - 5E-07x3 + 0.0002x2 - 0.0626x + 373.95R2 = 0.9961

355

357

359

361

363

365

367

369

371

373

375

0 100 200 300 400 500 600

Time (Sec)

Tem

p (K

) T3 (Theoretical)T3 (Average)Poly. (T3 (Theoretical))Poly. (T3 (Average))

Figure 4: Comparison of T3 - 5/20/02

T4 THEORETICAL VS. ACTUAL

y = 9E-09x3 - 2E-05x2 - 0.0008x + 376R2 = 1

y = 9E-15x6 - 2E-11x5 + 1E-08x4 - 4E-06x3 + 0.0007x2 - 0.0752x + 375.91R2 = 0.9967

358

360

362

364

366

368

370

372

374

376

378

0 100 200 300 400 500 600

Time (Sec)

Tem

p (K

) T4 (Theoretical)T4 (Average)Poly. (T4 (Theoretical))Poly. (T4 (Average))

Figure 5: Comparison of T4 - 5/20/02

13

T5 THEORETICAL VS. ACTUAL

y = 2E-11x4 - 3E-08x3 + 8E-06x2 - 0.0041x + 377.99R2 = 1

y = 6E-15x6 - 1E-11x5 + 9E-09x4 - 3E-06x3 + 0.0005x2 - 0.0577x + 377.82R2 = 0.9955

362

364

366

368

370

372

374

376

378

380

0 100 200 300 400 500 600

Time (Sec)

Tem

p (K

) T5 (Theoretical)T5 (Average)Poly. (T5 (Theoretical))Poly. (T5 (Average))

Figure 6: Comparison of T5 - 5/20/02

14

This table is a comparison of the percent errors between the actual average and

theoretical values for each temperature. Note that error increased as distance increased

away from the point of cooling. Largest error was 3.69%

We also created an ANSYS model in order to predict the cooling pattern of the

bar. The following pages have ANSYS results at various time intervals. It is worth

noting that our actual results are closer to the ANSYS results than those of the ideal

theoretical calculations. On average, actual temperatures were only off by approximately

5C from the ANSYS results.

15

Percentage ErrorStep P Time T1 T2 T3 T4 T5 T6

0 0 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%1 30 -0.49% -0.80% -0.47% -0.52% -0.50% 0.33%2 60 -0.57% -1.24% -0.67% -0.77% -0.47% -0.14%3 90 -0.21% -1.62% -0.87% -0.74% -0.71% -0.36%4 120 0.08% -1.69% -1.05% -0.98% -0.68% -0.59%5 150 0.30% -1.74% -1.50% -1.20% -0.92% -0.82%6 180 0.48% -1.78% -1.39% -1.43% -1.16% -1.06%7 210 0.61% -1.82% -1.55% -1.37% -1.14% -1.30%8 240 0.71% -1.86% -1.43% -1.58% -1.37% -1.54%9 270 0.78% -1.90% -1.57% -1.79% -1.61% -1.78%10 300 0.82% -1.94% -1.71% -1.99% -1.84% -2.02%11 330 0.84% -1.99% -1.85% -2.19% -1.80% -2.26%12 360 0.85% -1.77% -1.98% -2.11% -2.03% -2.24%13 390 1.12% -1.83% -2.12% -2.30% -2.26% -2.49%14 420 1.09% -1.90% -2.25% -2.49% -2.21% -2.73%15 450 1.05% -1.97% -2.11% -2.68% -2.43% -2.98%16 480 1.29% -1.77% -2.24% -2.59% -2.65% -2.96%17 510 1.22% -1.86% -2.38% -2.77% -2.87% -3.21%18 540 1.15% -1.95% -2.51% -2.96% -2.82% -3.46%19 570 1.35% -1.76% -2.65% -3.13% -3.03% -3.44%20 600 1.26% -1.86% -2.51% -3.04% -3.24% -3.69%

Time: 240 sec

16

Figure 7: 60 seconds

Figure 9: 240 seconds

Figure 8: 480 seconds

17

Figure 11: 540 seconds

Figure 10: 480 seconds

Discussion

During this experiment, a difference between experimental and theoretical data

was appreciated. The maximum difference was 12K in node #5. This divergence

between theoretical and experimental figures seemed to increase with distance from the

edge of the rod (4K in node 1, 6K in 2, 7K in 3, 11K in 4 and 12K in node 5).

What are the causes for this difference?

- Inadequate insulation: In almost all cases (except for the first two minutes in

node #1), the rod cools down quicker than our theoretical model. This can be

caused by the insulation not being perfect, even though we had three layers of

very tightly packed pipe insulation.

- The Delrin base acted as a heat sink. We tried to minimize the contact area with

the rod, but in order to prevent water leaks we had to keep a minimum contact

surface. This base provided extra cooling that was not accounted for in our

calculations.

- Poor thermocouple placement: In order not to alter the geometry of the bar, it was

decided not to drill holes for the thermocouples. We used a epoxi resin instead.

This resin has some thermal resistance and might have altered the temperatures

recorded by the thermocouple, although we believe this to be almost negligible.

- Uneven temperature distribution throughout the bar: We had some problems

heating the bar evenly. This was due mainly to the need to expose one end of the

bar and the somewhat temperamental heating wire. We made some changes to

minimize this, but a 10-15K difference in temperature is typical. We were forced

to enter the starting temperatures into our excel worksheet in order to get an

accurate prediction of the cooling process.

18

Diagrams and Schematics

19

Figure 12: Diagram showing nodes and point of convection

Figure 13: Schematic of setup

20

Figure 14: Photo of project from front

Cold water

h

Stainless steel rod

Insulation

Delrin water enclosure

Figure 15: Cutaway of fitting for rod. Note that bored hole is larger than hole for rod.

References

1. Fundamentals of Heat and Mass Transfer: 4th Edition. Incropera and DeWitt,

1996.

2. ME415 Lab manual

3. ME 404 class notes

4. ANSYS manual. Version 5.7, 1998

5. Fundamentals of Heat and Mass Transfer: 2nd Edition. Incropera and DeWitt,

1985.

21

Appendix

22

Equations Used

Vel. = velocityAV = volumetric flow rateRe = Reynolds numberV = velocityL = lengthDh = hydraulic diameterP = perimetera = heightb = widthNu = Nusselt numberPr = Prandtl numberh = convection heat transfer coefficientk = thermal conductivity

ρ = densityCp = specific heatα = diffusivityBi = Biot numberFo = Fourier number

Energy balance equation:m = massT = temperatureT = timeQ = energy transferq = heat transfer

23

Data for Each Trial

This experiment was run eleven times. Only the last three, numbers 9, 10 and 11

were used to compile results. The following pages contain the temperatures taken for

each trial. For completeness, hand-written data is also provided near the end of the

appendix. All trials are included except number one (it was lost).

24

Trial 9Date of Experiment: 5/11/02Time T1 T2 T3 T4 T5 T6

0 370 371 374 374 375 38230 365 368 372 373 375 38160 362 365 371 372 374 37990 361 364 370 372 374 378

120 360 362 369 371 373 377150 358 361 364 370 373 376180 357 360 367 369 372 375210 356 359 366 369 371 374240 355 358 365 368 371 373270 354 357 364 367 370 372300 353 356 363 366 369 372330 352 355 362 366 369 371360 351 355 362 365 368 370390 350 354 361 364 367 370420 349 353 360 363 366 369450 349 352 359 362 366 368480 348 351 358 362 365 367510 347 350 357 361 364 367540 346 349 356 360 363 366570 346 349 356 359 363 366600 345 348 355 359 362 365

25

Trial 10Date of Experiment: 5/11/02Time T1 T2 T3 T4 T5 T6

0 369 371 375 377 379 38230 365 368 374 376 378 38560 363 367 373 375 378 38490 362 365 372 375 377 383

120 360 364 371 374 377 382150 359 363 370 373 376 380180 358 362 369 372 375 380210 357 361 368 372 375 379240 356 360 368 371 374 378270 355 369 366 370 373 377300 355 358 366 369 372 376330 354 357 365 368 372 375360 353 357 364 368 371 375390 352 356 363 367 370 374420 351 355 362 366 370 373450 351 354 362 365 369 372480 350 353 361 365 368 372510 349 353 360 364 367 371540 348 352 354 363 366 370570 348 351 358 362 366 369600 347 351 358 362 365 368

Trial 11Date of Experiment: 5/11/02Time T1 T2 T3 T4 T5 T6

0 368 369 372 377 380 38530 362 364 370 373 376 38560 359 362 369 372 376 38390 357 361 368 371 375 382

120 356 360 367 370 374 381150 356 359 366 370 374 380180 355 358 365 369 373 379210 354 357 365 368 372 378240 353 356 364 368 372 377270 352 355 363 367 371 376300 352 355 362 366 370 375330 351 354 361 365 369 374360 350 353 361 365 369 374390 350 353 360 364 368 373420 349 352 359 363 367 372450 348 351 359 363 366 371480 348 351 358 362 366 370510 347 350 357 361 365 369540 346 349 357 361 365 369570 346 349 356 360 364 368600 345 348 355 359 363 367