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 }

Gesetz von Dalton: \sum\limits_i {{p_i}} = {p_{ges}}

Gesetz von Raoult: {p_i} = {x_i} \cdot p_i^0

Molarer Stoffstrom: \vec j_i^* = {n_i} \cdot \left( {{{\vec v}_i}-{{\vec v}^*}} \right)

Konvektiver Stoffübergang: {j_w} = \rho \cdot {h_m} \cdot \left( {{y_w}-{y_\infty }} \right)

Sherwood-Zahl: Sh = \frac{{{h_m} \cdot L}}{D}

Stanton-Zahl: S{t_m} = \frac{{Sh}}{{Re \cdot Sc}}