Water vapour transport through perforated foils

Transcription

1 Water apour transport through perforate foils Wim an er Spoel TU Delft, Faculty of Ciil Engineering an Geosciences, Section Builing Engineering, Steinweg, 68 CN Delft, The Netherlans ABSTACT: An analytical moel is presente that escribes the water apour permeability of a perforate foil in epenence of pinhole imension, egree of perforation an water apour permeability of the contacting meia. A comparison of analytical solutions with numerical results inicates a goo agreement when the egree of perforation is not too large. The ifferences are smaller than 5 % in case the ratio of pinhole surface area to total area is less than 5 %. Experimental ata obtaine by others are generally well escribe by the moel, but ifferences are seen for foils attache to a soli material. INTODUCTION In builing practice, foils are use for waterproofing in the builing phase, raiant barrier an apour retarer. In many cases, except the latter, the foil shoul be open for water apour transport to preent conensation. This can be accomplishe by perforation. Stanarize tests are performe to measure the water apour permeability of such foils. In these tests (e.g. DIN 53) the foil forms the separation between two air olumes at equal an constant temperature but at ifferent relatie humiity. In practise, the foil is generally loosely attache to another material or rests e.g. on the roof boaring. The practical apour permeability may therefore eiate from the measure alue. In this paper a simple analytical moel is presente for the water apour permeability of a perforate foil is in epenence of perforation characteristics an the water apour permeability of the contacting materials, an compare with measurement results. The influence of a small air layer between a less permeable material an the foil, air moement along an through the foil, water apour permeation an capillary water transport in a contacting material are iscusse. ANALYTICAL MODEL. Transport equations We limit the attention to isothermal iffusion of moisture in a material in the hygroscopic regime. The moisture transfer may then be escribe by Fick s law: δ j = p () with j the water apour flux ensity (kg m - s - ), δ the water apour permeability of air (, s at 0 C), the ratio of the water apour resistance of the material an air (also known as the -alue) an p the partial water apour pressure (Pa). Although the μ-alue may epen on the moisture content of a material, it is assume to be constant. Using the equation of continuity, we fin for stationary conitions: p = 0. () At an interface between two materials we hae: n j, = n j, (3) where n is the normal ector an j, an j, are the water apour flux ensity in the ajacent materials.. Problem omain We limit the attention to a apour-tight foil with perforations at regular positions. An example is shown in Figure where the typical istance between the gaps is r, which equals ( m) with m the number of gaps per unit surface area. For other gap istributions, we efine r ( m). The analytical moel may not be applie if the istance between the gaps

2 aries too much. For instance, it not suppose to be ali in case the istance between the rows in Figure iffers much from the istance between the columns. r r g Figure : Example of a regular istribution of gaps in a foil. The istance between the gaps is r an the raius of each gap is r g. z f z=0 r g r=0 meium r,s r,s meium r r foil Figure : Schematic representation of the cylinrical problem omain. A apour-tight foil with thickness f is situate between two meia with thickness (lower sie) an (upper sie). The foil has a cylinrical gap with raius r g. The problem omain raially extents to r, hale of the typical gap-to-gap istance of the perforate foil. For the moel eelopment, consier a hypothetical cyliner with raius r an its axis of symmetry perpenicular to the foil surface an through the mipoint of one of the gaps. We impose a cylinrical (r, z, ϕ) coorinate system, as shown in Figure for releant areas in the (r, z)-plane. The angle ϕ may be ignore ue to rotation symmetry. The bounaries in the z-irection correspon to the thickness of the (homogeneous) meium at a sie of the foil, enote as an for the lower an upper sie, respectiely. The thickness of the foil is f..3 Bounary conitions The problem omain is simplifie by isregaring transport in the region between the otte square an enclose circle in Figure. As a result, the hypothetical bounary at r = r in Figure is suppose to be impermeable, i.e. p/ r = 0 at r = r. The inuce error by isregaring this region is small an will be iscusse later. Furthermore, the partial water apour pressure at the lower (z = 0) an upper omain bounary (z = + f + ) is suppose to be constant at p, an p,, respectiely..4 Moel escription In analogy with an electrical circuit with a series of ohmic resistors, the water apour transport may be escribe by a series of regions each haing a specific resistance. This resistance equals the ratio of the water-apour pressure ifference oer the region an the corresponing apour flux ensity an epens on the geometry an -alue. Seen (homogeneous) regions are istinguishe in Figure : three regions at each sie of the foil an the gap itself. Because of symmetry, the lower sie is only consiere here. At a larger istance (>r ) from the gap, the transport in this first region is not much influence by the foil. The apour transfer is assume to be oneimensional (along z-axis) an may therefore be isregare for ealuating the apour resistance of the foil. The secon region is consiere to exist between two spheres with raius r g an r,s from the point (0, ) situate just below the centre of a gap. We efine r,s r when r, an r,s when < r. Assuming transport with spherical symmetry, the transport resistance is gien by, (4),s = πδ rg r,s where μ is the μ-alue of meium. The thir region lies between the spherical surface with raius r g an the cylinrical surface of the gap opening (0 < r < r g, z = ), where the transport eelops from spherical to cylinrical symmetry. A simple analytical expression for the resistance can therefore not be gien. Following the approach by Heiss (954) an Lange et al. (000), the resistance of this transition area is estimate at:,t =. (5) 3πδ rg

3 The two influencing regions at the upper sie of the foil are moelle in a similar way. Finally, the resistance of the cylinrical gap itself is escribe by =, (6) g f g πδ rg where μ g is the μ-alue of the gap an f the thickness of the foil. The total transport resistance tot of the region influence by the foil is gien by summation of fie resistances. After some rearranging, we fin 5 5 g f t = + + πδ 6rg r,s 6rg r,s rg.(7) Effectie μ-alue of a perforate foil For a simple one-imensional or numerical calculation of water apour transfer in a construction, it is practical to know the effectie apour resistance (μ per f ) of a perforate foil. In principle, for oneimensional transport we may write for the effectie total resistance: r + + r =. (8) per f tot,eff πδ r Equating with Equation (7) yiels: 5 r,s perf = r + 6rg r,s r 5 r,s g f + + 6rg r,s r rg. (9) Usually the gap-to-gap istance will be much larger than the gap raius, or r >> r g. If furthermore r an r (i.e. r,s = r,s = r ) a goo approximation is: r 5 5 g f perf = + +. (0) r g 6 6 r g where the thir term at the right-han sie may be neglecte for thin foils in proportion to the gap raius (an μ g = ). The effectie resistance of a thin foil is therefore ) proportional to the sum of the μ- alues of the meia at both sies of the foil, ) quaratically proportional to the gap-to-gap istance an 3) reersely proportional to the raius of the gaps. The first emonstrates that the apparent apour resistance of a perforate foil can not be regare inepenent of the apour resistance of the contacting meia. Howeer, in many cases the thir term at the right-han sie of Equation (0) shoul not be ignore. 3 COMPAISON WITH NUMEICAL CALCULATIONS To gain some insight in the aliity of the aboe iealise an simplifie analytical moel a comparison has been mae with numerical calculations of iffusion through a pinhole for the geometry as shown in Figure. The comparison focuses on the effectie resistance of the perforate foil an has been performe for seeral ariants, inoling the parameters μ, μ,,, r g an r. The consiere alues are gien in Table. The thickness of the foil is constant, f = 0.3 mm, an the gap is air-fille (μ g = ). Due to symmetry only 6 of the 6 ariants are principally ifferent. The ratio of the numerically an analytically calculate effectie resistance μ per f of these 6 ariants as calculate with Equation (9) is shown in Figure 3, sorte by this ratio. It is seen that for a small group the numerical alues are significantly larger than estimate with Equation (9). It appears that this group inoles all cases for which both r g = mm an r = mm. If we isregar this group, see inset in Figure 3, the ifferences between the analytical an numerical alue are within 9 %. Table. Values for μ, μ,,, r g an r use in the calculations. parameter alue # alue # alue #3 unit μ 30 μ mm 8 3 mm r g 0.5 mm r 8 3 mm ratio numerical an analytical alue for per f Variant, sorte Figure 3. atio of numerically an analytically calculate effectie iffusion resistance of perforate foil for the parameters shown in Table. Inset: enlargement of ertical axis for cases with r g = mm an r = mm left out. efinement of the numerical gri i not influence the results.

4 These results show that the analytical moel agrees well with the numerical moel, except when the gap iameter is not much smaller than the gapto-gap istance. In that case, more or less linear transport occurs instea of spherical transport as assume in the analytical moel. These cases are characterise by a large egree of perforation n, efine as the ratio of gap area to total area: π r n =, () ( r ) g for a istribution as shown in Figure. For r g = mm an r = mm the egree of perforation is 0 %. As the next largest egree of perforation in the preious set of calculations is. %, some aitional calculations were performe with 0.0 < n < 0.. These results, together with the preious ata, are shown in Figure 4 as a function of n. There is a clear correlation between the calculate ratio an the egree of perforation an it is seen that the ifferences are smaller than 5 % for n < 5 %. We may therefore conclue that Equation (9) is a goo escription for the effectie μ-alue of a perforate foil for a egree of perforation < 5%. ratio num. an anal. alue for per f egree of perforation / (-) Figure 4: atio of numerically an analytically calculate effectie iffusion resistance of a perforate foil as a function of the egree of perforation. Next to the ariants shown in Figure 3, aitional results are shown for which 0.0 < n < COMPAISON WITH EXPEIMENTAL ESULTS 4. Measurements by Schüle & eichart In this section we compare the analytical moel with experimental results obtaine by Schüle & eichart (980) who measure the apour flux through perforate metal plates for arying egree of perforation an gap iameter. Their 'first' set-up consiste of cups with silica gel, coere with either circular (90 mm iameter) or rectangular (0 x mm ) mm thick metal plates with, 5 or 9 gaps regularly istribute oer the plate, with gap iameters of, 5, 0 or 0 mm, place in a room at 3 C an 50 % H at practically still air (air elocity < 0.05 m s - ). The apour flux was erie from the rate of mass increase of the silica gel. The egree of perforation for reporte ata aries between 0.04 % an 0 %. In Figure 5 the moelle (left axis) an experimental water apour flux 3 normalise to Pa partial apour pressure ifference ( p, -p, = Pa) is plotte against each other (soli circles). For clarity, the relatie ifference (%, right axis) between these quantities is shown by open squares. The mean relatie ifference is -4 % with a maximum of 6 %. Although there is some tenency for a positie ifference (moel larger than experiment) for a smaller apour flux ensity an ice ersa, it may be conclue that the moel, consiering its accuracy of about 5 %, compares well with these experimental results. moelle flux / (kg s - Pa - ) experimental flux / (kg s - Pa - ) Figure 5. Left axis: analytically calculate water apour flux plotte against the measure flux (soli circles). ight axis: relatie ifference (%) between the two alues (open squares). A secon series of experiments focusse on the influence of the thickness of the metal plates f, i.e. on the gap epth. These measurements were one for,, 5 an 7 mm thick plates containing gap with a iameter of 5 mm. The comparison with the moelle apour flux is shown in Figure 6. Here a systematic ifference between moel an experiment is obsere. The moel starts to oerestimate the measure alues for larger plate thickness. For a 7 mm plate, the moelle alue is a factor.5 higher than measure. This obseration is quite opposite to expectations. As the apour transport in the gap is linear one-imensional an the iffusion resistance With = = 40 mm an = =. 3 Experimental uncertainty not reporte ifference / (%)

5 of the gap (last term at the right-han sie of Equation (9)) starts to ominate for gap epths > 5 mm, the iffusion resistance is more well-efine from the moelling point of iew. The cause of the ifference is therefore not clear an probably not ue to shortcomings of the moel. moelle flux / (0 - kg s - Pa - ) mm 5 mm mm mm moelle flux / (kg s - Pa - ) 0-0 mm 5 mm 0 - mm 0 mm 5 mm 0-3 mm experimental flux / (kg s - Pa - ) experimental flux / (0 - kg s - Pa - ) Figure 6. Analytically calculate water apour flux plotte against the measure flux (soli circles) through a 5 mm iameter gap for ifferent gap epth f, inicate in the graph. A thir series of experiments by Schüle an eichart concerne the effect of two isolating materials. Measurements were performe with 30 mm mineral wool (MW, =,6) an 0 mm expane polystyrene (PS, = 37) attache to one sie of a mm thick perforate plate containing 5 gaps with, 5 an 0 mm gap iameter. In Figure 7 the moelle flux is plotte against the measure alues for MW (soli circles) an PS (open circles). The ifferences between measurement an moel are consierable for mineral wool (<50 %) an large for polystyrene where ifferences of about a factor 4 to 5 are obsere for all ata points, the moelle alues being lower than measure. This may partly be explaine by a H-epenent -alue of the isolating material. A large part of the apour resistance is etermine by transport in the isolating material close to a gap where ry conitions (H < 0 %) preail, which may eiate from the humiity conitions at which the apour resistance of the material is etermine. The apour resistance is known to ecrease with ecreasing relatie humiity for some materials. Whether this hols for PS has not been erifie. Another explanation might be the presence of a thin air-layer between the metal plate an isolating material. Numerical calculations inicate that a 0.3 mm air layer (an no leakage at the sie of the cup) rises the apour flux by a factor 3 to 5. Unfortunately, these suppositions can not be erifie. Figure 7. Analytically calculate water apour flux plotte against the measure flux for a configuration with 30 mm mineral wool (soli circles) an 0 mm expane polystyrene (open circles) attache to a perforate metal plate with 5 gaps for ifferent gap iameter r g, inicate in the graph. 4. Perforate foils The imensional parameters r g, r an f of three commercially aailable perforate foils hae been measure an use in the analytical moel to calculate the water apour permeability (at p = 400 Pa). This preicte alue is compare with the measure alue cite by the representatie. The characteristics of the polyethylene foils an the preicte an measure water apour permeability are gien in Table. The preiction significantly unerestimates the measure alue for foil # whereas there is a reasonable corresponence for foil # an a slight oerestimation for foil #3. The uncertainty in the preicte alue for foil # is howeer large ue to a strong eformation of the polyethylene near the perforations which results in tortuous gaps that are much longer than the foil itself. This impees an accurate measurement. To a lesser extent this also accounts for foil #3, which may explain the oerestimation. The unerestimation for foil # may be cause by apour transport through the foil material. Table. Measure foil parameters an preicte water apour permeability of three commercial perforate foils. Uncertainty in preicte alue reflects uncertainty in gap raius an length. Foil r g m r mm f m measure permeab. preicte permeab. (g m - 4h - ) (g m - 4h - ) Foil # * ± Foil # # ± 5 Foil # $ ± 5 * 9000 gaps m -, # 3800 gaps m -, $ gaps m - rough estimates, foil thickness is 00 μm

6 5 DISCUSSION To what extent the simplifications in the analytical (an numerical) moel are permissible for application uner practical conitions, is iscusse. Iealisation of the problem omain Both the analytical an the numerical moel assume an impermeable bounary at the cylinrical wall r = r. This proceure isregars transport in the region between the (otte) square an the inner circle. The inuce error is howeer small for a perforation egree < 5 %. Instea of r as typical raius, we coul hae chosen r, corresponing to a circle with surface area equal to the otte square, i.e., r.3 r. For a perforation egree n = 5 % the effect on the total iffusion resistance is only %. The analytical moel also isregars the transport resistance of the area between the surface z = r,s an the hale sphere with raius r,s, see Figure (mutatis mutanis for the other sie of the foil). In this area the iffusie transport switches oer from linear to spherical symmetry. Since this transition area is correctly escribe by the numerical moel, there is no effect on the ata shown in Figure 3. It may howeer explain a calculate ratio < for a large part of the ata in this figure. Vapour permeability of the foil material In practice, synthetic foils are to some extent permeable to water apour whereas Equation (9) is only applicable if the apour flux through the foil material is much smaller than the flux through the gap. For a first-orer estimate of the ratio of these fluxes we assume two inepenent parallel transport paths through the foil material an the gap. If we impose the Dirichlet bounary conitions at the foil surface, the total effectie μ-alue of a perforate foil is approximate by eff = + per f, () where μ f is the μ-alue of the foil material. This is a lower-boun estimate of the effectie resistance, i.e. the true flux through the foil material will be somewhat lower. Air flow through the foil The moel has been erie for stagnant air. But, air will flow through the gaps in response to airpressure ifferences oer the foil. Typical air pressure ifference foun in practise ue to win an thermal buoyancy are of the orer of 0 Pa. To estimate the effect of an air flow through a perforate foil, we consier the Peclet number of the flow, characterising whether the water apour transport is iffusion ominate (Pe < ) or aection ominate (Pe > ), efine as charchar Pe = (3) D0 where char an char are a characteristic elocity an istance of the flow an D 0 the iffusion coefficient of water apour in air (approx m s - ). As yet, the foil is thought to be situate between air layers, i.e. μ = ; Consier again Figure. As a result of an airpressure ifference pa between the surfaces (z = 0) an (z = + f + ), a flow rate Q is inuce through the gap. The air flow through the foil can either be 'creeping' (iscous) or turbulent. In case of creeping flow, we hae to a ery goo approximation (Dagan et al., 98; Sisaath et al., 00): pa Q =, (4) 3 8f rg π r g where μ the iscosity of air ( Pa s). Viscous forces ominate for eynols number <, efine as ρcharchar e=, (5) where ρ is the air ensity (. kg m -3 ). Combining with Equation (3) gies ρd 0 - e = Pe = Sc Pe, (6) where Sc is the Schmit number (0.6 for air at 0 C) which shows that iffusion will ominate the transport of water apour for creeping flow conitions. It is howeer not so easy to specify the characteristic elocity an istance for the problem at han. At first instance, we might take the elocity in the gap an gap raius. But, since the transport process is largely goerne by the conerging an ierging flow fiel at some istance from the gap, we take, as a trae-off, the flow elocity at the spherical surface with 'characteristic' raius r g from a gap. The aerage flow elocity at this sphere is = Q, (7) char π ( rg ) an therefore, using Equation (4): ρ e= ρr p charchar g a = π f g ( + r ) 4 3 8( / ). (8) For example, iscous issipation an therefore apour iffusion will ominate at 0 Pa pressure ifference for gap raii smaller than 60 μm for a foil of thickness f = 00 μm. For these parameters an e.g.

7 a egree of perforation of 0.0 %, the flow rate through a unit area of perforate foil is 0. L s -, a reasonable alue from a practical point of iew. For larger gap raii conection may ominate epening on the pressure ifference oer the foil. For turbulent flow we may use the well-known orifice equation as an approximation (r >> r g ) to calculate the flow rate: p Q= Cπ rg. (9) ρ where C is the ischarge coefficient which is approximately 0.6 for high eynols number (Bir et al., 960). As the apour transport is ominate by conection in this flow regime, the apour flux J (kg s - ) through the gap may then be approximate by: p J = Q (0) T w where p is the partial water apour pressure at the high air-pressure sie of the foil, w the gas constant of water (46 J kg - K - ) en T the temperature. The aboe equations o not hol for foils attache (at one sie) to an air-permeable material. If we assume that Darcy's law hols for air-flow in such a meium: K qa = pa () where q a is the air flux ensity (m 3 m - s - ) an K (m ) the permeability of the meium, there is a mathematical corresponence with Equation (). If we further assume that the resistance to air flow is completely etermine by the material, we hae for the air-flow rate through the gap, following Equation (0) for r >> r g (an r ; r ): 6Krg p a q = () 5 so, using Equations (3) an (7), the Peclet number is 3 K pa Pe =, (3) 0π D0 For an extremely open meium (e.g. coarse san) K = m an μ, we fin Pe = 0. at 0 Pa pressure ifference. As for other enser materials, the prouct μk usually ecreases, it is conclue that iffusion will nearly always ominate the apour transport mechanism for foils attache to a soli material. Air flow along the foil For air flow along the foil we use the lateral elocity u x at a istance r g from the foil as the characteristic elocity. This elocity howeer strongly epens on the circumstances. For a foil expose to the climate, win gusts may locally impose large elocities in absence of a eelope bounary layer. On the other han, in practise perforate foils are usually installe in a caity an are thus sheltere from the win. To obtain some quantitatie iea, consier one-imensional laminar flow in a m wie caity for which the lateral flow elocity u x is escribe by y y ux = 4 umax ; y, (4) where y is the perpenicular istance from one of the sies an u max is the maximum elocity, at y = /. At a istance y = r g an r g <<, the Peclet number is 6umaxr Pe = g. (5) D0 For = 0.0 m an u max = 0. m s -, we fin Pe = for r g = 0,56 mm. This inicates that iffusion will ominate in most circumstances because the gap raius is usually smaller an higher elocities are not expecte to occur frequently. Exceptions may concern caities that are well entilate with outoor air. Note that air flow along an attache foil will normally hae no influence on the apour flux as the largest iffusion resistance lies in the soli material. Connection with the substrate In practise perforate foils are often loosely attache (staple) to a substrate material, which results in a small air layer (plenum) between the foil an the material. Such an air layer has a strong effect on the effectie apour resistance of the foil. Numerical calculations inicate that for a mm air layer between a foil an a material with μ = 30, the apour resistance is ecrease by about an orer of magnitue. Eiently, the analytical moel can not be use for such configurations. As moreoer in practise the air-layer thickness will ary oer the foil, the escription of the apour transfer becomes ery ifficult an three-imensional (a problem that oes not arise for non-perforate foils). An accurate escription of the apour transfer through a perforate foil can therefore only be gien for a perfect connection to a substrate or if the air layer thickness is a least about the size of the gap-to-gap istance. Water transport in the substrate The moel assumes that the moisture transport in the substrate is ominate by apour transfer, i.e. that the moisture content, for a porous material, is below the critical alue. If the moisture content is aboe that alue, the moisture transfer is ominate by capillary water transport which is normally much faster than apour transfer. In that case, an effectie

8 apour resistance of the foil may be erie base on the assumption that the moisture transport in the substrate oes not contribute to the total transport resistance. It is then allowe to use Equation (9) with the μ-alue of the substrate set to zero. It shoul howeer be remarke that this simple iew may not reflect reality. As the material may ry quickly near a gap (the apour flux ensity is locally higher than for an uncoere surface), it is conceiable that the capillary transport is not fast enough to keep up with the transfer rate. As a result, a new equilibrium may set in where the moisture content becomes sub critical near the gap. To what extent this happens epens on the moisture transport characteristics of the porous material. In any case, capillary water transport will lower the effectie resistance of the foil. 6 CONCLUSIONS A simple analytical moel has been eelope that preicts the water apour permeability of a perforate foil in epenence of the gap raius, gap-togap istance, foil thickness an the apour resistance of the substrate material (if present). Accoring to this moel, the effectie resistance of a thin perforate foil is, uner most circumstances, linearly proportional to the μ-alue of the substrate material(s), quaratically proportional with the gap-to-gap istance an reersely proportional to the raius of the gaps. A comparison with more precise numerical calculations of iffusie water apour transport shows that the moel preicts the permeability within 5 % if less than 5 % of the foil surface is perforate. A similar agreement is obsere in comparison with measurement results obtaine by others for water apour transport through thin perforate metal plates. Somewhat larger eiations are obsere for measurements with thicker perforate plates but this is probably not ue to weaknesses of the moel. Very large ifferences of about a factor 5 are howeer seen for perforate plates attache to a material with a μ-alue of 37. It is speculate that this might be cause by a thin air-layer between the plate an the material. Furthermore, the moel reasonably preicts the water apour permeability of one commercially aailable perforate polyethylene foil but unerestimates the permeability for another by a factor 3. This comparison shoul actually inclue more foils to obtain a better insight. There are circumstances in practise that iolate the moel assumption of pure iffusie transport. A key factor in this context concerns the air-flow elocity through an along the foil. Although iffusie transport will ominate in most cases, there are some configurations for which conectie transport will hae an influence epening on the climate conitions. Air flow along a perforate foil will always enhance the apour transport through the foil, whereas air flow through the foil may also ecrease the apour flow epening on the flow irection. Another outcome of the moel is the strong influence of the apour resistance of the substrate on the effectie apour permeability of the foil. As the water apour resistance of a perforate foil is usually measure with air at both sies of the foil, it is not allowe to use this alue for calculating the apour resistance of e.g. a wall or roof assembly if the foil a tightly attache to a substrate. EFEENCES Bir,.B., Stewart, W.E. & Lightfoot, E.N Transport Phenomena. John Wiley & Sons: New York. Dagan, Z., Weinbaum, S & Pfeffer,. 98. An infinite-series solution for the creeping motion through an orifice of finite length. J. Flui Mech. 5: Heiss, Verpackung feuchtigkeitsempfinlicher Lebensmittel. Springer: Berlin. Lange, J., Büsing, B., Hertlein, J., an Heiger, S Water Vapour Transport through Large Defects in Flexible Packaging: Moeling, Graimetric Measurement an Magnetic esonance Imaging. Packag. Technol. Sci. 3: Schüle, W. & eichart, I Wasserampfurchgang urch Öffnungen. Zeitschrift für Wärmeschutz, Kälteschutz, Branschutz. WKSB-Sonerausgabe August 980: -6. Sisaath, S., Jing, X., Pain, C.C, Zimmerman,.W. 00. Creeping Flow Through an Aximsymmetric Suen Contraction of Expansion. J. Fluis Eng. 4:

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