SOR
473 lines
20 KiB
Python
473 lines
20 KiB
Python
import math
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from decimal import Decimal
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"""
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Thermal Conductivity & Specific Heat Calculator (Steady-State Plate Method)
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导热系数与比热容计算器 (稳态平板法 — 一维无限大平板)
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实验原理:
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利用稳态平板法测量不良导体的导热系数。
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当加热板和样品达到热平衡（稳态）时，通过样品的热流密度 q_c 等于
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样品两面温差 ΔT 除以厚度 R 再乘以导热系数 λ。
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断电后由冷却曲线斜率计算样品比热容 c。
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核心公式:
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1. 热流密度:
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q_c = V² / (2 · F · r)
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其中 V = 加热电压, F = 有效加热面积, r = 加热电阻
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2. 导热系数:
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λ = q_c · R / (2 · ΔT)
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其中 R = 样品厚度, ΔT = 加热面温度 - 中心面温度
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3. 温度转换 (热电偶):
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ΔT = ΔU / (1000 · S)
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其中 S = 热电偶灵敏度系数 (mV/K), ΔU = U_加热面 - U_中心面 (单位: uV)
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4. 比热容 (冷却法):
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c = 2 · q_c · F / (m · |κ|)
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其中 m = 样品质量, κ = dT/dτ|_冷却 为冷却曲线在稳态温度处的斜率
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5. 不确定度传递 — 导热系数 λ = V²·R / (4·F·r·ΔT):
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u_r(λ) = √[ (2·u_V/V)² + (u_R/R)² + (u_F/F)² + (u_r/r)² + (u_ΔT/ΔT)² ]
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u(ΔU) 取稳态区间多点的 A 类 + 仪器 B 类合成; u_r(ΔT) ≈ u(ΔU)/|ΔU|
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6. 不确定度传递 — 比热容 c = V² / (r·m·|κ|):
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u_r(c) = √[ (2·u_V/V)² + (u_r/r)² + (u_m/m)² + (u_κ/|κ|)² ]
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u_κ 由冷却曲线线性拟合的斜率标准差给出
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"""
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from utils import (
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scientific_round,
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calculate_stats,
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linear_regression
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)
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# ──────────────────────────────────────────────
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# 数据输入辅助函数
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# ──────────────────────────────────────────────
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def input_voltage_series(name, n_points=18):
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"""输入 1~n_points 分钟的电压数据 (单位: uV)"""
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print(f"\n请输入 {name} 的 {n_points} 个电压值 (uV), 可用空格/逗号分隔一次输入:")
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data = []
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while len(data) < n_points:
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remaining = n_points - len(data)
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try:
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raw = input(f" [{len(data)+1}~{len(data)+remaining}] > ").strip()
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if not raw:
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continue
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tokens = raw.replace(',', ' ').split()
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for tok in tokens:
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data.append(Decimal(tok))
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if len(data) >= n_points:
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break
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except Exception:
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print(" 输入无效，请重新输入。")
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return data
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# ──────────────────────────────────────────────
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# 稳态判断
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# ──────────────────────────────────────────────
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def find_steady_state(voltages, window=3, threshold=Decimal("0.5")):
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"""
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判断电压序列何时达到稳态。
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使用滑动窗口：当连续 window 个读数的极差 < threshold (uV) 时认为稳态。
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返回稳态起始索引（从 0 开始），若未达到稳态则返回最后 window 个的起始。
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"""
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for i in range(len(voltages) - window + 1):
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w = voltages[i:i + window]
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if max(w) - min(w) < threshold:
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return i
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return len(voltages) - window
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# ──────────────────────────────────────────────
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# 核心计算函数
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# ──────────────────────────────────────────────
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def compute_thermal_conductivity(
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center_uV, # 中心面电压序列 (uV)
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heating_uV, # 加热面电压序列 (uV)
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V_heat, # 加热电压 (V)
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F, # 有效加热面积 (m²)
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r, # 加热电阻 (Ω)
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R_thickness, # 样品厚度 (m)
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S_sensitivity, # 热电偶灵敏度 (mV/K)
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material_name="" # 材料名称
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):
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"""计算单一材料的导热系数，并保留稳态 ΔU 列表供不确定度分析使用"""
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n = len(center_uV)
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time_min = list(range(1, n + 1)) # 1 ~ n 分钟
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# --- 1. 转换为温度差 ΔT(t) ---
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S_uV = S_sensitivity * Decimal("1000") # 将 mV/K 转换为 uV/K 以匹配 uV 数据
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delta_uV = [heating_uV[i] - center_uV[i] for i in range(n)]
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delta_T = [dv / S_uV for dv in delta_uV]
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# --- 2. 判断稳态 ---
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ss_idx_center = find_steady_state(center_uV)
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ss_idx_heating = find_steady_state(heating_uV)
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ss_idx = max(ss_idx_center, ss_idx_heating)
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# 稳态区间 ΔU 列表（A 类不确定度来源）
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steady_delta_uV = delta_uV[ss_idx:]
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delta_T_steady = sum(steady_delta_uV) / Decimal(len(steady_delta_uV)) / S_uV
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center_steady_avg = sum(center_uV[ss_idx:]) / Decimal(len(center_uV[ss_idx:]))
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heating_steady_avg = sum(heating_uV[ss_idx:]) / Decimal(len(heating_uV[ss_idx:]))
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# --- 3. 热流密度 q_c ---
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q_c = V_heat ** 2 / (Decimal("2") * F * r)
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# --- 4. 导热系数 λ ---
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lam = q_c * R_thickness / (Decimal("2") * delta_T_steady)
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# --- 5. 加热趋近速率 dT/dτ (用于验证是否达到稳态) ---
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if n >= 3:
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last_n = min(5, n)
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x_fit = [Decimal(str(t * 60)) for t in time_min[-last_n:]] # 秒
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y_fit = [center_uV[i] / S_uV for i in range(-last_n, 0)] # K (相对)
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dT_dtau, _, _, _ = linear_regression(x_fit, y_fit) # K/s
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else:
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dT_dtau = Decimal("0")
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return {
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"material": material_name,
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"q_c": q_c,
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"delta_T_steady": delta_T_steady,
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"lambda": lam,
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"ss_start_min": ss_idx + 1,
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"center_steady_uV": center_steady_avg,
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"heating_steady_uV": heating_steady_avg,
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"dT_dtau": dT_dtau,
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"time_min": time_min,
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"center_uV": center_uV,
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"heating_uV": heating_uV,
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"delta_uV": delta_uV,
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"delta_T": delta_T,
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"steady_delta_uV": steady_delta_uV, # 稳态区间 ΔU 列表，供不确定度分析使用
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}
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def compute_specific_heat(cooling_uV, S_sensitivity, q_c, F, m_sample):
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"""
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由冷却曲线计算比热容 c。
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cooling_uV : 断电后中心面电压序列 (uV), 每隔 60 s 一个点
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S_sensitivity: 热电偶灵敏度 (mV/K)
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q_c : 热流密度 (W/m²)
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F : 有效加热面积 (m²)
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m_sample : 样品质量 (kg)
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返回: (c, dT_dtau_cool, u_dTdtau)
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c — 比热容 J/(kg·K)
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dT_dtau_cool — 冷却速率绝对值 |κ| (K/s)
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u_dTdtau — |κ| 的不确定度 (K/s), 由线性拟合给出
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"""
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n = len(cooling_uV)
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S_uV = S_sensitivity * Decimal("1000") # 将 mV/K 转换为 uV/K 以匹配 uV 数据
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x_cool = [Decimal(str(i * 60)) for i in range(n)] # 秒
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y_cool = [uv / S_uV for uv in cooling_uV] # 相对温度 (K)
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k, _, _, u_k = linear_regression(x_cool, y_cool)
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dT_dtau_cool = abs(k) # 冷却时斜率为负，取绝对值
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u_dTdtau = abs(u_k)
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c = (Decimal("2") * q_c * F / (m_sample * dT_dtau_cool)) if dT_dtau_cool != 0 else Decimal("0")
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return c, dT_dtau_cool, u_dTdtau
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def compute_uncertainties(
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result, # dict from compute_thermal_conductivity
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dT_dtau_cool, # 冷却速率 |κ| (K/s, Decimal, 已取绝对值)
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u_dTdtau, # |κ| 的不确定度 (K/s)
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m_sample, # 样品质量 (kg)
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delta_m, # 质量仪器误差 (kg)
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V_heat, delta_V, # 加热电压及仪器误差 (V)
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F, delta_F, # 面积及仪器误差 (m²)
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r, delta_r, # 电阻及仪器误差 (Ω)
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R_thickness, delta_R, # 厚度及仪器误差 (m)
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delta_uV_inst, # 电压表仪器误差 (uV)
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S_sensitivity, # 热电偶灵敏度 (mV/K, 预留接口，暂不计入不确定度)
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):
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"""
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计算导热系数 λ 和比热容 c 的合成不确定度。
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λ = V²·R / (4·F·r·ΔT)
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u_r(λ) = √[ (2·u_V/V)² + (u_R/R)² + (u_F/F)² + (u_r/r)² + (u_ΔT/ΔT)² ]
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u(ΔU): 稳态区间多点的 A 类 + 仪器 B 类合成; u_r(ΔT) ≈ u(ΔU)/|ΔU_mean|
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c = V² / (2·r·m·|κ|)
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u_r(c) = √[ (2·u_V/V)² + (u_r/r)² + (u_m/m)² + (u_κ/|κ|)² ]
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u_κ 来自冷却曲线线性拟合
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"""
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sqrt3 = Decimal(str(math.sqrt(3)))
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lam = result["lambda"]
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q_c = result["q_c"]
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# --- u(ΔU): A 类 + B 类 ---
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steady_delta_uV = result["steady_delta_uV"]
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delta_U_mean, u_delta_U, u_a_dU, u_b_dU, _ = calculate_stats(
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steady_delta_uV, delta_uV_inst
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)
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# u_r(ΔT) ≈ u(ΔU) / |ΔU_mean|，忽略 S 的不确定度
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ur_deltaT = (u_delta_U / abs(delta_U_mean)) if delta_U_mean != 0 else Decimal("0")
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# --- 单次测量的 B 类相对不确定度 ---
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ur_V = (delta_V / sqrt3) / V_heat
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ur_F = (delta_F / sqrt3) / F
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ur_r = (delta_r / sqrt3) / r
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ur_R = (delta_R / sqrt3) / R_thickness
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ur_m = (delta_m / sqrt3) / m_sample
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# --- u_r(λ) ---
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ur_lam_sq = (Decimal("2") * ur_V)**2 + ur_R**2 + ur_F**2 + ur_r**2 + ur_deltaT**2
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ur_lam = Decimal(str(math.sqrt(float(ur_lam_sq))))
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u_lam = lam * ur_lam
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# --- u_r(c) ---
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ur_cool = (u_dTdtau / dT_dtau_cool) if dT_dtau_cool != 0 else Decimal("0")
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ur_c_sq = (Decimal("2") * ur_V)**2 + ur_r**2 + ur_m**2 + ur_cool**2
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ur_c = Decimal(str(math.sqrt(float(ur_c_sq))))
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c_val = (Decimal("2") * q_c * F / (m_sample * dT_dtau_cool)) if dT_dtau_cool != 0 else Decimal("0")
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u_c = c_val * ur_c
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return {
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"u_lam": u_lam,
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"ur_lam": ur_lam,
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"u_c": u_c,
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"ur_c": ur_c,
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"delta_U_mean": delta_U_mean,
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"u_delta_U": u_delta_U,
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"u_a_dU": u_a_dU,
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"u_b_dU": u_b_dU,
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"ur_deltaT": ur_deltaT,
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"ur_V": ur_V,
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"ur_F": ur_F,
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"ur_r": ur_r,
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"ur_R": ur_R,
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"ur_m": ur_m,
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"ur_cool": ur_cool,
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}
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# ──────────────────────────────────────────────
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# 输出函数
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# ──────────────────────────────────────────────
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def print_data_table(result, S):
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"""打印加热阶段数据表格"""
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n = len(result["time_min"])
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print(f"\n{'时间(min)':>10} {'中心面(uV)':>12} {'加热面(uV)':>12} {'ΔU(uV)':>10} {'ΔT(K)':>10}")
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print("-" * 60)
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for i in range(n):
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t = result["time_min"][i]
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uc = result["center_uV"][i]
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uh = result["heating_uV"][i]
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du = result["delta_uV"][i]
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dt = result["delta_T"][i]
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marker = " ← 稳态起始" if t == result["ss_start_min"] else ""
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print(f"{t:>10} {uc:>12} {uh:>12} {du:>10} {float(dt):>10.2f}{marker}")
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def print_full_result(result, c_val, dT_dtau_cool, u_result):
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"""打印含不确定度的完整计算结果"""
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lam = result["lambda"]
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lam_final, u_lam_final = scientific_round(lam, u_result["u_lam"])
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c_final, u_c_final = scientific_round(c_val, u_result["u_c"])
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W = 58
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print(f"\n{'='*W}")
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print(f" {result['material']} — 计算结果")
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print(f"{'='*W}")
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print(f" 稳态起始时间 = 第 {result['ss_start_min']} 分钟")
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print(f" 稳态温差 ΔT = {float(result['delta_T_steady']):.4f} K")
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print(f" 加热趋近速率 dT/dτ = {float(result['dT_dtau']):.6f} K/s")
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print(f" 冷却速率 |κ| = |dT/dτ| = {float(dT_dtau_cool):.6f} K/s")
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print(f" 热流密度 q_c = {float(result['q_c']):.4f} W/m²")
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print(f"-" * W)
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print(f" 稳态 ΔU (均值) = {float(u_result['delta_U_mean']):.3f} uV")
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print(f" u_A(ΔU) = {float(u_result['u_a_dU']):.4f} uV")
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print(f" u_B(ΔU) = {float(u_result['u_b_dU']):.4f} uV")
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print(f" u(ΔU) = {float(u_result['u_delta_U']):.4f} uV")
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print(f"-" * W)
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print(f" 不确定度贡献 (相对):")
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print(f" u_r(V) = {float(u_result['ur_V'])*100:6.3f}% (λ & c 共用)")
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print(f" u_r(F) = {float(u_result['ur_F'])*100:6.3f}% (λ)")
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print(f" u_r(r) = {float(u_result['ur_r'])*100:6.3f}% (λ & c 共用)")
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print(f" u_r(R) = {float(u_result['ur_R'])*100:6.3f}% (λ)")
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print(f" u_r(ΔT) = {float(u_result['ur_deltaT'])*100:6.3f}% (λ)")
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print(f" u_r(m) = {float(u_result['ur_m'])*100:6.3f}% (c)")
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print(f" u_r(κ) = {float(u_result['ur_cool'])*100:6.3f}% (c)")
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print(f"-" * W)
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print(f" u_r(λ) = {float(u_result['ur_lam'])*100:.2f}% → u(λ) = {float(u_result['u_lam']):.6f} W/(m·K)")
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print(f" u_r(c) = {float(u_result['ur_c'])*100:.2f}% → u(c) = {float(u_result['u_c']):.2f} J/(kg·K)")
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print(f"{'='*W}")
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print(f" ★ 导热系数 λ = ({lam_final} ± {u_lam_final}) W/(m·K)")
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print(f" ★ 比热容 c = ({c_final} ± {u_c_final}) J/(kg·K)")
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print(f"{'='*W}")
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# ──────────────────────────────────────────────
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# 主程序
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# ──────────────────────────────────────────────
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def main():
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# --- 1. 实验参数设置 ---
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W = 58
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print("=" * W)
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print(" 导热系数 & 比热容计算 (稳态平板法 — 一维无限大平板)")
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print("=" * W)
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# 默认实验参数
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V_HEAT_DEFAULT = "18.00" # 加热电压 (V)
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F_DEFAULT = "0.006885" # 有效加热面积 = 0.85 × 0.09 × 0.09 (m²)
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R_RESISTANCE_DEFAULT = "110" # 加热电阻 (Ω)
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R_THICKNESS_DEFAULT = "0.01" # 样品厚度 (m)
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S_SENSITIVITY_DEFAULT = "0.04" # 热电偶灵敏度 (mV/K)
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N_POINTS_DEFAULT = 18 # 加热段数据点数
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N_COOL_DEFAULT = 5 # 冷却段数据点数
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print(f"\n--- 实验参数 (按 Enter 使用默认值) ---")
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V_heat = Decimal(input(f"加热电压 V (V, 默认 {V_HEAT_DEFAULT}): ").strip() or V_HEAT_DEFAULT)
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F_input = input(f"有效加热面积 F (m², 默认 {F_DEFAULT} = 0.85×0.09×0.09): ").strip()
|
||
F = Decimal(F_input) if F_input else Decimal(F_DEFAULT)
|
||
|
||
r_input = input(f"加热电阻 r (Ω, 默认 {R_RESISTANCE_DEFAULT}): ").strip()
|
||
r = Decimal(r_input) if r_input else Decimal(R_RESISTANCE_DEFAULT)
|
||
|
||
R_input = input(f"样品厚度 R (m, 默认 {R_THICKNESS_DEFAULT}): ").strip()
|
||
R_thickness = Decimal(R_input) if R_input else Decimal(R_THICKNESS_DEFAULT)
|
||
|
||
S_input = input(f"热电偶灵敏度 S (mV/K, 默认 {S_SENSITIVITY_DEFAULT}): ").strip()
|
||
S = Decimal(S_input) if S_input else Decimal(S_SENSITIVITY_DEFAULT)
|
||
|
||
n_input = input(f"加热段数据点数 (默认 {N_POINTS_DEFAULT}): ").strip()
|
||
n_points = int(n_input) if n_input else N_POINTS_DEFAULT
|
||
|
||
n_cool_input = input(f"冷却段数据点数 (默认 {N_COOL_DEFAULT}): ").strip()
|
||
n_cool = int(n_cool_input) if n_cool_input else N_COOL_DEFAULT
|
||
|
||
# --- 2. 仪器误差输入 ---
|
||
print(f"\n--- 仪器误差 (按 Enter 使用默认值) ---")
|
||
delta_V = Decimal(input(" 加热电压仪器误差 δV (V, 默认 0.01): ").strip() or "0.01")
|
||
delta_F = Decimal(input(" 面积仪器误差 δF (m², 默认 0.00001): ").strip() or "0.00001")
|
||
delta_r = Decimal(input(" 电阻仪器误差 δr (Ω, 默认 0.1): ").strip() or "0.1")
|
||
delta_R = Decimal(input(" 厚度仪器误差 δR (m, 默认 0.0001): ").strip() or "0.0001")
|
||
delta_uV = Decimal(input(" 电压表仪器误差 δU (uV, 默认 2.0): ").strip() or "2.0")
|
||
|
||
# --- 3. 逐材料数据输入 ---
|
||
# 默认密度 (kg/m³); 样品为两块相同材料叠放
|
||
materials = [
|
||
("橡胶 (Rubber)", Decimal("1374")),
|
||
("有机玻璃 (PMMA)", Decimal("1196")),
|
||
]
|
||
results = []
|
||
extra_data = []
|
||
|
||
for mat_name, rho_default in materials:
|
||
print(f"\n{'#'*W}")
|
||
print(f" 材料: {mat_name}")
|
||
print(f"{'#'*W}")
|
||
|
||
# 加热阶段电压
|
||
center_uV = input_voltage_series(f"{mat_name} — 中心面电压 (加热阶段)", n_points)
|
||
heating_uV = input_voltage_series(f"{mat_name} — 加热面电压 (加热阶段)", n_points)
|
||
|
||
result = compute_thermal_conductivity(
|
||
center_uV=center_uV,
|
||
heating_uV=heating_uV,
|
||
V_heat=V_heat,
|
||
F=F,
|
||
r=r,
|
||
R_thickness=R_thickness,
|
||
S_sensitivity=S,
|
||
material_name=mat_name
|
||
)
|
||
|
||
print_data_table(result, S)
|
||
|
||
# 比热容测量 (冷却法) — 质量由密度与几何尺寸推算
|
||
print(f"\n--- {mat_name} — 比热容测量 (冷却法) ---")
|
||
rho_str = input(f" 材料密度 ρ (kg/m³, 默认 {rho_default}): ").strip()
|
||
rho = Decimal(rho_str) if rho_str else rho_default
|
||
# 两块样品叠放: m = 2 × ρ × F × R
|
||
m_sample = Decimal("2") * rho * F * R_thickness
|
||
print(f" → 样品质量 m = 2 × {rho} × {F} × {R_thickness} = {float(m_sample):.6f} kg")
|
||
# δm 由 δF、δR 传递: u_r(m) = √[(u_F/F)² + (u_R/R)²], delta_m_eff = m·√[(δF/F)²+(δR/R)²]
|
||
delta_m = m_sample * Decimal(str(math.sqrt(
|
||
float((delta_F / F)**2 + (delta_R / R_thickness)**2)
|
||
)))
|
||
|
||
# 冷却段中心面电压采用加热段中心面电压的最后几项反向填充（即从稳态开始降温）
|
||
cooling_uV = list(reversed(center_uV[-n_cool:]))
|
||
print(f" → 冷却阶段中心面电压 (反向填充自加热段): {[float(x) for x in cooling_uV]} uV")
|
||
|
||
results.append(result)
|
||
extra_data.append({"m_sample": m_sample, "delta_m": delta_m, "cooling_uV": cooling_uV})
|
||
|
||
# --- 4. 汇总计算与输出 ---
|
||
print("\n" + "=" * W)
|
||
print(" 汇 总 结 果")
|
||
print("=" * W)
|
||
|
||
q_c = results[0]["q_c"]
|
||
print(f"\n 加热电压 V = {V_heat} V")
|
||
print(f" 有效面积 F = {F} m²")
|
||
print(f" 加热电阻 r = {r} Ω")
|
||
print(f" 样品厚度 R = {R_thickness} m")
|
||
print(f" 热电偶灵敏度 S = {S} mV/K")
|
||
print(f" 热流密度 q_c = V²/(2·F·r) = {float(q_c):.4f} W/m²")
|
||
|
||
for res, extra in zip(results, extra_data):
|
||
m_sample = extra["m_sample"]
|
||
delta_m = extra["delta_m"]
|
||
cooling_uV = extra["cooling_uV"]
|
||
|
||
c_val, dT_dtau_cool, u_dTdtau = compute_specific_heat(
|
||
cooling_uV=cooling_uV,
|
||
S_sensitivity=S,
|
||
q_c=res["q_c"],
|
||
F=F,
|
||
m_sample=m_sample,
|
||
)
|
||
|
||
u_result = compute_uncertainties(
|
||
result=res,
|
||
dT_dtau_cool=dT_dtau_cool,
|
||
u_dTdtau=u_dTdtau,
|
||
m_sample=m_sample,
|
||
delta_m=delta_m,
|
||
V_heat=V_heat, delta_V=delta_V,
|
||
F=F, delta_F=delta_F,
|
||
r=r, delta_r=delta_r,
|
||
R_thickness=R_thickness, delta_R=delta_R,
|
||
delta_uV_inst=delta_uV,
|
||
S_sensitivity=S,
|
||
)
|
||
|
||
print_full_result(res, c_val, dT_dtau_cool, u_result)
|
||
|
||
# --- 5. 参考值对比 ---
|
||
print("\n" + "-" * W)
|
||
print(" 参考值 (室温):")
|
||
print(f" 橡胶: λ ≈ 0.13–0.16 W/(m·K), c ≈ 1000–2000 J/(kg·K)")
|
||
print(f" PMMA: λ ≈ 0.17–0.25 W/(m·K), c ≈ 1400–1500 J/(kg·K)")
|
||
print("-" * W)
|
||
|
||
input("\n按 Enter 退出...")
|
||
|
||
|
||
if __name__ == "__main__":
|
||
main()
|