06. 流动阻力和能量损失

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6.1 流动阻力和能量损失的两种形式

\qq{圆管沿程水头损失} h_f = \lambda \frac{l}{d} \frac{v^2}{2 g}

$l$ — 管长; $d$ — 管径; $v$ — 断面平均流速; $g$ — 重力加速度; $\rho$ — 流体密度; $\lambda$ — 沿程损失 (阻力) 系数

\qq{局部水头损失} h_j = \zeta \frac{v^2}{2 g}

$\zeta$ — 局部损失 (阻力) 系数; 一般由实验确定; $v$ — 断面平均流速; $\rho$ — 流体密度; $g$ — 重力加速度

\qq{雷诺数} Re = \frac{v d \rho}{\mu} = \frac{v d}{\nu} \\ \qq{临界雷诺数} Re_c = \frac{v_c d}{\nu} = 2000

Re < Re_c, \qq{流动为层流} \\ Re > Re_c, \qq{流动为紊流}

\qq{水力半径} R = \frac{A}{\chi}

$R$ — 水力半径; $A$ — 过流断面面积; $\chi$ — 过流断面上流体与固体边界接触部分的周长, 称为湿周

Re_{cR} = \frac{v R}{\nu} = 500

h_f = \frac{\tau_0 \chi l}{\rho g A} = \frac{\tau_0 l}{\rho g R} \\ \tau_0 = \rho g R \frac{h_f}{l} = \rho g R J

\tau = \frac{r}{r_0} \tau_0

\qq{阻力速度} v_* = v \sqrt{\frac{\lambda}{8}}

u = \frac{\rho g J}{4 \mu} \pqty{r_0^2 - r^2} \\ u_{\max} = \frac{\rho g J}{4 \mu} r_0^2 \\ v = \frac{1}{2} u_{\max}

h_f = \frac{32 \mu l v}{\rho g d^2} \\ h_f = \lambda \frac{l}{d} \frac{v^2}{2 g} = \frac{32 \mu l}{\rho g d^2} v = \frac{64}{Re} \frac{l}{d} \frac{v^2}{2 g} \\ \lambda = \frac{64}{Re}

u = \frac{v_*}{\kappa} \ln{y} + C

\qq{粘性底层理论厚度} \delta = 11.6 \frac{\nu}{v_*} = \frac{32.8 d}{Re \sqrt{\lambda}}

\qq{粘性底层厚度} \delta = 11.6 \frac{\nu}{v_*} \\ \frac{\delta}{k_s} = 11.6 \frac{\nu}{v_* k_s} = 11.6 \frac{1}{Re_*}

$k_s$ — 粗糙度; $Re_* = \frac{v_* k_s}{\nu}$ — 粗糙雷诺数

\qq{紊流光滑区} Re_* \leqslant 5 \qq{或} \frac{k_s}{\delta} \leqslant 0.4 \\ \qq{紊流过渡区} 5 < Re_* \leqslant 70 \qq{或} 0.4 < \frac{k_s}{\delta} \leqslant 6 \\ \qq{紊流粗糙区} Re_* > 70 \qq{或} \frac{k_s}{\delta} > 6

\lambda = 0.11 \pqty{\frac{k_s}{d} + \frac{68}{Re}}^{0.25}

\qq{当量直径} d_e = 4 R

h_j = \zeta \frac{v^2}{2 g}

$\zeta$ — 局部阻力系数; $v$ — 与 $\zeta$ 对应的断面平均流速

h_j = \pqty{1 - \frac{A_1}{A_2}}^2 \frac{v_1^2}{2 g} = \zeta_1 \frac{v_1^2}{2 g} \\ h_j = \pqty{\frac{A_2}{A_1} - 1}^2 \frac{v_2^2}{2 g} = \zeta_2 \frac{v_2^2}{2 g}