Heat transfer by conduction and convection, Lab report

OCEAN ENGINEERING LAB REPORT HEAT TRANSFER

AUTHOR: JOSE BETANCOURT

Lab Partners: Corey Page Isabella Pinos Michael Perry Ron

Florida Atlantic University, Boca Raton, Florida October 13th and 15th of 2015

Abstract In this report, three different heat transfer experiments are performed. The first experiment consists of 4 bars of different metals (2x brass, 1x aluminum and 1x stainless steel) which are heated by a common source, there are two temperature sensors embedded in the bars which allow to take temperature readings at two different points along the bar separated by a distance d, the rate of heat transfer by conduction in the four bars was computed and compared to determine which material it’s a better conductor. The two brass bars differ in cross section area and this allowed to identify the relation existing between the rate of heat transfer by conduction and the area, we concluded that the values obtained for the rate of heat transfer support the accepted values of thermal conductivity of the materials, that the rate of heat transfer is directly proportional to the cross section area and that insulation material should be placed on top of the bars in order to prevent the heat from being lost to the environment. The second experiment deals with the mechanism of convection, six cups filled with hot water are being cooled down in different ways and temperatures are taken at regular intervals by means of a digital thermometer, using this information it was possible to compare the effectiveness of every method in cooling down the water inside the cups, we concluded that the effectiveness of the rate of heat transfer by convection increases when a convective current is forced into the system (either so by blowing air unto the surface of the fluid, or by stirring the fluid or both), we also concluded that this experiment can be used to estimate the value of the convection coefficient h as long as there is not any insulating material on the cup preventing the surrounding air to be in contact with the cup or the fluid. In the third experiment we sought to build a device which allows us to measure the temperature using a thermocouple and adjusting its value by means of a potentiometer, a current which value depends on the position of the potentiometer was forced through a resistor to take advantage of the Joule’s effect to heat up the thermocouple, we concluded that it is possible to satisfactory control the increase of temperature by this method but when the temperature needs to be decreased we do not have any control on the rate of decay of the temperature.

Introduction Heat has always been perceived to be something that produces in us a sensation of warmth, and one would think that the nature of heat is one of the first things understood by mankind. But it was only in the middle of the nineteenth century that we had a true physical understanding of the nature of heat, thanks to the development at that time of the kinetic theory, which treats molecules as tiny balls that are in motion and thus possess kinetic energy. Heat is then defined as the energy associated with the random motion of atoms and molecules In 1701, Newton published (in Latin and anonymously) in the Phil. Trans. of the Royal Society a short article (Scala graduum Caloris), in which he established a relationship between the temperatures T and the time t in cooling processes. He did not write any formula but expressed verbally his cooling law: “The excess of the degrees of the heat were in geometrical progression when the times are in an arithmetical progression (by ‘‘degree of heat’’

Newton meant what we now call ‘‘temperature’’, so that ‘‘excess of the degrees of the heat’’ means ‘‘temperature difference’’). [1] In the formulation of his law, Newton shows his confusion, which was normal in his days, between heat and temperature. He spoke of heat loss and degree of heat and this means that for him a loss of heat was always proportionally accompanied by a decrease of ‘‘degree of heat’’. He wrote: “The heat which hot iron, in a determinate time, communicates to cold bodies near it, that is, the heat which the iron loses in a certain time is as the whole heat of the iron; and therefore (ideoque in Latin), if equal time of cooling be taken, the degrees of heat will be in geometrical proportion.” [1] Currently, Newton’s cooling law is usually given in terms of heat flux q, i.e., the rate of heat loss from a body q = dQ/dt: Although it was suggested in the eighteenth and early nineteenth centuries that heat is the manifestation of motion at the molecular level (called the live force), the prevailing view of heat until the middle of the nineteenth century was based on the caloric theory proposed by the French chemist Antoine Lavoisier (1743–1794) in 1789. [2] The caloric theory asserts that heat is a fluid-like substance called the caloric that is a massless, colorless, odorless, and tasteless substance that can be poured from one body into another. When caloric was added to a body, its temperature increased; and when caloric was removed from a body, its temperature decreased. When a body could not contain any more caloric, much the same way as when a glass of water could not dissolve any more salt or sugar, the body was said to be saturated with caloric. This interpretation gave rise to the terms saturated liquid and saturated vapor that are still in use today. The caloric theory came under attack soon after its introduction. It maintained that heat is a substance that could not be created or destroyed. Yet it was known that heat can be generated indefinitely by rubbing one’s hands together or rubbing two pieces of wood together. In 1798, the American Benjamin Thompson (Count Rumford) (1753–1814) showed in his

papers that heat can be generated continuously through friction. The validity of the caloric theory was also challenged by several others. But it was the careful experiments of the Englishman James P. Joule (1818–1889) published in 1843 that finally convinced the skeptics that heat was not a substance after all, and thus put the caloric theory to rest. Although the caloric theory was totally abandoned in the middle of the nineteenth century, it contributed greatly to the development of thermodynamics and heat transfer. [2] Background Heat Transfer Heat can be transferred in three different modes: conduction, convection, and radiation. All modes of heat transfer require the existence of a temperature difference, and all modes are from the high-temperature medium to a lower-temperature one [2]. Conduction is the transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones as a result of interactions between the particles. Conduction can take place in solids, liquids, or gases. In gases and liquids, conduction is due to the collisions and diffusion of the molecules during their random motion. In solids, it is due to the combination of vibrations of the molecules in a lattice and the energy transport by free electrons. The rate of heat conduction through a medium depends on the geometry of the medium, its thickness, and the material of the medium, as well as the temperature difference across the medium. Consider steady heat conduction through a large plane wall of thickness Dx= L and area A. The temperature difference across the wall is DT= T2-T1. Experiments have shown that the rate of heat transfer Q through the wall is doubled when the temperature difference DT across the wall or the area A normal to the direction of heat transfer is doubled, but is halved when the wall thickness L is doubled. Thus we conclude that the rate of heat conduction through a plane layer is proportional to the temperature difference across the layer and the heat transfer area, but is inversely proportional to the thickness of the layer.

Figure 1: Heat transfer by conduction through a wall

Q´cond=kA

T 1−T 2 ∆x

Where the constant of proportionality k is the thermal conductivity of the material, which is a measure of the ability of a material to conduct heat, a high value for thermal conductivity indicates that the material is a good heat conductor, and a low value indicates that the material is a poor heat conductor or insulator In the limiting case of Dx → 0, the equation above reduces to the differential form. Q´cond=−kA

dT dx

Here dT/dx is the temperature gradient, which is the slope of the temperature curve on a T-x diagram (the rate of change of T with x), at location x. The heat transfer area A is always normal to the direction of heat transfer. The kinetic theory of gases predicts and the experiments confirm that the thermal conductivity of gases is proportional to the square root of the absolute temperature T, and inversely proportional to the square root of the molar mass M. Therefore, the thermal conductivity of a gas increases with increasing temperature and decreasing molar mass. Convection is the mode of energy transfer between a solid surface and the adjacent liquid or gas that is in motion, and it involves the combined effects of conduction and fluid motion. The faster the fluid motion, the greater the convection heat transfer. In the absence

of any bulk fluid motion, heat transfer between a solid surface and the adjacent fluid is by pure conduction. The presence of bulk motion of the fluid enhances the heat transfer between the solid surface and the fluid, but it also complicates the determination of heat transfer rates. Consider the cooling of a hot block by blowing cool air over its top surface as shown in Figure 2. Energy is first transferred to the air layer adjacent to the block by conduction. This energy is then carried away from the surface by convection, that is, by the combined effects of conduction within the air that is due to random motion of air molecules and the bulk or macroscopic motion of the air that removes the heated air near the surface and replaces it by the cooler air. Convection is called forced convection if the fluid is forced to flow over the surface by external means such as a fan, pump, or the wind. In contrast, convection is called natural (or free) convection if the fluid motion is caused by buoyancy forces that are induced by density differences due to the variation of temperature in the fluid

Figure 2: Cooling of a hot block

Despite the complexity of convection, the rate of convection heat transfer is observed to be proportional to the temperature difference, and is conveniently expressed by Newton’s law of cooling as ´ Q=h A s (T s−T ∞ )

Where h is the convection heat transfer coefficient in W/m2 °C or Btu/h ft2 °F, As is the surface area through which convection heat transfer takes place, Ts is the surface temperature, and T_ is the temperature of the fluid sufficiently far from the surface. Note that at the surface, the fluid temperature equals the Surface temperature of the solid. The convection heat transfer coefficient h is not a property of the fluid. It is an experimentally determined parameter whose value depends on all the variables influencing convection such as the surface geometry, the nature of fluid motion, the properties of the fluid, and the bulk fluid velocity. Radiation is the energy emitted by matter in the form of electromagnetic waves (or photons) as a result of the changes in the electronic configurations of the atoms or molecules. Unlike conduction and convection, the transfer of energy by radiation does not require the presence of an intervening medium. In fact, energy transfer by radiation is fastest (at the speed of light) and it suffers no attenuation in a vacuum. This is how the energy of the sun reaches the earth. Fort the purposes of this experiment, heat transfer through radiation will not be considered. PID Controller The PID controller is the most common form of feedback. It was an essential element of early governors and it became the standard tool when process control emerged in the 1940s. In process control today, more than 95% of the control loops are of PID type, most loops are actually PI control. PID controllers are today found in all areas where control is used. The controllers come in many different forms. There are stand-alone systems in boxes for one or a few loops, which are manufactured by the hundred thousands yearly. PID control is an important ingredient of a distributed control system. The controllers are also embedded in many special-purpose control systems. PID control is often combined with logic, sequential functions, selectors, and simple function blocks to build the complicated automation systems used for energy production, transportation, and manufacturing. Many sophisticated control strategies, such as model predictive control, are also organized hierarchically. PID control is used at the lowest level; the multivariable controller gives the setpoints to the controllers at the lower level. The PID controller can thus be said to be the “bread and butter” of control engineering [3].

Proportional control is illustrated in Figure 3. The figure shows that there is always a steady state error in proportional control. The error will decrease with increasing gain, but the tendency towards oscillation will also increase. Figure 4 illustrates the effects of adding integral. The figure shows that the steady state error disappears when integral action is used.

Figure 3: Effect of proportional control

Figure 4: Effects of the addition of integral

Objective of the experiments In this report, 3 different heat transfer experiments will be addressed, one demonstrating the behavior of the conduction mechanism of heat transfer, one on convection, and a practical situation where reading the temperature and adjusting a variable to control the temperature will be required. The way this report is structured addresses first the materials and methods utilized during the realization of the experiments, the results obtained, a brief discussion of

the results and conclusions and it’s organized in sequence so that all the sections of one experiment are shown before moving on to the next experiment. The objective of this lab it’s to study the behavior of two of the mechanisms of heat transfer (conduction and convection), how the cross section area is related to the rate of heat transfer in conduction and how a convective current on a fluid can improve the rate of heat transfer.

Heat Bar Experiment (Conduction) Materials and methods Equipment List In order to perform this experiment the fallowing equipment has been provided: Pasco Xplorer GLX PS-2002 Pasco Xplorer 12v DC Power Transformer Pasco Heat Conduction Apparatus TD-8513 Pasco PasPort PS-2157 Power Supply Cable 15V DC 2A Power Supply Digital Camera Digital Chronometer Digital Caliper Equipment Description The Heat Conduction Apparatus has 4 bars with 8 temperature sensors embedded in the bars and designated by the numbers from T1 to T8, two sensors in every bar. The material properties of the bars were obtained from the manufacturer’s website and are shown below in Table 1. Table 1 Material properties of the bars [1]

Bars were measured using a digital caliper once the experiment concluded. The dimensions are shown in Table 2.

Table 2 Dimensions of the bars of the Heat Conduction Apparatus

Bar Aluminum Wide Brass Narrow Brass Stainless Steel

Width (mm) 11.8 +/- 0.1 11.7 +/- 0.1 7.9 +/- 0.1 11.9 +/- 0.1

Length (mm) 87.4 +/- 0.1 87.4 +/- 0.1 88.8 +/- 0.1 88.8 +/- 0.1

The location of the sensors is listed below in reference to the bar where they are embedded and the position respect to the heat sink located at the center of the board. T1= wide brass (far) T2= wide brass (close) T3= narrow brass (close) T4= narrow brass (far) T5= wide Aluminum (far) T6=wide Aluminum (close) T7=wide Stainless Steel (close) T8= wide Stainless Steel (far)

Experiment Setup All connections were done prior to the beginning of the experiment as shown in Figure 5. Pasco Xplorer was connected to the power source which was set to 6V and to the Heat Conduction Apparatus,

Figure 5: Xplorer GLX and Power Source

Figure 6: Heat Conduction Apparatus

The machine was set to the Cool position during 5 minutes and the temperatures of every sensor were recorded (See Table 3) Table 3 Stable Temperature of Sensors

Sensor Temp(˚C)

T1 26.54

T2 26.35

T3 25.96

T4 26.38

T5 25.70

T6 26.80

T7 26.68

T8 25.81

The machine was switched from the cool position to the heat position and temperature reading for every sensor were taken approximately every 30 seconds during the following 10 minutes by using a digital camera to capture the readings from the Xplorer GLX, the time can be read on the screen of the Xplorer as well. The readings have been plotted vs time for every bar by using Matlab and can be seen in Figure 7.

Results

Figure 7: Graph of Sensor Temperature vs Time for every bar

Discussion As expected, the temperature raises first at the sensor that is closer to the heat sink, and after some time, the temperature at the sensor farther apart from the heat sink reaches the value attained by the first sensor. This illustrates how the heat flows from the warmer section of the body to the coldest. Notice the behavior of the temperature in both sensors for every material; in the graph corresponding to the Aluminum bar, the temperature of the farther sensor gets very close to the temperature of the closer sensor, this means that the difference in temperature between these two points of the bar is very small compared to the other materials, so the aluminum would have values of temperature very close to each other along the bar, compared to the other bars. Doing the same analysis for the steel bar, it can be observed that the temperature in the farther sensor it’s distant from the temperature of the closer sensor, indeed, if we examined the data from Error: Reference source not found it can be observed that the DT of sensors T7 and T8 increases with time, meaning that the temperatures at these points get farther apart from each other as time passes. As for the Brass bar, it exhibits a behavior between

Aluminum and Steel, both temperatures being close to each other at the beginning but then start to separate as time passes. Two Brass bars of different cross sectional area. Heat transfer by conduction between two points of a solid body can be determined by the equation: ´ kA (T 2−T 1) Q= d

Where: ´ Q =It’s the rate of heat transfer (watts) k= thermal conductivity of the material (w/mK) T2 and T1= temperature of two points in the same body (K) A= area of the section perpendicular to the heat flow (m2) d= distance between point 1 and point 2 The temperature data collected from the wide brass bar and the narrow brass bar can be used to examine the behavior of the heat transfer as a function of the cross sectional area. The rate of heat transfer has been calculated and plotted vs time for both bars in Figure 8.

Figure 8: Comparison of Heat Transfer in the wide and narrow brass bars

From Figure 8, two remarkable aspects can be pointed out. 1- The greater the cross sectional area, the greater the rate of heat transfer by conduction. 2- The rate of heat transfer shows a similar behavior in both bars. Observing the behavior of both plots, the curve representing the rate of heat transfer of the wide bar (Qw) seems to be the same curve of the narrow bar (Qn) multiplied by some factor. If the curve corresponding to Qw is actually proportional to Qn, then the quotient of both at every data point should be the same constant value at every data point, or at least very close to each other, proving that one curve is proportional to the other. In order to demonstrate the validity of this hypothesis, the quotient of Qw/Qn has been performed and plotted for every data point obtaining the results shown in Figure 9.

Figure 9: t Vs Qw/Qn

If we compute the mean and standard deviation of these values, we obtain: ´x =1.5227 s=0.044

Now, if the above hypothesis it’s true, we could predict the behavior of Qw by multiplying Qn times

´x .

Figure 10: Qw actual Vs Qw predicted

Figure 11: Comparison of the actual and predicted Qw

Computing the mean error of the predicted values results in an error of 2%. Consider now Qw is the rate of heat transfer of the wide bar and Qn is the rate of heat transfer of the narrow bar. The quotient of both is given by: k A w (T 2−T 1)w ´ Qw dw = ´ n k A n (T 2−T 1)n Q dn Since both bars are of the same material, the points corresponding to T1 and T2 are spaced the same distance in both bars (dw = dn), the equation can be simplified as follows: ´ w A w (T 2−T 1)w Q = . ´ n An (T 2−T 1)n Q Where the sub-index ‘w’ corresponds to the wide bar, and ‘n’ corresponds to the narrow bar. Computing the quotient of DT of both bars from the collected data, it can be seen that its value at every instant is approximately 1 (See Figure 12), which allows us to state the following relation.

´ w Aw Q = ´ n An Q

This means that the factor previously calculated x=1.5227 it’s actually the area ratio of the two bars. If we compute the area ratio out of the known dimensions of the bars we obtain: A w ( 3.5 ± 0.1 )∗(11.7 ± 0.1) A = =1.4 < w