Radial Conduction 6v1c4n

Radial Conduction

1. Introduction: We have seen how different materials affect heat flow as well as how changes in crosssectional area affect heat flow. In this experiment, we shall examine how heat transfers radially by conduction using a cylindrical brass specimen..

2. Objective: To examine the temperature profile, and determine the rate of heat transfer resulting from radially steady conduction through the wall of a cylinder.

3.

Theory: When the inner and outer surfaces of a thick walled cylinder are each at a different uniform

temperature, heat flows radially through the cylinder wall. The disk can be considered to be constructed as a series of successive layers. From continuity considerations the radial heat flow through each of the successive layers in the wall must be constant if the flow is steady but since the area of the successive layers increases with radius, the temperature gradient must decrease with radius.

3.1

Equations:

Q   KA

dT dR

Where:

K W / m.K  is the thermal conductivity, QW  thermal energy, dRm Radius ,

  dT  C  the temperature difference. A m 2 the cross sectional area, o

 dT 

Q dR KA

To

Ro

Ti

Ri

  dT 

Q

 KA dr

Where: 1|Page

  T  C  is the inner temperature,

To oC is the outer temperature, o

i

Ro m  is the outer radius=0.055m,

Ri m is the inner radius=0.007m.

Ti  To 

Q ( Ro  Ri ) K * 2t

Q  2tK

(Ti  To ) log RRoi

Where: t is the thickness of the cylinder.

Q

Qavr 

i

i

Where: i represents several calculated values.

error % 

Qactual  Qavg Qactual

*100

Where:

error% is the error percentage,

Qavr W  is the average thermal energy, Qactual W  is the actual thermal energy. error % 

To ,exp  To ,theor To ,theor

*100

Where:

 is the experimental outer temperature,  C  is the theoretical outer temperature.

TO,exp o C TO,theor

o

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4. Apparatus:

A: Control Box B: Selector Box C: Power Regulator D: Heater Knob (CCW is off) E: Heater Power Cable conduction apparatus)

(from

either

Fig (1).

A: Linear Conduction Apparatus B: Radial Conduction Apparatus C: Thermistors D: Thermistors Cables E: Cooling Tubing

Fig (2).

3|Page

4.1 Apparatus process : Fig (1) shows the device which contains the electric circuit which provides the heat by an electric heater (the power source). Also, it contains a little screen which shows the reading of all the nine sensors as well as a power regulator to choose the desired power for the experiment. Fig (2) shows the brass cylinder on which we conducted our experiment. The center of the cylinder is heated by the electric heater. There are six heat sensors (1-2-3-7-8-9) positioned radially on the brass cylinder starting at the center with a space of 10 mm between any sensor and the next.

5. Procedure: 1. Make sure the Control Box (Fig 1A) and the Power Regulator (Fig 1C) are turned OFF. 2. Check that the Control Box is plugged into the Power Regulator. 3. Connect the Heater Power Cable (Fig 1E) from the desired conduction apparatus to the Control Box.

4. Take a brass sample (13 mm diameter) and coat each end of the sample and apparatus with conducting compound.

5. Insert the brass sample to the test unit. 6. Allow cold water to flow through the test unit. 7. Select an intermediate position for the heater power control and allow sufficient time for a steady state condition to be achieved at one end of the disc.

8. Record the temperature (T) at all six sensor points and the input power reading on the wattmeter (Q).

6. Calculation: 6.1

Lab Readings:

 

 

 

 

 

EXP #

Q (W)

T1 oC

T2 oC

 

T3 oC

T7 oC

T8 oC

T9 oC

1

15

34.4

30.8

26.8

24.8

23.3

22.3

2

20

38.9

33.8

28.8

26.2

24.2

22.7

TABLE 1 4|Page

6.2

Experimental and theoretical outer surface temperature:

From the temperature distribution sketches (1&2) see section (8) we find experimental outer temperature: at 15(w)

To,exp  21.3152 oC

at 20(w)

To,exp  21.5136 oC

We find theoretical temperature from

Q  2tK

To  Ti 

(Ti  To ) log RRoi

Q. log

Ro Ri

2tK

From TABLE 1 at 15W:

To,theor

055 15 * log 00..007  34.4   33.085o C 2 * 0.0032 *117

Repeat last operation with 20W. See TABLE 2.

6.3

Average thermal energy:

From TABLE 1 at 15W:

Qn  2tK i=1,2,3,7,8,9

(Ti  Ti 1 ) log RRii1 & n=1,2,3,4,5

5|Page

(Ti  T2 ) 34.4 - 30.8  2 *  * 0.0032 *117 *  23.7435 W R2 0.01 log Ri log 0.007

Q1  2tK

In the same way, we obtain Q2 , Q3 , Q4 , and Q5

Qavr 

Qavr 

Q

n

n

32.744  13.575  11.604  12.266  10.542  14.346  14.346 W 5

Repeat last operations at 20W. See TABLE 3.

6.4

Evaluation of the error percentage:

6.4.1

EXPERIMENTAL & THEORETICAL OUTER SURFACE TEMPERATURE .

Using TABLE 2 at 10W :

error % 

error % 

To ,theor  To ,exp To ,theor

*100

33.085 - 21.3152 *100  35.575 33.085

Repeat last operations at 20W.

6.4.2

THERMAL ENERGY.

From TABLE 3 at 15W:

error % 

error % 

Qactual  Qavg Qactual

*100

15  14.346 *100  4.36 15

Repeat last operations at 20W. See TABLE 4 .

6|Page

7. Tables:

EXP #

Q (watts)

To,exp

 C o

 

To,theor oC

1

15

21.31521

33.085

2

20

21.51361

37.147

TABLE 2

EXP#

Q1 W 

Q2 W 

Q3 W 

Q4 W 

Q5 W 

Qavr W 

1

23.744

13.575

11.604

12.266

10.542

14.346

2

33.637

16.969

15.085

16.354

15.813

19.571

TABLE 3

 

 

EXP#

To,exp oC

To,theor oC

error %

Qact (W )

Qavg (W )

error %

1

21.315

33.085

35.575

15

14.346

4.36

2

21.516

37.147

42.078

20

19.571

2.145

TABLE 4

7|Page

Temperature

8. Temperature distribution sketch:

Temperature distribution at 15 W

38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

Ti T = -5.953ln(r) + 4.0495

To

0

0.005

0.01

0.015

0.02

0.025

0.03

Radius (m)

0.035

0.04

0.045

0.05

0.055

0.06

8|Page

Temperature

Temperature distribution at 20 W

Ti

42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

T = -7.851ln(r) - 1.2577

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

To

0.04

0.045

0.05

0.055

0.06

Radius (m)

9. Discussion: 

The temperature decreases as we move away from the heat source.



As we can see from Table 2 To , exp did not change significantly when we changed the power source from 15W to 20W. This is due to not allowing the specimen to reach steady state.



The temperature error percentage increased when power input was increased as evident from Table 4 . However, error between average and actual thermal energy decreased when power

was increased.

All the above are effected by several factors such as:       

Human errors. Errors related to the device (old, out dated, and unmaintained). We ran the experiment before the cylinder reached a steady state. The power source was unstable. We neglected the resistance. Thermal expansion of cylinder neglected. Heat transfer from the cylinder to the water assumed to be by conduction only.

9|Page

10.

references: -

http://www.engr.iupui.edu/~mrnalim/me314lab/lab02.htm https://en.wikipedia.org/wiki/List_of_thermal_conductivities Textbook of laboratory 2.

Table of Contents 1.Introduction: ............................................................................................................................ 1

2.Objective: ................................................................................................................................. 1

3.Theory: .................................................................................................................................... 1

4.Apparatus: ............................................................................................................................... 3

5.Procedure: ............................................................................................................................... 4

6.Calculation: .............................................................................................................................. 5

7.Tables: ..................................................................................................................................... 7

10 | P a g e

8.Temperature distribution sketch: .............................................................................................. 8

9.Discussion: .............................................................................................................................. 9

10.References: .......................................................................................................................... 9

11 | P a g e