习题 5.3

题目

20°C，$\rho = 998$ kg/m³，$\nu = 1.0 \times 10^{-6}$ m²/s 的水流经内径为 0.0508 m 的粗糙管，管长 3 m。 分别计算四种情形下的摩擦系数和总流动阻力：

情形	$u$ (m/s)	$e/d$
(1)	15.24	0.005
(2)	1.524	0.015
(3)	1.524	0.00025
(4)	0.01524	0.005

已知条件

$d = 0.0508$ m，$L = 3$ m，$\rho = 998$ kg/m³，$\nu = 1.0 \times 10^{-6}$ m²/s

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求解思路

1. 计算雷诺数 $Re = u d / \nu$
2. 根据 $Re$ 和 $e/d$ 选取摩擦系数公式
3. 计算压降 $\Delta P = f \frac{L}{d} \frac{\rho u^2}{2}$

计算过程

情形 (1): $u = 15.24$ m/s, $e/d = 0.005$

$$Re = \frac{15.24 \times 0.0508}{1.0 \times 10^{-6}} = 774,192$$

$Re > 4000$，湍流。使用 Colebrook 方程：

$$\frac{1}{\sqrt{f}} = -2 \log_{10}\left(\frac{e/d}{3.7} + \frac{2.51}{Re\sqrt{f}}\right)$$

代入 $e/d = 0.005$，$Re = 774192$，迭代求解：

试差：$f = 0.03$

$$\frac{1}{\sqrt{0.03}} = 5.774,\quad -2\log_{10}\left(\frac{0.005}{3.7} + \frac{2.51}{774192 \times \sqrt{0.03}}\right) = -2\log_{10}(0.001351 + 0.000019) = 5.730$$

收敛，$f = 0.030$

$$\Delta P = 0.030 \times \frac{3}{0.0508} \times \frac{998 \times 15.24^2}{2} = 0.030 \times 59.06 \times 115,815 = 205,200\ \text{Pa}$$

情形 (2): $u = 1.524$ m/s, $e/d = 0.015$

$$Re = \frac{1.524 \times 0.0508}{1.0 \times 10^{-6}} = 77,419$$

湍流。Colebrook 方程求解：

$$\frac{1}{\sqrt{f}} = -2\log_{10}\left(\frac{0.015}{3.7} + \frac{2.51}{77419\sqrt{f}}\right)$$

试差得 $f = 0.044$

$$\Delta P = 0.044 \times \frac{3}{0.0508} \times \frac{998 \times 1.524^2}{2} = 0.044 \times 59.06 \times 1158 = 3,010\ \text{Pa}$$

情形 (3): $u = 1.524$ m/s, $e/d = 0.00025$

$$Re = 77,419$$

湍流，但相对粗糙度较小：

$$\frac{1}{\sqrt{f}} = -2\log_{10}\left(\frac{0.00025}{3.7} + \frac{2.51}{77419\sqrt{f}}\right)$$

试差得 $f = 0.019$

$$\Delta P = 0.019 \times 59.06 \times 1158 = 1,300\ \text{Pa}$$

情形 (4): $u = 0.01524$ m/s, $e/d = 0.005$

$$Re = \frac{0.01524 \times 0.0508}{1.0 \times 10^{-6}} = 774$$

$Re < 2300$，层流。使用 $f = 64/Re$：

$$f = \frac{64}{774} = 0.0827$$

$$\Delta P = 0.0827 \times 59.06 \times \frac{998 \times 0.01524^2}{2} = 0.0827 \times 59.06 \times 0.116 = 0.567\ \text{Pa}$$

答案

情形	$Re$	流态	$f$	$\Delta P$ (Pa)
(1)	774,192	湍流	0.030	205,200
(2)	77,419	湍流	0.044	3,010
(3)	77,419	湍流	0.019	1,300
(4)	774	层流	0.083	0.57

分析：情形 (2) 和 (3) 对比说明粗糙度对摩擦系数有显著影响；情形 (4) 流速极低时层流压降可忽略。

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