Paraphrasing a report

EXPERIMENT2 - RADIAL HEAT CONDUCTION

ME 315 – HEAT & MASS TRANSFER

A: A Steady State Heat Conduction

B: The Fourier Rate Equation for Radial Heat Transfer

Table of Contents: Abstract 5 Objectives 6 Theory 6 Procedure 8 Complete Results 9 Results of the Measurements: 9 Temperature vs Radius Plots: 9 (T – r) Plot at 13 V: 10 (T – r) Plot at 14 V: 10 (T – r) Plot at 16 V: 11 Results of the Experimental Temperature (T3) for Each Electrical Power Input. 11 Experimental Temperature (T3) at 13 Volts: 11 Experimental Temperature (T3) at 14 Volts: 12 Experimental Temperature (T3) at 16 Volts: 12 Results of the Reading and Calculations of the and : 13 The Percentage Error for Heat Transfer: 17 Conclusion 18 References 19

List of figures:

In this laboratory, we will calculate the temperature using different voltage values and a cylindrical wall in a stable state with a centered heat point. We will use various readings, one of which will be unknown, and we will calculate it using Fourier's law and excel graphs before measuring electrical heat transfer. We will use the calculated voltage and the current from the test unit, Q ̇_Elec will be defined as the Q ̇_Cond will be measured using the Fourier law. Readings will be measured to help with the application of calculations. In addition, students have to calculate the error value to test whether the experiment was successful or not. At the end of our experiment, we are going to compare the theoretical temperature with the experimental to know our percentage error. Finally, in our report we are going to list the steps of our experiment and list the procedure, theory, objectives, calculations, the graphs and the conclusion.

Some of the objectives for forming the experiment are: 1) Use the test readings and calculations to determine what is needed 2) Use excel to locate and determine the unknown temperature T3. 3) Measure the heat transfer of the conduction Q

In various modes of conductivity, convection and radiation heat transfer. You must see heat transfer in the conduction in this experiment. Leading of gasses, liquids and solids may take place. The conductivity in solids develops due to molecular vibrations, collision, and diffusion in gasses and liquids.

Heat conduction generally occurs in a number of scientific applications and can be measured by the Fourier Heat Conductivity Law:

𝑄̇𝑐𝑜𝑛𝑑 = −𝑘 A 𝑑𝑇/𝑑𝑟

The heat transfer in radial direction to the medium layers for a stable conductive feature in an enclosed cylindrical wall (disk), will depend on the radius of cylinder layers and the temperature of each layer. Take the cylinders surface 2αrL and the correct form of the general heat conduction will then be determined in radial direction by The following:

𝑄̇𝑐𝑜𝑛𝑑 = 2𝜋𝑘𝐿 (𝑇1 − 𝑇2) / ln ( 𝑟2 𝑟1)

Where: r1 and r2 are the internal and external surfaces, and L is the total cylinder length.

As in (Figure 1), the apparatus shows a metal disk with thermocouples at different radii and heat flow out of the center to the periphery. As a result, heat flow and the distribution of temperature can be investigated. There are six Thermocouples, which are placed on the disk and their purpose is to measure temperature gradient from the heated section (central core) to the cooled section.

As listed below we have the Fourier’s Law and the derived Fourier’s Law:

· K: the thermal conductivity.

· A: the area.

· : Temperature gradient.

By taking the area of the cylinder to be , and obtain another form of Fourier’s Law which is listed as the following:

Also, we used this formula:

· I: Current.

· V: Voltage.

1- Just continue your experiment after your teacher provides you with operating and security practices.

2- Make sure to supply water to the cold area.

3- Make sure the versatile water link to the drain.

4- Ensure every thermocouple is connected to the socket at the service unit component (K).

5- Make sure the switch (B) to the manual unit is set.

6- Make sure on the measurement selection switch (E) you pick V.

7- Switch on the key reserve switch (A).

8- See the readings of voltage in the top meter (D) panel (as V in step 6 was selected).

9- The voltage rises slowly to 13V from zero voltage.

10- Change the Measurement Set (E) to I at 13 V and read the corresponding current measurement value at meter (D).

11- Ensure you are located on T1 at the temperature switch (G) and change it later.

12- Watch the temperature on the adjacent panel meter (J) for 10min (for a steady state condition).

13- Take the read T1 and change the temperature reading switch to show further temperatures after reaching a steady state.

14- Fill in the table in different layers below the temperatures, and remember to mention the units. Remember that you cannot assess T3 from the third layer, this is measured later.

15- Repeat step 9 with 14 and 16V voltage change.

In the results part, we determined the experimental and theoretical third temperature. For the theoretical, we calculated it from the formula, but for the experimental we found by a logarithmic equation which is obtained from the plot drawn in Excel. After that, we found their percentage of error. We repeated these steps three times for 12 volts, 15 volts and 17 volts.

We have the obtained results from the unit service at the three different values for the voltage.

For this section, we took the five obtained temperatures values and used the Microsoft Excel software to plot them. After that, from excel features you can get the logarithmic pattern for the drawn plot to get an equation to calculate the missing data, in our case it is the experimental temperature.

Figure 2: (T – r) Plot at 13 V.

From the plot above, we obtained the following logarithmic equation.

T=-11.43ln(r)+73.43

Figure 3: (T – r) Plot at 14 V.

From the plot above, we obtained the following logarithmic equation.

T=-13.19ln(r)+80.761

From the plot above, we obtained the following logarithmic equation.

T=-17ln(r)+97.379

We will show the calculations for the values of the T3 which are the experimental ones. The results will be obtained by using in the logarithmic equations. The radius is given to be (r3 = 20 mm).

T, exp = -11.43ln (20) + 73.43 = 39.189

T, exp = -17.17ln (20) +97.379

Hand calculations for the and :

and Calculations at 13V:

For

and Calculations at 14 V:

For :

For :

and Calculations at 16 V:

For :

For :

We listed the values for the temperatures theoretically and experimentally, as for the theoretical values we got them by using Fourier Rate Equation which is shown as the following:

Error% =

At 13 V: Error% =

Error% = 3.21 %

At 14 V: Error% =

Error% = 3.69 %

At 16 V: Error% =

Error% = 5.02 %

Error% =

At 13 V: Error% =

Error% = 9.47%

At 14 V: Error% =

Error% = 9.94 %

At 16 V: Error% =

Error% = 11.08%

The possible source of errors for this experiment are clarified in details down below:

· Human errors: we might have miscalculated in any part of the experiment.

· Equipment: the tools that have been used could have been damaged, broken or has some flaws.

· Environmental effects: the weather could have some moisture that would affect the temperature sensors.

We were able to learn how to calculate theoretical and experimental measurements and calculations at the end of our study. After we have the temperature values, we can use Fourier's law to calculate the conduction heat transfer and the voltage and current values to calculate the electrical heat transfer. In our experiment we had to perform it three different times with three different voltage values (13V, 14V, 16V) and in each time we will have different calculations. We studied two methods for calculating heat transfers: the first is to multiply voltage by current, and the second is to use Fourier's law. We also learned how to measure the third temperature using the Fourier theory. Furthermore, after entering the experimental values, we plot the graphs on Excel to obtain the logarithmic equations. In addition, we were able to understand the relationship between heat transfer and conduction that we studied in class and what we did in the laboratory.