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D
Jarl-Gunnar Salin
Prediction of Heat and Mass
Transfer Coefficients for Individual
Boards and Board Surfaces.
A Review
Paper presented at the 5* International lUFRO
Wood Drying Conference, Quebec City, Canada,
August 13-17,1996
Trätek
P R E D I C T I O N O F H E A T AND MASS T R A N S F E R C O E F F I C I E N T S F O R I N D I V I D U A L B O A R D S A N D B O A R D S U R F A C E S . A R E V I E W
Paper prescFited at the 5"^ International l U F R O Wood Drying Conference, Quebec City, Canada, August 13-17, 1996
Trätek, Rapport 1 9702017 ISSN 1102- 1071 ISRN TRÄTEK - R - - 97/017 - - S E Nyckelord air flow heat transfer kiln stack mass transfer liiiiher drying Stockholm februari 1997
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Reports issued by the Swedish Institute for Wood Technology Research comprise complete accounts for research results, or summaries. sur\'eys and studies. Published reports bear the designation I or P and are numbered in consecutive order together with all the other publications from the Institute. Extracts from the text may be reproduced provided the source is acknowledged.
Trätek — Institutet för träteknisk forskning — be-tjänar de fem industrigrenarna sågverk, trämanu-faktur (snickeri-, trähus-, möbel- och övrig träför-ädlande industri), trätlberskivor, spånskivor och ply-wood. Ett avtal om forskning och utveckling mellan industrin och Nutek utgör grunden för verksamheten som utförs med egna, samverkande och externa re-surser. Trätek har forskningsenheter i Stockholm. Jönköping och Skellefteå.
The Swedish Institute for Wood Technology Re-search serves the five branches of the industry: sawmills, manufacturing (joinery, wooden hous-es, furniture and other woodworking plants), fibre board, particle board and plywood. A research and development agreement between the industry and the Swedish National Board for Industrial and Technical Development forms the basis for the Institute's activities. The Institute utilises its own resources as well as those of its collaborators and other outside bodies. Our research units are located in Stockholm. Jönköping and Skellefteå.
Sammanfattning
Torkningsprocessen kan ses som två resistanser i serie. Den interna resistansen är kopplad till fliktrörelsen mot ytan. Den externa resistansen består av värme- och fuktöverföringen mellan ytan och torkluftens huvudström. Den interna resistansen beskrivs med difiusionskoefficienter och har studerats utförligt, medan den externa delen har givits mycket mindre uppmärksamhet i trävetenskapen. Det är mycket begränsad information som finns om värme- och massöver-föringskoefficienter för luft strömmande genom ett normalt ströpaket. Emellertid innehåller den omfångsrika litteraturen om värmeöverföring en mängd information rörande mer eller mindre liknande flödesgeometrier. Denna artikel presenterar sådan införmation för flera fall. Först ges en översikt över fiillt utbildat flöde mellan parallella plattor. Det framgår emellertid att inloppsfenomen dominerar och att fullt utbildat flöde sällan uppnås. En metod att förutsäga värmeöverföringskoefficienter för individuella virkesbitar (horisontella ytor) längs flödesvägen har utvecklats. Metoden beaktar också inverkan av springan mellan intilliggande bitar. Vidare studeras värmeöverföringskoefficienterna för vertikala ytor.
Ströpaketets ändar, där varannan virkesbit saknas, studeras slutligen genom jämförelse med en tubsats-värmeväxlare.
Massöverföringskoeflficienten beräknas normalt ur värmeöverföringskoefficienten, genom att använda analogin mellan värme- och massöverföring. Rapporterade mätningar indikerar emel-lertid att analogin inte gäller för träytor. Detta fenomen belyses och en förklaring samt intro-duktion av en korrektionsfaktor föreslås.
Prediction of Heat and Mass Transfer Coefficients for
Individual Boards and Board Surfaces. A Review
J - G . Salin Swedish Institute for Wood Technology Research
P.O.B. 5609, S-114 86 Stockholm, Sweden
ABSTRACT
The drying process can be viewed as two resistances in series. The internal resistance is connected to tlie migration of moisture towards the surface. The external resistance consists of heat and moisture transfer between the surface and the drying air bulk flow. The internal resistance is described by diffusion coefficients and has been extensively studied, whereas the external part has been given much less attention in wood science. There are very limited direct information on heat and mass transfer coefficients for air flowing through a normal kiln stack. How-ever, the vast heat transfer literature contains a lot of information on more or less similar flow geometries. This paper presents such information for several cases.
First fully developed flow between parallel plates is reviewed. It is however found that entrance phenomena dominate and that fnlly developed flow is seldom achieved. A method to predict the heat transfer coefficient for individual boards (horizontal surfaces) along the flow path is thus developed. The method takes also into account the influence of the gap between adjacent boards. Further, heat transfer coefficients for vertical surfaces are studied.
Finally the ends of the stack, where every second board is missing, is studied by comparison with a tube bundle heat exchanger.
The mass transfer coefficient is normally calculated from the heat transfer coefficient, using the analogy between heat and mass transfer. Reported measurements indicate, however, that the analogy is not valid for wooden sur-faces. This phenomenon is reviewed and an explanation and introduction of a correction factor is suggested
INTRODUCTION
The wood drying mechanism can be viewed as two drying resistances in series. First, the moisture has to migrate from the centre part of the board towards the board surface - internal resistance. Second, the mois-ture that evaporates at the board surface has to be transferred to the bulk of the drying air flow - external resistance. This requires a lot of heal which in turn is transferred from the bulk flow to the board surface. The external resistance is thus composed of two coupled parts, heat and mass transfer. The internal resistance is
described by moisture diffusion coefficients and the external resistance is described by convective heat and mass transfer coefficients.
In wood science the internal resistance has been the dominating area of interest and tlie external resistance is practically ignored. The rapidly increasing number of drying simulation models developed in many countries (Kainke et al, 1994) has required formulas for heat and mass transfer prediction. The formulas used are nor-mally very simple, such as
h = (1) the centre pari of a stack where every second board is located flush with one end of the stack. where the air velocity, v, has to be expressed in in/s and
h is the heat transfer coefficient, or theoretically more correct versions, such as
Nu = 0.0228 Re"-^ Pr*^^ (2) where Nu, Re and Pr are the Nusselt, Reynolds and Prandtl numbers respectively. When these equations (or similar) are used, if is assumed that the heat transfer cocfTicicnt. for a given total air flow, is constant throughout the timber stack and that heat and mass transfer from horizontal surfaces only, need to be con-sidered. This is far from the actual situation, as will be shown in this review.
The internal resistance is proportional to the square of the board thickness, whereas the external is directly proportional to the thickness. This means that for thick dimensions the internal resistance is dominating. For thin dimensions, however, i.e. < 25 mm (often con-taining much easily dried sapwood), the external resis-tance will dominate and a more sophisticated approach than Eq.(l-2) is needed for modelling drying of the whole stack.
Perhaps the first serious attempt to analyse the lit-erature on flow pattern and corresponding transfer coefficients in a timber stack was done by Ashworth (1977). It was found that
• Entrance effects in the flow duct between board layers should not be neglected.
• The geometry represented by the kiln stack has received little attention in the literature. However, a number of similar situations have been studied and could be used for prediction of transfer coefficients. The aim of the present work is to extract results from the vast literature on heat and mass transfer, on situations that are similar enough to the kiln stack geometry and present these results in a usable fonn. The presentation will concentrate on heat transfer - the coupling between heat and mass transfer is briefly mentioned at the end of the paper.
BASIC KILN STACK GEOMETRIC CONFIGU-RATION
The basic configuration is illustrated by Figure I . The drying air flows between layers of boards that are stacked edge to edge. This is the nonnal situation in a stack where all boards are of equal length and also in
mm
Figure 1.
The drying air flow chaimel is a rectangular duct between two board layers and two stickers. The aspect ratio of this rectangle is, however, very large - nonnally 20...50 - and the situation can thus also be viewed as flow between parallel plates.
We will first consider fully developed flow in this duct, i.e. it is assumed that all entrance effects have disappeared.
Fully Developed Turbulent Flow
The traditional way to predict heat transfer coeffi-cients for flow in non-circular ducts, is to define an "equivalent" characteristic length, dh , called hydraulic diameter, and then use correlations for circular tubes. The hydraulic diameter is defined as four times the cross section area divided by the "wetted" perimeter. For a circular tube the hydraulic diameter is thus equal to the actual diameter. The heat transfer coefficient is then calculated, for instance, from the Dittus-Boelter equation (Incropera et al 1985)
Nu = 0.023 Re^-^ Pr^"^ (3) where Nu = Nusselt number = hd/ X
Re = Reynolds number = vd/ v
Pr = Prandtl number (equal to 0.70-0.71 for air) h '- heat transfer coefficient
X = air thermal conductivity V = mean air velocity in duct V = air kinematic viscosity
In this case the characteristic length d is put equal to tlie hydraulic diameter.
For flow between parallel plates the hydraulic diameter will be equal to twice the sticker thickness. It is however more logical to use the sticker thickness as such, as the characteristic length and if the almost constant Pr-number is also incorporated into the pro-portionality factor the result is
Nus = 0.018Res-^ (4) where index s indicates that Nu and Re are based on the sticker thickness.
For circular tubes the transition from laminar to turbulent flow occurs for Re about 2300. If this value is adopted for this non-circular case also, it is found that we nonnally can assume turbulent flow for air veloci-ties above I mis. Eq.(3) is considered vahd for Re >
10000 (> 3...4 mis) but can be used with acceptable accuracy for lower values also.
Direct measurements for large aspect rectangular ducts are found in (Haynes et al 1980).
Both theoretical and experimental values for fully developed flow between parallel plates are found in (Özisik et al 1989, Hatton et al 1964, Kays et al 1963) and in references listed in these. Heat transfer in non-circular ducts is reviewed in (Rohsenow et al 1972).
Kayihan (1984) suggests the following correlation
Nu = 5.5 Nu = 0.5029(RePr) Nu = 0.1244(RePr) RePr < 300 0.4194 300 < RePr < 2000 0.6032 2000 < RePr (5) This correlation gives however rather low values (Milota 1994, Salin 1990) and is thus not recom-mended for normal drying air velocities. Correlating values reported in references mentioned above gives for air between parallel plates
Nu = 4.3+ 0.0195 Re Entry Region
0.79 lO"* < R e < 10^ (6)
The bluff entrance and sudden contraction of flow into the stack will cause disturbances. Actually three transients are superimposed, development of the veloc-ity profile and the temperature and humidveloc-ity profiles. This will result in enhanced heat transfer within the entry region. Information on the entry length (parallel plates) is found in (Özisik et al 1989, Hatton et al 1963,
1964) and reviews in (Ashworth 1977, Rohsenow et al 1972). It seems that these disturbances penetrate at least 10 hydraulic diameters into the stack - in some cases 20...30 diameters. For a sticker thickness of 25 mm, this corresponds to 0.5-1.5 m. It is thus obvious
that assuming fully developed flow will not give an
acceptable result.
The flow pattern at the entrance depends certainly also on the thickness of the boards (the board/sticker thickness ratio determines the flow contraction). It is tltus worth to study heat transfer for parallel flow over a flat plate with a blunt leading edge.
Flat Plate(s) in Parallel Flow
The flow pattern past a single truncated slab is indi-cated in Figure 2. A stationary eddy is formed just be-hind the leading edge. The eddy causes large heat transfer coefficients at the point of reattachment "A", which is located 1 ...4 board thicknesses from the edge.
lurbtHenc* , . 01 Bounaory
A Point of r « a t t a c h m e n l
Figure 2. Flow pattern over a tmncated slab
The single plate case has been investigated by (Sörensen 1969, Ota et al 1974,1979, Kottke et al 1977a,b, Danckwerts et al 1962, Malmquist 1964) among others. (In several cases the mass transfer coef-ficient has been measured, but a conversion to a heat transfer coefficient can be done based on the analogy between heat and mass transfer). Results for a single plate are valid for a real stack only close to the leading edge. Further in, the board layers above and below will influence the flow pattern.
Measurements for an array of timber boards have been reported by (Kho et al 1989. Langrish et al 1992,1993, Milota 1994, Langrish 1994, Keey 1994, Keey et al 1994, Pang et al 1995, Pang 1996). Figure 3 is an example from tlie first mentioned paper. As is seen, there is a maximum heat transfer value some distance from tlie leading edge, in the same way as for a single plate. There is also an increase at each gap between adjacent boards, which will be analysed in more detail below.
The heat transfer coefficient for a single truncated slab was thoroughly investigated by Sörensen (1969) and correlaUon equations were established. It is Uius of interest to apply the same methods of correlation also to
« MO no 100
Figure 3. Local mass transfer coefficients as a function of the distance from the leading edge for the first three boards. (Khoetal 1989)
the results obtained for multiple slabs arranged in a way similar to a kiln stack. Such an approach gives the following result
should be mentioned. Only one naphthalene coated "board" was used in the test, and this board was moved from location to location. Figure 3 thus consists of three different measurements. This means that the mass transfer maximum after the gap (Fig.3) has two differ-ent reasons:
1. Increased turbulence caused by the gap and/or by
boards not being exactly level.
2. Undeveloped naphthalene concentration profile. If for example the naphthalene board is in the third location then the velocity profile has developed over two boards before the concentration profile starts to develop.
In a kiln stack, velocity, temperature and air humidity profiles all start to develop at the leading edge of the stack. Only the first of the mentioned transfer enliancement reasons is tlius valid in practice. Figure 3 thus gives an exaggerated picture of the influence of the gap between boards.
0.182 RePr'^^
( R e ^ - R e ^ y for Re^ / Re^ > 1.5
(7)
Re A =29.7ReS^^
where RCz is Reynolds number based on the distance from the leading edge, Rcd is based on board thickness and Re^ is based on the distance from the leading edge to the point of maximal heat transfer. The velocity used is the air mean velocity in the gap between board layers. Eq.(7) are valid for distances from the leading edge that are above about 1.5 times the distance to the reattachment point. For shorter distances the method proposed by Sörensen can be used.
In this way a good fit is obtained with measured values of the local Nusselt number across the first board. This is probably true also for much wider boards than those used in the experiments. However, as is seen from the mathematical form of Eq.(7), the heat transfer coefficient decreases towards zero for very long dis-tances from the leading edge. This is of course not correct as the limiting value is given by Eq.(6). This property is however no limitation when the influence of the gap between adjacent boards is taken into account.
Influence of Gap Between Adjacent Boards
As is seen in Figure 3, the local heat (mass) transfer coefficient increases substantially just after each gap between boards. One important detail regarding meas-urement values, as for instance presented in Figure 3,
By estimating the part caused by undeveloped pro-files (Incropera et al 1985) the remaining interesting part can be calculated. The result was tliat the gap between adjacent boards causes a 6% increase in the heat transfer coefficient. This is an average value - a wide gap will have higher impact.
Now this "6% rule" should be incorporated into Eq.(7). The following procedure is suggested. The distance from the leading edge can in Eq.(7) be laken as a measure of how far the development of profiles has proceeded. The disturbance caused by the gap between boards thus implicate that the effective distance has decreased by a factor approximately equal to 1.06'"°^''^ = 0.845. Eq.(7) can thus still be used if the actual dis-tance from the leading edge is replaced by an effective distance, which is corrected when a gap between two adjacent boards is passed. It is found that tliis method after a few boards in tlie direction of the air flow will reach an "equilibrium ' so that the local heat transfer coefficient variation will repeat itself for the next board. This is in accordance with results presented in (Langrish et al 1993,1994, Keey et al 1994, Langrish
1994, Pangetal 1995).
In this way the local heat transfer coefficient can be estimated for horizontal surfaces along the air flow path for boards stacked edge to edge. By integration a mean value for each board can be obtained and ftirther a mean value for the whole kiln stack also. A computer code, HTCOEFF, has been developed for these simple but time consuming calculations.
As an example, calculated mean heat transfer coef-ficients for the horizontal surfaces of individual boards in the flow direction are given in Table 1. The stack width is 1.5 m, sticker thickness 25 mm and tempera-ture level 60°C.
Table 1. Calculated heat transfer coefficients in the centre part of the stack.
Dimension 25 X 100 75 X 150 75 X 150 Velocity m/s 3 3 4 W/m-°C W/m-°C W/m-T Board no: 1 41.1 33.2 39.4 2 25.0 22.0 26.4 3 21.0 18.3 22.0 4 19.0 16.6 19.9 5 17.8 15.5 18.7 6 16.9 14.8 17.8 7 16.4 14.2 17.2 8 15.9 13.9 16.7 9 15.6 13.6 16.4 10 15.3 13.3 16.1 11 15.1 12 14.9 13 14.8 14 14.7 15 14.6 Mean 18.5 17.5 21.1 In conclusion the following results should be men-tioned
• The first 2-3 boards have considerably higher heat transfer coefficients than the stack mean value. Many laboratory tests with small stacks arc thus not representing the situation in a full scale kiln slack. • The gaps between adjacent boards have a marked
influence which increase with gap width. It should be observed that the gap increases during drying due to wood shrinkage. If there are several stacks in a load, then the gap between stacks is normally centimetres wide, not millimetres, and it is thus suggested that each kiln stack is considered as being alone when heat transfer values are estimated. • There seems to be no information on heat transfer
coefficients for vertical board surfaces in the gap between boards. Values for outer vertical surfaces of the front and trailing boards are investigated below. • The trailing board seems to have slightly higher
heat transfer values for the horizontal surfaces, probably due to exit flow phenomena. The flow at the exit is not necessarily symmetric with respect to
the flow centre line, as has been found by (Cho et al 1994).
FLOW THROUGH ENDS OF KILN STACK
The centre part of the kiln stack has been investi-gated above. At the ends of the kiln stack every other board is missing, because the boards are all normally shorter than the stack length. There are then two possible geometric configurations aligned and staggered -as shown in Figure 4. We have of course re-asons to suspect that the heat transfer coefficients for these board surfaces differ from the centre part of the stack.
Figure 4. Aligned and staggered board configuration. Figure 4 easily gives an association that tliis flow pattern has to resemble the situation in a normal tube bundle heat exchanger, where, however, the circular tubes are replaced by rectangular "tubes". This gives a basis for an estimation of these coefficients.
Especially in the staggered case, the distance between boards in the front row is rather big. The flow pattern around a board in the front row is probably not influenced very much by other boards in that case. Heat transfer to a single board in cross flow is thus investi-gated first.
Single rectangular cylinder in cross flow
Heat transfer coefficients for flow perpendicular to a rectangular cylinder have been measured in several cases. Most cases are however for square cross sections. (Hilpert 1933, Jakob 1949) wliich report mean heat transfer coefficients and (Igarashi 1985,1986, Goldstein et al 1990, Yoo et al 1993, Tatsutani et al 1993) which give local values also. For rectangles with width/thickness ratios up to 1.5 there is one report (Igarashi 1987) and another for an aspect ratio of 6
(Test et al 1980) that both present local values. Trans-fer values for an impinging jet on rectangular cylinders have been reported, but are obviously not applicable in this case (Hossain et al 1993a,b)
According to these reports, the mean heat transfer coefficient for the vertical front surface of a single board may be calculated from
Nu = 0.71Re^-^ Pr'^^ (8) where board thickness is used as characteristic length and velocity is the approaching velocity of air. Eq.(8) seems to be valid for l700<Re<53000 and for a square, rectangle and infinite plate (Kottke 1977a).
The heat transfer coefficient for the leeward vertical surface may in the same way be estimated from
N u - C R e ^ ^ ^ Pr'^^ (9) where for aspect ratios 1.0, 1.33 and 1.5 the factor C is equal to 0.198, 0.183 and 0.153 respectively. Eq.(9) is valid for Re-numbers from 7500 (aspect ratio 1.5) to 53000 (square). As is found from Eqs.(8) and (9). for Re-numbers above 10000 the heat transfer coefficient is Iiiglier for the rear surface than for the front surface, which perhaps is surprising
Measured values for the horizontal surfaces of a single rectangular cylinder may be expressed by
Nu = 0.126Re^'^ Pr''^ (10) which seems to be valid for aspect ratios 1.33 - 6 and 7500<Re<46000.
Values for all four surfaces of a square cross section for a low velocity (Re=592) have been reported by (Tatsutani et al 1993). These values are considerably lower than corresponding values obtained from
Eqs.(8-10) This indicates that these equations should not be used outside the given limits.
It is assumed in all cases above that the air flow approaches the stack perpendicular to the vertical front surface of the board. This is often not the case in prac-tice. The influence of a different angle of attack may be estimated from results found in (Igarashi 1985,1986, Test et al 1980, Yoo et al 1993).
Multiple rectangular cylinders in cross flow
A few papers have been published on heat transfer to bundles of rectangular cylinders in cross flow. These are reviewed in the following.
Cho et al (1994) have investigated a situation rather similar to the first row in the aligned configuration case (Figure 4). The rectangles are however rotated 90°. A comparison of measured values to those predicted by Eqs.(8-10) show that values for the array are considera-bly higher than for a single rectangle on all four sur-faces. (The ranges of valid Re-numbers are however not overlapping.) This indicates that ail coefficients are influenced as the air is forced into a smaller duct and the velocity is increased.
It is thus suggested that transfer coefficients for boards in the first row of an aligned configuration (Figure 4), where boards are one sticker thickness apart, are estimated as follows. Horizontal surfaces from Eq.(7), vertical front surface and vertical leeward surface from a judgement based on both Cho et al (1994) and Eq.(8) or Eq.(9). In the case of a staggered configuration, where boards in the first row are 2 sticker thicknesses + 1 board thickness apart, the sug-gestion is that Eqs.(8-I0) are used.
There are a few reports on transfer coefficients for two or several rectangular cylinders arranged in a col-linear array in the air flow direction. Cur et al (1978) consider two plates (with rather low thickness/widUi ratios) in series and Cur et al (1979) extend this to eight plates (same aspect ratios). Sparrow et al (1980) investigate several parallel arrays of 10 plates each (still the same aspect ratios). Finally Tatsutani et al (1993) investigate the case of two aligned square cylin-ders for low velocities.
These four reports could be used to estimate tlie influence of upstream cylinders on heat transfer coeffi-cients for downstream cylinders. The results are how-ever contradictory. According to Cur et al (1978,1979) will the second cylinder have a mean transfer coeffi-cient only about 96% of the value for the first cylinder. Sparrow et al (1980) and Tatsutani et al (1993) report, on the contrary, transfer enhancement of 33% and 66% respectively. Influence on the third and subsequent rows is reported to be small.
It is well known that in bundles with circular tubes, there will be a heat transfer enhancement in the flow direction (Incropera et al 1985). The second row has a 30-60% higher value and the third row 50-70% higher
value than the first row, depending on geometric con-figuration etc. There are reasons to believe that the enhancement is lower for rectangular cylinders, but the correct level is thus not known.
It is suggested that transfer coefficients calculated for all four surfaces of boards in the first row, are increased by about 15% (aligned configuration) and 25% (staggered) for the second and all subsequent rows.
It should finally be mentioned tliat bundles of tubes with other cross sections than circular or rectangular have also been studied (Merker et al 1992)
AIR VELOCITY DISTRIBUTION IN THE KILN STACK
The correlations and methods given above, will hopefully provide a basis for estimation of heat transfer coefficients both in the centre and end parts of the kiln stack. Air velocity is a main input value in these calcu-lations. However, air velocities in the centre and end parts are not independent of each other. We can assume that the pressure is constant in the direction of the stack, on both sides of the stack. The air flow through different parts of the stack is thus determined by the flow resistance in each part. To determine the air velocity distribution we have to estimate tliese flow resistances.
In the centre part we can describe the situation as flow between parallel plates with corresponding flow entrance and exit losses. The total pressure drop can be calculated by introduction of the hydraulic diameter and by using standard methods for flow in circular tubes. The end part of the stack is analysed in the same way. We can still describe this case as flow between parallel plates (board layers) although parts of tlie duct walls are missing. Instead we have several duct con-tractions and enlargements tliat cause pressure drop. A reasonable accuracy for the total flow pressure drop can be achieved in this way (Öhman 1996).
Results from such pressure drop calculations show that air velocity (mis) in the stack end region is nor-mally lower than in the centre part. This may seem surprising as there is much more free space towards the stack end, but it is in agreement with measured veloci-ties in full scale stacks (Esping 1977). The air flow (mVs) through the end part depends on the configura-tion (Figure 4). For the aligned configuraconfigura-tion the flow is lower than in the centre. For the staggered configu-ration the flow is equal or higher than in the centre part.
When the velocity distribution has been calculated, then simultaneous heat transfer coefficients may be estimated for different locations in the kiln stack, by using the methods suggested earlier. The calculated example presented in Table 1 may now be extended to the end part of the stack. Table 2 presents such a cal-culation done for the same conditions as in Table 1. (It should be noticed that air velocity here refers to velocity between board layers in the centre part of the stack). Table 2. Calculated heat transfer coefficients in the end part of the stack.
Dimension 25 x 100 75 x 150 75 X 150 Velocity mis 3 3 4 W/m^"C W/m'°C W/m^°C Board no: Staggered configuration 1 20.2 11.8 14.2 2- 25.3 14.7 17.8 mean 25.0 14.4 17.4 Aligned configuration 1 31.2 26.0 31.0 2- 35.9 29.9 35.6 mean 35.3 29.1 34.6 There is a rather clear difference between the stag-gered and aligned configurations, when compared for an equal pressure drop over the stack. The author is not aware of any direct full scale measurements of drying
rates that support or contradict this theoretical result.
MASS TRANSFER COEFFICIENTS
Convective heat and mass transfer can be viewed as two different forms of the same fundamental phenome-non, i.e. the diffusion of heat or mass through a bound-ary layer close to the solid surface. There is thus a cou-pling between the heat and mass transfer coefficients also. The analogy between heat and mass transfer can, in its simplest form be expressed by
p = h / c , (11)
where |3 is the mass transfer coefficient and Cp tlie specific volumetric heat capacity in the boundar>' layer. This analogy is described in detail in all textbooks on the subject. The analogy was used above to transform
measured mass transfer coefficients for naphthalene sublimation, to heat transfer coefficients.
When drying processes are modelled, the heat transfer coefllcient is normally known approximately and then the mass transfer coefficient is calculated from Eq.(ll). This is the normal procedure for all materials and it has been used extensively for wood drying also. However, there is increasing experimental evidence that the analogy is not valid for wooden sur-faces (Plumb et al 1985. Gong 1992, Laurila 1993. Cloutier et al 1992. Morcn et al 1992, Wu 1989. Milota 1994, Samuelsson et al 1994. Rosenkilde et al 1996, Siau et al 1996). These results are reviewed in more detail in (Salin 1994.1996a,b). The observed deviation from the analogy is substantial - one order of magni-tude, in some cases even more. This means that Eq.(l 1) cannot be used for wood. By introducing a correction factor into Eq.(ll). this problem can be partly solved, and this method has been used in a few wood drying models.
The reason for the observed deviation is not known. A promising hypothesis is described as follows. If the air in contact with the surface is not in equilibrium with this surface, then the analogy may still be valid
within the boundary layer, but as we assume
equilib-rium, an apparent deviation is seen for the global transfer process. It has been shown by (Salin
I994,l996a,b) tliat even a small deviation from equilib-rium will have a strong influence on the apparent mass transfer coefficient. This hypothesis is satisfactory as we can still assmne that the fundamental analogy prin-ciple is valid, but ue are instead faced with the problem of how to describe the degree of non-equilibrium.
If the non-equilibrium hypothesis is correct, then one implication is that the mass transfer process has a dynamic nature, i.e. the (apparent) mass transfer coeffi-cient depends on the drying rate itself The correction factor approach is thus not theoretically correct. At the moment, due to the lack of information, the correction factor method seems, however, to be the only possible practical solution.
The so called surface emission coefficient (S) has sometimes been used instead of the mass transfer coef-ficient, especially in wood science. A list of measured surface emission coefficient values reported in the lit-erature has been collected and published in (Söderström et al 1993). This list shows no obvious correlation between measured S-value and desorption/absorption conditions. None of these measurements have been done for flow patterns relevant for a normal kiln stack.
The surface emission coefficient is defined only for steady state conditions and is thus unusable in ad-vanced wood drying models. It is strongly suggested that the surface emission coefficient concept should be avoided in wood drying modelling.
REFERENCES
Ashworth, i . e . 1977: The mathematical simulation of batch-drying of softwood timber. PhD Thesis. University of Canterbury, New Zealand, p.26-38 Cho, H.H.; Jabbari, M.Y; Goldstein R.J. 1994: Mass
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