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408 CHAPTER 23 ft to be independent of vapor pressure (and temperature) along the flow path. Equation 4 would have been obtained, but with ft replaced by ft. The coefficient p is therefore an average permeability coefficient applicable to the varying conditions along the flow path of length l, while the coefficient ft is the spot or differential permeability. Equation 4 may be rewritten and units assigned: W- (5) where W = total weight of vapor transmitted, grains. A n area of cross-section of the flow path, square feet. 0 = time during which the transmission occurred, hours. Ap = difference of vapor pressure between ends of the flow path, inches of mercury. 1 = length of flow path, (or thickness of specimen), inches. P average permeability, grains per (hour) (square foot) (inch of mercury vapor pressure difference per inch of thickness). See Table 1. The basic units given are those now favored by the build ing industry. Whenever it is convenient to deal with a material of a stated or implied thickness other than the unit thickness to winch it or p refer, use may be made of the permeance co efficient M, where M =* p/L The designation perm for the unit of permeance is now widely used, and is a convenient substitute for the unit, 1 grain per (hour) (square foot) (inch of mercury vapor pressure difference).* The corresponding unit of permeability is perm-inch, since it is the permeance of unit thickness. 'Hie corresponding flow equation is: W - MAOCtp (6) where M = permeance coefficient, perms, or grains per (hour) (square foot) (inch of mercury vapor presure difference). See Table 1 for typical values of M and p. Resistance to vapor flow provided by a sheet or board is the reciprocal of the permeance, and correspondingly, the overall vapor resistance of an assembly (like a wall) of ma terials in series is the sum of the resistances of its component parts. The overall permeance of the assembly may be found from the permeances of the individual components4** in a manner paralleling that used in calculating the overall co efficient of thermal conductance from the individual con ductances: . 1965 Guide And Data Book Example 1: A wood frame wall is exposed to indoor conditions 70 F and 50 percent relative humidity (0.37 in. Hg vapor pres sure) and outdoor conditions 0 F and 80 percent relative humidity (0.03 in. Hg vapor pressure). The wall consists of painted plaster on gypsum lath on the inside over 2 X 4 in. studs, mineral.wool nil between studs, 1 in. wood exterior sheathing, paper, and pine lap siding. Check for possible condensation. To simplify the example, consider the paint, plaster, and lath as a tingle dement having a permeance M - 1.0 penns and a thermal conductance C -- 2.4, and the exterior sheathing, paper, aiding, and paint as another single element for which M TM 2.0 perms ana C " 0.50. From Table 1 the value for the permeability of mineral wool fill may be found as p = 116 perm-inches. Die thermal conductivity for mineral wool, i = 0.27. Solution: In this particular wall, insulated and with moder ate warm-tide relative humidities, condensation is unlikely to occur until the sheathing ia reached. To' check directly for condensation on the warm tide of the exterior sheathing (des ignated as plane X -- X in Fig. 2 for convenient reference) proceed as follows: Calculate according to the method of Chapter 24 the tempera ture at plane X -- A. This is found to be 9 F. The saturation vapor pressure at this temperature is 0.06 in. Hg. If condensa tion is imminent or occurring at X -- X, the vapor pressure there will be 0.06 in. Hg. Calculate the permeance for the portion of the wall from the warm tide to X -- X, and the vapor flow rate to X -- X as follows: Permeance of wall to X -- X =* ------- " 0.97 penns. To + 116 Vapor measure drop to X -- X " 0.37 -- 0.06 = 0.31 in. Hg. Vapor flow to X -- X = 0.97 X 0.31 - 0.30 grains per (hr) (aqft). Calculate the vapor flow rate from X -- X to outdoors as Permeance of wall from X -- X to outdoors TM 2.0 perms. Vapor pressure drop from X -- X to outdoors o 0.06 - 0.03 - 0.03 in. Hg. Vapor flow rate to outdoors = 2.0 X 0.03 " 0.06 grains per (hr) (sq ft). It is apparent that continuity of vapor flow is not possible, since at the highest vapor pressure at X -- X permitted by the temperature, mere is indicated a greater flow to plane X -- X than from it to outdoors. Condensation is indicated ,at a rate of 0-30 -- 0.06 ** 0.24 grains per (hr) (sq ft). Whether this condensation rate will be serious must still be decided, since it might readily be absorbed by the Bheathing PLASTER MINERAL WOOL SHEATKNS CM LATH BETWEEN STUDS PAPER 5 SXOUM X Mi (7) k) This simple theory for vapor flow, as in the case of the cor responding simple heat-flow theory, assumes conditions of unidirectional, steady-state flow. Useful calculations can be made for an assembly or sub-assembly for which the inflow and outflow of vapor are equal (a condition at which no con densation occurs) if a permeance applicable to actual condi tions can be assigned to each component part. Overall per meances, vapor pressures and vapor flow can be calculated, and in conjunction with thermal calculations, relative hu midities can be determined, and the imminence of condensa tion predicted. (See Example 1, and Fig. 2. See also Chapter 85 of the 1964 Guide And Data Book for use of vapor-flow calculations.) fig. 2 .... Temperatures and Vapor Pressures under Vapor Flow Conditions in the Insulated Frame ' Wall of Example 1. Moisture in Building Construction ' during the condensation period without excessive wetting, the temperature at X -- X is below freezing, as in this the condensation will be in the form of frost which may acnimnlp** until released over a short period upon a rise in outdoor temperature. Condensation will be reduced or avoided if the permeance of the warm side of the wall can be reduced so that toe flow to V _ X is limited to 0.06 grains per (hr) (sq ft). Permeance 0.06 required for this 37 _ q qq " *19 perms or less- A more resistant pint 61TM on the plaster, reducing the paint-plastcrUtbpermeance to 0.19 perms would accomplish this. When the critical plane for condensation is unknown, or for a more informative, graphical representation of toe situation throughout the wall, temperatures and vapor pressures may he and plotted as in Fig. 2. The vapor pressures throughout the wall, for continuity of flow, are calculated in a wnnw similar to toe temperatures; the external vapor prwairw are given, and the vapor pressure drops across each dement are taken in proportion to resistance to vapor flow. The curve for saturation vapor pressures at toe various temperatures throughout the wall is also shown in Fig. 2 and b seen to fall below the curve for vapor pressures with con tinuity of flow, toward the outer portions of the wail. This indjTM*** that with the given temperatures and vapor pres sure, continuity of flow b not possible and that condensation will occur. Condensation on the sheathing b indicated as a definite possibility^ and the new vapor-pressure curve can be constructed for this condition as shown. Dus new curve does not rise above the saturation-vapor-pressure curve, thereby confirming that the critical plane. for condensation was cor rectly assumed. With toe vapor pressures thus established, the relative humidities may be found, by reference to the saturation vapor pressures. Die permeances originally assigned to toe various dements may then be re-examined in the light of the service conditions of temperatures and relative humidities indicated, and toe analysis repeated,.if necessary, using more appropriate permeance values. In a more detailed analysis, individual values might be assigned to the elements forming the outer portion of too wall which b here dealt with as a composite, homogeneous element. The transmission of water vapor as outlined is based on the assumption of a diffusion process. The possibility of vapor being transferred as part of a moving air stream has thus far been ignored, except in Example 1 in which it is implied that the air circulation on either side of the wall will be sufficient to eliminate surface-film resistances to vapor flow. Differences in total pressure of the air may result in a transfer of vapor with air, augmenting and at times over-riding the effects of RELATIVE HUMIDITY- % Rg. 3----- Relation between Dry and Wet Cup Tests and the Spot Permeability for a Material such as Wood 409 the flow produced by vapor-pressure gradients alone. This can be particularly important in the transfer of vapor through cracks and pinholes or through air-permeable building con structions. Similar effects can be obtained with air-permeable materials themselves.**1 This means of vapor transfer is similar to that of transfer of heat by air leakage in and through building constructions, and requires, for purposes of calculation, information on the nature and amount of the air leakage. It will seldom be im portant in constructions without air spaces and having parged or plastered surfaces. It may, however, be an important means of vapor transfer through constructions lacking hi air tightness, and may contribute to condensation difficulties, since the mechanism of condensation is not dependent upon the way in which the vapor is transferred. PERMEANCE AND TESTING The simplest method of finding the permeance of a speci men is to seal it over the top of a cup containing desiccant of water, placing it in a controlled atmosphere, and weighing it periodically. The steady rate of weight gain or loss is nor mally the water vapor transfer. When the cup contains a desiccant the procedure is called the dry-cup method and when toe cup contains water, the wet-cup method. Usually the sur rounding atmosphere is held at 50 percent relative humidity, thus providing, in either method, substantially the am< dif ference of vapor pressure, but the results obtained by the two methods for the same specimens are likely to be much differ ent, the wet-cup method producing the higher values. Die relationship between these values ftan best be understood by reference to Fig. 3, which shows a typical variation of spot permeability with relative humidity at one particular tem perature (isothermal conditions) for a material such as wood. The vapor permeability is shown to vary only moderately at low humidities, but to increase at an increasing rate as higher humidities are reached. The dry-cup test of this material carried out with 0 percent relative humidity on one tide and 50 percent on the other, will experience a variation in spot per meability throughout its thickness, because of the variation in relative humidity. The average permeability p, is by definition ,/JJ pdp (Equation 3), given by----------- , and since at a fixed tem perature there is a linear relationship between vapor pressure and relative humidity, this expression can be seen to cor respond to the mean height of the area under the spot per meability curve, between the appropriate relative humidity limits. The average permeability as found for the dry-cup conditions should therefore have the value pi. Similarly for the wet-cup test between 50 percent and 100 percent rela tive humidity, the value should be ft. It is not uncommon for these values for wood and wood-fiber materials to be in the ratio of 1 to 3, or higher (See Table 1.) The average permeabilityp for any other relative humidities at a particular temperature is given by the mean height of toe area under the curve of spot permeability for the material at that temperature, between the appropriate limits of rela tive humidity. Only average permeabilities (or permeances) are measurable directly in practical tests. However, if several average permeabilities at different relative humidities are known, and can be plotted as for the wet-cup and dry-cup tests shown in Dg. 3, it is possible to construct, by trial and error, a spot permeability curve which will satisfy the condi tion that the average height of- the curve between the ap propriate limits for each test must equal the value found in each test. Separate curves are required for each temperature so that a large number of permeance cup tests would be re-