Fourier’s Law Applied to the Heat Equation forConduction of Heat Through a Concrete Slab

James RobinsonENGR 322/326Professor Willis

December 7, 2007

Abstract

A one-dimensional, transient heat transfer model is solved numerically to examine thetemperature change in a homogeneous concrete slab. In the concrete slab, hydronic floorheating elements uniformly heat up the under surface of the concrete floor, and heat convectsand radiates from the top surface. The lower boundary condition is defined as constantusing the Dirchlit condition. The upper surface has a heat flux described by the Neumanncondition. The model is a partial differential equation that uses Fourier’s Law. The model issolved numerically by a FORTRAN 90 program approximated via second order finite differencemethod. Following this, a Monte Carlo Simulation is performed to test the effects of uncertaintyin thermal conductivity and specific heat. Results found the upper surface of a 10cm concretefloor to reach 25 ◦C in 24 seconds with a greater uncertainty in specific heat.

i

Variable Dictionary

c2 Thermal diffusivity kσρ

( m2

◦Cs)

Cp Specific heat ( kJ◦Ckg

)

g Gravity (ms2 )

Gr Grashof Number

h Heat coefficient ( W◦C m2 )

H Height of concrete slab (m)

k Thermal conductivity ( W◦Cm2 )

L = AP

P=perimeter, A=area ( 1m

)

n Sample/population size

N Number of runs

Nu Nusselt number (dimensionless)

Pr Prandtl Numbers

q Heat transfer ( Wm2 )

Ra Rayleigh number

t Time (s)

TH Temperature at z=H

T∞ Ambient temperature

z Coordinate direction opposite gravity (m)

α Thermal diffusivity (m2

s)

β Thermal expansion coefficient (1θ)

ε Emissivity Constant (dimensionless)

µ Viscosity (NSm2 )

ρ Density, ( kgm3 )

σ Stefan-Boltzmann constant 5.67× 10−8 (dimensionless)

θ Temperature (◦C)

ξ Truncation error of Talyor’s Series

1

Contents

1 Introduction 4

2 Problem Formulation 4

3 Literature Review 5

3.1 Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2 Effect of Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Model Formulation and Development 9

4.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 Finite difference method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Model Application 13

5.1 Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Results and Discussion 17

6.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.2 Probabilistic Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.2.1 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.2.2 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7 Conclusion 28

8 Future Recommendations 29

9 References 30

10 Task List 33

11 Estimated Costs 34

12 Appendix A: History 34

13 Appendix B: Model Validation and Verification 35

14 Appendix C: Fortran Program 36

2

15 Appendix D: FA1NEW Results 41

15.1 D.1: Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 41

15.2 D.2 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

List of Figures

1 Floor plan of radiant tubing and concrete slab . . . . . . . . . . . . . . . . 5

2 Flow chart of Monte Carlo Simulation (Makarov, et al., 1973) . . . . . . . 14

3 % error between analytical and numerical solution . . . . . . . . . . . . . . 16

4 % error between analytical and numerical solution . . . . . . . . . . . . . . 16

5 Multiple node numbers at the same time step. . . . . . . . . . . . . . . . . 16

6 Steady-State temperature gradient. . . . . . . . . . . . . . . . . . . . . . . 18

7 Analytical solution to temperature gradient over 16.7 minutes . . . . . . . 18

8 Numerical solution to temperature gradient over 16.7 minutes. . . . . . . . 18

9 Temperature gradient for all time steps . . . . . . . . . . . . . . . . . . . . 19

10 ∆t = 0.2 sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

11 ∆t = 10 sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

12 Effect of spatial discretization on time for upper surface to reach 25 ◦C . . 22

13 Ratio # of nodes:time to reach 25◦C . . . . . . . . . . . . . . . . . . . . . 22

14 Effect of temporal discretization on temperature of upper surface . . . . . 23

15 Histogram of population of temperature values based on normally dis-

tributed thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 24

16 Normally distributed Conductivity Population . . . . . . . . . . . . . . . . 25

17 PDF of normal distribution for thermal conductivity . . . . . . . . . . . . 26

18 Probability of temperature between 38 and 39 ◦C is 65.73% . . . . . . . . 27

19 PDF based on specific heat as a random variable . . . . . . . . . . . . . . 27

List of Tables

1 Table of values for spatial sensitivity analysis over 1000 seconds, ∆t = 1 sec 19

2 Table of values for temporal sensitivity analysis with 120 nodes . . . . . . 21

3 Table of values for sensitivity analysis when upper boundary temperature

equals 25 ◦C, with 120 nodes . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Statistical Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Task List and Schedule for Completing the Engineering 326 Final Project

and Report (Woo, 2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Estimated Costs for Engr 326 Final Project and Report (Woo, 2007) . . . 34

3

1 Introduction

Temperature gradients are ubiquitous in daily life. One interesting example happens

when one picks up an ice cube. The temperature gradient created from the transfer of

heat from a hot finger to the extreme cold of the ice, pulls a person’s finger to the ice.

Another example is the feeling of warmth on the outside of a coffee mug, even though

all the heat from its contents is contained within it. The phenomena of heat transport is

observed over all media phases, over a given interval of time. Three types of heat transfer

are possible (Carslaw and Jaeger, 1986):

• Conduction transfers heat through a solid medium.

• Convection transfers heat through gaseous or liquid media.

• Radiation transfers heat via electromagnetic waves, such as the sun.

Heat transfer is an important consideration when designing a house. The optimum tem-

perature conducive for comfort in a residential building has been determined to range

from 19 − 29◦ C (ISO, 1994). Many benefits arise from homes that are built using ef-

ficient energy design (USDE, 2005; Good, et al., 2005). These benefits include a more

uniform temperature, a comfortable temperature range for occupants and energy/cost

savings (Good, et al., 2005).

2 Problem Formulation

This study examines the conductive transfer of heat in concrete slab radiant flooring

to heat a room (Figure 1). The objectives are to:

1. Describe heat energy transfer by approximating the heat equation using Fourier’s

Law via finite differences.

2. Determine the temperature gradient through a concrete slab.

3. Determine the time needed to heat a cold concrete slab to a comfortable temperature.

4. Assess the impacts of spatial and temporal discretization on the solution of the

model.

5. Use Monte Carlo Simulation to determine the probability of uncertainty in param-

eters.

4

Figure 1: Floor plan of radiant tubing and concrete slab

6. Develop a probability density function (PDF) to classify the uncertainty of the

thermal conductivity constant.

The change in temperature of a concrete slab foundation is described over time and

space using a one-dimensional second order, linear, transient system of partial differential

equations. A review of professional literature will develop an understanding of a reason-

able approach to the problem and typical values used in the model. This type of second

order equation can be described as parabolic (Faires and Burden, 1998). This model is

approximated using a finite difference method. Fortran 90 solves a system of linear equa-

tions to obtain the numerical solution.

Research on statistical and probabilistic methodologies will add significance to the

model. Among other probabilistic techniques, FA1NEW (Kite, 1978) tests the model for

goodness-of-fit to six different distributions. A Monte Carlo Simulation to characterize

the uncertainty of the thermal conductivity is accomplished by generating a population

of uniformly distributed random variables. If a distribution from FA1NEW or another

type of distribution fits the model, a PDF can be obtained to find the parameter with the

highest probability of occurrence. Otherwise, a Cumulative Frequency Diagram (CFD)

must be constructed to accomplish the same task.

3 Literature Review

3.1 Process

Fourier’s Law sufficiently describes most conduction phenomena (Loh, et al., 2007).

However, Fourier’s Law does not apply well during rapid heating. Loh’s study compares

the finite element method to solve the Fourier approximation against an asymptotic wave-

form evaluation (AWE) to solve the non-Fourier equation. In the case of rapid heating,

the latter was found to be more accurate.

For most materials that conduct heat, there is an observable effect from thermoelas-

ticity. Elastic solids such as malleable metals exhibit thermal conductivity that depends

5

on the internal stress distribution (McGraw-Hill, 2005). Most models assume this effect

to be negligible; however, this is contrary to physical observations (Aouadi, 2005). The

speed of thermal disturbances can be determined using a thermal relaxation time, which

changes the governing equation to one of hyperbolic shape, treating heat transfer more like

a wave, rather than the more common parabolic shape. In an idealized solid, the transport

of thermal energy is due to the quantized movement of free electrons. A relaxation time

is associated with the average communication time between collisions of electrons.

Among other proofs, Fourier demonstrated a linear relationship between the tempera-

ture gradient and the subsequent current (Garriga and Ritort, 2001). Conduction of heat

acts similarly to conduction of electricity, but the difference is how the medium affects the

particles involved. While electricity is caused by free electrons moving through an ideal

solid, heat conduction results from less energetic vibrating of whole atoms in a solid. “It

has long been recognized that thermal transport and atomic transport follow analogous

phenomenological laws” (Belova and Murch 2004).

The above studies describe common heat transfer phenomena in very specific cases.

Fourier’s law of heat transfer describes linear phenomena; heat transfer is modeled using

a parabolic second order Partial Differential Equation (PDE) (Faires and Burden, 1998;

Marsden, et al., 2005). Thus, obtaining an exact solution to describe heat through a solid

is highly unlikely (Khattabi and Steinhagen, 1993). PDEs are frequently solved using a

numerical approximation such as finite differences.

Although most PDEs describe continuous physical phenomena, the use of computer

tools to solve them requires a discrete approximation of the function (Haveraaen, et al.,

2004). In the Haveraaen (2004) study, PDEs were solved using tensor and scalar fields.

Although the work did not suggest any computable scalar fields, standard approaches

were presented. Accuracy was obtained via the resolution of the discretization, and less

so from accuracy of the real numbers. Alternatively, and more commonly, PDEs can be

approximated by truncating the infinite Taylor Series (Parker and Sochacki, 2000) and

either using finite difference (Marsden, et al., 2005; Faires and Burden, 1998) or finite

element methods (Nicholson and Lin, 1997; Quadir, et al., 2002).

To solve the two-dimensional problem from Aouadi (2005) using the finite element

method, the equation had to either be modeled at one instant in time by the Laplace

transform, or modeled with only one internal node; otherwise, the computer time was

excessive. Aouadi’s approach combined these two methods one at a time and obtained

accurate results based on previous studies. The study showed that “the generalized heat

6

conduction mechanism is completely different from the classical Fourier’s in essence.”

Recall from (Loh, et al., 2007) that this is one of the few conduction situations not de-

scribed by the Fourier equations due to the speed of transfer. A slower process, such

as conduction of heat through a concrete slab of a home, follows Fourier’s law closely

(Bonacina, et al., 2003).

Studies on radiant heating are important for cold climates such as Canada (Good et al.,

2005), where radiant heating was modeled using ESP-r, a Unix-based modeling tool that

assesses the feasibility of a low emission residential heating system. Complicated methods

of analysis using ESP-r allow comparisons of energy savings between radiant heating and

forced air systems. The conclusion of the study was that radiant floor heating has many

benefits, including its ability to provide an even temperature profile, its ability to maintain

optimal comfort level for occupants, and its energy saving strategy of setback control at

night despite reheating of the thermal mass. The latter showed an 11.2% end-use energy

savings over forced air.

For a typical radiant system in a common household, a performance test can be eval-

uated using a two-dimensional model written in a simple imperative computing language,

or in a more sophisticated program such as DOE-2. The former method can help cali-

brate the latter (Ihm and Krarti, 2004). Ihm and Krarti (2004) studied a control volume

approach using implicit finite difference techniques to solve the unsteady-state tempera-

ture field of a slab-on-grade foundation. The implicit method has the advantage of being

unconditionally stable.

3.2 Effect of Uncertainty

All parameters in engineering contain some degree of inherent uncertainty. However,

most models can be calibrated accurately enough to obtain valid and useable results.

Every parameter based on measurements has reliability only if a quantitative statement

of uncertainty accompanies it (Labudova and Vozarova, 2002).

Thermal conductivity has a narrow range of variation for most pure substances (ASHRAE,

2005; McQuiston, et al., 2005; Hirsch, et al., 2006), however most substances are far from

pure. For example, the thermal conductivity of liquid water is 25 times greater than it is

for air (Bonacina, et al., 2003). Because many substances can easily contain water vapor

that condenses within the aggregate, thermal conductivity becomes unpredictable. The

degree of thermal conductivity uncertainty of lightweight concrete was analyzed using the

heat flux as a function of the thermal transmissivity (Bonacina, et al. 2003). Samples

7

of concrete were subjected to various levels of heat and humidity. Concrete with higher

relative humidity was determined to have higher thermal transmissivity, and thus, higher

heat transfer rates and thermal conductivity.

Another experiment tested an industrial ceramic thermal-barrier coating designated

PWA 266. The the material was analyzed using a steady-state thermal conductivity

technique. The conclusion was that thermal conductivity was “relatively independent of

temperature” (Slifka and Filla, 2003). The greatest variation resulted, instead, from the

small thickness of some coatings which increased the uncertainty of thermal conductivity.

To address the uncertainty of thermal conductivity, probabilistic analyses of the value

can be applied within an interval of uncertainty that is most likely to occur; the Monte

Carlo method is frequently used. The Monte Carlo method allows the analysis of space-

dependent thermal conductivity by constructing a two-dimensional random walk (Burmeis-

ter, 2002). A random walk is a term used in probability to describe processes where the

next movement can not be predicted based on the last move. The predictions agree

closely with exact answers following finite difference approximations. Without the use

of the Monte Carlo method, the same procedure would have been restricted to assume

constant thermal conductivity.

Frenkel (2004) used the Monte Carlo method in a computer experiment, to model

molecular dynamics. The Monte Carlo method simulated random numbers based on the

position of a polymer particle in a sample. Similarly, Metropolis, et al. (1953) followed

a similar, rudimentary procedure when computers were first being used in Los Alamos

to construct a random walk in such a way that the probability of a particle visiting a

particular point is proportional to the Boltzmann factor (approximately 1/temperature).

This factor is the basis for radiative heat transfer.

One final example of the Monte Carlo and the random walk pertaining to thermal con-

ductivity is demonstrated by Belova and Murch (2004). The atomic transport, which is

analogous to heat transport, was modeled by two-phase systems in three dimensions using

calculations of thermal conductivity. Close to a million ‘particles’ were released, one at a

time, from a random location in the lattice. Each had, at every intersection, six possible

choices of direction. After a set number of time steps, the displacement indirectly indicated

the thermal conductivity. Longer time durations showed a linear relationship between the

mean square displacement and time. This process is best described by Maxwell’s equation.

The studies examined show a variation in the thermal conductivity constant for con-

crete. This variation is due to the composition and type of concrete. The interest of

8

this paper is to determine the uncertainty of thermal conductivity for concrete. Radiant

heating is optimized at higher thermal conductance rates. Therefore, ignoring lightweight

concrete that has a low conductivity, limits the range of uncertainty. This paper will focus

on heavy concrete with higher conductance. The range of values, based on the literature

is 1.1 to 1.8 WmK

(Hirsch, et al., 2006; Franco, 2007; McQuiston, et al., 2005)

A review of the literature has shown that conduction through a concrete slab is linear

and can be described by Fourier’s laws. Thermal conductivity is a parameter with a high

degree of uncertainty. Although thermal conductivity changes greatly with minute changes

in composition, it does not change greatly with variation in temperature. Many others

have researched and expanded on Fourier’s law, heat transfer through solids, radiant floor

heating, the finite difference method, and the Monte Carlo method. Thermal conductivity

has been shown to be relatively constant over normal temperature ranges but harbors

great uncertainty based on the composition of the material. This paper considers these

uncertainties.

4 Model Formulation and Development

The conduction of heat through solids is described using a partial differential equation

(PDE)(SCBR, 1983). Fourier’s equation describes the conductive aspect of heat through

concrete over some period of time (SCBR, 1983).

∂θ

∂t= c2∇2θ (1)

where:

θ(x, t)=temperature (◦C)

t=time (hrs)

c2 = kCpρ

( m2

◦Cs)

k=thermal conductivity ( W◦Cm

)

Cp = specific heat ( kJ◦Ckg

)

ρ =density( kgm3 )

The assumptions are:

1. The model can be reduced to a one dimensional problem where all conduction is

assumed in the upward (z) direction through pure, homogeneous, limestone concrete.

2. The room is assumed to be perfectly sealed and insulated, reducing the complexity

of a convective current.

3. No flooring overlay assists or retards conduction.

9

4. Any convection is assumed to be between pure concrete and pure air within the

room.

5. The heat below the slab is assumed to be a uniform, constant temperature from

hydronic heating elements.

Using these assumptions, the model reduces to:

∂θ

∂t= c2 ∂2θ

∂z2(2)

The model, congruent with most PDEs, is solved numerically by first approximating

the function using Taylor’s series, then solved via finite difference method using FOR-

TRAN 90. Finally, once results are gathered, they are compared to a population of

results based on the uncertainty of thermal conductivity for concrete. This technique is

called a Monte Carlo simulation. Each of the methods and equations used are specified

in the same order described above.

4.1 Boundary Conditions

For a concrete slab with a heat source on the bottom surface and a heat sink on the

top surface, two boundary conditions exist, Dirchlit and Neumann respectively (Figure 1).

Assume that the bottom surface has a uniform, constant temperature due to the hydronic

tubing. Thus at the datum z=0, θ0 = 70◦C. At the top surface, the heat fluctuates due to

the convection and radiation of the open surface. Newton’s law of cooling is a convective

parameter and Stefan-Boltzmann law is a radiative parameter. Both contribute to heat

loss from the surface of the concrete floor to the ambient surroundings within the room.

Heat flux at z=H (Figure 1) can be described by:

dθ

dz|z=H = −1

kqtot (3)

where:

qtot = qconv + qrad

= h(∆θ) + εσ∆θ4

h=heat coefficient

h=kairNu

L( W◦Cm2 )

kair =Conductivity of air ( W◦Cm

)

(Nu)= Nusselt number (dimless)

L = AP(L); P=perimeter, A=area

10

For natural convection, the Nusselt number is a function of the Rayleigh number. The

Rayleigh number is, in turn, a function of the Grasof number and Prandtl number.

Nu=f(Ra)

Ra=Gr× Pr; Grashof and Prandtl Numbers

Gr = βgρ2|∆T |L3

µ2

Pr = Cp µkair

; Cp=specific heat ( kJ◦Ckg

), µ=viscosity (NSm2 ) ∆T = TH − T∞

Newton’s law of cooling describes convection (qconv) from a solid surface (Thomas,

1999).

qconv = h(θ(x,t) − θamb) (4)

The Stefan-Boltzmann equation describes radiative heat (qrad) from the surface of a solid

to the surrounding environment.

qrad = εσ((θ(x,t)4 − θ4amb) (5)

where:

ε =emissivity (dimensionless)

σ =Stefan-Boltzmann constant=5.67× 10−8 (dimensionless)

4.2 Finite difference method

One method of solving a partial differential equation is the finite difference method.

This method is derived from the infinite Taylor series approximation to create a system

of linear equations (Kreyszig, 1993). The Taylor series is:

f(x + ∆x) ≈ f(x) +df

dx|x∆x +

d2f

dx2|x

∆x2

2!+ ... + fn|x

∆xn

n!+ ξ (6)

Backward difference from the Taylor series:

f(x−∆x) ≈ f(x)− df

dx|x∆x +

d2f

dx2|x

∆x2

2!+ ξ

For a partial differential of the form: af ′′ + bf ′ = 0

a(f(x + ∆x)− 2f(x) + f(x−∆x)

∆x2) + b(

f(x)− f(x−∆x)

∆x) = 0 (7)

In short hand:

a(fi+1 − 2fi + fi−1

∆x2) + b(

fi − fi−1

∆x) = 0 (8)

11

where, i refers to the internal node of the discretized system.

Specifically for this project, a steady state system of equations is solved using Thomas’

algorithm.

The applied finite difference method is:

0 = c2(θi+1 − 2θi + θi−1

∆z2) (9)

with an error order, O(∆z2) the corresponding vector matrix form is:

Aθ + f = 0

A backward difference approximation for the transient case at a node is represented

by:θt+1 − θt

∆t

From these, a system of equations is developed with the form:

(A− I

∆t)θt = −f t−1 − θ

∆t

t−1

(10)

with error O(∆z2, ∆t)

4.3 Monte Carlo Simulation

The Monte Carlo Simulation method (MCS) is a powerful tool for estimating proba-

bilistic systems. In the case of numerical analysis and modeling, MCS is used to verify

the confidence interval of the results generated from the model (Ang and Tang, 2007).

First, a set of n randomly generated numbers called a population is simulated within an

interval of uncertainty for a model parameter (Figure 2).

Steps of the MCS:

1. Build a validated model

2. Generate a population of n random variables

3. Choose a spatial point at a given time

4. Run the model n times for each value

5. Analyze n state variables at a given space and time for probability of occurrence

The model is run N times for a sample size n. The greater the sample size, the higher

the accuracy (Ang and Tang, 2007). FA1NEW is used to analyze this population sample,

12

and sometimes identify a distribution that fits the model solved using n random variables.

From this process, a coefficient of variance (c.o.v.= σP) is generated based on the sample

variance (σ) of the sample mean P by:

σ2P =

P (1− P )

n

c.o.v.(P ) =

√(1− P )

nP(11)

The percent error can also be computed from using:

%error = 200

√1− P

nP(12)

In addition, to give the desired accuracy, the sample size n required can be determined

given a specified % error. If a goodness-of-fit test applies within a confidence interval, a

probability density function (PDF) is generated that displays the population value that

has the highest probability of occurrence, i.e., the lowest uncertainty.

5 Model Application

5.1 Heat Conduction

The model to be solved is based on Fourier’s Equation. The governing equation is:

∂θ

∂t= c2 ∂2θ

∂z2(13)

This heat conduction equation is applied to a concrete slab 10cm thick. The system is

bounded by two surfaces. The boundary conditions are θ(0,t)=70◦C and θ(h,t)=∂θ∂z|z=H =

0. The given initial condition of the slab is θ(z,0)=15◦C.

The system, modeled using FORTRAN 90, is built in a number of steps. First, the

steady-state solution is obtained. Next, the transient solution has to work before addi-

tional manipulation can be implemented. The steady-state condition is validated using

Scilab to solve the model with three, internal, spatial nodes and the transient condition

is validated using an analytical approximation (Equation 14) to be sure the results are

appropriate. After the transient model is working, the effects of discretization are tested.

13

Figure 2: Flow chart of Monte Carlo Simulation (Makarov, et al., 1973)

Analytical solution: (Carslaw and Jaeger, 1986)

T (z, t) = (To − Ts)erf(z√4αt

) + Ts (14)

where:

14

T(z,t) unknown temp at node n and time t

Ts T(1,t)=70◦C

To temp at z →∞; equals initial condition

=15 ◦C

erf =error function, which is intrinsic to FORTRAN

erf =2√π

∫ z

0

e−u2

du (15)

where:u argument of erf; in this case u = z√

4αt

z position in the slab foundation

α diffusivity constant

t timeThe error can be calculated between the analytical and numerical solution by two

methods. The least squares method (Equation 17) and the percent error method (Equation

16). Due to the large amount of data generated by the model, the square root of the sum is

calculated for a given time step (Equation 18). Both methods gave similar plots (Figures

3 and 4).

%error =SN − SA

SN

× 100 (16)

LS = (SA − SN)2 (17)

SSDt =√

Σni=1LS(n) (18)

where:SA Analytical solution

SN Numerical solution

LS Least square method

SSD Sum of squared difference

5.2 Optimization

Using the least squares method (Equation 19), the optimal number of spatial nodes

can be determined in order to provide a clear picture of the temperature gradient while

minimizing the amount of data generated. The easiest way to accomplish this optimization

is to calculate the LS of the state variable at multiples of the initial discretization. The

initial spatial discretization of 5 nodes means that for the 10cm concrete slab, 2cm , 4cm,

..., 10cm was compared to the corresponding positions in a 10 node system (Figure 5). The

criteria for optimization is to provide a clear picture of the heat transport while limiting

the amount of data that is generated.

15

60

50

60

40

50

30

ge % error

20

Averag

10

0

0 100 200 300 400 500 600 700 800 900 1000

Time (sec)

Figure 3: % error between analytical and numerical solution

170

160

170

150

differen

ce

140

of squ

ared

 d

130

 roo

t of sum

 

110

120

Squa

re

100

110

0 100 200 300 400 500 600 700 800 900 1000

Time (Sec)

Figure 4: % error between analytical and numerical solution

Figure 5: Multiple node numbers at the same time step.

16

LSt = Σni=1(P (i)−Q(2i))2 (19)

After the model is chosen, validated and optimized, the model is run to apply the de-

sign criteria. The system is tested to determine at what time the upper surface reaches a

comfortable temperature of 25 ◦C. Probabilistic analyses attempt to verify the usefulness

of the values acquired at this point. Finally, a Monte Carlo Simulation is added to the

model to determine the effects of uncertainty.

5.3 Monte Carlo Simulation

MCS is performed using two random variables. Each is tested for the probability of

uncertainty one at a time. Both parameters, thermal conductivity (k) and specific heat

at constant pressure(Cp) are tested by assuming both a normal distribution. A popula-

tion of values for specific heat is between 0.84 and 0.92 kJ◦Ckg

with a mean of 0.881 and

standard deviation of 0.0404 kJ◦Ckg

. The conductivity constant has a population of normaly

distributed values between 1.1 and 1.8 W◦Cm2 based on a review of published values (Hirsch,

et al., 2006; Franco, 2007; McQuiston, et al., 2005). A normal distribution of the variable

has a mean of 1.142 and a standard deviation of 0.0363 W◦Cm2 .

Each population of 100 realizations is read into the FORTRAN program to obtain

two sets of 100 state variables. The state variable, θ is read into FA1NEW to attempt

to fit a probability distribution. In addition, a cumulative distribution function may be

created to find either the value with the highest probability of occurrence, or the highest

probability of a certain value occurring.

6 Results and Discussion

The steady-state solution was found to be similar to the initial condition of the system

in the transient case. The solution has a linear trend everywhere in the domain for all

changes in nodes with a negative slope (Figure 6). The transient model was created and

validated. Using a Dirichlet boundary condition below the slab, and a Neumann boundary

condition with a zero temperature gradient above the slab, the upper surface approaches

the temperature of the bottom surface (Figure 8).

The results of the analytical verification of the transient case display a similar trend

to that of the numerical solution, although the analytically derived results (Figure 7) are

shown to be only an approximation of the numerical solution (Figure 8). The analytical

solution is closer to the form of a numerical solution modeled using a large ∆t (Figures 7

17

0 50 100 150 200 250 300 350 400 450 50050

52

54

56

58

60

62

64

66

68

70

node

Tem

pera

ture

(° C)

Figure 6: Steady-State temperature gradient.

and 10). Once the model had been verified, the optimal node numbers were determined

to be approximately 120 nodes (Figure 9).

0 50 100 150 200 250 300 350 400 450 50010

20

30

40

50

60

70

node

Tem

pera

ture

(° C)

Figure 7: Analytical solution to temperature gradient over 16.7 minutes

0 50 100 150 200 250 300 350 400 450 50010

20

30

40

50

60

70

Node

Tem

pera

ture

(° C)

Figure 8: Numerical solution to temperature gradient over 16.7 minutes.

18

40000

50000

60000

70000

ared

 differen

ce

0

10000

20000

30000

Sum of Squ

Number of Nodes

Figure 9: Temperature gradient for all time steps

6.1 Sensitivity Analysis

Although the number of nodes was found, the effects of spatial discretization on the

model must be determined. The system, if run long enough, will reach a uniform temper-

ature profile of 70 ◦C. The sensitivity analysis shows that for this condition to happen,

the number of spatial nodes should be greater (Table 1) and ∆t should be higher, (Table

2) connoting that the greater time duration and also fewest time steps allows the upper

boundary to be closest to the lower boundary. However, increasing ∆t (Figures 10 and

11) sacrifices resolution of the temperature profile and displays a more linear trend that

is more reminiscent of the analytical approximation (Figure 7).

Table 1: Table of values for spatial sensitivity analysis over 1000 seconds, ∆t = 1 secRun Nodes Boundary value(◦C)

1 10 26.423 30 47.182 60 56.573 100 61.304 120 62.606 150 63.967 240 66.09

The aim of the design aspect of this paper is to assess the best way to run the model

so that the upper surface temperature is comfortable to the occupants of the room. Thus,

19

0 20 40 60 80 100 12010

20

30

40

50

60

70

node

Tem

pera

ture

(° C)

Figure 10: ∆t = 0.2 sec

??0 20 40 60 80 100 120

10

20

30

40

50

60

70

node

Tem

pera

ture

(° C)

Figure 11: ∆t = 10 sec

20

Table 2: Table of values for temporal sensitivity analysis with 120 nodesTotal Time(sec) # of time steps ∆ t Boundary value(◦C)

500 500 1 62.611000 1000 1 62.60500 1000 0.5 56.601000 2000 0.5 56.60250 500 0.5 56.071000 500 2 66.101000 100 10 69.181000 5000 0.2 43.89

Table 3: Table of values for sensitivity analysis when upper boundary temperature equals25 ◦C, with 120 nodes

Run Total Time(sec) ∆ t Boundary value(◦C)

1 24 0.006 25.002 77 0.1 25.123 34 0.2 25.014 19 0.5 25.145 13 1 25.376 19 2 25.147 11 10 25.08

the closest the upper surface can get to 25 ◦C in the shortest amount of time is reached

by calculations using a ∆t of 0.006 (Table 3).

The model was tested with a limit placed on the upper surface of the slab of 25 ◦C.

This serves two purposes: (1) the simulation is more realistic when the system shuts off

at a comfortable temperature for inhabitants and (2) an end point can be reached af-

ter a finite amount of time. The latter is a test of the effect of discretization on upper

surface temperature. Several trials resulted in a trend where increasing the number of

spatial nodes increased the total time elapsed before the upper surface of the slab reached

a comfortable temperature of 25 ◦C (Figure 12). This trend is regular enough that an

equation can generally represent the results of the trials at all except the first spatial node:

t = n10

+ 16.

A ratio of the number of nodes:time taken to reach 25 ◦C, created from this trend goes

to 0.1, as the number of nodes approaches infinity (Figure 13). This means that this ratio

has the most stable solution. If the number of spatial nodes has been optimized to 120,

the time taken to reach 25 ◦C should be 1200 seconds or 20 minutes. Using these two

numbers to graph the number of time steps (Figure 14), the exponential trendline shows

a temperature of 25 ◦C reached after 2 ×105 time steps. This connotes a ∆t of 0.006

seconds.

21

1400

1200

1400

1000

)

800

f tim

e (sec)

y = 0.0997x + 17.75R² 1

600

Amou

nt of

R² = 1

200

400

0

200

0 2000 4000 6000 8000 10000 12000 14000

Number of Nodes

Figure 12: Effect of spatial discretization on time for upper surface to reach 25 ◦C

0 7

0.6

0.7

0.5

0.4

odes:tim

e)

0.3

Ratio (no

0.1

0.2

0

0.

50 100 200 300 500 1000 2000 3000 4000 5000 9000 10000 12000 12500

Number of Nodes

Figure 13: Ratio # of nodes:time to reach 25◦C

22

y = 45.70e‐3x

R² = 0.949

30

40

50

60

rature at z=H

 (°C)

0

10

20

30

0.00E+00 2.00E+05 4.00E+05 6.00E+05 8.00E+05 1.00E+06 1.20E+06

Surface tempe

r

Number of Time Steps

Figure 14: Effect of temporal discretization on temperature of upper surface

6.2 Probabilistic Analyses

6.2.1 Thermal Conductivity

Many simulations using the Monte Carlo Method were performed to fit a distribution

to the data. A population of 100 normally distributed random variables were generated

for both thermal conductivity (k) and specific heat (Cp). However, after many trials, no

distribution from FA1NEW fit the population of state variables generated from running

the random variables through the model.

However, a histogram with a cumulative distribution overlay (Figure 15) was created

for the data based on thermal conductivity as the random variable. The histogram ap-

peared to be based on a normal PDF. When the state variables were plotted against the

population of conductivity, a data point that appeared to be an outlier stood apart from

the data (Figure 16).

A Q-Test was performed to determine statistically if this truly is an outlier (Equation

20).

Q =X2 −X1

Xn −X1

(20)

From this it was determined that Qtheoretical > Qcritical ⇒ 0.9110 > 0.5168 therefore, the

23

120%16

100%

120%

14

16

80%

100%

12

60%8

10qu

ency

40%6

Freq Frequency

Cumulative %

20%2

4

0%0

2

37 2 37 4 37 6 37 8 38 38 2 38 4 38 6 38 8 39 39 2 39 4 39 6 M37.2 37.4 37.6 37.8 38 38.2 38.4 38.6 38.8 39 39.2 39.4 39.6 More

Temperature (°C)

Figure 15: Histogram of population of temperature values based on normally distributedthermal conductivity

data point is rejected with a confidence of 99%.

Similarly, the outlier can be proven to be rejected by finding the probability of occurrence

using the standard normal variate S. Using Table A.1 from Ang and Tang, (2007) the

standard normal probability φ can be calculated.

P (θ < x) = φ(S)

= φ(x− µ

σ)

P (θ < 32.85) = φ(32.85− 38.362

0.5067)

= φ(−10.88)

= 1− (1− φ(10.88))

= 0

(21)

Once this outlier was removed, the population of state variables was run through

FA1NEW to fit a distribution. The results were that a population of 99 normally dis-

tributed state variables fit a normal (Gaussian) distribution and both method of moments

and method of maximum likelihood for two parameter lognormal, and only the method

of moments for type I extremal and Pearson type III. To claim parsimony allows the data

analysis to incorporate the normal distribution only, as the simplest of the distributions.

For a normal distribution the Chi Squared test passes for a confidence interval of 80%

however accepting the null hypothesis that the state variable population fits this distri-

24

40

42

44

46

48

ture  (Deg

 C)

30

32

34

36

38

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Tempe

rat

Conducitivty Constant (k)

Figure 16: Normally distributed Conductivity Population

bution is a possible type II error. Allowing α to equal 0.05 still means the null hypothesis

is accepted at a confidence interval of 95% (Appendix D). Based on this satisfactory level

of acceptance, a normally distributed PDF can be generated using a population mean, µ

of 38.362 ◦C and a standard deviation, σ of 0.5067. A confidence interval of the mean is

calculated using Table A.3 (Ang and Tang, 2007) to verify that this distribution can be

accepted at the 95% confidence level by:

µ1−α = (x + tα2

,n−1σ√n

; x + t1−α2

,n−1σ√n

)

= (38.362± 1.6449(0.5067√

99))

= (38.278; 38.446)

(22)

By plugging each random state variable into the standard normal PDF (Equation 23)

a graph is obtained with a characteristic bell-shaped curve (Figure 17):

fX(x) =1

σ√

sπexp (−1

2(x− µ

σ)2) (23)

The results of the Monte Carlo Simulation (MCS) are shown to be useful in determin-

ing the probability of occurrence and the probability of uncertainty in the state variables

obtained. A way to quantify the potential error in this scenario is to calculate the percent

error (Equation 24). For example, if the probability of the temperature that lies between

38 ◦C and 39 ◦C is 65.73% (Figure 18) then the percent error of the MCS can be calcu-

25

0.5

0.6

0.7

0.8

0.9

fX(x)

N(38.362 , 0.5067)

0

0.1

0.2

0.3

0.4

37 37.5 38 38.5 39 39.5

f

Temperature (° C)

Figure 17: PDF of normal distribution for thermal conductivity

lated as follows (Ang and Tang, 2007):

c.o.v.(P ) =

√(1− P )

nP

=

√(1− 0.6573)

100× 0.6573.

= 0.0722

error(in%) = 200(c.o.v.)

= 14.44%

(24)

From this last equation, the minimum sample size n, needed to give a specified percent

error can be determined. For an error of 1% the minimum sample size is:

n =1− P

P ( 1200

)2=

1− .6573

0.6573( 1200

)2

= 20855

(25)

6.2.2 Specific Heat

Specifying a mean of 0.88 kJ◦Ckg

and a standard deviation of 0.04 kJ◦Ckg

to get a normally

distributed population of 100 random variables, results in 100 random state variables

after each specific heat value is run through the model. Using an analysis similar to that

for thermal conductivity, the outlier is removed from the population, and the 99 state

26

Figure 18: Probability of temperature between 38 and 39 ◦C is 65.73%

variables are run through FA1NEW. Temperatures at the last node of the concrete slab

(z=H) for a time duration of 60 sec, using 120 spatial nodes and a ∆t of 0.006, fit a

normal distribution and pass the method of moments and method of maximum likelihood

for two-parameter lognormal and the method of moments only for extremal type I and

Pearson type III. A Gaussian PDF is created (Figure 19) and any probability of occurrence

of a state variable can be determined as well as the state variable value given a specified

probability.

0.3

0.4

0.5

0.6

X(x)

N(37.43 , 0.7207)

0

0.1

0.2

35 35.5 36 36.5 37 37.5 38 38.5 39 39.5

fX

Temperature (°C)

Figure 19: PDF based on specific heat as a random variable

27

A summary of the mean, standard deviation, confidence interval of the mean and

skewness coefficient is given (Table 4).

Table 4: Statistical SummaryVariable temp t(sec) µ σ skewness µ0.95

Temp – 60 38.36 0.5076 9.37E-02 38.278;38.446Temp 25 – 24.96 0.09114

t 25 – 24.06 1.048Temp from Cp – 60 37.45 0.7207 2.35E-03 37.331;37.569

7 Conclusion

Results of this study demonstrated:

1. A working model has been written, tested and validated for the steady-state and

transient case using a Dirchlit and zero Neumann condition.

2. The optimal number of spatial nodes used to discretize the 10cm thick concrete slab

is about 120.

3. The more spatial nodes there are, the greater the time required to bring the upper

surface of the concrete to a comfortable temperature.

4. When using normally distributed random variables k and Cp for various spatial and

temporal discretization, the Monte Carlo Simulation resulted in a corresponding

normal distribution of the state variable in both cases.

5. A probability density function as well as a cumulative distribution function can be

used to determine: (1) the maximum probability of a state variable equating to a

certain value and (2)The maximum value of the state variable given a probability.

6. For the upper surface of the slab to equal a value with a probability of 65%, the

error estimation of the Monte Carlo Simulation for a population of 100 is 14.44%.

7. For an error of less than 1%, a sample size of at least 20855 is needed to run in the

MCS.

28

8 Future Recommendations

The most important recommendation is to create a model based on a Cauchy boundary

condition that accounts for convection and radiation at the upper surface of the concrete

slab. This will also be the next first step to create a cyclic system of hydronic heating.

Ideally, similar to most all refrigeration and heating systems that are designed, the heat

source should turn on at a given temperature and turn off again at another given temper-

ature. In addition the outlier of the MCS needs to be explained so the problem can be

rectified if possible. Any number of other probability analyses will provide a better idea

of the significance of the model.

29

9 References

1. Alterovitz, S.A., Y. Shapira, (1980), Application of a moments method and of

Laplace transforms to heat transfer experiments. Journal of Thermal Analysis and

Calorimetry, vol. 18 (3) pp. 477-491

2. Ang, A. H-S. and W.H. Tang, (2007), Probability Concepts in Engineering, Chapter

5:Computer-Based Numerical and Simulation Methods in Probability

3. Aouadi, M. (2007) Hybrid Laplace transform-finite element method to a generalized

electromagneto-thermoelastic problem. Applied Mathematical Modeling. Vol.31 (4),

pp.712-726

4. ASHRAE Handbook - Fundamentals (2005) Chapter 3: Heat Transfer. ASHRAE

Press

5. Belova, I.V., and G.E. Murch (2004), Monte Carlo simulation of the effective con-

ductivity in two-phase material, Journal of Materials Processing Technology, Vol.

153-154 (10), pp. 741-745

6. Bermeister, L.C. (2002) The effect of space-dependant thermal conductivity on the

steady central temperature of a cylinder Journal of Heat Transfer, Vol. 124 (1),

pp.195-197

7. Bonacina, C., M. Campanale, L. Moro (2003) Analytical and Experimental Investi-

gations on the Heat Transfer Properties of Light Concrete. International Journal of

Thermophysics, Vol. 24 (5) pp. 1407-1414

8. Carslaw, H.S., Jaeger, J.C. (1986) Conduction of Heat in Solids. Clarendon Press,

Oxford

9. Faires, J.D., R. Burden (1998) Numerical Methods. Numerical Methods for Partial

Differential Equations: Finite Difference Methods for Parabolic Problems. Brooks/Cole

Publishing Co. pp.478-489

10. Franco, A. (2007), An apparatus for the routine measurement of thermal conduc-

tivity of materials for building application based on a transient hot-wire method,

Applied Thermal Engineering, Vol. 27(14-15), pp. 2495-2504

11. Frenkel, D. (2004), Introduction to Monte Carlo Methods, Computational Soft Mat-

ter: From Synthetic Polymers to Proteins, John von Neumann Institute for Com-

puting, Vol. 23 pp. 29-60

30

12. Garriga, A. and F. Ritort (2001) Heat transfer and Fourier’s law in off-equilibrium

systems. European Physics Journal B. Vol:21 pp:115-120

13. Good, J., V.I. Ugursal, A. Fung (2005) Simulation strategy and sensitivity analysis of

an in-floor radiant heating model. Ninth International IBPSA Conference, Montreal,

Canada. pp:341-348

14. Haveraaen, M., H.A. Friis, H. Munthe-Kaas (2004) Computable scalar fileds: A basis

for PDE software. Journal of Logic and Algebraic Programming. Vol: 65, pp:36-49

15. Hirsch, J.J., Addison, M., Criswell, S., Gates, S. and Madison, K. (2006) Doe 2.2

Library:Building Energy Use and Cost Analysis Program, Libraries and reports, Vol.

4 pp. 9-10

16. Ihm, P., M. Krarti (2004) Implementation of two-dimensional foundation model for

radiant floors into EnergyPlus. SimBuild2004 IBPSA-USA National Conference,

Boulder, CO.

17. ISO Standard 7730. (1994) Moderate thermal environments determination of the

PMV and PPD indices and specification of the conditions for thermal comfort.

Geneva: International Organization for Standardization.

18. Khattabi, A., P. Steinhagen (1993) Analysis of transient nonlinear heat conduction

in wood using finite-difference solutions. Holz als Roh- und Werkstoff. Vol.51 pp.

272-278

19. KITE (1978), FA1NEW FORTRAN Program, IMSL Inc.

20. Kreyszig, Erwin.(1993) Advanced Engineering Mathematics(7th ed.). John Wiley

and Sons, Inc., U.S.A. pp:552,645-646

21. Labudova, G. and Vozarova, V. (2002) Uncertainty of the Thermal Conductivity

Measurement Using the Transient Hot Wire Method.” Journal of Thermal Analysis

and Calorimetry. Vol. 67 (1), pp: 257-265

22. Loh, J.S., K.N. Seetheramu, G.A. Quadir (2007) Fast transient analysis of Fourier

and non-Fourier heat conduction International Journal of Heat and Mass Transfer,

Vol. 50 (21) pp.4400-4408

23. Makarov, A.M., K.B. Pavlov, S.L Simkhovich (1973) Use of Monte Carlo method

for solution of time-dependent problems in the hydrdynamics of viscoplastic media,

Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 18 (6), pp. 1116-1121

31

24. Marsden, J.E., L. Sirovich, M. Golubitsky, W. Jger (2005) Introduction to Partial

Differential Equations A Computational Approach: Finite Difference Schemes for

the Heat Equation. Springer Berlin Heidelberg. pp. 117-158

25. McGraw-Hill, (2005), Sci-Tech Dictionary, McGraw-Hill publishing, Inc.

26. McQuiston, F.C., Parker, J.D., Spitler, J.D. (2005) Heating, Ventilation and Air

Conditioning. John Wiley and Sons, Inc pp:126-127

27. Metropolis, N., A.W. Rosenbluth, M.N. Rosenbluth, A.N. Teller, and E. Teller.

(1953) Equation of state calculations by fast computing machines. Journal of

Chemical Physics, Vol. 21, pp. 1087-1092

28. Nicholson, D.W., B. Lin (1997) Finite element method for thermomechanical re-

sponse of near-incompressible elastomers Acta Mechanica. Vol. 124 (1-4) pp. 181-

198

29. O’ Conner, J.J., E.F. Robertson (1997) Joseph Fourier http://www-groups.dcs.st-

and.ac.uk/%7Ehistory/Mathematicians/Fourier.html (Oct. 18th 2007)

30. Parker G.E., J.S. Sochaki (2000), A Picard-Maclaurin theorem for initial value PDEs

Abstract and Applied Analysis Vol. 5 (1) pp. 47-63 doi:10.1155/S1085337500000063

31. Quadir, G.A., G.M. Krishnan, K.N. Seetharamu (2002), Modeling of wire-on-tube

heat exchangers using finite element method Finite Elements in Analysis and Design,

Vol. 38 (5) pp. 417-434

32. Slifka, A. J., and B. J. Filla (2003), Thermal Conductivity Measurement of an

Electron-Beam Physical-Vapor-Deposition Coating, Journal of Research of the Na-

tional Institute of Standards and Technology, Vol. 108 (2) pp. 147-150

33. Swedish Council for Building Research (SCBR). (1983) Calculation Methods to Pre-

dict Energy Savings in Residential Buildings. International Energy Agency, Annex

III. Stockholm, Sweden pp. 19,89,98

34. Thomas, Linden C.(1999) Heat Transfer Professional Edition (2nd ed.). Capstone

Publishing, Tulsa, OK. pp:26

35. USDE (2005) A Consumer’s Guide to Energy Efficiency and Renewable Energy U.S.

Dept of Energy.

36. Woo, S. (2007) Assignment: Request [Demand] for Proposals For an Engineering

326 Final Project and Report, Classroom handout

32

10 Task List

Table 5: Task List and Schedule for Completing the Engineering 326 Final Project andReport (Woo, 2007)Task Description Deadline1. Search literature for project idea. Developproject statement.

Done√

2. Obtain Roberts approval of project Done√

3. Write Intro and Problem Formulation Friday, Oct 19, 2007√

4. Write Lit Review, Model Formulation and De-velopment, and References sections

Friday, Oct 19, 2007√

5. Write program. Achieve working program Friday, Oct 26, 2007√

6. Submit 1st Half (Intro and Problem Formu-lation, LR, Model Formulation and Developmentsections) to Sheri for a grade

Friday, Nov 2, 2007noon

√

7. Finish runs and sensitivity analyses Friday, Nov 9√

8. Start graphics, tables, and/or figures Friday Nov 16√

9. Finish draft of Model Application, Results,Conclusions, and Suggestions for Further Researchsections

Friday, Nov 23√

10. Finish graphics, layout entire report Thursday, Nov 29√

11. Submit entire draft report to Sheri for editing Friday, Nov 30√

12. Give FP presentation Tuesday and Thursday,Dec 4 and 6

√

13. Revise draft given Sheris editing Thursday, Dec 6√

14. Submit revised Final Project Report to Robertand submit copy to Sheri

Friday, Dec 7, 2007 noon(1200 PST), at House 18inbox

√

33

11 Estimated Costs

Table 6: Estimated Costs for Engr 326 Final Project and Report (Woo, 2007)Task Description Estimated Staff Hours Estimated

Costs,$

1. Search literature for project idea. Developproject statement.

Staff engr 6 hr @$100/hr $600

2. Obtain Roberts approval of project Sen engr 1 hr@$200/hr $200Staff engr 1 hr @$100/hr $100

3. Write Intro and Problem Formulation Staff engr 10 hr @$100/hr $1,0004. Write Lit Review, Model Formulation, and Ref-erences sections

Staff engr 33 hr @$100/hr $3,300

5. Write program. Achieve working program Staff engr 20 hr @$100/hr 2,0006. Submit 1st Half (Intro and Problem Formu-lation, LR, Model Formulation and Developmentsections) to Sheri for a grade

Staff engr 7 hr @$100/hr $700

Tech editor 4 hr @$110/hr $4407. Finish runs and sensitivity analyses Staff engr 10 hr @$100/hr $1,0008. Start graphics, tables, and/or figures Staff engr 20 hr @$100/hr $2,0009. Finish draft of Model Application, Results,Conclusions, and Suggestions for Further Researchsections

Staff engr 6 hr @$100/hr $600

10. Finish graphics, layout entire report Staff engr 10 hr @$1000/hr $1,00011. Submit entire draft report to Sheri for editing Tech editor 2 hr @$110/hr $22012. Give FP presentation Staff engr 10 hr @$100/hr $100013. Revise draft given Sheris editing Tech editor 4 hr @$110/hr $440

Staff engr 2 hr @$100/hr $20014. Submit revised Final Project Report to Robertand submit copy to Sheri

Sen engr 2 hr@$200/hr $400

Tech editor 4 hr @$110/hr $440Staff engr 2 hr@$200/hr $400Subtotal EstimatedCost

$16,040

Project Management*@ 15% $ 2406Total Estimated Cost $18,446

12 Appendix A: History

The heat equation is well researched. Heat transfer is caused by energy in transit due

to differences in temperature (ASHRAE, 2005). Many ways of describing heat flow are

possible. Of these, Fourier’s equation and the Laplace transform are the most common

34

(Alterovitz and Shapira, 1980). Fourier was a student of Laplace, Lagrange and Monge

(O’Conner and Robertson, 1997). In 1807, Fourier submitted his famous memoir On the

Propagation of Heat in Solid Bodies. Although this document was controversial at the

time, it is now highly regarded. In 1811, the institute began a math prize for the con-

duction of heat through infinite solid bodies. Fourier’s was one of two submissions that

was reviewed by Lagrange, Laplace, Malus, Hauy, and Legendre, who awarded the prize

to Fourier. His work won fame and inspired much more research on conduction.

13 Appendix B: Model Validation and Verification

Model Run from FORTRAN 90:

74.5

59.0

43.5

Validation using Scilab:

-->diary validation

-->af=[-1.272321,.636161,0,-57.254465;.636161,-1.272321,.636161,.913;0,.636161,-1.272321,-17.8125]

af =

- 1.272321 0.636161 0. - 57.254465

0.636161 - 1.272321 0.636161 0.913

0. 0.636161 - 1.272321 - 17.8125

-->rref(af)

ans =

1. 0. 0. 73.782531

0. 1. 0. 57.564985

0. 0. 1. 42.78252

-->diary(0)

35

14 Appendix C: Fortran Program

Program Floorheating

!This version of the program is for

verification of the model. This is the version I am most likely to

use again in the future.

!======================================================================================

!This program solves a Partial Differential Equation describing the

transfer of heat in a !limestone concrete slab. Modeling radiant

heating in a floor. Approximates the PDE at n nodes !using a

backward difference approximation. THe system of linear equations is

then solved !using Thomas Algorithm. !Variables used:(in SI units)

!a=area !alpha=alpha !k=thermal conductivity !cp=specific heat

!rho=density !len=length !diam=diameter !Kw=thermal conductivity

constant !Nu=Nusselet number !num=counter !n=number of

equations/internal nodes !z=length between nodes !eqn=row=i=number

of rows in array !y=col=j=number of columns !temp=temperature

!tempL=left hand boundary condition !tempR=right-hand boundary

condition !Tb=time at the bottom !Tt=Time at the top

!========================================================================================

implicit none character(len=50)filec,filen,results

integer::EOF,n,i,j,t,IOSTAT

doubleprecision::alpha,k,cp,rho,len,deltat,temp,tempB,tempT,thick,time,totaltime,Tt,Tb,z,e,w,sum,sss

doubleprecision, dimension(:,:),allocatable::coeff doubleprecision,

dimension(:),allocatable::junk,a,b,c,d,r,tempprev,v,LS

interface

subroutine thomas(a,b,c,d,r,n)

double precision,dimension(:),intent(out)::r

double precision,dimension(:),intent(in)::a,b,c,d

integer,intent(in)::n

end subroutine

end interface

!Files used============================= !results - Final file to

plot !results-1 - Intermediate step to check !results-s -

Steady-state values !open file to save data=================

36

open(unit=15,file="results-s") open(unit=16,file="resultsv")

open(unit=17,file="coeff") open(unit=18,file="junk")

open(unit=19,file="analytical") open(unit=20,file="resultsvLS")

!assign number of nodes==================

Write(*,*)"What is

thickness of floor in centimeters?"

read(*,*)thick !thick=10

thick=thick/100

write(*,*)"How many nodes are used for

discretization?"

read(*,*)n !n=120 allocate(coeff(n,n))

allocate(junk(n),a(n),b(n),c(n),d(n),r(n),tempprev(n),v(n),LS(n))

!assign time duration====================== Write(*,*)"How many

seconds are you interested in seeing the sytem running?"

read(*,*)totaltime !totaltime=1200 Write(*,*)"How many timesteps?"

read(*,*)time !2E^5 deltat=totaltime/time

write(*,*)"delta

t=",deltat

!Assign values to constants===============

k=1.14!!!RANDOM

DISTRIBUTION BETWEEN 1.1-1.8 IN DO LOOP

cp=.84

rho=1920.

!Find all variables=====================

alpha=k/(rho*cp)

z=thick/(n+1)

write(*,*)"z=",z

write(*,*)"alpha=",alpha

!Specifiy boundary conditions========== TempB=70 TempT=25 !=r(n-1)

in do loop Tt=5 Tb=0 a=0 b=0

c=0

d=0

coeff=0

37

junk=0

coeff(1,1)=(-2*alpha)/(z**2)

coeff(2,1)=alpha/(z**2)

coeff(n,n)=(-2*alpha)/(z**2)

coeff(n-1,n)=alpha/(z**2)

!------------------------------ !For steady state

initialconditions============ junk(1)=((alpha*tempB)/(z**2))

junk(n)=((alpha*tempT)/(z**2)) a(1)=0

c(n)=0

b(1:n)=(-2*alpha)/(z**2) a(2:n)=alpha/(z**2)

c(1:(n-1))=alpha/(z**2)

d=-junk

call thomas(a,b,c,d,r,n)

Do i= 1,n

write(17,"(1000f10.6)")(coeff(i,j),j=1,n)

write(18,"(f10.6)")junk(i)

write(15,"(f10.6)")r(i)!!!!

end Do

!Transient loop======================================== tempprev=15

DO t=1,totaltime

!VERIFICATION ANALYTICAL SOLUTION=========== w=(z*i)/sqrt(4*alpha*t)

Do i=1,n v(i)=(15-70)*erf((z*i)/sqrt(4*alpha*t))+70 end do

write(19,"(1000f10.5)")(v(i), i=1,n) tempt=r(n-1)

junk(1)=((alpha*tempB)/(z**2)-tempprev(1)/deltat)!!!!!!!!!!!!!!!!!!

junk(n)=((alpha*tempT)/(z**2)-tempprev(n)/deltat)!!!!!!!!!!!!!!!!!!!

coeff(1,1)=((-2*alpha)/(z**2)-1/deltat)!!!!!!!!!!!!!!!!!!

coeff(2,1)=alpha/(z**2)

coeff(n,n)=((-2*alpha)/(z**2)-1/deltat)!!!!!!!!!!!!!!!!!!

coeff(n-1,n)=alpha/(z**2)

Do i=2,n-1

junk(i)=tempprev(i)/deltat

38

end Do

a(1)=0

c(n)=0

a(2:n)=alpha/(z**2)

b(1:n)=((-2*alpha)/(z**2)-1/deltat)

c(1:(n-1))=alpha/(z**2)

d=-junk

call thomas(a,b,c,d,r,n)

!+++++Write outputs++++++++++++++++++++

!write(16,"(a,i3)")"Time====",t

! write(18,"(a,i3)")"Time====",t !!!!Don’t need to write junk or

coeff!!!!! ! Do i= 1,n

!write(17,"(1000f10.6)")(coeff(i,j),j=1,n)

! write(18,"(f15.9)")junk(i)

!end Do

!++++++++++++++++++++++++++++++++++++++

!Option to end before totaltime elapses+++++

!if((r(n)-r(n-1)) .lt.

.0001)exit !+++++++++++++++++++++++++++++++++++++++++++

Do i=1,n!!!!!!!!!!

tempprev(i)=r(i)!!!!!!!!!!!!!!!

end Do!!!!!!!!!!!!!!

!LEAST SQUARES option +++++++++++++++++++++++

!sum=0

!Do i=1,n

!LS(i)=(v(i)-r(i))**2

!sum=sum+LS(i)

!end Do

!SSS=sqrt(sum)

!write(20,"(f10.5)")SSS

!write(20,"(1000f10.5)")(LS(i),i=1,n)

!% ERROR option+++++++++++++++++++++++++

sum=0

39

Do i=1,n

LS(i)=(((v(i)-R(i))/r(i))*100)

sum=sum+LS(i)

end do

SSS=sum/n

write(20,"(1000f10.5)")SSS

!===================================

write(16,"(1000f10.5)")(r(i), i=1,n) end DO!

write(*,*)"tempprev(n)==",tempprev(n) write(*,*)"r(n)==",r(n)

write(*,*)"t==",(t-1) end program

subroutine thomas(a,b,c,d,r,n)

!======================================================================================

!This program solves any tridiagonal system of linear equations

using Thomas’ Algorithm !The matrix must be augmented to be

compatible. For instance, if you have !a matrix of the form Ax=f,

pass to Thomas’ a negative array of ’f.’ !Variable Description:

!a=lower triangular matrix !b=primary diagonal matrix !c=upper

triangular matrix !d=f matrix (see above description)

!e,f=placeholders !r=resulting matrix ! !See Wikipedia for more

information on Thomas’ Algorithm

!=======================================================================================

implicit none

double precision,dimension(:),intent(out)::r

double precision,dimension(:),intent(in)::a,b,c,d

integer,intent(in)::n

double precision,dimension(n)::e,f

integer::i,j

e(1)=d(1)/b(1)

f(1)=c(1)/b(1)

do i=2,n

f(i)=c(i)/(b(i)-a(i)*f(i-1))

e(i)=(d(i)-a(i)*e(i-1))/(b(i)-a(i)*f(i-1))

end do

40

r(n)=e(n)

do i=n-1,1,-1

r(i)=e(i)-f(i)*r(i+1)

end do

end subroutine

15 Appendix D: FA1NEW Results

15.1 D.1: Thermal Conductivity

A summary of the normal distribution results from FA1NEW follow:

NUMBER OF DATA = 99

DATA STATISTICS

STATISTIC BIASED UNBIASED

(*LEAST BIASED)

SUM ------ 0.37979E+04

SQUARED SUM ------ 0.14572E+06

CUBED SUM ------ 0.55922E+07

MEAN ------ 0.38362E+02

MEDIAN ------ 0.38389E+02

VARIANCE 0.25392E+00 0.25651E+00

STANDARD DEVIATION 0.50390E+00 0.50647E+00 *

COEFFICIENT OF VARIATION 0.13135E-01 0.13202E-01 *

SKEW 0.90901E-01 0.93722E-01

SKEWNESS COEFFICIENT 0.71043E+00 0.72141E+00

COEFFICIENT OF EXCESS -0.13482E+03 -0.14035E+03*

(KUTOSIS-3 ; PEAKEDNESS)

41

LN(DATA) STATISTICS

STATISTIC BIASED UNBIASED

(*LEAST BIASED)

SUM ------ 0.36105E+03

SQUARED SUM ------ 0.13168E+04

CUBED SUM ------ 0.48024E+04

MEAN ------ 0.36470E+01

MEDIAN ------ 0.36478E+01

VARIANCE 0.17724E-03 0.17905E-03

STANDARD DEVIATION 0.13313E-01 0.13381E-01 *

COEFFICIENT OF VARIATION 0.36505E-02 0.36690E-02 *

SKEW -0.34122E-04 -0.35181E-04

SKEWNESS COEFFICIENT -0.14461E+02 -0.14684E+02

COEFFICIENT OF EXCESS 0.94973E+04 0.98951E+04*

(KUTOSIS-3 ; PEAKEDNESS)

1

- NORMAL -

TITLE: k-normal-99observations

====================================

I I

I 2 I

I 1 -(X-A) I

I P(X)= ------------ EXP ------- I

I B*SQRT(2*PI) 2 I

I 2*B I

I I

====================================

42

A = 0.38362E+02

B = 0.50647E+00

THEORETICAL SORTED EVENTS

NOTE: PROBABILITY IN PARENTHESIS = RANK/(N+1)

(.0100)0.37184E+02 (.0200)0.37322E+02 (.0300)0.37410E+02 (.0400)0.37476E+02

(.0500)0.37529E+02 (.0600)0.37575E+02 (.0700)0.37615E+02 (.0800)0.37651E+02

(.0900)0.37683E+02 (.1000)0.37713E+02 (.1100)0.37741E+02 (.1200)0.37767E+02

(.1300)0.37792E+02 (.1400)0.37815E+02 (.1500)0.37838E+02 (.1600)0.37859E+02

(.1700)0.37879E+02 (.1800)0.37899E+02 (.1900)0.37918E+02 (.2000)0.37936E+02

(.2100)0.37954E+02 (.2200)0.37971E+02 (.2300)0.37988E+02 (.2400)0.38005E+02

(.2500)0.38021E+02 (.2600)0.38037E+02 (.2700)0.38052E+02 (.2800)0.38067E+02

(.2900)0.38082E+02 (.3000)0.38097E+02 (.3100)0.38112E+02 (.3200)0.38126E+02

(.3300)0.38140E+02 (.3400)0.38154E+02 (.3500)0.38168E+02 (.3600)0.38181E+02

(.3700)0.38195E+02 (.3800)0.38208E+02 (.3900)0.38221E+02 (.4000)0.38234E+02

(.4100)0.38247E+02 (.4200)0.38260E+02 (.4300)0.38273E+02 (.4400)0.38286E+02

(.4500)0.38299E+02 (.4600)0.38312E+02 (.4700)0.38324E+02 (.4800)0.38337E+02

(.4900)0.38350E+02 (.5000)0.38362E+02 (.5100)0.38375E+02 (.5200)0.38388E+02

(.5300)0.38401E+02 (.5400)0.38413E+02 (.5500)0.38426E+02 (.5600)0.38439E+02

(.5700)0.38452E+02 (.5800)0.38465E+02 (.5900)0.38478E+02 (.6000)0.38491E+02

(.6100)0.38504E+02 (.6200)0.38517E+02 (.6300)0.38530E+02 (.6400)0.38544E+02

(.6500)0.38557E+02 (.6600)0.38571E+02 (.6700)0.38585E+02 (.6800)0.38599E+02

(.6900)0.38613E+02 (.7000)0.38628E+02 (.7100)0.38643E+02 (.7200)0.38657E+02

(.7300)0.38673E+02 (.7400)0.38688E+02 (.7500)0.38704E+02 (.7600)0.38720E+02

(.7700)0.38737E+02 (.7800)0.38753E+02 (.7900)0.38771E+02 (.8000)0.38789E+02

(.8100)0.38807E+02 (.8200)0.38826E+02 (.8300)0.38846E+02 (.8400)0.38866E+02

(.8500)0.38887E+02 (.8600)0.38910E+02 (.8700)0.38933E+02 (.8800)0.38958E+02

(.8900)0.38984E+02 (.9000)0.39012E+02 (.9100)0.39042E+02 (.9200)0.39074E+02

(.9300)0.39110E+02 (.9400)0.39150E+02 (.9500)0.39196E+02 (.9600)0.39249E+02

(.9700)0.39315E+02 (.9800)0.39403E+02 (.9900)0.39541E+02

STANDARD ERROR STATISTIC = 0.46626E-01

KOLMOGOROV-SMIRNOV STATISTIC =0.0529

CHI-SQUARE G-O-F STATISTICS:

CHI-SQUARE= 2.182

DEGREES OF FREEDOM= 4.

43

SIGNIFICANT AT 1-ALPHA=0.7024

Table A.4

80 % confidence 2.182 .lt. 5.9886 (χ2 variate, 2.182 represents the tail of the distribution)

95 % 2.182 .lt. 9.477

Accept null hypothesis that state variables resulting from a population of 99 normally

distributed thermal conductivity values fit a normal distribution also at a 95% confidence

interval (Ang and Tang, 2007)

15.2 D.2 Specific Heat

DATA STATISTICS

STATISTIC BIASED UNBIASED

(*LEAST BIASED)

SUM ------ 0.37053E+04

SQUARED SUM ------ 0.13873E+06

CUBED SUM ------ 0.51963E+07

MEAN ------ 0.37428E+02

MEDIAN ------ 0.37425E+02

VARIANCE 0.51415E+00 0.51940E+00

STANDARD DEVIATION 0.71704E+00 0.72069E+00 *

COEFFICIENT OF VARIATION 0.19158E-01 0.19256E-01 *

SKEW 0.22811E-02 0.23519E-02

SKEWNESS COEFFICIENT 0.61875E-02 0.62831E-02

COEFFICIENT OF EXCESS -0.37017E+01 -0.37311E+01*

(KUTOSIS-3 ; PEAKEDNESS)

LN(DATA) STATISTICS

STATISTIC BIASED UNBIASED

(*LEAST BIASED)

SUM ------ 0.35860E+03

SQUARED SUM ------ 0.12990E+04

CUBED SUM ------ 0.47054E+04

44

MEAN ------ 0.36222E+01

MEDIAN ------ 0.36223E+01

VARIANCE 0.37493E-03 0.37875E-03

STANDARD DEVIATION 0.19363E-01 0.19462E-01 *

COEFFICIENT OF VARIATION 0.53456E-02 0.53728E-02 *

SKEW -0.53052E-04 -0.54699E-04

SKEWNESS COEFFICIENT -0.73078E+01 -0.74207E+01

COEFFICIENT OF EXCESS 0.18328E+04 0.19096E+04*

(KUTOSIS-3 ; PEAKEDNESS)

1

- NORMAL -

TITLE: cp-normal

====================================

I I

I 2 I

I 1 -(X-A) I

I P(X)= ------------ EXP ------- I

I B*SQRT(2*PI) 2 I

I 2*B I

I I

====================================

A = 0.37428E+02

B = 0.72069E+00

THEORETICAL SORTED EVENTS

NOTE: PROBABILITY IN PARENTHESIS = RANK/(N+1)

(.0100)0.35751E+02 (.0200)0.35947E+02 (.0300)0.36072E+02 (.0400)0.36166E+02

(.0500)0.36242E+02 (.0600)0.36307E+02 (.0700)0.36364E+02 (.0800)0.36415E+02

45

(.0900)0.36461E+02 (.1000)0.36504E+02 (.1100)0.36544E+02 (.1200)0.36581E+02

(.1300)0.36616E+02 (.1400)0.36649E+02 (.1500)0.36681E+02 (.1600)0.36711E+02

(.1700)0.36740E+02 (.1800)0.36768E+02 (.1900)0.36795E+02 (.2000)0.36821E+02

(.2100)0.36847E+02 (.2200)0.36871E+02 (.2300)0.36895E+02 (.2400)0.36919E+02

(.2500)0.36942E+02 (.2600)0.36964E+02 (.2700)0.36986E+02 (.2800)0.37008E+02

(.2900)0.37029E+02 (.3000)0.37050E+02 (.3100)0.37071E+02 (.3200)0.37091E+02

(.3300)0.37111E+02 (.3400)0.37131E+02 (.3500)0.37150E+02 (.3600)0.37170E+02

(.3700)0.37189E+02 (.3800)0.37208E+02 (.3900)0.37227E+02 (.4000)0.37245E+02

(.4100)0.37264E+02 (.4200)0.37282E+02 (.4300)0.37301E+02 (.4400)0.37319E+02

(.4500)0.37337E+02 (.4600)0.37355E+02 (.4700)0.37374E+02 (.4800)0.37392E+02

(.4900)0.37410E+02 (.5000)0.37428E+02 (.5100)0.37446E+02 (.5200)0.37464E+02

(.5300)0.37482E+02 (.5400)0.37500E+02 (.5500)0.37518E+02 (.5600)0.37536E+02

(.5700)0.37554E+02 (.5800)0.37573E+02 (.5900)0.37591E+02 (.6000)0.37610E+02

(.6100)0.37629E+02 (.6200)0.37647E+02 (.6300)0.37666E+02 (.6400)0.37686E+02

(.6500)0.37705E+02 (.6600)0.37725E+02 (.6700)0.37744E+02 (.6800)0.37764E+02

(.6900)0.37785E+02 (.7000)0.37805E+02 (.7100)0.37826E+02 (.7200)0.37847E+02

(.7300)0.37869E+02 (.7400)0.37891E+02 (.7500)0.37914E+02 (.7600)0.37936E+02

(.7700)0.37960E+02 (.7800)0.37984E+02 (.7900)0.38009E+02 (.8000)0.38034E+02

(.8100)0.38060E+02 (.8200)0.38087E+02 (.8300)0.38115E+02 (.8400)0.38144E+02

(.8500)0.38175E+02 (.8600)0.38206E+02 (.8700)0.38239E+02 (.8800)0.38275E+02

(.8900)0.38312E+02 (.9000)0.38351E+02 (.9100)0.38394E+02 (.9200)0.38440E+02

(.9300)0.38491E+02 (.9400)0.38548E+02 (.9500)0.38613E+02 (.9600)0.38690E+02

(.9700)0.38783E+02 (.9800)0.38908E+02 (.9900)0.39105E+02

STANDARD ERROR STATISTIC = 0.82210E-01

KOLMOGOROV-SMIRNOV STATISTIC =0.0927

CHI-SQUARE G-O-F STATISTICS:

CHI-SQUARE= 9.394

DEGREES OF FREEDOM= 4.

SIGNIFICANT AT 1-ALPHA=0.0520

Table A.4

95 % 9.394 .lt. 9.477

Accept null hypothesis that state variables resulting from a population of 99 normally

distributed thermal conductivity values fit a normal distribution also at a 95% confidence

interval (Ang and Tang, 2007)

46