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Lab report for transient heat conduction
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MAE 338 Fall 2020

Transient Heat Conduction

Prepared By: Date Performed: 16th September 2020 Group #: L05 Lab Partners: Chowdhury, R. Richardson, A DeFilippo, A Chen, Stephen Cordova, Robert Huang, Jef

1. Objective The objective of this experiment is to compare the transient heat conduction in different materials by comparing the experimental temperature of the tubes and the theoretical temperature of the tubes found using two different methods. This experiment helps us compare the validity of the analytical method and find sources of error, if any. This experiment is used to develop powerful analytical tool for transient solutions of components, such as casting moulds and turbine blades, which are difficult to model when employing the conventional domain based methods[2].

2. Background and Theory In this experiment, solution to one-dimensional transient heat conduction in a thin rod is used to find. Analytical solution for one dimensional transient heat conduction in a thin rod is given by T ( t ) −T bath −bt =e T 0−T bath

(1)

where T(t) is the temperature of the end of the rod as a function of time, T 0 is the initial uniform temperature of the rod, T bath is the ice water bath temperature, t is time and b is given by b= where,

ρ is density,

h ρ c p Lc

c p is specific heat capacity,

(2) Lc is the length of the rod

Biot number is given by the formula, Bi=

h Lc k

(3)

where, h is heat transfer coefficient and k is thermal conductivity of the metal

When the Biot number is larger than 0.1, lumped systems can not be assumed and in that case, the solution is given by 4 sin (β n) T ( t ) −T bath τ ∞ e−β cos(β n ) =Σ n=1 2 β +sin (2 β ) T 0−T bath n n 2 n

where,

τ=

αt L2

(4)

and α is thermal diffusivity

The series solution converges rapidly with increasing time, the above equation is approximated to

T ( t ) −T bath τ −β τ = A 1 e− β + A 2 e T 0−T bath 2 1

Where

2 2

(5)

β n are the roots of equation (6) β n tan ( β n )=Bi

(6)

The Value of A1 and β1 depend on the Biot number A table of values for A 1 and β1 is contained in the table below. Table 1: A1 and β1 based on Biot number

3. Experimental Setups and Procedures 1. A bowl was filled with water and enough ice was added to bring the temperature of the ice bath close to 0℃. 2. Three tubes (figure 1 made of different material (Aluminum, Copper and Brass) were selected from the room temperature bathtub, with only one being tested at once.

Figure 1: Internal structure of the tubes being tested 3. One of the selected tubes was put on the ring stand and connected to the data logger (Figure 2).

Figure 2: Experimental setup

4. Hoboware software was launched on the computer and connection to the data logger was confirmed. 5. The logger was launched, and all the relevant settings done in the software(Figure 3). The “Push Button” was clicked for the logger to start logging (Figure 4).

Figure 3: Relevant settings

Figure 4: Logger

6. Once the status on the computer screen turned to “Awaiting Button Start”, the left button on the thermocouple data logger was held for 5 seconds for it to start logging the data and the status changes to “Launched, Logging” on the computer screen. 7. The ring stand was then lowered such that the tube was slightly submerged in the water bath. 8. The data was logged for approximately 5 minutes and then the logging was stopped. The data was then saved from the readout as an excel file. 9. The same procedure was repeated for the remaining two tubes and data saved. 10. Once the relevant data for all three tubes was saved, the tubes were placed back into the room temperature water bath, the water from the ice bath discarded and the table cleaned up.

4. Results and Discussion

1. Total heat transfer rate Time (s) 1 20 40 60 80 100 120 140 160 180 200 220 240 260 280

Bath Temperature (C) 1.97 1.35 1.68 1.50 1.65 1.83 1.84 1.64 1.61 1.68 1.66 1.70 1.68 1.64 1.62

Q e and Q a

Al Temperature (C) 23.66 23.53 23.47 23.13 22.84 22.57 22.24 21.96 21.60 21.28 20.97 20.72 20.43 20.12 19.90

Exp(-bt) 0.999508 0.990203 0.980503 0.970897 0.961385 0.951967 0.942641 0.933406 0.924262 0.915207 0.906241 0.897363 0.888572 0.879867 0.871247

Lump Method Temp (C) 23.79 23.58 23.37 23.15 22.94 22.74 22.54 22.32 22.12 21.92 21.72 21.53 21.34 21.14 20.94

Tau 0.002422 0.048450 0.096899 0.145349 0.193798 0.242248 0.290697 0.339147 0.387596 0.436046 0.484495 0.532945 0.581394 0.629844 0.678293

A1*exp(-Beta1² x tau) 1.048052882 1.029605653 1.01053819 0.991823841 0.973456067 0.955428449 0.937734688 0.920368601 0.90332412 0.886595288 0.870176262 0.854061302 0.838244778 0.822721164 0.807485034

Table 2: Experimental vs Calculated Aluminum tube temperature

Sample Calculation Lumped method T ( t ) −T bath −bt =e T 0−T bath b=

h ρ c p Lc

= 4.92 ×10−4

T ( t) −275.12 =e−4.92 ×10 296.81−275.12

−4

∗1

T(t) = 296.94 K = 23.79 ℃

1st Term App Temp (K) 24.85 24.46 24.03 23.62 23.21 22.82 22.43 22.04 21.65 21.29 20.93 20.57 20.22 19.87 19.53

1st Term Approximation T ( t ) −T bath = A 1 e− β T 0−T bath

2 1

τ

−5

τ=

αt = 9.7 × 10 × 1=0.00242 0.1522 L2

T ( t) −275.12 =1.049× e−β × 0.00242 296.81−275.12 2 1

β n tan ( β n )=0.21 β 1=0.621 T ( t) −275.12 × 0.00242 =1.049 × e−0.621 296.81−275.12 2

T ( t )=298 K=24.85℃

Table 3: Experimental vs Calculated Brass tube temperature Tim e (s) 1 20 40 60 80 100 120 140 160 180 200 220 240 260 280

Bath Temperature (C) 1.93 1.36 1.27 1.47 1.09 1.44 1.07 1.42 1.10 1.43 1.47 1.51 1.16 1.44 1.09

Brass Temperature (C) 21.40 21.39 21.47 21.47 21.48 21.38 21.28 21.13 21.07 20.92 20.76 20.60 20.43 20.30 20.14

Exp(-bt) 0.999508 0.990203 0.980503 0.970897 0.961385 0.951967 0.942641 0.933406 0.924262 0.915207 0.906241 0.897363 0.888572 0.879867 0.871247

Lump Method Temp (C) 23.79 23.58 23.36 23.15 22.92 22.73 22.50 22.31 22.08 21.90 21.71 21.51 21.28 21.11 20.88

Tau 0.002422 0.048450 0.096899 0.145349 0.193798 0.242248 0.290697 0.339147 0.387596 0.436046 0.484495 0.532945 0.581394 0.629844 0.678293

A1*exp(-Beta1² x tau) 1.048052882 1.029605653 1.01053819 0.991823841 0.973456067 0.955428449 0.937734688 0.920368601 0.90332412 0.886595288 0.870176262 0.854061302 0.838244778 0.822721164 0.807485034

1st Term App Temp (C) 24.85 24.46 24.04 23.62 23.20 22.80 22.38 22.02 21.61 21.26 20.90 20.55 20.14 19.84 19.43

Table 4: Experimental vs Calculated Copper tube temperature Time (s) 1 20 40 60 80 100 120 140 160 180 200 220 240 260

Bath Temperature (K) 1.99 1.80 1.82 1.72 1.66 1.68 1.64 1.60 1.60 1.60 1.58 1.58 1.59 1.59

Copper Temperature (K) 20.96 20.74 20.51 20.33 20.06 19.87 19.62 19.44 19.21 19.10 18.81 18.64 18.44 18.32

Exp(-bt) 1.000000 0.999997 0.999994 0.999991 0.999988 0.999985 0.999982 0.999979 0.999976 0.999973 0.999970 0.999967 0.999964 0.999962

Lump Method Temp (K) 23.80 23.80 23.80 23.80 23.80 23.80 23.80 23.80 23.80 23.80 23.80 23.80 23.80 23.80

Tau 0.002422 0.048450 0.096899 0.145349 0.193798 0.242248 0.290697 0.339147 0.387596 0.436046 0.484495 0.532945 0.581394 0.629844

A1*exp(-Beta1² x tau) 1.048732563 1.04304261 1.037086533 1.031164467 1.025276218 1.019421592 1.013600398 1.007812445 1.002057542 0.996335502 0.990646136 0.984989258 0.979364683 0.973772225

1st Term App Temp (K) 24.86 24.75 24.62 24.49 24.36 24.23 24.10 23.97 23.85 23.72 23.59 23.47 23.34 23.22

Temperature Distribution for Aluminum 30.00

Temperature (°C)

25.00 20.00 Expt. Al Temp. (°C) Lump Al Temp (°C) 1st Term App. Al Temp. (°C) Bath Temp. (°C)

15.00 10.00 5.00 0.00 0

50

100

150

200

250

300

Time (Sec)

2.

Figure 7: Temperature of the Aluminum Tube In the graph above, the experimental temperature of Aluminum is compared to the theoretical temperature of the tube, found using Lump method and 1st term approximation. The theoretical approximations of the transient heat conduction for the three tubes comes out to be very close to the experimental value, however, the 1 st term approximation method is closer to the experimental value. The discrepancy in the difference in the Lump Aluminum temperature might be because of the Biot number being greater than 0.1

Temperature Distribution for Brass 30.00

Temperature (°C)

25.00 20.00 Expt. Brass Temp. (°C) 15.00 Lump Brass Temp (°C) 1st Term App. Brass Temp. (°C)

10.00 5.00 0.00

Bath Temp. (°C)

0

50

100

150

200

Time (Sec)

Figure 8: Rotameter Reading vs. Flow rate

250

300

In the graph above, the experimental temperature of Brass is compared to the theoretical temperature of the tube, found using Lump method and 1st term approximation. The theoretical approximations of the transient heat conduction for the three tubes comes out to be slightly different to the experimental value in the beginning but as the experiment goes on, the values converge. However, the 1st term approximation method is closer to the experimental value. The discrepancy in the difference in the Lump Aluminum temperature might be because of the Biot number being greater than 0.1.

Temperature Distribution for Brass 30.00

Temperature (°C)

25.00 20.00 Expt. Copper Temp. (°C)

15.00

Lump Copper Temp (°C) 1st Term App. Copper Temp. (°C) Bath Temp. (°C)

10.00 5.00 0.00 0

2

4

6

8

10

12

14

16

Time (Sec)

Figure 9: Turbine Meter Reading vs. Flow Rate In the graph above, the experimental temperature of Copper is compared to the theoretical temperature of the tube, found using Lump method and 1st term approximation. The Lump method temperature is slightly less in the beginning of the experiment and goes on to be slightly more by the eld of the experiment. The 1st term approximation method is closer to the experimental value. The discrepancy in the difference in the Lump Copper temperature might be because of the Biot number being greater than 0.1.

The temperature is greater than the experimental value at the beginning for the first term approximation method because as the time goes on, the subsequent value after the first term start getting smaller and smaller.

τ Al=110 τ Brass=100 τ Copper =120

5. Conclusions The objectives of the experiment were reached, the data was plotted, and relevant measurements extracted from them. The experimental temperature was compared to the theoretical temperature found using two different methods and the results were as expected from theory. The result has a variation from the theory because of the uncertainty in the data calculation and the various error during the measurement of the data, like error in 1 st term approximation method where we ignore all the terms after the first term. Another source for error can be the theoretical method used to find the temperature might not be suitable for the Biot number of the given material. 6. References 1. Khan, J.R. and Sabato, J., 2020, "Transient Heat Conduction", Proceeding of ASME, September 13-16, 2020, Buffalo, NY, USA.

2.

Dargush, G.F. and Banerjee, P.K. (1991), Application of the boundary element method to transient heat conduction. Int. J. Numer. Meth. Engng., 31: 1231-1247. doi:10.1002/nme.1620310613...