NBS

PUBLICATIONS

NBSIR 84-2906

NATL INST. OF STAND & TECH

A11107 3TDflED

Thickness Effect in Low- Density

Insulation

Department of Mechanical Engineering

Colorado State University

Fort Collins, CO 80523

August 1984

Prepared for Department of Energy

Office of Building Energy Research and Development

Building Energy Research and Development

Building Systems Division

Washington, DC 20585

and

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

QQ )ter for Building Technology

thersburg, MD 20899

national bureau

OF STANDARDS

library

NBSIR 84-2906

* * * THICKNESS EFFECT IN LOW-DENSITY

INSULATION fo

P. J. Burns /

Department of Mechanical Engineering

Colorado State University

Fort Collins, CO 80523

August 1 984

Prepared for Department of Energy

Office of Building Research and Development

Building Energy Research and Development

Building Systems Division

Washington, DC 20585

and

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Gaithersburg, MD 20899

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
-i#' 'V' f,,S ,L.
PREFACE

This report is one of a series documenting NBS research and analysis efforts to

support the Department of Energy/National Bureau of Standards' Measurements

Program. The work reported in this document was performed cooperatively and

supported by DoE/NBS Task Order A008 under Interagency Agreement No. DE-AIOl76PR06010.

iii
ABSTRACT

A discussion is presented of theory of heat transfer in low-density, glassfiber insulation via conduction, convection, and radiation. It is concluded

that the primary modes of heat transfer in this material are air conduction and

radiation. An analysis of NBS data of measured apparent thermal conductivity

for different thicknesses results in a parameter estimate of the optical

extinction coefficient. This parameter determines the amount of change in

apparent thermal conductivity as a function of sample thickness. This phenomena

is referred to as the "thickness effect."

Keywords: conduction; convection; guarded-hot-plate; low-density insulation;

thermal conductivity; thermal radiation; thickness effect.

IV
ACKNOWLEDGMQIT

The present work was supported by the Department of Energy and National Bureau

of Standards. The National Bureau of Standards also provided high quality

data necessary as input to the present approach. Many thanks are also due to

Brian Rennex for his valuable comments and practical expertise. Finally, the

author wishes to thank Frank Powell, who made the entire project possible.

V
Nomenclature

A = area (m^)

;

e = blackbody emissive power, oT‘^(W»m~2);

fy = volume fraction, defined by equation (3);

g = gravitational constant, 9.8 (m»s“2);

= non-dimensional temperature integral defined in equation (25);

K = permeability (m^);

k = thermal conductivity (W*m“^

;

L = thickness of insulation (m);

n = iteration counter;

Nu = Nusselt number;

P = porosity;

6 = heat flow (W);

q = heat flux (W*m~2)j

= ratio of conductivities, k^/kg;

Ra* = modified Rayleigh number, defined by equation (5);

r2 = goodness of fit parameter, defined by equation (9);

T = temperature (K);

AT = temperature difference, T]^ - T2 (K) ;

X = non-dimensional spatial coordinate, X'/L;

x’ = dimensional spatial coordinate (m)

;

= first unknown parameter (kg»m );

f Xj^ = linearized (old) value of the first parameter (kg*m );

y^ = measured value;

Greek Symbols

= stagnant thermal diffusivity (m *s );

vi
3 = mass attenuation coefficient (m^'kg or coefficient of cubical

expansion (K~l);

Y

= second unknown parameter (W*m"l*K~l);

5 = deviation;

e = emissivity;

V = kinematic viscosity (m-*s“M;

p = density (kg*m~3);

a = standard deviation;

Tq = optical depth, defined by equation (12).

Subscripts

a denotes air or "in the air"

c denotes convection or critical

e denotes effective

g denotes glass

i denotes insulation

j denotes j

* value

k denotes conduction or k’^^^ iteration

1 . s

.

denotes least squares

m denotes metering section

r denotes radiation

t denotes total (i.e. the entire 48" x 48

w denotes at the wall

0 denotes stagnant

1 denotes the hot surface

2 denotes the cold surface

sample)

vii
Table of Contents

Page

PREFACE ill

ABSTRACT iv

ACKNOWLEDGMENT v

NOMENCLATURE vi

1. THICKNESS EFFECT IN LOW-DEJISITY INSULATION 1 1.1 INTRODUCTION 1

2. HEAT TRANSFER OVERVIEW 4

2.1 THEORETICAL FORMULATION 4

2.2 CONDUCTION 4

2.3 CONVECTION 5 2.4 RADIATION 7

3. ANALYSIS 9

3.1 PARAMETER ESTIMATION 9

3.2 SEOUENTIAL FORMULATION 10

3.3 APPLICATIONS 12

4. CONCLUSIONS AND RECOMMENDATIONS 19

5. REFERENCES 21

APPENDIX A A-1

viil
1. THICKNESS EFFECT IN LOW-DENSITY INSULATION

1.1 INTRODUCTION

Determining the heat transfer through porous insulating materials has long been

a subject of concern. In particular, judicious engineering design requires the

minimization of the total cost associated with the purchase of insulation.

This total cost includes the base cost of the insulation and the total cost of

the energy that flows through the insulation. The base cost consists of the

material cost and the installation cost. Furthermore, the cost of the energy

that is transmitted through the insulation must be amortized over the lifetime

of the insulation, and it is convenient to perform this minimization in terms

of net present worth.

Indeed, to effect the minimization, it is necessary to characterize the

performance of the insulation with parameters that can be associated with cost.

Thus, the thermal performance of insulating materials is presently characterized

versus the density of the sample and the mean diameter of the fibers comprising

the porous matrix [1], Moreover, such has been the practice for some time, as

many have analytically addressed [2-5] the problem of modeling the thermal performance of insulating materials, and the experimental investigations are

legion [4-8] also. Recently, the research emphasis has been dedicated towards

determining the "thickness effect" [8-11], that is, the change in the effective

thermal conductivity with specimen thickness. In particular, a very detailed

discussion of the thickness effect may be found in Reference 11. In connection

with this, it is noteworthy that the effective thermal conductivity can be wellcharacterized [12] versus percent light transmission.

1
To date, the problem has been addressed only deterministically in that physical

mechanisms have been assumed, ranges of transfer coefficients have been

selected, and solutions have been obtained under a wide variety of simplifying

assumptions. This technique yields excellent insight when determining the

relative effects of various physical mechanisms, but it is sometimes fruitless

to attempt to accurately predict the actual heat transfer occurring in a specific physical system because the predicted property (in this case the effective

thermal conductivity) may vary much more under the range of input parameters

than does the observed experimental data. Actually, this can be attributed

both to the variation due to the approximate nature of the solutions, and due

to the lack of knowledge of the specific input parameters. In particular, the

solid conduction contribution will change depending upon the assumptions made

as to fiber orientation, and the radiative parameters (phase function and

extinction and single scattering coefficients) are not well known for general

situations. The adhesive binder is a complicating factor of great physical

importance that is usually neglected in theoretical analyses, and there is

likely to be large spatial variations in binder content in any one batch of

insulation. There is also likely to be large global variations in binder content from batch to batch. Thus, the law of diminishing returns is operating

in that further deterministic analyses will be very tedious and will yield

less practical information than those of the past.

In this light, a stochastic approach will be undertaken herein that will

circumvent many of these difficulties and yield some new insight into the

experimental determination of the effective thermal conductivity. In short,

relationships for heat transfer modes will be assumed in terms of unknown

2
constants. These unknown constants will, in effect, be the uncertain parameters

mentioned above. Since these parameters will be determined from the data, they

will automatically be the "best-fit” values consistent with the assumed model

and the scatter in the data. The true test of this method will entail comparison of the overall scatter in the data against the scatter in the predicted

values. Also, enough data points must be used to ensure a statistically significant result. Both of these facets of the present approach will be quantified.

Because the parameters will be derived from experimental data obtained at the

NBS in Gaithersburg, MD, a few comments concerning the quality and the material

variability of the insulation samples are in order. Figure 1, which was abstracted

from Reference 13, shows that both density and fiber diameter are important

parameters in determining the effective thermal conductivity of fibrous insulations.

The data used herein are NBS-measured values taken from the calibrated transfer

samples prepared for the NBS by the Johns Manville Corporation. These samples

vary widely in thickness and density, yet the fiber diameters were controlled

very carefully to have a mean value of 3.8 x 10“^ m for all of the samples.

Further detail may be found in Reference 1. Thus, we shall proceed realizing

that the fiber diameter is a parameter which is held fixed in the present

analysis. The most that can be done, then, is to predict the thickness effect

at a constant fiber diameter. For completeness. Table 1 includes values of

the datum points used.

The present analysis will proceed in the following vein. Physical mechanisms

for the heat transfer will be examined and proposed, and detailed discussions

of past models are given. A brief description of the sequential least squares

3
algorithm follows. The data are then filtered through a variety of models,

and the present work ends with conclusions and recommendations.

2. HEAT TRANSFER OVERVIEW

2.1 THEORETICAL FORMULATION

The problem under consideration is to predict the heat flow through low density

insulation given the boundary conditions (the boundary temperatures). Of

course, the basic material properties must also be known as they are required

input parameters. The heat flow will be assumed to be one-dimensional as the

NBS GHP (Guarded Ifot Plate) satisfies very stringent criteria in this regard

(about 0.1 percent accuracy). In this case, the heat flow may be conceptualized

as follows;

Q

= + Qc + 0^ 0]^c ^r ^c

where separate contributions have been included for conduction (k)

,

convection

(c), and radiation (r). Also, interactions between all three modes of heat

transfer have been allowed as separate terms and it is presumed that these terms

are of smaller magnitude than the ones involving the basic modes of heat

transfer. The validity of this assumption will be demonstrated by the match between

thing and experiment. The terms will be addressed one-by-one.

2.2 CONDUCTION

The conduction terra, Q[^, involves the gaseous and solid conduction through the

porous medium. There is also an interaction term associated with these two

modes of conduction. This can be conceptualized as follows: since the contact

areas between fibers are small, the constriction resistance is large and

little heat flows through the solid matrix directly. However, if the fibers do

contact or if they come close together, heat will flow from one fiber through the

4
intervening gas to another fiber. A careful analysis of this process for packed

spheres [14] shows that this, in fact, is a major transfer mechanism for heat

flowing through the solid. Indeed, a number of analytical investigators [2-5,

9, 10, 14-16] have pursued this subject. Experimental treatments of this topic

[2-8] confirm most of the theories for low density porous insulation wherein

the solid matrix is so widely dispersed (porosity = 0.995) that only a small portion of heat flow can be attributed to the fibers. This is readily evident by

viewing Figure 2, which has been abstracted from the review article by Bankvall

[5]. The model used herein is one proposed by Bhattacharyya [10]:

1 -_

1 - P

1 4- 2 f^

1 + Rk

+ (l-\) [1 -

1 - R,

( 2 ) 1 + fv( 1 + 5R^)

3(1 + Rk)

where, for air at 287K and glass fibers, the properties are taken as

follows: kg = 1.04 Wm~^‘K~^, k^ = 0.026 W*m^ • K“^ , and Pg = 2580 kg*m"^,

p_ = 1.04 kg*m . The volume fraction is computed as follows: 3

2.3 CONVECTION

f = (3)

The convective heat transfer in horizontal porous media has been the subject of

extensive investigation for several decades [17-30]. The critical modified

•k Rayleigh number, Ra^ , below which no convection can occur in a horizontal

porous medium heated from below has been determined to be as follows [17, 18]:

Ra* = 4 r2 2 39.5 (4)

where the modified Rayleigh number qualitatively represents the ratio of the

buoyancy force to the drag force of the porous matrix, and is given as:

5
( 5 ) = g g at K L vao

wherein the effective thermal conductivity of the stagnant porous medium is

used to calculate the thermal diffusivity, o.q, A number of experimental

investigations [20, 22-24] have verified the relationship given by equation (4).

Elder [24], gives the following formula for the non-dimensional steady convective heat transfer in porous media heated from below:

• • Q + Q

* * Nu = S Ji = 1 + (Ra - Ra(,)/40 (6)

where Nu = 1 if Ra ^ ^c*

It is of interest to estimate the magnitude of the free convective heat transfer

in porous media using equations (4) and (5). First the modified Rayleigh number is calculated. The following values of the parameters are used:

g

= 9.8 m-s“^, 8 = T“^ = 3.37 x 10“^ K“^, v = 1.56 x lO"^ m^*s"^

k* = 0.045 W*m“^*K“^, p = 1.16 kg»m“^, Cp = 1.011 J*kg“^»K“^, =

3.84 X 10“5 L = 0.1562 m, K = 5 x 10~® m^.**

These values result in a modified Rayleigh number of:

Ra* (AT) = 0.419 AT(K)"1 (7)

For the AT = 27. 8K used in the GHP, the modified Rayleigh number is:

Ra* (27.8) = 11.6 (8)

which is below the critical value, so all of the terms associated with

convection in equation (1) are nil. Indeed, setting equation (7) equal to

* Taken from measurements made using the NBS Guarded Ifot Plated [1].

** Taken from Fournier and Klarsfeld [7], Fig. 5, F = 3.6 (typical of the

3.8 pm diameter fibers in the measurement system of [1]).

6
equation (4) results in a AT = 95. 9K for the onset of convective flow. Thus,

free convection is seen to be a significant contributor to the heat transfer

for porous insulations only at very large thicknesses and very large temperature

differences. Indeed, for all practical cases of thermal insulation at room

temperature, free convection heat transfer does not occur in the horizontal

geometry

.

2.4 RADIATION

As seen by viewing Figure 2, radiation heat transfer contributes significantly

to the total heat transfer in low density fibrous insulation. Recent theoretical calculations neglecting the phenolic binder [31] show that the scattering

coefficient is roughly a factor of 2 to 4 times larger than the absorption

coefficient, but this conclusion has not been conclusively supported with

experimental data. Indeed, there is some indication that the phenolic binder

is highly absorptive. Moreover, it is extremely difficult to measure absorption

and scattering coefficients separately [32-33], especially in the midst of a

matrix of solid fibers. Despite these limitations, a brief review of past

work in this area will be presented for illustrative purposes.

The pioneering work was done by Hamaker [34] and Larkin and Churchill [35] who

applied the two flux model for gray absorbing and scattering media. The approximation of radiative equilibrium used does not allow free interaction between

the radiation and conduction which limits the general applicability of their

results. Viskanta subsequently pursued the problem including isotropic scattering from a more rigorous standpoint for the condition of radiative equilibrium

[36], and later coupled with conduction [37], It is mentioned in passing that

7
Usiskin and Sparrow [38], and Heaslet and Warming [39] presented a solution for

a purely absorbing medium. Furthermore, the diffusion technique has also been

applied to the problem [46] of radiation heat transfer in a purely absorbing

medium.

Some further analyses into the problem at hand have attempted to remove the

assumption of isotropic scattering [41-43], enhance the applicability of the

two-flux assumption [44, 45] or include transients [46]. Further analytical

work may be found in Reference 47, wherein a large parameter variation was

performed. All of the investigations mentioned so far assume gray material

properties

.

Recently, Tong and Tien [31] have applied a two-flux model to the specific

problem involving glass fibers wherein the spectral solution of Maxwell's equations have been used to determine the absorption and scattering coefficients.

While this represents a large step in the right direction, the questions of

fiber orientation and the effect of the adhesive binder are still to be included

in a more rigorous analysis. Furthermore, it is questionable whether such an

approach involving an approximate solution technique can ever be predictive to

within, say, two percent as is necessary to predict the thickness effect.

It is therefore concluded that the research community has yet to accurately

solve the generalized equation of transfer for the specific problem of radiant

heat transfer in fibrous insulations with binder present. Furthermore, the

property measurements that have been made [e.g., extinction coefficient] must

be augmented by further measurements including the single scattering albedo

[32-33] to be generally applicable to the present problem. Indeed, this is

8
not an easy problem either as the question of the separation of effects when

measuring quantities that can be attributed to diverse physical mechanisms is

an elusive one [48].

Rather, the approach adopted here will consist of taking the measured data for

heat transfer rates, assuming physical models and filtering the measured data

through the assumed model to attain "best-fit" values for the thermal parameters.

This approach has the advantages of yielding the best thermal parameters that

pertain to the data at hand consistent with the noise level of the data, the

assumed model and, of course, the minimization of the L-2 norm. However, the

approach cannot truly be called deterministic as the material properties are

derived from performance data rather than vice versa, but this is a philosophical

and not a technical issue and, consequently, will not be further addressed.

Instead, the pitfalls inherent in such an attempt will be discussed, and it

will become evident that the range of applicability of the generated parameters

will be pronouncedly affected by the specific models assumed and by the specific

data filtered.

3. ANALYSIS

3.1 PARAMETER ESTIMATION

When parameter estimation is employed to determine thermal parameters, two

general rules must be separately observed in order to obtain meaningful results.

Firstly, the model must be representative of the physical processes occurring

in the experimental system. For example, problems are sometimes encountered

when free convection (T^/^ dependence) and radiation (T^ dependence) are erroneously lumped into the same heat transfer coefficients in a model. This type

9
of model will not allow modes to be identified. Also, all terms of significance must be included in the model. If convection is indeed an important

mode of heat transfer, its effects will be contained in the data, and when the

data are analyzed, these effects will be attributed to some parameters, and

incorrectly so if convection is not represented in the model.

Secondly, the combination of the noise level of the data and the number of data

samples taken must be such as to render the results statistically Tiomogeneous

.

The general rule of thumb is that the noise level of the data should be much less

than the signal level of the physical mechanism that is to be identified. This

stringent criterion may be relaxed as larger numbers of datum points become

available. Other problems that are specific to each individual problem under

investigation [47] will not be addressed here.

3.2 SEQUENTIAL FORMULATION

The formulation of the sequential linear least squares algorithm is well-developed

elsewhere [48] and will not be repeated here. Instead, the use of the technique

will be presented by example. Suppose that j parameters xj are to be identified

from the measured values A, which multiply the parameters and y, which appear

separately. Mathematically,

Akj Xj = yk (9)

where k denotes the k’th measurement. The system becomes overdeterrained when k

becomes greater than j, and no general solution is then possible. Rather, the

system must be solved for the parameters that satisfy some global criterion.

It is convenient to choose this criterion to be the minimization of the L-2

norm of the errors in the parameters.

10
When identifying j parameters, the linear system of equations becomes complete

when j experimental observations have been added to the system. The equations

can be solved at this time by Gaussian elimination, yielding a set of parameters

that are representative of the measurements taken. When these measurements

contain significant noise, the parameters are likely to be substantially in

error at this point. In order to yield physically realistic parameters,

additional data points are processed.

However, as one more observation is added to the set, the system becomes

overdetermined. Then, least squares procedures are used. Least squares data

processing will yield a new set of parameters each time an additional observation

is processed. This is possible to do with ordinary least squares algorithms by

reformulating and reducing the A matrix each time. The sequential least squares

algorithm performs this function at a fraction of the computer time because the

matrix does not get reformulated and reduced for each data point. Thus,

sequential least squares yields values for the parameters after each data

observation is added.

This is advantageous in two ways. First, by observing the parameter trajectories,

it can usually be determined whether enough data points have been processed for

the noise to be filtered out. Secondly, it can sometimes be determined whether

there are imperfections in the model. To explain this, consider the situation

where data have been taken on a 0.05 m specimen. Suppose that the initial noise

in the parameters has died out, and the parameter trajectories have been observed

to become flat. Suppose, further, that data from a 0.1 m sample are next added.

If the parameter trajectories are observed to suddenly start to change and

continue to change steadily, it can be deduced that there is error in the model

11
as it does not fit all the different physical situations in a similar fashion.

This is termed "model noise," Thus, the model should be reformulated and the

data reprocessed until a uniformily valid set of parameters is obtained.

In summary, a sequential observation of the parameter trajectories can be useful

in detecting model noise and in determining when a sufficient number of data

observations have been processed to overcome the experimental noise level. The

sequential algorithm is given for reference in the Appendix,

3,3 APPLICATIONS

The problem at hand is formulated in terms of the effective thermal conductivity

as follows:

^e ~ ^r (10)

where is the term in brackets in equation (2), It is for pure conduction

thru the solid and air.

The radiation conductivity is taken to be as follows:

2 2 a(T, + T-)(T, + T-)L

kj. = i ± ± (11)

L+L -1+3t/4

£l S2

where all of the quantities in equation (11) are assumed to be measured except

for Tq, the effective optical depth. This parameter is termed effective in that

it incorporates the effects of absorption, anisotropic scattering, spectral

properties, fiber orientation and the binder material. It is standard practice

to determine the properties of fibrous insulation (i.e, the effective thermal

conductivity and the permeability) as a function of the material density, p.

Since the scattering of radiation is proportional to the number of scattering

12
particles within the medium, it is reasonable to assume that:

Tq = 3 pL (12)

Further, it is known [1] that:

L + - 1 «±^o (13)

£2 ^

Equations (11) and (12) may be combined together with the condition stipulated

as equation (13) and the binomial theorem to yield:

4-(X? + T?)(Ti + T^) . 4(1/£^ + 1 /e2 - 1).

- >^1 ^°

3P

^ J

where the term has been linearized by the inclusion of the old value of the

parameter Also:

(14)

= 3"^ (15)

It is apparent that it is necessary to iterate to determine a "thickness effect"

f as it only shows up in equation (14) as a product involving the old value, x^,

f of the radiation parameter. Thus, is set to zero (the first old value),

f Xj^, is determined; the old value x^^ is next set to this value of Xj^ ; the

filtering is performed again yielding a new value of xi and the procedure

continues until convergence is achieved. Finally, it is noted that equation

(11) is similar in form to the derivations of Verschoor and Greebler [2],

Bankvall [5], van der Held [50], Hager and Steere [51], and Strong et. al

.

[52],

The geometry for the first problem is shown in Figure 3. At the low densities

at which the present data exist, solid conduction plays very little role in

the heat transfer, and, therefore, the conductive contribution was modeled as

in equation (2), and taken as a known quantity (i.e., included in the righthand side) .

13
The linearized multiplier for the radiative parameter is explicitly:

4o(Ti + i)(Ti +

i

2 ) r, - I’l

Akl

3^

11 1=1 J

Finally, the known quantity including the effective thermal conductivity,

is explicitly:

yk = ke ~ ^ L/(AT A)

where the second term (= k]^) is calculated using equation (2).

(16)

ke >

(17)

Figure 4 shows the variation of X]^ , with the number of datum points, n. Recall

that a sequential filter generates the "best-fit" value for the parameter being

estimated for each sequence of datum points up to and including the present one.

The data are organized from left to right as the 0.0254 m (1") set, the 0.0381 m

(1.5") set, the 0.0762 m (3") set and the 0.1524 m (6") set. Local variations

are seen to be extreme below n = 10 and are observed to damp out for n>20 due

to the averaging of a larger number of points. Thus, it may be concluded that

there are more than enough data points to render local fluctuations (measurement

noise) insignificant, However, the global variation (model noise) of the

parameter manifested as a decrease in x^ with an increase in thickness (negative

slope) is disturbing. This suggests that the proposed model for the effective

thermal conductivity is of the wrong form else the parameter, X]^ , would have

leveled off. Obviously, relative to the value of x^, at a thickness of 0.0254

m (1"), the value of x^, for a thickness of 0.1524 m (6") should be lower.

Thus, although the family of predicted curves should nestle close to the average

of the experimental data, the thickness effect will not be well represented.

14
This is indeed observed in Figure 5 which presents the predicted curves and the

experimental data points for the four thicknesses of interest. While the average values have been predicted well, the thickness effect has been overpredicted.

The 0.0254 m (1") curve lies on or below all of the 0.0254 m (1") data but one

point. The 0.0381 m (1.5") curve is statistically unimportant as there are

only 3 data points appertaining thereto. The 0.0762 m (3") curve appears to

be a good fit for its data and the 0.1524 m (6") curve lies above all of its

associated data points but one. For the data shown, the thickness effect is

eyeballed to be about 3.5 percent at p = 9kg»m“3) while the predicted

curves indicate a thickness effect of about 6 percent.

Several comments are appropriate here. First of all, the value of 3 = 39.5

m2*kg“l is the best value of 3 consistent with the present model for the given

experimental data. Secondly, even though the scatter in the data is substantial,

it is apparent from Figure 6 that the thickness effect is overpredicted.

Finally, the additional correction required to bring the predictions in line

with the data is small (about 2.5 percent), and, therefore, may be attributed

to a variety of factors. The approach taken is to attribute the discrepancy

to the most obvious physical mechanism. The merit of this assumption can then

be judged by the final results obtained.

Insight may be gained by viewing the predicted temperature profiles. Since

radiative and conductive heat transfer have been treated separately, the temperature profiles are plotted separately with the caveat that since about

half the heat transfer is due to radiation and half is due to conduction, the

initial predicted temperature profile may be roughly taken as the average of

the two. The conduction temperature profile, T]^ , is taken as linear from

15
T2 = 283.0 K to Ti = 310.3 K as shown in Figure 6. The radiation temperature

profile was derived by Deissler [40], and will be presented here in brief.

The jump in emissivity at surface 2 is given as:

®a2 = ^w2 - e„2)(l/£2 ~ 0.5) (18)

The blackbody emissivity power then varies linearly (with x) through the medium,

viz

e^(x) = e^o + X 0.75p3L (^1-^2)

^ 0.75p8L + 1/e^ +1/S2-1

(19)

Combining the two previous equations, the temperature profile in the medium is

determined as

Tr = { "w2 ^®wl - ^w2 ^ |~ 1 _ Q 3 +

0. 75pSL X

^ 0.75p3L + 1/cj^ + l/e2

( 20 )

The profiles of Tj- are shown in Figure 6 for the four thicknesses of interest

where the parameters have been taken as = 9 kg»m and = C 2 “ 0.9. The

jump boundary conditions result in an increase above T]^^ at x = 0 and a decrement

below T^ at x =1. There is clearly a significant increase in the average temperature when radiation is considered, and it is observed that the radiation profiles

are nonlinear. Conduction will have the following two effects on the radiative

temperature profile: (1) it will eliminate the jump at the boundaries, and (2)

it will tend to smooth the overall profile by transferring heat through the insulation. This second effect will serve to reduce the magnitude of the temperature

difference from conduction and, hence, will reduce the thickness effect. This

is precisely the type of mechanism needed to enhance the present theory.

16
Attention is therefore focused upon the coupling between the radiative and the

conductive profiles. Since the effects are small, the base solutions as given

above will be used, similar to perturbation theory, to determine the coupled

heat transfer. The initial approach taken was to integrate the radiative temperatures excess (as a function of position) divided by the conduction length to

the cold wall. This approach did not bear fruit as the temperature jump at the

cold wall rendered the integrand and subsequently the integral infinite. Since

the conduction will eliminate the jump, the mathematics is not inconsistent

with the physics of the situation. Next, it was reasoned that the contribution

to the effective thermal conductivity of the coupling between radiation and

conduction should be proportional to the absolute magnitude of the average

radiative temperature excess, i.e.:

x=l

kcr = T /

x=0

(25)

This integral, denoted as the non-dimensional temperature integral, I'p, is shown

in Figure 7 , as a function of density for the four thicknesses of interest with

2 —1

= ^2 “ 3 = 4 m »kg . The graph appears to be consistent with

the observed data, i.e., the curves decrease with increasing density and the

effects are more pronounced at lower thicknesses.

The approach taken is to apply the sequential least squares filtering technique

to estimate y in equation (21). Since this term is likely to be small, the

filtering is done for this term alone to avoid possible computational "conditioning" problems. Thus, the conduction and the primary radiation terms are taken

to the right-hand-side of the equations as knowns , with an assumed 3. The

sequential filtering is then done for a variety of values of 3, resulting in

17
a variance, o^, for each assumed 3. This corresponds to hunting around

in the two space of (Yj 3) for the overall minimum, and is illustrated

graphically in Figure 8. The minimum is achieved for 8 = 44.0 m^'kg”!,

and the corresponding "best-fit" value of Y if Y = 0,0740 W*m”^»K“^.

Recall that a correction of about 0.0015 W*m“l»K“^ is being sought,

and the integral, I'p, was shown to have a value of about 0.025. Thus,

Y appears to be of the correct magnitude.

The new relationship is presented together with the experimental data in

Figure 9 for the four thicknesses. The predicted thickness effect is now seen

to be consistent with the experimental data. Moreover, it appears that the

thickness effect is delineated about as much as possible given the scatter of

the data. Two datum points from the curve fit of Rennex [11] at the two thicknesses of 0,0254 m (1") and 0.1524 m (6") are shown for 3 = 44 m2»kg~l as

open and closed hexagons, respectively. The thickness effect is about the

same as is predicted by the present analysis, but these values lie well

below the present data. The lower values are attributed to the fact that the

value of 3 used in the calculation is different from the one that would be

obtained by filtering using his curve fit. This illustrates that the "best-fit"

parameters for one model are not necessarily the "best-fit" values for another

model. Using the previous value of 3 = 39.5 m-*kg~^, the open hexagons

are plotted for all densities and at the thicknesses of 0.0254 m (1") and

0.1524 m (6") with a double-dashed line and a triple-dashed line, respectively.

From this, it is observed that the Rennex-curve fit slightly overpredicts the

thickness effect for the present data set, but the overall agreement is good.

18
4. CONCLUSIONS AND RECOMMENDATIONS

Conduction has been shown to be essentially that of stagnant air, and

convection has been demonstrated to be nonexistent in horizontal building

insulation under the usual circumstances encountered in practice. Radiation

was found to contribute approximately one-half to the total heat transfer. A

relation of the diffusion approximation genre was found to be inappropriate in

form to predict the thickness effect. The inclusion of a term to account for

the conductive-radiative interaction was found to represent the data well.

The present approximation was found to be slightly better for the present

data set than the curve fit of Rennex [11]. For completeness, the entire

relationship is presented as follows:

^c = + ^r + TIt (22)

where the conductive contribution is given as [10]:

^c ~ ^g *^k (

1 -

1 +

) 2_Mv

+ (1-cti^) (1

1 -

1 +

Rr

3(1 + R^)

(23)

and the radiative term is as follows [40]:

_ ^^^1^ ^ T2^)(T^ + T2)L

^ ~

3 P3L/4 + l/e^+l/£2-l

The values of the constants are given as follows:

3 = 44.0 m^'kg ^ (25)

Y = 0.0740 W*m“^*K~3 (26)

2^k = 0.5 C7

19
The present approach predicts the thickness effect very well from a firm fundamental

basis. The universal merit of the present result can only be assessed when

the technique has been applied to an inclusive data set including different

fiber diameters (most especially), different surface temperatures, and a wider

variation of densities. Also, the variation in types and quantities of binders

is a complicating factor.

20
5. REFERENCES

1. B. G. Rennex, "Low-Density Thermal Insulation Calibrated Transfer SamplesA Description and a Discussion of the Material Variability," NBSIR-82-2538

(1982)

.

2. J. D. Verschoor and P. Greebler, "Heat Transfer by Gas Conduction and

Radiation in Fibrous Insulations," ASME Trans

.

, 7

4

, p. 961 (1952).

3. R. H. E. Braiiey, "feat Flow Through Insulations Used in the Chemical Process

Industry," presented at the Symposium on Glass and Related Materials, II,

AICHE Preprint 32E, Phila. PA (1968).

4. C. Bankvall, "feat Transfer in Fibrous Materials," J. Testing' and Evaluation , J_, p. 235 ( 1973).

5. C. Bankvall, "Mechanisms of feat Transfer in Permeable Insulation and Their

Investigation in a Special Guarded Hot Plate," Heat Transmission Measurements in Thermal Insulations , ASTM STP 544 , p. 34 (1974).

6. C. M. Pelanne, "Experiments on the Separation of feat Transfer Mechanisms in

Low-Density Fibrous Insulation," Proc. Eighth Thermal Conductivity Conf . Plenum Press, New York (1969).

7. D. Fournier and S. Klarsfeld, "Some Recent Experimental Data on Glass Fiber

Insulating Materials and Their Use for A Reliable Design of Insulations at

Low Temperatures," feat Transmission Measurements in Thermal Insulations , ASTM STP 544 , p. 223 (1974).

8. C, M, Pelanne, "Experiments to Separate the 'Effect of Thickness' from

Systematic Equipment Errors in Thermal Transmission Measurements," presented at the ASTM C 1600 Thermal Insulation Conference, Tampa, FLA,

(Oct. 1978).

9. C. J. Shirtliffe, "Effect of Thickness on the Thermal Properties of Thick

Specimens of Low-Density Thermal Insulation," Thermal Insulation Performance, ASTM STP 718 , p. 36 (1980).

10. R. K. Bhattacharyya , "feat Transfer Model for Fibrous Insulations," Thermal

Insulation Performance , ASTM STP 718 , p. 272 (1980).

11. B. G, Rennex, "Thermal Parameters as a Function of Thickness for Combined

Radiation and Conduction in Low-Density Insulation," Journal of Tliermal

Insulation , p. 283 (1981).

12. C. M. Pelanne, "Thermal and Physical Characteristics of Glass Fiber

Insulation Produced for the National Bureau of Standards." Johns Manville

Report 436-T-1528 (April 1981).

13. ASHEIAE Handbook of Fundamentals (1981), p. 205 .

21
14. Y. Ogniewicz and M, M, Yovanovich, "Effective Conductivity of Regularly

Packed Spheres: Basic Cell Model with Constriction," AIAA paper 77-188

(1977).

15. Luikov, et. al . , "Thermal Conductivity of Porous Systems," Int. J. Heat

Mass Transfer , 11 , p. 117-140 (1968).

16. M. E. Stephenson and M. Mark, "Thermal Conductivity of Porous Materials,"

ASHRAE Journal , pp . 75-81 (1961).

17. A. Pellew and R. V. Southwell, "On Maintained Convective Motion in a Fluid

Heated from Below," Proc. Royal Soc. , A176 , p. 312-343 (1940).

18. C. W. Horton and F, T. Rogers, "Convective Currents in a Porous Medium,"

J. Appl. Physics , 16 , p. 367-370 (1945).

19. E. R. Lapwood , "Convection of a Fluid in a Porous Medium," Proc. Camb.

Phil. Soc. , p. 508-521 (1948).

20. H. L. Morrison, F. T. Rogers and C. W. Horton, "Convective Currents in Porous

Media, II. Observation of Condition at Onset of Convection," J. Appl.

Physics , p. 1027-1029 (1949).

21. F. T. Rogers, H. L. Morrison, "Convection Currents in a Porous Media, III.

Extended Theory of the Critical Gradient," J. Appl. Physics, 21, p. 1177-

1180 (1950).

22. F. T. Rogers, L. E. Schilberg and H. L. Morrison, "Convection Currents in a Porous Media, IV. Remarks on the Theory," J. Appl. Phys . , 22, p. 1476-

1479 (1951).

23. Y. Katto and T. Masuoka, "Criterion for the Onset of Convective Flow in a Fluid in a Porous Medium," Int. J. Heat Mass Transfer, 10, p. 297-309

(1967).

24. J. W. Elder, "Steady Free Convection in a Porous Medium Ifeated Flow Below,"

J. Fluid Mech ., p. 29-48 (1967).

25. J. W. Elder, "Transient Convection in a Porous Medium," J. Fluid Mech ., 27 , 609-623 (1967).

26. E. Palm, J. E. Weber and 0. Kvernvold, "On Steady Convection in a Porous

Medium," J. Fluid Mech ., 54 , p. 153-161 (1972).

27. F. H. Busse and D. D. Joseph, "Bounds for Heat Transport in a Porous Layer,"

J. Fluid Mech. , 2^, p. 521-543 (1972).

28. J. M. Straus, "Large Amplitude Convection in Porous Media," J. Fluid Mech. , 64, p. 51-63 (1974).

22
29. D. R. Kassoy and A. Zebibe, "Variable Viscosity Effects on the Onset of

Convection in Porous Media," Phys . Fluids , 18 , p. 1649-1651 (1976).

30. R. J. Ribando and K. Torrance, "Natural Convection in a Porous Medium:

Effects of Confinement, Variable Permeability, and Thermal Boundary

Conditions," Journal Rt . Transfer , 98 , p. 42-48 ( 1976).

31. T. W. Tong and C. L. Tien, "Radiation Pfeat Transfer in Fibrous Insulation

- Part I, Analytical Study," ASME Paper No. 81 - HT-42 (also see T.W.

Tong, 0. 3. Yong and C. L. Tien,” Radiative Beat Transfer in Fibrous

Insulation - Part II, Experimental Study," ASME Paper 81-KT-43.

32. J. Reichman, "Determination of Absorption and Scattering Coefficients for

Monhomogeneous Media. l:Therory," Applied Optics , 12 , p. 1811-1815 (1973).

33. W. G. Egan, T. Rilgenan and J. Reichman, "Determination of Absorption and

Scattering Coefficients for Nonhomogeneous Media. 2 : Experiment , " Applied

Optics , p. 1816-1823 (1973).

34. H. C. Ifemaker, "Radiation and Beat Conduction in Light Scattering Materials,

Phillips Research Reports , _3, p. 55-67 (1947).

35. B. K. Larkin and S. W. Churchill, "Ifeat Transfer by Radiation Through Porous

Insulations," AICHE Journal , _5, p. 467-474 (1959).

36. R. Viskanta and R. J. Grosh, "Beat Transfer in a Thermal P^adiation Absorbing

and Scattering Medium," International Dev. in KA Transfer, IV ASME, New

York, p. 820-828 (1961).

37. R. Viskanta, "Beat Transfer by Conduction and Radiation in Absorbing and

Scattering Materials, Journal Et . Transfer , 87 , p. 143-150 ( 1965).

38. C. M. Usisken and E. M. Sparrow, "Thermal Radiation Between Parallel Plates

Separated by an Absorbing-Emitting Nonisothermal Gas," Int. J. Beat Mass

Transfer , J_, p. 28-36 (1960).

39. M. A, Beaslet and R. F. Warming, "Radiative Transport and Wall Temperature

Slip in an Absorbing Planar Medium," Int. J. Beat Mass Transfer, 3,

p. 979-994 (1965).

40. R. G. Deissler, "Diffusion Approximation for Thermal Radiation in Gasses

with Jump Boundary Condition," Journal Ht . Transfer , 86, p. 240-246 ( 1964).

41. L. B. Evans, C, M. Chu and S. W. Churchill, "The Effect of Anisotropic

Scattering on Radiant Transport," Journal Ht . Transfer, 87, d. 381-387

(1965).

42. H. M, Ksiu and T. J. Love, "Radiative Heat Transfer Between Parallel Places

Separated by a Nonisothermal Medium with Anisotropic Scattering," Journal

Ht . Transfer, 89, p. 197-204 ( 1967).

23
43. A, Dayan and C, L. Tien, ’’ Hsat Transfer in a Gray Planar Medium with Linear

Anisotropic Scattering," Journal Ht. Transfer , 97 , p. 391-396 (1975).

44. S. C. Traugott and K. C. Wang, "On Differential Methods for Radiant Ifeat

Transfer," Int« J. Heat Mass Transfer , 7^, p. 269-270 (1964).

45. J. B. Berquam and R. A. Seban, "Hsat Transfer by Conduction and Radiation in

Absorbing and Scattering Materials," Journal Ht. Transfer, 93, p. 236-239

(1971).

46. C. C. Lii and M. N. Ozisik, "Transient Radiation and Conduction in an Absorbing, Einitting, Scattering Slab with Reflecting Boundaries," Int . J.

Heat Mass Transfer , 15 , p. 1175-1179 (1972).

47. H. A. Fine, S. H. Jury, D. W. Yarbrough and D. L. McElroy, "Analysis of

Heat Transfer in Building Thermal Insulation," Oak Ridge Report ORNL/TM-7481

(1980)

.

48. D. V. Pryor, "Parameter Estimation for teat Transfer During Itypertherraia

Treatment," Ph-D. Thesis, Colorado State University (in preparation).

49. G. J. Bierman, Factorization Methods for Discrete Sequential Systems , Academic Press, New York, pp. 8-20 (1973).

50. E. F. M. van der teld, "The Contribution of Radiation to the Conduction of

teat. III. Boundary Conditions," Appl. Sci . Res. , A4 , pp. 77-99 ( 1953).

51. N. E. teger, Jr. and R. C. Steere, "Radiant teat Transfer in Fibrous Thermal

Insulation," J. Appl. Phys . , 38 , pp. 4463-4668 (1967).

52. H. M. Strong, F. P. Bundy and H. P. Bovenkerk, "Flat Panel Vacuum Thermal

Insulation," J. Appl. Phys., 31, pp. 39-50 (1960).

24
Table 1 . Experimental Data

Thickness

(mm) (W.^Hr-I)

Pm

(kg*m“3) PT

(kg*m J)

25.27 0.04470 10.03* 10.03

25.46 0.04542 9.47 9.52

25.35 0.04618 9.14 8.92

25.35 0.04617 9.11 9.04

25.35 0.04557 8.83 8.78

25.35 0.04619 9.25 9.98

25.35 0.04810 8.09 8.67

25.35 0.04362 10.67 10.39

25.35 0.04443 9.92 9.91

25.35 0.04427 9.17 10.17

25.38 0.04487 10.56* 10.56

25.35 0.04512 9.13 9.09

25.37 0.04577 9.11 9.39

25.35 0.04568 9.01 9.22

25.38 0.04527 9.63 9.78

25.38 0.04457 10.22 10.79

25.77 0.04377 11.13 11.14

25.77 0.04364 10.88 11.07

38.08 0.04715 9.51 8.64

38.08 0.05030 7.14 7.34

38.15 0.04272 11.40 11.24

76.26 0.04949 8.33 8.42

76.91 0.04721 9.20 8.99

76.91 0.04968 8.22 8.34

76.15 0.04647 9.27 9.25

76.15 0.04777 8.85 8.56

76.15 0.04554 9.75 9.28

76.15 0.04736 8.89 9.06

76.22 0.04800 8.78* 8.78

76.15 0.04632 9.51 9.07

76.22 0.04717 9.10 9.18

76.18 0.04746 8.95 8.97

152.35 0.04752 8.96 9.06

152.35 0.04768 8.94 9.07

152.35 0.04798 9.03 9.00

152.35 0.04680 9.36 9.00

152.35 0.04880 8.47 8.92

152.35 0.04780 8.95 9.14

152.35 0.04788 8.93 9.16

152.42 0.04882 8.94* 8.94

152.35 0.04808 8.78 8.88

152.37 0.04772 8.97 8.88

152.35 0.04824 8.69 8.67

152.42 0.04713 9.27 9.32

152.34 0.04803 8.93 9.31 152.35 0.04752 8.96 9.06

152.39 0.04677 9.16 8.97

* No metering sections densities were available for these points, so the metering section density was taken as the total specimen density.

25
t.u,

;s«t:

kU i Tfi .<S44. vrt^i:# Otrt ' rntMn pkvii i iv^ > J«- iSy^.- ,1 1C;1 ^;r’ ^.

•

?tfSTur^*«.v ,, -H

^

<».»T

Aft’S,

s

It If .

• ‘ m

*'

‘ i <‘**L

'

'1 >

': 'V.0 Fl.-<

-)*.»48|it»> ,' A» A‘> >7

*! . ®

Sc’ .*,*-»> ett.sifti ,, tifxmu

0»*>rV

AfUM*' ti.aj;

ct>s«'

•li' Ml,«t

a' irf.J.U

,-'f ‘t*i .V

i'tii'.ij rr.,t'“|

CC.S<1

trwv'v

• W/k.U

_„* •/ i *

jja • » ’ ’ B •’'1'4.. -

^ #’1i t»l<liS4A4’ « 4J»,. ^ol 1 >!<• yA tT«94Ui <M *

W%#lij>i^ »A,# «A « .) M* o, . wh
Effective

Tiiermal

Conductivity,

k

(IJTllH

• 1

N

• ET

0.1 0.2 0.5 1 2 5 10

Density, p C‘<g'° )

FIGURE 1 THE VARIATION OF THE EFFECTIIT] THERMAL CONDUCTIVITY.

WITH THE DENSITY, D, .^ND THE FIBER DI.-^DIETER, 5 (ABSTRACTED FROM THE A5HRAE HANDBOOK OF FUNDAMENTALS'
Thermal

Conductivity

(W-m

1.00 0.99 0.98 0.97

Porosity, P

till I ] i I t i 0 10 20 30 40 50 60 70 80 90

Density, o (kg-ni )

FIGURE 2 THERMAL CONDUCTIVITY VERSUS POROSITY, P, AND

DENSITY, 0 (Abstracted from Bankvall [7])
X = 0 X = 1

T = T9 = 283.0 X T = = 310.3 K

FIGURE 3 THE PHYSICAL SYSTEM
First

Parameter,

Xi

(kg*in

0.0265

CN

I

0.0260

0.0255

0.0250

0.0245

0.0240

FIGURE 4 THE VARIATION OF THE FIRST PARAMETER,

WITH THE NUMBER OF DATUTI POINTS, N
0.052 o

(

‘ AiTATiionpuoo

Fir.llHI.

,

rilK

VAKIATION

OK

Tllli

KKKKCT

I

VK

THKKMAl,

CONOIJCTI

VITY

,

Ic

,

WITH

OMNSITY.

p,

AND

THICKNIiSS

KOK

niK

i)NK

I'AKAMK'l'KK

MODl

i.
315

310

305

300

295

290

285

280

T T T T

L (m)

0.0254

0.0381

0.0762

0.1524

Pure Conduction,

-3

= 9k2*m

= £9 = 0.90

0.2 0.4 0.6 0.8

Non-Dimensional Position, X

FIGURE 6 TEMPERATURE PROFILES FOR VARIOUS THICKNESSES
Non-Dimensional

'['emperature

Integral

,

FIGURE 7 THE VARIATION OF TEE NON-DIMENSIONAL TE:-E’ER-\TURE

INTEGRAL WITH DENSITY THICKNESS
I

04

I

04

I c=

04

o

X

04O

04U

c

03

)-< 03 >

O ** J- Mass Attenuation Coefficient, 8(m"^*kg )

FIGURE 8 THE VARIATION OF THE VARIANCE, , WITH

THE MASS ATTENUATION COEFFICIENT, 6
KKNNKX

/

• 9 (_ >l ‘.-.aTAT^-npUOQ 5ATi33j;2

L~ ... . . -

IliaiKK

>J

Mil;

VAKIATION

OK

TllK

K

KFKCT

I

VK

Tlll'KMAI.

CONDUCTIVITY,

k

WITH

DKNSITY,

p

AND

TllICKNl-SS

ro!<

llli:

IWiMl'AKAMKIliU

MiiDM,

‘‘

APPENDIX A

LEAST SQUARES DEVELOPMENT

APPENDIX A. LEAST SQUARES DEVELOPMEIT

A.l INTRODUCTION

A formal mathematical development of the sequential least squares algorithm [43]

used in this work is presented in this appendix. In the following discussion,

matrices will be devoted by upper case letters, vectors by lower case letters

with a bar above, and scalars by lower case letters with no bar. All subscripts

will refer to order in time or in sequence, although in the general setting of

linear systems no reference to time is necessary.

A. 2 GENERAL DEVELOPMENT

The general over-determined linear system is given by:

where A is k x n (k>n) and x and y are of dimension n and k respectively. In

general, no solution exists which satisfies equations (A.l), so it is common

practice to choose as a "solution" that value of x, denoted Xj^ g, which

minimizes the functional

I.e. j is the sum of the squared residuals.

The derivation of the least squares solution to equation (A.l) is

straightforward, using the following identities for an arbitrary matrix B and

vector z:

Ax = y. (A.l)

j = (Ax-y)T (Ax-y) (A. 2)

(A. 3)

3(z^B) ^

9z (A. 4)

3(3z) = bT

9z

A-1
To minimize equation (A. 2), j is differentiated with respect to x and the

9t

derivative ~ is set equal to zero,

3x

_ 9(x'^a'^Ax) _ 9 (x’^a'^y) _ 9(y^Ax) _ 9 (y'^y)

9x ~5x 9x 9x 9x

= 2x\'^Ax- A^y - (y^A)'^

= 2aTax- 2A^y

= 0 ^ (A. 6)

Therefore

:

x^^g = (A^A)"^ A^y (A. 7)

A. 3 SEQUENTIAL LEAST SQUARES

It is convenient to perform the operations specified by equations (A. 7) in a

sequential manner, without requiring excessive amounts of computer storage and

without necessitating the performance of a matrix inverse as each new row of

the system given by equation (A.l) is obtained. To this end, the Woodbury

matrix inversion lemma is introduced:

(F+ BCD)"1 = F"1 - F^1b(C"1 + DF^1b)“1dF~1 (A, 8)

It is now useful to rewrite equations (A.l) as a block system as follows:

XI

^2

•

X2

•

= ^2

•

•

•

•

Xn

•

^k

; or A^ X = y,^. (A. 9)

A-2
where represents the i— row of the A matrix and represents the A matrix

before the (k+1)^ row has been adjoined to the system.

Assume that for some value of k, the system, equation (A. 9), has been solved

for

s ^^-en the information and become available,^ can

then be updated to

^ using equation (A. 8) .

First, define and by:

^k " \ " ^k^k (A. 10)

Then it is seen that for any value of k

''^l.s,k = ^k^k* (A. 11)

To update and to P^+l and note that

1 _ aT 3

T

Ak

^k+1

\+l

~ ^k \ ^k+1 ^k+1

" \

^ ^k+1 ^k+1

with C = 1 in equation (A. 8), = ^^k+1 found as follows:

\+l " \ ~ \ ^k+1*^^ ^k+1 ^k ^k+1^

^ ^k+1 ^k*

(A. 12)

(A. 13)

Note in particular that the quality (1 + ai,^j^ P^ ^k+1^ ^ scaler, so that

updating Pj^ to Pj^+i requires no matrix inversion.

The vector z is also easily updated as follows:

^k+1 " ^k+1 ^k+1

aT ^k

^k+1

\+l

A-3
(A. 14)

= Ak 7k + 4+1 7k+l

" 4+1 ^k+1

Storage requirements for the above sequential least squares scheme are minimal.

The entire A matrix and y vector need not be saved; only the relatively small P

matrix (n x n) and the z vector (dimensional n) need to be saved. As new data

arrives it is absorbed by the current P and z, and the least squares estimate

of X is kept current.

A-4
SBS-lUA =>E.

1 _ . E . 0 F CCMW. 1. PUBLICATION -OR 2. Performing Organ. Report No. 3. Puoilcation Date BIBLIOGRAPHIC DATA

SHEET See instructions^

REPORT NO. NBSIR 84-2906 August 1984

4. TITLE AND SUBTITLE

Thickness Effect in Low-Density Insulation

5. AUTHOR(S)

Patrick J. Burns

6. PERFORMING ORGANIZATION (If Joint or other than N BS, see in struction s)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON. D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS .'Street, City. Stote, ZIP)

National Bureau of Standards

Department of Energy

10. SUPPLEMENTARY NOTES

I

' Document describes a computer program; SF-185, t-lPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most sign i fi cant information. If document includes a significant

bibliography or literature survey, mention it here)

A discussion is presented of theory of heat transfer in low-density, glass-fiber

insulation via conduction, convection, and radiation. It is concluded that

the primary modes of heat transfer in this material are air conduction and

radiation. An analysis of NBS data of measured apparent thermal conductivity

for different thicknesses results in a parameter estimate of the optical

extinction coefficient. This parameter determines the amount of change in

apparent thermal conductivity as a function of sample thickness. This

phenomena is referred to as the "thickness effect."

12. KEY WORDS (S/x to twelve entries; alphabetical order; capitalize only proper names; and separate <ey words py sem co : Conduction; convection; guarded-hot-plate; low-density insulation; thermal

conductivity; thermal radiation; thickness effect.

13. AVAILABILITY

XX] Unl imited

I

i For Official Distribution. Do Not Release to NTIS

1 ; Order From Superintendent of Documents, U.S. Government Printing Office, Washington. D.C. 20402.

[Xj Order From National Technical Information Service (NTIS), Springfield. VA. 22161

14. NO. OF

PRINTED PAGES

•16

15. P' crpi .'>)