butler belcher german

7/27/2019 Butler Belcher German

http://slidepdf.com/reader/full/butler-belcher-german 1/27

1

Unsteady State Heat

Transfer

RJ Butler Donald Belcher Carrie German

CHE 4002Project #2 Pre-plan

2/21/2013



2

Executive Summary

An experiment was developed and carried out for the analysis of unsteady state heattransfer in the unit operations lab. The objective was to create a unit for unsteady stateheat transfer analysis, and to then use this unit to determine the thermal diffusivity of a

slab of test material. This diffusivity was compared to a range of reference thermaldiffusivity values. Comparison of these values demonstrated the uncertainty of thermaldiffusivity of concrete. Uncertainty in thermal diffusivity of concrete could lead toheating and cooling issues in construction.

This project was conducted in two phases completed in a simultaneous manner. The first phase was to build the unit for the heat transfer experiment. The team decided that due tothe high temperature of the heating element, a ceramic box should be used. The teamcould only find ceramic tiles, so the team encased the tiles in an aluminum crate. Theteam suspended the aluminum/ceramic box inside of an outer wooden box, and placedfiber glass insulation in between the two.

Once the basic build of the crate was complete, experiments were started inside the unit.During the experimental trails, the crate did not have a lid. While the experiment wasrunning, the lid was built for the crate. The lid was completed for future use, but had noinvolvement in the initial experiment. The lid was completed with the intention of placingthermocouples through the top, to be placed into a slab of test material inside the unit asto minimize error in the experiment.

The experiment was conducted using a 1 ft. X 1 ft. X .027in. slab of concrete. Onedimensional unsteady state heat transfer analysis was performed by placing fivethermocouples into drilled wells of equal elevation. The heating source was placed along

one side of the concrete slab and the rheostat was set to level 3. Temperature data wasrecorded at 2 minute intervals for ten minutes, 5 minute intervals for an additional 50minutes, 10 minute intervals for an additional 30 minutes, and one fifteen minute interval,for a total of 105 minutes.

A finite difference method was used to calculate theoretical temperatures at each point atgiven times. Regression of theoretical and experimental temperature differences wasused to determine the slabs thermal diffusivity. This determined thermal diffusivity waslower than the reference range of thermal diffusivity values [2].

Ideally, multiple slabs of uniform composition will be used. Error in thermal diffusivity

calculation would be minimized because variation in density, heat capacity, and thermalconductivity would decrease. Analysis differences in thermal diffusivity of differentslabs of test material of singular composition should be investigated.

$863 of the provided $15000 budget has currently been used. The recommendations provided will be funded by the remaining budget.



3

Table of Contents

Unsteady State Heat Transfer ............................................................................................. 1 Executive Summary ............................................................................................................ 2 Objective ............................................................................................................................. 4 Rationale ............................................................................................................................. 4 Overview ............................................................................................................................. 4 Experimental Equipment .................................................................................................... 5 Environmental, Health, and Safety ..................................................................................... 7 Theory ................................................................................................................................. 8 Data Processing/Required Measurements ......................................................................... 10 Evaluation ......................................................................................................................... 13 Experimental Plan ............................................................................................................. 13 Results ............................................................................................................................... 14 Conclusion ........................................................................................................................ 16 Recommendations ............................................................................................................. 16 References ......................................................................................................................... 19 Appendix 1 ........................................................................................................................ 20 Appendix 2 ........................................................................................................................ 21



4

Objective

The objective of this project is to analyze unsteady state heat transfer. A slab of testmaterial of known composition, thermocouples, a heating source, and a ResistanceTemperature Detector (RTD) will be required to run this experiment. These products willallow us to model the dimensional series of differential equations used to describeunsteady state heat transfer in order to determine the thermal diffusivity of the slab.

Rationale

The addition of this new experiment to the Unit Operations Lab will provide students anopportunity to better understand heat transfer. Students will gain experience via a hands

on learning technique for equipment that may be encountered in their career path after graduating.

Heat transfer analysis is necessary in the design of boilers, condensers, evaporators,heaters, refrigerators, and heat exchangers. The amount and rate of heat transfer must beconsidered during design in order to provide accurate instrument performance parameters, as well as to avoid instrument destruction. Heat transfer analysis isimperative in the design of electronic components, electric machines, and transformers toavoid the overheating and damage of equipment [4].

Overview

A slab of test material of desired composition will be chosen. Wells will be drilled into aslab of test material for a predetermined number of thermocouples. Well location will bedetermined based off of a template that lines up with predrilled holes for thermocouple placement through the lid (See Appendix 1).

The heat source will be connected to the rheostat and attached to the slab of test materialin the desired position. (Note: The power supply should not be plugged into the wallyet.) The slab will be placed inside of a prebuilt, insulated box with the drilled wellsfacing upward.

The thermocouples will be connected to the RTD and carefully placed into the slabswells. (Note: The lid will not be in place for this experiment.) The RTD will be pluggedin and initial slab temperatures will be found for each thermocouple. A thermocouplewill be placed at a point where the heat source and slab of test material meet in order tomonitor the temperature of the heating pad.



5

Once the initial temperatures are obtained, the rheostat will be plugged into the wall to begin heating. The temperature at each thermocouple will be recorded at designated timeintervals. Analysis of the thermal diffusivity of the slab will be performed.

Experimental Equipment

Insulated Box: This equipment will be used to hold the slab of mass, heat source, andthermocouples while gathering temperature data. Insulation inside of the box will reduce error by reducing heat transfer to or from outside sources.(Insulation is not shown, but will be placed in between the aluminuminner box and the wooden outer box.)

Wooden Box Dimensions: 22” X 22” X 21”

Aluminum Box Dimensions: 13” X 13” X 13”

Figure 1 - Insulated Box- shown above is the insulated box the team has designed for the experiment. The box

has a wooden exterior with a aluminum and ceramic interior

Heat Source: This equipment will be a flexible heating pad that can be placed almost

anywhere on the slab of test material and capable of heating up to 450 ⁰ F.The heating pads flexibility allows for placement on corners as well as flatsurfaces. These pads have an adhesive side, ensuring that the heatingsource is immobile and secure. These heating pads are available in circular or rectangular shapes and numerous dimensions. A rectangular, 5” X 5”

heating pad will be used in this experiment.



6

Figure 2-Heat source[2]

Shown above are heating elements coated with a silicon outer layer to prevent damage to

the heating coils.

Rheostat (power source): This instrument will supply power to the selected heat source.The rheostat does not have the ability to set the temperature. The rheostat provides cyclical power to maintain a constant

Figure 3 – Rheostat- Shown above is a rheostat used for controlling the current and cycles provided to a powersource.

Thermocouple: This instrument will determine the internal temperature of the slab by producing a voltage differential. Five thermocouples were used in thisexperiment for one dimensional, unsteady state heat transfer analysis.

.Figure 4-Thermocouple Example [4]- shown above is a typical J type thermocouple without quick disconnect

attachments.



7

Resistance Temperature Detector (RTD): The RTD outputs the temperature reading atthe tip of the thermocouple in degrees Fahrenheit. The dial on the right,seen in Figure 5, will be turned to find the temperature of thermocouplesat different locations throughout the slab.

Figure 5 - Resistance Temperature Detector whoen above is an RTD with 10 possible outputs. The Temperature

scale for the RTD is in Fahrenheit.

Slab of mass: This will be a slab of test material of known composition. Materials suchas stone, wood, metal could be used. For this experiment, a1 ft. X 1 ft. X3.25 in. slab of concrete was used.

Figure 6-Examples of slabs of mass[4] The slabs of mass shown above are of varying material of granite, wood

and metal.

Environmental, Health, and Safety

There are no environmental or health hazards associated with this experiment.

There are three safety hazards:1. Caution should be taken to avoid liquid spills around the electrical

components of the experiment, as this could lead to electrocution.

2. The heating source may be extremely hot, so caution should be taken to avoid being burned or igniting a fire.

3. Because of the need to drill wells for the thermocouples in a slab(s), only anindividual with experience should drill and caution should be taken to avoidcutting or puncturing oneself.



8

Theory

Transient Energy Conduction Theory

Heat transfer occurs with the exchange of thermal energy and heat through the means of

convection, conduction, and radiation. Of importance in heat transfer is the conduction of energy through solid materials. Conduction occurs when energy is exchanged by theintermolecular contacts in the material established by a temperature gradient.

In the case of transient conduction heat is added to the system by an imposed change intemperature at some point in the system. This change in temperature establishes agradient in the system facilitating heat transfer from the high to the low regions. Thesetemperature gradients show the energy differentials in the system as presented in Figure7. This differential results in a transfer of energy that varies as a function of the energydifferential. This transfer of energy decreases over time as the differentials decreases andthe system approaches an equilibrium condition. When the equilibrium conditions are

met, the system is no longer in the transient condition. Equilibrium will never beachieved.

Figure 7- Transient heat conduction in a rectangular coordinate system: The above figure shows a

rectangular shell with heat transfer occurring from a heat source across the slab. The arrows

indicate a gradient of the heat transfer at a transient condition. (This figure was created by Drew

Belcher using Microsoft Paint)

For a solid three dimensional object, the equation for heat conduction is stated as thedifferential temperature generation with respect to time as a function of the Fourier function of the temperature gradients with respect to the system volume [1].



9

(1)

Where:

ρ ≡ density of the solid (lbm/in3 )

C p ≡ specific heat (Btu/(lbm* ⁰ R))

T ≡ Temperature ( ⁰ R)

T ≡ Time (min)k ≡ Thermal conductivity( ⁰ R /(in*Btu/min))

*Thermal conductivity, specific heat, and density values will be obtained from literature for this experiment

Assuming the thermal conductivity and density are independent of temperature andlocation, equation 1 can be reduced to the form [1]:

(2)

Where:

: Thermal diffusivity (in2

/min)

Transformation of equation (2) into rectangular coordinates gives:

(3)

Where: x≡ x coordinate in Length(in)

y≡ y coordinate in Width (in)

z ≡ z coordinate in Height (in)

The following dimensionless numbers for both temperature and length are used tointerpret equation (3) [1].

(4)

Where:

Θ ≡ Nondimensionalized temperature

T 0≡ initial temperature ( ⁰ R)

T 1≡ Steady state temperature ( ⁰ R)

(5)

Where: β≡ Nondimensionalized Length

η≡ Nondimensionalized Widthγ≡ Nondimensionalized Height

L ≡ Length of slab (in)

W ≡ Width of slab (in) H ≡ Height of slab (in)



10

By substituting the terms in equations (4 & 5) into equation (3), the heat transfer equationfor rectangular coordinates can be transformed such that [1]:

(

) (6)

Thermocouple Theory

Thermocouples function by using two dissimilar metals to create a voltage when heated.This voltage varies as a function of temperature that is defined by the equation [5]:

∑ (7)Where:

an≡ Coefficients based on thermocouple materials obtained from

databases(K n /V

n )

V n≡ Voltage generated from thermocouple (V)

For temperature measurements within a range specified by the thermocouplemanufacturer, a linear approximation can be used to describe the temperature as a

function of voltage such that [5]∷

(8)

a≡ Correcting coefficient.( ⁰ R /V)

Data Processing/Required Measurements

Temperature data (°F) will be collected at selected time intervals (min) for the duration of the run or until the system reaches steady state, depending on initial temperature and slabcomposition.

Finite Difference Method

To approximate the one dimensional temperature gradient for the slab of test materialwith variable boundary conditions the team can utilize the finite difference method. Thereare three forms of the finite difference method utilized to numerically solve thedifferential equations.

Forward difference:

(9)

Backward difference:

(10)

Central Difference



11

(11)

Where:h ≡ step size

f(x) ≡ function of x

In the one dimensional transient heat conduction system, the forward difference equationcan be transformed into the following form:

(12)

By eliminating constant terms from the heat conduction equation 6 the conductionequations can be reduced such that:

(

) (13)

Therefore it is possible to transform equation 13 into the form:

(14)

Where:

n ≡ step position in first dimension

i ≡ step in second dimension

For the boundary condition at x=β it is necessary to use the Direchlet-Neumann boundarycondition such that:

(15)

Where:

N ≡location where β = 1

The boundary condition at the β = 0 is defined as:

(16)Where: Polynomial(t) ≡ equations obtained from Excel generated system.

Regression

To determine the thermal diffusivity of the slab the team will utilize nonlinear regression.The process of nonlinear regression compares theoretical values with experimental valuesto determine the value of independent variables. By using the least squares method todetermine the value of α for the equation.

∑ (17)



12

Where:

S ≡ sum of the squares of residuals.

and the residual:

(18)Where:

α ≡ independent constant

xi≡ location in x at i f()≡ theoretical value

yi≡ experimental data

The team can estimate a value for the thermal diffusivity by using Solver in Excel to findthe minimum of the least squares by varying the value of the thermal diffusivity. Byexamining the residuals of the experimental and theoretical data, the team can determinewhere the experimental data varies from the proposed theoretical data for the estimated

thermal diffusivities.

F Test

The F test is used to identify if a model fits a population of sampled data. By interpretingthe data using the null hypothesis the F test can compare with the F-distribution tocompare statistical models. The test statistic is a ratio of the sum of squares taking intoaccount sources of variability. The value obtained for the F test can be generated from thefollowing equation.

(19)

Where: F≡ F statistic

S 1≡ Residual sum of squares (Equation 17) X

2≡ Chi squared test statistic

n ≡ number of observations

p ≡ parameters.

Standard ErrorTo determine the effect of probably error on the precision of the observed values thegroup chose to utilize the 95% confidence interval. Using this interval the group chose to

utilize the error analysis such that:

(20)Where:ε95%: Probable error

1.96: Value determined by the 95% Z statistical analysis of the normal Gaussian

distribution.σ : standard deviation of the sample



13

Where the standard deviation is defined as

√ ∑ (21)

Where:σ : Standard deviation.

N: Number of observed values. xi: Observed value.

: Theoretical value

Evaluation

Before the initial time, t = 0, the temperature across the slab will be constant. As the heat

source is added, the temperature at the contact point will increase, while the rest of thesystem remains unaffected. As time progresses, the temperature gradient throughout theslab will increase. Thermal diffusivity will remain relatively constant throughout theconcrete slab as the density, specific heat, and thermal conductivity are assumed toremain constant. The team expects the model to correspond to the experimental data.

Experimental Plan

Fabrication1. Choose composition of slab(s)2. Measure diameter of thermocouples for drill bit sizing3. Decide/mark thermocouple/heat source placement4. Drill holes for thermocouples using drill press (3/4 depth of slab)5. Assemble insulated box from components to desired configuration

Start-up1. Place heat source on slab in desired position2. Place slab into insulated box3. Place thermocouples in pre-drilled holes to desired depth4. Make sure wiring of the thermocouples are connected to the RTD5. Make sure wiring of the heat source is connected to the rheostat

Run-time1. Choose a time interval to record temperatures2. Turn on heat source3. Record temperature of each thermocouple at set time intervals4. Allow slab to cool5. Repeat steps 1 – 4 for each slab

Shut down



14

1. Turn off heat source2. Let slab cool down3. Remove heat source/thermocouples4. Remove slab from box

Emergency shut down1. Turn off heat source2. Extinguish any fires if necessary3. If no signs of fire remove lid from box4. Remove heat source5. Use a lab approved cooling agent to rapidly cool slab and interior of box6. Remove slab from box

Results

The team found the temperature values at the heating element to vary as a function of time such that during the interval between 2 minutes and 60 minutes the equation to

describe the temperature is . For the timeinterval from 70 minutes to 105 minutes the equation to describe the temperature as a

function of time at the boundary condition is . There is adiscontinuity approximately 100 degrees in magnitude for the temperature between the60 minute and 70 minute intervals as shown in Figure 9. The team also found that at t 0 the temperature across the slab was a constant temperature of 78.2.

Modeling the theoretical data the team constructed a temperature gradient system with h= 1 in and Δt=0.5 with X= 12 in. This data is constructed as shown in Table 2 andgraphically demonstrated below in Figure 7.



15

Figure 7- The figure above shows the temperature gradients for various times from t=0 to t= 105. The distance is

expressed in β which is defined as fractional distance.

The team found the values of the thermal diffusivity to be 0.027 in2/min for the slab takenfrom the time interval from 2 and 105 minutes. At the time interval from 2 minutes to 60minutes the thermal diffusivity at 2 inches from the heat source was 0.037 in 2/min withthe thermal diffusivity at locations greater than or equal to 4 inches being 0.027 in2/min.The values for the calculations are shown in tables 3 and 4.

By utilizing statistical analysis the team determined the standard error and F test value for each of the thermocouples with the model equation as shown in the table below.

Table 1- the table below shows the results from various statistical analyses of thermocouple data against the

theoretical model.

Tx1 (α=0.037) Tx1 (α=0.027) Tx2 Tx3 Tx4 Tx5

F Test 0.83 0.94 0.79 0.37 0 0

R^21.00 0.94 0.96 0.95 0.68 0.52

Standard Error 0.95 5.58 0.70 0.54 0.52 0.47

Tables 5-10 listed in the appendix show the residual plots from experimental data againstthe theoretical model generated in excel.



16

Conclusion

The experimental thermal diffusivity value of 0.027in2/min did not fall within the valuerange of 0.048in2/min – 0.144in2/min for regular concrete found in literature [6]. Ascalibration for the thermocouples, rheostat, heat pads, and RTD have not been performed;

this may be a source of error in this experiment. Also, the flexible heat pad was not placed directly on the concrete slab, but rather adhered to a ceramic tile and placed firmlyup against the concrete slab. This may have allowed air to gather in between the heatsource and concrete slab, altering thermal diffusivity. The lack of insulation and runningof the experiment without the box lid would also be another source of error, as this wouldallow convective forces and external temperature gradients to affect experimental data.An example of the effect of external temperature is found in the similarities in theresiduals shown in tables 9 and 10. These phenomena were not accounted for intheoretical calculation which would have an affect on the determination of the thermaldiffusivity through regression. Another source of error may arise from assuming thethermal diffusivity is constant throughout the slab. This assumes constant density,

thermal conductivity, and heat capacity throughout the slab. The concrete slab used wascomprised of more than one type of material, so the assumption that these propertiesremained constant was not valid for this case.

For the construction of the experiment the team was given a budget of $15000 for materials and equipment. As shown in Table 11 the team has only spent $863at the timeof this report (not including the cost of shipping). Future uses of the remaining budget aredetailed in the recommendations section below.

Recommendations

Slabs of different composition: Right now the only slab provided is a 1ft. X 1ft. X

0.27ft. block of concrete. This was suitable for the initial experimentation of

unsteady state heat transfer due to time constraints of this project. The downside

to concrete is the fact that is has rocks incorporated into the slab. These rocks

have different densities and therefore different heat transfer coefficients, which

make theoretical calculations for this slab difficult. New slabs, with a consistent

composition throughout, will be of better use for this experiment. The new slabs

should not exceed the current length and width, as the insulated box would not

support such dimensions. Slabs such as; wood, cement, glass, ceramic, stone, or

metal would be excellent choices for this experiment, and would provide futurestudents the opportunity to study heat transfer through multiple materials.

Thermocouples: The current thermocouples in use have one data collection point

at the tip of the thermocouple. There are thermocouples available that possess

multiple data collection points throughout the length of the thermocouple. This



17

can provide the possibility for more precise results in 2-dimensional and 3-

dimensional data collection through the slabs.

More flexible heating pads: There are 4 different heating pads available at the

time being. There is one 1in. X 12in. and three 5in. X 5in. flexible heat sources.

The three 5in. X 5in. heating pads have adhesive on the backside. This adhesive

is beneficial to the experiment so there isn’t the possibility of the heating pad

falling off of the slab. Ordering more flexible heating pads with multiple

dimensions, and an adhesive backside would be advantageous in future

experiments. The new heating pads would provide options that would best fit the

slab and analysis for a multitude of set ups.

Finish the lid of the Unsteady State Heat Transfer experiment: There is a gap

between the lid and the base of the box. Aside from making the box more

aesthetically pleasing, attaching a lip or overhang to the lid will close the gap

between the lid and base, while giving the box complete insulation from its

surroundings. t will also provide an additional safety measure by preventing

anything from the surroundings to enter the box.

Protective barrier: Inside the box, between the outside wooden box and the inner

aluminum box, is a layer of fiberglass insulation. This insulation is open to the

surroundings, and contact is possible. Constructing a protective barrier over the

showing insulation will provide safety measures to prevent contact with the

insulation and possible skin irritation. The recommended material for the

protective barrier is wood.

New data collection unit: During the initial experiment, two resistance

temperature detectors (RTDs) were used in the data collection process. Since the

RTDs are only capable of displaying one thermocouple output at a time, reading

measurements at specific time intervals is difficult if more than one thermocouple

is in use. Also, each RTD is only capable of holding 10 thermocouple inputs at a

time. It is suggested that a new RTD unit be purchased for this experiment, to

ensure more precise measurements and to accommodate all 25 possible

thermocouple wells. Specifications of the new RTD should be capable of holding

at least 30 thermocouple inputs and simultaneously displaying all thermocoupleoutputs. If possible, it is suggested that the RTD to be purchased be capable of

transferring the thermocouple output values to a computer, using software such as

LabView. An RTD of this nature would provide the ability to record

instantaneous measurements for each thermocouple at the same time, delivering

more precise results.



18

CFD (computational fluid dynamics) program: A CFD program uses numerical

methods and algorithms to solve and analyze problems that involve fluid flows

such as heat transfer through an object. The program will use input equations for

the system, solve them simultaneously, and then output results in a desired

format. The two options that will benefit this experiment the most are the tabular

and graphical forms. The tabular option will list the results in a table that is easy

to read. The graphical method will display a diagram that represents the heat

transfer through the slab (Figure _). The graphical method will be most useful in

2-dimensional and 3-dimensional analysis.

Figure 8-Example of the CFD graphical method for 2-dimensional heat transfer

New power source: The power source provided doesn’t have the capability of

controlling the output temperature of the heating pad. It is recommended that a

power source with a computer controlled temperature gauge be used in future

experiments. Ideally, this temperature gauge should be controlled through the

same software that used by the new data collection unit.

High temperature silicon insulation caulk: Using this, the team can seal all cracks

and cervices inside both the aluminum and wooden box. Also, a high temperature

resistant silicon caulking should be used to prevent melting at the high

temperatures inside the box during the experiment. It will also provide extra

protection from sharp edges and the fiberglass insulation.

Outer wrappings for thermocouples: These wrappings will decrease any error in

measurements taken during the experiment.

Calibrating the heating element: Calibration of the heating pads and

thermocouples will make data collection more precise, while also providing an

http://www.google.com/url?sa=i&rct=j&q=heat+transfer+cfd&source=images&cd=&cad=rja&docid=J_50u1CGCQQLxM&tbnid=08CMcXOSaA1FLM:&ved=0CAUQjRw&url=http://www.fnb.maschinenbau.tu-darmstadt.de/de/software/database/index.php&ei=wfxAUaapF8Tm2AWvqYH4Dg&bvm=bv.43287494,d.b2I&psig=AFQjCNEX4sR6F69SfXUHUACb2DsfjfMywQ&ust=1363299816911446



19

accurate value for boundary conditions used in calculating theoretical results of

the experiment.

References

1. Bird, Byron, Warren Stewart, and Edwin Lightfoot. Transport Phenomena. 1.

John Wiley and Sons Inc., 1960. 352-360. Print.

2. http://www.omega.com/pptst/SRFR_SRFG.html

3. http://ptumech.loremate.com/ht/node/4

4. http://www.google.com/images

5. Scervini, Michele. Thermocouples; The Operating Principle. University of Cambridge. 2009.http://www.msm.cam.ac.uk/utc/thermocouple/pages/ThermocouplesOperatingPrinciples.html

6. "Loremate: Heat Transfer." Loremate. Loremate. Web. 13 Mar 2013.<http://ptumech.loremate.com/ht/node/4>.

http://www.msm.cam.ac.uk/utc/thermocouple/pages/Thermocouples%20OperatingPrinciples.html







20

Appendix 1

Template for Thermocouple Wells

Figure 9- Diagram for thermocouple layout on a ceramic tile. The circles mark where holes will be drilled to

allow the thermocouple to be inserted into the box.



21

Appendix 2

Figure 10- This figure shows the temperature as a function of time for the heat source as well as equations

established from excel generated analysis.

Table 2-This table shows the format and values of the gradient table used to calculate the transient heat

conduction system using forward difference methods with variable thermal conductivity.

0 1 2 3 4 5 6 7 8 9 10 11 12

α 0.027 in2/min β 0.00 0.08 0.17 0.25 0.33 0.42 0.50 0.58 0.67 0.75 0.83 0.92 1.00

T0 78 t (min) T (F) T T T T T T T T T T T T

0.0 79.0 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

a b 0.5 131.7 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 A 0.00E+00 0. 00E+00 1.0 132.8 79.9 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

B 0.00E+00 0. 00E+00 1.5 134.0 81.6 78.3 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

C 0.00E+00 0. 00E+00 2.0 135.1 83.1 78.4 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

D 0.00E+00 0. 00E+00 2.5 136.2 84.7 78.6 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

E - 2.02E- 02 0. 00E+00 3.0 137.3 86.1 78.8 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

F 2.35E+00 1.44E-01 3.5 138.5 87.5 79.0 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

E 1.30E+02 2. 62E+02 4.0 139.6 88.9 79.3 78.3 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

X 12 in 4.5 140.6 90.2 79.6 78.3 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

ΔX 1 in 5.0 141.7 91.5 80.0 78.3 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

Δt 0.5 min 5.5 142.8 92.7 80.3 78.4 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

. . . . . . . . . . . . . .

α1 1.19 . . . . . . . . . . . . . .

α2 1.37 . . . . . . . . . . . . . .

α3 1.19 100.5 276.3 219.8 172.5 135.6 110.3 95.3 86.8 82.3 80.1 79.0 78.5 78.3 78.3

α4 1.00 101.0 276.4 220.1 172.9 136.0 110.6 95.4 86.9 82.4 80.1 79.0 78.5 78.3 78.3α5 1.00 101.5 276.4 220.4 173.3 136.4 110.9 95.6 87.0 82.5 80.1 79.0 78.5 78.3 78.3

α6 1.00 102.0 276.5 220.7 173.6 136.7 111.1 95.8 87.2 82.5 80.2 79.1 78.5 78.3 78.3

α7 1.00 102.5 276.6 221.0 174.0 137.1 111.4 96.0 87.3 82.6 80.2 79.1 78.6 78.3 78.3

α8 1.00 103.0 276.7 221.3 174.4 137.5 111.7 96.2 87.4 82.7 80.2 79.1 78.6 78.3 78.3

α9 1.00 103.5 276.7 221.5 174.8 137.8 112.0 96.3 87.5 82.7 80.3 79.1 78.6 78.3 78.3

α10 1.00 104.0 276.8 221.8 175.1 138.2 112.3 96.5 87.6 82.8 80.3 79.1 78.6 78.4 78.3

α11 1.00 104.5 276.9 222.1 175.5 138.5 112.5 96.7 87.7 82.8 80.3 79.1 78.6 78.4 78.3

α12 1.00 105.0 276.9 222.3 175.8 138.9 112.8 96.9 87.8 82.9 80.4 79.2 78.6 78.4 78.3

constants

Temperature equation

Theoretical model

Thermal Diffusivity Modifiers



22

Table 3- This table shows the regression of the experimental and theoretical data for the time range of 2 minutes

to 60 minutes. As the values from Tx3 – Tx5 have an innsugnificant variance in this range, only Tx1

and Tx2 are used to determine the values for the thermal conductivity.

Figure 11- The above figure shows the difference between the experminental and theoretical data for the

thermocouples X1 and X2 for the interval 0-60 minutes. The relations between these values are found in tables 6

and 7.

t (min) Tx1 Tx2 Tx3 Tx4 Tx5 t (min) Tx1 Tx2 Tx3 Tx4 Tx5

2 79.3 78.6 78.5 78.1 78.2 2 78.39 78.20 78.20 78.20 78.20 0.84 0.16 0.09 0.01 0.00 X1 0.83

4 79.7 78.8 78.6 78.4 78.3 4 79.31 78.20 78.20 78.20 78.20 0.15 0.36 0.16 0.04 0.01 X2 0.44

6 80.6 78.8 78.6 78.4 78.3 6 80.72 78.21 78.20 78.20 78.20 0.01 0.34 0.16 0.04 0.01 X3 0.81

8 81.9 78.9 78.7 78.5 78.3 8 82.43 78.25 78.20 78.20 78.20 0.28 0.43 0.25 0.09 0.01 X4 0.00

10 83.3 79.0 78.8 78.7 78.3 10 84.31 78.31 78.20 78.20 78.20 1.03 0.48 0.36 0.25 0.01 X5 0.00

15 88.7 79.4 78.9 78.7 78.9 15 89.38 78.65 78.21 78.20 78.20 0.47 0.57 0.48 0.25 0.49

20 93.4 80.2 78.8 78.6 78.2 20 94.59 79.28 78.23 78.20 78.20 1.41 0.85 0.33 0.16 0.00

25 99.0 81.2 79.0 78.5 78.6 25 99.71 80.20 78.28 78.20 78.20 0.51 1.01 0.51 0.09 0.16

30 104.1 82.5 79.5 78.7 78.9 30 104.68 81.37 78.39 78.21 78.20 0.34 1.28 1.24 0.24 0.49 mod C

35 109.0 84.0 79.7 78.7 79.0 35 109.45 82.75 78.55 78.22 78.20 0.20 1.56 1.33 0.23 0.64 αx1 0.037 1.37

40 113.4 85.2 80.0 78.7 78.9 40 113.97 84.30 78.78 78.23 78.20 0.32 0.80 1.50 0.22 0.49 αx2 0.027 1

45 116.8 86.0 79.8 78.1 78.4 45 118.21 85.99 79.08 78.26 78.20 1.99 0.00 0.52 0.03 0.04 αx3 0.027 1

50 119.8 86.9 79.8 77.8 78.1 50 122.15 87.76 79.45 78.31 78.21 5.53 0.75 0.12 0.26 0.01 αx4 0.027 1

55 122.4 87.7 80.1 77.6 77.8 55 125.76 89.61 79.90 78.37 78.21 11.28 3.64 0.04 0.59 0.17 αx5 0.027 1

60 125.2 89.0 80.6 77.7 77.6 60 129.01 91.48 80.42 78.45 78.22 14.53 6.17 0.03 0.57 0.38

38.88 16.61 7.74 3.00 2.91

Experimental data Theoretical data

Regression F Test



23

Table 4- the below table shows the theoretical and expermintal data taking the assumption that the thermal

diffusivity is constant throughout the slab.

Figure 12- temperature vs. time for the thermocouple at x = 2 in for variable thermal conductivities from the

time range 0 to 60 minutes. A further analysis in the data is shown in the residual analysis in the tables 5 and 6.

t (min) Tx1 Tx2 Tx3 Tx4 Tx5 t (min) Tx1 Tx2 Tx3 Tx4 Tx5

2 79. 3 78. 6 78. 5 78. 1 78. 2 2 78. 32 78. 20 78. 20 78. 20 78. 20 0. 97 0. 16 0. 09 0. 01 0. 00

4 79.7 78.8 78.6 78.4 78.3 4 78.91 78.20 78.20 7 8.20 7 8.20 0.62 0.36 0.16 0.04 0.01 X1 0.94

6 80.6 78.8 78.6 78.4 78.3 6 79.88 78.21 78.20 7 8.20 7 8.20 0.52 0.35 0.16 0.04 0.01 X2 0.79

8 81.9 78.9 78.7 78.5 78.3 8 81.10 78.23 78.20 7 8.20 7 8.20 0.64 0.45 0.25 0.09 0.01 X3 0.3710 83. 3 79. 0 78. 8 78. 7 78. 3 10 82. 51 78. 26 78. 20 7 8. 20 7 8. 20 0. 63 0. 54 0. 36 0. 25 0. 01 X 4 0. 00

15 88. 7 79. 4 78. 9 78. 7 78. 9 15 86. 50 78. 48 78. 20 7 8. 20 7 8. 20 4. 84 0. 86 0. 48 0. 25 0. 49 X 5 0. 00

20 93. 4 80. 2 78. 8 78. 6 78. 2 20 90. 83 78. 90 7 8. 22 7 8. 20 7 8. 20 6. 59 1. 70 0. 34 0. 16 0. 00

25 99. 0 81. 2 79. 0 78. 5 78. 6 25 95. 28 79. 55 78. 25 78. 20 78. 20 13. 88 2. 73 0. 56 0. 09 0. 16 mod

30 104.1 82.5 79.5 78.7 78.9 30 99.71 80.42 78.32 78.20 78.20 19.31 4.34 1.39 0.25 0.49 αx1 0.027 1

35 109.0 84.0 79.7 78.7 79.0 35 104.05 81.48 78.43 78.21 78.20 24.49 6.35 1.60 0.24 0.64 αx2 0.027 1

40 113.4 85.2 80.0 78.7 78.9 40 108.26 82.72 78.60 78.22 78.20 26.44 6.17 1.96 0.23 0.49 αx3 0.027 1

45 116.8 86.0 79.8 78.1 78.4 45 112.28 84.09 78.82 78.24 78.20 20.41 3.64 0.95 0.02 0.04 αx4 0.027 1

50 119.8 86.9 79.8 77.8 78.1 50 116.09 85.58 79.11 78.27 78.20 13.77 1.73 0.47 0.22 0.01 αx5 0.027 1

55 122.4 87.7 80.1 77.6 77.8 55 119.64 87.16 79.46 78.32 78.21 7.60 0.29 0.41 0.52 0.17

60 125.2 89.0 80.6 77.7 77.6 60 122.92 88.80 79.87 78.38 78.21 5.22 0.04 0.53 0.46 0.38

70 131.2 92.2 82.1 78.7 78.6 70 130.34 92.19 80.88 78.56 78.24 0.74 0.00 1.50 0.02 0.13

80 136.1 95.1 83.8 79.7 79.2 80 142.78 95.91 82.09 78.83 78.28 44.67 0.66 2.92 0.75 0.85

90 141.0 98.8 86.1 81.1 80.3 90 153.95 100.35 83.53 79.20 78.34 167.83 2.41 6.62 3.61 3 .83

105 143.2 100.1 86.5 80.8 80.0 105 167.01 107.80 86.17 79.95 78.51 566.76 59.30 0.11 0.72 2.22

925.91 92.08 20.85 7.97 9.93

Experimental data Theoretical data

Regression

F Test



24

Table 5- This table shows the residual plot and other statistical analysis for the thermocouple at x = 2 in with a

thermal diffusivity of 0.027.

Table 6- This table shows the residual plot and other statistical analysis for the thermocouple at x = 2 in with a

thermal diffusivity of 0.037.

TX1 Thermal Diffusivi ty = 0.027

Regression Statistics

Multiple R 0.97

R Square 0.94

Adjuste d R Square 0.94Standard Error 5.58

Observations 19.00

RESIDUAL OUTPUT

Observation Predicted Y Residuals

1 84.57 -5.27

2 84.97 -5.27

3 85.63 -5.03

4 86.49 -4.59

5 87.49 -4.19

6 90.42 -1.72

7 93.71 -0.31

8 97.16 1.84

9 100.68 3.42

10 104.18 4.8211 107.62 5.78

12 110.95 5.85

13 114.13 5.67

14 117.13 5.27

15 119.92 5.28

16 126.02 5.18

17 135.80 0.30

18 145.01 -4.01

19 156.22 -13.02

-15.00

-10.00

-5.00

0.00

5.00

10.00

0 .0 0 2 0.0 0 4 0.0 0 6 0.0 0 8 0.0 0 1 00. 00 1 20. 00 1 40. 00 1 60. 00 18 0. 00

R e s i d u a l s

T (X1)

T(X1) Residual Plot [Thermal Diffusity = 0.027]

TX1 (Thermal Diffusivity = 0.037)

Regression S tatistics

Multiple R 1.00

R Square 1.00

Adjuste d R Square 1.00

Standard Error 0.95

Observations 15.00

RESIDUAL OUTPUT


1 79.13 0.17

2 79.89 -0.19

3 81.08 -0.48

4 82.55 -0.65

5 84.19 -0.89

6 88.73 -0.037 93.51 -0.11

8 98.31 0.69

9 103.04 1.06

10 107.61 1.39

11 111.99 1.41

12 116.13 0.67

13 120.01 -0.21

14 123.59 -1.19

15 126.84 -1.64

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

0.00 20.00 40.00 60.00 80.00 100.00 120.00 140.00 R e s i d u a l s

T (X1)

T (X1) Residual Plot [Thermal Diffusivity = 0.037]



25

Table 7- This table shows the residual plot and other statistical analysis for the thermocouple at x = 4 in.

Table 8- This table shows the residual plot and other statistical analysis for the thermocouple at x = 6 in

T X2


Multiple R 0.98

R Square 0.97

A dj us te d R Squ are 0.96

Standard Error 0.70Observations 15.00

RESIDUAL OUTPUT


1 79.28 -0.68

2 79.28 -0.48

3 79.29 -0.49

4 79.31 -0.41

5 79.36 -0.36

6 79.61 -0.21

7 80.09 0.11

8 80.80 0.40

9 81.72 0.7810 82.81 1.19

11 84.06 1.14

12 85.42 0.58

13 86.86 0.04

14 88.38 -0.68

15 89.92 -0.92

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

76.00 78 .00 80.00 8 2.00 84.0 0 86.00 88.0 0 90.00 9 2.00 94.00 R e s i d u a l s

T (X2)

T (X2) Residual Plot

TX3


Multiple R 0.98

R Square 0.95

Adj usted R Square 0. 95

Standard Error 0.54

Observations 19.00

RESIDUAL OUTPUT


1 78.97 -0.47

2 78.97 -0.37

3 78.97 -0.37

4 78.97 -0.27

5 78.97 -0.17

6 78.97 -0.07

7 78.99 -0.19

8 79.02 -0.02

9 79.10 0.40

10 79.22 0.48

11 79.40 0.6012 79.65 0.15

13 79.96 -0.16

14 80.35 -0.25

15 80.80 -0.20

16 81.90 0.20

17 83.23 0.57

18 84.80 1.30

19 87.68 -1.18

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

7 7.0 0 7 8.00 79 .0 0 8 0.00 81 .0 0 8 2.00 83 .0 0 8 4.00 85 .0 0 8 6.00 87 .0 0 R e s i d u a l s

T (X3)




26

Table 9- This table shows the residual plot and other statistical analysis for the thermocouple at x = 8 in

Table 10 This table shows the residual plot and other statistical analysis for the thermocouple at x = 10 in

T (X4)


Multiple R 0.83

R Square 0.69

Adj usted R Square 0. 68

Standard Error 0.52Observations 19.00

RESIDUAL OUTPUT


1 78.34 -0.24

2 78.34 0.06

3 78.34 0.06

4 78.34 0.16

5 78.34 0.36

6 78.34 0.36

7 78.34 0.26

8 78.34 0.16

9 78.35 0.35

10 78.36 0.34

11 78.38 0.32

12 78.41 -0.31

13 78.46 -0.66

14 78.54 -0.94

15 78.64 -0.94

16 78.95 -0.25

17 79.40 0.30

18 80.02 1.08

19 81.28 -0.48

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

78.00 78.20 78.40 78.60 78.80 79.00 79.20 79.40 79.60 79.80 80.00 80.20 R e s i d u a l s

T (X4)


T X5

Regression S tatistics

Multiple R 0.74

R Square 0.54Adjusted R Square 0.52

Standard Error 0.47

Observations 19.00

RESIDUAL OUTPUT


1 78.42 -0.22

2 78.42 -0.12

3 78.42 -0.12

4 78.42 -0.12

5 78.42 -0.12

6 78.42 0.48

7 78.42 -0.22

8 78.42 0.18

9 78.42 0.4810 78.42 0.58

11 78.43 0.47

12 78.43 -0.03

13 78.45 -0.35

14 78.47 -0.67

15 78.51 -0.91

16 78.66 -0.06

17 78.92 0.28

18 79.36 0.94

19 80.44 -0.44

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

78.15 78.20 78.25 78.30 78.35 78.40 78.45 78.50 78.55 R e s i d u a l s

T (X5)




Table 11- cost analysis of components obtained for construction of the lab.

Product Store/site model # item # count cost per item total cost

Pine Sheathing Plywood Lowes 12246 2 $14.77 $29.54

Zinc Plated Wood Screws Lowes 40015 427593 1 $3.94 $3.94

Chrome Cabinet Pull Bar Lowes S305-3/136-PC 59609 2 $5.97 $11.94

Radiant Barrier Lowes BP24025 13357 1 $25.07 $25.07

Fiberglass Insulation Roll Lowes B1284 31116 1 $11.20 $11.20

thermocouples www.omega.com JQ55-116G-12 25 $21.25 $531.25

Heat Source (1"x12") www.omega.com SRFG-112/110 1 $20.00 $20.00

Heat Source (5"x5") www.omega.com SRFG-505/10-P 3 $39.00 $117.00

Fiberglass Sheets www.acpsales.com FS-17 6 $19.00 $114.00

Total $863.94

butler belcher german

Documents