Heat transfer by conduction and convection, Lab report

Lab Partners:

Corey Page Isabella Pinos Michael Perry Ron

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

AUTHOR: JOSE BETANCOURT

OCEAN ENGINEERING LAB REPORT

HEAT TRANSFER

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=kAT1−T2 ∆ 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=−kAdT 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

´

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 Width (mm) Length (mm)

Aluminum 11.8 +/- 0.1 87.4 +/- 0.1

Wide Brass 11.7 +/- 0.1 87.4 +/- 0.1

Narrow Brass 7.9 +/- 0.1 88.8 +/- 0.1

Stainless Steel 11.9 +/- 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 T1 T2 T3 T4 T5 T6 T7 T8

Temp(˚C) 26.54 26.35 25.96 26.38 25.70 26.80 26.68 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: ´ Q=kA (T 2−T 1) 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 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:

´ Q_w ´ Q_n= k Aw(T2−T1)w d_w k An(T2−T1)n d_n

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:

´ Qw ´ Qn = Aw A_n . (T2−T1)w (T₂−T₁)_n

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.

´ Qw ´ Qn = Aw A_n

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:

Aw An =(3.5 ± 0.1)∗(11.7± 0.1) (3.5 ± 0.1)∗(7.9± 0.1)=1.4 < Aw An <1.6

And the value of x=1.5 lies in the expected interval of area ratio.

Now concerning the thermal conductivity of the three materials used in this experiment shown in Table 1, where: k of aluminum = 150, k of brass = 115 and k of Stainless Steel = 14, we expect that aluminum shows a more effective conduction of heat than brass, and brass would exhibit a more effective conduction of heat than steel. This assumption is confirmed by the experimental data obtained since the temperature in both sensors of the aluminum bar remain very close to each other. Brass exhibits a behavior similar to the aluminum but with bigger differences between the temperature of both sensors, while the temperatures in the sensors of stainless steel trend to separate from each other with time.

Two problems on heat transfer

1- What would be the volume of the ice cube you would need to bring a 500 mL cup of coffee from 80 ˚C down to 60˚C if the ice cube is 0˚C?

Assuming that the coffee is mostly water, the properties of water will be used to represent those of coffee, we will assume that the heat transfer to the surroundings is negligible.

Energy lost by the coffee = Energy gained by ice (Conservation of energy)

[

]

[

]

Vice=

−

_[

]

V_ice= −

[

]

[

]

2- If you have a 200mW heat source inside a sealed container that is completely insulated against any transfer of heat, what is the maximum temperature that will be reached in the box after a very long time?

Conclusions

Given the evidence pointed above, we can conclude that the ratio of heat transferred by conduction of two bodies of the same material and constant cross section it’s equal to the ratio of the cross section area of the two bodies as long as they share the same heat source and the effects of other mechanisms of heat transfer upon the system can be neglected or assumed to be equal for both bodies.

Figure 12: DTw/DTn

Some errors during the realization of this experiment would include the absence of an insulating material on the bars to ensure a purely conductive behavior of the bars. Since the bars were exposed to the ambient air, part of the heat was lost to the ambient which could have altered the measurements taken.

Coffee cup station (1st _attempt)

Materials list

Large Coffee Maker 6 coffee cups with lids 1 Cup insulating sleeve Stirring sticks

1 Digital Thermometer

Experiment Setup and procedure

The coffee carafe was filled with water and poured into the coffee maker, in the meantime, the six coffee cups were identified by letters using a marker to write on them and were set in different configurations as shown in Figure 13. The description of the cups is as follows: A=Lid on with insulating jacket

B=Lid on C=No lid

D= No lid and blown on

E=No lid and stirred with a stick constantly

F= No lid and stirred with stick constantly and blown on

Figure 13: Set up of the coffee station experiment

Six different timers were set, one for each coffee cup, then the hot water was poured in every cup at a depth of 57.7 mm into each cup. The digital thermometer was introduced in

the first cup and once the reading was stable, the timer corresponding to that cup was started, the same procedure was repeated for all six cups, every member of the team was responsible for setting the timer corresponding to that cup, and blowing/stirring depending on the case. Readings were taken during 10 minutes for every cup and were recorded on a table (time and temperature, see Figure 14). Once the 10 minutes had passed, one of the cups was measured using a caliper and the following dimensions were obtained.

Diameter on top= 3 ½ in

Diameter on the bottom= 2 ½ in Wall thickness= 1/16 in

Height of the cup= 4 1/8in

Also the temperature of the room was measured using the same thermometer of the experiment and it was found to be 77 ˚F.

Results

Figure 15: Temperature measurements of coffee cup experiment, first attempt

Discussion

The data obtained from this experiment shows a very erratic behavior, there are peaks of temperature that raise above the previous values of temperature for the same cup, this can be observed in the graph of Figure 15, the curve corresponding to No lid and blown descends until a value of 62 Celsius and then increases to a value of 66.6 Celsius in the next reading, as if heat was being input to the system (which was not the case). A similar situation is notice on the purple line corresponding to the No Lid and blown scenario. This untypical behavior of the results could be due to the fact that the same thermometer was being used to measure the temperature of all six cups, so by the moment the thermometer is put in the next cup, it is already hot given it was already put into another cup whose temperature could be higher or lower, and if not enough time is allowed for the temperature to stabilize, the reading would be inaccurate, also, the time limitations to do the experiment didn’t allow to take enough readings to create a more reliable database.

Given the unreliability of this data, it was decided to redo the experiment doing some small modifications to the procedure.

Coffee cup station (2nd _attempt)

Materials list

1 lid

1 stirring stick 1 timer

1 Digital Thermometer 1 Coffee Maker

1 Cup insulating jacket

Figure 16: first coffee cup (lid on and insulating sleeve)

Figure 17: Digital thermometer

Experiment Setup and procedure

Before the beginning of the experiment, the air conditioner of the room was shut down so that it will not affect the measurements when the a/c turns on.

1st _{Lid on and insulating sleeve (A)} 2nd _{Lid on (B)}

3rd _{No Lid (C)}

4th _{No Lid and blown (D)} 5th _{No lid and stirred (E)}

6th _{No lid and stirred and blown (F)}

Hot water was poured into the cup, the digital thermometer was introduced in the cup and once the reading was stable the timer was started, taking readings every 30 seconds during the following 10 minutes and recording the data in a table, the same procedure was repeated for every setup until complete the 6 different configurations.

Results

Figure 18: Temperature readings from the different coffee cups

Discussion

Given the six configurations discussed above, we wanted to compere how the heat transfer increases or decreases in every situation compared to the cup that only has a lid.

Since the hot water is surrounded by a fluid media (air) and there is a difference in temperature between the water and the air, we know that we are dealing with a situation of

heat transfer by convection [2], so we can apply Newton’s cooling law to address the problem of finding the rate of heat transfer. Recalling the formula of heat transfer by convection: dQ dt =hA (Ts−T∞) and dQ=−m c_pdT Substituting we have −mc_pdT dt =hA (Ts−T∞) dT dt = −hA mc_p(Ts−T∞)

Since the term _mchA_p is a constant, we can call it k which yields: dT

dt =−k (Ts−T∞)

Solving the differential equation we obtain: Ts=T∞+(Ti−T∞)e

−kt

Were Ts is the temperature of the surface at the instant t and Ti is the initial temperature of the surface. Since in our experiment the value of temperature of the water is known at every instant, it is possible to solve for the constant k of every configuration.

k =

ln

(

T_s−T_∞

)

Calculating the k at every instant for every configuration and computing the mean and corresponding standard deviation using Matlab, the mean value of k for every case was obtained.

Table 4:

Value of constant k for every situation

Cup k std A 0.0002 0.0001 B 0.0002 0.0001 C 0.0005 0.0001 D 0.0012 0.0003 E 0.0006 0.0001 F 0.0015 0.0004

If we predict the values of temperature based on the obtained k using the equation Ts=T∞+(Ti−T∞)e

−kt

and plot the predicted values and the actual experimental values vs time we obtain the following graph.

From Figure 19 it can be noticed that the predicted value it’s very close to measured values when the situation it’s primarily convective (i.e. stirring, blowing, blowing and stirring). And it’s more inaccurate when insulators are added to the system (Lid and jacket, and just Lid).

Now it is possible to determine the h for every configuration: h=mk cp

A

Where m is the mass of water, A is the area of convection and cp is the specific heat of the water. So in conclusion, this experiment setup it’s an accurate way to estimate the value of the convection coefficient as long as the convection is the predominant mechanism of heat transfer and the effects of radiation and conduction can be neglected.

The error found from case 1 and case 2 it’s due to the considerable effect of the insulators, for example, when the cup is covered with the lid, the heat it’s transferred first to the air inside the cup and then it’s transferred by conduction through the lid to the environment, where finally the heat is dissipated by natural convection.

Table 5:

Convection area, h and comparison of rate of heat transfer

Cup Convection Area

(m2) h (Q- QB )/QB*100 A 0.0059 0.0377 -26.9% B 0.0183 0.0165 0 C 0.0229 0.028 112.6% D 0.0229 0.0682 416.8% E 0.0229 0.0332 151.3% F 0.0229 0.0834 532.1%

Referring to Table 5, the area of convection changes in every situation knowing that this area it’s the one in contact with the surroundings, in this case we haven’t considered the variations in temperature at the wall of the cup due to the thermal conductivity of the

material of the cup, so in cup A (jacket and Lid), the area of convection it’s just the portion of area of the level of liquid that raises above the insulating jacket, since the surface of the liquid itself it’s not in contact with the surroundings provided the lid is on, on the other hand one the lid and jacket are removed, this increases the area of the convection since now the wall area in contact with the liquid it’s exposed to the ambient and the surface area of the liquid it’s also exposed. Surprisingly the coefficient h in case A it’s greater than the one in case B (no jacket), but the product hA it’s smaller, this is what makes the rate of heat transfer in situation A being smaller than B

Figure 20: Comparison of the rate of heat transfer vs time

Thermocouple Experiment Materials List

1 Arduino Due OE Lab Board 1 OE Lab Board power adapter 1 Dell Laptop

1 Dell Laptop power adapter Arduino software on the laptop DataPlot3 software on the laptop 1 micro USB-serial cable

1 type K Thermocouple with 45 cm leads

1 Short solid 22ga wire stripped 5 mm on each end Transistor Unit

11.3 Ω Heat Resistor 1 Clothes pin

Experiment setup and procedure

The laptop and OE lab board were plugged into the wall, and the USB cable was connected from the OE lab Board USB port named PROG USG to the Laptop, then the yellow wire of the thermocouple was inserted in the pin +BRG on jumper J3, the other wire of the thermocouple was inserted in pin –BRG with a short jumper cable connected to the ground pin, then the alligator wire was connected to a ground pin on J3 as shown in Figure 21.

Figure 21: Connection of the thermocouple and jumper cable to the OE lab Board

Now the resistor end of the transistor unit was attached to pin D/P 8 on the board and the black alligator clip was attached to one of the leads of the heater resistor (Figure 23)

Figure 22: Connection of the transistor’s unit resistor

J3 Jumper wire Alligator wire Thermocouple Yellow wire of thermocouple Transistor unit’s resistor Transistor unit

Figure 23: Connection of the heater resistor

The red alligator clip from the transistor unit was then connected to the fuse on the OE lab Board as shown in Figure 24, and the thermocouple was held in contact with the heater resistor using the clothes pin (Figure 23).

Figure 24: Transistor unit connected to fuse

The main idea of this experiment it’s to prove that it is possible to adjust the temperature of a room, boiler, tank, etc, by moving a potentiometer to a different location, which controls the intensity of current that passes through a heater resistor. The thermocouple in contact with the heater resistor provides the data read by the board which is converted in the code

Alligator clip coming from ground Thermocouple Heater resistor Alligator coming from transistor unit Transistor’s Alligator clip connected to fuse

to a value of temperature, also the position of the potentiometer is read and written to the current to increase, stop or decrease the intensity depending on the case, the readings were collected through the Arduino’s serial monitor and pasted into a new text document for storage and further analysis.

Results

Figure 25: Temperature and potentiometer position vs time

Discussion

Commented code used in the lab:

// set up global scope variables for the moving average calculation int i_dataPosition =0;//initialize variable

float a_f_Readings[500]; void setup(){

Serial.begin(9600);// initialize serial comunication at 9600 bauds analogReadResolution(12);//set resolution on analog read to 12 pinMode(8,OUTPUT);// declare pin number 8 as an output }

void loop(){ //start loop

// Read the voltage at the thermocouple amplified input int sensorValue=analogRead(A6);

//Convert it to volts with zero referenced to the zero point of the amplifier

float f_TC_Volts=((((float(sensorValue)-2034.839/4095.0)*2.97)/99.0); //2034 is center of bridge input.

//Read the voltage at the temperature sensor input int tempValue=analogRead(A7);

//convert the DAQ reading into a value in degrees Celsius float f_CJ_Temp=((float(tempValue)/4095.0)*2.97)-0.5849; f_CJ_Temp=(f_CJ_Temp/0.00625);// in degrees Celsius

// Extremely simplified Thermocouple formula

float f_Temp=(24.375*f_TC_Volts*1000.0)+0.438+f_CJ_Temp; // Use a moving average

//Set one element in the array to the current reading a_f_Readings[i_dataPosition]=f_Temp;

//Get total of the reading float f_TempTotal=0;

for(int i_count=0; i_count<500; i_count++){ f_TempTotal+=a_f_Readings[i_count]; }

//increment the array position i_dataPosition++;

//loop the array back to the start when it overruns if(i_dataPosition>499)i_dataPosition=0;

f_TempTotal=f_TempTotal/500.0;

float f_setPoint=analogRead(A9);// read voltage from potentiometer

f_setPoint=20.0*(f_setPoint/4096.0)+35.0;// convert voltage into a scaled value Serial.print(f_TempTotal,3);// print temperature to serial monitor

Serial.print(",");

Serial.println(f_setPoint,3);// print potentiometer's position to serial monitor

float f_outputpower=(f_setPoint-f_TempTotal)*10.0; // Proportional control statement int i_Out = constrain(f_outputpower+((f_setPoint-35.0)*3.7),0.0,255.0);

analogWrite(8,i_Out);// Takes a vulue between 0 and 255 to heat the resistor delay(10); // wait 10 ms

The oscillations exhibited by the temperature are typical of a proportional control such as the one that has been implemented in the code, according to [3], this error can be eliminated by the implementation of an integral control which will eliminate the steady state error derived from the proportional control.

In Figure 25 it can be seen that a variation on the position of the potentiometer has a direct effect on the temperature read by the thermocouple, this happens because when the position of the potentiometer it’s changed, its voltage it’s read by the microcontroller board and used to write a new value of voltage on the output pin connected to the resistor which will produce a variation on the current that passes through the resistor. Here we take advantage of Joule’s effect to generate heat in the resistor and heat up the thermocouple. However, when the temperature needs to be reduced to the minimum, the only thing that can be done it’s to stop the current going to the resistor and let it cool down by convection, which implies that we lack control on the cooling process.

After the discussion shown above we can conclude that an integral control loop it’s necessary to reduce the oscillations of the readings derived from the use of a proportional control, however the desired effect was achieved and this method represents a simple way to address the problem of temperature control. This methodology provides effective control on the increase of temperature, but the cooling process will depend on the surroundings of the system, since the resistor will cool down by convection and conduction if it is in contact with another body, and not because of the effect of a control variable.

Works Cited

[1] U. Besson, The History of Cooling Law: When the search for simplicity can be an obstacle, Pavia, Italy: Springer, 2010.

[2] Y. Cengel, Heat and mass transfer, 2nd edition, Reno: McGraw Hill, 2007. [3] K. J. Astrom, Control System Design, 2002.

[4] "pasco.com," 07 11 2015. [Online]. Available: www.pasco.com/prodCatalog/TD/TD-8513_heat-conduction-apparatus.