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--- en:grundlagenhandbuch:aufschmelzberechnung [2026/05/24 09:12] – [Melting Model for Compact Solids] neelest
+++ en:grundlagenhandbuch:aufschmelzberechnung [2026/09/04 09:40] (aktuell) – [Temperature of the Solids at the Location of the First Melt] paal
@@ Zeile 2: / Zeile 2: @@
 ===== Phenomenology=====
-The feet material is held in underneath the opening by the screw-elements and is transported in the screws as a solid in the direction of the screw-tips. Through contact with the heated cylinder wall, the particles in the proximity of the cylinder wall begin to sinter and melt on until finally a broad standing connected molten film over the solid tailback in the intermeshing area is formed. This procedure is supported through in the practice conventionally carried-out reductions of the short pitch of the screw elements. These are found in front of the geometric plastification zone, through which an extra compression of the solids occurs and thus a better contact with the heated cylinder wall can be achieved. At the same time, granule particles lying on the underside of this layer are heated through convection.
+The feed material is held in underneath the hopper opening by the screw-elements and is transported in the screws as a solid in the direction of the screw-tips. Through contact with the heated cylinder wall, the particles in the proximity of the cylinder wall begin to sinter and melt on until finally a continuous molten film over the solid tailback in the intermeshing area is formed. This process is supported in practice by reducing the pitch of the screw elements.. These are found in front of the geometric plastification zone, through which an extra compression of the solids occurs and thus a better contact with the heated cylinder wall can be achieved. At the same time, granule particles lying on the underside of this layer are heated through convection.
-Through the formation of the molten film, the friction conditions in the mesh area and in the solid tailback change behind the intermeshing area. Through this process, the forced conveying of solids in a intermeshing region breaks down, and material is forced through the nip region. As a result of the 3-dimensional speed profile in the intermeshing area, an intensive mixture of the materials occurs. If the proportion of ready-fused materials is sufficient, a dispersion from the existing granules is formed from the solid phase and from the existing viscous phase polymer melt.
+Through the formation of the molten film, the friction conditions in the mesh area and in the solid tailback change behind the intermeshing area. Through this process, the forced conveying of solids in an intermeshing region breaks down, and material is forced through the nip region. As a result of the 3-dimensional speed profile in the intermeshing area, an intensive mixture of the materials occurs. If the proportion of ready-fused materials is sufficient, a dispersion from the existing granules is formed from the solid phase and from the existing viscous phase polymer melt.
-If the melted share for a completely formed dispersion is not yet sufficient an agglomeration of granule particles occurs. These are then further fused through the dissipation of the already existing melt, and through heat conduction of the surrounding walls. Also the agglomerates are further more molten and trans-ported during their transportation through convection and through passing of further intermeshing areas in the already stated condition of dispersion.
+If the melted share for a completely formed dispersion is not yet sufficient an agglomeration of granule particles occurs. These are then further fused through the dissipation of the already existing melt, and through heat conduction of the surrounding walls. Also the agglomerates continue to melt and are transported during their transportation through convection and through passing of further intermeshing areas in the already stated condition of dispersion.
-After the achievement of this condition, the further melting of the not yet molten Polymer particles can only be successful through the heat conduction from the hot melt to the solid granule particles. This is because the granule particles no longer have direct contact with the heated cylinder walls, i.e. they are longer touching the heated screw surface.
+After the achievement of this condition, the further melting of the not yet molten Polymer particles can only be successful through the heat conduction from the hot melt to the solid granule particles. This is because the granule particles no longer have direct contact with the heated cylinder walls, i.e. they no longer touch the heated screw surface
-A condition for the described melting behavior is a sufficiently long solids-conveying section. In practice, inserted screw configurations point out that with a short conveying section, combinations of plastic elements with subsequent dust particles, results in only a partial filling in the plastification zone.
+A condition for the described melting behavior is a sufficiently long solids-conveying section. However, the screw configurations used in practice feature a combination of kneading and restrictor elements after a short conveying zone. This results in the plasticising zone being partially filled.
 If the available length of conveying section to the molten film over the solid tail-back does not suffice, so the forced conveying of the solids remains up to the location where the first filling with melt took place. Principally, if the solid that holds a significantly larger speed component in the axial direction than the melt, reaches the melt filled area, and melt can penetrate in the cavities of the granules, then the forced conveying will collapse. In front of the location of the first filling up with melt, through a sufficiently large mass throughput, originates one with a granule filled screw area, in which, the granules rotate around the screw like in the figure.
@@ Zeile 83: / Zeile 83: @@
 For the training of a dispersed particle distribution at the location of the first filling, a minimum share of the already present melt $S_0$ is necessary. This is shown by:
-$$S_0 = 1 - F_0 \tag{4.3}$$
+$$S_0 = 1 - F_0 \tag{11}$$
 Whereby $F_0$ represents the solid section. This solid section is identical to the bulk density and can be calculated using the assumptions of a central, cubic-faced particle formation.
@@ Zeile 102: / Zeile 102: @@
 The coupling of the energy theorem and kinetics gives the differential calculus for the spherically symmetric temperature field.
-$$\frac{\partial \theta}{\partial \tau} = \frac{1 \delta}{\xi^2 \delta \xi}\left(\xi^2 \frac{\delta \theta}{\delta \xi}\right) \tag{1}$$
+$$\frac{\partial\theta}{\partial\tau} = \frac{1}{\xi^2}\frac{\partial}{\partial\xi}\left(\xi^2\frac{\partial\theta}{\partial\xi}\right)\tag{12}$$
-whereby these standardization's:
+whereby these standardizations:
-$$\theta = \frac{T - T_0}{T_0 - T_U} ; \tau = \frac{at}{r_0^2} ; \xi = \frac{r}{r_0} \tag{2}$$
+$$\theta = \frac{T - T_0}{T_0 - T_U} ; \tau = \frac{at}{r_0^2} ; \xi = \frac{r}{r_0} \tag{13}$$
-were introduced. For the case of a unique erratic temperature change from the starting temperature $T_0$ up to the environment temperature $T_U$, the solution of the differential calculus (2) is given by  [[en:grundlagenhandbuch:aufschmelzberechnung#references|[NN88]]]. For sufficient lengths of time t in-side particles of finite expansion, similar temperature profiles can be expected. They are then described by the location function $f(x)$, which with extra time, experiences scaled reductions  [[en:grundlagenhandbuch:aufschmelzberechnung#references|[NN88]]]:
+were introduced. For the case of a unique erratic temperature change from the starting temperature $T_0$ up to the environment temperature $T_U$, the solution of the differential calculus 13 is given by  [[en:grundlagenhandbuch:aufschmelzberechnung#references|[NN88]]]. For sufficient lengths of time t in-side particles of finite expansion, similar temperature profiles can be expected. They are then described by the location function $f(x)$, which with extra time, experiences scaled reductions  [[en:grundlagenhandbuch:aufschmelzberechnung#references|[NN88]]]:
-$$\theta = g(\tau) \cdot f(\xi) \tag{3}$$
+$$\theta = g(\tau) \cdot f(\xi) \tag{14}$$
 If one puts this product approach into the differential calculus so with the first order conditions one gets:
-$$\theta(\xi, 0) = 1 \tag{4}$$
+$$\theta(\xi, 0) = 1 \tag{15}$$
 and the boundary conditions
-$$\left(\frac{1}{Bi} \frac{\partial \theta}{\partial \xi} + \theta\right)_{\xi=1} = \begin{cases}1 \text{ für } \tau \leq 0\\0 \text{ für } \tau > 0\end{cases} \tag{5}$$
+$$\left(\frac{1}{Bi} \frac{\partial \theta}{\partial \xi} + \theta\right)_{\xi=1} = \begin{cases}1 \text{ for } \tau \leq 0\\0 \text{ for } \tau > 0\end{cases} \tag{16}$$
 as a solution of the average caloric temperature of the particle:
-$$\bar{\theta} = \sum_{i=1}^{\infty} c_i(m_i)D_i(m_i)e^{-m_i^2\tau} \tag{6}$$
+$$\bar{\theta} = \sum_{i=1}^{\infty} c_i(m_i)D_i(m_i)e^{-m_i^2\tau} \tag{17}$$
 with:
-$$m_i = \left(1 - \frac{\alpha_L \cdot r_0}{\lambda}\right)\frac{\cos(m_i)}{\sin(m_i)} \tag{7}$$
+$$m_i = \left(1 - \frac{\alpha_L \cdot r_0}{\lambda}\right)\frac{\cos(m_i)}{\sin(m_i)} \tag{18}$$
-$$c_i(m_i) = 2\frac{\sin(m_i) - m_i \cos(m_i)}{m_i - \sin(m_i)\cos(m_i)} \tag{8}$$
+$$c_i(m_i) = 2\frac{\sin(m_i) - m_i \cos(m_i)}{m_i - \sin(m_i)\cos(m_i)} \tag{19}$$
-$$D_i(m_i) = 3\frac{\sin(m_i) - m_i \cos(m_i)}{m_i^3} \tag{9}$$
+$$D_i(m_i) = 3\frac{\sin(m_i) - m_i \cos(m_i)}{m_i^3} \tag{20}$$
-For fair calculation, it is sufficient to consider the first four elements of the sum function. The constants $m_i$ are the roots of the transcended eqn. (7). With the absolute Biot number
+For fair calculation, it is sufficient to consider the first four elements of the sum function. The constants $m_i$ are the roots of the transcended eqn. 18. With the absolute Biot number
-$$Bi = \frac{\alpha_L \cdot r_0}{\lambda} \tag{10}$$
+$$Bi = \frac{\alpha_L \cdot r_0}{\lambda} \tag{21}$$
 The first four roots can be described through the following approximation equations:
-$$m_1 = \frac{a}{\left(1 + \frac{b}{Bi}\right)^c} \tag{11}$$
+$$m_1 = \frac{a}{\left(1 + \frac{b}{Bi}\right)^c} \tag{22}$$
-$$m_{2-4} = a \cdot \tanh\left(\frac{\ln Bi}{\ln b} - c\right) + d \tag{12}$$
+$$m_{2-4} = a \cdot \tanh\left(\frac{\ln Bi}{\ln b} - c\right) + d \tag{23}$$
 The constants a-d are stated in the table.
@@ Zeile 152: / Zeile 152: @@
 **Table:** Constants of the determination of the temperature function
-The figure shows the value of the first four roots taken from  [[en:grundlagenhandbuch:aufschmelzberechnung#references|[NN88]]] and the approximate values of those with the equations (11) and (12).
+The figure shows the value of the first four roots taken from  [[en:grundlagenhandbuch:aufschmelzberechnung#references|[NN88]]] and the approximate values of those with the equations 22 and 23.
 {{ :en:grundlagenhandbuch:aufschmelzberechnung:en_sigma150_dlg_grundlagenhandbuch_aufschmelzberechnung_004.svg?500%nolink |}}
@@ Zeile 160: / Zeile 160: @@
 The dimensionless Bios number is dependent upon the external heat transfer coefficient $\alpha_L$, which can be determined with the help of the dimensionless Nusselt number.
-$$Nu = \frac{\alpha_L \cdot \pi r_0}{\lambda_L} \tag{13}$$
+$$Nu = \frac{\alpha_L \cdot \pi r_0}{\lambda_L} \tag{24}$$
 The Nusselt number exposes itself from a laminate and turbulent section. For spherically formed single particles it is:
-$$Nu = 2 + \sqrt{Nu_{lam}^2 + Nu_{tur}^2} \tag{14}$$
+$$Nu = 2 + \sqrt{Nu_{lam}^2 + Nu_{tur}^2} \tag{25}$$
 with:
-$$Nu_{lam} = 0,664\sqrt{Re}\sqrt[3]{Pr} \tag{15}$$
+$$Nu_{lam} = 0,664\sqrt{Re}\sqrt[3]{Pr} \tag{26}$$
-$$Nu_{tur} = \frac{0,037Re^{0,8}Pr}{1 + 2,443Re^{-0,1}(Pr^{0,66} - 1)} \tag{16}$$
+$$Nu_{tur} = \frac{0,037Re^{0,8}Pr}{1 + 2,443Re^{-0,1}(Pr^{0,66} - 1)} \tag{27}$$
 The dimensionless Reynolds number considers the air flow in the screw channel.
@@ Zeile 178: / Zeile 178: @@
 It results in the air velocity being proportional to the conveying velocity of the particles.
-$$Re = \frac{w\pi r_0}{v} \tag{17}$$
+$$Re = \frac{w\pi r_0}{v} \tag{28}$$
 whereby $n_0$ is the screw speed and $t$ is the lead of the screw. The collection of solids in the screw channel reduces the flow effective channel cross section. This results in a rise in the flow velocity proportionate to the solid section $F_0$.
-$$w = \frac{n_0 t}{F_0}$$
+$$w = \frac{n_0 t}{F_0}\tag{29}$$
-The Prandtl number contained in eqn. 15 and 16 is a ratio of material data. Using the here used consideration with air as a surrounding, round-flowing medium, the Prandtl number can be approximated by ignoring the pressure de-pendency using
+The Prandtl number contained in eqn. 26 and 27 is a ratio of material data. Using the here used consideration with air as a surrounding, round-flowing medium, the Prandtl number can be approximated by ignoring the pressure de-pendency using
-$$\Pr(T)3,545 \cdot 10^{-7}T^2 - 1,309 \cdot 10^{-4}T + 0,7169 \tag{18}$$
+$$\Pr(T)3,545 \cdot 10^{-7}T^2 - 1,309 \cdot 10^{-4}T + 0,7169 \tag{30}$$
 Whereby the temperature has to be filled in with degrees Celsius.
@@ Zeile 208: / Zeile 208: @@
 The flow effective channel depth is thus a function of the dimensionless solid content:
-$$h(x,F) = (1 - F)h(x) \tag{1}$$
+$$h(x,F) = (1 - F)h(x) \tag{31}$$
 The calculation of the melt temperature development should be based on the following simplifications:
@@ Zeile 220: / Zeile 220: @@
 With these assumptions, the general energy equation reduces to a description of the temperature field.
-$$\frac{\partial T}{\partial z} = -\frac{1}{\rho_m c_p \bar{v}_z} \frac{\partial q_y}{\partial y} + \frac{(\tau\dot{\gamma})_j}{\rho_m c_p \bar{v}_z} e^{-\beta(T-T_j)} \tag{3}$$
+$$\frac{\partial T}{\partial z} = -\frac{1}{\rho_m c_p \bar{v}_z} \frac{\partial q_y}{\partial y} + \frac{(\tau\dot{\gamma})_j}{\rho_m c_p \bar{v}_z} e^{-\beta(T-T_j)} \tag{32}$$
 The heat flow $q_y$ is represented by the Fourier heat conduction equation.
-$$q_y = -\lambda \frac{\partial T}{\partial z} \tag{4}$$
+$$q_y = -\lambda \frac{\partial T}{\partial z} \tag{33}$$
 When including the considered characteristic value, the described differential equation becomes:
-$$\frac{\partial \theta}{\partial \zeta} = \frac{1}{Gz} \frac{\partial^2}{\partial \xi^2} + \frac{Br}{Gz} e^{-b T_z \theta} \tag{5}$$
+^                    ^                                  |
+| $\frac{\partial \theta}{\partial \zeta} = \frac{1}{Gz} \frac{\partial^2 }{\partial \xi^2} + \frac{Br}{Gz} e^{-\beta T_Z \theta}$        |  |
+|  $\theta = \frac{T - T_j}{T_z}$                                   | $\theta_0 = \frac{T_0 - T_j}{T_z}$ |
+|  $\xi= \frac{y}{h}$  | $\xi= \frac{z}{\Delta z}$    |
+| $Br = \frac{\overline{\tau \cdot \dot{y}} \, \bar{h}^2}{\lambda T_z}$ |    $Gz = \frac{c_p \rho \overline{h} \, }{\lambda \overline{b} \Delta z} \dot{V}_z$
 **Table:** Dimensionless characteristics of the calculation of the melt temperature
-For the reason of the exponential function, equation (5) is not solvable without further simplification. In order to make the problem solvable, the exponential function has been approximated by Potente  [[en:grundlagenhandbuch:aufschmelzberechnung#references|[Pot90]]] using a set of linear equations:
+For the reason of the exponential function, the equation is not solvable without further simplification. In order to make the problem solvable, the exponential function has been approximated by Potente  [[en:grundlagenhandbuch:aufschmelzberechnung#references|[Pot90]]] using a set of linear equations:
-$$e^{-b(T-T_0)} \approx C_1 - T_z \beta(T - T_0) \tag{6}$$
+$$e^{-b(T-T_0)} \approx C_1 - T_z \beta(T - T_0) \tag{34}$$
 The constants $C_1$ and $C_2$ are asserted for the specified areas by $\beta \cdot \Delta T$.
@@ Zeile 254: / Zeile 258: @@
   * No heat transfer to the screws occurs:
-$$\frac{\partial \theta_0}{\partial \xi} = 0 \text{ für } \xi = 0 \tag{7}$$
+$$\frac{\partial \theta_0}{\partial \xi} = 0 \text{ für } \xi = 0 \tag{35}$$
   * On the cylinder wall there is a given temperature $T_Z$:
-$$\theta_0(\xi = 0) = \theta_z = \frac{T_z - T_0}{T_z} \tag{8}$$
+$$\theta_0(\xi = 0) = \theta_z = \frac{T_z - T_0}{T_z} \tag{36}$$
   * The average mass temperature of the melt at the beginning of the calculation section is known:
-$$\int_0^1 \theta_0(\xi)d\xi = \theta_z = \frac{T_z - T_0}{T_z} \tag{9}$$
+$$\int_0^1 \theta_0(\xi)d\xi = \theta_z = \frac{T_z - T_0}{T_z} \tag{37}$$
   * An extreme should be permitted. This can be perceived by a large dissipation in the screw channel, or more specifically, in the melting zone.
@@ Zeile 268: / Zeile 272: @@
 A possible function that satisfies these conditions, is the superimposition of an exponential function with a second degree polynomial.
-$$\theta_0(\xi) = 0.0154\theta_z(1 - e^{1-\xi})^8 - 0.1232\theta_z(\xi - 1)^2 - 0.04704\theta_z \tag{10}$$
+$$\theta_0(\xi) = 0.0154\theta_z(1 - e^{1-\xi})^8 - 0.1232\theta_z(\xi - 1)^2 - 0.04704\theta_z \tag{38}$$
-The figure shows diagrammatically the starting temperature distribution for various cylinder wall temperatures according to equation (10).
+The figure shows diagrammatically the starting temperature distribution for various cylinder wall temperatures according to equation 38.
 {{ :en:grundlagenhandbuch:aufschmelzberechnung:en_sigma150_dlg_grundlagenhandbuch_aufschmelzberechnung_008.svg?600%nolink |}}
@@ Zeile 278: / Zeile 282: @@
 With this base temperature profile as well as the acceptance that the change in heat flow in the deep channel direction can be estimated by means of the starting temperature profile:
-$$\frac{\partial^2 \theta}{\partial \xi^2} \approx \frac{\partial^2 \theta_0}{\partial \xi^2} \tag{11}$$
+$$\frac{\partial^2 \theta}{\partial \xi^2} \approx \frac{\partial^2 \theta_0}{\partial \xi^2} \tag{39}$$
-you get the stated solution to the differential equation (5) in the table.
+you get the stated solution to the differential equation in the table.
@@ Zeile 295: / Zeile 299: @@
 $$+ 0,8624\theta_z e^{3(1-\xi)} - 1,0780\theta_z e^{4(1-\xi)} + 0,8624\theta_z e^{5(1-\xi)} - 0,4312\theta_z e^{6(1-\xi)}$$
-$$\left.+ 0,1232\theta_z e^{7(1-\xi)} - 0,0154\theta_z e^{8(1-\xi)} + 0,1232\theta_z\xi^2 - 0,2464\theta_z\xi\right)$$
+$$\left.+ 0,1232\theta_z e^{7(1-\xi)} - 0,0154\theta_z e^{8(1-\xi)} + 0,1232\theta_z\xi^2 - 0,2464\theta_z\xi\right)\tag{40}$$
-with $$\varepsilon = \frac{\theta_z}{Br C_2 \beta T_z}$$
+with $$\varepsilon = \frac{\theta_z}{Br C_2 \beta T_z}\tag{41}$$
 Since only the channel height averaged mass temperature is of interest, this can be calculated by:
-$$\bar{\theta} = \int_0^1 \theta_0(\xi)d\xi = \theta_z = \frac{T_z - T_0}{T_z} \tag{12}$$
+$$\bar{\theta} = \int_0^1 \theta_0(\xi)d\xi = \theta_z = \frac{T_z - T_0}{T_z} \tag{42}$$
 The figure shows a comparison of the documented solutions in Table against the publicized approaches of Potente  [[en:grundlagenhandbuch:aufschmelzberechnung#references|[Pot90]]] and Ansahl  [[en:grundlagenhandbuch:aufschmelzberechnung#references|[Ans93]]]. For small Graetz numbers, the publicized approach of Ansahl tends to infinity, whilst both of the other curves converge on a different threshold value. At this point, the energy supplied from shearing equals the energy expelled from the heat conduction. For large Graetz numbers, subtract the deviation in the dimensionless temperature between the solutions.
@@ Zeile 314: / Zeile 318: @@
 The neglecting of the real channel geometry makes the analytical solution of the reduced energy equation ($\frac{\partial \theta}{\partial \zeta} = \frac{1}{Gz} \frac{\partial^2 \theta}{\partial \xi^2} + \frac{Br}{Gz} e^{-bT_z\theta}$) possible. For the reason of self-cleansing by the co-rotating twin screw extruder, the channel geometry varies greatly from the shallow channel. Through these simplifications, it can lead to big mistakes in the calculation of the average dissipated output per unit of volume $(\overline{\tau\dot{\gamma}})_j$, which moreover, still possess a marked dependency upon the local degree of filling. For the rectangular channel, the average dissipated output can be estimated by:
-$$(\overline{\tau\dot{\gamma}})_j = \frac{K(T_j)v_0^{1+n}}{\bar{h}^{1+n}} \tag{1}$$
+$$(\overline{\tau\dot{\gamma}})_j = \frac{K(T_j)v_0^{1+n}}{\bar{h}^{1+n}} \tag{43}$$
 The changed averaged dissipated energy through the real channel geometry, as compared with the shallow channel, is taken into account by the correction factor $C_K$.
-$$(\overline{\tau\dot{\gamma}})_j = C_K \frac{K(T_j)v_0^{1+n}}{\bar{h}^{1+n}} \tag{2}$$
+$$(\overline{\tau\dot{\gamma}})_j = C_K \frac{K(T_j)v_0^{1+n}}{\bar{h}^{1+n}} \tag{44}$$
 For the determination of corrective factors, you have to approximate the real channel geometry using a step function.
@@ Zeile 324: / Zeile 328: @@
 Every interval i of the step function should possess, like the flat channel, a linear velocity distribution. If one lets the number of intervals tend towards infinity and the interval widths tend towards zero, it results in the average dissipated output:
-$$(\overline{\tau\dot{\gamma}})_j = \frac{1}{x_f + \frac{b_{max}}{2}} \int_{-\frac{b_{max}}{2}}^{x_f} \frac{K(T_j)v_0^{1+n}}{h(x)^{1+n}} dx \tag{3}$$
+$$(\overline{\tau\dot{\gamma}})_j = \frac{1}{x_f + \frac{b_{max}}{2}} \int_{-\frac{b_{max}}{2}}^{x_f} \frac{K(T_j)v_0^{1+n}}{h(x)^{1+n}} dx \tag{45}$$
 $x_f$ represents the position of the flow-front in the partially filled channel sections, assuming an ideal perpendicular flow-front. With the average effective channel depth
-$$\bar{h}(x_f) = \frac{1}{x_f + \frac{b_{max}}{2}} \int_{-\frac{b_{max}}{2}}^{x_f} h(x)dx \tag{4}$$
+$$\bar{h}(x_f) = \frac{1}{x_f + \frac{b_{max}}{2}} \int_{-\frac{b_{max}}{2}}^{x_f} h(x)dx \tag{46}$$
 solving for $C_K$:
-$$C_K = \frac{1}{x_f + \frac{b_{max}}{2}} \int_{-\frac{b_{max}}{2}}^{x_f} \left(\frac{\bar{h}(x_f)}{h(x)}\right)^{1+n} dx \tag{5}$$
+$$C_K = \frac{1}{x_f + \frac{b_{max}}{2}} \int_{-\frac{b_{max}}{2}}^{x_f} \left(\frac{\bar{h}(x_f)}{h(x)}\right)^{1+n} dx \tag{47}$$
-To avoid a null division for location $x=-b_{max}/2$ in equation (5), for the analogue to the output calculation, the practical channel profile should be used. Because of the segment definition of the channel profile, the determination of the intervals proves itself difficult, which is the reason for the integration being processed numerically. Equation (4) describes the filling degree dependent aver-age effective channel depth that has to be used for the plane channel model.
+To avoid a division by zero in the equation at the point $x=-b_{max}/2$, the practical channel profile must be used here as well, in the same way as for the power calculation. Determining the integral proves difficult due to the segmented definition of the channel profile, which is why the integration was performed numerically. The equation describes the flow depth-dependent mean effective channel depth, which is to be used for the flat channel model.
 ==== Calculation of the Solid Bed Reduction Along the Melt Path ====
@@ Zeile 340: / Zeile 344: @@
 For the physical mathematical description of the melting of single particles in a polymer melt, according to prerequisite 2, changing effects between adjacent particles should be neglected. The energy equation in sphere coordination is reduced with the consideration of stationary relationships and natural heat at constant solid data to:
-$$\rho c \frac{\partial T}{\partial t} = -\frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \dot{q}_r) \tag{1}$$
+$$\rho c \frac{\partial T}{\partial t} = -\frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \dot{q}_r) \tag{48}$$
 To obtain the description of heat conductivity, Fourier's differential equation has to be used:
-$$\dot{q}_r = -\lambda \frac{\partial T}{\partial r} \tag{2}$$
+$$\dot{q}_r = -\lambda \frac{\partial T}{\partial r} \tag{49}$$
 Put equation 2 into 1, under the assumption of constant material value to get:
-$$\rho c \frac{\partial T}{\partial t} = \frac{\lambda}{r^2}\left(2r \frac{\partial T}{\partial t} + r^2 \frac{\partial^2 T}{\partial r^2}\right) \tag{3}$$
+$$\rho c \frac{\partial T}{\partial t} = \frac{\lambda}{r^2}\left(2r \frac{\partial T}{\partial t} + r^2 \frac{\partial^2 T}{\partial r^2}\right) \tag{50}$$
 {{ :en:grundlagenhandbuch:aufschmelzberechnung:en_sigma150_dlg_grundlagenhandbuch_aufschmelzberechnung_010.svg ?300%nolink |}}
@@ Zeile 354: / Zeile 358: @@
 **Figure:** Sphere co-ordination on a solid particle
-From the mass balance on the sphere face (see figure) it results in a change in the mass of the sphere per unit of time (Equation 4) equal to the change of the mass of the melt per unit of time (Equation (5).
+The mass balance at the surface of the sphere shows that the change in the mass of the sphere per unit of time is equal to the change in the mass of the melt per unit of time.
 Solid:
-$$\frac{\partial m_f}{\partial t} = -4\pi r_G^2 \rho_f \frac{\partial r_G}{\partial t} \tag{4}$$
+$$\frac{\partial m_f}{\partial t} = -4\pi r_G^2 \rho_f \frac{\partial r_G}{\partial t} \tag{51}$$
 Melt:
-$$\frac{\partial m_s}{\partial t} = -4\pi r^2 \rho_s \frac{\partial r}{\partial t} \tag{5}$$
+$$\frac{\partial m_s}{\partial t} = -4\pi r^2 \rho_s \frac{\partial r}{\partial t} \tag{52}$$
-By equating (4) and (5) one gets:
+Setting the two equal gives:
-$$\frac{\partial r}{\partial t} = -\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \frac{\partial r_G}{\partial t} = -\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \frac{\partial r_G}{\partial_z} \frac{\partial_z}{\partial t} \tag{6}$$
+$$\frac{\partial r}{\partial t} = -\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \frac{\partial r_G}{\partial t} = -\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \frac{\partial r_G}{\partial_z} \frac{\partial_z}{\partial t} \tag{53}$$
-Solve equation (6) with respect to $\partial t$ and include the average flow velocity in the channel using $\left(\bar{v} = \frac{\partial z}{\partial t}\right)$ to get:
+Solve equation with respect to $\partial t$ and include the average flow velocity in the channel using $\left(\bar{v} = \frac{\partial z}{\partial t}\right)$ to get:
-$$\partial t = \left(-\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \right)^{-1} \partial r \tag{7}$$
+$$\partial t = \left(-\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \right)^{-1} \partial r \tag{54}$$
-Now put equation (7) through (3) with:
+Substituting gives:
-$$a_s = \frac{\lambda_s}{\rho_s c_p} \tag{8}$$
+$$a_s = \frac{\lambda_s}{\rho_s c_p} \tag{55}$$
 With the definition of the constants:
-$$\frac{\partial^2 T}{\partial r^2} + \left[\frac{1}{a_s}\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} + \frac{2}{r}\right] \frac{\partial T}{\partial r} = 0 \tag{9}$$
+$$\frac{\partial^2 T}{\partial r^2} + \left[\frac{1}{a_s}\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} + \frac{2}{r}\right] \frac{\partial T}{\partial r} = 0 \tag{56}$$
-From equation (8):
+From equation 55:
-$$A = \frac{1}{a_s} r_G^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \tag{10}$$
+$$A = \frac{1}{a_s} r_G^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \tag{57}$$
 The double integration of
-$$\frac{\partial^2 T}{\partial r^2} + \left[\frac{A}{r^2} + \frac{2}{r}\right] \frac{\partial T}{\partial r} = 0 \tag{11}$$
+$$\frac{\partial^2 T}{\partial r^2} + \left[\frac{A}{r^2} + \frac{2}{r}\right] \frac{\partial T}{\partial r} = 0 \tag{58}$$
 results in the following equation:
-$$T(r) = \frac{C_1}{A}e^{-\frac{A}{r}} + C_2 \tag{12}$$
+$$T(r) = \frac{C_1}{A}e^{-\frac{A}{r}} + C_2 \tag{59}$$
 Whereby C1 and C2 are the integration constants. With the following boundary conditions:
-$$T(r = \infty) = T_m \tag{13}$$
+$$T(r = \infty) = T_m \tag{60}$$
-$$T(r = r_G) = T_{fl} \tag{14}$$
+$$T(r = r_G) = T_{fl} \tag{61}$$
 It results in the integration constants:
-$$C_1 = A \frac{T_m - T_{fl}}{\exp\left(\frac{A}{r_G}\right) - 1} \tag{15}$$
+$$C_1 = A \frac{T_m - T_{fl}}{\exp\left(\frac{A}{r_G}\right) - 1} \tag{62}$$
-$$C_2 = T_m + \frac{T_m - T_{fl}}{\exp\left(\frac{A}{r_G}\right) - 1} \tag{16}$$
+$$C_2 = T_m + \frac{T_m - T_{fl}}{\exp\left(\frac{A}{r_G}\right) - 1} \tag{63}$$
 The solution of the differential equation results in:
-$$\frac{T_m - T(r)}{T_m - T_{fl}} = \frac{1 - \exp\left(\frac{A}{r}\right)}{1 - \exp\left(\frac{A}{r_G}\right)} \tag{17}$$
+$$\frac{T_m - T(r)}{T_m - T_{fl}} = \frac{1 - \exp\left(\frac{A}{r}\right)}{1 - \exp\left(\frac{A}{r_G}\right)} \tag{64}$$
 Which is equilibrium of heat flows.
@@ Zeile 413: / Zeile 417: @@
 On the interface of the sphere, every time period $t \geq t_0$ has to be applied:
-$\dot{q}_f|r_G = \dot{q}_s|r_G \tag{17}$
+$\dot{q}_f|r_G = \dot{q}_s|r_G \tag{65}$
 Wherein $\dot{q}_s$ is the melt-side heat-flow on the interface, and $\dot{q}_f$ is the heat-flow in the sphere on the interface.
-For the heat-flow on the interface in the melt, with equation (16) the following is valid:
+The heat flux at the interface within the melt is given by:
-$\dot{q}_s = -\lambda \frac{\partial T}{\partial r} = \frac{\lambda(T_m - T_{fl})}{1 - \exp\left(\frac{A}{r_G}\right)}\left(-\frac{A}{r^2}\exp\left(\frac{A}{r}\right)\right) \tag{18}$
+$\dot{q}_s = -\lambda \frac{\partial T}{\partial r} = \frac{\lambda(T_m - T_{fl})}{1 - \exp\left(\frac{A}{r_G}\right)}\left(-\frac{A}{r^2}\exp\left(\frac{A}{r}\right)\right) \tag{66}$
 At the position r = $r_G$, for heat-flow the following is valid:
-$\dot{q}_{s|r_G} = \lambda(T_m - T_{fl})\frac{A}{r_G^2} \frac{\exp\left(\frac{A}{r_G}\right)}{1 - \exp\left(\frac{A}{r_G}\right)} \tag{19}$
+$\dot{q}_{s|r_G} = \lambda(T_m - T_{fl})\frac{A}{r_G^2} \frac{\exp\left(\frac{A}{r_G}\right)}{1 - \exp\left(\frac{A}{r_G}\right)} \tag{67}$
-Then extend equation (20) with $\frac{\exp(-A/r_G)}{\exp(-A/r_G)}$ solving for heat flow results in:
+Then extend equation (68) with $\frac{\exp(-A/r_G)}{\exp(-A/r_G)}$ solving for heat flow results in:
-$\dot{q}_{s|r_G} = \lambda(T_m - T_{fl})\frac{A}{r_G^2} \frac{1}{\exp\left(\frac{-A}{r_G}\right) - 1} \tag{20}$
+$\dot{q}_{s|r_G} = \lambda(T_m - T_{fl})\frac{A}{r_G^2} \frac{1}{\exp\left(\frac{-A}{r_G}\right) - 1} \tag{68}$
 Now substitute the constant A.
-$A = \frac{1}{a_s} r_G^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \tag{21}$
+$A = \frac{1}{a_s} r_G^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \tag{69}$
 With the constant A':
-$A' = \frac{1}{a_s} \frac{\rho_f}{\rho_s} \bar{v} \tag{22}$
+$A' = \frac{1}{a_s} \frac{\rho_f}{\rho_s} \bar{v} \tag{70}$
 So the heat-flow on the melt-side of the interface results in:
-$\dot{q}_{s|r_G} = \lambda(T_m - T_{fl})A' \frac{\partial r_G}{\partial z} \frac{1}{\exp\left(-A'r_G \frac{\partial r_G}{\partial z}\right) - 1} \tag{23}$
+$\dot{q}_{s|r_G} = \lambda(T_m - T_{fl})A' \frac{\partial r_G}{\partial z} \frac{1}{\exp\left(-A'r_G \frac{\partial r_G}{\partial z}\right) - 1} \tag{71}$
 For the heat-flow in the solid on the interface is valid:
-$\dot{q}_{f|r=r_G} = \rho_f \bar{v}\Delta h \frac{\partial r_G}{\partial z} \tag{24}$
+$\dot{q}_{f|r=r_G} = \rho_f \bar{v}\Delta h \frac{\partial r_G}{\partial z} \tag{72}$
-Then put equations (23) and (24) into equation 17 to get:
+Substituting gives:
-$\rho_f \bar{v}\Delta h = \frac{\lambda(T_m - T_{fl})A'}{\exp\left(-A'r_G \frac{\partial r_G}{\partial z}\right) - 1} \tag{25}$
+$\rho_f \bar{v}\Delta h = \frac{\lambda(T_m - T_{fl})A'}{\exp\left(-A'r_G \frac{\partial r_G}{\partial z}\right) - 1} \tag{73}$
 Solve this equation with respect to $\frac{\partial r_G}{\partial z}$ to get:
-$r_G \frac{\partial r_G}{\partial z} = -\frac{1}{A'}\ln\left[1 + \frac{\lambda A'(T_m - T_{fl})}{\rho_f \bar{v}\Delta h}\right] \tag{26}$
+$r_G \frac{\partial r_G}{\partial z} = -\frac{1}{A'}\ln\left[1 + \frac{\lambda A'(T_m - T_{fl})}{\rho_f \bar{v}\Delta h}\right] \tag{74}$
-For the changing of the radius $r_0$ on a length $\Delta z$, through the integration of equation (26), the following is valid:
+For the changing of the radius $r_0$ on a length $\Delta z$, through the integration of equation (74), the following is valid:
-$r_{G,i+1} = \sqrt{r_{G,i}^2 - \frac{2}{A'}\ln\left[1 + \frac{\lambda A''(T_m - T_{fl})}{\Delta h}\right]\Delta z} \tag{27}$
+$r_{G,i+1} = \sqrt{r_{G,i}^2 - \frac{2}{A'}\ln\left[1 + \frac{\lambda A''(T_m - T_{fl})}{\Delta h}\right]\Delta z} \tag{75}$
-include the constant A' in (27) to get:
+include the constant A' in (75) to get:
-$r_{G,i+1} = \sqrt{r_{G,i}^2 - \frac{2\lambda}{\rho_f \bar{v}c_s}\ln\left[1 + \frac{c_s(T_m - T_{fl})}{\Delta h}\right]\Delta z} \tag{28}$
+$r_{G,i+1} = \sqrt{r_{G,i}^2 - \frac{2\lambda}{\rho_f \bar{v}c_s}\ln\left[1 + \frac{c_s(T_m - T_{fl})}{\Delta h}\right]\Delta z} \tag{76}$
-$r_{0,i}$ is the radius at the beginning of the considered channel section. Equation (28) makes the calculation of the particle radius along the screw direction possible.
+$r_{0,i}$ is the radius at the beginning of the considered channel section. Equation 76 makes the calculation of the particle radius along the screw direction possible.
 The solid section gives:
-$F(r_G) = \frac{N_{p,ges} \frac{4}{3}\pi r_0^3}{\Delta z A_{channel}} \tag{29}$
+$F(r_G) = \frac{N_{p,ges} \frac{4}{3}\pi r_0^3}{\Delta z A_{channel}} \tag{77}$
 {{ :en:grundlagenhandbuch:aufschmelzberechnung:en_sigma150_dlg_grundlagenhandbuch_aufschmelzberechnung_011.svg?550%nolink |}}
@@ Zeile 485: / Zeile 489: @@
 The heat flow which exists at the interface particle/melt in case of a pure thermal conduction is corrected by a thermal conduction correction factor $f_lh$.
-$$f_{lh,sim} = 1,30525 + 1,98091 \cdot \frac{d}{h}$$
+$$f_{lh,sim} = 1,30525 + 1,98091 \cdot \frac{d}{h}\tag{78}$$
 The correction factor is valid in the range 0,2 < d/h <0,9 and is based on an approximation of the dimensionless temperature field. It shows a good match with the CFD results, and compares the middle temperature gradient at the particle surface with finite channel height to the middle temperature gradient with infinite expansion.
@@ Zeile 497: / Zeile 501: @@
 The influence of the convection on the melt process in twin screw extruders was until here neglected. That is why a factor fk to consider the convective heat transfer is introduced in the modified melting model. This factor is also based on CFD simulations. Based on a single rotating particle surrounded by melt in an unwound twin screw channel different relations of the pellet diameter to the channel height dp/h as well as different Péclet numbers were used. The latter balances the ratio of the convection and the thermal conduction and describes the amount of the convection occurring in the flow.
-$Pe = \frac{\rho_m \cdot c_{pm} \cdot v_{0z} \cdot d_p}{\lambda_m}$
+$$Pe = \frac{\rho_m \cdot c_{pm} \cdot v_{0z} \cdot d_p}{\lambda_m}\tag{79}$$
 The simulation results show that for high convective shares a considerably faster melting is to be expected. Mathematically, the correction factor can be described by:
-$f_k = 1 + \frac{(b_1 \cdot \kappa + b_2 \cdot \kappa^2 + b_3 \cdot \kappa^3 + b_4 \cdot \kappa^4) \cdot Pe}{(b_5 \cdot \kappa + b_6 \cdot \kappa^2 + b_7 \cdot \kappa^3 + b_8 \cdot \kappa^4) \cdot \left(\frac{3}{4} \cdot \sqrt{Pe} + b_9\right)}$
+$$f_k = 1 + \frac{(b_1 \cdot \kappa + b_2 \cdot \kappa^2 + b_3 \cdot \kappa^3 + b_4 \cdot \kappa^4) \cdot Pe}{(b_5 \cdot \kappa + b_6 \cdot \kappa^2 + b_7 \cdot \kappa^3 + b_8 \cdot \kappa^4) \cdot \left(\frac{3}{4} \cdot \sqrt{Pe} + b_9\right)}\tag{80}$$
 with: $\kappa = 1 - e^{-\frac{dp}{h}}$
@@ Zeile 507: / Zeile 511: @@
 and it compares the heat flow from the analytical calculation (without considering the convection) with the heat flow from the simulations. Therefore the following is true:
-$\dot{q}_r = \left(-\lambda_m \frac{\partial T}{\partial r}\right) \cdot f_k \cdot f_{lh}$
+$$\dot{q}_r = \left(-\lambda_m \frac{\partial T}{\partial r}\right) \cdot f_k \cdot f_{lh}\tag{81}$$
 The regression constants b1 to b9 can be taken from the following table.
@@ Zeile 530: / Zeile 534: @@
 In the modified melting model the flow in the melt-solid mixture is formed with the help of an adjustment of the viscosity. For this purpose the correction of the power law consistency with the factor $f_Φ$ is in introduced.
-$$K_{sm} = K \cdot f_\phi$$
+$$K_{sm} = K \cdot f_\phi\tag{82}$$
-$$f_\phi = \frac{\eta_{sm}}{\eta_0} = 1 + \frac{5}{2} \cdot \phi_{V,s}$$
+$$f_\phi = \frac{\eta_{sm}}{\eta_0} = 1 + \frac{5}{2} \cdot \phi_{V,s}\tag{83}$$
 The correction factor $f_Φ$ is dependent on the volume content of the solid $Φ_{V,s}$ and compares the viscosity of the fluid-solid mixture $η_{sm}$ with the viscosity of the pure fluid $η_0$. This correction is based on Einstein but for large solid shares the viscosities calculated deviate too much. A further development according to Guth and Simah yields better results. Therefore the following is true for the correction factor:
-$$f_\phi = \frac{\eta_{sm}}{\eta_0} = 1 + \frac{5}{2} \cdot \phi_{V,s} + \frac{141}{10} \cdot \phi_{V,s}$$
+$$f_\phi = \frac{\eta_{sm}}{\eta_0} = 1 + \frac{5}{2} \cdot \phi_{V,s} + \frac{141}{10} \cdot \phi_{V,s}\tag{84}$$
 A conversion of the volume content on the mass content yields:
-$$f_\phi = 1 + \frac{5}{2}\left(\frac{\phi_{M,s}}{\phi_{M,s} + \frac{\rho_s}{\rho_m}(1 - \phi_{M,s})}\right)\phi_{V,s} + \frac{141}{10}\left(\frac{\phi_{M,s}}{\phi_{M,s} + \frac{\rho_s}{\rho_m}(1 - \phi_{M,s})}\right)^2$$
+$$f_\phi = 1 + \frac{5}{2}\left(\frac{\phi_{M,s}}{\phi_{M,s} + \frac{\rho_s}{\rho_m}(1 - \phi_{M,s})}\right)\phi_{V,s} + \frac{141}{10}\left(\frac{\phi_{M,s}}{\phi_{M,s} + \frac{\rho_s}{\rho_m}(1 - \phi_{M,s})}\right)^2\tag{85}$$
 If the $d_p/h$-ratio approaches 1 the particle diameter considered corresponds approximately to the channel height, and the influence of the particle size on the flow has to be considered. Following Pape  [[en:grundlagenhandbuch:aufschmelzberechnung#references|[Pap06]]] another correction of the power law consistency is introduced.
-$$K_{sm} = K \cdot f_\phi \cdot f_{dh}$$
+$$K_{sm} = K \cdot f_\phi \cdot f_{dh}\tag{86}$$
-$$f_{dh} = 1 + \frac{\frac{d_p}{h} \cdot \left(1 + \left(\frac{d_p}{h}\right)^2\right)}{\left(\frac{\rho_s}{\rho_m} - \frac{1 - \Phi_{M,s}}{\Phi_{M,s}}\right) \cdot \left[\left(\frac{d_p}{h}\right)^2 + \frac{d_p}{h} + 4\right] \cdot \left(1 - \frac{d_p}{h}\right)}$$
+$$f_{dh} = 1 + \frac{\frac{d_p}{h} \cdot \left(1 + \left(\frac{d_p}{h}\right)^2\right)}{\left(\frac{\rho_s}{\rho_m} - \frac{1 - \Phi_{M,s}}{\Phi_{M,s}}\right) \cdot \left[\left(\frac{d_p}{h}\right)^2 + \frac{d_p}{h} + 4\right] \cdot \left(1 - \frac{d_p}{h}\right)}\tag{87}$$
 Based on a solid particle in a Newtonian melt the flow conditions around this particle are considered in order to determine the correction. After this the viscosity for the flow of an equivalent volume flow of pure melt is determined. The ratio of the viscosity of this consideration and the viscosity of the solid-melt mixture leads to the correction with the factor $f_{dh}$.
@@ Zeile 558: / Zeile 562: @@
   * The influence of the convection on the heat flow in radial direction $f_k$.
-by the assumptions made.
+The calculation is based on the general energy equation, which, given the assumptions made, simplifies to:
+$$\rho c \frac{\partial T}{\partial t} = -\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\dot{q}_r\right)\tag{88}$$
 With the influence of the correction factors flh and fk the heat flow in radial direction is given by the following equation (cf: [[en:grundlagenhandbuch:aufschmelzberechnung:aufschmelzmodell_fuer_dispers_verteilte_fuellstoffe#Calculation_of_the_solid_bed_reduction_along_the_melt_path]]):
-$\dot{q}_r = -\lambda_m \cdot \frac{\partial T}{\partial r} \cdot f_k \cdot f_{lh}$
+$$\dot{q}_r = -\lambda_m \cdot \frac{\partial T}{\partial r} \cdot f_k \cdot f_{lh}\tag{89}$$
-These factors are used in the whole calculation process which was introduced in the previous chapter. Therefore the following is true for the change of the particle radius in the interval considered:
+These factors govern the entire calculation process described in the previous chapter, so that the change in particle radius over the interval under consideration is given by:
-$r_{i+1} = \sqrt{r_i^2 - \frac{2 \cdot \lambda_m \cdot f_{lh} \cdot f_k}{c_m \cdot \rho_s \cdot \bar{v}} \cdot \ln\left(1 + \frac{c_m \cdot (T_m - T_{flow})}{\Delta h \cdot f_{lh} \cdot f_k}\right) \cdot \Delta z}$
+$$r_{i+1} = \sqrt{r_i^2 - \frac{2 \cdot \lambda_m \cdot f_{lh} \cdot f_k}{c_m \cdot \rho_s \cdot \bar{v}} \cdot \ln\left(1 + \frac{c_m \cdot \left (T_m - T_{flow}\right)} {\Delta h \cdot f_{lh} \cdot f_k}\right) \cdot \Delta z}\tag{90}$$
-The factors fdh and fΦ, which reflect the influence of the particles on the flow, are anchored in the calculation of the middle flow velocity $\bar{v}$.
+The factors $f_{dh}$ and $f_\Phi$, which reflect the influence of the particles on the flow, are incorporated into the calculation of the mean flow velocity $\bar{v}$.
 ===== 2D disperse melting model =====
@@ Zeile 586: / Zeile 592: @@
 The filling degree of the components melt and solid is the decisive criterion for going on with the calculation according to the modified disperse melting model. In this model, the spherical particles in the screw channel after the "deformation zone" are almost closely spaced packed. It is model assumption the disperse melting starts when the voids between the spheres are filled with melt. The packing density of a densest packing is
-$$\frac{\pi}{3\sqrt{2}} \sim 0,74048 = 74,048$$
+$$\frac{\pi}{3\sqrt{2}} \sim 0,74048 = 74,048\tag{91}$$
 which means that the free volume occupies 25.952 %. As soon as the degree of molten material exceeds the melt rate of 25.952 %, the subsequent melting is calculated by the modified disperse melting model. This modeling takes into account the melting by a successive reduction of the radius due to convective heating in a finite channel geometry.