Mathematical modelling of capillary micro-flow through woven fabrics

Transcription

1 Composites: Part A 31 (2000) Mathematical modelling of capillary micro-flow through woven fabrics S. Amico, C. Lekakou* The School of Mechanical and Materials Engineering, University of Surrey, Guildford, GU2 5XH, UK Received 17 July 1999; accepted 2 February 2000 Abstract The present work considers the capillary flow of Newtonian fluids along a single fibre yarn and through a plain-woven fabric. In the first case, one-dimensional Darcy s flow is considered through the micro-pores of the fibre yarns. In the second case, two-dimensional in-plane network infiltration is considered through the micro-pores of the network of fibre yarns in the fabric. In both cases predictions include the infiltration length as a function of time, the apparent permeability and the capillary pressure. The latter case also includes the number of unsaturated transverse yarns and the degree of saturation. All predictions are compared to experimental measurements and the agreement is very good Elsevier Science Ltd. All rights reserved. Keyword: Capillary micro-flow 1. Introduction * Corresponding author. Tel.: ; fax: address: c.lekakou@surrey.ac.uk (C. Lekakou). Resin transfer moulding (RTM) has been increasingly used in the processing of composites. In this process, a dry reinforcement or fibre preform is placed in the cavity of a mould, the mould is closed and resin is forced to flow through the fibrous reinforcement. Modelling the flow of resin during the infiltration stage of the RTM process is justified by the importance of this stage on the manufacturing of high quality parts. Furthermore, reliable modelling of mould filling with an accurate flow description has the potential of enabling a proper choice of mould and process parameters, addressing critical issues in the process, such as dry spots and voids [1]. Understanding of the flow requires the knowledge of the permeability of the reinforcement, which is correlated to the velocity of the flow front and the pressure gradient. Flow models usually consider the flow as Darcy s flow [2] through a homogeneous porous medium. Although the validity and applicability of this law through a structured reinforcement have been questioned [3 5], Darcy s law is still by far the most used model to describe the flow of resin in reinforcements during the processing of composites. Permeability is a major property regarding the infiltration of fibre reinforcement in RTM and, hence, many studies in the literature have focused on experiments for the measurement of permeability and the analysis of experimental data in these experiments. Amongst other factors, depending on process conditions during the experiment of permeability measurement, the influence of capillary pressure on the overall flow can be of relative importance. In fact, many of the discrepancies in the literature [6] about using Darcy s law to model the infiltration process in permeability experiments may be due to neglecting the effects of microscopic flow phenomena in the interpretation of the macroscopic flow. Regarding the low injection pressure cases, attempts have been made to explain some of the discrepancies from Darcy s law by using generalised flow models including non-newtonian effects [7,8], capillary pressure effects [9] and separation of the flow into macro- and micro-flow [9 11]. In the latter studies of fibre reinforcements, micro-flow is considered to be the flow inside the fibre yarns and macroflow is considered to be the flow in the macro-pores between fibre yarns. Darcy s law was employed for both macro- and micro-flow [9 11], taking also into account capillary pressures and wetting effects [9]. Some published papers focus on microscopic studies of void formation in composites manufacturing. Void formation has been related to resin impregnation and more specifically to viscous flow and wetting properties of the resin related to the reinforcement. It has also been observed that the measured global permeability may vary, depending on the infiltrating fluid and its wetting properties [12,13]. In flow visualisation studies [14 16] it has been observed that at a low flow rate micro-flow was leading whereas at a high flow rate macro-flow was leading X/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S X(00)

2 1332 S. Amico, C. Lekakou / Composites: Part A 31 (2000) Nomenclature A y cross-sectional area of the yarn (m 2 ) D f fibre filament diameter (m) dh/dt height change with time (m/s) dp/dh pressure gradient (Pa/m) F shape or form factor in the Young Laplace equation g acceleration of gravity (m/s 2 ) H Height of fluid rise (m) H e equilibrium height due to P c (m) H ff height of the flow front (m) H i height of a particular junction i (m) K o Kozeny constant P pressure (Pa) P c capillary pressure (Pa) P i local pressure at a junction i (Pa) Q a,i axial flow rate at junction i (m 3 /s) Q t,i transverse flow rate at junction i (m 3 /s) Q tot total flow rate (m 3 /s) u superficial fluid velocity (m/s) t time (s) X ff horizontal position of the flow front of the transverse flow (m) Greeks a a,i axial coefficient of flow at a network junction i a t,i transverse coefficient of flow at a network junction i DX ff,i horizontal distance between the transverse flow front and a network junction i (m) e porosity of the fibrous preform u contact angle between the fluid and the solid k permeability of the medium (m 2 ) k mi,a micro-axial permeability of the fibre yarn (m 2 ) k mi,t micro-transverse permeability of the fibre yarn (m 2 ) m fluid viscosity (Pa s) n interstitial velocity (m/s) r density of the fluid (kg/m 3 ) s surface tension of the wetting fluid (Pa m) The prediction of permeability and wetting properties of a fibre reinforcement would be a major advance for the design stage of the processing of a composite product, especially for RTM. This study focuses on woven fabrics. As several layers of fabric are laid in an RTM mould, in-plane flow during the infiltration stage is expected both in the macropores between fibre yarns and in the micro-pores within fibre yarns. A general macro- and micro-infiltration model, such as that suggested by Lekakou and Bader [9], is expected to predict global permeability but such models so far can provide only qualitative trends and need further refinement and detailed validation to provide accurate predictions of permeability. Because of the complex nature of such models, which consist of a few sub-models, each sub-model needs to be considered, refined and validated, first, independently. The present study focuses on the micro-flow through a woven fabric. A mathematical model has been developed regarding micro-flow through a two-dimensional (2D) network of fibre yarns in woven fabrics. The term network flow models as encountered in the literature so far, Refs. [17 19] amongst others, describes the flow through a network of tubes and corresponds to pore level simulations. The present study considers Darcy s network flow where the bonds of the flow network consist of fibrous porous medium. It corresponds to micro-flow simulations at a level higher than the micro-pores. The aim is to apply the developed Darcy s network micro-flow model for the prediction of the permeability and capillary pressure in flows through woven fabrics and to validate the model with experimental flow data. In order to be able to isolate the micro-flow through a network of fibre yarns, capillary flow experiments through a single layer of a plain-woven fabric were designed for model validation. This report follows an earlier study [20] which describes the capillary experiments to follow the capillary impregnation of an E-glass fibre yarn and a plain-woven fabric. The experimental procedure and the obtained results have been described in this earlier study. Apart from being

3 S. Amico, C. Lekakou / Composites: Part A 31 (2000) Fig. 1. Photograph of the infiltration of the fabric; (a) at the flow front and (b) close to the liquid surface of the container. one of the sub-models for infiltration of an assembly of woven fabrics in RTM, the model is potentially important in RTM applications with high fibre volume fraction where macro-pores are expected to be very small and the flow through the network of fibre yarns in each fabric layer may be a decisive factor in the infiltration of fabric reinforcement. 2. Modelling of Darcy s flow through a fibre yarn In this model, the fibre yarn is considered as a tow of unidirectional fibres. The aim is to numerically evaluate the vertical advancement of the fluid front along the axial yarn direction. Darcy s law is used to model the flow through the yarn micro-pores, correlating flow-rate and pressure drop according to Eq. (1). u ˆ e dh ˆ k dp 1 dt m dh or n ˆ k me P c rgh H For resin systems which show highly non-newtonian behaviour, a viscosity shear rate relationship such as the power law must be employed [21]. The theoretical value for the capillary pressure is estimated from the Young Laplace equation as suggested by Ahn and Seferis [22]: P c ˆ F 1 e s cos u 3 D f e where F is a dimensionless shape or form factor which depends on the flow direction. For axial flow, it has been 2 derived [23] that F ˆ 4 on the basis of the principle of hydraulic radius of a fibre bed. The micro-pores are assumed to be uniform or with an averaged pore size distribution and fibre twist in the yarn and any defects at the fibre surface were neglected. The analytical solution of Eq. (2) is rather complex [20], requiring an iterative numerical procedure to evaluate the height of fluid rise as a function of time. A procedure based on the finite difference method is employed for the solution of Eq. (2) in this study. The fluid is considered to advance in length steps during the corresponding time steps. At every new step, the time is increased by a constant value and the length step is evaluated. Error estimation monitors whether the length step is appropriate, i.e. error smaller than a given tolerance. Otherwise, the length is re-evaluated until numerical convergence. The model was implemented in a simple computer code written in Fortran77. This program allows for different Newtonian fluids by inputting the density and viscosity values. The fibre radius, micro-porosity (i.e. porosity of the yarn), micro-axial permeability and capillary pressure are the other input parameters. The Carman Kozeny equation [23], given by relation (4), is generally used to estimate the value of micro-axial permeability on the basis of the value of micro-porosity. k ˆ D f 2 e 3 16K o 1 e 2 4 The value used for the Kozeny constant, K o, is discussed in Section 4.1. It is important to remember at this point that Gauvin et al. [24] stated that flow predictions are as good as the accuracy of the permeability value entered in the simulation.

4 1334 S. Amico, C. Lekakou / Composites: Part A 31 (2000) Fig. 2. Simplified representation of the network flow in the plain-woven fabric. The value of capillary pressure was best estimated from the Young Laplace equation (3), for specific values of micro-porosity, the surface tension of the liquid (measured by using a DuNuoy ring apparatus [20]) and the contact angle between the liquid and the E-glass fibre (measured by the single fibre pull-out test using a dynamic contact angle analyser, DCA-322 CAHN, which measures the force exerted on a fibre slowly lowered in a beaker containing the fluid [20]). The capillary pressure is the driving force responsible for the rise of the liquid in the yarn. As the infiltration starts, the weight of the column of liquid in the yarn acts against the capillary pressure, slowing down the rate of infiltration (dh/dt). The numerical time step used in the computer simulations was usually of 1 s. However, if the final equilibrium position of the flow is required, this time step may be increased so that less computational effort is used. 3. Modelling the vertical capillary flow through a plainwoven fabric It is necessary to understand the flow pattern in the plainwoven fabric in order to realistically model the flow. Fig. 1 shows an experiment of capillary infiltration of a vertical piece of plain-woven fabric, Y0212 supplied by Fothergill Engineered Fabrics, at the flow front (Fig. 1a) and close to the liquid surface of the container (Fig. 1b). Three major macroscopic features can be noticed: 1. The flow front is not homogeneous, with some yarns showing further flow advancement. Since other factors concerning the flow are supposed to be constant, this is likely to happen due to characteristics of the individual yarns, especially the pore distribution of individual yarns. 2. The flow progresses axially in the vertical yarns whereas the horizontal yarns are filled in such a way that there are at least three partially filled transverse yarns at any time. The precise number cannot easily be estimated by visual observation of the flow alone. 3. There are clear unfilled gaps between the yarns (macropores). These gaps can be seen even in the portion of the fabric closest to the liquid surface of the container (Fig. 1b) and were confirmed by careful visual observation of the infiltrated fabric. Therefore, the vertical capillary infiltration process through a plain-woven fabric can be modelled considering the macro-flow to be absent. Since the model only considers the presence of microflow, the reinforcement is represented as a network of porous fibre yarns linked at junctions. Fig. 2 shows a schematic representation of the fabric with its repeating basic column. Fig. 3 describes the pressures acting at a single junction i. Flow in the transverse yarn is driven by the capillary pressure, acting at the flow front at a horizontal distance X ff. Flow in the axial yarn is driven by the capillary pressure acting at the flow front (H ff ) but, in this case, the weight of the column of liquid above the junction originates a counter pressure P i at this i junction (H i ), where P i ˆ rg H ff H i : The axial flow-rate, Q a,i, and transverse flow-rate, Q t,i,ata given junction i will then, respectively, be: Q a;i ˆ A y k mi;a m Q t;i ˆ A y k mi;t m P c rg H ff H i H ff H i P c rg H ff H i DX ff;i 5 6

5 S. Amico, C. Lekakou / Composites: Part A 31 (2000) Fig. 3. Flow advancement and pressures at a given junction i. where DX ff;i is the horizontal distance from the flow front to the junction i and H ff H i is the height difference between the flow front and the junction. At every junction the flow splits between axial and transverse yarns according to the coefficients defined by Eqs. (7) and (8), where a a;i a t;i ˆ 1: a a;i ˆ Q a;i Q a;i Q t;i Q t;i a t;i ˆ 8 Q a;i Q t;i A similar analysis is carried out for each unsaturated junction (with a transverse yarn that has been reached by the axial flow front but has not been completely filled yet) throughout the duration of the experiment. For instance, if for a given time there are three unsaturated junctions (Fig. 4) according to mass balance, the total flow rate at that time will be: Q tot ˆ Q t;i Q t;i 1 Q t;i 2 Š Q a;i 2 9 Thus, it can be said that for a junction i the flow split will be given by: Q a;i ˆ Q a;i 1 a a;i 10 Q t;i ˆ Q a;i 1 a t;i 11 The transverse flow will be responsible for filling the transverse yarns whilst the axial flow will proceed until the next junction is reached and at this time a new flow split will occur. A few models can be proposed, regarding the unknowns a and a t. Model 1: a is the same at all unsaturated junctions throughout the infiltration process; this model ignores the effect of the flow advancement in the axial and transverse direction on the flow split. A value for a is selected at the beginning of a computer simulation and the flow-rate is split at junctions according to Eqs. (10) and (11). This is the simplest of the three models and a is considered really as an empirical constant, being estimated on the basis of the values of the axial and transverse micro-permeabilities as a first approximation. Accurate values of a can only be provided from the fitting of the final infiltration predictions 7 Fig. 4. Schematic representation of the flow distribution for three unsaturated junctions, junctions i, i 1 and i 2: to the experimental data. The use of this model in this study is to improve the understanding of the Darcy s network flow through the woven fabric. In the following Models 2 and 3, the coefficients relevant to the flow split at junctions are evaluated on the basis of relations (12) and (13) which can be derived directly from Eqs. (5) (8): a a;i ˆ k mi;a DX ff;i k mi;a DX ff;i k mi;t H ff H i 12 a t;i ˆ 1 a a;i 13 More specifically: Model 2: a is the same at all junctions but varies with infiltration time; this model considers an averaged flow advancement length, expressed by average values of H ff H i and DX ff;i over the region of unsaturated junctions near the flow front. This model is expected to give good predictions of a if the length of the unsaturated region near the flow front is small in comparison to the total length of infiltration in the axial direction. Model 3: a varies depending on junction and time according to Eq. (12), thus, taking detailed account of all effects on flow split. This model is the most comprehensive model of all three models for a and is expected to give very good predictions. Models 2 and 3 require a, and consequently a t,tobe estimated regarding current local pressures and flow distances as illustrated by Figs. 3 and 4. An iterative procedure is involved in Models 2 and 3, where for an estimated flow advancement, the coefficient of flow split is evaluated according to Eq. (12), the flow-rate is split according to Eqs. (10) and (11) and new lengths of the flow advancement in the axial and transverse directions are calculated on the basis of the estimated flow-rate values. The procedure is then repeated until numerical convergence has been reached. The model of Darcy s network micro-flow was

6 1336 S. Amico, C. Lekakou / Composites: Part A 31 (2000) implemented in numerical form in a Fortran77 computer program described in Fig. 5. The yarns are considered to be uniform and equally spaced. Capillary pressure and permeability values are estimated according to experimental data [20] and the Young Laplace and Carman Kozeny equations, respectively. The overall porosity is considered to be constant, since the intricate architecture of the fabric is expected to restrict variations in the cross-sectional area of the yarns and the gap between them. Thus, the porosity of the yarn has the same value for all runs, independently of the infiltrating fluid. A few other parameters, namely the density of the liquid, the radius of the yarn, the transverse micropermeability, the distance between two junctions and the length of the transverse yarn are also used as input values. The transverse micro-permeability of a fibre yarn is estimated as 19 times lower than the axial micro-permeability [25]. In this program, the transient infiltration process is regarded as a numerical sequence of steady state solutions at each time step where the finite difference method is employed. If there are junctions with unfilled transverse fibre yarns (unsaturated junctions), the flow is split between the axial and transverse yarns and a new height and transverse position of the flow front is calculated. If new junctions are reached, the number of unsaturated junctions is increased. If a junction is fully filled in the transverse direction, that junction is considered saturated. This process is repeated for each time step until equilibrium is reached. At any time, the number of unsaturated junctions, the time necessary for an individual junction to saturate, the position of the flow front in the transverse unsaturated yarns and the height of the axial flow front are known, allowing the description of the flow. Fig. 5. Flowchart of the computer program used to model the impregnation of the woven fabric. 4. Results and discussion 4.1. Numerical studies for a single yarn Figs. 6 and 7 present a sensitivity analysis for the vertical capillary flow of a model Newtonian fluid in an E-glass fibre yarn. Fig. 6 illustrates the influence of the permeability value on the final curve. As expected, an increase in k increases the rate of the flow height rise. The experiment takes less time to approach equilibrium. It is important to mention that the three curves in this figure will eventually reach the same equilibrium position, which is dependent only on P c. Likewise, if k is kept constant, an increase in P c also shifts the flow height curve upwards (Fig. 7). Fig. 8 shows typical predicted flow curves for different Fig. 6. Vertical axial capillary flow in a single yarn: curves for different permeability values; P c ˆ 4013 Pa for all curves.

7 S. Amico, C. Lekakou / Composites: Part A 31 (2000) Fig. 7. Vertical axial capillary flow in a single yarn: curves for different P c values; k ˆ 5:5 mm 2 for all curves. pairs of permeability and capillary pressure values as well as experimental data from a capillary experiment [20]. In the capillary flow experiments [20], a fibre yarn from an E- glass, plain woven fabric, Y0212 supplied by Fothergill Engineered Fabrics, was suspended over the surface of a Newtonian liquid with the lower end of the yarn, attached to a small weight, was immersed in the liquid. The experimental data included the weight increase of the fibre yarn being infiltrated and the progress of the flow front of the liquid as it infiltrated the fibre yarn in vertical (upwards) flow under the action of capillary pressure. From the relation between the corresponding curves of weight and fluid height rise with time [20], the area fraction of the pores of the fibre yarn available for fluid flow was estimated and was considered to be equal to the micro-porosity of fibre yarn. The experimental value of micro-porosity was then used in simulations where the infiltration predictions were compared to the corresponding experimental data. The predicted curve in Fig. 8, which used data derived from the experiments, P c ˆ 352:3 Pa and k ˆ 78:9 mm 2 ; gives a fair representation of the experimental points. As expected, whenever the values for P c and k obtained from a regression analysis of the experimental data are used as the input values for simulation runs, the modelled curve reasonably represented the data for the duration of the experiment. If purely theoretical values from the Young Laplace equation for capillary pressure P c ˆ 4443 Pa and the Carman Kozeny equation for permeability k ˆ 12:1 mm 2 are used where F ˆ 4 and K o ˆ 0:30; the curve overestimates the experimental data. This is attributed to the fact that insufficient time has been allowed for the experiment, resulting in misleading estimations of P c and k. If one considers that the theoretical value of P c from the Young Laplace equation was correct, i.e. e, F, s and D f were correctly estimated or measured, a permeability value of about 3.4 mm 2 could be found in order to match the data. Fig. 8. Short-term vertical axial capillary flow of silicone oil in a single yarn: predictions for different pairs of capillary pressure and permeability values.

8 1338 S. Amico, C. Lekakou / Composites: Part A 31 (2000) Fig. 9. Vertical axial capillary flow of a Newtonian epoxy resin in a single yarn long-term experiment: predictions for different pairs of capillary pressure and permeability values. On the other hand if k from the Carman Kozeny equation was correct, i.e. e, K o and R f were correctly estimated or measured, a P c of about 1300 Pa could be predicted. Other experimental data were analysed using the procedure of changing the input values in an analogous way and similar findings were reached. Thus, it can be seen that several pairs of P c and k can satisfy the experimental data. Thus, for an accurate estimate of P c, for example, the permeability value k must be previously known. The possible combinations of P c and k are much more restricted if a long-run experiment is analysed (see Fig. 9). In this case, it can be seen that the pair of P c ˆ 4019 Pa and k ˆ 5: m 2 results in a curve very close to the experimental data [20] and virtually coincident to the pair P c ˆ 4013 Pa and k ˆ 5: m 2 derived from the regression analysis of the experimental data. This estimated P c value agrees with the P c value derived from the Young Laplace equation where F ˆ 4: A combination of short- and long-run experiments of capillary infiltration of the E-glass fibre yarn used in this study and the comparison of experimental data with infiltration predictions yielded the conclusion that the best values of P c can be derived from the Young Laplace equation, where F ˆ 4; and the best values for the micro-axial permeability can be derived from the Carman Kozeny equation, where K o ˆ 1:58 for e in the range of and K o ˆ 5:77 for e in the range of Considering the presented data it could be said that if the experiment is not given sufficient time to approach Fig. 10. Five short-term silicone oil experiments through a plain-woven fabric and predictions of Model 1 for different values of a a ( ).

9 S. Amico, C. Lekakou / Composites: Part A 31 (2000) Fig. 11. Five short-term silicone oil experiments through a plain-woven fabric and predictions of Models 2 and 3. equilibrium, the proposed model is likely to lead to an underestimation of P c, consequently increasing k. Although a wide number of combinations of these parameters can fit the experimental data for short infiltration time, such combinations are more restricted for longer infiltration times, making the fitting procedure more reliable Numerical studies for the woven fabric The network Darcy s flow model for the infiltration of a fabric layer was validated by the experimental data of experiments [20] of the capillary flow of a Newtonian fluid through an E-glass plain-woven fabric, Y0212 supplied by Fothergill Engineered Fabrics. In these experiments, a rectangular piece of fabric was suspended over the surface of the Newtonian liquid, with one edge of the fabric immersed in the liquid. The progress of the overall upward flow of the liquid was monitored in the infiltration of the fabric under the action of capillary pressure (also see Fig. 1). The height versus time data for five experiments [20] carried out for up to 5.5 h of infiltration are shown in Fig. 10. The experimental points are reasonably close and minor deviations can be attributed to inherent characteristics of the fabric or its handling prior to the experiment. Fig. 10 also shows curves predicted by Model 1, when a a was pre-set at 0.01, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 0.99, and Besides, it is shown what the flow curve would be if the flow was modelled as through a single yarn with the same porosity as that of the fabric (upper curve in Fig. 10). The curves for a a in the range of are practically coincident. This can be explained by the fact that the faster the axial flow (for a higher a a ) the more junctions will be reached, and hence, the axial flow becomes further reduced, specially as a t is low and there are many unsaturated junctions. In the runs for a lower a a the flow will not only reach junctions at a lower rate but also, due to its higher a t, saturate those junctions faster. Curves for a a in the range of shift from the others due to the fact that the vast majority of the flow is being used in the axial direction, tending to neglect the presence of transverse flow, especially for a a ˆ 0:999: Hence, by the end of s all junctions reached by the flow are unsaturated. Models 2 and 3 are shown in Fig. 11. Model 2 (the same a a for all junctions for a specific time) slightly overestimates the experimental data whilst Model 3 shows good agreement with the experiment for most of the time. When the time reaches around s, Model 2 has reached 55 junctions, with 4 5 of them still unsaturated at the flow front and an a a within the range of The determination of a single a a for all unsaturated junctions (see Eq. (12)) requires the use of an average value for the difference between the height of the axial flow front and the height of a particular unsaturated junction H ff H i : The same is required for the position of the transverse flow front at each unsaturated junction (X ff ). The averaged values will be responsible for keeping the a a varying in a particularly narrow range ( ). The number of unsaturated junctions, consequently, fluctuates between 4 and 5. As could be expected, the use of a different a a for each junction, according to Eq. (12), generates results closer to the experimental ones for the curve of Model 3. Values of a a were in the range of depending on the position of the junction. By the end of the experiment, there were between 3 and 4 unsaturated junctions which agrees with the photograph of the flow in Fig. 1a. In order to verify the findings, a long-run experiment was also analysed according to the three models. In the first model, a a was pre-set to 0.01, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 0.99 and for each run and the resulting curves can be seen in Fig. 12. Here again, the

10 1340 S. Amico, C. Lekakou / Composites: Part A 31 (2000) Fig. 12. Long-term silicone oil experiment through a plain-woven fabric and predictions of Model 1 for different values of a a ( ). curves for a a in the range of are practically coincident and, for higher a a values, the curve shifts upwards. The histogram in Fig. 13 shows the number of unsaturated junctions by the time the run was interrupted. As expected, the higher a a is, the more unsaturated junctions are present. Runs for a low a a use most of the total flow to close the unsaturated junctions, and at the moment of closing an unsaturated junction, the axial flow receives a boost because there will be one less unsaturated junction. On the long term, both factors tend to counter-balance, being responsible for generally coincident flow behaviour for runs with a dissimilar a a. For a a ˆ 0:99999 the number of unsaturated junctions is equal to the number of open junctions, which means that no junction was closed due to saturation in the transverse direction. If the findings from the photographs (Fig. 1) were to be compared to this histogram, a value for a a of around 0.70 could be considered the best one in making the model agree with the experimental observations (three unsaturated junctions at the flow front). The results of Model 2 (Fig. 14) show a maximum height error of approximately 10% in comparison with the experimental data over the period covered (approximately 19 days), 4 5 unsaturated junctions at the flow front and a value for a a in the range of The curve predicted by Model 3 is presented in Fig. 15, Fig. 13. Histogram showing the number of unsaturated junctions for different values of a a.

11 S. Amico, C. Lekakou / Composites: Part A 31 (2000) Fig. 14. Long-term silicone oil experiment through a plain-woven fabric and predictions of Model 2. where 342 junctions were reached. The agreement was excellent and the error was in the range of less than 1% in comparison with the experimental data. As expected, the use of an individual a a for each junction, which considers the flow distance from that specific junction instead of an average value for all unsaturated junctions, generates results closer to the experimental data. Values of a a were in the range of depending on the position of the junction and by the end of the experiment, there were between 3 and 4 unsaturated junctions. This is in accordance to what was expected from the analysis of the photographs of the actual flow (Fig. 1a) which was used to predict that at least three junctions were partially filled at the flow front. In order to illustrate how a a varies with time in Model 3, data from the computer program for three consecutive junctions at three different time steps are shown in Table 1. The number associated with the junction represents the time step at which the junction was reached by the axial flow front. The lower the number of the junction, the closer it is to the liquid surface of the container and therefore, the earlier it was reached. Analysis of the table shows that for a particular time, the further a junction is from the flow front, the higher a a is, e.g. for the time step number 5764, a a;5161 ˆ 0:79 and a a;5631 ˆ 0:67: This characteristic cannot be so promptly derived from Eqs. (7) and (8) since both Q a,i and Q t,i depend on the height of the junction. Actually, both Q a,i and Q t,i increase with the height of the junction but the percentile increase of Q t,i is larger than Q a,i. The overall effect is to decrease a a for the upper junctions. Another feature is that a a increases with time for a junction, e.g. a a,5161 increases from 0.79 to Since the total Fig. 15. Long term silicone oil experiment through a plain-woven fabric and predictions of Model 3.

12 1342 S. Amico, C. Lekakou / Composites: Part A 31 (2000) Table 1 Values of a a for three different junctions at three different time steps Junction a a Time step 5764 Time step 5765 Time step Table 2 Range of a a for three consecutive junctions and the time when the junctions were open and closed Junction Opening Closing Time interval (s) a a Time (s) a a Time (s) flow rate decreases with time according to Eq. (1), both Q a,i and Q t,i decrease with time and the percentile decrease of Q t,i is larger than Q a,i, generating this effect. Table 2 presents the same three junctions and their respective values of a at the time they were open and closed. As above, a junction presents an increase in its a with time. Another expected result from this table is that the further the junction is from the surface of the liquid of the container, the longer it takes to be fully saturated (time interval data). The values for time interval throughout the duration of the modelling vary from 168 s for the first saturated junction closest to the container to s for the 338th junction (last saturated junction). The predicted curve of capillary flow of the silicone oil through the woven fabric up to equilibrium is shown in Fig. 16. In this graph, the final equilibrium position can be seen, in this case, H e ˆ 1:098 m corresponding to P c ˆ 9216 Pa; along with the experimental data. By the end of the computational run 718 junctions had been reached by the fluid, taking approximately 156 days to saturate the last junction (the 714th one). This curve allows one to predict the time necessary for the equilibrium to be reached, when the final position (H ff )is an arbitrary percentage of the expected equilibrium height (Table 3). This table illustrates a well-known feature of capillary experiments, the required long duration of experiments. Furthermore, the experimental data has to be collected for a minimum period so that reliable P c and permeability values can be extracted [20,26]. In this case, for 90% of the equilibrium height (H e ) to be reached, the experiment would take approximately 156 days, and this number would increase to around 403 days for H ff ˆ 0:99H e : Perhaps one of the most important parametric studies, due to its practical value, is the influence of the viscosity on the final curve. Fig. 17 shows three curves generated at different input viscosity values (0.108, and mpa s). As it is expected due to Darcy s law, the higher the viscosity, the slower the flow. It is important to mention that all three curves will eventually reach the same final position the equilibrium height, which is dependent only on P c. Taking this into consideration, it is interesting to notice that, for the studied time interval, a change of 10% in viscosity is responsible for a shift of the curve of up to 22 mm (4.1%) in height. Moreover, if a considerable change in viscosity of the infiltrating fluid is likely to happen during experiments, monitoring of viscosity has to be carried out to provide appropriate data for the computer program for correct modelling of the process. 5. Conclusions The scope of this study has been to develop and validate a model for a network Darcy s micro-flow in the in-plane infiltration of woven fabrics. After the model was implemented in an algorithm, the first task when running the algorithm was to provide appropriate input values for the micro-permeability and capillary pressure of the yarns of the fabric for a known micro-porosity. The methodology of these estimations was concluded from the simulations of the infiltration of a single yarn. The suggested numerical model for the axial capillary flow through a single yarn was validated by demonstrating that estimated values for P c and k from the previously reported capillary experiments do represent the experimental infiltration data for the duration of the experiment. Although different combinations of P c and k can fit the data for the initial portion of the flow curve, the possibilities are restricted for long run experiments, becoming more accurate and proving the importance of the duration of the experiment when mathematical curve fitting is used. The conclusion from the single yarn infiltration simulations was that the capillary pressure of a single fibre yarn can be estimated from the Young Laplace equation, with F ˆ 4 for axial flow in the yarn, and the axial micro-permeability can be estimated from the Carman Kozeny equation, with Table 3 Range of a a for three consecutive junctions and the time when the junctions were open and closed H ff ˆ 0:90H e H ff ˆ 0:99H e H ff ˆ 0:999H e H ff ˆ 0:9999H e Time (s) 1: : : : Time (days)

13 S. Amico, C. Lekakou / Composites: Part A 31 (2000) Fig. 16. Modelled curve showing the final equilibrium position. K o ˆ 1:58 for e ˆ 0:60 0:47 and K o ˆ 5:77 for e ˆ 0:20 0:30: The Darcy s network micro-flow model was tested by using experiments where a sample of fabric was impregnated by a Newtonian fluid in upwards, vertical flow under the action of capillary pressure. This type of experiment was selected because it produced no macro-flow, so that the micro-flow model could be tested independently. Three models of different degree of accuracy regarding the evaluation of flow split coefficient, a a, at the network junctions were tested. Model 1, which employed a constant pre-set input value for a a, showed that the main effect of accuracy of a a lies in the prediction of the number of unsaturated junctions behind the flow front: an a a value in the range of provided predictions close to the experimental data. Model 2, in which a a was evaluated inside the algorithm while making the assumption that the unsaturated region by the flow front is small compared to the whole infiltrated region of fabric, yielded predictions which differed by as much as 10% in comparison with the infiltration data of the experiment, for 4 5 unsaturated junctions at the flow front. Model 3, in which a a was evaluated inside the algorithm by taking all the terms of the model into account, showed the best agreement with the experimental results, since it allows a more appropriate distribution of the flow, with a varying flow split factor, a a, in the range of , which was in accordance with experimental data and photographs describing the number of unfilled transverse yarns at junctions by the flow front. The successful validation of the network Darcy s model for woven fabrics means that one of the sub-models of the infiltration of an assembly of woven fabrics in RTM is Fig. 17. Sensitivity analysis regarding fluid viscosity (m values in Pa s).

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