paraphrasing

ABSTRACT

The objective of this experiment is to understand the relationship between the change of temperature of an object and its surroundings. The Newtonian Cooling says that the temperature of an object is proportional to the temperature of the surrounding. The reason for the experiment is to make an experiment that measures temperature using a transducer of our own choice, understand the heat transfer and determine the overall heat transfer coefficient. After finishing the experiment and calculating the data and graph them, we concluded that the area of the crucible was 33.53 by using either AutoCad or SolidWorks. Furthermore, the lead undergoes two phases and between these two phases there is a transition phase where the temperature stays constant for a period of time, which is between the 2nd and the 5th minute of the experiment. Point () turned out to be (17.0315,126.6141). And point () is (6.3648,301.5625). Moreover, then we got the heat transfer coefficient which is h = 2.6019.

TABLE OF CONTENTS

Abstract ……………………………………………………………………………2

Table of Contents…………………………………………………………………..3

Introduction and Theory……………………………………………………….......4-9

Procedures………………………………………………………………………..10-11

Summary of Important Results…………………………………………………….12

Sample Calculations and Error Analysis…………………………………………...13

Discussion and Conclusion…………………………………………………………14

References…………………………………………………………………………..15

Appendix…………………………………………………………………………16-19

INTRODUCTION AND THEORY

In this experiment, a mass of lead in a crucible will be heated to its melting point and a transducer will be inserted in the lead. The heating is then ceased and the data of temperature versus time are accumulated by some data-acquisition system of your choice.

It is known from thermodynamics that when two bodies at different temperatures are in contact, heat will flow from the hotter body to the cooler one in a process known as Heat Transfer. The rate of this heat flow depends upon the temperature difference and thermal resistances in much the same way that electric current depends upon the potential difference (voltage) and electrical resistances. In solids, heat transfer occurs by molecular motion in a process called conduction, whereas in fluids, such as air and water, heat is transferred by fluid motion in a process called convection. In addition, heat is also transferred by electromagnetic radiation in transparent substances, or in a vacuum.

Consider a solid object in contact with air. If the surface temperature of the body, , is higher than the air temperature, , then there will be heat transferred from the object to the air. Newton proposed that the rate of this heat transfer, q, is proportional to the surface area of the object, A, and the temperature difference, :

(IV-1)

where the constant of proportionality, h, is called the heat transfer coefficient. Equation (IV-1) is known as the Newton’s rate equation.

In Newton’s time, the actual mechanism whereby this heat transfer occurred was not well understood. Today, however, it is known that heat transfer from a surface involves convection and radiation, and that these two mechanisms occur in parallel. As a result, the total rate of heat transfer from the surface, q, is the sum of the parts due to convection, , and radiation,. Thus, from Eq. (IV-1),

(IV-2)

where,

= convective heat transfer coefficient

= effective radiative heat transfer coefficient

In convection, if the surface temperature of the body, , is higher than the air temperature, , then the air particles near the surface will also take on a higher temperature. This will reduce their density so they, in effect, become “lighter” than the air particles farther removed from the body; in turn, this causes them to drift upward and be replaced by cooler, more dense particles. In this way, the heat is transferred from the hotter surface to the cooler air by fluid motion in a process called free convection (as opposed to forced convection where the air is blown over the surface). For a sphere of diameter D, the convective heat transfer coefficient can be estimated from:

(IV-3)

where,

= conductivity of the air = 0.024

= average

T = average absolute surface temperature

= length parameter (0.5mm)

In radiation, the rate of heat transfer is actually proportional to the difference of temperatures to the fourth powers. However, as an approximation, the effective radiative heat transfer coefficient can be estimated from

(IV-4)

where,

T = average absolute surface temperature

= Stefan-Boltzman constant, 5.67E-8

= emissivity (0.8)

In terms of thermal resistance, Eq. (IV-1) can be written as

where is the convective and radiative thermal resistance. In reality, the heat transfer coefficients vary considerably over the surface area. In the above formulation, h represents the average heat transfer coefficient.

From the interior of the solid, heat is transferred to the surface by conduction. For a one-dimensional system this process is described by the Fourier Rate Equation:

(IV-5)

where k is the thermal conductivity of the material and x is the coordinate distance. If L is a certain characteristic length or dimension of the body, then Eq. (IV-5) can be written approximately as

(IV-6)

where = L/kA is the conductive thermal resistance, and Δ is a temperature difference within the body, where a typical temperature distribution is plotted above the body. The flow of heat from an interior point to the air can thus be represented by a series thermal network.

If the thermal conductivity of the body is quite high, as it is for metals, then would tend to be small compared to. This relative effect of conduction and convection is measured by the Biot Number, :

(IV-7)

Thus, if the Biot Number is much smaller than one, that is <<1, then << this implies that for all practical purposes the temperature, T, at an interior point can be considered the same everywhere in the body and on the surface. This observation, when Bi << 1, leads to an important analytical simplification is known as “lumped mass” analysis.

Consider the body having temperature T at some time t. The internal energy, U, of the body is then

U = mcT (IV-8)

where m is the mass of the body, and c is the specific heat. If q is the heat (or energy) leaving the body per unit time, then by the first law of thermodynamics, the (negative) time rate of change of U must equal q:

(IV-9)

But, according to Eq. (IV-1), q leaves the surface by convection and radiation, and by the assumption of “lumped mass” analysis, T = Ts). Hence, the substitution of Eqs (IV-1) and (IV-8) into Eq. (IV-9) yields

(IV-10)

which is the lumped-mass differential equation describing the change of temperature with time. If To is the initial temperature, then the initial condition for Eq. (IV-10) is T(0) = To. Notice that the left side of Eq. (IV-10) is in units of temperature over time; therefore, the right side must also have these units. Since Ta is a constant, Eq. (IV-10) can be written as

where tc = mc/hA is known as the time constant. The above equation can be written as follows:

(IV-11)

Integration of Eq. (IV-11) yields

or, after evaluating the constant of integration

(IV-12)

where T=T0 when t  0. Thus, the temperature decays exponentially

(IV-13)

From Eq. (IV-12) it can be seen that if a body undergoes Newtonian cooling (h is a constant), then a plot of ln (T - Ta)/(To - Ta) vs. time will produce a straight line with slope -1/tc . From Eq.(IV-13) it is seen that when t = tc, T - Ta will have changed from To - Ta by a factor of 1/e  37%, and that when t = 5tc, T has been reduced to within 1% of room temperature.

PROCEDURES

Lead

Figure: 1

First, we obtain the following equipment in order to setup our experiment: crucible, pipe clay triangle, tripod, Bunsen burner, lead and a heat resistant mat. After setting up the experiment as shown in figure 1, we double click to open the LabView software that was on the desktop. On LabView, click the run arrow. We measured the diameters and heights of the crucible to help us later calculate the volume of the crucible. Then we heat the crucible with the lead in it to a temperature of 400C. When a temperature of 400C is reached, we begin calculating data as the crucible is cooled by the room temperature. The experiment took from 20 to 30 minutes to be completed as the lead reached to a temperature of 100C. When the maximum temperature is reached, click the DATA box on LabView, to stop the time data collection. After finishing calculating data, we saved them into our flash drives to later use them.

SUMMARY OF IMPORTANT RESULTS

Figure 2: Temperature vs Time that shows the important phases and points

Figure 3: Plot that shows ln(T-T0)/(T0-Ta) vs. Time

SAMPLE CALCULATIONS AND ERROR ANALYSIS

New Values:

–Point 41

Time constant: (minutes) ≈ 640.002 (seconds)

Determined Average Heat Transfer Coefficient, h:

, mass of crucible (83grams) =0.083kg, Area (33.53)=0.021632, ,

h=

DISCUSSION AND CONCLUSION

After finishing the experiment and calculating the data and graph them, we concluded that the area of the crucible was 33.53. We determined this value by using SolidWorks or AutoCad. Also, from the data obtain from the experiment, we noticed that the lead undergoes two phases and between these two phases there a transition phase where the temperature stays constant for a period of time which is between the 2nd and the 5th minute of the experiment. Point () turned out to be (17.0315,126.6141) and point () is (6.3648,301.5625). Moreover, then we got the heat transfer coefficient was h = 2.6019.

Errors in any experiment are unavoidable. One can only try to reduce them as much as possible. Therefore, the errors in this experiment could be due to wrong calculations, mistake in taking measurements, or a wrong setup for the sample in the machine. These are the errors that could happen by the one doing the experiment. However, there are errors that are not related to the one doing the experiment. The machine itself could be old or hasn’t had the proper maintenance and that could cause errors. Not only the machine could cause errors, but also the samples could be not manufactured well.

REFERENCES

[1] CSUF EGME 306A Lab Manual