Heat Transfer Handbook part 128 pptx

BOOKCOMP, Inc. — John Wiley & Sons / Page 1267 / 2nd Proofs / Heat Transfer Handbook / Bejan PROCESSING OF POLYMER-MATRIX COMPOSITE MATERIALS 1267 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1267], (37) Lines: 943 to 963 ——— 5.72406pt PgVar ——— Normal Page PgEnds: T E X [1267], (37) through application of an external pressure (or vacuum) cycle on the laminate. The process is influenced by the resin viscosity, which is a function of the temperature and degree of cure in the laminate, and the applied pressure. The resin viscosity decreases initially with temperature; however, as the cross-linking reactions proceed, the vis- cosity sharply increases near the gel point, at which the viscosity is theoretically infinite. Thus, the cure pressure cycle is designed to take advantage of the window during processing when the viscosity is low. Mathematically, the consolidation process is described as thatofresinflow through a porous bed formed by the reinforcing fibers. Models for the consolidation process have been presented by Springer (1982), Davè et al. (1987), and Gutowski et al. (1987), among others. The models by Davè et al. (1987) and Gutowski et al. (1987) consider the composite to contain a deformable fiber network in which resin flow in all directions is governed by Darcy’s law. The fiber network also takes part in carrying the load due to the applied pressure (or vacuum) during processing. The total force σ acting on the porous fiber bed is balanced by the sum of the force due to the springlike behavior of the fiber network p and the hydrostatic force due to the pressure of the resin in the layup, P. The hydrostatic pressure is obtained as the solution of the governing equation for consolidation of a porous bed within a given time interval with three-dimensional flow and a one-dimensional confined compression condition (no boundary motion in the x and y directions), given by (Davè et al., 1987) 1 µm v  ∂ ∂x  κ x ∂P ∂x  + ∂ ∂y  κ y ∂P ∂y  + ∂ ∂z  κ z ∂P ∂z  = ∂P ∂t (17.41) where µ is the viscosity of the resin in the porous fiber bed; κ x , κ y , and κ z are the specific permeabilities in the x, y, and z directions, which depend on the stress level; t is time; and m v is the coefficient of volume change, which describes the stress–strain behavior of a body in confined compression. For a porous medium, it is the ratio of change in porosity to axial (normal) stress for confined compression of the porous body with its vertical side faces constrained from any motion (Kardos, 1997). Solution of the foregoing equation requires information on the anisotropic perme- ability tensor and its variation as the fiber bed consolidates during processing. Several theoretical and experimental studies have been conducted to determine this informa- tion (Gutowski et al., 1986; Lam and Kardos, 1991; Adams and Rebenfeld, 1991a,b; Skartsis et al., 1992; Skartsis and Kardos, 1990), which is provided predominantly in terms of the Carman–Kozeny relationship κ = ϕ 3 (1 − ϕ) 2 1 sk o (17.42) where ϕ is the resin volume fraction; k o is the Kozeny constant, determined empiri- cally; and s is the specific surface of the fibers and is related to the fiber radius r f as 4/r 2 f . The Kozeny constant is reported to remain relatively constant over a wide range of resin volume fractions, except near the extreme limits of ϕ. For flow parallel to the BOOKCOMP, Inc. — John Wiley & Sons / Page 1268 / 2nd Proofs / Heat Transfer Handbook / Bejan 1268 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1268], (38) Lines: 963 to 993 ——— 0.41615pt PgVar ——— Normal Page PgEnds: T E X [1268], (38) fibers, values of k o in the range 0.5 to 0.7 have been reported, while for transverse flow the values range from 11 to 18 (Gutowski et al., 1987; Lam and Kardos, 1988). Viscoelastic characteristics of the resin, such as near the gel point, are accounted for through the use of a pseudo-Kozeny constant, k ∗ o , which is a function of both fluid and fiber bed properties (Skartsis et al., 1992). Resin viscosity is another important parameter that affects resin flow and transport of voids during the consolidation process. The viscosity is a function of the temper- ature and the degree of cure and is often given by an empirical correlation of the form µ = µ ∞ exp  E RT + λε  (17.43) where µ ∞ is a constant, E the activation energy for viscosity, and λ a constant that is independent of temperature, all of which are determined empirically. For the Hercules 3501–6 resin, Lee et al. (1982) reported the values of the model parameters as: µ ∞ = 7.93 × 10 −14 Pa · s,E = 9.08 × 10 4 J/mol, and λ = 14.1 ± 1.2. Gutowski and co-workers (Gutowski, 1985; Gutowski et al., 1987) modeled fiber deformation by assuming that a composite is a porous, nonlinear elastic medium that is filled with a viscous resin. They modeled the deformation of bundles of fibers as beams bending between multiple contact points, and derived an expression for the stiffness p as p = 3π E β 4  v f /v i − 1   v a /v f − 1  4 (17.44) where E is the bending stiffness of the fiber and v a , v f , and v i are the maximum fiber volume fraction, the instantaneous fiber volume fraction, and the initial fiber volume fraction, respectively. The term β is a geometric parameter that is related to the fiber architecture and fiber diameter. For well-aligned fiber bundles, the parameters, β, v i , and v a , determined by fitting experimental measurements to the model, were given by Gutowski and co-workers as β = 225, v i = 0.50, and v a = 0.829. The values increase with increasing fiber alignment. A model for diffusion-controlled void growth and dissolution in an epoxy resin system during consolidation was developed by Kardos et al. (1986). The model provides the size of a void located in an infinite isotropic medium as a function of the processing parameters and identifies conditions under which void growth can be prevented or voids can be made to collapse during the cure process. Based on their model, they proposed that the resin pressure (atm) at any point within the laminate being cured must be greater than a minimum value as given below: P ≥ P min P min = 4.962 × 10 3 exp  − 4892 T  ω o (17.45) where P is the pressure (atm) in the resin, obtained as solution of eq. (17.41), P min the minimum resin pressure required to prevent void growth by moisture diffusion at BOOKCOMP, Inc. — John Wiley & Sons / Page 1269 / 2nd Proofs / Heat Transfer Handbook / Bejan PROCESSING OF POLYMER-MATRIX COMPOSITE MATERIALS 1269 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1269], (39) Lines: 993 to 1013 ——— -0.073pt PgVar ——— Normal Page PgEnds: T E X [1269], (39) any time during cure, w o the relative humidity (%) to which the resin in the prepreg is equilibrated prior to processing, and T the temperature (K). Design of the cure process calls for selecting the optimum temperature and pres- sure cycles so as to minimize the fabrication time and simultaneously minimize the void content and processing-induced stresses. Optimization of cure cycles utilizing the physical models has been reported by Pillai et al. (1994), Rai and Pitchumani (1997a,b), Diwekar and Pitchumani (1993), and Pitchumani and Diwekar (1994). 17.7.3 Processing of Thermoplastic-Matrix Composites Unlike thermosetting resins, thermoplastic melts have significantly higher viscosity, which renders fabrication of quality composites via impregnation of a net-shaped fiber structure as a single step impractical. Continuous fiber-reinforced thermoplastic composites are commonly fabricated in two stages, as shown schematically in Fig. 17.22. In the first stage, called prepregging, thin reinforcement layers are impregnated with the thermoplastic to form prepregs (short for preimpregnated reinforcements). Because the reinforcement layers in prepregs are usually about 150 to 200 µmin thickness, a high degree of impregnation can be achieved under controlled conditions. Thermoplastic prepregs are commercially available in a variety of widths, ranging from small-width ribbons or tows to wider tapes and sheets, and have a low void con- tent with a fairly uniform fiber distribution. Processes for the fabrication of prepregs are outside the scope of this chapter (discussed in detail in Pitchumani, 2002). The focus of this section is on the second stage, referred to as consolidation, in which the prepregs are stacked to the desired shape and thickness and fusion- bonded and solidified to obtain the final composite product. Consolidation of prepreg layers is achieved by a number of processes, including prepreg layup with autoclave Stage 1: Prepregging Stage 2: Consolidation Fiber Matrix Prepregging Prepreg Prepreg Heating + Pressure Fusion bonding Composite Product T t Solidification (crystallization) Figure 17.22 Two main steps in thermoplastic composite processing. BOOKCOMP, Inc. — John Wiley & Sons / Page 1270 / 2nd Proofs / Heat Transfer Handbook / Bejan 1270 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1270], (40) Lines: 1013 to 1020 ——— 1.097pt PgVar ——— Normal Page PgEnds: T E X [1270], (40) V z y Prepreg supply spool Hot gas torch F Roller Tool Figure 17.23 Tow placement/tape laying process with in situ consolidation. consolidation, and pultrusion, which are similar to processes described in the context of thermosetting matrix composites processing, with the notable difference that there is usually a cooling step to solidify the consolidated composite. Thermoplastic tape laying and automated tow placement, illustrated in Fig. 17.23, are processes based on incrementally laying down and continuously consolidating prepreg layers to build the composite product. The nip point formed by the incoming tow and the substrate layer at the entry to the roller region is heated by an appropriate means (such as the hot-gas torch shown in Fig. 17.23). The mechanisms leading to consolidation and bonding take place under the roller, and the fully consolidated structure emerges ready to form the substrate for new tows added during subsequent passes of the process. The rollers and torches, along with the supply spool, are mounted on a common frame, called the tow-placement head, which is translated with a line speed V during the process. Unlike the autoclave process, which involves batch consolidation of tow layers, tow placement and tape-laying processes involve consolidation in situ and offer the potential for rapid fabrication. These processes can be used to produce part geometries with intricate features and are particularly suited for fabrication of large structures such as aircraft wing skins and the fuselage (Lamontia et al., 1992, 1995). Tape laying and tow placement differ principally in the size of the prepregs used; while wider tapes are used in tape laying, the prepreg tows used in tow placement are smaller (on the order of about 0.25 in. wide). In this process, a thermoplastic impregnated tow is passed under a lay-down roller onto a substrate formed of previously deposited and consolidated tows on a cylindrical or flat tool. Fabrication of composites from prepregs using the aforementioned processes is based on the principle of fusion bonding, which fundamentally consists of application of heat and pressure at the interface of two layers of thermoplastic prepregs in contact, BOOKCOMP, Inc. — John Wiley & Sons / Page 1271 / 2nd Proofs / Heat Transfer Handbook / Bejan PROCESSING OF POLYMER-MATRIX COMPOSITE MATERIALS 1271 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1271], (41) Lines: 1020 to 1028 ——— 0.097pt PgVar ——— Normal Page PgEnds: T E X [1271], (41) Interface softening Pressure q . Intimate contact Tow compaction/ void reduction Healing Bonded material Figure 17.24 Mechanisms involved in the fusion bonding process. and subsequently, cooling down of the interface to obtain a bonded product. The mechanisms involved in fusion bonding of two thermoplastic surfaces are illustrated schematically in Fig. 17.24. The applied high temperatures cause the interface to soften, and the simultaneous application of pressure serves to flatten the surface asperities and establish an area contact at the interface, referred to as the intimate contact process. Further, the elevated temperatures cause an interdiffusion of polymer molecules, termed autohesion or healing, across the interfacial areas in intimate contact, resulting in the development of bond strength in the laminate. Polymer degradation refers to the cumulative effect of exposure of the polymer matrix to high temperatures during the process. The cooling down of the bonded layers causes solidification of the molten thermoplastic matrix, which in the case of semicrystalline thermoplastics, influences the crystalline morphology of the polymer matrix in the composite product. The dominant transport mechanisms involved in the process and their relation- ship to the pressure and temperature cycles applied are shown in Fig. 17.25. As the material is heated, when its temperature reaches a certain value known as the glass transition temperature T g , the crystal structure of semicrystalline thermoplastics be- gins to break, and material softening takes place. At temperatures exceeding T g , the crystalline structure disintegrates progressively until the material melting point T mp is reached, whereupon all crystallinity is lost and the polymer is fully molten. For amorphous thermoplastics, owing to the absence of any significant crystalline struc- ture, the glass transition temperature and the melting point are nearly identical. As seen in Fig. 17.25, the material temperature exceeding the glass transition temperature is a prerequisite for most of the mechanisms of fusion bonding, while heat transfer and polymer degradation occur throughout the process. Healing takes place as long as the temperature is above the melting point, and polymer crystallization accompa- nying solidification occurs when the material is cooled down from the melting point BOOKCOMP, Inc. — John Wiley & Sons / Page 1272 / 2nd Proofs / Heat Transfer Handbook / Bejan 1272 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1272], (42) Lines: 1028 to 1028 ——— * 61.927pt PgVar ——— Normal Page PgEnds: T E X [1272], (42) Temperature Temperature Pressure Time, t T mp T c T g Applied pressure Heat transfer 1 2 3 4 5 Void dynamics/tow compaction Polymer crystallization (Solidification) Polymer degradation Interfacial bonding Healing ( > )TT mp Intralaminar void growth (=0; > )pTT g Intimate contact (>0; > )pTT g Intralaminar void reduction tow compaction (>0; > )pTT g Figure 17.25 Dominant mechanisms during thermoplastic composites processing and their relationship to the temperature and pressure cycles. BOOKCOMP, Inc. — John Wiley & Sons / Page 1273 / 2nd Proofs / Heat Transfer Handbook / Bejan PROCESSING OF POLYMER-MATRIX COMPOSITE MATERIALS 1273 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1273], (43) Lines: 1028 to 1042 ——— -1.62994pt PgVar ——— Normal Page PgEnds: T E X [1273], (43) past the crystallization temperature (T c ) to the glass transition point. Intimate contact, intralaminar void reduction (i.e., reduction of voids present within prepregs), and tow compaction processes prevail during the application of pressure, whereas intralami- nar void growth occurs in the absence of applied pressure and while the temperature is greater than the glass transition point. The mechanisms are discussed in detail next by considering the tape-laying and tow-placement processes. HeatTransfer Heat transfer during thermoplastic composites processing has been studied extensively by many investigators [see Pitchumani (2002) for a detailed list] in the context of autoclave, filament winding, tape/tow placement, and pultrusion processes. The goal of heat transfer analysis is to predict the transient temperature field within the tow layers, which is utilized in the analysis of the other mechanisms involved in the process. The thermal model formulation consists of the energy equa- tion in an appropriate coordinate system with a source term corresponding to the heat of crystallization. The material domain modeled is usually two-dimensional along the fiber length (the y direction) and through the thickness of the tow layers (the z di- rection). Considering a tape laying or tow-placement process for rectangular product geometries, the governing equations for the transient temperature field in the com- posite may be written as follows: ∂ ∂t (ρcT ) + ∂ ∂y (ρcV T ) = ∂ ∂y  k y ∂T ∂y  + ∂ ∂z  k z ∂T ∂z  + ρ m (1 − v f ) ∆H c d ˆc dt (17.46) where ρ and c are the density and specific heat of the composite medium, T the temperature, t the time, V refers to the tow velocity in the direction of travel, and v f refers to the fiber volume fraction in the composite. The anisotropic conductivities of the composite medium, k y and k z , are evaluated as the conductivities of an equivalent homogeneous medium. The longitudinal conductivity in the fiber direction (k y ) is determined using the rule of mixtures, which is simply a volume average of the fiber and the matrix conductivities, while the transverse conductivity (k z ) is obtained by any of a number of analytical models available [see Hashin (1983), Han and Cosner (1981), and Pitchumani (1999) and references therein]. The last term in eq. (17.46) denotes the source/sink effects due to the crystalliza- tion, in which ∆H c is the heat of crystallization, ρ m the matrix density, and d ˆc/dt the crystallization rate. Generally, the magnitude of the crystallization source term is very small in comparison to the other terms in the energy equation, during the heat- ing process, and it has been customary to neglect the source term in the heat transfer analysis of the process during the heating stage. Crystallization, however, plays a significant role in the cooling-down step, which is discussed later in the section on solidification. The governing equations are subject to initial conditions on the tem- perature and appropriate boundary conditions, specific to the process and the product geometry under consideration. For example, in the tape-laying or tow-placement pro- cesses (Pitchumani et al., 1996), the bottom surface of the prepreg layers in contact BOOKCOMP, Inc. — John Wiley & Sons / Page 1274 / 2nd Proofs / Heat Transfer Handbook / Bejan 1274 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1274], (44) Lines: 1042 to 1055 ——— 0.0pt PgVar ——— Normal Page PgEnds: T E X [1274], (44) with a tool surface is subject to a prescribed temperature either with perfect contact or with a finite contact resistance. The top surface condition is, in general, a combination of convection to the ambient, exposure to the heat source, and contact with the roller or tool/die surfaces. The heat transfer coefficients and contact resistance values used in the problem formulation are generally determined from experiments, or assumed empirically. The equations are solved numerically using a finite difference or finite element method. Void Dynamics The application of a pressure cycle to a prepreg tow heated to above its glass transition temperature causes deformation, referred to as tow com- paction, due to flow of the softened material, whereby the tow thickness decreases accompanied by a corresponding increase in the width. During the process, owing to the high viscosity of thermoplastics, the fibers in the prepregs move along with the thermoplastic resin rather than relative to the resin (Pitchumani et al., 1996; Ran- ganathan et al., 1995). Therefore, the process may best be described as a squeeze flow instead of a Darcy flow. The fiber–resin–voids mixture is modeled as an equivalent homogeneous fluid with the rheological properties of the continuum dependent on the temperature, the fiber volume fraction, and the void content. The compaction process is accompanied by void reduction, and the regions outside the compaction zone subject to high temperature are areas of void growth. The void reduction and void growth mechanisms are collectively referred to in this discussion as void dynamics. Several mechanisms contribute to void dynamics, including void migration, void compression and expansion, void coalescence, gas diffusion from the void to the melt, and void bubbling (Ranganathan et al., 1995). The dominant consolidation-related void dynamics mechanisms in thermoplastics processing are those of void migration along with resin and void compression due to the effects of cooling, and compaction under the applied pressure. The diffusion of gases across the void–tow melt interface may be assumed to be negligible in the analysis (Pitchumani et al., 1996; Ranganathan et al., 1995), owing to the poor solubility of the gases in the thermoplastic melt. The effect of void migration is accounted for in a macroscopic flow model, while the void compression effects are considered in a microscopic void dynamics model. These two models are coupled and necessitate a simultaneous solution for the void fraction in the composite. The macroscopic flow model further yields the pressure field in the consolidation region, which governs the intimate contact process discussed in a later subsection. Pressure Field in the Consolidation Region (Macroscopic Model) Pitchumani et al. (1996) and Ranganathan et al. (1995) presented models for tow compaction for consolidation under a roller as encountered in tow/tape placement, and filament winding processes. Figure 17.26 shows an enlarged view of the modeling domain, which is the region under the compaction roller. A tow of a given height h i and width w i enters the region under the roller with a specified line speed V , and in the compaction process, its height reduces to h f while its width increases to w f . Since the tow dimension in the y direction is much larger than the x and z dimensions, flow in the y direction may be neglected. Further, owing to the high viscosity of the BOOKCOMP, Inc. — John Wiley & Sons / Page 1275 / 2nd Proofs / Heat Transfer Handbook / Bejan PROCESSING OF POLYMER-MATRIX COMPOSITE MATERIALS 1275 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1275], (45) Lines: 1055 to 1067 ——— 2.51701pt PgVar ——— Normal Page PgEnds: T E X [1275], (45) z Fiber-resin mixture Voids Section A-A ()a ()b ()c h 1 h f h v y z L c A Consolidation region Ty,z() y x x w 1 w f Compaction roller D r A Figure 17.26 Region under the consolidation roller illustrating the tow compaction process. matrix resin, the inertial effects may be neglected and the consolidation process may be treated as a creeping flow problem. The fluid motion under the compaction roller is governed by the continuity and momentum equations in Cartesian coordinates, which may be simplified by utilizing the fact that the tow thicknesses (typically about 0.006 in.) are small relative to the width and length. Further, because the quantity of interest in the consolidation region is the pressure field under the rollers and not the actual velocity profiles, the governing equations may be cast in an integral form, as described in Pitchumani et al. (1996). The resulting integral equation is given by eq. (17.47), which constitutes the macroscopic governing equation for determining the pressure distribution under the rollers. h ∂ρ ∗ ∂τ + ρ ∗ dh dτ + ∂ ∂x  ρ ∗  h 0  v x (0) + dp dx  z 0 ξ µ dξ + C 1 (x)  z 0 1 µ dξ  dz  = 0 (17.47) where the process is assumed to be at steady state, ξ is a dummy variable of inte- gration, C 1 (x) is a constant of integration, ρ ∗ is the density of the fiber–resin–voids mixture scaled with respect to the density of the mixture in the absence of voids, and τ is the Lagrangian time, which is related to the line speed V and the location under the roller (measured from the entrance to the roller region) y,asτ = y/V. The term h BOOKCOMP, Inc. — John Wiley & Sons / Page 1276 / 2nd Proofs / Heat Transfer Handbook / Bejan 1276 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1276], (46) Lines: 1067 to 1091 ——— -0.30486pt PgVar ——— Normal Page PgEnds: T E X [1276], (46) is the instantaneous thickness of the tow under the compaction roller, which is related to the under-roller distance y, based on geometric considerations (Pitchumani et al., 1996), and v x (0) is the width-wise velocity component at the tow–substrate interface (z = 0 in Fig. 17.26). The boundary conditions on the pressure p correspond to the tow being uncon- strained along its width, which implies that p(x =±w/2) = p atm , the atmospheric pressure. Furthermore, the unknowns C 1 (x) and v x (0), which result from integration of the continuity equation across the instantaneous tow thickness, are determined us- ing the no-slip condition at the tow–substrate interface and a partial slip condition at the tow–roller interface (Pitchumani et al., 1996). Void Compression (Microscopic Model) The time derivative of the dimension- less density, ∂ρ ∗ /∂τ, appearing in eq. (17.47) is evaluated from a microscopic model which accounts for void compression (one of the void reduction mechanisms) during the consolidation process. A typical void at any location may be approximated by a sphere of radius R and is assumed to be surrounded by a concentric spherical resin shell of outer radius S. The ratio of R and S is determined by the void fraction at the location under consideration. Void growth or collapse is governed by a balance between the pressure inside and outside the void and the surface tension σ and resin viscosity µ. Using conservation of the incompressible resin mass, and the ideal gas law for the gas within the void, the microscopic void dynamics equation may be writ- ten as (Pitchumani et al., 1996) 4µ  R ∗3 S ∗3 o + R ∗3 − 1 − 1  dR ∗ dτ +  p go R ∗3 T T o − p f  R ∗ µ − 2σ µR o = 0 (17.48) where the bubble and outer shell radii are scaled with respect to the initial radius of the void, R o , which is determined based on the void fraction in the tow at the entrance to the consolidation roller. In eq. (17.48) p g and p go are the instantaneous and initial pressures inside the voids, respectively, and the fluid pressure surrounding the void is labeled p f . Further, for the concentric spherical shell description, the rate of change of ρ ∗ with respect to time can be expressed in terms of the rate of change of the nondimensional radius of void as (Ranganathan et al., 1995) ∂ρ ∗ ∂τ = −3R ∗2  S ∗3 o − 1   S ∗3 o − 1 + R ∗3  2 dR ∗ dτ (17.49) For a set of given initial conditions on the void radii, density, pressure, and temper- ature, the rate of change of the nondimensional density with respect to time may be obtained using eqs. (17.48) and (17.49). This expression may be used in eq. (17.47) to compute the pressure distribution in the consolidation region. Equation (7.14) can then be used to compute the change in radius of the voids and update the void radius at various locations in the domain. Similarly, eq. (17.49) may be integrated numerically to obtain the local densities in the tow as a function of x and τ. . parallel to the BOOKCOMP, Inc. — John Wiley & Sons / Page 1268 / 2nd Proofs / Heat Transfer Handbook / Bejan 1268 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1268],. processing. BOOKCOMP, Inc. — John Wiley & Sons / Page 1270 / 2nd Proofs / Heat Transfer Handbook / Bejan 1270 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1270],. melting point BOOKCOMP, Inc. — John Wiley & Sons / Page 1272 / 2nd Proofs / Heat Transfer Handbook / Bejan 1272 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1272],