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Permeability estimation algorithm to simultaneously characterize the distribution media and the fabric preform in vacuum assisted resin transfer molding process
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 65 (2005) 2129–2139 www.elsevier.com/locate/compscitech
Permeability estimation algorithm to simultaneously characterize the distribution media and the fabric preform in vacuum assisted resin transfer molding process Ali Gokce a, Mourad Chohra a
a,b
, Suresh G. Advani
a,b,*
, Shawn M. Walsh
c
Center for Composite Materials, Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA b Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA c Army Research Laboratory, Aberdeen Proving Grounds, MD, USA Received 3 November 2004; received in revised form 21 April 2005; accepted 2 May 2005 Available online 29 June 2005
Abstract In vacuum assisted resin transfer molding (VARTM), available pressure for resin injection is limited to 1 atm. Therefore, layers of high-permeability fabrics, called the distribution media (DM), are included in the lay-up to enhance the resin flow. During injection, resin flows initially in-plane saturating the DM, and subsequently through the thickness impregnating the bulk preform. Therefore, the DM permeability governs the planar resin flow pattern and lead length, which is defined as the difference between the flow position in the DM and in the bottom preform layer adjacent to the tool surface. In this study, the DM permeability was investigated as a function of the preform lay-up. A new permeability estimation method, called permeability estimation algorithm (PEA), that can estimate permeability values of the DM and the preform simultaneously from a single experiment, was developed and validated. Using PEA, it has been shown that the DM permeability does vary with the preform lay-up. 2005 Elsevier Ltd. All rights reserved. Keywords: E. Resin transfer molding; A. Fabrics/textiles; A. Hybrid compounds; B. Modelling; Permeability
1. Introduction Vacuum assisted resin transfer molding (VARTM), a low-cost process to manufacture composite structures with moderate dimensional tolerances, is a subclass of liquid composite molding (LCM) processes. In LCM, layers of fiber reinforcements are placed in a mold cavity and a liquid resin is injected into the mold to impregnate the empty spaces between the fibers. The VARTM process employs a single sided mold tool. The fiber reinforcement is placed on top of the tool and is covered by a distribution medium (DM) with a high in-plane permeability to accel*
Corresponding author. Tel.: +1 302 831 8975; fax: +1 302 831 3619. E-mail address: [email protected] (S.G. Advani). 0266-3538/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2005.05.012
erate the in-plane flow. Thus, a typical VARTM lay-up comprises fiber reinforcements, surface or inter-laminar DM, chemical agents/textiles to minimize defects and facilitate de-molding. A vacuum bag is used to seal the assembly as shown in Fig. 1. Vacuum is drawn through the vacuum line to remove air from the mold cavity, induce fiber compaction under atmospheric pressure and draw resin into the mold cavity through the injection line. After the resin arrives at the vent, the injection is discontinued but the vacuum is maintained until the resin cures allowing one to de-mold the part. Although its origins can be traced back to early 1950s [1], VARTM and similar processes were developed primarily in the 1980s. Researchers investigated several issues of the process such as resin flow [2–4], preform compaction [5,6] and flow control [7–9].
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Fig. 1. Schematics of VARTM.
Complete saturation of the bulk preform is critical for an acceptable part. The resin flow through the porous media is governed by Darcys law: hui ¼
K rhP if ; g
ð1Þ
where hui is the volume-averaged velocity vector, K is the second order permeability tensor of the porous medium, g is the viscosity and $hPif is the gradient of the pressure averaged over the fluid volume. Insertion of Eq. (1) into the continuity equation ($ Æ hui = 0) results in an elliptic partial differential equation. For constant pressure injection, the governing equations can be solved for one-dimensional (1D) flow as, x2 K ðDP Þt; ¼ 2 g/
ð2Þ
where x is the flow front location, DP is the pressure difference between the injection gate and the flow front, / is the porosity and t is the time. However, analytical solution is difficult for 2D and 3D flows, hence numerical methods are employed. In this study, liquid injection molding simulation (LIMS) [10], a software package developed at the University of Delaware, is used as the process model. LIMS uses a finite element/control volume approach to simulate the resin flow through porous media in liquid composite molding processes using Darcys Law and mass and energy conservation equations [11]. LIMS has been integrated into MATLAB to create a virtual experimentation environment, where the studies presented in this paper were conducted. Permeability, K in Eq. (1), is used to quantify the resistance of a porous medium to fluid flow. Extensive research has been conducted to predict and measure the permeability of the porous media used in LCM processes through analytical, numerical or experimental methods. In analytical studies, the fluid dynamics equations are solved for a representative domain of an idealized porous medium [12–14]. The use of analytical models is limited to idealized geometries of cylindrical/ elliptical arrays, which are approximations to the actual preform architectures. Numerical methods can be used to investigate complex porous structures, by solving the flow equations in the flow medium [15–19]. In this
method, problem complexity and computational time becomes an issue when a satisfactory performance is needed for an actual preform. Carefully conducted unidirectional, 2D or 3D experimental permeability measurements can return reliable results [20–27], but only for the given experimentation conditions. In addition, experimental methods are susceptible to disturbances such as racetracking [28,29]. None of the existing methods addresses the situations where the LCM lay-up includes dissimilar porous media, hence multiple unknown permeability values, such as in VARTM can be calculated from the same experiment. Although [4] offers a closed form flow solution for VARTM, it has limited applicability due to the assumptions made, and for a certain window in the parameter space its result include significant error. In reported experimental permeability measurement methods, the flow data from a set of experiments are input into analytical solutions such as Eq. (2), to solve for the permeability values. Since analytical solutions are available only for 1D flow, such methods have limited use and are susceptible to disturbances such as racetracking. In this paper, a new permeability estimation method that replaces the analytical step of the existing methods with numerical methods is proposed. This permeability estimation algorithm (PEA) was used to calculate the in-plane DM permeability and the preform permeability in the through thickness direction from the same experiment in this study. DM permeability is usually measured separately and the result is used to simulate resin flow in VARTM. However, DM permeability in a VARTM lay-up with preform underneath it and a vacuum bag on top could be significantly different from the one measured separately due to the dynamics of compression of the DM, fiber reinforcements and the vacuum bag. PEA allows in situ measurement of the DM permeability in the VARTM setup, and investigation of the interactions between various materials that form the lay-up. In the following section, PEA will be described and validated in the virtual domain. Also, its sensitivity to input errors will be studied. In Section 3, experimentation setup will be introduced, and the experimental results will be presented and analyzed.
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2. Permeability estimation algorithm 2.1. Procedure In a typical experimental permeability measurement operation, the inputs include the processing parameters such as injection pressure, resin viscosity, and material parameters such as preform porosity and dimensions. Flow data, which is defined as the resin flow front location as a function of time, can be obtained using point, lineal or optical sensors. In this study, the resin flow was taped using camcorders from the top through the transparent vacuum bag, and from the bottom through the transparent acrylic table, as shown in Fig. 2. The goal is to estimate the permeability values using the inputs and the flow data. No proven analytical relationship is available for use in this study, where the objective is to find the permeability of the distribution media and through-the-thickness permeability of fiber preforms. Developing a
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numerical formulation to solve for the permeability values directly is not straightforward. Therefore, LIMS, an established process model for resin flow, was incorporated into an iterative algorithm to estimate the permeability values as shown in Fig. 3. In this approach, the experimental VARTM setup is modeled in the simulation environment for use by LIMS. Each material in the VARTM lay-up is represented as a region. A sensory system is designed to track resin flow in each region using point, lineal or optical sensors. Once data is acquired on resin flow from the two regions, it is input to the PEA, which then searches iteratively for the permeability values that will reproduce the experimental flow behavior in the simulation environment. Starting with initial permeability values, at iteration j, resin flow is simulated with the permeability values returned from the previous iteration. Simulated resin flow is compared with the experimental flow data and the permeability values for the next iteration are estimated as follows. If point sensors are used
Fig. 2. Data acquisition setup used in this study.
Fig. 3. Flowchart of permeability estimation algorithm (PEA).
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porosity Resin pressure distribution
Mold cavity thickness
Resin flow field permeability
Fig. 4. Interaction of process parameters in VARTM process during resin injection.
K i;jþ1
p ti;exp ¼ K i;j ; ti;j
i ¼ 1; 2; . . . ; n
ð3Þ
where Ki,j+1 is the permeability value of region i for the next iteration, Ki,j is the current permeability value of region i, ti,exp is the experimental resin arrival time at the point sensor in region i, ti,j is the resin arrival time at the sensor in the simulation, p is the update power constant and n is the number of regions in the lay-up, and if lineal sensors are used p xi;exp K i;jþ1 ¼ K i;j at t ¼ tk ; i ¼ 1; 2; . . . ; n ð4Þ xi;j where xi,exp is the experimental flow position in region i at time t = tk and xi,j is the corresponding flow position in the simulation. In this study, the flow data was taken from the video recordings of the experiments using appropriate software in a way that mimics point sensors, hence Eq. (3) was used as the update law. The iterations stop either when maximum number of iterations has passed or when simulated flow data is sufficiently close to the experimental flow data: 8i; ¼
jti;j ti;exp j 6 e; ti;exp
i ¼ 1; 2; . . . ; n;
ð5Þ
where e is the convergence criterion. In case of online PEA use, iterations terminate when the next set of flow data arrives from the sensory system. The frequency of data acquisition depends on user preferences and sensory system capabilities. PEA can be used to find the average permeability of the entire wetted section of the regions or the sections that were wetted since the last data acquisition. In either case, only the essential period of the resin flow is simulated for computational efficiency. In this study, a single data set acquired towards the end of the filling operation was used to estimate the average permeability values of the regions. For higher dimensional flows, it is not possible to prove rigorously that the left hand side of Eq. (5) will converge to e using the update law in Eq. (3). Assuming that the permeability of a region has the most influence
on the resin flow through that region among all permeability parameters in the setup, it is clear from Eqs. (1) and (2) that the update law in Eq. (3) will drive permeability estimates in the right direction. For all our cases, it was observed that this assumption was justified. Note that several properties of a porous medium interact during injection as shown in Fig. 4. In this study, averaged values are used for all parameters except for the permeability, and the variations in other parameters were lumped into the permeability value. As long as consistent values are used in the calculations and the simulations, this approach is justifiable. 2.2. Validation The PEA was tested and validated in the virtual domain using flow data acquired from simulated VARTM experiments. The same material and process parameters as the virtual experiments were input to the models used in PEA. Since the exact permeability values used in the virtual experiments were known, it was possible to quantify the accuracy and efficiency of the PEA permeability estimates. In addition, permeability values for 1D flow were estimated by both PEA and Eq. (2), and the comparison of both methods is presented in Section 3.2. Since rectangular plate geometries were injected with negligible variation along the width direction, the crosssection of the VARTM setup was modeled as a 2D geometry as shown in Fig. 5. The mold cavity was divided into two regions: the perform layers as region I and the DM as region II. Two virtual point sensors were used to track resin flow in the regions: sensor I at the tool surface on data collection line I to track resin flow in region 1, and sensor II on the lay-up surface on data collection line II to track resin flow in the DM. Since sensor locations changed from one virtual experiment to the next, they are not included in Fig. 5. The DM was modeled using bar (1D) elements with one permeability component, KDM, which is associated with the flow data from sensor II, because it has a stronger influence on the flow along data collection line II compared
Fig. 5. Cross-section of the VARTM setup.
A. Gokce et al. / Composites Science and Technology 65 (2005) 2129–2139
ei ¼
K i;est K i;exp ; K i;exp
ð6Þ
where Ki,est is the permeability estimate of region i, and ei is the estimation error associated with the region. Five e values were tested: 0.1, 0.01, 0.001, 0.0001 and 0.00001. log (e) vs. |ei| · 100 and is plotted in Fig. 6 along with the number of iterations. The figure shows that the estimation errors drop quickly as e is reduced and the number of iterations is linearly related to log(e). In this study, e was chosen to be equal to 0.001. In 1D flow, p in Eq. (3), is equal to 1 according to Eq. (2) and the permeability estimation will take a single iteration. For higher dimensional flows, selection of a p value becomes complicated. Five p values were tried to find an optimal value for this study: 0.67, 0.93, 1.2, 1.53 and 2. The convergence behavior of the permeabilTable 1 The parameters of the primary virtual VARTM experiment Injection pressure, Pi (kPa) Resin viscosity, g (Pa s) DM porosity, /DM DM permeability, KDM (m2) Preform porosity, /p Preform permeability, x, Kxx (m2) Preform permeability, z, Kzz (m2) Mold length (m) Mold width (m) Preform thickness (m) DM thickness (m)
97 0.55 0.876 2 · 108 0.486 1010 8.33 · 1012 0.3556 0.1524 0.0036 2.13 · 104
80
error(%) and # of iterations
to other permeability values. The preform was modeled with 2D elements with two permeability components, Kxx andKzz, assuming principal permeability axes are aligned with the coordinate system. Since there is only one data input for the flow location in the preform -sensor I-, it is not possible to estimate both parameters independently. Since, in the preform, the resin flow is primarily through the thickness, Kxx was used as an input and Kzz was estimated using PEA. Kxx of the preform is much easier to measure separately in advance using the standard 1D experimental method. The material and process parameters of the primary virtual VARTM experiment are listed in Table 1. PEA was validated using this VARTM setup and its variations. In the testing and validation process, two major PEA parameters, p in Eq. (3) and e in Eq. (5), were varied to find their optimal values. Next, KDM and Kzz in Table 1 were varied to evaluate the performance of PEA for various virtual experiments. In the next section, PEA sensitivity to input errors will be addressed. In all of these studies, the permeability estimates returned by PEA will be compared with the permeability values used in the virtual experiments. Unless otherwise noted, the estimation error will be calculated using the equation below in the rest of the paper:
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70
K_DM error (% ) K_zz error(% )
60
iterations
50 40 30 20 10 0 0
-1
-2
-3
-4
-5
log (ε) Fig. 6. Permeability estimation errors and the number of iterations as a function of convergence criterion (p = 1.2).
ity estimates during PEA iterations were formulated by the ratio of the estimated values to the experimental values at each iteration (KDM,est/KDM,exp and Kzz,est/ Kzz,exp), and plotted against iteration number in Fig. 7 for several p values. As p is increased, the permeability estimates converge faster. However, oscillatory component appears at p = 1.53 and it intensifies at p = 2. p vs. number of iterations plot in Fig. 8 shows that the number of iterations decreases until oscillatory behavior appears, after which the number of iterations increase. In this study p was chosen to be as 1.2. PEA is not sensitive to initial permeability values due to its algorithm, which has been confirmed by the observations during the testing and validation process. Therefore, it was not studied in a systematic manner. After selection of e and p, PEA was used with various values for the DM and the preform in the virtual VARTM experiment to explore the accuracy and limitations of the algorithm. New virtual experiments were created and simulated by taking different values for KDM and Kzz in Table 1; the remaining parameter values were kept the same. Five different values were used for each permeability: K1 = KDM/Kxx = {1, 50, 200, 1000, 5000} and K2 = Kxx/Kzz = {1, 5, 12, 40, 120}, which creates a 5 · 5 configuration matrix C = {cij}, where i corresponds to K1 value and j corresponds to K2 value. Only the configurations on the diagonals of the matrix, ci,i and ci,5 i + 1 (i = 1 ,. . ., 5), a total of 9 configurations, have been addressed in order to observe the interaction of KDM and Kzz with less computation. The flow patterns, ei values, numbers of iterations and approximate data collection times are tabulated in Table 2. The estimate errors for configuration c15 are significantly high. Since, for KDM 6 Kxx, the DM does not have any influence on the resin flow except for acting as a sink due to high porosity, any KDM value lower than Kxx will satisfy the flow data from sensor I, resulting
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K_zz convergence
K_DM convergence 4
(K_ DM, est ) / (K_ DM, exp )
(K_ zz, est ) / (K_ zz, exp )
10000
1000
100
10 1 1
6 11 16 21 26 31 36 41 46 51 56 61 66 71 76
3 2.5 2 1.5 1 0.5 0 1
0.1
iteration number p=0.67
3.5
p=0.93
p=1.2
p=1.53
6 11 16 21 26 31 36 41 46 51 56 61 66 71 76
iteration number p=2
p=0.67
p=1.2
p=2
Fig. 7. Convergence behavior of the permeability estimates as a function of p. Kzz,est/Kzz,exp is plotted on a logarithmic scale to highlight the convergence behaviour from a much larger initial guess. KDM,est/KDM,exp is plotted on a linear scale and the number of lines are reduced by two to highlight the convergence behaviour around the equilibrium state.
large parameter space and the permeability estimates converge to the selected values asymptotically (Fig. 7).
80
# of iterations
75 70 65 60
2.3. Sensitivity to errors
55
In the actual experimentation environment, there are errors involved with the inputs; hence it is critical to study the sensitivity of PEA to input errors. Three parameters were selected for sensitivity study: Kxx, flow data (t1,exp and t2,exp), and injection pressure DP. Selected level of error was introduced for each of these parameters (Table 1) before it was supplied as input to the PEA. Subsequently, the effects of the input error on the permeability estimation errors (Eq. (6)) were studied. Due to the presence of DM, majority of the flow in the preform is through the thickness. However, there is a planar component due to Kxx, and it becomes stronger away from the DM layer, which causes the semi-elliptic flow front shape (Table 2). In PEA, Kxx is measured separately and entered as an input value. Since there are inherent variations in the permeability measurements, there will be some error introduced with this input. Five error levels were introduced in Kxx: E1 = Kxx,PEA/Kxx = {0.1, 0.3, 1, 3, 10}, where Kxx,PEA is the value that was used in the VARTM model of PEA, and the permeability estimations were found based on the incorrect value of Kxx. log (E1) vs. 100 · ei is plotted in Fig. 9. In general, if a lower Kxx value was used as input, PEA overestimated KDM and Kzz; in case a higher value was used, the permeability values were underestimated, which is in agreement with the physics of the process. As Kxx approaches zero on the left of the figure, the planar flow component in the VARTM model becomes more and more insignificant
50 45 40 35 30 0.6
1.1
1.6
2.1
p Fig. 8. Number of iterations as a function of update power constant, p.
in large estimation error. The error in Kzz is high because the resin flow is primarily planar with insignificant flow through the thickness, hence the relationship between the resin arrival time at sensor I and Kzz is very weak. Since the sensitivity of arrival times at sensors I and II are weak to the changes in Kzz and KDM, respectively, case c15 is an ill-conditioned reverse problem. To verify these conclusions, configuration c25 (KDM/Kxx = 50, Kxx/Kzz = 120) was also studied. The estimation errors dropped sharply with respect to c15, with eK zz ¼ 0.35% and eDM = 0.03% (Eq. (6)). All other estimation errors were less than 2.1%, many below 0.1%. The c55 case gives the best results because the sensitivity of arrival times at sensors I and II are very strong to the changes in Kzz and KDM, respectively. The numbers of iterations listed in Table 2 decrease as K1 and K2 increase. The results presented in this section illustrate that PEA can estimate permeability values accurately for a
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Table 2 Flow patterns of virtual VARTM experiments and PEA permeability estimation errors
The height of the mold is increased 5 times in the contour plots to provide better visibility.
40
percentage error
20 0 0.1
1
10
-20 -40 K_zz K_DM
-60 -80 -100
log(K_xx,PEA / K_xx) Fig. 9. Permeability estimation errors due to input error in Kxx.
and ei plateau towards the left. As Kxx increase towards the right, Kzz and KDM drop increasingly faster to reproduce the experimental flow data. In the limit, where Kxx is so high that resin arrives at the sensor locations earlier than experimental flow data through planar flow, there is no solution to the problem. The plots show that, as Kxx,PEA was varied two orders of magnitude, the perme-
ability estimates varied hardly one order of magnitude, hence it can be concluded that PEA is not very sensitive to error in Kxx,PEA. Note that it is safer to choose Kxx,PEA values from the lower end of the range, if the option exists. Due to hardware problems, low resin conductivity and noise, it is possible to introduce error into the flow data. Five error levels were introduced into flow data from sensors I and II: Ti = 100 Æ (ti,PEA ti,exp)/ ti,exp = {5, 1, 0, 1, 5} (i = 1, 2), where ti,PEA is the flow data value input to VARTM model in PEA, which yields a 5 · 5 configuration matrix A = {aij}, where i and j correspond to T1 and T2 values, respectively. Instead of all 25 configurations, only 9 configurations along the diagonals, ai,i and ai,5 i + 1 (i = 1, . . ., 5), have been studied to observe the interaction of both parameters with less computations. 100 · ei is plotted in Fig. 10. The error level in KDM is around the same magnitude as T2 at worst, since resin flow through the DM resembles 1D flow, and K 1/t in 1D flow (Eq. (2)). On the other hand, resin arrival at sensor I depends both on KDM and Kzz. Therefore, in order to match t1,PEA, Kzz varies significantly more than KDM for the same level of input error, creating larger error estimates. The results show that eK DM is larger for configurations ai,i, whereas eK zz
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K_zz error (%)
K_DM error (%) 6
90 80
5 70
4
60 50
3 40
2
30 20
1
T1=0
T1=-1
T1=-5
T2=5
T2=5
T2=1
T1=1
T2=0
T1=-5
T1=5
T2=-1
T1=-1
0
T2=-5
T1=0
T2=0
T1=1
T2=-1
T1=5
T2=-5
0
T2=1
10
Fig. 10. Estimation errors for various input error configurations in flow data.
25 20 Permeability error (%)
is larger for configurations ai,5 i + 1. Note that the lead length is very sensitive to Kzz and vice versa. Since t1,PEA and t2,PEA were perturbed at the same levels in configurations ai,i, the lead lengths remained unchanged; hence the small eK zz values in these configurations. However, in ai,5 i + 1 configurations, t1,PEA and t2,PEA were perturbed in opposite directions, which PEA interpreted as large variations in the lead length and responded with significant differences in Kzz estimations. For example, in configuration a15 (T1 = 5 and T2 = +5), due to the perturbations at the sensor data, the resin appeared to arrive earlier at sensor I and later at sensor II. In response, PEA overestimated Kzz by 85% to match early resin arrival at sensor I. Due to the increased Kzz, DM lost more resin to the preform on the flow front, causing a slowdown in flow in DM, which PEA addressed by increasing KDM slightly (0.125%) to match the resin arrival time at sensor II. It can be concluded that the Kzz estimates are sensitive to input errors in flow data; hence the sensory system must be designed with care. Note that, for a typical fill time of 300 s, 5% error in monitoring of the flow data is equal to a measurement error of around 15 s, which is high and not likely to happen with careful maintenance of the sensory hardware. During injection, there is some level of pressure loss in the tubing; the vacuum level may not be perfect and the pressure gage may have imprecision, which may create some input error in the injection pressure, Pi. Seven error levels were introduced into the pressure: P ¼ 1 0 0 ðP i;P E A P i Þ=P i ¼ f 1 5; 1 0; 5; 1; 0; 1; 5; 10; 15g, where Pi,PEA is the injection pressure used in the VARTM model. P vs. 100 Æ ei plot in Fig. 11 is consistent with K 1/$P relationship in Eq. (1), since $(cP) = c$P, where c is a constant.
K_zz error K_DM error
15 10 5 0 -20
-10
-5
0
10
20
-10 -15 -20 Pressure error (%)
Fig. 11. Permeability estimation errors as a function of input errors in injection pressure.
It is clear from Eqs. (1) and (2) that permeability estimation errors due to g and /, are proportional to the errors involved with the measurement of these parameters. In summary, PEA is quite stable against input error. During evaluation of the actual experimentation results, maximum potential permeability estimation errors will also be calculated.
3. Experiments 3.1. Design of experiments S51 Shade Cloth from Roxford Fodell is commonly used as a DM in VARTM. Three VARTM setups have
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Fig. 12. Snapshots from the top (left) and bottom (right) recordings from an experiment. The flow front distance data is taken from the recording and the time information is taken from the software.
been designed for injection and each experiment has been repeated three times: (I) DM only, (II) DM with 4 layers of 24-ounce 5 · 4 woven E-glass, and (III) DM with 6 layers of the same preform. Other common fabric layers that are used to facilitate processing were omitted to prevent flow disturbances. Injection pressure Pi was measured using pressure gages mounted on the vacuum buckets and viscosity g was measured using Brookfield viscometer. To estimate the porosity of a fabric, its solid thickness h was estimated as h = (mass per area)/(material density) first. Next, porosity was estimated by / = 1h/(thickness in layout). The in-plane permeability of the preform, Kxx, was measured separately through 1D injection using Eq. (2). Different numbers of preform layers were used in the experiment. Three separate measurements were done for each configuration. The average of all the measurements was used as input in PEA. Flow was taped from the top and the bottom of the mold, and flow data was acquired manually in a way that mimics point sensors. Fig. 12 shows snapshots from the video of an experiment: the flow front location is measured using the metric drawn on the mold surface and time information is taken from the software. The Table 3 The parameter value ranges measured during the VARTM experiments Injection pressure, DP (kPa) Resin viscosity, g (Pa s) DM porosity /DM Preform porosity /p *
Preform permeability, x Kxx (4 layers) Kxx (6 layers)
84.7–96.5 0.48–0.64 0.778–0.792 0.430–0.526 1.66 · 1011 2.63 · 1011
Mold length (m) Mold width (m) * DM thickness (m)
0.381–0.508 0.127–0.178 8.08 · 104
Total lay-up thickness (m) 4 layers 6 layers
3.22–3.28 (·103) 4.18–4.23 (·103)
Parameters preceded by an (*) sign are obtained through separate measurements.
flow data was entered into the PEA along with other material and processing parameters, whose value ranges are tabulated in Table 3. 3.2. Results and discussion Three 1D experiments were conducted for each of 4layer and 6-layer lay-ups with preform only, and the average in-plane permeability values (Kxx) were found as 1.66 · 1011 and 2.63 · 1011, respectively. The measurements are consistent with the fact that as the number of layers increase, the permeability increases due to reduced compaction (lower fiber volume fraction). Three experiments were carried out to measure the DM permeability without placing any preform layers under it (Setup I). PEA and Eq. (2) were used to estimate KDM. The results from both methods have been tabulated in Table 4. The comparison of the results shows that PEA results are accurate. For the set of experiments conducted with the VARTM lay-up that contains preform layers under the distribution media (Setup II and III), the average Kzz and KDM values are listed in Table 5. Note that KDM values are nearly the same for VARTM setups II and III, which are around 45% lower than the measurement from setup I. The standard deviation for Kzz estimations is 8.02 · 1014, which is a small value. The results in Table 5 show that DM permeability depends on the VARTM lay-up. Using typical parameter values along with KDM value measured by PEA, it took 586 s to fill the VARTM part. If PEA was not available to measure KDM in situ, and the value from setup I was used, the fill time would be predicted as 336 s. Such a discrepancy is likely to create problems with larger molds. The overall estimation error range can be estimated by accumulating the effects of all input errors as shown below for KDM: K DM ¼ f ðK xx ; time=sensors; P ; g; /Þ. Also, eK DM ¼ f ðeK xx ; esensors ; eP ; eg ; e/ Þ.
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Table 4 Comparison of permeability estimates by Eq. (2) and PEA Exp. ID 1 2 3
KDM, Eq. (2) (m2) 9
4.998 · 10 4.338 · 109 4.467 · 109
KDM, PEA (m2) 9
4.995 · 10 4.408 · 109 4.446 · 109
Standard deviation 2.12 · 1012 4.95 · 1011 1.48 · 1011
Table 5 Permeability estimates as found by PEA using data from VARTM experiments Setup ID
Kzz (m2)
KDM (m2)
I II III
N/A 9.90387 · 1013 8.7702 · 1013
4.60096 · 109 2.5033 · 109 2.6441 · 109
Assuming the effects of various input errors to permeability estimation errors are not coupled, that is, error in one input does not amplify estimation error because of error in another input, one can estimate the maximum cumulative error in the permeability estimation as follows: DM DK DM K ¼ ð1 ^eK DM Þ DM K ð1 ^eK xx Þð1 ^esensors Þð1 ^eP Þ ð1 ^eg Þð1 ^e/ Þ;
ð7Þ
DM is the average permeability estimate, DKDM where K is the maximum error range, ^eK DM is the maximum cumulative error level and ^ej (j = Kxx, sensors, P, g, /) is the maximum error in KDM estimate due to the input error in j. Eq. (7) applies to Kzz estimation as well. The sensitivity of permeability estimation error to input errors was investigated in Section 2.3, which will provide some of the variables in Eq. (7). The maximum ratio between a Kxx measurement and the mean of all Kxx measurements were 1.4. Fig. 9 shows that, such an input error will cause 3.5% error in K DM ð^eK xx ¼ 0.035Þ, and 10.1% error in K zz ð^eK xx ¼ 0.101Þ. The flow location and time was taken from snapshots when flow was at least more than 1300 away from the injection line after 320 s later. Within these conditions it is reasonable to take the error due to human eye as 2%, which will induce 2% maximum error in K DM ð^esensors ¼ 0.02Þ, and around 29% maximum error in Kzz estimation ð^esensors ¼ 0.29Þ according to Fig. 10. The measurement error in the pressure gages are taken as 2% as no data could be found ð^eP ¼ 0.02Þ. The measurement error in the viscometer is given as 1% by the manufacturer ð^eg ¼ 0.01Þ. The porosity measurement includes working with water, scales and caliper s. Since all tools used are precise, cumulative error can be safely predicted as 3% ð^e/ ¼ 0.03Þ. Then, maximum cumulative error in KDM estimation can be found as 11%, and 40% for Kzz
estimation. Note that errors due to numerical solutions and simulations are not in the scope of this study. The computational overhead of PEA depends on the size of the finite element model used. PEA saves computational time by simulating the resin flow only through the mold section whose permeability values are being estimated. In this experimental study, the VARTM setup was modeled with 1206 nodes and 1200 elements. PEA took around 40 iterations to converge with e = 0.001, where 85% of the mold was filled at each iteration. The estimation took around 20 s on PC with a Pentium 4 – 3.06 GHz CPU and 1 GB RAM.
4. Summary and conclusions Permeability Estimation Algorithm (PEA), a new experimental permeability estimation method that uses a numerical process model instead of analytical relationships, was presented. Subsequently, it was used to find preform permeability in the thickness direction and the in-plane distribution media permeability from one experiment using the flow front data from an experiment. The accuracy of PEA was validated in the virtual environment. PEA was also used for permeability measurements in 1D flow experiments and its results were verified by Eq. (2). PEA, unlike existing experimental permeability measurement methods, allows one to calculate permeability of different materials in the same lay up. As PEA uses a numerical simulation and an iterative solver, it can be adjusted to accommodate various configurations and the flow does not have to be 1D. For example, it can be used to measure the permeability of a preform despite the presence of racetracking; or it can be used to find the principal permeability values of a preform using flow data from a radial injection. The results presented in Section 2 show that PEA converges on accurate permeability estimates asymptotically, hence it is stable. In this study, PEA has been successfully applied to permeability measurements in VARTM, revealing the fact that DM permeability depends on the VARTM lay-up; hence it should be measured in situ. Many VARTM systems of interest are increasingly using a combination of materials (e.g., graphite and glass) which makes permeability assessment difficult from a purely closed-form analytical approach. The PEA prescribes a means by which such hybridized systems can be accommodated and a reasonable estimate of the permeability field obtained. Note that numerical methods inherently include some level of error. PEA finds the permeability estimates that reproduce the actual flow data. If the same finite element model is used for both PEA and other tasks (optimal design, process control, etc.), then numerical error introduced into the solution will be reduced.
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Acknowledgments We thank Niki Frangakis for assisting in the experimental work. The financial support, provided by the Office of Naval Research (ONR) under Grant N0001403-1-0891 for the Advanced Materials Intelligent Processing Center at the University of Delaware, and the Center for Materials Technology established by Army Research Laboratory at the University of Delaware made this work possible.
References [1] Potter KD. The early history of the resin transfer moulding process for aerospace applications. Composites: Part A 1996;30:619–21. [2] Advani SG, Sozer MS. Process modeling in composites manufacturing. New York: Marcel Dekker; 2003. [3] Han K, Jiang S, Zhang C, Wang B. Flow modeling and simulation of SCRIMP for composites manufacturing. Composites A: Appl Sci Manufac 2000;31(1):79–86. [4] Hsiao K-T, Mathur R, Advani SG, Gillespie Jr JW, Fink BK. A closed form solution for flow during the vacuum assisted resin transfer molding process. J Manuf Sci Eng 2000;122:463. [5] Correia NC, Robitaille F, Long AC, Rudd CD, Simacek P, Advani Suresh G. Use of resin transfer molding simulation to predict flow, saturation and compaction in the VARTM process. J Fluid Eng 2004;126:201–15. [6] Lomov SV, Verpoest I. Compression of Woven resinforcements: a mathematical model. J Reinforced Plastics Compos 2000;19(16):1329–50. [7] Hsiao KT, Gillespie Jr JW, Advani SG, Fink BK. Role of vacuumpressure and port locations of flow front control for liquid composite molding processes. Polym Compos 2001;22(5):660–7. [8] Heider Dirk, Hofmann C, Gillespie Jr JW. Automation and control of large-scale composite parts by VARTM processing. In: International SAMPE symposium and exhibition (proceedings), vol. 45 (II); 2000. p 1567–75. [9] Johnson RJ, Pitchumani R. Enhancement of flow in VARTM using localized induction heating. Compos Sci Technol 2003;63(15):2201–15. [10] Simacek Pavel, Advani SG. Desirable features in mold filling simulations for liquid molding processes. Polym Compos 2004;25(4):355–67. [11] Bruschke MV, Advani SG. A finite element/control volume approach to mold filling in anisotropic porous media. Polym Compos 1990;11:398–405. [12] Kozeny J. Sitzungsberichte Abt IIa. Wiener Akademie der Wissenschaft; 1927.
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[13] Carman PC. Fluid Flow Through Granular Beds. Transactions 15, Institution of Chemical Engineers; 1937. [14] Simacek P, Advani SG. Permeability model for a woven fabric. Polym Compos 1996;17:887–99. [15] Spaid M, Phelan F. Lattice-boltzman methods for modeling microscale flow in porous media. Phys Fluid 1997;9(9). [16] Gokce A, Advani SG. Permeability estimation with the method of cells. J Compos Mater 2001;35(8):713–28. [17] Nedanov Pavel B, Advani Suresh G. Numerical computation of the fiber preform permeability tensor by the homogenization method. Polym Compos 2002;23:758–70. [18] Belov EB, Lomov SV, Verpoest I, Peters T, Roose D, Parnas RS, et al. Modelling of permeability of textile reinforcements: lattice Boltzmann method. Composites Science and Technology 2004;64(7–8):1069–80. [19] Dimitrovova Z, Advani Suresh G. Analysis and characterization of relative permeability and capillary pressure for free surface flow of a viscous fluid across an array of aligned cylindrical fibers. J Colloid Interf Sci 2002;245:325–37. [20] Parnas RS, Salem AJ. A comparison of the unidirectional and radial in-plane flow of fluids through woven composite reinforcements. Polym Compos 1993;14(5). [21] Adams KL, Russel WB, Rebenfeld L. Radial penetration of a viscous liquid into a planar anisotropic porous medium. Int J Multiphase Flow 1988;14(2). [22] Nedanov Pavel B, Advani Suresh G. A method to determine 3D permeability of fibrous reinforcements. J Compos Mater 2002;36(2):241–54. [23] Hoes K, Dinescu D, Sol H, Vanheule M, Parnas RS, Luo Y, et al. New set-up for measurement of permeability properties of fibrous reinforcements for RTM. Composites A: Appl Sci Manufac 2002;33(7):959–69. [24] Stadfeld HC, Erninger M, Bickerton S, Advani SG. An experimental method to continuously measure permeability of fiber preforms as a function of fiber volume fraction. J Reinfoced Plastics Compos 2002;21:879–900. [25] Laval N, Sozer EM, Advani SG. Characterization of permeability around a 90o corner. Adv Compos Lett 2000;9(1): 17–24. [26] Slade J, Sozer M, Advani S. Effect of shear deformation on permeability of fiber preforms. J Reinforced Plastics Compos 2000;19:552–68. [27] Luce TL, Advani SG, Howard JG, Parnas RS. Permeability characterization. Part 2: flow behavior in multiple-layer preforms. Polym Compos 1995;16:447–58. [28] Lawrence Jeffrey M, Barr John, Karmakar Rajat, Advani Suresh G. Characterization of preform permeability in the presence of race tracking. Composites Part A: Appl Sci Manufac 2004;35:1393–405. [29] Devillard Mathieu, Hsiao Kuang-Ting, Gokce Ali, Advani Suresh G. On-line characterization of bulk permeability and racetracking during the filling stage in resin transfer molding process. J Compos Mater 2003;37:1525–41.
Permeability estimation algorithm to simultaneously characterize the distribution media and the fabric preform in vacuum assisted resin transfer molding process Ali Gokce a, Mourad Chohra a
a,b
, Suresh G. Advani
a,b,*
, Shawn M. Walsh
c
Center for Composite Materials, Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA b Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA c Army Research Laboratory, Aberdeen Proving Grounds, MD, USA Received 3 November 2004; received in revised form 21 April 2005; accepted 2 May 2005 Available online 29 June 2005
Abstract In vacuum assisted resin transfer molding (VARTM), available pressure for resin injection is limited to 1 atm. Therefore, layers of high-permeability fabrics, called the distribution media (DM), are included in the lay-up to enhance the resin flow. During injection, resin flows initially in-plane saturating the DM, and subsequently through the thickness impregnating the bulk preform. Therefore, the DM permeability governs the planar resin flow pattern and lead length, which is defined as the difference between the flow position in the DM and in the bottom preform layer adjacent to the tool surface. In this study, the DM permeability was investigated as a function of the preform lay-up. A new permeability estimation method, called permeability estimation algorithm (PEA), that can estimate permeability values of the DM and the preform simultaneously from a single experiment, was developed and validated. Using PEA, it has been shown that the DM permeability does vary with the preform lay-up. 2005 Elsevier Ltd. All rights reserved. Keywords: E. Resin transfer molding; A. Fabrics/textiles; A. Hybrid compounds; B. Modelling; Permeability
1. Introduction Vacuum assisted resin transfer molding (VARTM), a low-cost process to manufacture composite structures with moderate dimensional tolerances, is a subclass of liquid composite molding (LCM) processes. In LCM, layers of fiber reinforcements are placed in a mold cavity and a liquid resin is injected into the mold to impregnate the empty spaces between the fibers. The VARTM process employs a single sided mold tool. The fiber reinforcement is placed on top of the tool and is covered by a distribution medium (DM) with a high in-plane permeability to accel*
Corresponding author. Tel.: +1 302 831 8975; fax: +1 302 831 3619. E-mail address: [email protected] (S.G. Advani). 0266-3538/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2005.05.012
erate the in-plane flow. Thus, a typical VARTM lay-up comprises fiber reinforcements, surface or inter-laminar DM, chemical agents/textiles to minimize defects and facilitate de-molding. A vacuum bag is used to seal the assembly as shown in Fig. 1. Vacuum is drawn through the vacuum line to remove air from the mold cavity, induce fiber compaction under atmospheric pressure and draw resin into the mold cavity through the injection line. After the resin arrives at the vent, the injection is discontinued but the vacuum is maintained until the resin cures allowing one to de-mold the part. Although its origins can be traced back to early 1950s [1], VARTM and similar processes were developed primarily in the 1980s. Researchers investigated several issues of the process such as resin flow [2–4], preform compaction [5,6] and flow control [7–9].
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Fig. 1. Schematics of VARTM.
Complete saturation of the bulk preform is critical for an acceptable part. The resin flow through the porous media is governed by Darcys law: hui ¼
K rhP if ; g
ð1Þ
where hui is the volume-averaged velocity vector, K is the second order permeability tensor of the porous medium, g is the viscosity and $hPif is the gradient of the pressure averaged over the fluid volume. Insertion of Eq. (1) into the continuity equation ($ Æ hui = 0) results in an elliptic partial differential equation. For constant pressure injection, the governing equations can be solved for one-dimensional (1D) flow as, x2 K ðDP Þt; ¼ 2 g/
ð2Þ
where x is the flow front location, DP is the pressure difference between the injection gate and the flow front, / is the porosity and t is the time. However, analytical solution is difficult for 2D and 3D flows, hence numerical methods are employed. In this study, liquid injection molding simulation (LIMS) [10], a software package developed at the University of Delaware, is used as the process model. LIMS uses a finite element/control volume approach to simulate the resin flow through porous media in liquid composite molding processes using Darcys Law and mass and energy conservation equations [11]. LIMS has been integrated into MATLAB to create a virtual experimentation environment, where the studies presented in this paper were conducted. Permeability, K in Eq. (1), is used to quantify the resistance of a porous medium to fluid flow. Extensive research has been conducted to predict and measure the permeability of the porous media used in LCM processes through analytical, numerical or experimental methods. In analytical studies, the fluid dynamics equations are solved for a representative domain of an idealized porous medium [12–14]. The use of analytical models is limited to idealized geometries of cylindrical/ elliptical arrays, which are approximations to the actual preform architectures. Numerical methods can be used to investigate complex porous structures, by solving the flow equations in the flow medium [15–19]. In this
method, problem complexity and computational time becomes an issue when a satisfactory performance is needed for an actual preform. Carefully conducted unidirectional, 2D or 3D experimental permeability measurements can return reliable results [20–27], but only for the given experimentation conditions. In addition, experimental methods are susceptible to disturbances such as racetracking [28,29]. None of the existing methods addresses the situations where the LCM lay-up includes dissimilar porous media, hence multiple unknown permeability values, such as in VARTM can be calculated from the same experiment. Although [4] offers a closed form flow solution for VARTM, it has limited applicability due to the assumptions made, and for a certain window in the parameter space its result include significant error. In reported experimental permeability measurement methods, the flow data from a set of experiments are input into analytical solutions such as Eq. (2), to solve for the permeability values. Since analytical solutions are available only for 1D flow, such methods have limited use and are susceptible to disturbances such as racetracking. In this paper, a new permeability estimation method that replaces the analytical step of the existing methods with numerical methods is proposed. This permeability estimation algorithm (PEA) was used to calculate the in-plane DM permeability and the preform permeability in the through thickness direction from the same experiment in this study. DM permeability is usually measured separately and the result is used to simulate resin flow in VARTM. However, DM permeability in a VARTM lay-up with preform underneath it and a vacuum bag on top could be significantly different from the one measured separately due to the dynamics of compression of the DM, fiber reinforcements and the vacuum bag. PEA allows in situ measurement of the DM permeability in the VARTM setup, and investigation of the interactions between various materials that form the lay-up. In the following section, PEA will be described and validated in the virtual domain. Also, its sensitivity to input errors will be studied. In Section 3, experimentation setup will be introduced, and the experimental results will be presented and analyzed.
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2. Permeability estimation algorithm 2.1. Procedure In a typical experimental permeability measurement operation, the inputs include the processing parameters such as injection pressure, resin viscosity, and material parameters such as preform porosity and dimensions. Flow data, which is defined as the resin flow front location as a function of time, can be obtained using point, lineal or optical sensors. In this study, the resin flow was taped using camcorders from the top through the transparent vacuum bag, and from the bottom through the transparent acrylic table, as shown in Fig. 2. The goal is to estimate the permeability values using the inputs and the flow data. No proven analytical relationship is available for use in this study, where the objective is to find the permeability of the distribution media and through-the-thickness permeability of fiber preforms. Developing a
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numerical formulation to solve for the permeability values directly is not straightforward. Therefore, LIMS, an established process model for resin flow, was incorporated into an iterative algorithm to estimate the permeability values as shown in Fig. 3. In this approach, the experimental VARTM setup is modeled in the simulation environment for use by LIMS. Each material in the VARTM lay-up is represented as a region. A sensory system is designed to track resin flow in each region using point, lineal or optical sensors. Once data is acquired on resin flow from the two regions, it is input to the PEA, which then searches iteratively for the permeability values that will reproduce the experimental flow behavior in the simulation environment. Starting with initial permeability values, at iteration j, resin flow is simulated with the permeability values returned from the previous iteration. Simulated resin flow is compared with the experimental flow data and the permeability values for the next iteration are estimated as follows. If point sensors are used
Fig. 2. Data acquisition setup used in this study.
Fig. 3. Flowchart of permeability estimation algorithm (PEA).
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porosity Resin pressure distribution
Mold cavity thickness
Resin flow field permeability
Fig. 4. Interaction of process parameters in VARTM process during resin injection.
K i;jþ1
p ti;exp ¼ K i;j ; ti;j
i ¼ 1; 2; . . . ; n
ð3Þ
where Ki,j+1 is the permeability value of region i for the next iteration, Ki,j is the current permeability value of region i, ti,exp is the experimental resin arrival time at the point sensor in region i, ti,j is the resin arrival time at the sensor in the simulation, p is the update power constant and n is the number of regions in the lay-up, and if lineal sensors are used p xi;exp K i;jþ1 ¼ K i;j at t ¼ tk ; i ¼ 1; 2; . . . ; n ð4Þ xi;j where xi,exp is the experimental flow position in region i at time t = tk and xi,j is the corresponding flow position in the simulation. In this study, the flow data was taken from the video recordings of the experiments using appropriate software in a way that mimics point sensors, hence Eq. (3) was used as the update law. The iterations stop either when maximum number of iterations has passed or when simulated flow data is sufficiently close to the experimental flow data: 8i; ¼
jti;j ti;exp j 6 e; ti;exp
i ¼ 1; 2; . . . ; n;
ð5Þ
where e is the convergence criterion. In case of online PEA use, iterations terminate when the next set of flow data arrives from the sensory system. The frequency of data acquisition depends on user preferences and sensory system capabilities. PEA can be used to find the average permeability of the entire wetted section of the regions or the sections that were wetted since the last data acquisition. In either case, only the essential period of the resin flow is simulated for computational efficiency. In this study, a single data set acquired towards the end of the filling operation was used to estimate the average permeability values of the regions. For higher dimensional flows, it is not possible to prove rigorously that the left hand side of Eq. (5) will converge to e using the update law in Eq. (3). Assuming that the permeability of a region has the most influence
on the resin flow through that region among all permeability parameters in the setup, it is clear from Eqs. (1) and (2) that the update law in Eq. (3) will drive permeability estimates in the right direction. For all our cases, it was observed that this assumption was justified. Note that several properties of a porous medium interact during injection as shown in Fig. 4. In this study, averaged values are used for all parameters except for the permeability, and the variations in other parameters were lumped into the permeability value. As long as consistent values are used in the calculations and the simulations, this approach is justifiable. 2.2. Validation The PEA was tested and validated in the virtual domain using flow data acquired from simulated VARTM experiments. The same material and process parameters as the virtual experiments were input to the models used in PEA. Since the exact permeability values used in the virtual experiments were known, it was possible to quantify the accuracy and efficiency of the PEA permeability estimates. In addition, permeability values for 1D flow were estimated by both PEA and Eq. (2), and the comparison of both methods is presented in Section 3.2. Since rectangular plate geometries were injected with negligible variation along the width direction, the crosssection of the VARTM setup was modeled as a 2D geometry as shown in Fig. 5. The mold cavity was divided into two regions: the perform layers as region I and the DM as region II. Two virtual point sensors were used to track resin flow in the regions: sensor I at the tool surface on data collection line I to track resin flow in region 1, and sensor II on the lay-up surface on data collection line II to track resin flow in the DM. Since sensor locations changed from one virtual experiment to the next, they are not included in Fig. 5. The DM was modeled using bar (1D) elements with one permeability component, KDM, which is associated with the flow data from sensor II, because it has a stronger influence on the flow along data collection line II compared
Fig. 5. Cross-section of the VARTM setup.
A. Gokce et al. / Composites Science and Technology 65 (2005) 2129–2139
ei ¼
K i;est K i;exp ; K i;exp
ð6Þ
where Ki,est is the permeability estimate of region i, and ei is the estimation error associated with the region. Five e values were tested: 0.1, 0.01, 0.001, 0.0001 and 0.00001. log (e) vs. |ei| · 100 and is plotted in Fig. 6 along with the number of iterations. The figure shows that the estimation errors drop quickly as e is reduced and the number of iterations is linearly related to log(e). In this study, e was chosen to be equal to 0.001. In 1D flow, p in Eq. (3), is equal to 1 according to Eq. (2) and the permeability estimation will take a single iteration. For higher dimensional flows, selection of a p value becomes complicated. Five p values were tried to find an optimal value for this study: 0.67, 0.93, 1.2, 1.53 and 2. The convergence behavior of the permeabilTable 1 The parameters of the primary virtual VARTM experiment Injection pressure, Pi (kPa) Resin viscosity, g (Pa s) DM porosity, /DM DM permeability, KDM (m2) Preform porosity, /p Preform permeability, x, Kxx (m2) Preform permeability, z, Kzz (m2) Mold length (m) Mold width (m) Preform thickness (m) DM thickness (m)
97 0.55 0.876 2 · 108 0.486 1010 8.33 · 1012 0.3556 0.1524 0.0036 2.13 · 104
80
error(%) and # of iterations
to other permeability values. The preform was modeled with 2D elements with two permeability components, Kxx andKzz, assuming principal permeability axes are aligned with the coordinate system. Since there is only one data input for the flow location in the preform -sensor I-, it is not possible to estimate both parameters independently. Since, in the preform, the resin flow is primarily through the thickness, Kxx was used as an input and Kzz was estimated using PEA. Kxx of the preform is much easier to measure separately in advance using the standard 1D experimental method. The material and process parameters of the primary virtual VARTM experiment are listed in Table 1. PEA was validated using this VARTM setup and its variations. In the testing and validation process, two major PEA parameters, p in Eq. (3) and e in Eq. (5), were varied to find their optimal values. Next, KDM and Kzz in Table 1 were varied to evaluate the performance of PEA for various virtual experiments. In the next section, PEA sensitivity to input errors will be addressed. In all of these studies, the permeability estimates returned by PEA will be compared with the permeability values used in the virtual experiments. Unless otherwise noted, the estimation error will be calculated using the equation below in the rest of the paper:
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70
K_DM error (% ) K_zz error(% )
60
iterations
50 40 30 20 10 0 0
-1
-2
-3
-4
-5
log (ε) Fig. 6. Permeability estimation errors and the number of iterations as a function of convergence criterion (p = 1.2).
ity estimates during PEA iterations were formulated by the ratio of the estimated values to the experimental values at each iteration (KDM,est/KDM,exp and Kzz,est/ Kzz,exp), and plotted against iteration number in Fig. 7 for several p values. As p is increased, the permeability estimates converge faster. However, oscillatory component appears at p = 1.53 and it intensifies at p = 2. p vs. number of iterations plot in Fig. 8 shows that the number of iterations decreases until oscillatory behavior appears, after which the number of iterations increase. In this study p was chosen to be as 1.2. PEA is not sensitive to initial permeability values due to its algorithm, which has been confirmed by the observations during the testing and validation process. Therefore, it was not studied in a systematic manner. After selection of e and p, PEA was used with various values for the DM and the preform in the virtual VARTM experiment to explore the accuracy and limitations of the algorithm. New virtual experiments were created and simulated by taking different values for KDM and Kzz in Table 1; the remaining parameter values were kept the same. Five different values were used for each permeability: K1 = KDM/Kxx = {1, 50, 200, 1000, 5000} and K2 = Kxx/Kzz = {1, 5, 12, 40, 120}, which creates a 5 · 5 configuration matrix C = {cij}, where i corresponds to K1 value and j corresponds to K2 value. Only the configurations on the diagonals of the matrix, ci,i and ci,5 i + 1 (i = 1 ,. . ., 5), a total of 9 configurations, have been addressed in order to observe the interaction of KDM and Kzz with less computation. The flow patterns, ei values, numbers of iterations and approximate data collection times are tabulated in Table 2. The estimate errors for configuration c15 are significantly high. Since, for KDM 6 Kxx, the DM does not have any influence on the resin flow except for acting as a sink due to high porosity, any KDM value lower than Kxx will satisfy the flow data from sensor I, resulting
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K_zz convergence
K_DM convergence 4
(K_ DM, est ) / (K_ DM, exp )
(K_ zz, est ) / (K_ zz, exp )
10000
1000
100
10 1 1
6 11 16 21 26 31 36 41 46 51 56 61 66 71 76
3 2.5 2 1.5 1 0.5 0 1
0.1
iteration number p=0.67
3.5
p=0.93
p=1.2
p=1.53
6 11 16 21 26 31 36 41 46 51 56 61 66 71 76
iteration number p=2
p=0.67
p=1.2
p=2
Fig. 7. Convergence behavior of the permeability estimates as a function of p. Kzz,est/Kzz,exp is plotted on a logarithmic scale to highlight the convergence behaviour from a much larger initial guess. KDM,est/KDM,exp is plotted on a linear scale and the number of lines are reduced by two to highlight the convergence behaviour around the equilibrium state.
large parameter space and the permeability estimates converge to the selected values asymptotically (Fig. 7).
80
# of iterations
75 70 65 60
2.3. Sensitivity to errors
55
In the actual experimentation environment, there are errors involved with the inputs; hence it is critical to study the sensitivity of PEA to input errors. Three parameters were selected for sensitivity study: Kxx, flow data (t1,exp and t2,exp), and injection pressure DP. Selected level of error was introduced for each of these parameters (Table 1) before it was supplied as input to the PEA. Subsequently, the effects of the input error on the permeability estimation errors (Eq. (6)) were studied. Due to the presence of DM, majority of the flow in the preform is through the thickness. However, there is a planar component due to Kxx, and it becomes stronger away from the DM layer, which causes the semi-elliptic flow front shape (Table 2). In PEA, Kxx is measured separately and entered as an input value. Since there are inherent variations in the permeability measurements, there will be some error introduced with this input. Five error levels were introduced in Kxx: E1 = Kxx,PEA/Kxx = {0.1, 0.3, 1, 3, 10}, where Kxx,PEA is the value that was used in the VARTM model of PEA, and the permeability estimations were found based on the incorrect value of Kxx. log (E1) vs. 100 · ei is plotted in Fig. 9. In general, if a lower Kxx value was used as input, PEA overestimated KDM and Kzz; in case a higher value was used, the permeability values were underestimated, which is in agreement with the physics of the process. As Kxx approaches zero on the left of the figure, the planar flow component in the VARTM model becomes more and more insignificant
50 45 40 35 30 0.6
1.1
1.6
2.1
p Fig. 8. Number of iterations as a function of update power constant, p.
in large estimation error. The error in Kzz is high because the resin flow is primarily planar with insignificant flow through the thickness, hence the relationship between the resin arrival time at sensor I and Kzz is very weak. Since the sensitivity of arrival times at sensors I and II are weak to the changes in Kzz and KDM, respectively, case c15 is an ill-conditioned reverse problem. To verify these conclusions, configuration c25 (KDM/Kxx = 50, Kxx/Kzz = 120) was also studied. The estimation errors dropped sharply with respect to c15, with eK zz ¼ 0.35% and eDM = 0.03% (Eq. (6)). All other estimation errors were less than 2.1%, many below 0.1%. The c55 case gives the best results because the sensitivity of arrival times at sensors I and II are very strong to the changes in Kzz and KDM, respectively. The numbers of iterations listed in Table 2 decrease as K1 and K2 increase. The results presented in this section illustrate that PEA can estimate permeability values accurately for a
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Table 2 Flow patterns of virtual VARTM experiments and PEA permeability estimation errors
The height of the mold is increased 5 times in the contour plots to provide better visibility.
40
percentage error
20 0 0.1
1
10
-20 -40 K_zz K_DM
-60 -80 -100
log(K_xx,PEA / K_xx) Fig. 9. Permeability estimation errors due to input error in Kxx.
and ei plateau towards the left. As Kxx increase towards the right, Kzz and KDM drop increasingly faster to reproduce the experimental flow data. In the limit, where Kxx is so high that resin arrives at the sensor locations earlier than experimental flow data through planar flow, there is no solution to the problem. The plots show that, as Kxx,PEA was varied two orders of magnitude, the perme-
ability estimates varied hardly one order of magnitude, hence it can be concluded that PEA is not very sensitive to error in Kxx,PEA. Note that it is safer to choose Kxx,PEA values from the lower end of the range, if the option exists. Due to hardware problems, low resin conductivity and noise, it is possible to introduce error into the flow data. Five error levels were introduced into flow data from sensors I and II: Ti = 100 Æ (ti,PEA ti,exp)/ ti,exp = {5, 1, 0, 1, 5} (i = 1, 2), where ti,PEA is the flow data value input to VARTM model in PEA, which yields a 5 · 5 configuration matrix A = {aij}, where i and j correspond to T1 and T2 values, respectively. Instead of all 25 configurations, only 9 configurations along the diagonals, ai,i and ai,5 i + 1 (i = 1, . . ., 5), have been studied to observe the interaction of both parameters with less computations. 100 · ei is plotted in Fig. 10. The error level in KDM is around the same magnitude as T2 at worst, since resin flow through the DM resembles 1D flow, and K 1/t in 1D flow (Eq. (2)). On the other hand, resin arrival at sensor I depends both on KDM and Kzz. Therefore, in order to match t1,PEA, Kzz varies significantly more than KDM for the same level of input error, creating larger error estimates. The results show that eK DM is larger for configurations ai,i, whereas eK zz
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K_zz error (%)
K_DM error (%) 6
90 80
5 70
4
60 50
3 40
2
30 20
1
T1=0
T1=-1
T1=-5
T2=5
T2=5
T2=1
T1=1
T2=0
T1=-5
T1=5
T2=-1
T1=-1
0
T2=-5
T1=0
T2=0
T1=1
T2=-1
T1=5
T2=-5
0
T2=1
10
Fig. 10. Estimation errors for various input error configurations in flow data.
25 20 Permeability error (%)
is larger for configurations ai,5 i + 1. Note that the lead length is very sensitive to Kzz and vice versa. Since t1,PEA and t2,PEA were perturbed at the same levels in configurations ai,i, the lead lengths remained unchanged; hence the small eK zz values in these configurations. However, in ai,5 i + 1 configurations, t1,PEA and t2,PEA were perturbed in opposite directions, which PEA interpreted as large variations in the lead length and responded with significant differences in Kzz estimations. For example, in configuration a15 (T1 = 5 and T2 = +5), due to the perturbations at the sensor data, the resin appeared to arrive earlier at sensor I and later at sensor II. In response, PEA overestimated Kzz by 85% to match early resin arrival at sensor I. Due to the increased Kzz, DM lost more resin to the preform on the flow front, causing a slowdown in flow in DM, which PEA addressed by increasing KDM slightly (0.125%) to match the resin arrival time at sensor II. It can be concluded that the Kzz estimates are sensitive to input errors in flow data; hence the sensory system must be designed with care. Note that, for a typical fill time of 300 s, 5% error in monitoring of the flow data is equal to a measurement error of around 15 s, which is high and not likely to happen with careful maintenance of the sensory hardware. During injection, there is some level of pressure loss in the tubing; the vacuum level may not be perfect and the pressure gage may have imprecision, which may create some input error in the injection pressure, Pi. Seven error levels were introduced into the pressure: P ¼ 1 0 0 ðP i;P E A P i Þ=P i ¼ f 1 5; 1 0; 5; 1; 0; 1; 5; 10; 15g, where Pi,PEA is the injection pressure used in the VARTM model. P vs. 100 Æ ei plot in Fig. 11 is consistent with K 1/$P relationship in Eq. (1), since $(cP) = c$P, where c is a constant.
K_zz error K_DM error
15 10 5 0 -20
-10
-5
0
10
20
-10 -15 -20 Pressure error (%)
Fig. 11. Permeability estimation errors as a function of input errors in injection pressure.
It is clear from Eqs. (1) and (2) that permeability estimation errors due to g and /, are proportional to the errors involved with the measurement of these parameters. In summary, PEA is quite stable against input error. During evaluation of the actual experimentation results, maximum potential permeability estimation errors will also be calculated.
3. Experiments 3.1. Design of experiments S51 Shade Cloth from Roxford Fodell is commonly used as a DM in VARTM. Three VARTM setups have
A. Gokce et al. / Composites Science and Technology 65 (2005) 2129–2139
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Fig. 12. Snapshots from the top (left) and bottom (right) recordings from an experiment. The flow front distance data is taken from the recording and the time information is taken from the software.
been designed for injection and each experiment has been repeated three times: (I) DM only, (II) DM with 4 layers of 24-ounce 5 · 4 woven E-glass, and (III) DM with 6 layers of the same preform. Other common fabric layers that are used to facilitate processing were omitted to prevent flow disturbances. Injection pressure Pi was measured using pressure gages mounted on the vacuum buckets and viscosity g was measured using Brookfield viscometer. To estimate the porosity of a fabric, its solid thickness h was estimated as h = (mass per area)/(material density) first. Next, porosity was estimated by / = 1h/(thickness in layout). The in-plane permeability of the preform, Kxx, was measured separately through 1D injection using Eq. (2). Different numbers of preform layers were used in the experiment. Three separate measurements were done for each configuration. The average of all the measurements was used as input in PEA. Flow was taped from the top and the bottom of the mold, and flow data was acquired manually in a way that mimics point sensors. Fig. 12 shows snapshots from the video of an experiment: the flow front location is measured using the metric drawn on the mold surface and time information is taken from the software. The Table 3 The parameter value ranges measured during the VARTM experiments Injection pressure, DP (kPa) Resin viscosity, g (Pa s) DM porosity /DM Preform porosity /p *
Preform permeability, x Kxx (4 layers) Kxx (6 layers)
84.7–96.5 0.48–0.64 0.778–0.792 0.430–0.526 1.66 · 1011 2.63 · 1011
Mold length (m) Mold width (m) * DM thickness (m)
0.381–0.508 0.127–0.178 8.08 · 104
Total lay-up thickness (m) 4 layers 6 layers
3.22–3.28 (·103) 4.18–4.23 (·103)
Parameters preceded by an (*) sign are obtained through separate measurements.
flow data was entered into the PEA along with other material and processing parameters, whose value ranges are tabulated in Table 3. 3.2. Results and discussion Three 1D experiments were conducted for each of 4layer and 6-layer lay-ups with preform only, and the average in-plane permeability values (Kxx) were found as 1.66 · 1011 and 2.63 · 1011, respectively. The measurements are consistent with the fact that as the number of layers increase, the permeability increases due to reduced compaction (lower fiber volume fraction). Three experiments were carried out to measure the DM permeability without placing any preform layers under it (Setup I). PEA and Eq. (2) were used to estimate KDM. The results from both methods have been tabulated in Table 4. The comparison of the results shows that PEA results are accurate. For the set of experiments conducted with the VARTM lay-up that contains preform layers under the distribution media (Setup II and III), the average Kzz and KDM values are listed in Table 5. Note that KDM values are nearly the same for VARTM setups II and III, which are around 45% lower than the measurement from setup I. The standard deviation for Kzz estimations is 8.02 · 1014, which is a small value. The results in Table 5 show that DM permeability depends on the VARTM lay-up. Using typical parameter values along with KDM value measured by PEA, it took 586 s to fill the VARTM part. If PEA was not available to measure KDM in situ, and the value from setup I was used, the fill time would be predicted as 336 s. Such a discrepancy is likely to create problems with larger molds. The overall estimation error range can be estimated by accumulating the effects of all input errors as shown below for KDM: K DM ¼ f ðK xx ; time=sensors; P ; g; /Þ. Also, eK DM ¼ f ðeK xx ; esensors ; eP ; eg ; e/ Þ.
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Table 4 Comparison of permeability estimates by Eq. (2) and PEA Exp. ID 1 2 3
KDM, Eq. (2) (m2) 9
4.998 · 10 4.338 · 109 4.467 · 109
KDM, PEA (m2) 9
4.995 · 10 4.408 · 109 4.446 · 109
Standard deviation 2.12 · 1012 4.95 · 1011 1.48 · 1011
Table 5 Permeability estimates as found by PEA using data from VARTM experiments Setup ID
Kzz (m2)
KDM (m2)
I II III
N/A 9.90387 · 1013 8.7702 · 1013
4.60096 · 109 2.5033 · 109 2.6441 · 109
Assuming the effects of various input errors to permeability estimation errors are not coupled, that is, error in one input does not amplify estimation error because of error in another input, one can estimate the maximum cumulative error in the permeability estimation as follows: DM DK DM K ¼ ð1 ^eK DM Þ DM K ð1 ^eK xx Þð1 ^esensors Þð1 ^eP Þ ð1 ^eg Þð1 ^e/ Þ;
ð7Þ
DM is the average permeability estimate, DKDM where K is the maximum error range, ^eK DM is the maximum cumulative error level and ^ej (j = Kxx, sensors, P, g, /) is the maximum error in KDM estimate due to the input error in j. Eq. (7) applies to Kzz estimation as well. The sensitivity of permeability estimation error to input errors was investigated in Section 2.3, which will provide some of the variables in Eq. (7). The maximum ratio between a Kxx measurement and the mean of all Kxx measurements were 1.4. Fig. 9 shows that, such an input error will cause 3.5% error in K DM ð^eK xx ¼ 0.035Þ, and 10.1% error in K zz ð^eK xx ¼ 0.101Þ. The flow location and time was taken from snapshots when flow was at least more than 1300 away from the injection line after 320 s later. Within these conditions it is reasonable to take the error due to human eye as 2%, which will induce 2% maximum error in K DM ð^esensors ¼ 0.02Þ, and around 29% maximum error in Kzz estimation ð^esensors ¼ 0.29Þ according to Fig. 10. The measurement error in the pressure gages are taken as 2% as no data could be found ð^eP ¼ 0.02Þ. The measurement error in the viscometer is given as 1% by the manufacturer ð^eg ¼ 0.01Þ. The porosity measurement includes working with water, scales and caliper s. Since all tools used are precise, cumulative error can be safely predicted as 3% ð^e/ ¼ 0.03Þ. Then, maximum cumulative error in KDM estimation can be found as 11%, and 40% for Kzz
estimation. Note that errors due to numerical solutions and simulations are not in the scope of this study. The computational overhead of PEA depends on the size of the finite element model used. PEA saves computational time by simulating the resin flow only through the mold section whose permeability values are being estimated. In this experimental study, the VARTM setup was modeled with 1206 nodes and 1200 elements. PEA took around 40 iterations to converge with e = 0.001, where 85% of the mold was filled at each iteration. The estimation took around 20 s on PC with a Pentium 4 – 3.06 GHz CPU and 1 GB RAM.
4. Summary and conclusions Permeability Estimation Algorithm (PEA), a new experimental permeability estimation method that uses a numerical process model instead of analytical relationships, was presented. Subsequently, it was used to find preform permeability in the thickness direction and the in-plane distribution media permeability from one experiment using the flow front data from an experiment. The accuracy of PEA was validated in the virtual environment. PEA was also used for permeability measurements in 1D flow experiments and its results were verified by Eq. (2). PEA, unlike existing experimental permeability measurement methods, allows one to calculate permeability of different materials in the same lay up. As PEA uses a numerical simulation and an iterative solver, it can be adjusted to accommodate various configurations and the flow does not have to be 1D. For example, it can be used to measure the permeability of a preform despite the presence of racetracking; or it can be used to find the principal permeability values of a preform using flow data from a radial injection. The results presented in Section 2 show that PEA converges on accurate permeability estimates asymptotically, hence it is stable. In this study, PEA has been successfully applied to permeability measurements in VARTM, revealing the fact that DM permeability depends on the VARTM lay-up; hence it should be measured in situ. Many VARTM systems of interest are increasingly using a combination of materials (e.g., graphite and glass) which makes permeability assessment difficult from a purely closed-form analytical approach. The PEA prescribes a means by which such hybridized systems can be accommodated and a reasonable estimate of the permeability field obtained. Note that numerical methods inherently include some level of error. PEA finds the permeability estimates that reproduce the actual flow data. If the same finite element model is used for both PEA and other tasks (optimal design, process control, etc.), then numerical error introduced into the solution will be reduced.
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Acknowledgments We thank Niki Frangakis for assisting in the experimental work. The financial support, provided by the Office of Naval Research (ONR) under Grant N0001403-1-0891 for the Advanced Materials Intelligent Processing Center at the University of Delaware, and the Center for Materials Technology established by Army Research Laboratory at the University of Delaware made this work possible.
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