Z 01 – Erweiterungen Formelsammlung Wärme- und Stofftransport

Instationäre Wärmeleitungsgleichung

$\rho \cdot c \cdot \frac{{\partial T}}{{\partial t}} = \frac{1}{{{r^n}}} \cdot \frac{\partial }{{\partial r}}\left( {k \cdot {r^n} \cdot \frac{{\partial T}}{{\partial r}}} \right)$

Platte: $n = 0$

Zylinder: $n = 1$

Kugel: $n = 2$

Rippen und Nadeln

Rippenwirkungsgrad: $\eta = \frac{{{{\dot Q}_{tats}}}}{{\dot Q_{ges}^{\max }}}$

Rippeneffektivität: $\varepsilon = \frac{{{{\dot Q}_{ges}}}}{{{{\dot Q}_{ohne\;Rippe}}}}$

Exakte Lösung für die Ebene Platte mit Randbedingung 3. Art

wust-fs-exakte-loesung-ebene-platte-randbedingung-3-art

$\theta = \frac{{T-{T_U}}}{{{T_0}-{T_U}}}$

$\xi = \frac{x}{L}$

$\theta = \sum\limits_{k = 1}^\infty {{C_k} \cdot \exp \left\{ {-\delta _k^2 \cdot {\tau ^+}} \right\} \cdot \cos \left( {{\delta _k} \cdot \xi } \right)}$

${C_k} = \frac{{2 \cdot \sin \left( {{\delta _k}} \right)}}{{{\delta _k}+\sin \left( {{\delta _k}} \right) \cdot \cos \left( {{\delta _k}} \right)}}$

Bestimmungsgleichung für die Eigenwerte: $\tan \left( {{\delta _k}} \right) = \frac{{Bi}}{{{\delta _k}}}$

Näherungslösung für lange Zeiten

$\theta = {C_1} \cdot \exp \left\{ {-\delta _1^2 \cdot {\tau ^+}} \right\} \cdot \cos \left( {{\delta _1} \cdot \xi } \right)$

Gültig für ${\tau ^+} > {\tau ^*} = 0,25$ für die ebene Platte.

Zahlenwerte für die Näherungslösung der Platte $\left( {{\tau ^*} = 0,25} \right)$ :

$\begin{array}{*{20}{c}} {\frac{1}{{Bi}}}&\vline & {{C_1}}&\vline & {{\delta _1}} \\ \hline {{\text{0}}{\text{,0}}}&\vline & {{\text{1}}{\text{,2732}}}&\vline & {{\text{1}}{\text{,5708}}} \\ {{\text{0}}{\text{,1}}}&\vline & {{\text{1}}{\text{,2620}}}&\vline & {{\text{1}}{\text{,4289}}} \\ {{\text{0}}{\text{,2}}}&\vline & {{\text{1}}{\text{,2402}}}&\vline & {{\text{1}}{\text{,3138}}} \\ {{\text{0}}{\text{,5}}}&\vline & {{\text{1}}{\text{,1784}}}&\vline & {{\text{1}}{\text{,0769}}} \\ {{\text{0}}{\text{,8}}}&\vline & {{\text{1}}{\text{,1379}}}&\vline & {{\text{0}}{\text{,9308}}} \\ {{\text{1}}{\text{,0}}}&\vline & {{\text{1}}{\text{,1191}}}&\vline & {{\text{0}}{\text{,8603}}} \\ {{\text{2}}{\text{,0}}}&\vline & {{\text{1}}{\text{,0701}}}&\vline & {{\text{0}}{\text{,6533}}} \\ {{\text{5}}{\text{,0}}}&\vline & {{\text{1}}{\text{,0311}}}&\vline & {{\text{0}}{\text{,4328}}} \\ {{\text{8}}{\text{,0}}}&\vline & {{\text{1}}{\text{,0199}}}&\vline & {{\text{0}}{\text{,3464}}} \\ {{\text{10}}{\text{,0}}}&\vline & {{\text{1}}{\text{,0161}}}&\vline & {{\text{0}}{\text{,3111}}} \\ {{\text{20}}{\text{,0}}}&\vline & {{\text{1}}{\text{,0082}}}&\vline & {{\text{0}}{\text{,2218}}} \\ {{\text{50}}{\text{,0}}}&\vline & {{\text{1}}{\text{,0033}}}&\vline & {{\text{0}}{\text{,1410}}} \\ {{\text{80}}{\text{,0}}}&\vline & {{\text{1}}{\text{,0021}}}&\vline & {{\text{0}}{\text{,1116}}} \\ {{\text{100}}{\text{,0}}}&\vline & {{\text{1}}{\text{,0017}}}&\vline & {{\text{0}}{\text{,0998}}} \end{array}$

Näherungslösung “halbunendlicher” Körper

Dimensionslose Orts-Zeit-Koordinate: $\eta = \frac{x}{{\sqrt {4 \cdot \alpha \cdot t} }}$

Randbedingung 1. Art: $\frac{{T-{T_W}}}{{{T_0}-{T_W}}} = erf\left( \eta \right)$

Randbedingung 2. Art: $T-{T_0} = \frac{{{{\dot q}_W}}}{k}\left[ {2 \cdot \sqrt {\frac{{\alpha \cdot t}}{\pi }} \cdot \exp \left( {-{\eta ^2}} \right)-x \cdot \left( {1-erf\left( \eta \right)} \right)} \right]$

Randbedingung 3. Art: $\frac{{T-{T_U}}}{{{T_0}-{T_U}}} = erf\left( \eta \right)+\exp \left\{ {{{\tilde B}^2}+2 \cdot \tilde B \cdot \eta } \right\} \cdot \left[ {1-erf\left( {\tilde B+\eta } \right)} \right]$

mit $\tilde B = \frac{{h \cdot \sqrt {\alpha \cdot t} }}{k}$

Fehlerfunktion: $erf\left( \eta \right) = \frac{2}{{\sqrt \pi }}\int\limits_0^\eta {\exp \left\{ {-{\zeta ^2}} \right\}d\zeta }$

Zahlenwerte der Gaußschen Fehlerfunktion:

$\begin{array}{*{20}{c}} \eta &\vline & {erf\left( \eta \right)}&\vline & \eta &\vline & {erf\left( \eta \right)}&\vline & \eta &\vline & {erf\left( \eta \right)} \\ \hline {{\text{0}}{\text{,00}}}&\vline & {\text{0}}&\vline & {{\text{0}}{\text{,65}}}&\vline & {{\text{0}}{\text{,642029}}}&\vline & {{\text{1}}{\text{,6}}}&\vline & {{\text{0}}{\text{,976348}}} \\ {{\text{0}}{\text{,05}}}&\vline & {{\text{0}}{\text{,056372}}}&\vline & {{\text{0}}{\text{,70}}}&\vline & {{\text{0}}{\text{,677801}}}&\vline & {{\text{1}}{\text{,7}}}&\vline & {{\text{0}}{\text{,983790}}} \\ {{\text{0}}{\text{,10}}}&\vline & {{\text{0}}{\text{,112463}}}&\vline & {{\text{0}}{\text{,75}}}&\vline & {{\text{0}}{\text{,711156}}}&\vline & {{\text{1}}{\text{,8}}}&\vline & {{\text{0}}{\text{,989091}}} \\ {{\text{0}}{\text{,15}}}&\vline & {{\text{0}}{\text{,167996}}}&\vline & {{\text{0}}{\text{,80}}}&\vline & {{\text{0}}{\text{,742101}}}&\vline & {{\text{1}}{\text{,9}}}&\vline & {{\text{0}}{\text{,992790}}} \\ {{\text{0}}{\text{,20}}}&\vline & {{\text{0}}{\text{,222703}}}&\vline & {{\text{0}}{\text{,85}}}&\vline & {{\text{0}}{\text{,770668}}}&\vline & {{\text{2}}{\text{,0}}}&\vline & {{\text{0}}{\text{,995322}}} \\ {{\text{0}}{\text{,25}}}&\vline & {{\text{0}}{\text{,276326}}}&\vline & {{\text{0}}{\text{,90}}}&\vline & {{\text{0}}{\text{,796908}}}&\vline & {{\text{2}}{\text{,2}}}&\vline & {{\text{0}}{\text{,998137}}} \\ {{\text{0}}{\text{,30}}}&\vline & {{\text{0}}{\text{,328627}}}&\vline & {{\text{0}}{\text{,95}}}&\vline & {{\text{0}}{\text{,820891}}}&\vline & {{\text{2}}{\text{,4}}}&\vline & {{\text{0}}{\text{,999311}}} \\ {{\text{0}}{\text{,35}}}&\vline & {{\text{0}}{\text{,379382}}}&\vline & {{\text{1}}{\text{,00}}}&\vline & {{\text{0}}{\text{,842701}}}&\vline & {{\text{2}}{\text{,6}}}&\vline & {{\text{0}}{\text{,999764}}} \\ {{\text{0}}{\text{,40}}}&\vline & {{\text{0}}{\text{,428392}}}&\vline & {{\text{1}}{\text{,10}}}&\vline & {{\text{0}}{\text{,880205}}}&\vline & {{\text{2}}{\text{,8}}}&\vline & {{\text{0}}{\text{,999925}}} \\ {{\text{0}}{\text{,45}}}&\vline & {{\text{0}}{\text{,475482}}}&\vline & {{\text{1}}{\text{,20}}}&\vline & {{\text{0}}{\text{,910314}}}&\vline & {{\text{3}}{\text{,0}}}&\vline & {{\text{0}}{\text{,999978}}} \\ {{\text{0}}{\text{,50}}}&\vline & {{\text{0}}{\text{,520500}}}&\vline & {{\text{1}}{\text{,30}}}&\vline & {{\text{0}}{\text{,934008}}}&\vline & {{\text{3}}{\text{,5}}}&\vline & {{\text{0}}{\text{,999999}}} \\ {{\text{0}}{\text{,55}}}&\vline & {{\text{0}}{\text{,563323}}}&\vline & {{\text{1}}{\text{,40}}}&\vline & {{\text{0}}{\text{,952285}}}&\vline & {{\text{4}}{\text{,0}}}&\vline & {{\text{1}}{\text{,000000}}} \\ {{\text{0}}{\text{,60}}}&\vline & {{\text{0}}{\text{,603856}}}&\vline & {{\text{1}}{\text{,50}}}&\vline & {{\text{0}}{\text{,966105}}}&\vline & {{\text{5}}{\text{,0}}}&\vline & {{\text{1}}{\text{,000000}}} \end{array}$

Periodische Temperaturänderungen im “halbunendlichen” Körper:

Wellenlänge: $\Lambda = 2\sqrt {\pi \cdot \alpha \cdot {t_0}}$

Eindringtiefe: $\frac{1}{B} = \exp \left\{ {\frac{{-2 \cdot \pi \cdot {x_n}}}{\Lambda }} \right\}$ ( $\overset{\wedge}{=}$ Abfall auf den B-ten Anteil)

Schmelzen und Erstarren

schmelzen-erstarren-wasser-phasengrenze

1. HS: $0 = {\dot Q_{zu}}-{\dot Q_{ab}}+\Delta {h_s} \cdot \frac{{\partial {m_{fest}}}}{{\partial t}}$

Phasenzahl: $Ph = \frac{1}{{St}} = \frac{{\Delta {h_s}}}{{{c_E} \cdot \left( {{T_S}-{T_K}} \right)}}$

Kompressible Strömungen

Schallgeschwindigkeit: $a = \sqrt {\kappa \cdot R \cdot T} = \sqrt {{c_p} \cdot \left( {\kappa -1} \right) \cdot T}$

Ideale Staupunktströmung $\left( {\Pr = 1} \right)$ : ${T^0} = {T_1}+\frac{{v_1^2}}{{2 \cdot {c_p}}} = {T_1}+\frac{{\kappa -1}}{2} \cdot M{a^2} \cdot {T_1}$

Wandwärmestrom: ${\dot q_w} = h \cdot \left( {{T_w}-{T_r}} \right)$

Recovery Factor: $r = \frac{{{T_r}-{T_1}}}{{{T^0}-{T_1}}}$

laminare Strömung: $r \cong \sqrt {\Pr }$ für $0,6 < \Pr < 15$

turbulente Strömung: $r \cong \sqrt[3]{{\Pr }}$ für $0,25 < \Pr < 10$

Referenztemperatur: ${T_{ref}} = {T_1}+0,5 \cdot \left( {{T_w}-{T_1}} \right)+0,22 \cdot \left( {{T_r}-{T_1}} \right)$

Stofftransport

Ideales Gasgesetz: $p = \rho \cdot R \cdot T$ bzw. ${p_i} = {\rho _i} \cdot {R_i} \cdot T$

$R = \frac{\Re }{{\hat M}}$

Massenbruch: ${y_i} = \frac{{{\rho _i}}}{\rho } = \frac{{{m_i}}}{m}\quad ;\quad \sum\limits_i {{y_i}} = 1$

Molenbruch: ${x_i} = \frac{{{n_i}}}{n} = \frac{{{N_i}}}{N}\quad ;\quad \sum\limits_i {{x_i}} = 1$

Umrechnung Massenbruch – Molenbruch:

${y_a} = \frac{{{x_a} \cdot {{\hat M}_a}}}{{{x_a} \cdot {{\hat M}_a}+{x_b} \cdot {{\hat M}_b}+ \ldots }}$

${x_a} = \frac{{\frac{{{y_a}}}{{{{\hat M}_a}}}}}{{\frac{{{y_a}}}{{{{\hat M}_a}}}+\frac{{{y_b}}}{{{{\hat M}_b}}}+ \ldots }}$

1. Fick’sches Gesetz:

${j_i} = -{D_{ij}} \cdot \frac{{\partial {\rho _i}}}{{\partial z}}$

$j_i^* = -{D_{ij}} \cdot \frac{{\partial {n_i}}}{{\partial z}}$

Diffusionsgleichung für $\rho \approx const$ .: $\frac{{\partial {y_i}}}{{\partial t}} = \frac{\partial }{{\partial z}}\left( {D \cdot \frac{{\partial {y_i}}}{{\partial z}}} \right)+\frac{{{\omega _m}}}{\rho }$