A theory for the effects of convective air flow through fibrous thermal insulation

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ASHRAE Transactions, 72, 2, pp. 111.2.1-111.2.9, 1967-07-01

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A theory for the effects of convective air flow through fibrous thermal

insulation

Wolf, S.

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Ser

TH1

N21r2

no.

321

c. 2 BLDG

NATIONAL RESEARCH COUNCIL O F CANADA

CONSEIL NATIONAL DE RECHERCHES DU CANADA

A

THEORY FOR THE EFFECTS O F CONVECTIVE AIR FLOW

THROUGH FIBROUS THERMAL INSULATION

by

S. Wolf

Reprinted from

TRANSACTIONS

American Society of Heating, Refrigerating and Air-Conditioning Engineers

Vol.

72, Part 11, 1966

p.

111.2.1 -

111.2.9 Research Paper No. 321

of the

Division of Building Research

OTTAWA

July 1967

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THEORlE RENDANT COMPTE DES EFFETS DES COURANTS

D'AIR DE CONVECTION AU TRAVERS DES ISOLANTS THERMIQUES

FIBREUX

S O M M A I R

E

L'auteur pr6sente l a de'veloppernent th6orique d'une hypothese rendant cornpte d u transfert d e c h a l e u r a u t r a v e r s d'un i s o l a n t poreux p a r u n e cornbinaison d e conduction par l ' a i r e t d e convection d e l'air. E n v u e d e rhaliser c e t t e etude, l'auteur c h o i s i t un i s o l a n t fibreux aux deux f a c e s permkables 5 l'air. I1 a n a l y s a theoriquernent l e cornporternent d e deux rnoddles: l e premier b6n6ficiait d e ternpgratures uniforrnes s u r c h a q u e f a c e d e l ' i s o l a n t , t a n d i s que l ' a u t r e e t a i t sournis & un gradient regulier d e temperatures 5 c h a q u e f a c e . L'auteur r g a l i s a d e s rnesures d e transfert d e chaleur au travers d e l ' i s o l a n t fibreux d'un panneau normal d e 4 p i e d s d e haut e t d e f a i b l e densitk. L e s r 6 s u l t a t s exphrimentaux o n t 6 t 6 cornpar6s a v e c c e u x fournis par l'etude du premier mod6le.

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S. WOLF

A

Theory for the Effects of Convective Air Flow

through Fibrous Thermal Insulation

Low density fibrous insulation usually posses- s e s more than 95% void space which in ordinary application is occupied by a i r . The thermal conductivity of this material is a complex property because the threebasic mechanisms of heat trans- f e r , conduction, convection and radiation, play a part. The insulating value results from the fact that the gas contained in the pores is a t rest, o r that any convective motion present i s small. Sev- eral papers have been written on the thermal properties of fibrous i n ~ u l a t i o n , ' , ~ however, the interaction between fluid conduction and convection has not been fully investigated.

Verschoor and ~ r e e b l e r

'

found that in guarded hot plate tests, where the insulation fills the space between plates, gas conduction is the mechanism which contributes the largest component of the heat transfer. A situationcould a r i s e in practice, however, where a i r spaces exist in contact with one o r both sides of insulation. The insulation would then act a s a permeable membrane, allowing a i r movement by natural convection between the spaces, a s a result of the pressure difference created by different temperature conditions. This a i r movement would decrease the total heat transfer by fluid conduction and increase convective heat transfer. The resulting overall effect would be an increase in the apparent thermal conductance of the insulation.

~ o t z ~ detected the presence of convection currents in several applications, including insulation with neither face, one face, orbothfacesperrpeable

S. Wolf i s a Research Assistant, Thermosciences Div, Stanford University, Stanford, Calif. He was formerly R es e arc hof ficer, Building S e n i c e s Section, Div of Build- ing Research, National Research Council, Ottawa, Ont., Canada. This paper was prepared for presentation at the ASHRAE 73rd Annual Meeting in Toronto, Ont., Canada, June 27-29, 1966.

to a i r flow. Negligible effects on the total heat transfer were observed f o r the f i r s t two cases; however, the last one showed a significant increase in heat transfer.

Two simplified models with both insulation faces permeable to a i r were analyzed to determine the heat transmission r a t e by combined fluid conduction and convection. Experimental results a r e included to substantiate the theoretical development.

THEORY

Model 1

The situation to be analyzed is schematically rep- resented in Fig. 1. Two sealed cavities a r e sep- Fig . 1 Natural cocvection through fibrous insulation

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arated by a vertical partition of fibrous insulation, having an air flow coefficient X

.

The air densities and temperatures a r e assumed uniform over the insulation a r e a

at

p and p w, and Tc and

C

Tw '

Since no net mass flow occurs a c r o s s the insulation, a neutral axis must exist where the ab- solute p r e s s u r e is everywhere equal. The pres- s u r e difference between the two a i r spaces a t any level can be expressed as

The fluid velocity can be related to the p r e s - s u r e difference a c r o s s the insulation. Assuming that the flow is laminar, the apparent velocity is given by the basic relationship of Darcy's Law as

A P

v

z -

CIL (2

According to the work of several investigators (e. g.,

arm an^),

the value of A, based on both capillary and particle models, i s a function of fiber diameter and porosity. F o r highporosities, a s encountered in lightweight fibrous insulations, no r e - liable analytical prediction of A was available and an experimental method was used to determine an a i r infiltration coefficient, which in turn allowed determination of A. The infiltration t e s t s showed that f o r the p r e s s u r e differences in a normal application, the volume flow rate, and hence the velocity, is directly proportional to the pressure difference. This validates the assumption of laminar flow. The infiltration coefficient is defined a s a proportionality constant between flow rate p e r unit a r e a and pressure difference across unit thickness of the insulation. Hence, it follows that

A = F X (3)

Combining Eqs (I), (2) and (3), the velocity may be e m r e s s e d as

The convective air flow, Q, f r o m the warm side to the cold side through the top half of the insulation of width Wand height H, and the equal flow in the opposite direction through the bottom half can be found by integrating Eq (4) from 0 to H/2.

Ignoring conduction effects, the predicted heat transfer associated with this a i r interchange would be

This can also be expressed in t e r m s of a conductance coefficient C such that

C then

-

c X A P H ~ B H = p

cc

= P cp

8

L

kc

(7)

The heat transfer due to fluid flow a s expressed above is modified, however, by conduction in the a i r in the insulation. Similarly, the heat transfer by conduction in the a i r b the insulation is modified by the air flow. This interaction is expressed by ~ r o w n ~ in the following heat consemationequa- tion:

This equation assumes conduction in the a i r only in the direction normal to the plane of the insulation and neglects conduction in other directions. Inte- gration and substitution yields the following equation in differential form:

H H

Integrating from y =

-

to

+

-

gives:

2 2 where if we write

x

=BLH/4a 32 a 2

1 _x

coth

x

dx = K B ~ L ~ H ~ CC 0 (10) then qcc

-

- -

B H ~ P Cp

W

(T W

-

Tc) Kc, o r

-

(1 1) qcc - q c Kcc (12) An apparent conductance C can be used to ex-

CC

p r e s s the combined conduction

-

convection heat flow : Ccc = qcc/ WH(Tw

-

Tc) (13)

-

- - BH cp

T-

Kcc (14)

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F i g . 2 Values of Kc, and Cc v s

A H

f t 5 ! h ~ , Ib force

and

The heat transfer by the conduction-convection mechanism is therefore the transfer that would be predicted on the b a s i s of convective flow alone multiplied by Kcc

The value of Kcc i s function of

BLH

= 40

X A P Hg

40 gc and Cc is equal to

P

cP A P H ~ 8 L gc

In a given application, the heat transfer is there- f o r e influenced primarily by X

,

H and A p

.

The density difference is a function of temperature difference so that both Kcc and Cc become a function of X

,

H and

A

T. At low values of these t h r e e p a r - a m e t e r s , K is very large and Cc small. Athigh

CC

values, K becomes asymptotic to 1.0, while Cc

C C

increases in direct proportion to the product of the parameters.

Values of K were determined by numerical

CC

integration of Eq (lo) and a r e shown in Fig. 2. This figure shows Kcc vs X H for various tempera- t u r e differences and applies only to a i r a t about 70 F on the warm side. Values of Cc a r e a l s o shown f o r the s a m e conditions.

When X H A T is very small, the heat transfer is by pure conduction. This can best be shown by

Eq (8) which, when V approaches zero, reduces to:

o r q = WHZ (T k

-

Tc).

W

The heat transfer coefficient Ccc can there- f o r e range from a minimum of k/L t o a maximum of Cc depending on the magnitude of X H A T.

By rearranging and adding t e r m s in Eq (13)

the conductance factor C can be expressed in

C C

t e r m s of the followina dimensionless arouos:

where the t e r m G can be regarded a s a geometri- cal factor peculiar to the system.

Model 2 (17)

In the second situation analyzed there is a uniform vertical temperature variation in the cavities on either side of the insulation. The neutral axis will be located a t the horizontal centerline of the insulation, s o that p r e s s u r e difference, a i r velocity and flow r a t e will be the same a s f o r model 1. The temperature in the cavities may be expressed in t e r m s of the centerline temperature a s follows:

-

and Tc

-

Tmc + Dy

where

D

= Tt

-

Tb

H

and

-

The total heat-transfer r a t e due to fluid conduction and convection is obtained by substituting Eqs (18) and (19) into Eq (8), combining with Eq (4) and in- tegrating from y =

-

~ / 2 t o y = + ~ / 2

(7)

where

2

-

Ktv = 3

Comparing this result with that obtained f o r model 1, it i s evident that the only modification i s the ad- dition of the temperature-variation variable,

Ktv, to the K

-

factor. As beforefor the pureconduc-

CC

tion case, when the factor BLH/4a i s very s m a l l , the heat-transfer r a t e defined j n Eq (2 1) reduces to

= W H ~( T ~

-

T ~ ) (22)

F o r the pure convection case, the total heat-trans- f e r r a t e becomes:

_.-,

It is evident that K - a c t s to increase only thecon- tv

vective mode of heat transfer. The Ktv variable may be thought to occur because the significant temperature difference f o r convective heat t r a n s - f e r is between the temperature of the fluid enter- ing and leaving through a r e a s locatedan equal dis- tance above and below the neutral axis.

The apparent heat-transfer coefficient due to fluid conduction and convection, Ccc, may now be defined s o that

The final correlation among dimensionless groups then becomes

TEST WALL AND TEST PROCEDURE

Glass fiber insulatioll nlanufactured wlthout paper backing, of 2-in. nominal thickness, having a density between 0 . 8 and 0 . 9 lb/cu ft and a n average fiber diameter of 0. 4 mil w a s used in the experimental program. A standard 4-ft batt was in- stalled in a 14 3/16- by 48-in. opening with the long dimension vertical; this was centrally located in a n 8-ft sq partition fabricated f r o m 2-in. thick foamed polystyrene glued to 1/2-in. thick plywood backing (Fig. 2). The insulation was maintained a t the desired 2-in, thickpess by 1-in. hexagonal- netted w i r e mesh stretched a c r o s s both sides.

To isolate the effect of convective a i r flow on heat transfer, t e s t s w e r e performed with and without polyethylene sheeting over the insulation surface. T e s t s w e r e conducted with the w a r m box maintained a t about 72 F , while the cold boxwas varied in s t e p s down to about -20 F . Temperatures of inside and outside a i r and insulation s u r f a c e s w e r e measured by copper-constantanthermocouples. Temperature readings under steady-state conditions a r e consid- e r e d accurate to within

+

0 . 1 F , and heat t r a n s f e r measurements to

+

1%.

The heat transmitted through the insulation was obtained by subtracting the heat transmitted through the partition f r o m the total measured. The heat transmitted through the partition was calculated from partition-conductance values determined in previous tests.

To compare the t e s t r e s u l t s with the theory just presented, the heat t r a n s f e r by radiation must be known. An apparent radiation coefficient was determined f r o m measurements with a n 18-in. EXPERIMENTAL PROGRAM

F i g . 3 T e s t wall The experimental program w a s designed to demon-

s t r a t e the increase in heat transmission of fibrous insulation due to convective a i r flow and, in addi- tion, to verify the theoretical correlation among dimensionless groups.

TEST APPARATUS

Heat transmission r a t e s w e r e measured with the wall panel test unit which has previously been de- scribed. Basically, the apparatus consists of a cold-box/hot-box combinaiion; each box is 4 f t deep with a n 8-ft s q opening. Panels lining the i n - side walls a r e heated in the w a r m box and cooled in the cold box by circulating liquid. Additional surface a r e a f o r cooling is provided in the cold box by finned tubing. The warm-box liquid can be controlled between 65 and 75 F to within 2 0.01 F. The cold-box temperature can be varied down to about -25 F.

Heat t r a n s f e r to the test wall on t h e w a r m side i s by natural convection. The s u r f a c e conductance a t the w a r m side of the test panel simulates that obtained in practice and a g r e e s closely withvalues published in the ASHRAE Guide And Data Book. Facilities f o r forced circulation a r e available in the cold box but f o r this t e s t s e r i e s natural convection conditions w e r e provided.

rn

11 Guard

P a r t i tion

1,' *, *.-* e

'-;

* # *,

-

I<--,

b

# ,

.;

.

*

-'#-*:

I

n s u l a t i o n

I;~,':

2 i n . t h i c k

0,- *,

-

0 I &

I'

e

(8)

T E M P E R A T U R E , F

Fig. 4 . Vertical temperature variations with poly- ethylene covering/: (I) warm-side air; (2) warm sur- face; (3) cold s u r f a c e ; (4) cold-sid.e air; (5) cold-box air

guarded hot-plate conductivity apparatus by subtracting the calculated heat transfer by fluid conduction and solid fiber conduction from the total measured. Measurements on a 2-in. thick sample of 0.90 lb/cu f t density a t a mean temperature of 55 F and a temperature difference of 40 F result- ed in a total conductivity of 0.255 and aradiation conductivity of 0.079

Btu/

h r , s q ft, F/ in. The radiation conductivity may be expressed a s :

The factor l / c is anopacity factor dependent upon temperature. F o r the limited temperature range encountered in the experiment, 30 F < T < 55 F,

m

Fig. 6 Total heat-transfer c o e f f i c i e n t ( C t ) o f insula! tion with andwithout polyethylene covering a s a func- tion o f over all temperature d i f f e r e n c e

I I I I I I I

-

7 -

-

e ; Q

-

3 -

"

;

-*-* -0.

--

-

I n s u l a t i o n w i t h P o l y e t h y l e n e c o v e r i n g I I I I I I I I I , - 2 0 - 1 0 0 10 2 0 3 0 4 0 50 6 0 70 T E M P E R A T U R E , F

Fig. 5 Vertical temperature variations

-

without polyethylene covering: (I) warm-side air; (2) warm s u r f a c e ; (3) cold s u r f a c e ; (4) cold-side air; (5) cold box air

the radiation conductivity (applicable a t a bulk in-

sulation density of 0.85 pcf) was determined a s

T E S T RESULTS AND DISCUSSION

Figs. 4 and 5 show vertical variations in temper- a t u r e a t the centerline of the insulation section, with and without a polyethylene cover over the insulation surface. In Fig. 5, the occurrence of convective a i r flow is indicated by the decreased tem- p e r a t u r e over the bottom half of the warm s u r f a c e

of the insulation and the increased temperature

Fig. 7 Apparent thermal conductance 01 insulation due to gas conduction and convection v s air tempera- ture d i f f e r e n c e

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over the top half of the cold surface. Thesevaria- tions can be explained with reference to the a i r movement. Cold a i r moves from the cold side to the warm side below the neutral axis, thereby low

-

ering the warm side surface temperature; an opposite a i r movement and temperature effect occurs above the axis.

The variation in surface temperature of the insulation section with polyethylene covering is due to convection currents within the insulation. Su,ch internal convection currents were also noticed by Lotz who found, however, that the average heat- transfer coefficient was practically the same a s t+at determwed by guarded hot-plate tests.

The total heat-transfer coefficients for the insulation Ct, with and without the polyethylene cover

-

ing, were obtained by dividing the total heat trans- f e r through the insulation by

its

a r e a and the over

-

all a i r temperature difference. These coefficients, plotted a s a function of overall temperature difference in Fig. 6, demonstrate the effect of convection inincreasing the total heat transfer. At an overall temperature difference of 90 F. the apparent thermal conductance of the insulation has Increased by a factor of about four.

The insulation heat-transfer coefficients due to fluid conduction and convection only, C were

cc' determined by subtracting the calculated heat transfer due to radiation and solid fiber conduction from the measured total. These values a r e plotted in Fig. 7, and a r e compared with the theoretical curves calculated from Eqs (7) and (13) applicable to model 1, which simulates the actual test conditions more closely than model 2. Values obtained from Eq (7) a r e for heat transfer by convective a i r flow only, while those from Eq (13) a r e for com-

I

-

E x p a r l m o n l t l

,.

/

-

V a l u e s l o r

-

- / /

'

-

_.'

Fig. 8 C o n v e c t i v e air flow through fibrous insula- tion in a v e r t i c a l partition

bined conduction and convection. It is seen that gas conduction predominates a t low-temperature differences, and gas convection at high-temperature differences. In Fig. 8, the Nu numbersbased on test values of C a r e plotted against the cor-

CC

responding values of Pr Gr

K

based on average cc'

fluid properties and a measured value of the a i r flow coefficient, X

,

of 460 cu ft/hr, sq ft, lb/sq f t pressure difference/ft of thickness; definitions a r e in accordance with Eq (1 5). Also shown is a theoretical Nu curve based on values of G calcu-

(10)

lated from Eq (17). The theoretical correlation according to Eq (15) then becomes

Agreement between the experimental and theoretical correlation is within a few p e r cent. The heat flow attributable to the insulation in the tests was a relatively small part of the total measured. Even small percentage e r r o r s in the total therefore r e - sult in relatively large percentage e r r o r s in the insulation heat flow.

The factor K is useful in predicting whether C C

convective a i r flow will be a problem in a practical installation, Eq (14) can be rearranged to read:

C

In this form, l/Kcc represents the ratio of the convective flow heat-transfer potential to the total heat-transfer coefficient (exclusive of radiation transfer and fiber conduction). As l/Kcc approaches zero, C also approaches z e r o and Ccc

C

approaches the fluid conduction coefficient k/L. At low values of 1/K Eq (29) can be expressed

c c J approximately as:

- Cc

l/Kcc k/L

+

Cc

The value of K required to limit the increase in

CC

heat flow due to convective a i r flow t o any specified ratio of the total conduction-convection through the insulation can be calculated from Eq (30). For example, if the increase in heat flow due to convective a i r flow a t a specific temperature difference is to be limited to lo%, then the value of Kcc must be 10 o r greater. If the insulation properties will not providea sufficiently high K value a t the design

CC

condition, then a n a i r b a r r i e r must be incorporated into the system.

The value of X H for the insulation used in the test was 460 x 4 = 1840 which gives Kcc values of about 2.5 a t 10 F temperature difference and close to 1.0 for the 90 F temperature difference. The a i r leakage coefficient required f o r a K greater

CC

than 10.0 i s very low; X H must be lower than about 35 for 90 F temperature difference and, if

H

were 4 ft, h must be l e s s than 9.

CONCLUSION

Theoretical relationships have been developed to describe the heat transfer by combined fluid conduction-convection through air-permeable insulation with vertical a i r spaces adjacent to both s u r - faces. The fluid conduction-convection is shown to be a function of fluid properties, a i r flow coefficient of the insulation, insulationheight and thickness, and temperature difference; a correlation in t e r m s of dimensionless groups has been derived. Results of measurements on a 4-ft high insulation specimen over a temperature difference range

from 30 to 90 F w e r e in agreement with the theory. The relationships suggest that very low a i r flow coefficients a r e required to minimize convection effects and, in practice, a n a i r b a r r i e r in contact with the insulation i s required if a i r spaces occur on both sides.

ACKNOWLEDGMENT

The author i s grateful to Dr. W. G. Brown for ad- vice in connection with the theoretical development and to Messrs. A. G. Wilson and K. 8. Solvason f o r their assistance with the experimental program and the preparation of the paper. This paper is a contribution from the Div of Building Research, National Research Council, Canada, and i s published with the approval of the Director of the Di- vision.

NOTATION

E n g l i s h L e t t e r S y m b o l s

A , B, C , D = constants defined when introduced

C

P = specific heat of fluid g = acceleration of gravity gc = gravitational constant

H = height of insulation

C t = total heat-transfer coefficient for insulation

Cc = apparent heat-transfer coefficient due to convective a i r flow

Ccc = apparent heat-transfer coefficient due to fluid conduction and convection

k = conductivity of air

kr = apparent radiation conductivity through in- _sulation

Kcc = conduction-convection variable

Ktv = temperature-variation variable

L = thickness of insulation

Lf = mean f r e e path at very low p r e s s u r e s f o r molecule-fiber collisions

P

-

- p r e s s u r e

4 = total heat-transfer r a t e

T = temperature

Tie = equivalent warm side temperature Tmc = bulk mean air temperature on cold s i d e Tmw = bulk mean a i r temperature on warm side V = fluid velocity through insulation

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Y = vertical co-ordinate

x = variable defined i n Eq (9)

G r e e k L e t t e r S y m b o l s CY = thermal diffusivity A = property difference

E = fraction of incident radiant energy absorbed by a fiber

p = fluid density = mean fluid density

X = a i r infiltration coefficient/unit thickness

= fluid viscosity

4 = Stefan-Boltzmann radiation constant

D i m e n s i o n l e s s G r o u p s

C = Geometricfactor

XI"

8gc HL

G r = Grashof number gpAp ~ ~ / p ~ Nu = Nusseltnumber _Ccc

P r = Prandtl number CpP

/k

S u b s c r i p t s

b = air on one side a t bottom

c = cold-side air i = insulation m = mean of surfaces p = partitions r = radiation s = solid fiber

t = air on one side a t top

w = warm-side air

REFERENCES

2. Thigpen, J. J. and B. E. Short, Apparent Thermal Conductivity of Fibrous Materials, ASME Paper S9A293, 1959.

3. Lotz, W. A., Heat and Air Transfer in Cold Storage Insulation, ASHRAE Transactions, Vol. 70, pp. 181-188, 1964.

4. Carman, P . C. , Flow of Gases Through Porous Media, Academic P r e s s Inc., New York, 1956.

5. Natural Convection Through Rectangular Openings in Partitions, P a r t 1: Vertical Partitioqs by W.G. Brown and K. R. Solvason; P a r t ll Horizontal partitions by W. G. Brown, Int. Jour. Heat Mass Transfer, Vol. 5, pp. 859- 868 and pp. 869-881 ($962).

6. Solvason, K. R. , Large-scale Wall Heat-flow Measur- ing Apparatus, ASHRAE Transactions, Vol. 65, pp. 541- 550(1959).

APPENDIX A

The conduction-convection variable i n Eq (10) may be re- written as 2 E = -

1

x c o t h x d x C C E 2 0 where E i s defined a s BLY

-

AA P Hg E = - 4 a g c

Values of Kc, a s a function of the single dimensionless group E were determined by numerical integration and a r e shown in Fig. A-1. The application of this figure i s not restricted a s that in Fig. 2, but rather applies to any fluid a t any temperature.

F i g . A-1 Kc, a s a function o f E

1. Verschoor, J.D. and P. Greebler, Heat Transfer by Gas Conduction and Radiation in Fibrous Insulations, ASME Transactions, Vol. 74, No. 6, pp. 961-968, 1952.

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B. E. ENO, Brookings, S. D. (Written): I would like f i r s t to question the r e s u l t s indicated in Fig. 7. The C curve comes f r o m Eq. (7) where C

C

i s shown a s linearly proportionate to Ap which in turn i s linear t o AT a c r o s s the insulation. Cc in Fig. 7 i s not a straight line. Also the gap between the C and Ccc curves should represent conduction.

C

If the AT a c r o s s the insulation i s increasing it does

P

L

s e e m reasonable that the r a t i o

2

_{= K} _{should de-}

C c CC

c r e a s e but it does not s e e m reasonable that the conduction effect reduces to zero. The latter should s t i l l be a n effect superimposed upon the convection effect.

Secondly, Table I s e e m s to indicate that radiation heat t r a n s f e r i s considerably g r e a t e r than conduction. Since the technique for calculating radiation heat t r a n s f e r i s not a v e r y exacting one, I would be suspicious of the accuracy of predictions on conduction.

Finally, it a p p e a r s that only one type and density of insulation was chosen. This may not be sufficient to prove the theory although it readily shows the convection effect being predominant. I think the grand conclusion is simply to keep polyethylene c o v e r s on.

AUTHOR WOLF, (Written): The apparent heat- transfer coefficient due to convective air flow, C

,

is by Eq. (7) linearly proportional to j>Ap which i s in turn, by assuming the application of the perfect

2

g a s law, proportional to (I/T:

-

1 / T ). Hence,

W

C is not a linear function of (T

-

T ) and the C curve in Fig. 7 should not be a straight line. The gap between the C and C cc curves in Fig. 7 d e c r e a s e s a s A T i n c r e a s e s , and in the limit, a s AT approaches infinity, the ratio C cc

/

-

C

-

K cc approaches one, a s i s a l s o shown in

Fig. A-1. This trend is a s expected, since it will be recognized that the heat-transfer, conduction- convection equation, Eq. (8) o r (9), reduces to the pure convection form for highvelocity o r low Ap o r high AT. Similarly, f o r low velocity o r low AT, the equation reduces to the pure conduction form, s o that a s AT approaches zero, C cc approaches k/L.

In Table I, Tests 11 to 1 5 were performed without polyethylene sheeting over the insulation. The r a t e of heat transfer f o r the insulation due to combined fluid conduction and convection, (qi

-

qr q ), varied f r o m 78 to 95% of the total insulation heat-transfer rate, q ₁. Regarding the effect of a n e r r o r in q upon the accuracy of the fluid con-

r

duction-convection heat-transfer r a t e , q

C C

'

it

will be recognized from Table I, considering for example t e s t 15, that a 20/0 e r r o r in q will result _r in an e r r o r in q of only about one-tenth of 1°/O.

C C

The l a s t point of the discussion i s well taken. Only one type of insulation was tested t o demon- s t r a t e the usefulness of the theory. However, the theory was further proven by application to a different type of insulation, as reported in the companion paper by Wolf, Solvason and Wilson (1966).

G. D. DAVIS, Providence, R. I. : What s t e p s did you take to ensure that there was no condensation o r freezing of the moisture o r the a i r , if one hap- pens t o have sub-zero t e m p e r a t u r e s ?

K. R. SOLVASON, Ottawa, Ont., Canada: T h i s did not become a problem because the dew point in the e n t i r e apparatus very quickly came t o a dew point in the cold box. Both boxes, in effect, came down to the dew point temperature in the cold box. The samples were considerabl) above the cold box o r the dew point temperature s o that there was no possibility of any condensation in the system. These boxes were sealed well enough s o that t h e r e was no infiltration into the boxes.

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REPRINTED FROM THE ASHRAE TRANSACTIONS, 1966, VOLUME 72, PART II, BY PERMISSION OF THE AMERICAN SOCIETY OF HEATING, REFRIGERATING, AND AIR-CONDITIONING ENGI- NEERS, INC. ASHRAE DOES NOT NECESSARILY AGREE WITH OPINIONS EXPRESSED AT ITS MEETINGS OR PRINTED IN ITS PUBLICATIONS.