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en:grundlagenhandbuch:aufschmelzberechnung:aufschmelzmodell_fuer_dispers_verteilte_fuellstoffe [2026/01/30 11:00] – [Melt Temperature Development] deppe2en:grundlagenhandbuch:aufschmelzberechnung:aufschmelzmodell_fuer_dispers_verteilte_fuellstoffe [2026/02/05 11:03] (aktuell) – [Temperature of the Solids at the Location of the First Melt] pka
Zeile 24: Zeile 24:
 If you consider the increase in temperature starting form a moving coordinate system (within a particle), the temperature rise poses an unsteady procedure. The essential heat transmitting mechanisms are heat conduction and convection. If you consider the increase in temperature starting form a moving coordinate system (within a particle), the temperature rise poses an unsteady procedure. The essential heat transmitting mechanisms are heat conduction and convection.
  
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 **Figure:** Temperature trends within a spherically formed particle **Figure:** Temperature trends within a spherically formed particle
  
-In the area of unsteady heat transmittance, the heat increase starting from single particles [1, 2], or through solid beds [2, 3, 4], is considered. Since according to assumption 2, the melting of single particles should be assumed, it is also sensible to assume single particles in the solid conveying area. The figure shows diagrammatically the temperature trends inside a spherically formed particle. $\bar{T}$ is the average caloric temperature of the particle.+In the area of unsteady heat transmittance, the heat increase starting from single particles [[en:grundlagenhandbuch:aufschmelzberechnung:literatur|[1, 2]]], or through solid beds [[en:grundlagenhandbuch:aufschmelzberechnung:literatur|[2, 3, 4]]], is considered. Since according to assumption 2, the melting of single particles should be assumed, it is also sensible to assume single particles in the solid conveying area. The figure shows diagrammatically the temperature trends inside a spherically formed particle. $\bar{T}$ is the average caloric temperature of the particle.
  
 The coupling of the energy theorem and kinetics gives the differential calculus for the spherically symmetric temperature field. The coupling of the energy theorem and kinetics gives the differential calculus for the spherically symmetric temperature field.
Zeile 38: Zeile 38:
 $$\theta = \frac{T - T_0}{T_0 - T_U} ; \tau = \frac{at}{r_0^2} ; \xi = \frac{r}{r_0} \tag{2}$$ $$\theta = \frac{T - T_0}{T_0 - T_U} ; \tau = \frac{at}{r_0^2} ; \xi = \frac{r}{r_0} \tag{2}$$
  
-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 [2]. 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 [2]:+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:literatur|[2]]]. 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:literatur|[2]]]:
  
 $$\theta = g(\tau) \cdot f(\xi) \tag{3}$$ $$\theta = g(\tau) \cdot f(\xi) \tag{3}$$
Zeile 82: Zeile 82:
 **Table:** Constants of the determination of the temperature function **Table:** Constants of the determination of the temperature function
  
-The figure shows the value of the first four roots taken from [2] 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:literatur|[2]]] and the approximate values of those with the equations (11) and (12).
  
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 **Figure:** Roots for the calculation of the average caloric particle temperature **Figure:** Roots for the calculation of the average caloric particle temperature
Zeile 120: Zeile 120:
 Whereby the temperature has to be filled in with degrees Celsius. Whereby the temperature has to be filled in with degrees Celsius.
  
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 **Figure:** Average caloric particle temperature as a function of location **Figure:** Average caloric particle temperature as a function of location
Zeile 132: Zeile 132:
 The dispersed solid particles increase the yielded dissipation energy, since in this area of particles with ignored particle rotation, the shear gradient tends to zero. The consideration of these effects is conceived according to the figure, whereby the dispersion phase is seen as continual phase on the screw base. The dispersed solid particles increase the yielded dissipation energy, since in this area of particles with ignored particle rotation, the shear gradient tends to zero. The consideration of these effects is conceived according to the figure, whereby the dispersion phase is seen as continual phase on the screw base.
  
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 **Figure:** Model of the consideration of the shear inflation **Figure:** Model of the consideration of the shear inflation
Zeile 168: Zeile 168:
 The constants $C_1$ and $C_2$ are asserted for the specified areas by $\beta \cdot \Delta T$. The constants $C_1$ and $C_2$ are asserted for the specified areas by $\beta \cdot \Delta T$.
  
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 **Figure:** Approximation of the exponential function through a set of linear equations **Figure:** Approximation of the exponential function through a set of linear equations
Zeile 202: Zeile 202:
 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 (10).
  
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 **Figure:** Absolute base temperature distribution for various cylinder wall temperatures **Figure:** Absolute base temperature distribution for various cylinder wall temperatures
Zeile 228: Zeile 228:
  
  
-with $\varepsilon = \frac{\theta_z}{Br C_2 \beta T_z}$+with $$\varepsilon = \frac{\theta_z}{Br C_2 \beta T_z}$$
  
 Since only the channel height averaged mass temperature is of interest, this can be calculated by: Since only the channel height averaged mass temperature is of interest, this can be calculated by:
Zeile 236: Zeile 236:
 The figure shows a comparison of the documented solutions in Table against the publicized approaches of Potente [4] and Ansahl [6]. 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. The figure shows a comparison of the documented solutions in Table against the publicized approaches of Potente [4] and Ansahl [6]. 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.
  
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 **Figure:** Dimensionless mass temperature as a function of the Graetz number **Figure:** Dimensionless mass temperature as a function of the Graetz number
Zeile 254: Zeile 254:
 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: 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:
  
-$$(\overline{\tau\dot{\gamma}})_j = \frac{1}{\bar{h}^{1+n}} \int_{-\frac{b_{max}}{2}}^{x_f} \frac{K(T_j)v_0^{1+n}}{h(x)^{1+n}} dx \tag{3}$$+$$(\overline{\tau\dot{\gamma}})_j = \frac{1}{x_f + \frac{b_{max}}{2}} \int_{-\frac{b_{max}}{2}}^{x_f} \frac{K(T_j)v_0^{1+n}}{h(x)^{1+n}} dx \tag{3}$$
  
 $x_f$ 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 $x_f$ 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
Zeile 280: Zeile 280:
 $$\rho c \frac{\partial T}{\partial t} = \frac{\lambda}{r^2}\left(2r \frac{\partial T}{\partial t} + r^2 \frac{\partial^2 T}{\partial r^2}\right) \tag{3}$$ $$\rho c \frac{\partial T}{\partial t} = \frac{\lambda}{r^2}\left(2r \frac{\partial T}{\partial t} + r^2 \frac{\partial^2 T}{\partial r^2}\right) \tag{3}$$
  
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 **Figure:** Sphere co-ordination on a solid particle **Figure:** Sphere co-ordination on a solid particle
Zeile 294: Zeile 294:
 By equating (4) and (5) one gets: By equating (4) and (5) one gets:
  
-$\frac{\partial r}{\partial t} = -\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \frac{\partial r_G}{\partial t} = -\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \frac{\partial r_G}{\partial_z} \frac{\partial_z}{\partial t} \tag{6}$+$$\frac{\partial r}{\partial t} = -\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \frac{\partial r_G}{\partial t} = -\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \frac{\partial r_G}{\partial_z} \frac{\partial_z}{\partial t} \tag{6}$$
  
 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 (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:
  
-$\partial t = \left(-\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \frac{\bar{v}}{\partial z}\right)^{-1} \partial r \tag{7}$+$$\partial t = \left(-\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \right)^{-1} \partial r \tag{7}$$
  
 Now put equation (7) through (3) with: Now put equation (7) through (3) with:
  
-$a_s = \frac{\lambda_s}{\rho_s c_p} \tag{8}$+$$a_s = \frac{\lambda_s}{\rho_s c_p} \tag{8}$$
  
 With the definition of the constants: With the definition of the constants:
  
-$\frac{\partial^2 T}{\partial r^2} + \left[\frac{1}{a_s}\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \frac{\bar{v}}{\partial z} \frac{\partial r_G}{\partial z} + \frac{2}{r}\right] \frac{\partial T}{\partial r} = 0 \tag{9}$+$$\frac{\partial^2 T}{\partial r^2} + \left[\frac{1}{a_s}\left(\frac{r_G}{r}\right)^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} + \frac{2}{r}\right] \frac{\partial T}{\partial r} = 0 \tag{9}$$
  
 From equation (8): From equation (8):
  
-$A = \frac{1}{a_s} r_G^2 \frac{\rho_f}{\rho_s} \frac{\bar{v}}{\partial z} \frac{\partial r_G}{\partial z} \tag{10}$+$$A = \frac{1}{a_s} r_G^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \tag{10}$$
  
-The double integration results in the following equation:+The double integration of
  
-$\frac{\partial^2 T}{\partial r^2} + \left[\frac{A}{r^2} + \frac{2}{r}\right] \frac{\partial T}{\partial r} = 0 \tag{11}$+$$\frac{\partial^2 T}{\partial r^2} + \left[\frac{A}{r^2} + \frac{2}{r}\right] \frac{\partial T}{\partial r} = 0 \tag{11}$$ 
 + 
 +results in the following equation: 
 + 
 +$$T(r) = \frac{C_1}{A}e^{-\frac{A}{r}} + C_2 \tag{12}$$
  
 Whereby C1 and C2 are the integration constants. With the following boundary conditions: Whereby C1 and C2 are the integration constants. With the following boundary conditions:
  
-$T(r = \infty) = T_m \tag{12}$+$$T(r = \infty) = T_m \tag{13}$$
  
-$T(r = r_G) = T_{fl} \tag{13}$+$$T(r = r_G) = T_{fl} \tag{14}$$
  
 It results in the integration constants: It results in the integration constants:
  
-$C_1 = A \frac{T_m - T_{fl}}{\exp\left(\frac{A}{r_G}\right) - 1} \tag{14}$+$$C_1 = A \frac{T_m - T_{fl}}{\exp\left(\frac{A}{r_G}\right) - 1} \tag{15}$$
  
-$C_2 = T_m + \frac{T_m - T_{fl}}{\exp\left(\frac{A}{r_G}\right) - 1} \tag{15}$+$$C_2 = T_m + \frac{T_m - T_{fl}}{\exp\left(\frac{A}{r_G}\right) - 1} \tag{16}$$
  
 The solution of the differential equation results in: The solution of the differential equation results in:
  
-$\frac{T_m - T(r)}{T_m - T_{fl}} = \frac{1 - \exp\left(\frac{A}{r}\right)}{1 - \exp\left(\frac{A}{r_G}\right)} \tag{16}$+$$\frac{T_m - T(r)}{T_m - T_{fl}} = \frac{1 - \exp\left(\frac{A}{r}\right)}{1 - \exp\left(\frac{A}{r_G}\right)} \tag{17}$$
  
 Which is equilibrium of heat flows. Which is equilibrium of heat flows.
Zeile 357: Zeile 361:
 Now substitute the constant A. Now substitute the constant A.
  
-$A = \frac{1}{a_s} r_G^2 \frac{\rho_f}{\rho_s} \frac{\bar{v}}{\partial z} \frac{\partial r_G}{\partial z} \tag{21}$+$A = \frac{1}{a_s} r_G^2 \frac{\rho_f}{\rho_s} \bar{v} \frac{\partial r_G}{\partial z} \tag{21}$
  
 With the constant A': With the constant A':
Zeile 393: Zeile 397:
 $F(r_G) = \frac{N_{p,ges} \frac{4}{3}\pi r_0^3}{\Delta z A_{channel}} \tag{29}$ $F(r_G) = \frac{N_{p,ges} \frac{4}{3}\pi r_0^3}{\Delta z A_{channel}} \tag{29}$
  
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 **Figure:** Comparison of calculated and measured melt trends **Figure:** Comparison of calculated and measured melt trends