SOR
292 lines
11 KiB
Plaintext
292 lines
11 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 蓖麻油粘滞系数计算 (斯托克斯定律)\n",
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"\n",
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"本项目用于计算落球法测量蓖麻油粘滞系数实验的数据处理与不确定度评定。\n",
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"\n",
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"---\n",
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"### 实验原理与计算公式\n",
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"\n",
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"#### 1. 粘滞系数计算公式（考虑管壁修正）\n",
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"小球在液体中受重力、浮力与斯托克斯粘滞阻力作用达到终端匀速下落速度 $v = l/t$。考虑到圆筒玻璃管壁的有限边界效应，引入管壁修正因子 $(1 + 2.4 d/D)$：\n",
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"$$\\eta = \\frac{(\\rho - \\rho_0) g d^2 t}{18 l \\left(1 + 2.4 \\frac{d}{D}\\right)}$$\n",
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"\n",
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"* $\\rho$：小球密度（钢球通常取 $7800\\,\\mathrm{kg/m^3}$）\n",
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"* $\\rho_0$：液体（蓖麻油）密度（室温下约 $960\\,\\mathrm{kg/m^3}$）\n",
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"* $g$：重力加速度（取 $9.80\\,\\mathrm{m/s^2}$）\n",
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"* $d$：小球直径\n",
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"* $D$：玻璃管内径\n",
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"* $l$：小球下落距离\n",
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"* $t$：下落平均时间\n",
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"\n",
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"#### 2. 雷诺数（$Re$）判断与奥辛（Oseen）修正\n",
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"斯托克斯公式成立的条件为层流状态且惯性力远小于粘性阻力，其雷诺数定义为：\n",
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"$$Re = \\frac{\\rho_0 v d}{\\eta}$$\n",
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"* 当 $Re \\le 0.1$ 时，斯托克斯定律完全适用，无需修正。\n",
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"* 当 $0.1 < Re \\le 1.0$ 时，流场出现微小扰动，采用奥辛公式进行一级修正：\n",
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"$$\\eta_{\\mathrm{corrected}} = \\frac{\\eta}{1 + \\frac{3}{16} Re}$$\n",
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"\n",
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"#### 3. 不确定度传递公式\n",
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"假定 $\\rho, \\rho_0, g$ 的误差可忽略不计，由误差合成法则，粘滞系数的相对不确定度传递公式为：\n",
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"$$u_r(\\eta) = \\frac{u_\\eta}{\\eta} = \\sqrt{ \\left(2 \\frac{u_d}{d}\\right)^2 + \\left(\\frac{u_t}{t}\\right)^2 + \\left(\\frac{u_l}{l}\\right)^2 + \\left(\\frac{u_D}{D}\\right)^2 }$$\n",
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"\n",
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"绝对不确定度为：\n",
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"$$u_\\eta = \\eta \\cdot u_r(\\eta)$$\n",
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"\n",
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"其中各物理量 $X$ 的不确定度按 A 类与 B 类合成：\n",
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"$$u_X = \\sqrt{u_{A,X}^2 + u_{B,X}^2}, \\quad u_{A,X} = \\frac{s_X}{\\sqrt{n}}, \\quad u_{B,X} = \\frac{\\Delta_{\\mathrm{inst}}}{\\sqrt{3}}$$"
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],
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"id": "e5127401d3a8f15d"
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},
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{
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"cell_type": "code",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2026-09-22T07:54:14.369907400Z",
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"start_time": "2026-09-22T07:54:14.243997100Z"
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}
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},
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"source": [
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"import math\n",
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"from decimal import Decimal\n",
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"from python.utils import scientific_round, calculate_stats\n",
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"\n",
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"print(\"工具函数与模块加载完成。\")"
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],
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"id": "ca5f57b71e825238",
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"工具函数与模块加载完成。\n"
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]
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}
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],
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"execution_count": 1
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 1. 实验参数与数据输入\n",
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"> **提示**：可直接在下方单元格中修改小球下落时间测量值 `t_raw_data` 以及各项仪器参数，无需命令行逐次敲入。"
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],
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"id": "4f13dab8eefdafa4"
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},
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{
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"cell_type": "code",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2026-09-22T07:54:26.075852500Z",
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"start_time": "2026-09-22T07:54:26.004272300Z"
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}
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},
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"source": [
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"# --- 物理常数与材料密度 ---\n",
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"G = Decimal(\"9.80\") # 重力加速度 (m/s^2)\n",
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"RHO_BALL = Decimal(\"7800\") # 小球(钢球)密度 (kg/m^3)\n",
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"RHO_OIL = Decimal(\"960\") # 蓖麻油密度 (kg/m^3)\n",
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"\n",
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"# --- 几何与仪器参数 (单位: mm / cm) ---\n",
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"D_mean = Decimal(\"20.0\") # 玻璃管直径 D (mm)\n",
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"d_mean = Decimal(\"1.0\") # 小球直径 d (mm)\n",
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"l_mean = Decimal(\"20.0\") # 下落距离 l (cm)\n",
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"\n",
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"# --- 仪器允差 (仪器误差) Δ_inst ---\n",
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"delta_D = Decimal(\"0\") # D 的仪器误差 (mm)，如未单独评估可设为 0\n",
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"delta_d = Decimal(\"0\") # d 的仪器误差 (mm)，如未单独评估可设为 0\n",
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"delta_l = Decimal(\"0\") # l 的仪器误差 (mm)，如未单独评估可设为 0\n",
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"delta_t = Decimal(\"0.01\") # 秒表仪器误差 (s)\n",
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"\n",
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"# --- 5次下落时间测量值 (s) ---\n",
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"# 请在此填入实际实验数据：\n",
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"t_raw_data = [Decimal(\"12.45\"), Decimal(\"12.38\"), Decimal(\"12.50\"), Decimal(\"12.42\"), Decimal(\"12.48\")]\n",
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"\n",
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"print(f\"参数设置完成:\")\n",
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"print(f\"- 玻璃管直径 D: {D_mean} mm\")\n",
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"print(f\"- 小球直径 d: {d_mean} mm\")\n",
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"print(f\"- 下落距离 l: {l_mean} cm\")\n",
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"print(f\"- 时间测量值 t: {[float(x) for x in t_raw_data]} s\")"
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],
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"id": "b222d81b56a956be",
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"参数设置完成:\n",
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"- 玻璃管直径 D: 20.0 mm\n",
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"- 小球直径 d: 1.0 mm\n",
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"- 下落距离 l: 20.0 cm\n",
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"- 时间测量值 t: [12.45, 12.38, 12.5, 12.42, 12.48] s\n"
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]
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}
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],
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"execution_count": 2
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 2. 测量数据统计与不确定度计算"
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],
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"id": "b96c99013eae004b"
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},
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{
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"cell_type": "code",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2026-09-22T07:54:28.891288400Z",
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"start_time": "2026-09-22T07:54:28.810903200Z"
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}
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},
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"source": [
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"# 长度单位换算为 mm 以便统一不确定度计算\n",
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"l_mean_mm = l_mean * Decimal(\"10\")\n",
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"\n",
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"# 单次测量参数的 B 类不确定度 (u = Δ_inst / √3)\n",
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"D_u = delta_D / Decimal(str(math.sqrt(3)))\n",
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"d_u = delta_d / Decimal(str(math.sqrt(3)))\n",
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"l_u = delta_l / Decimal(str(math.sqrt(3)))\n",
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"\n",
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"# 5次时间测量的统计量 (平均值、合成不确定度、A类、B类)\n",
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"t_mean, t_u, t_ua, t_ub, t_s = calculate_stats(t_raw_data, delta_t)\n",
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"\n",
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"print(\"--- 时间统计分析 ---\")\n",
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"print(f\"时间平均值 t_mean : {t_mean:.4f} s\")\n",
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"print(f\"样本标准差 s(t) : {t_s:.4f} s\")\n",
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"print(f\"A 类不确定度 u_A(t): {t_ua:.4f} s\")\n",
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"print(f\"B 类不确定度 u_B(t): {t_ub:.4f} s\")\n",
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"print(f\"合成不确定度 u(t) : {t_u:.4f} s\")"
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],
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"id": "b609644356d56555",
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"--- 时间统计分析 ---\n",
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"时间平均值 t_mean : 12.4460 s\n",
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"样本标准差 s(t) : 0.0477 s\n",
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"A 类不确定度 u_A(t): 0.0214 s\n",
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"B 类不确定度 u_B(t): 0.0058 s\n",
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"合成不确定度 u(t) : 0.0221 s\n"
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]
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}
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],
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"execution_count": 3
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 3. 粘滞系数计算、雷诺数校验与修约输出"
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],
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"id": "652642328be1ab03"
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},
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{
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"cell_type": "code",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2026-09-22T07:54:30.748035800Z",
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"start_time": "2026-09-22T07:54:30.683789300Z"
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}
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},
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"source": [
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"# 统一转换为国际单位制 (m, kg, s)\n",
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"D_m = D_mean / Decimal(\"1000\")\n",
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"d_m = d_mean / Decimal(\"1000\")\n",
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"l_m = l_mean_mm / Decimal(\"1000\")\n",
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"\n",
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"# --- 1. 斯托克斯粘滞系数计算 (带管壁修正) ---\n",
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"correction_factor = Decimal(\"1\") + Decimal(\"2.4\") * (d_m / D_m)\n",
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"numerator = (RHO_BALL - RHO_OIL) * G * (d_m**2) * t_mean\n",
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"denominator = Decimal(\"18\") * l_m * correction_factor\n",
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"eta_val = numerator / denominator\n",
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"\n",
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"# --- 2. 下落终端速度与雷诺数 Re 校验 ---\n",
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"v_val = l_m / t_mean\n",
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"Re = RHO_OIL * v_val * d_m / eta_val\n",
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"\n",
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"# --- 3. 雷诺数判断与 Oseen 修正 ---\n",
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"if Re > Decimal(\"0.1\"):\n",
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" eta_corrected = eta_val / (Decimal(\"1\") + Decimal(\"3\") / Decimal(\"16\") * Re)\n",
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" oseen_applied = True\n",
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"else:\n",
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" eta_corrected = eta_val\n",
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" oseen_applied = False\n",
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"\n",
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"# --- 4. 相对不确定度与绝对不确定度 ---\n",
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"# 各相对不确定度平方和项: (2*ud/d)^2 + (ut/t)^2 + (ul/l)^2 + (uD/D)^2\n",
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"term_d = (Decimal(\"2\") * d_u / d_mean) if d_mean != 0 else Decimal(\"0\")\n",
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"term_t = (t_u / t_mean) if t_mean != 0 else Decimal(\"0\")\n",
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"term_l = (l_u / l_mean_mm) if l_mean_mm != 0 else Decimal(\"0\")\n",
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"term_D = (D_u / D_mean) if D_mean != 0 else Decimal(\"0\")\n",
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"\n",
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"rel_u_sq = term_d**2 + term_t**2 + term_l**2 + term_D**2\n",
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"rel_u = Decimal(str(math.sqrt(float(rel_u_sq))))\n",
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"eta_u = eta_corrected * rel_u\n",
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"\n",
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"# --- 5. 按照科学修约规范对齐保留有效数字 ---\n",
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"eta_final, u_final = scientific_round(eta_corrected, eta_u)\n",
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"\n",
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"# --- 6. 实验报告格式化输出 ---\n",
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"print(\"=\" * 45)\n",
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"print(\" 实 验 计 算 结 果 \")\n",
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"print(\"=\" * 45)\n",
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"print(f\"下落平均时间 t : {t_mean:.3f} ± {t_u:.3f} s\")\n",
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"print(f\"小球下落速度 v : {v_val:.5f} m/s\")\n",
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"print(\"-\" * 45)\n",
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"print(f\"雷诺数 Re : {Re:.4f}\")\n",
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"if not oseen_applied:\n",
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" print(\"判定结论 : Re <= 0.1，满足斯托克斯层流条件，无需 Oseen 修正。\")\n",
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"else:\n",
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" print(\"判定结论 : Re > 0.1，已自动应用 Oseen 修正公式。\")\n",
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" print(f\"修正前粘滞系数 η0 : {eta_val:.4f} Pa·s\")\n",
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"print(\"-\" * 45)\n",
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"print(f\"相对不确定度 u_r : {rel_u * 100:.2f}%\")\n",
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"print(f\"绝对不确定度 u_η : {eta_u:.4f} Pa·s\")\n",
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"print(\"-\" * 45)\n",
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"print(f\"最终结果表示 : η = {eta_final} ± {u_final} Pa·s\")\n",
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"print(\"=\" * 45)"
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],
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"id": "5abf7dd1e1769bf6",
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"=============================================\n",
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" 实 验 计 算 结 果 \n",
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"=============================================\n",
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"下落平均时间 t : 12.446 ± 0.022 s\n",
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"小球下落速度 v : 0.01607 m/s\n",
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"---------------------------------------------\n",
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"雷诺数 Re : 0.0746\n",
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"判定结论 : Re <= 0.1，满足斯托克斯层流条件，无需 Oseen 修正。\n",
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"---------------------------------------------\n",
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"相对不确定度 u_r : 0.18%\n",
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"绝对不确定度 u_η : 0.0004 Pa·s\n",
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"---------------------------------------------\n",
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"最终结果表示 : η = 0.2069 ± 0.0004 Pa·s\n",
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"=============================================\n"
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]
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}
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],
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"execution_count": 4
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}
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],
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"metadata": {
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"language_info": {
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"name": "python"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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