HEAT TRANSFER THROUGH LOW-DENSITY CELLULAR M A T E R I A L S R
.H,HA R DING,
C'nion Carbide Cor@., South Charleston, W . V a .
Commercial acceptance of rigid foams blown with fluorocarbons i s motivated largely by their superior thermal insulating characteristics. However, because the conductivities of such materials tend to increase, long-term performance of foam-containing products has been defined only by experience.
An obvious need
for a predictive function led to the synthesis of the model proposed here, which describes foam conductivity in terms of its complex fundamentals. Potential refinements are anticipated, since detailed confirmation awaits the further development of foam technology In its present form, the model can be used to evaluate product designs empirically.
FOAMED are currently the subject of intensive study PLASTICS
and expanding application. Although these materials offer unusual combinations of properties, the stimulus for their development probably centers about the excellent thermal insulation they provide a t low weight. This is particularly true of unicellular rigid foams generated using low-conductivity vapors as blowing agents. Rapid commercialization of foam-insulated products has naturally led to different opinions about long-term foam performance characteristics. Several theoretical (6, 74, 76, 77) and empirical (7, 70, 77, 75) analyses have attempted to clarify the role of cellular plastics as thermal insulation. These treatments have not, however, related theory and test data to product design. The present analysis attempts to advance toward this goal by accounting for the variable effects of cell structure and gas composition, in addition to the basic heat transfer mechanisms (73) which the reader has already mastered. Geometry
Let S equal the volume fraction of solids in a foam, and, therefore, in its average cell. Then, following present nomenclature:
s
=
(V
S
=
- V,)/V
(la)
Dr/Dp
(1b)
if the specific gravity of enclosed gases equals that of air. For rigid urethane polymers based on the reaction of polyether polyols with toluene diisocyanate, D , = 70 to 75 pounds per cubic foot ( 9 ) . A unit volume of foam must contain 1 / V = (1 - S ) / V , average cells. I t has been shown ( 8 ) that a specimen removed from this foam will contain the volume fraction:
0 , = F(l
-
S)
+ 1/2 RpVpl13 ( 1 - F)'13 ( 1 -
$)*I3
revealed that membranes are often nearly equilateral b u t seldom equiangular: mechanical restriction of free expansion causes cells to elongate in the direction they flow during manufacture. Figure 1 presents a simplified version of this phenomenon (dodecahedra project as hexagons), but does not reveal the accompanying distortion of cell faces. Models were constructed ( 9 ) to portray this geometry in three dimensions and expedite some aspects of analysis. T h e elongation, E , of a hypothetical average cell approximates unity in the ideal structure and 2 in the fully elongated structure. .4 plane of unit area intersects ( E / V ) 2 ' 3cells when 3 passed passed laterally through a foam and ( l / V 2 E ) 1 /when through the same foam's long cell dimension [frontal planes for heat and mass transfer parallel ('I) and perpendicular (I) to the foam rise direction, respectively]. Microscopic inspection has several disadvantages. I t is time-consuming, requires skilled interpretation, usually involves too small a sample to be representative, and assigns no value to F . T h e air-displacement approach of Equation 2 circumvents these limitations but provides no value of E. A mechanical analog effectively supplements the latter tcchnique, for it has been observed (9) that: E
E ( R , + 2)/3
(3)
It was found that the gas-solid interfacial arca within a closed cell approximates 16E0.2Vg2/3/3. Average membrane thickness is obviously affected by foam density as well as cell shape and size:
X , = 3SV,'/3/8E0.20 (1 - S ) FOAM
FOAM R I S E
RISE1
(4)
I
(2)
of open cells. Note that R, = 1 / X or 2 / X for semi-infinite slabs exposing one or two cut faces, respectively, and that this equation can be used to determine average cell size and true open cell content. Theories of surface chemistry predict that the cells of lowdensity foams will be regular dodecahedra ( 3 ) . Microscopic studies ( 9 ) confirmed that, while four- and six-sided cell faces (polymer membranes in unicellular foams) are normally present, the pentagon dominates as expected. The microscope also
Figure 1. Two-dimensional sketch of (left) regular (isotropic) and (right) elongated (anisotropic) foam cells VOL. 3
NO. 2
APRIL
1964
117
- 0.10E
when S is low.
The heights of an average foam cell “layer”:
and
and
11 xo= [v,E~/(I I xo= [v,/E(I -
IIE. = L E , = 0.75 when the foam is isotropic ( E = l ) , and E , falls generally between the values predicted by simple
s)]1/3
(5a)
s)11/3
(5b)
are illustrated in Table I. The interfacial areas separating adjacent “layers” of closed cells are:
11 A ,
= 8E0.8’ (1
IA ,
and
=
8(1
- S)2/3/3
- S)*’3/3E0.’3
(6b)
Vr = Vig1‘3E2‘3(1 - S)2/3
(74
per unit of foam cross-section (any F). The above equations constitute the geometric model used to characterize low-density foams for subsequent analysis. Conduction
Steady-state conductive transfer is determined by three major resistances operating approximately in parallel. Solid Phase. Coefficients of thermal conductivity for polymers are not always available in current literature. A typical coefficient for rigid urethane polymers based on polyether pol>-olsand toluene diisocyanate (9) is about
K,
=
1.2
+ O.O0024(T - 74) B.t.u./(hr.)(sq. ft.)(” F./in.)
(8)
Early derivations (6, 76), recognizing that a simple material balance does not describe the actual heat-flow path through foamed solids, have proposed that cells are arrayed as spheres or cubes. T h e path around a regular dodecahedron obviously falls betTveen these extremes. Figure 1 shows that its effective length can increase or decrease with E, depending on whether transfer is perpendicular or parallel to the foam rise. The relative conductive efficiency of foamed polymers was therefore described ( 9 ) by a composite of the inclinations of membranes relative to the appropriate transfer plane. Results indicated that:
11 E ,
= 0.60
+ 0.15E Table 1.
V,a 70-4 cc. ‘/3
E
70-6 cu. in.
2
a11 1
118
18
(;
Solid-phase conduction can now be defined as:
Q.
=
K,AE,S(Th
-
T,)/X
(10)
and can be evaluated by supplementing Equations 9a and 9b with Equations l b , 3, and 8 when more exact data are not available. Open-Cell Phase. Coefficients of thermal conductivity for common gases appear in the literature and are theoretically independent of pressure. T h e following values represent atmospheric gases near ambient temperatures in units of B.t.u./(hour)(square foot)(’ F./inch):
+ O.O0033(T - 74) K ( H 2 0 vapor) = 0.127 + 0.00040(T - 74) K(C02) = 0.119 + 0.00044( T - 74) K(dry air) = 0.182
(114 (llb) (114
A foam slab of thickness X contains X/Xo cell “layers” (see Table I). The thickness X0/2 represents the depth of air-filled cells at each cut face. These layers contain the volume fraction (1 - S) of air, while the foam below contains the smaller fraction F ( l - S). The over-all open cell content of the slab is the 0, computed from Equation 2. Accurate description of conduction through the air in a foam’s open cells thus involves three systems in series. Assume that those polymer solids which lie directly across the path of heat transferring through air neither retard nor accelerate heat flow (Le., K , approximates K,, or S is low enough to be neglected for this aspect of analysis). It can then be shown that:
Q.
= Kdo,(Th
-
Tc)/X
(12)
when K , is essentially linear in T . Since the S, V,, and F of this foam are known, Equation 12 can be solved by use of Equation 2 supported by Equation l l a if exact data are not available. Closed-Cell Phase. Coefficients of thermal conductivity for pure gases which might be used as blowing agents appear
Cell “Layers” in Foam Slobs of Varying Structure XlXoC When X = 0.5 in. 1 in. X Q ,In. ~ 0.25 in. 12 25 50 0,0200 20 40 80 0.0126 25 50 100 0.0100 34 - . 0.0290 9 17 55 0.0183 14 27 69 0.0145 17 34
II )
I&EC PROCESS D E S I G N A N D DEVELOPMENT
(9b)
2/3).
6 12 0.0418 10 19 0.0263 12 24 2(1) 0.0209 4 8 0.0602 9 7 13 0.0379 8 17 0.0301 3 6 0,0868 9 5 0,0546 6 12 0.0434 2 4 0.1112 81 493 2( II 1 4 7 0.0700 1 5 9 2(11 0.0556 Average crll volume. Assuming S = 0 when calculating Xo, so tabulated values are minimal. 3
5
(94
= 0.85
material balance ( E , = 1) and some cubical models ( E , =
(64
per unit cross-section of foam. The gas volumes associated with one cell “layer” are:
11
1E.
c
24 38 48 17 26 33 12 18 23 9 14 18 To closest whole number.
2 in. 100 160 200 69 109 138 48 76 96 33 53 66 23 37 46
18 29 36
4 in. 200 320 400 138 ~ .
21 8 276 96 152 191 66 106 133 46 73 92 36 57 72
.
Table II.
local Transient Fractions of Gas Transferred into Foam Slabs _
_
F,, = 0 _ When ~F -
~
.-
-- ~ ..-
A nY XIXQ, 1 =
o,x/xQ 0 5 10 15 20 25 30 35 40 45 50 55 60 70 80 90 100 120 140 160 180 200
1
m
1,000 1.000 1.000
1.000 1.000 1.000 1,000 1,000 1.000 1.000 1,000 1.000 1.000 1.000 1.000 1.000 1.000 1,000 1.000 1,000 1.000 1.000
'
0 0.688 0.754 0.804 0.824 0.845 0.856 0.869 0 876 0 886 0.892 0.900 0.905 0.916 0.926 0.934 0.942 0.955 0 965 0.972 0.978 0.983
1
1.000
0 0.031 0.109 0.210 0.264 0.329 0,367 0.416 0.446 0.487 0.512 0,548 0.570 0.620 0.664 0.704 0.738 0.796 0.840 0.875 0.903 0.924
I
I
1 .o"oo
l.o"O0
i=70 0 0 0.002 0.015 0.053 0.087 0.142 0.179 0.234 0,270 0.322 0.354 0.400 0,470 0.531 0,586 0,634 0.714 0.777 0.826 0.864 0,894
1
1.000
i = 7 0 0.688 0.754 0,804 0.824 0.845 0.856 0.868 0.875 0,883 0.888 0.894 0.898 0.905 0.912 0.917 0,922 0.931 0.938 0.945 0.950 0,956
1
1.000
i = 5
i = 70
i
0 0.031 0.109 0.210 0.263 0.327 0.362 0.405 0.430 0.462 0.480 0.505 0,520 0.553 0.581 0.606 0,629 0.669 0.704 0.735 0,763 0.787
0
0
0 0.001 0.007 0.027 0.043 0.071 0.090 0.119 0.138 0.165 0.184 0.210 0.252 0,292 0.330 0.366 0.432 0.491 0.544 0.591 0.634
0 0 0.000 0.001 0.005 0.010 0.023 0.033 0.052 0.066 0.088 0.104 0.144 0.186 0.227 0.266 0,341 0.409 0.470 0.526 0,575
1.000
1.000
1
1.000
in the literature. Probably the most widely used vapor is trichlorofluoromethane, whose conductivity is only: K(CCI3F) = 0.057
15
1
(13)
The data which permit solution of Equation 12 are sufficient for Equation 13 with one exception: K , has been defined only for foams blown by air, in which case K , = K,. LVhen the nominal blowing agent is any other gas, its conductivity generally does not define K , since a potential for mass transfer is established wherever membranes separate unlike gases. If transfer occurs, K , changes with the composition of entrapped vapors. Conductivities of gas mixtures are not generally available in the literature and are difficult to predict accurately. Values derived from the coefficients of component gases tend to be high when calculated assuming simple molar mixing and low when calculated assuming reciprocal molar mixing ( 4 ) . Either mixing rule could, however, provide a working relationship. O n this basis, assume that dry air is diffused into the closed cells of a foam blown with CCIIF. Combining Equations 1l a and 1I d by the simple mixing rule suggests that the coefficients of resulting mixtures will be:
- 0.12511~9+ 10-5 (33 - 28Me)(T - 74)
7 0 0.688 0.754 0.804 0.824 0.845 0.856 0,868 0.875 0,883 0,888 0.894 0.897 0,905 0.911 0.916 0.920 0.928 0.933 0.938 0.942 0.946
1
1
1.000
i = 15
0 0 0,001 0.007 0.027 0.043 0.071 0.090 0.117 0.135 0.161 0.177 0.200 0.235 0.267 0.296 0.322 0.370 0.411 0.448 0.483 0.514
0
0 0 0.000 0.000 0.002 0,005 0.011 0.017 0.026 0.033 0.045 0,053 0.075 0,098 0.121 0.145 0.193 0,239 0,283 0.326 0.366
0 0 0 0 0 0 0
1
1
1.000
1,000
1.000
2(J
1
0
(14)
Vapor Composition
T o evaluate A$, the mole fraction of fluorocarbon in the enclosed gases, consider a closed foam cell immersed in dry air a t temperature T , for e days. This cell contains a gas mixture a t total pressure P,, whose mole fractions of blowing agent and
P , = P,(1
and
- M,)
=
000 000 00u
001 003 0 005 0 009 0 013 0 020 0 034 0 051 0 071 0 032 0.137 0 184 0 229 0 274 0 316
u
1
1 000
The partial pressuies
Pt, = PtA4e (lld)
When discussion of the preceding section is extended, it is evident that the slab thickness ( X - XO)contains the volume fraction (1 - F ) ( 1 - S) of gases within its closed cells. Following the treatment applied to the open-cell phase :
Q, = K,A(l - S - O , ) ( T h - T , ) / X
i = 10
i = 5 0 0.031 0.109 0.210 0.263 0.327 0,362 0.405 0.430 0.461 0.480 0,504 0.519 0,550 0.576 0.599 0.618 0,650 0.676 0.698 0.718 0,736
i =
air are Me and (1 - M o ) ,respectively. of blowing agent and air are :
+ O.O0005(T - 74) B.t.u./(hr.)(sq. ft.)(" F./in.)
K , = 0.182
1
=
(154
P,
- P,,
(1 51))
So long as these vapors obey the perfect gas law under the constant volume conditions existing within f o a m , the ratios Pb/(Ta 460) and P , / ( T , 460) remain constant, and 7h influences heat transfer only through the conductivity co-. efficients However, blowing agents are usually not "permanent" gases and will begin to condense a t some environmental temperature TO. Reduction of T , below To thus superimposes a rising mole fraction of air upon the nornial temperature sensitivity of gas conductivities ( 7 4 ) . Values of A4, a t temperatures below Te can be calculated from Equations 15a and l 5 b by letting Pt, equal the vapor pressure of the liquid. (Both P, and Pb must, of course, be known a t some T , before this approach can be pursued.) Vapor pressures of many compounds can be presented in the general Antoine form:
+
+
In P , = 13.975 - [ 6 0 2 0 / ( T ,
+ 460)]
(1 6 )
where the given coefficients apply specifically to CC1,F. Initial. Foams expanded by nonatmospheric gases nurnially incorporate some air, which may have been entrained during the manufacturing process or injected deliberately as a nucleating agent. I t is, therefore, unlikely that the mole fiactioii of blowing agent within the product's closed cells ever equals 1. Because foams expand a t elevated temperatures, it is also unlikely that To !vi11 equal the blowing agent's normal boiling point. For example, K for a newly prepared foam, blown with CC1$ (normal boiling point = 75' F.) in a l-footthick mold, reached a minimum near 50" F. (9). Equatiun 16 shows that Pb = P , = 9 p.s.i.a. a t 51' F. Since data have been presented for Pt, and TO, they might he used to illustrate the preceding comnients. Figure 2 incorilorates the arbitary assumption that P, = 3 p.s.i.a. a t T a = 51 O .!I VOL. 3
NO.
2
APRIL 1 9 6 4
119
p.s.i.0. lo
t.
2 0.7
Me
0.6
0.5
0 10
K9 0
20
40
Te 60
80
100
010
120
0
200
100
Ta, ‘If
e
FOAM AGE,
300
DAYS AT 7
4
400
~
Figure 2. Typical effect of environmental temperature on vapor in closed foam cells
Figure 3. K vs. 0 functions predicted for example (Table VII)
Terminal. Gases on opposite sides of permeable membranes must ultimately become identical in composition, for not until then do mass transfer potentials disappear. For the case of foam immersed in air, Me must approach zero as e increases. Thus, TO will also be reduced as B increases. The knowledge that any foam will ultimately contain only a i r is academic because the transition requires infinite time. Products are designed for some finite service life, so the practical criterion is that level to which MOfalls during this period. Intermediate. Fick’s law states that the quantity of “perfect” gas crossing a membrane depends only on film thickness and area and on gas permeability through and pressure drop across the material:
Equation 18 can always be solved, although a digital computer can be used to advantage. The number of solutions can be reduced by substituting F i j for Pij: by definition F i j equals zero within the foam initially and unity around the foam at any time. Equation 18 can be evaluated on this basis for subsequent conversion to partial pressures :
I n freshly prepared foam slabs, partial pressure gradients exist across the walls of closed cells adjacent to cut surfaces or open cells. As soon as gas composition changes in the first layer, transfer into and from the second begins. Applying Fick’s law to any gas in the ith layer of closed cells after j time increments have elapsed, it can be shown (9) that:
P,,,+A= ( P s - ~ , i when
+ f‘i+i,,)/2
A = 9SV,2/3/128RK,Eo.4(1
- S ) ( T + 460)
(18) (19)
Direct application of Equation 19 is hindered by a dearth of reliable K , data for foam gases through foam polymers. Implicit restrictions include : first, the assumption that products are unicellular and, second, the fact that effective membrane area and thickness will be less than the A, and X , of the geonietiic model if the foaming liquid flows toward the intersections of adjacent cell walls. Commercial rigid foams normally approximate the first restriction. Since wall thickness and area must both decrease in the second restriction a t constant V gand S, drainage effects are partially self-compensating and should not affect A significantly if moderate. A fundamental exception clearly exists if T , < Te,since Pi, = P, while a gas contacts its condensate. However, since liquid vaporizes to maintain P, constant, the condensed phase gradually disappears, and the analysis regains its validity. Whether this temporary deviation is significant is not presently known. 120
I & E C P R O C E S S DESIGN A N D D E V E L O P M E N T
(Pb)ij = Pb,(l
- Fij)
(20a)
for nonatmospheric blowing agent transferring out of the foam, and
(f‘a)ij
= Fij(B
- Pao)+ P o o
@Ob)
for atmospheric gas transferring into the foam. Selected values of Fij for semi-infinite unicellular slabs of thickness X appear in Table 11. The largest i tabulated for each X / X o represents the central cell “layer” of that slab. By symmetry, Fij is the same in layers i and ( X / X , ) - i when equal opportunity exists for transfer across both slab faces. If one face is covered by a perfect mass transfer barrier, the given fractions are correct for slabs of thickness X/2Xo and i equals unity independently o f j only a t the surface i = 0. Recognizing that A may be different for each gas present, M e could be determined in each cell layer, and conductive transfer might be evaluated by analysis over X / X Oresistances in series. However, since Equations 18 and 19 imply constanttemperature aging, a mean F ; j can be defined as: ( X - X 0 ) 1x0 Fj
=
xo
Fij/(X -
~
0
)
(21)
so Equations 20a and 20b can be rewritten: P b
Po
=
=
Pb,(l
Fj(B
- Fj)
+
- PG‘J
(224 Pa,
(22b)
Selected values of Fj for semi-infinite unicellular slabs of thickness X / X o appear in Table 111. As before, the given X / X o must be halved if one face is covered by a perfect barrier. Applying Table I11 or equivalent solutions of Equation 18 to each gas present, MB can be estimated from Equations 15, 19, and 22 when the foam and its environment are defined and when j ’ s corresponding to the same 0 are known. Substituting these mole fractions into an appropriate form of Equation 14
Table 111.
Average Fractional Completion of Gas Transfer into or out of Foam Slabs F ; When F = 0
0
= 20 0
5 10 15 20 25 30 35 40 45 50 55 60 70 80 90 100 120 140 160 180 200
0,145 0.219 0.278 0.328 0.372 0.411 0.448 0.481 0.513 0.542 0.570 0.596 0.643 0.685 0.721 0.754 0.808 0.850 0.883 0.909 0.929
$
m
4
1.000
= 40
0 0.070 0.107 0.136 0.160 0.181 0.200 0.218 0.235 0.250 0.265 0.279 0.293 0.318 0.342 0.364 0.384 0.424 0.459 0.492 0,523 0.552
1
1.000
0
0
0
0
0
0.047 0.071 0.090 0.106 0.120 0.132 0.144 0.155 0.166 0.175 0.184 0.193 0.210 0.226 0,240 0.254 0.280 0.304 0,326 0,346 0.366
0.035 0.053 0.067 0.079 0.089 0.099 0.108 0.116 0.124 0.131 0.138 0.144 0.157 0.169 0,180 0.190 0.209 0.227 0.243 0.259 0.273
0.017 0.026 0.033 0,039 0.044 0.049 0.054 0.058 0.061 0.065 0.068 0.072 0.078 0.084 0.089 0.094 0.104 0.113 0.121 0.128 0.136
4
1
0.023 0.035 0,044 0.052 0.059 0.066 0.072 0.077 0.082 0.087 0.092 0.096 0.104 0.112 0.119 0.126 0.139 0.150 0.162 0.172 0.182
0.014 0,021 0,027 0,031 0.036 0.039 0.043 0.046 0.049 0.052 0.055 0,057 0.062 0.067 0.071 0.075 0.083 0.090 0.097 0.103 0.108
1.000
4
$
favorable conditions for convection as might reasonably exist within a foam. T h e average cell volume of rigid urethane foam is normally in the range of 5 to 25 X 10-6 cubic inches. Table I shows that the largest X, associated with a foam whose V , = 5 X cubic inches is 0.11 inch. By contrast, Table IV demonstrates that the existence of convection currents within closedcell foam is a truly remote possibility. Consequently, Q c can be neglected. Radiation
Radiant transfer is theoretically dependent on absolute temperature and the number, disposition, and effective emissivities of the surfaces present. Larkin and Churchill (72) determined that the net radiant heat flux is:
0’173A Qr
4
1.000 1.000 1.000 1.000
makes available the time- and temperature-sensitive contribution of a foam’s closed cell contents to thermal conductivity. Convection
Temperature gradients tend to promote natural circulation of any normally stagnant fluid contained within thermal insulators. Once such convective flow is established, the fluid will transfer Q c B.t.u. per hour in excess of that transferred by random molecular motion (conduction). While convection is stimulated thermally, it is retarded by surfaces which interfere with the development of mass flow patterns. Doherty et al. (5) concluded that this resistance prevents convection within foams until:
Xo4(Th- T , j / X
When available properties of CC1,F and air a t 1 atm. and 75’ F. are used, the right side of Equation 23 equals 0.03 and 1.O cubic inches x ’F., respectively. These values represent as
> 17Og(td)(kv)/g(ce)
T h e numerical constant is dimensionless, g is the gravitational acceleration constant, and td, ku, and ce represent the thermal diffusivity, kinematic viscosity, and volumetric coefficient of expansion of enclosed vapors. This derivation assumes that thermal gradients are essentially linear and considers only the mass flow resistance and driving force presented by opposed surfaces a t different temperatures. Since the side walls of foam cells increase resistance, Equation 23 is conservative, and convection should not begin until the inequality becomes large.
Gas at 1 atm. and 75’ F.
Applied Gradient, O F./In.
10 20 40 60 80 100 a
Equation 23.
Minimum Air
N
0.56 0.47 0.40 0.36 0.33 0.32
=
0.23 0.20 0.17 0.15 0.14 0.13
( ~ E / E )-
(24)
1 f NX
8 R,E,(~ - S ) E0.20/3
(25)
for low-density unicellular foam. Ideally, R , and E, could be evaluated from electromagnetic theory if the detailed “optical geometry” of a foam, and the refractive indices of contained gases and solids, were known as functions of wave length.
Table V.
Estimation of Effective R,EJ for Rigid Foamsa
Direction
v6’ 9
10-0
Test Of
s
Cu. In.
E Urethane
I/
110 110 110
1
0.016 0.016 0.016 0.016 0.024 0.024 0.024 0.024
1.4 1.7 1.5 1.3 1.4 1.2 1.1 1.0
]I I]
0.016 0.024
30 80
11
11 11 I
110
30 30 30 30
XO,In.,for Convection of CCl3F
>’ ( iR60”)’] - Tc
100
+
through slabs of nonabsorbing, porous insulation faced with at T I parallel, opaque surfaces having emissivities e h and and T,, respectively. Absorption would greatly complicate this relationship, but might reasonably be ignored in most lowdensity foams containing no dyes or fillers. This hypothesis is based on the fact that a cubic inch of typical foam contains only about 0.02 cubic inch of solid (plus low-pressure heteropolar gas) which might absorb radiation, but about 200 square inches of “reflective” interfacial surface. N defines a foam’s ability to retard transfer by reradiation from internal discontinuities in refractive index. When it is recognized a foam’s internal geometric cross-section is that of its cell walls, it can be shown (9) that:
1 Threshold Cell Heights for Vapor Circulation within Closed Foam Cellsa
[(
(1/Eh)
1 Table IV.
=
+ 460
T h
AQI/A, B.t.u./ (Sq. F t . ) (Hr.)
0.16 0.22 0.24 0.16 0.14 0.11 0.08
0.16
N, In.-’
290 210 190 290 335 415 590 290
Styrene 1.0
0.20 230 1.2 0.30 150 Basis: Q / A (at 54” F./inch) - 6 Q / A ( a t 9’ F./inch)
Indicated RsEf
5.0 3.3 3.2 5.0 3.7 4.8 6.9 3.4 2.7 2.4
a = AQJA B.t.u./(sq. f t . ) (hr.). Since apparatus is operating near limit of its sensitioity under these conditions, each Q / A measurement replicated to improve precision. From Equation 24, N = [ 4 8 . 0 / ( A Q , / A ) ] - 9.5 in.-’ across 2-inch specimen thickness at 74’ F. mean when Eh = eC 0.1. From Equation 25, RsEf = 3NV,’la/8( 1 - S)EQ.20.
VOL. 3
NO. 2
APRIL 1 9 6 4
121
T o provide a guide when these data are not available, experiments were conducted in the apparatus described he urrthane foams studied were coarsepreviously (70). 'I celled, CC1,F-blown products based on the reaction of toluene diisocyanate with a commercial polyethrr pol~-olblend. The styrene foams were commercial materials. l'able V summarizes the method used to calculate I?.€,, suggesting that average values of 4.4 and 2.6 represent urethane and styrene foams, respectively. Note that higher density would have inc-reased R,E, if absorption controlled radiant transfer, but no such effect was obtained. Further, although radiation coiitiibuted little to over-all heat transfer under the conditions of test, this need not always be the case. When use of the given I?*€, suggests that radiation may be significant, the product should be reevaluated for the foam ' system and thermal environment under consideration.
where e = jA(CC1,F) = njA(air). F, is related to j by Equations 18 and 21, and Table VI presents this function for the foam slabs described above. The pressures R, Pbo, and Pao are assumed to equal 14.7, 9.0, and 1.7 p.s.i.a., respectively. Since Equation 28 contains two unknown time elements, it cannot be solved directly. Values of both A, or of either A and n, must be assumed. Table VI1 summarizes the application of data from Table V I to Equation 28 a t two reasonable levels of n. Each M is converted to the K defined by Equations 27a and 27b but expressed now as a function of j and nj rather than 8. These K values were plotted in Figure 3 letting A(air) = 1 day. Quantitative agreement between the synthetic n = 50 curves and the actual behavior of similar 1-inch-thick rigid foam slabs (70) is satisfactory. Thus, while the proposed model cannot be validated completely without K , data, its quality appears sufficient to justify further consideration of the variables affecting K .
Results
The model now indicates that Q, the over-all rate of heat transmission through foam, is the sum of Equations 10, 12, 13, and 24:
Q / A = K,E,S(?'h
- ?',)/X
+ KaOc(7h- T c ) / X + - ?',)/A' +
K,(1 - S - O , ) ( T h
Trial solutions should illustrate the model's proper use and demonstrate some of the K m. e relationships it predicts. A typical cell structure for molded, CC1,F-blow n rigid urethane foam is S = 0.026, V , = 16 X 1 0 F cubic inches, F = 0.01, and E = 1.5. Immediately after foaming, specimens can be cut for parallel and perpendicular K measurements a t 74" F. mean and iinder an 18" F. per inch gradient applied by plates whose emissivity is 0.1. Each K specimen consists of two large 1-inch-thick slabs aged a t T, = 74' F. between measurements. From this information, it can be determined from Equation 2 that 0, = 0.034. Equations 5a and 5b show that liX/Xo = 30 and l X / X , = 45 cells per slab, while Equation 25 suggests that N = 490 inch.-' Dividing each term by ( z ' h - Ztc)/X and using values of K , ? E,, K " , and K , provided by Equations 8, 9, 1l a , and 14, Equation 26 may be recast in the form:
+ 0.94 K , = 0.205 = 0.030 + 0.94 K , = 0.201
1/K = 0.034
-
0.1175 M ,
I K
-
0.1175
M,
(27a)
Discussion
Formulation. Polymer structures affect E,. K,, and K , directly. Since radiation and solid-phase conduction are normally minor factors, foam insulating performance would not be changed appreciably by adjusting K , or € 1 . Permeability can, however, vary greatly with polymer structure. If all K , values were halved by modification of the polyols and/or isocyanates used to produce urethanes, design criteria would improve because the time required for a given change in gas composition would double. Small quantities of catalyst and surfactant are formulated into foams. These ingredients might theoretically affect K , by modifying polymer structures and introducing new chemical groups, but such possibilities would be masked by the control these components exert over V , and F. T h e last essential foam ingredient is the blowing agent. The amount formulated determines the minimum D, which might be realized. The blowing agent's chemical structure determines its P, and conductivity coefficient, as well as its K , through a given solid. Since these factors help define K , and its rate of change, certain blowing agent adjustments have major effects on K behavior. Process. No manufacturing variables appear directly in the model. However, mixing conditions can affect V , and F ~~
Table VI. Average Fractional Completion of Gas Transfer across 1-Inch-Thick Slabs Cut from Typical Foam V o and E are such that
(27b) 3
Since M I ultimately approaches zero, it is apparent that K will eventual y equal 0.2. Evaluating the important intermediate K is more complex. Consider the various gases which may be present within a foam. Their unit transfer increments are related generally by some multiplying factor- e.g,, A(CC1,E') = n A ( 0 , ) . Equation 1 9 reveals that A = l .47/10"K, day for any gas in the foam selected as a n example. Since appropriate K , are not available, further analysis of this example utilizes several approximat ions. Assuming for convenience that only two molecular species are present in the vapor phase, Equations 1 5 and 22 can be com bined to read :
122
I L E C P R O C E S S DESIGN A N D DEVELOPMENT
~
0 1 2 3 4 5 6 7 8 10 15 20 25 30 35 40 45 50 55 60
!I
F,
0 0.034 0.052 0.069 0.082 0.095 0.106 0.116 0,126 0.144 0.182 0.215 0.243 0.269 0.293 0.316 0.337 0.356 0,376 0,394
1
X / X o = 30 and
I Fi
i
0 0.023 0.034 0.045 0,054 0.063 0.070 0.077 0.083 0.095 0.120 0.142 0.161 0.178 0.194 0,208 0.222 0.235 0.248 0,260
65 70
IX/Xo II Fi
80 90 100 120 140 160 180 200 250 300 350 400 450 500
0.411 0.428 0.444 0.459 0.488 0.516 0.567 0.612 0.652 0.689 0.721 0.788 0.839 0.878 0.907 0.929 0.946
.1
.1 1 .QOO
75
m
= 45
IF , 0.271 0.282 0.293 0.303 0.322 0,341 0.376 0.407 0.437 0.464 0.491 0.550 0.602 0.647 0.688 0.724
I
1.000
Table VII.
Calculating Rates of K Change in Examplea M =
10.7
9(1 - Fj) 13Fnj - 9Fj
-
0
2 5 10 15 20 40 60 80
0
0
0.841
0.052 0 095 0 144 0 182 0 215 0.316 0.394 0.459
0.144 0 243 0 356 0 444 0 516 0.721 0.839 0.907
0.705 0 626 0 549 0 496 0 457 0.357 0.302 0.265
1 2 3
a
0.034 0.052 0.069 4 0.082 5 0.095 6 0.106 7 0.116 8 0.126 Table VI. Ste also Equations
0.106 n = 5 0.122 o 131 0 141 0 147 0 151 0.163 0.170 0.174
0.356 0.579 0.516 0.504 0,633 0.458 0.721 0.427 0.788 0.406 0.839 0.390 0.878 0.378 0.907 0.368 27 and 28 and example described in
0 0.034 0 063 0 095 0 120 0 142 0.208 0.260 0 303
0.095 0.161 0.235 0.293 0.341 0.491 0.602 0.688
0.748 0.690 0.631 0.590 0.557 0.469 0.411 0.371
0.113 0.120 0.127 0.132 0.136 0.146 0.153 0.157
0.235 0.341 0.422 0.491 0.550 0.602 0,647 0,688
0.649 0.586 0.545 0.513 0.488 0.468 0.451 0.437
0.125 0.132 0.137 0.141 0.144 0.146 0.148 0.150
n = 50 0.137 0.023 0.146 0.034 0.151 0.045 0.155 0 054 .. ... 0.157 0.063 0.159 0.070 0.161 0.077 0.162 0.083 latter half of Results section.
(8),Pao,and product uniformity. Molding operations, which restrict foam expansion, can affect E, S,Pb,,and uniformity. Thus, while many relationships are poorly defined a t present, the total manufacturing environment influences K significantly through cell structure and initial gas composition. Density. One component of cell structure influences both cost and K i n foam systems. When other factors are constant, the solid-phase and radiant components of heat transmission are proportional to S and about 1/(1 - S), respectively. Vapor-phase transfer is nearly proportional to (1 - S) a t fixed K,. Since A is proportional to S/(1 - S), foam density also affects rates of K , change. Table VI11 relates these predictions to the typical foam described above. T h e initial K advantage of lower-density products is largely nullified by their greater rate of change, although all K ZJS.e curves remain members of the general family illustrated by Figure 3. Table VIII.
Effects of Density on K Components on l-lnchThick Rigid Foam Slab V , = 16 X
Density, lb./cu. ft. Corresponding S A(air), days Effective 74 F. K of: Polymer solids ( E = 1.5), /I I Air in open cells ( F = 0.01) Vapor in closed cells, initial Ultimate Radiation (18" F./in.) Foam 11 ( X / X o = 30), initial
cu. in.
1 0.013 0.49
2 0,026 1 .oo
3 0.039 1.52
0.013 0.011 0.006 0.073 0,173 0.002 0.094
0,026 0,022 0.006 0.072 0.171 0.002 0.106
0.039 0.033 0.006 0.071 0.169 0.002 0.118
0.134 0.143 0.147 0.194 0.192
0.137 0.146 0.151 0.205 0.102
0.143 0.152 0.158 0.216 0.112
0.120 0.130 0.136 0.192
0.125 0.132 0.137 0.201
0.132 0.138 0.142 0.210
O
( 0 = 0)
Aged 50 days" 100 days" 150 days" Ultimate Foam I ( X / X o = 45), initial
(e
=
0)
Aged 50 daysa 100 daysa 150 daysu Ultimate a
Assuming n = 50 at 74' F.
I to Foam
Transfer Fni 0
-~
i
Rise M 0.841
74°F. K 0.102
Unicellularity. Completely open-celled foams are naturally air-filled from the time they are made. "Ultimate" K's in Table VI11 thus constitute minima when F = 1. If cell walls were absent, rather than merely cracked or pinholed, convection and/or increased radiant transmission (lower R,) could easily generate still higher K. Such cases fall beyond the region in which the model yields quantitative predictions. The F of rigid foams intended for use as thermal insulation approaches zero in practice. Intermediate levels (0 < F < 1) increase initial K , and tend to reduce A by providing direct contact between the environment and the slab's interior. When other factors are constant, the open- and closed-cell components of heat transmission are proportional to 0, and about (1 - Oc), respectively. Table I X imposes these effects on the 2-pound-per-cubic-foot sample from Table V I I I , simultaneously incorporating the reasonable assumption that A is proportional to (1 - F ) .
Table IX.
Effects of Open Cells on K Components of l-lnchThick Rigid Foam Slab Assumed F 0.01 0.10 0.20 0, ( V , = 16 X 10-6 cu. in.) 0,034 0.120 0,216 A(airj, days 1.00 0.91 0.81 Effective 74" F. K of: Polymer solids (S = 0.026), I/ 0.026 0.026 0,026 0.022 0.022 0,022 -4 Air in open cells 0.006 0.022 0 039 Vapor in closed cells, initial 0.072 0.065 0.058 Ultimate 0.171 0.155 0.138 Radiation (18' F./in.) 0.002 0.002 0.002 Foam 11 ( X / X o = 30), initial 0.106 0,115 0,125
(e = 0) Aged50 daisa 100 days" 150 days" Ultimate Foam I ( X / X , = 45), initial (e = 0 ) Aged 50 days" 100 days" 150 days" Ultimate a
0.137 0.146 0.151 0.205 0,102
0.144 0.152 0.157 0.205 0.111
0.152 0,160 0.164 0.205 0,121
0.125 0.132 0.137 0.201
0.132 0.139 0.144 0.201
0.141 0.148 0.152 0,201
Assuming n = 50 at 74' F .
VOL. 3
NO. 2
APRIL
1964
123
Cell Size and Shape. V , and E desciibe the distiibution of pol)inei solids in space. Together they influence Equation 26 through the variables 6 and K , Equations 9a and 9b indicate that E contributes independently to solid-phase conduction LVith other factors constant, radiant transmission is approximately pioportioiial to V,O 3 3 / E 0 . 2 0 and A ic directly proportional to the q u a r e of this ratio. However, Equations 5a and Sb show that V uand E alqo define Xo Since K Oincrements are proportional to X O / X at low 8, the net effects of changing polymei distribution ale complex T o illustrate, let E = 1 in the 2-pound-per-cubic-foot foam desciibed by Table L'III. The perfoimance of this modified product, summarized in the fiist column of 'Table X, falls betv een correcponding pal allel and perpendicular values for the parent foam as expected NOM,let L*O increase tenfold and consttuct the second column of Table X Since K incremrnts during finite 0 are similar to those in the first column, these changes in A and ,Yo nearly compensate for each other Ideally, then, large V g variations could be tolei ated in applications \$hrre moderate thermal gradienrs piovide negligible radiant tiansfer. I t must, however, be recalled that the model assumes unifoiin membiane thickness. LVhile it is possible to approach this requirement. coaise-celled foams seen] particularly susceptible to "capillary" flow. The last column of Table X demonstrates the accelerated K rise which follows when effectike membrane thickness is halved by this phenomenon. The model's predicthe functions might be improved by supplementing VO and E with parameters describing polymer distribution in greater detail Since representative A , and X, are not now measuied accurately and ioutinely, A can be multiplied by a fraction representing the "geometiic e a ciency" of available solids. Interactions. Although the factors mentioned above are independent, this theory is not alwabs supported by current practice. For example, Vu and F rise as D, is ieduced below some limit characteristic of the foam system and process. Such combinations explain aspects of normal foam behavior ( 7 7 , 75) whose sources are not obvious. Product Design. Examples presented thus far have involved 1-inch-thick slabs exposed to the atmusphere and subjected to 18' 1.. gradients a t 74' F. mean \Vhile these or similar conditions are commonly used in the laboratory, they are seldom encountered in practice The foam described by Table X. column 1, was chosen to illustiate some effects of changing physical and thermal environmeiits. A large 2-inch slab of this product (X,"Yo = 80),faced on one side with sheet metal, ic considered in Table XI It was assumed for convenience that temperature had no effect on any k', and that no vapors condensed Since F, values have been derived for two exposed faces, the first column of Table X I was calculated as though X/"Uo were twice the actual thickness. This assembly required 800 days to reach the K achieted by a cut 1-inch-thick slab of the same foam in SU days. Both faces of a n insulating wall are normally protected from the atmosphere A transfer harrier, effective as 0.5 inch of this foam (.Y/.Yo = 20), was placed over the exposed surface to produce the second column Physically, this bai iier might have been the natural uiethane skin formed duiing molding operations, a plastic liner, or a wood veneer. More than 1300 days will elapse before thir assembly reaches the K achieved by a n inch of cut foam in SO days. The first t u o Lolumns of Table X I reflect K aging rates in the thermal environment investigated prcviously. Now, build 124
l&EC PROCESS DESIGN A N D DEVELOPMENT
Effects of Polymer Distribution on K Components of 1-Inch-Thick Rigid Foam Slab VB, 1O--fl CII.in. 16 160 160 1.08 5.45 1.08 A( air), days 40 18 18 A/'& ( E = 1 ) Effective 74' F. K of Polymer solids (S = 0.026) 0.023 0,023 0,023 Air in open cells (0, = 0.034) 0.006 0.006 0.006 Vapor i n closed cells, initial 0.072 0.072 0.072 Ultimate 0.171 0.171 0,171 Radiation (18' F./in.) 0.002 0.005 0,005
Table X.
Foam 1; or I, initial Aged 50 days" 100 days" 150 days" Ultimate Assumin,g n
=
(e
=
0)
0.103 0.127 0.135 0.140 0 202
0,106 0.128 0.136 0,142 0 205
0,106 0.149 0.157 0,161 0.205
50 at 74' I;.
an oven and a freezer using the second assembly. Let inner skin temperatures of the insulating panels equal 225' F. and -So F. in the respective units. Let outer skin temperatures equal ?So I'. in both cases. The third and fourth columns review K changes of these foam-cored panels as the appliances operate under steady-state conditions. The data might be extrapolated by noting that projected rates of K change will not exceed 0.001 per 100 days. O n this conservative basis, oven and freezer wall K's might approximate 0.17 and 0.14, respectively, after 10 years of continuous service. Conclusions
Foam conductivities depend on the original composition of contained gases and the permeability of each through the solid. Because consideration of these factors was limited by lack of appropriate data, potential refinements in the model can only be noted and verification of its absolute accuracy invites further work. A practical objective was achieved independently of this limitation, however: K aging curves for cut test slabs can be used to characterize the performance of definable end products containing the same foam.
Table XI.
Selected Environmental Influences on ponents of 2-Inch Rigid Foam Slab V , = 16 X
TeInu..
a
Com-
cu. in.
F.
Mkin (Ti, f T,)/2 Gradient ( T h - 7 ; ) / X Effectivr ,U/X, for transfer O,(F = 0 01, S = 0 026) Effective operating K of: Polymer solids ( E = 1 ) Air in open cells Vapor i n closed cells Initial Ultimate Radiation ( N = 450 in. -1) Foam, initial (e = 0 ) Aged" 50 days 100 days 150 days 200 days 400 days 600 days 800 days 1000 days Assuming A ( a i r )
rangp.
K
=
74 74 150 35 18 18 75 40 160 200 200 200 0 016 0 010 0 010 0.010 0.023 0.003
0.023 0.002
0.024 0.002
0.023 0.002
0.074 0.174 0.002
0.074 0.175 0.002
0.081 0.200 0.004
0.071 0.163 0.002
0.101 0.106 0.109 0,111 0.112 0.116 0 120 0.122 0.124 = 50 over
0.111 0.118 0.121 0.123 0.125 0.130 0.134 0.137 0.139
0.098 0.103 0.105 0.107 0.108 0.112 0.115 0.117 0.119
0,102 0.108 0.111 0.114 0.116 0.121 0.124 0.127 0.129 1.08 days and n
entire temferature
Nomenclature
R,
A = foam cross-sectional area, sq. ft. A , = gas-solid interfacial surface per unit A per cell “layer” B = barometric (absolute atmospheric) pressure, p.s.i.a. D, = foam density (21, Ib./cu. ft. D, = polymer density, lb./cu. ft. E = cell clongation (average height/width ratio) E, = dimensionless efficiency factor for heat conduction through foam solids F = volume fraction of interconnected (open) cells within foam F i j = fractional accomplishment of total permissible partial pressure change in foam’s ith layer of cells a t specimen agc j F , = average fractional partial pressure change of gas within foam slab X / X Ocells thick a t a g e j i = number of cell “layers” separating given cell from reference surfaice of foam specimen j = age of foam spc!cimen measured as multiple of A, equals 8/A for specific gas K = over-all heat transfer coefficient, B.t.u./(hr.)(sq. ft.) (” F,,’in.) K , = thermal conductivity coefficient of air in foam’s opencellrd phase K, = thermal conductivity coefficient of gas mixture in foam’s closrd-cell phase a t age 8 K, = coefficient for permeability of a gas through polymer film a t T , F.:, (lb. mole)/(sq. ft.)(day)(p.s.i./in.) K, = thermal conductivity coefficient of foam solids Me = mole fraction of blowing agent in foam’s closed cells at specimen age 8 n = A(CCI,F)/A(air) in examples N = radiant back-scattering cross-section per unit foam volume, in. -l 0, = total volume fraction of open cells in cut foam specimen P = pressure, p.s.i.a. Pa = average partial -presrure of atmospheric gas within foam‘s closed cc,lls, a function of 0, p.s.i.a. Pa”= original partial pressure of atmospheric gas within foam’s closed cells, p.s.i.a. Pb = average partial pressure of blowing agent within foam’s closed cells, a function of 8. p.s.i.a. Pb0 = original partial pressure of blowing gas within foam’s closed cells, p.s.i.a. Pi, = partial pressure (of gas within closed foam cell in ith “layer” at specimen agej, p.s.i.a. P, = total pressure of g,ases within closed foam cell a t age 0, p.s.i.a. P, = vapor pressure of gas, p.s.i.a. Q = over-all rate of heat transfer, B.t.u./hr. Q. = rate of heat transfer by conduction through air in open cells. B.t.u./hr, Q, = rate of heat transfer by convection, B.t.u./hr. Q, = rate of heat trarrjfer by conduction through vapor in closed cells, B.i . u . j h r . Qm = transfer rate of gas through polymer film, lb. mole/day Q7 = rate of heat transfer by radiation, B.t.u./hr. Q8 = rate of heat transfer by conduction through foam solids, B.t .u./hr. R = universal gas constant R, = measured ( 7 ) dimensionless ratio of foam compressive strcngth paralld to that perpendicular to the foam rise
R,
= gross geometric surface/volume ratio of foam specimen,
in.-’ dimensionless ratio of radiant scattering to geometric cross-sections within foam S = volume fraction of solids in foam T = temperature, ” F. T, = ambient temperature, O F. T, = temperature a t cooler side of foam slab, O F. Th = temperature a t warmer side of foam slab, ” F. To = temperature a t which gas begins to condense within closed foam cell a t specimen age 8, F. V = total volume of average foam cell, cu. in. V , = volume of gas in the average foam cell, cu. in. V, = total gas volume per unit foam A per crll “layer,” cu. in. X = foam slab thickness, in. X, = average wall (membrane) thickness within a closed-cell foam, in. X O = total thickness of an average cell “layer,” in. A = mass transfer time increment specific to gas-solid combination, days E, = emissivity of cooler opaque surface bounding a foam slab E, = fraction of incident radiant energy back-scattered a t each gas-solid interface E~ = emissivity of warmer opaque surface bounding a foam slab e = foam specimen age, days [ I = measurement parallel to the foam rise direction 1 = measurement perpendicular to the foam rise direction =
Acknowledgment
J. C. Cugini and C. J. Hilado calculated the F f j and F3 values literature Cited (1) Am. SOC. Testing Materials, Philadelphia, Pa., “ASTM Standards on Plastics,” 12th ed., p. 281, D1621-59T, 1961. (2) Ibid., p. 643, D1622-59T.
(3) Bikerman, J. J., “Foams: Theory and Industrial Applications,” Rrinhold, New York, 1953. (4) Brokaw, R. S., Znd. Eng. Chem. 47, 2398 (1955). (5) Doherty, I). J., Hurd, R . , Lester, G . K., Chem. G’ Ind. (London) 1962, p. 1340. (6) Gorring, R. L., Churchill, S. W., C h m . Eng. Pray. 57, No. 7, 53 (1961). (7) Guenther, F. O., SPE Trans. 2, No. 3 , 243 (1962). (8) Harding, R. H., Mod. Plastics 37, 156 (June 1960). (9) Harding, R. H., Chemicals Division, Union Carbide Corp., South Charleston, \V. Va., unpublished work, 1959--61. (10) Harding, R. H., James, B. F., M o d . Plastics 39, 133 (March 1962).
(11) K n o x , K. E., A S H R A E J . 4, No. 10, 43 (1962). (12) Larkin, B. K., Churchill, S. \V., A.2.Ch.E. J . 5, 467 (Decrmber 1959). (13) McAdams, \V. H . , “Heat Transmission,” McGraw-Hill, Nrw York. 1954. (14) Patten,’G. A . , Skorhdopolr, R. E., iifod. Plastics 39, 149 (July 1962). (15) Skochdopole, R. E., Chem. Eng. P r o p . 57, No. 10, 55 (1961). (16) Stephenson, M. E., Mark, M., ASHRAE J . 3, No. 2, 75 ~
,,n,.\
(1YUl).
(17) Topper, L., 2nd. Eng. ChPm. 47, 1377 (1955) RECEIVED for review April 17, 1963 ACCEFTE.D August 16, 1963
VOL. 3
NO. 2
APRIL
1964
125
