DIFFWSION PHROUGH VAPOR-BARRIER. GAPS EN HOUSE WALLS r I‘ i ‘ Thesis for flu Doqmo of M. 5.- MECHIGAN STATE UNIVERSITY 1 David Alan Norman 1959 1142a. LIBRA RY L1 h/fichiganilmtc a Univcmt Y A DIFFUSION THROUGH VAPOR-BARRIER GAPS IN HOUSE WALLS by DAVID ALAN NORMAN AN ABSTRACT Submitted to the College of Agriculture Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Agricultural Engineering 1959 Approved by _222322§: jflu53u¢¢~f:z;/ ii ABSTRACT The vapor flow rate through gaps in a vapor barrier has been obtained by a conformal—mapping solution of the two-dimensional diffusion equation, the effects of convec- tion being neglected. The flow rate in grains/hour-foot of crack length for a crack of width f in an impermeable material of thickness g is given approximately by D (92- pl) w 2 ln(4y/f) + 1Tg/f where D is the permeability of the surrounding medium in grains/hour-foot-inch of mercury, and p2 and p1 are the partial vapor pressures (in Hg) at a distance y on either side of the barrier. An analogous formula is given for a lap. Since p2 and pl, and the expression for flow rate vary slowly with y at distances far from the gap, the point of measurement of p2 and p1 is not critical. The expressions obtained are found to be consistant with published measurements. Calculations by the expressions obtained show that some gaps occurring commonly in practice may allow a damaging amount of vapor to pass through the barrier. DIFFUSION THROUGH VAPOR-BARRIER GAPS IN HOUSE WALLS by DAVID ALAN NORMAN A THESIS Submitted to the College of Agriculture Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Agricultural Engineering 3 , A I ‘ , “7 If“! ‘ ‘ o a /DJ 1v ACKNOWLEDGEMENTS Dr. Merle L. Esmay, as major professor, provided the leadership necessary to keep the research moving toward an attainable end while providing freedom in the means. Dr. Donald J. Montgomery, of the Department of Physics and Astronomy, as minor professor suggested ways of making the thesis clearer and more useful. Dr. Ross D. Brazee, while a fellow graduate student, presented a timely introduction to conformal mapping. Funds for an assistantship were provided by the North Central Region Farm Housing Research Committee NC 9. Mrs. Joanne Norman gave an abundance of a wife's patience and faith. TABLE OF CONTENTS THE NEED FOR A STUDY OF GAPS IN A VAPOR BARRIER LIMITING THE THEORETICAL PROBLEM A SLIT IN A THIN VAPOR BARRIER A CRACK IN A THICK VAPOR BARRIER A LAP IN A THIN VAPOR BARRIER COMPARISON OF THE THEORY NITH PUBLISHED MEASUREMENTS APPLICATION OF THE THEORY SUGGESTIONS FOR FURTHER STUDY APPENDIX A. THE PERMEABILITY OF AIR APPENDIX B. THE PERMEANCE OF BUILDING MATERIALS APPENDIX C. THE EXACT SOLUTION FOR A CRACK REFERENCES Page 16 20 28 34 39 41 43 47 LIST OF FIGURES Figure U14>v1m 10. ll. 12. 15. 14. Cross-sections of a wall showing typical vapor barriers Vapor flow through a slit Mapping of a slit into an infinite strip Flow and constant-pressure lines for a slit Dependence of vapor pressure on distance perpendicular to a slit for several values of the slit wiith f. Arbitrary units. Vapor flow through a crack Dependence of V/LD on 2f/y with 2g/y as parameter. Dimensionless quantities Mapping of a lap into the upper half—plane Mapping of a polygon into the upper half-plane Flow and constant-pressure lines for a lap The relation between equivalent g/f and k/h Four types of vapor—barrier gaps Mapping of a slit into a rectangle, the upper half-plane, and an infinite strip Error in the approximate equation for flow rate through a crack vi 12 13 16 18 20 20 24 27 35 48 49 THE NEED FOR A STUDY OF GAPS IN A VAPOR BARRIER 'The purpose of a vapor barrier in a house wall is to prevent an excessive amount of water vapor from entering the wall from the inside of the house. The vapor-barrier material might be a paint coating, a polyethylene or aluminum foil, or an aSphalt-ooated paper. Figure 1 shows typical ways of installing vapor-barrier material as an integral part of blanket insulation. In spite of the name vapor barrier, some vapor passes into the wall through the vapor—barrier material or through gaps. The amount of vapor passing per unit time, wall area, and Vapor—pressure difference is the permeance of the vapor barrier. Experience has shown that a permesnce of about one BEER, defined as one grain per hour—siuare foot-inch of mercury, is the maximum value allowable in a vapor barrier that will prevent ccnlensation. A vapor barrier of prOper permeance for a given type of construction, and given inside and outside temperatures and vapor pressures, can be designed with the aid of available data for the heat conductance and vapor permeance of the materials on both sides of the vapor barrier anl the results of this thesis. Both he vapor flow rate through a gap and the vapor pressure near a gap need consideration. The flow rate is important Nhen there is a lot of gap length per square foot as is the case when the vapor barrier is made up of sheathing blanket insulation \ \ r l plasterboard vapor-barrier material gap ?____ " , i E W K" 7 K ‘ S z I ‘1) ‘ T 4 I i f vapor-barrier material 7 3 l 7 2 vapor—barrier material Figure l. Cross-sections of a wall showing typical vapor barriers narrow strips. For a 16-inch stud spacing there are about lB—inches of gap length for each square foot of wall. Even though the average permeance of a vapor barrier may be acceptable, a gap may cause local conden- sation depending on the closeness of a cold surface and how the vapor pressure changes with distance from the gap. The harmfulness of vapor passing through gaps has been shown by experiment. Dill (Ref. 1) reported on tests of walls with eight arrangements of insulation and, vapor barriers. Conditions were 70°F and 30%RH (70 degrees Fahrenheit and 30 percent relative humidity) on one side of the wall and —5°F on the other. The tests were run 100 hours. Frost was gathered from the sheathing and weighed. A wood-fiber fill insulation arrangement, with a 0.56-perm vapor barrier turned and sealed against the frame with Scotch tape, had more than three times as much frost as a similar arrangement with no vapor barrier. With the vapor barrier the frost occurred near the corners of the panels. Without the vapor barrier the frost was evenly distributed over the sheathing, and the insulation and sheathing accumulated more moisture. A rock-wool fill arrangement had similar results. A double thickness of one-inch blankets enclosed by a 8.87-perm envelope placed between the studs with an air space on both sides had no frost on the sheathing or siding. An actual weather test at Pennsylvania State College was reported by Reichel (Ref. 2). A test house of 48 panels of 22 different constructions was built outdoors. Inside conditions were 70°F and 40%RH. The siding of all the panels had three coats of conventional exterior white house paint. Observations were made of paint blistering, mold growth on the sheathing, and moisture in the Sheath- ing and siding. Paint blistering started on one panel on January 16 after one month of exposure. This panel had a 0.32-perm barrier on a one—inch blanket with a S/lé—inch gap between the barrier and the top and bottom plates. By March this panel had more blistering than the other panels which had no barrier or where the barrier flaps were attached to the studs and plates. Most of the blisters had water between the first and second paint layers. No mold occurred where a vapor barrier was used. Heavy mold occurred where fill insulation only was used. Moisture in the sheathing and siding was generally higher when no barrier was used. The highest siding moisture content occurred in the panel with a gap at the top and bottom of the vapor barrier. Other experiments have found the flow rate through Various slits, cracks, laps, and holes (Ref. 5, 4, and 5). The flow-rate data cannot, in general, be used to design gaps. All these tests show that gaps in vapor barriers can allow passage of damaging amounts of water vapor. The tests results however are not very helpful in designing minimum dimensions for gaps. For this, theory is needed. Then perhaps vapor—barrier failures can be better analyzed and engineers can'be more specific about what gaps may be allowed. LIMITING THE THEORETICAL PROBLEM The vapor—flow rate through gaps in vapor barriers is a function of many variables. Some of the variables such as the dimensions of the gap, the pressure difference across the vapor barrier, and the permeability of the medium surrounding the gap, will be considered mathemat- ically. Cases which have a vapor source or sink will not be considered. Some of the other variables will be assum- ed to have a negligible effect on the flow rate, such as the end effect for a long narrow gap, the variation of the vapor pressure with time, or the variation of the permea— bility due to temperature or relative humidity. A factor left undetermined will be the amount of vapor flowing due to a difference in air buoyancy caused by temperature and vapor—pressure differences. This flow by convection could conceivably exceed the flow by diffusion. The total flow rate for a wall is probably at least as much as that caused by diffusion and therefore gaps should be designed at least to limit diffusion to a safe amount. An experi- mental check of derived vapor-diffusion equations for a gap in air could best be made by having the lightest air above a horizontal Vapor barrier. By considering only diffusion, without any vapor sources or sinks, or any variation in the vapor pressure with time or with the direction along the length of the gap, the vapor pressure, p, will be a harmonic function and the vapor flow rate can be found by solving Laplace's equation in two dimensions, that is, 2 2 9._§+.a__%=0’ 5x by subject to the boundary conditions apprOpriate to the gap. The gaps chosen, the slit, crack, and lap, are real— istic and yet mathematically manageable. A SLIT IN A THIN VAPOR BARRIER The slit, of width f, can be drawn and the boundary conditions stated as follows: / 1) 2) ,Y vapor flow line constant pressure line p2 x vapor barrier Pi Figure 2. Vapor flow through a slit 52p 52p -—5 + ——E = 0 (all x and y); Ox by Op __ _—, o (x<-—“/2, x>f/2, y=0); 5y 5) the vapor pressures, pl and p,, are Specified L for two of the constant-pressure curves appearing in the solution of the problem. The problem can be solved using conformal mapping. The slit, constant vapor pressure and vapor flow lines as shown in the complex z-plane are mapped by an analytic function into simpler lines in the w—plane. z—plane w—plane Figure 5. Mapping of a slit into an infinite strip The pressure, gradient, and vapor flow rate are simple functions of u and v, and these may be transformed by the analytic function into the less simple functions of x and y. Some of the properties of conformal mapping described by Churchill (Ref. 6) are: l) 2) 3) A harmonic function, H(u,v), remains harmonic under a conformal transformation, 2 = F(w), where F(w) is analytic and dz/dw # C. A boundary condition H = c, a constant, in the w—plane transforms to H = c in the z-plane. If the normal derivative, dH/dn, along some curve in the w-plane equals zero, then dH/dn = 0 along the corresponding curve in the z-plane. lO 4) The absolute value of the gradient of H(x,y) equals the product of the absolute values of dw/dz and the gradient of H(u,v). 5) Rate of flow across corresponding curves in the z-plane and the w-plane is the same. The analytic functions 2 = sinh w, z = cosh w, z = sin w, and z = cos w, one or more of which are usually illustrated and discussed in a book on complex-variable theory, could be used to solve this problem. The function 2 = cosh w will give the particular orientation shown in Figure 3. A transformation function for a lap, which may not be listed, will be derived later by means of the Schwarz—Christoffel transformation. If the mapping function is z = %f cosh w = sf cosh(u + iv) = %f(cosh u)(cos v) + éfi(sinh u)(sin v), then x = §f(cosh u)(cos v), y = éf(sinh u)(sin v), x2 #12 %fgcosh2u + fifZSinhZu = 1’ and X2 - 2 = 1 ~ 2 2 . if cos v if Sln v 11 Lines «here u is a constant are semiellipses in the z-plane and lines where v is a constant are hyperbolas in the z-plane. Figure 4 is an example of the constant- pressure and flow lines for vapor diffusing through a slit. In the w-plane the pressure p is related to u and the points where the pressure is known by p-p p-p . l = 2_ ui or, u‘ui 1“2 1 P ‘ P pl“ P p = u u2- ul — ul u2— u1 + pl' 2 l 2 1 For an ellipse of constant pressure the major axis is f cosh u, and the foci are at f/2 and -f/2. The variable u in the above equation may be replaced by a function of x and y for u = cosh.l % [WV/(x+§f)2+ y2 + W//(x—%f)2+ y2 ]. For x = 0, v = “/2, y = %f sinh u and u = sinh-12y/f = ln [2y/f + W//(2y/f)2+ l ]. In Figure 5 the pressure, p, is plotted versus y, the distance perpendicular to the vapor barrier measured from the center of the slit, for three values of the slit width f. 12 2 "/4 1% “/8 Figure 4. Flow and constant-pressure lines for a slit 13 pfian map on asHSOfinsompwm mocmpmfic H o _ _ r . w u _ a W . . l‘il‘l.l‘l .I. 'II‘II."I "I‘i‘LF’l ‘. IIJIII’QII It"! 'I‘t it.l.ll'.'.ll\l’luv.lzllilal.\ll."i |+.!||V.An II.1|.. ,‘ . ‘ I I. ‘alt. ‘ .tt‘l v- 1":.|.|.1TI‘ ‘Ofiél‘;.lt. Ilfllll'l...‘..:-’u'll|v" Hflflhufinfllafllumhamumm .mpfins mpmupflnp< .m Suva; pfiam esp mo mosHm> Hapo>mm «om paam s cu hmH50accmmnmm monoumwu so onsmmmnm poms> no mommccmmmm .m mpswfim Tm eJnsseJd JodsA 14 The vapor pressure changes rapidly near a narrow slit and then slowly farther away. If the point at which a particular pressure occurs is known only generally, on either side of a narrow slit, a point can be assumed with little error in the calculated pressure at any other point. The vapor in a stud wall air Space on one side of a vapor barrier having only a narrow slit will be at about the same pressure, except near the slit. Thus the required permeance of a vapor barrier, calculated for plane flow through the wall, will be correct for a vapor barrier when some of.the vapor flows through a narrow slit. The absolute value of the gradient, bp Op dw _+i__ Ox 6y P1 u - ul H v N I dz -l [%f sinh w] [22- ] ‘tl, becomes infinite as z approaches %f or —%f, so that |P2- pl “1 ‘pZ' pl 'u2’ u1 vapor flows faster near the edges of the slit than near the center. This suggests that reducing the width of the slit will not reduce in direct proportion the vapor flow through the slit. 15 The vapor flow rate in the w—plane depends on the permeability of the medium, the gradient, and the width of the flow region. Thus (with sample units) P ' P % = D Egt‘fil n, where 2 l % = flow rate per unit length of slit (gr/hr-ft), D = permeability of the medium (gr—in/hr-ft2-in Hg), = gradient (in Hg/in), and W = width (ft). Flow rate across corresponding curves in the two complex planes is the same. In the z-plane this can be expressed in terms of the slit width, f, and the vapor pressures, pl and p2, at the points yl and y2 taken perpendicular to the vapor barrier from the center of the slit.‘ Thus W D (p2' pl) M... . L... 2 2 . 2Y2 4y2 4y1 2y1 f f f f If y2 = -y1 and both are called y, and if 1 is negligible compared to (2y/f)2 then 2 ln(4y/f) is: L 16 A CRACK IN A THICK VAPOR BARRIER Flow rate through a crack in a vapor barrier of thickness g, as shown in Figure 6, can be approximated by adding a resistance for the nearly plane flow between the edges of the vapor barrier to the resistance of the nearly hyperbolic flow on either side. An exact solution is discussed in Appendix C. [\‘J ..,___.‘_OQ “'1‘.“ C< __.__... << H r.” Figure 6. Vapor flow through a crack Let V be defined by the equation M/L = (V/L)(p2— pl). Then Luz +_§_+B_1_, V "52 D3 le where u2 and ul may be found for y2, yl, and f by u = ln [ay/f + W//(2y/f)2+ l J. 17 D D and D5 are the permeabilities of the materials on 1’ 2’ either side of and between the edges of the crack. If they are all equal they can be called D. Then if y2 equals yl and they are called y, the specific flow rate for a crack, l % ln [Zy/f 4 W//(2y/f)2+ l ] + g/f X. LD Figure 7 is a plot of this equation. For the range of variables of Figure 7 the error in V/LD will be less than about 2% (see Appendix C). To find the vapor flow rate through a crack of width f, in a vapor barrier of thick- ness g, when the pressures p2 and pl are assumed at equal distances y on either side of the crack, find V/LD from Figure 7 for the known values of 2f/y and 2g/y. Then M/L = (V/LD)(p2— pl) . Figure 7 makes finding the flow rates through slits and cracks easy and also aids in a general discussion of these gaps. First, the thickness of a thin sheet has little effect on the flow rate through a slit unless the slit width is about the same or less than the sheet thickness. Second, if it is necessary to make V/LD less than say 0.2, bringing the butt edges of a thin vapor barrier close together is not a very practical method. On the other hand, making g large is a good way to reduce flow. This can be done by lapping the sheets, as will be 3oz) Ad-.q:o:cd\>c u ._\2 18 62:23.5 32:32:50 £32.28 3 {om £3 {N co o...\> 8 3:09.300 N 959E a \*N 8. .2... o. a} v\_ .898 39.8 35.8% 9539... .. .a-~a .352. 8.28:... 3 £38.58 a v :2 no: 3:33 o..\> £93. .696 :5. .2. 29. Io: u ._\z &'-—>~—>'°-‘—>~—~E if. -., , _.. 7 .. - 22..) A.a-ua:oZan\>v ._\z shown theoretically correct in the next chapter, or by placing an inch or so of the vapor barrier flat against the stud, sill or plate so that there are no gaps of large f and small g. Third, for V/LD le‘s than say 0.1 for a particular crack, y may vary over quite a wide range without changing V/LD much. For example, if f is 1/16- ‘inch and g is 1-inch, then if l/B-inch, V/LD 0.058, and if y 0.050. y 8—inches, V/LD 20 A LAP IN A THIN VAPOR BARRIER Vapor flow through a lap, whose cross-section is shown in the z-plane of Figure 8, can be found by means of a conformal mapping into the upper half of the t-plane. z-plane t—pldne z oo4 k+ih 22 Z1 Z4 0 a: a) L 0 l a) 23 22 t4 t1 t2 t3 t4 Figure 8. Mapping of a lap into the upper half-plane The analytic function required can be found by the Schwarz-Christoffel transformation, by which the interior of a polygon is mapped into the upper half-plane. z—plane t—plane Figure 9. Mapping of a polygon into the upper half-plane 21 The Schwarz-Christoffel transformation is (Ref. 6) -a -—a. -a _ l 2 n-l where A and B are complex constants, t1, t2, . . . tn-l are points on the real axis of the t plane corresponding to 21’ 22, . . . zn-l’ the successive vertices taken so that the interior of the polygon is to the left when moving around the boundary, and Hal, na2, . . . "an—l are the exterior angles. The image of An is tn=co. Two of the constants t1, t2, . . . tn—l can be chosen arbitrarily. The remaining n-3 constants and A and B must be determined to fit the polygon. Now going back to Figure 8, the constants t2 and t. 5 are arbitrarily O and 1, t4 is at cm, and t1 is to be determined. The exterior angles at 21 and 23 are both —n, and the angle at 22 is 2n, which is the change in direc— tion required to pass from the direction 212 at y = h, 2 to the direction 2225 at y = 0, when going around the boundary of the polygon at Z2. 22 Thus the required transformation is N I - A/r(t-tl)(t)'2(t-l) dt + B -l A [t - (tl+1)ln t - tlt ] + B A [t - (tl+l)lnlt|— i(tl+l)arg t - tlt—l] + B. When 2 is positive, real, and infinite, then t is positive, real, and infinite. Therefore the imaginary part of A and so also of B must be zero. When 2 = O, t = 1, and A [l - t1] + B. C II When 2 = k+ih, then t = t1, arg t = w, and so k = A [tl- (tl+l)ln|tl|- l] + B, and Thus -h h(l—tl) A =-———————, B = ———————, and "(tl+l) "(tl+l) h t tlt'l 1-121 2 = —’ ln t - + + , w tl+l tl+l tl+l where tl is related to h and k by the equation, nk (—tl+l) —— = inltll— 2 ———————- h (—tl-l) The function t = ew maps the upper half of the t-plane into the region OSEVSQW in the w-plane in which constant-pressure lines can be represented by u equals a constant and vapor—flow lines by v equals a constant. Expressing z in terms of w and t1, which from here on is called t, _ w -w _ h, _ e te t 1 Z ‘ w _w t+l + t+l ' l ] ' a > _ h _ e cos v + i sin v 7 n _u + 1v t+l — te'u(cos v — i sin v) _ t—l ] t+l t+l Thus, x = g [u + 02:1v(te u_ eu) - %§% ] and = h v _ sin v2(-y1> (—t)h2 Now a lap can be compared with a crack. Flow through a crack is M _ 13(92‘ pl)" L ’ u2+ ul + ng/f' If y2 and y1 are greater than 3f, then with error less than 1% 16(y )(y ) , fig _ 2 1 fig u2+ u1 + f — ln f2 -+ f . 26 If (y2)(yl)/(f2) for a crack equals (y2)(-yl)/(h2) for a lap, then for equal flow rate through both gaps 2 2 fig _ W C:t'1) r " 1“ l6(—t) -’ where t is related to k and h by wk _ —t+l T-lnl‘tI-Zfi. Figure 11 is a plot of (g/f - k/h) versus k/h for equivalent cracks and laps. To the right of k/h = l, (-t) becomes very large and (g/f - k/h) approaches =l|!\> n 2 _ ‘When (g/f — k/h) = -k/h, at k/h = -O.8, then g/f = o. ZNegative values of g/f are not allowed. When the equivalent g/f is found for a lap, then ZEdgure 7 can be used to find the flow rate. Conversely, ‘the dimensions of a lap may be found which will limit 11he flow rate to an acceptable level. 27 492 I i i 192 I x2 5 ' l T y2 ; I ' i T l + <— --—> h g *f" T k i“ I ' T 1 i y1 i P1 ipl g/f f k/h _1_1_ 1 ,1 MW. '0 1,- m ___L...” ___ k/h -1 _ f 1 lFEigure 11. The relation between equivalent g/f and k/h 28 COMPARISON OF THE THEORY WITH PUBLISHED REASUREM NTS Flow through slits in aluminum foil was reported by Babbitt (Ref. 3). The foil was between two pieces of plasterboard or fiberboard or was backed on one side with plasterboard. Tests were made in a chamber used to test permeances of building materials. Air in the chamber was circulated with a fan so that the vapor pressure would be known at the surface of the material being tested. The units used by Babbitt have been changed to those used in the rest of this thesis. Calculated flow rates are compared with experimental flow rates in Table I. For the tests of slits between two sheets of Islasterboard, the permeability of the plasterboard, D, vvas l2.8-perm-in; the vapor pressure difference, p2- pl, evas 1.05-in Hg; and the thickness of the plasterboard, y, {V38 0.4l-in. The theoretical flow rate is given by w D (92' pl) 2 1n [2y/f + 'V/(zy/f)a + 1 ] "(12.8—gr-in/hr-ft2-in Hg)(ft/12-in)(l.05—in Hg) 2 1n [0.82/f + WV/(o.82/f)2 + 1 ] For the slits between two sheets of fiberboard, D tflE l waaes 30—perm-in, p2- pl was 1.05—in Hg, and y was 0.51-in. For the slits backed on only one side with plaster- Table I. Flow rates through slits as reported by Babbitt (Ref. 3) compared with theoretical values width, .009 .017 .028 .045 .072 .103 .147 .004 .019 .034 .045 .056 .065 .086 .120 .155 .016 .031 .047 .063 .079 in flow rateLgr/hr-ft experimental .71 .85 .98 1.09 1.20 1.56 1.46 1.03 1.37 1.44 1.42 1.45 1.65 1.82 1.78 2.36 1.31 1.51 1.67 1.80 1.90 theoretical .33 .38 .43 .48 .56 .63 .73 a slit between two sheets of fiberboard .64 .87 1.00 1.07 1.14 1.19 1.29 1.44 1.58 .77 090 1.01 1.09 1.17 exp.[theo. a slit between two sheets of plasterboard NNNNNNN OU‘INNWNH HHHHHHHHH mm-p-e-wwbmox a slit backed on one side with plasterboard HHHHH ova-s~rq 29 30 board, pd- pl was 1.07-in Hg. Values for permeability and thickness were not stated but will be assumed to be 'l2.8—perm-in and 0.4l-in as before. In these tests y1 = 0, ya = 0.41-in, and "(12.8)(1/12)(l.07)-gr/hr-ft 1n [c.e2/r + ‘\//(o.s2/f)2 + 1 J The values of f/y would have to be increased about HIS ten times, for the narrow slits, to make theoretical values in Table I agree with experimental values, so error in the measurement of f and y are probably not the cause of the discrepancy. From the variation in published measurements for the permeabilities of various building materials (see Appendix B), it is possible that the permeabilities of the fiberboard and the plasterboard could have been higher by a factor of 1% or 2. Vapor leakage was reported by Joy (Ref. 4) through cracks in painted plaster and through laps in sheet steel. The test cells were ones used to determine permeance of 12-in diameter building—material specimens. The air was static and at a temperature of 70.7°F. The vapor pressure was measured about three inches away from the specimens. Leakage, which includes diffusion and convection, was reported for a pressure difference of one inch of mercury. A painted, l/2—inch thick plaster panel, which had a permeance of 0.47-perm, was broken and reassembled with 31 the two halves separated l/l6—in. Leakage reported was 2.04—gr/hr for the crack horizontal and 2.55-gr/hr for the crack vertical, which was stated to be more because of convection. The permeability of air (from Appendix A) is 143/12-perm-ft or 11.9-gr/hr—ft-in Hg. So 2(1/16)/3 = 0.042, 2(1/2)/5 = 0.33, 2f/y Zs/y and from Figure 7, V/LD = 0.088. Thus for the crack, M aV/LD)(L>1-75, and the flow rate, M = K A dc/dn, where A is the area perpendicular to the gradient of the concentration, dc/dn. The value listed for K is for the diffusion of air and water vapor into each other. For vapor concentrations small compared with the concentration of air, K is correct for water vapor diffusing into stationary air (Ref. 11). Vapor pressure can be used instead of concentration to find flow rate, by substituting p/RT for c, where R is the gas constant for water vapor. Then M = K/RT A dp/dn. Let D = K/RT be the permeability of air to water vapor. The units for D can be changed to perm-in, the units often used for building materials, as follows: At At Ko pdl 42 2 0.220-cm lb—T 5600—sec ft sec 85.8-ft-lb hr (50.5)2-cm2 7000—gr lb/in2 l728—in5 1b 2.056-1n Hg rt5 59000-gr—in-T = 59000-perm-in-T. hr-ftz-in Hg 460°R : 0°F, and for standard total pressure, K/RT = Ko/RT = 59000—perm—in-T/460—T 128-perm-in. 530°R = 70°F, , / I . ‘— K/RT = (K./RT)(T/T.)1 7’ (59000/550)(530/460)1‘75-perm-in 143-perm—in. 45 APPENDIX B. THE PERMEANCE OF BUILDING MATERIALS The permeance of vapor barriers is of significance mainly in comparison with the permeance of other building materials. If condensation is to be prevented at a point in a wall, the rate of vapor flow toward and away from the point must balance. Since the vapor-pressure difference across the siding is smaller than across the rest of the wall, the rest of the wall must have a lower permeance than the siding. The following list shows why about one perm is the maximum allowable permeance of an applied vapor barrier. The unit for permeance is perm, which is a short name for grain/hour—square foot-inch of mercury. The permeability in perm-in is the permeance in perms multiplied by the thickness in inches. In the column headed by humidity, the relative humidity is given for the air on either side of the material as it was tested, first for the high vapor pressure side andthen for the low. Some of the tests included a temperature difference across the material. The references selected give a description of the test. 44 .mH .mH .ma .Nn .mo .mm .Hm .mNH .ONH :fiIEpmm .mom hpaaanmmahmm monomaamm xpfidfissm .ma N.H .HH m.m .MH H.H O.H m. .om .mm .NH .Nm o.v .om .mH Shmm mmlov OMIOOH Onlooa omIOOH OMIOOH OOHIOm oumn OIOm 030m 010m OIOm Olmb OmIOOH OMIOOH Olmh Olmh Oimh m¢IO® omnooa OmImb Dumb omnooa mnlmm w ¢M.H mum. mum. mun. mum. Fm. mm nmo£¥oane Sana Edmmzw co umpmsam nocflnm\a pCHMQ Ham; pmHm mumoo 03p .Hmaanq ”moo one mafia npma admmhm no pmpmmHm pcwmm assaazan mpmoo 03p mafia npmH coo; do nopmmaq npma 6003 no hmpmmam nocfits\n Hmpmwan pqfimm wean omen Hwo pmoo one asqfiasHm moanmpmfi pmoo mac mafia psfimm pea“ omen Hfio psoo ego .HmHMmm can Hmaapm psoo mqo mafia pnwmm mmmn Hmpnsh mpmoo 0:» mafia upwonaam; admmzw whmonhmpmmam unamq Hams pmam manoo 0;» mafia pmoo mafinq mac .unmonpmnHM oomMASm one no vamnmmm msam euaopumnfle omszMm Hoe; goon mofiw .:30© soam nomm> .momqm Ham assesses ¢H NH. 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No. ¢.v moo. mm. mm. o.H O.H ma. m.m .mm .wm .om anon monomaumm Hlooa Humm omlooa elem atom Onlooa omlooa OmIOOH 010m mumw Omlooa omlooa Omlmb Re apflaaesm ma ma Om wm mm Om Om m» Os om N pm100m\na phase; boom .noefinsoo.o moom moooa moom womb momm .eoefiumoo.o .aafio memflaepmaaoa noefinsoo.o Momma so .nosfinmooo.o momma so .zocfinmmooo.o .Hfiom Bdnfiasam Momma wcflnpmmsm umpmoo was ompMHSPMm uaonmms amass mcfizuanm paom uopmcmonmefi pamnmmm Amman wcflnpwmnm pmmam umuflm camOH Momma msflnpnmsm qflmop cop Hsflnmpos APPENDIX C. THE EXACT SOLUTION FOR A CRACK An exact solution for the vapor flow rate through a crack can be found by using transformations derived by Davy (Ref. 18). Figure 13 shows corresponding points and lines for the transformations: _ ais ai w Lon s)(dn 5) Z ' b ' K _ 2E—m,K [2Z(°) + (sn 5) J’ t = ns 8 w = ln t Figure 14 indicates the error incurred when the flow rate equation for a crack found on page 17 is used. Several values of 3 along the negative imaginary axis were chosen for each of several values of m. Correspond- ing values of w, z, and b/a = g/f were found using numerical values from references 19 and 20. The values of exact/approximate were found by dividing the difference between values of w by the corresponding values of u2 + u1 + ug/f assuming flow along straight lines between the edges of the crack and flow along semihyperbolas on either side of the crack. 4s z-plane s—plane F J B = -G = b+ia A = J = O I = —B = K g (l+m)K'-2E' a 2(2E—m,K) G = K-iK' t—plane w—plane E D C B A I I I I I I I I I I l l A J F G H I J l i _ = mE H = m4 G = ln m2 I = l B = -l I = O B = 1" Figure 15. Mapping of a crack into a rectangle, the upper half—plane, and an infinite strip 49 scape m smsopzp mush scam you soapmzdm opmafixonmmm esp ea nonpm .sa onsmwm ‘ _ - ' 0,4 \ >3 N —Ir—R I I I0 N I I I I + O flI—H I : i NH® O H moo.m ¢om. owm. sad. memooo o .LIH.H m\m pcfloq mamafixomeM\pomxm OQIX l) 2) '3) 4) 6) 8) 9) 10) x. 1 Q T vs T r. «.1. 'fi ‘0 REFERENCES Dill, R.S. and H. V. Cottony. Laboratory observa— tions of condensation in wall specimens. U. S. Dept. of Commerce, National Bureau of Standards, Report EMS 106. 1946. Reichel, R. C. Moisture and the durability of wood- frame walls. Housing and Home Finance Agency, Division of Housing Research, Housing Research Paper No. 16. 1951. abbitt, J. D. The diffusion of water vapour through bi a slit in an impermeable membrane. Can. J. Res., Sect. A, 19: 42. 1941. Joy, F. A., E. R. Queer, and R. E. Schreiner. Water vapor transfer through building materials. Penn. State 001., Eng. Exp. Sta. Bul. 61. 1948. Yeaple, F. D., Jr. The water vapor transmission test cell. Unpublished M. S. Thesis. State College, Penn., Penn. State Col. Library. 1948. Churchill, R. V. Introduction to complex variables and applications. New York, McGraw-Hill Book Co., Inc. 1948. Babbitt, J. D. Osmotic pressure, semipermeable membranes, and the blistering of paint. Can. J. Tech. 32: 49. 1954. Kuzmak, J. M. and P. J. Sereda. The blistering of paint in the presence of water. Can. J. Tech. 33: 67. 1955. iite, S. C. and J. L. Bray. Research in home humid- ity control. Purdue Univ. Eng. Exp. Sta., Research Series No. 106. 1948. Boynton, W. P. and W. H. Brattain. Interdiffusion of gases and vapors. International Critical Tables. 13) 14) 15) 16) 17) 18) 19) 2O 5: 62. 1929. Pfalzner, P. M. On the flow of gases and water vapor through wood. Can. J. Res., Sect. A, 28: 389. 1950. Teesdale, L. V. Comparative resistance to vapor transmission of various building materials. Trans., Amer. Soc. Heat. Vent. Eng. 49: 124. 1945. Babbitt, J. D. The diffusion of water vapour through various building materials. Can. J. Res., See. A, 17: 15. 1939. Barre, H. J. The relation of wall construction to moisture accumulation in fill-type insulation. Iowa State 001., Agr. Exp. Sta., Res. Bul. 271. 1940. Teesdale, L. V. Condensation problems in modern buildings. U. S. Dept. Agr., Forest Service, Forest Products Laboratory, Report 1196. 1955. Bell, E. R., M. G. Seidl, and N. T. Krueger. Water— vapor permeability of building papers and other sheet materials. Trans., Amer. Soc. Heat. Vent. Eng. 57: 287. 1951. Doty, P. M., W. H. Aiken, and H. Mark. Temperature dependence of water vapor permeability. Ind. Eng. Chem. 38: 788. 1946. Davy, N. The field between equal semi-infinite rectangular electrodes or magnetic pole pieces. Lond. Edin. Dublin Phil. Mag., Ser. 7, 35: 819. 1944. Milne—Thomson, L. M. Jacobian elliptic function tables. New York, Dover Pub., Inc. 1950. Airey, J. R. Toroidal functions and the complete elliptic integrals. Lond. Edin. Dublin Phil. Mag., Ser. 7, 19: 177. 1935. ROOM USE ONLY.