Temperature rise in a verging annular die

3.1.1 Physical intuition

Figures 2 and 3 show that the molten plastic moves in the r-direction only:

Figure 2:

Velocity profile for pressure-driven flow through converging coapical concentric conical annulus (κ=513,θ_i =11°, θ_o =31°).

Figure 3:

Velocity profile for pressure-driven flow through diverging coapical concentric conical annulus (κ=513,θ_i =11°, θ_o =31°).

where:

for converging flows, and:

for diverging flows.

3.1.2 Equation of continuity in spherical coordinates (r, θ, φ)

From the equation of continuity (Eq. B.4-3 of [2]), for constant density and using Eq. (7) we get:

and differentiating gives:

which we will use to simplify the rr-component of the rate of deformation tensor, and also, to simplify the r-component of the equation of motion.

3.1.3 Equations of motion in spherical coordinates (r, θ, φ)

The r-component of the equation of motion (Eq. B.5-7 of [2]), after applying Eq. (7) and ϕ symmetry, reduces to:

Each relevant component of the rate of deformation tensor (§B.1 of [2]), simplified with Eqs. (8), (9), (10) or (11) then gives:

Hence τ_θθ =τ_φφ, so that combining Eq. (12) with Eqs. (10) and (11) yields:

3.1.4 Eliminating stresses

Combining Eqs. (13) and (14) with the constitutive equation for Newtonian fluids:

yields:

We can eliminate the stresses from Eq. (15), and since C depends on θ only:

Neglecting both fluid inertia and gravity we get:

for which the variables easily separate:

and thus:

which is easily integrated to give the (nonlinear) pressure profile:

from which we learn that the pressure depends at most on r, and which is subject to the pressure boundary conditions:

where, for converging flows, the outer pressure must exceed the inner:

and for diverging:

Solving Eqs. (23) and (24):

where κ is defined in Eq. (2). Hence:

Returning to the right side of Eq. (20) gives:

which has the general real solution:

where:

and where C[=]C₂[=]C₃[=]C₁/μ[=]L³/t.

Eq. (31) is subject to the no-slip boundary conditions:

and hence:

where C₁ is given by Eq. (27), and where:

Solving Eqs. (35) and (36) and substituting the results into Eq. (31) gives:

Substituting this into Eq. (10) we get the exact solution for the velocity profile in spherical coordinates for pressure-driven flow through conical coapical coaxial annulus:

be it converging or diverging. Differentiating Eq. (42) and solving for θmin,vr we get:

For converging (and diverging) flows, θmin,vr<(θi+θo)/2, with θ_i, θ_o ∈[0; π/2], and thus the minimum (and maximum) velocity cones are nearer the inner surface than the outer. This is why shear rates γ˙rθ (and thus the shear stresses τ_rθ ) on the inner cone are higher than on the outer (for both converging and diverging flows).

We now turn our attention to the ϕ-component of the equation of motion (Eq. B.5-9 of [2]) which, following Subsections 3.1.3 and 3.1.4, and since we are neglecting gravity, gives:

Eliminating the stresses using:

we get,

from which we learn that the pressure does not depend on ϕ.

Similarly, for the θ-component (Eq. B.5-8 of [2]), by following Subsections 3.1.3 and 3.1.4 we get:

and then using:

to eliminate the stresses gives:

and integrating yields:

Substituting Eq. (29) into Eq. (50) gives:

Solving for C₅ at θ=θ_o we get:

Substituting Eqs. (27), (41) and (52) into Eq. (50) yields the two dimensional nonlinear pressure profile:

with:

For most plastics extrusion die designs:

Therefore Eq. (53) reduces to Eq. (29) and our assumption that p depends on r only (see Subsection 3.1.4) is confirmed.

3.1.5 Nondimensional velocity and pressure

We begin our nondimensionalization by rearranging:

from Table 3, to give:

which, substituted into Eqs. (32) and (33), yields:

Next, in Eq. (41), we let:

so that:

Substituting Eq. (27) into Eq. (61), we get:

Using this with Table 3 gives:

and thus:

which we will use below. Also letting:

so that:

Substituting Eqs. (64) and (66) into Eq. (10), rearranging, and using Table 3, we get the dimensionless velocity profile:

where κ is defined in Eq. (2).

Using Table 3 with Eq. (29) and rearranging gives the dimensionless pressure profile:

3.1.6 Nondimensional 𝒬/Δ𝒫

Integrating Eq. (42) twice, with respect to θ and ϕ, and then rearranging gives:

where:

with a_o , b_o , a_i and b_i defined in Eqs. (37)–(40). Equation (70) is the first main result of this paper. Using Tables 2 and 3, we adimensionalize Eq. (69) to get:

Figure 10 illustrates Eq. (71) and shows how the dimensionless flow rate per unit pressure drop, f_s, changes with the extrusion die shape, θ_i and θ_o . Practitioners can use Figure 10 to get the volumetric flow rate graphically, or the pressure drop (see worked example in Section 5).

In Figure 11, we will compare the previous approximate solutions of Parnaby and Worth [3] [Eq. (1)] and of Liang [6] [Eq. (5)] with our exact solution [Eq. (71)]. For this, we must nondimensionalize Eqs. (1) and (5) in the same way as our Eq. (71). For Parnaby and Worth [3] [Eq. (1)], we get:

where:

where Ω is defined in Eq. (3), and for Liang [6] [Eq. (5)], we get:

with f_L defined in Eq. (6).

3.2.1 Physical intuition

Figures 4 and 5 show how the temperature profile evolves in the dies for converging and diverging flows. When viscous dissipation matters, the temperature rises as the melt flows from r_o to r_i (r_i to r_o) for converging (diverging) flows. Not unlike v_r (r, θ), the temperature also depends on both r and θ, T(r, θ). Thus, for converging flows through isothermal die walls we have:

Figure 4:

Temperature rise profile for pressure-driven flow through converging coapical concentric conical annulus (κ=513,θ_i =11°, θ_o =31°).

Figure 5:

Temperature rise profile for pressure-driven flow through diverging coapical concentric conical annulus (κ=513,θ_i =11°, θ_o =31°).

and:

whereas for diverging flows we have:

and:

For both converging and diverging flows, we define:

which defines the hottest surface in the flowing melt, which is at most a function of r.

3.2.2 Viscous dissipation term in spherical coordinates (r, θ, φ)

We begin by combining the general expression for the viscous dissipation in spherical coordinates (Eq. (C) of Table A.7-3 of [2]) with our physical intuition for the velocity, Eq. (7):

which when combined with the velocity profile and its derivative, Eqs. (10) and (11), gives:

from which we eliminate the stresses using Eqs. (17), (45), τ_rr =-2τ_θθ and τ_φφ=τ_θθ to get:

which we will use in Subsection 3.2.3, and where the exact solution for C is given by Eq. (41).

3.2.3 Equation of energy in spherical coordinates (r, θ, φ)

We begin by simplifying the equation of energy written in terms of temperature for an incompressible fluid (Eq. B.8-3 of [2]), with ϕ symmetry for the longitudinal heat flux, q_φ, gives:

Eliminating the heat fluxes using Fourier’s law (Eqs. B.2-7 and B.2-8 of [2]) we get:

Now, if we assume that axial conduction is negligible (with respect to radial convection):

then:

where α≡k/ρC^p is the thermal diffusivity (see Table 2). Substituting Eqs. (10) and (83) into (87) and rearranging gives:

which is subject to the boundary conditions given by Eqs. (77) and (79).

3.2.4 Dimensionless temperature rise

Using Tables 2 and 3, we rewrite Eq. (88) as:

which uncovers the characteristic time:

so that:

where we have also uncovered the Péclet number:

Further, using Table 3, yields:

which we will use presently.

In this Subsection, we begin by neglecting conduction in the θ-direction, and finish with an exact analytical solution for the temperature rise that is consistent with isothermal, inner and outer die walls, Eqs. (77) and (79). Neglecting conduction in the θ-direction (Pé≪1), Eq. (93) reduces to:

and has the exact solution:

subject to the entrance condition, which for converging flows is:

so that:

and for diverging flows is:

so that:

Using Eq. (133), proven in Appendix 7.2, we get:

which, when applied to Eqs. (97) and (99) gives:

where the condition A≪1 [Eqs. (54) and (55)] on our pressure and velocity [Eqs. (29) and (42)], upon which Eqs. (101) and (102) are based, is already sufficient for A²≪1. The reader is reminded that, for most plastics extrusion die designs, A≪1.

Since ℂ(θ_o)=ℂ(θ_i)=0, the temperature rise on the die surfaces is also zero, Θ(θ_o)=Θ(θ_i)=0. In other words, Eqs. (101) and (102) are exactly consistent with the isothermal wall boundary conditions Eqs. (77) and (79), which we have yet to use.

Using Table 3, Eqs (101) and (102) become:

with A already defined in Eq. (54). We next differentiate Eqs. (103) and (104) to get the hottest surface in the flowing melt:

which is conical, and which matches the velocity minimum cone, θ=θmin,vr, given by Eq. (43). We thus find that the temperature maximum cone θ=θ_max,T is nearer the inner surface than the outer.

Using Table 3 to rewrite Eqs. (103) and (104) we get:

Eqs. (106) and (107) give the nondimensional temperature rise for converging and diverging flows, respectively, and these are thus the second main result of this work. Equations (106) and (107) are exactly consistent with the isothermal wall boundary conditions Eqs. (77) and (79). In Section 4, we will differentiate Eqs. (106) and (107) to calculate the corresponding cooling requirement for conical die surfaces.

4.1.1 Converging flows

The accelerating velocity for converging die flows is negative, and the pressure drop in Eq. (42), is positive. Figure 6 illustrates the velocity profile shape for a particular converging die design. We see that the minimum velocity cone is nearer the inner surface. Using Eq. (43) for the specific die considered, this minimum velocity cone is θmin,vr≃12.42°, which is just inside the midcone, θ=12.5°.

4.1.2 Diverging flows

The decelerating velocity for diverging die flows is positive, and the pressure drop in Eq. (42) is negative. Figure 7 illustrates the velocity profile shape for a specific diverging die design. We see that the maximum velocity cone is nearer the inner surface. Using Eq. (43) for the specific die considered, this maximum velocity cone is θmax,vr≃12.42°, which is just inside the midcone, θ=12.5°.

4.2.1 Converging flows

In Figure 8, we plot the dimensionless temperature rise per unit pressure drop, Θ(𝓡, ξ)/ℙ which shows the shape of the temperature rise for a specific converging die design. The surface of maximum temperature rise is conical θmax,T=θmin,vr≃12.42°, which matches the minimum velocity cone. Figure 8 also shows that the wall temperatures are constant at T₀ and thus the walls are isothermal. To achieve this condition, to compensate for viscous heating, the walls must be cooled.

The expression for the required heat flux for cooling is given by (Eq. B.2-8 of [2]):

Substituting Eq. (106), into Eq. (108), and evaluating the result for the inner surface gives the cooling load requirement:

and for the outer surface: