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en:grundlagenhandbuch:strangabkuehlung [2026/05/28 10:04] – deppe2en:grundlagenhandbuch:strangabkuehlung [2026/08/17 11:01] (aktuell) – [Determination of the temperature field] paal
Zeile 79: Zeile 79:
 After a few simplifications, the following expression results: After a few simplifications, the following expression results:
  
-$$T_{i+1,j} = T_{i,j} + \frac{\Delta t \cdot a}{R^2} \left[(T_{i,j+1} - 2T_{i,j} + T_{i,j-1}) \cdot N^2 + \frac{N}{j} \cdot \frac{T_{i,j+1} - T_{i,j-1}}{2}\right] \tag{13}$$+$$T_{i+1,j} = T_{i,j} + \frac{\Delta t \cdot a}{R^2} \left[(T_{i,j+1} - 2T_{i,j} + T_{i,j-1}) \cdot N^2 + \frac{N^2}{j} \cdot \frac{T_{i,j+1} - T_{i,j-1}}{2}\right] \tag{13}$$ 
  
 In addition, the time difference Δt can be described by the ratio of the length to the pull-off speed: In addition, the time difference Δt can be described by the ratio of the length to the pull-off speed:
  
-$$T_{i+1,j} = T_{i,j} + \frac{\Delta x \cdot a}{v_{ab} \cdot R^2} \left[(T_{i,j+1} - 2T_{i,j} + T_{i,j-1}) \cdot N^2 + \frac{N^2 T_{i,j+1} - T_{i,j-1}}{j \cdot 2}\right] \tag{14}$$+$$T_{i+1,j} = T_{i,j} + \frac{\Delta x \cdot a}{v_{ab} \cdot R^2} \left[(T_{i,j+1} - 2T_{i,j} + T_{i,j-1}) \cdot N^2 + \frac{N^2}{j} \cdot \frac{T_{i,j+1} - T_{i,j-1}}{2}\right] \tag{14}$$
  
 $$T_{i+1,j} = T_{i,j} + \frac{\Delta x \cdot a \cdot N^2}{v_{ab} \cdot R^2} \left[(T_{i,j+1} - 2T_{i,j} + T_{i,j-1}) + \frac{1}{j} \frac{T_{i,j+1} - T_{i,j-1}}{2}\right] \tag{14.1}$$ $$T_{i+1,j} = T_{i,j} + \frac{\Delta x \cdot a \cdot N^2}{v_{ab} \cdot R^2} \left[(T_{i,j+1} - 2T_{i,j} + T_{i,j-1}) + \frac{1}{j} \frac{T_{i,j+1} - T_{i,j-1}}{2}\right] \tag{14.1}$$
Zeile 127: Zeile 127:
 The following minimum value is specified for the axial number of segments: The following minimum value is specified for the axial number of segments:
  
-$$M_{min} = N^2 \cdot 2 \cdot \frac{\lambda}{\rho \cdot c_p} \cdot \frac{L}{v_{ab} \cdot R^2} \tag{19}$$+$$N_{min} = \frac{\alpha \cdot R}{2\lambda} \tag{19}$$ 
 + 
 +$$M_{min} = N^2 \cdot 2 \cdot \frac{\lambda}{\rho \cdot c_p} \cdot \frac{L}{v_{ab} \cdot R^2} \tag{20}$$
  
 ===== Calculation of the output ===== ===== Calculation of the output =====
Zeile 135: Zeile 137:
 Core temperature: Core temperature:
  
-$$T_{i,Mitte} = T_{i,1} \tag{20}$$+$$T_{i,Mitte} = T_{i,1} \tag{21}$$
  
 Surface temperature: Surface temperature:
  
-$$T_{i,OF} = \frac{(T_{i,N} + T_{i,Hilfs.})}{2} \tag{Equation 19}$$+$$T_{i,OF} = \frac{(T_{i,N} + T_{i,Hilfs.})}{2} \tag{22}$$
  
 Mean temperature: Mean temperature:
  
-$$T_{i,Mittel} = \frac{2}{N^2 + N} \cdot \sum_{j=1}^N j \cdot T_{i,j} \tag{Equation 20}$$+$$T_{i,Mittel} = \frac{2}{N^2 + N} \cdot \sum_{j=1}^N j \cdot T_{i,j} \tag{23}$$
  
 ===== References ===== ===== References =====