import math from decimal import Decimal """ Thermal Conductivity & Specific Heat Calculator (Steady-State Plate Method) 导热系数与比热容计算器 (稳态平板法 — 一维无限大平板) 实验原理: 利用稳态平板法测量不良导体的导热系数。 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 样品两面温差 ΔT 除以厚度 R 再乘以导热系数 λ。 断电后由冷却曲线斜率计算样品比热容 c。 核心公式: 1. 热流密度: q_c = V² / (2 · F · r) 其中 V = 加热电压, F = 有效加热面积, r = 加热电阻 2. 导热系数: λ = q_c · R / (2 · ΔT) 其中 R = 样品厚度, ΔT = 加热面温度 - 中心面温度 3. 温度转换 (热电偶): ΔT = ΔU / (1000 · S) 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) 4. 比热容 (冷却法): c = 2 · q_c · F / (m · |κ|) 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 5. 不确定度传递 — 导热系数 λ = V²·R / (4·F·r·ΔT): u_r(λ) = √[ (2·u_V/V)² + (u_R/R)² + (u_F/F)² + (u_r/r)² + (u_ΔT/ΔT)² ] u(ΔU) 取稳态区间多点的 A 类 + 仪器 B 类合成; u_r(ΔT) ≈ u(ΔU)/|ΔU| 6. 不确定度传递 — 比热容 c = V² / (r·m·|κ|): u_r(c) = √[ (2·u_V/V)² + (u_r/r)² + (u_m/m)² + (u_κ/|κ|)² ] u_κ 由冷却曲线线性拟合的斜率标准差给出 """ from utils import ( scientific_round, calculate_stats, linear_regression ) # ────────────────────────────────────────────── # 数据输入辅助函数 # ────────────────────────────────────────────── def input_voltage_series(name, n_points=18): """输入 1~n_points 分钟的电压数据 (单位: uV)""" 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:") data = [] while len(data) < n_points: remaining = n_points - len(data) try: raw = input(f" [{len(data)+1}~{len(data)+remaining}] > ").strip() if not raw: continue tokens = raw.replace(',', ' ').split() for tok in tokens: data.append(Decimal(tok)) if len(data) >= n_points: break except Exception: print(" 输入无效，请重新输入。") return data # ────────────────────────────────────────────── # 稳态判断 # ────────────────────────────────────────────── def find_steady_state(voltages, window=3, threshold=Decimal("0.5")): """ 判断电压序列何时达到稳态。 使用滑动窗口：当连续 window 个读数的极差 < threshold (uV) 时认为稳态。 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。 """ for i in range(len(voltages) - window + 1): w = voltages[i:i + window] if max(w) - min(w) < threshold: return i return len(voltages) - window # ────────────────────────────────────────────── # 核心计算函数 # ────────────────────────────────────────────── def compute_thermal_conductivity( center_uV, # 中心面电压序列 (uV) heating_uV, # 加热面电压序列 (uV) V_heat, # 加热电压 (V) F, # 有效加热面积 (m²) r, # 加热电阻 (Ω) R_thickness, # 样品厚度 (m) S_sensitivity, # 热电偶灵敏度 (mV/K) material_name="" # 材料名称 ): """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""" n = len(center_uV) time_min = list(range(1, n + 1)) # 1 ~ n 分钟 # --- 1. 转换为温度差 ΔT(t) --- S_uV = S_sensitivity * Decimal("1000") # 将 mV/K 转换为 uV/K 以匹配 uV 数据 delta_uV = [heating_uV[i] - center_uV[i] for i in range(n)] delta_T = [dv / S_uV for dv in delta_uV] # --- 2. 判断稳态 --- ss_idx_center = find_steady_state(center_uV) ss_idx_heating = find_steady_state(heating_uV) ss_idx = max(ss_idx_center, ss_idx_heating) # 稳态区间 ΔU 列表（A 类不确定度来源） steady_delta_uV = delta_uV[ss_idx:] delta_T_steady = sum(steady_delta_uV) / Decimal(len(steady_delta_uV)) / S_uV center_steady_avg = sum(center_uV[ss_idx:]) / Decimal(len(center_uV[ss_idx:])) heating_steady_avg = sum(heating_uV[ss_idx:]) / Decimal(len(heating_uV[ss_idx:])) # --- 3. 热流密度 q_c --- q_c = V_heat ** 2 / (Decimal("2") * F * r) # --- 4. 导热系数 λ --- lam = q_c * R_thickness / (Decimal("2") * delta_T_steady) # --- 5. 加热趋近速率 dT/dτ (用于验证是否达到稳态) --- if n >= 3: last_n = min(5, n) x_fit = [Decimal(str(t * 60)) for t in time_min[-last_n:]] # 秒 y_fit = [center_uV[i] / S_uV for i in range(-last_n, 0)] # K (相对) dT_dtau, _, _, _ = linear_regression(x_fit, y_fit) # K/s else: dT_dtau = Decimal("0") return { "material": material_name, "q_c": q_c, "delta_T_steady": delta_T_steady, "lambda": lam, "ss_start_min": ss_idx + 1, "center_steady_uV": center_steady_avg, "heating_steady_uV": heating_steady_avg, "dT_dtau": dT_dtau, "time_min": time_min, "center_uV": center_uV, "heating_uV": heating_uV, "delta_uV": delta_uV, "delta_T": delta_T, "steady_delta_uV": steady_delta_uV, # 稳态区间 ΔU 列表，供不确定度分析使用 } def compute_specific_heat(cooling_uV, S_sensitivity, q_c, F, m_sample): """ 由冷却曲线计算比热容 c。 cooling_uV : 断电后中心面电压序列 (uV), 每隔 60 s 一个点 S_sensitivity: 热电偶灵敏度 (mV/K) q_c : 热流密度 (W/m²) F : 有效加热面积 (m²) m_sample : 样品质量 (kg) 返回: (c, dT_dtau_cool, u_dTdtau) c — 比热容 J/(kg·K) dT_dtau_cool — 冷却速率绝对值 |κ| (K/s) u_dTdtau — |κ| 的不确定度 (K/s), 由线性拟合给出 """ n = len(cooling_uV) S_uV = S_sensitivity * Decimal("1000") # 将 mV/K 转换为 uV/K 以匹配 uV 数据 x_cool = [Decimal(str(i * 60)) for i in range(n)] # 秒 y_cool = [uv / S_uV for uv in cooling_uV] # 相对温度 (K) k, _, _, u_k = linear_regression(x_cool, y_cool) dT_dtau_cool = abs(k) # 冷却时斜率为负，取绝对值 u_dTdtau = abs(u_k) c = (Decimal("2") * q_c * F / (m_sample * dT_dtau_cool)) if dT_dtau_cool != 0 else Decimal("0") return c, dT_dtau_cool, u_dTdtau def compute_uncertainties( result, # dict from compute_thermal_conductivity dT_dtau_cool, # 冷却速率 |κ| (K/s, Decimal, 已取绝对值) u_dTdtau, # |κ| 的不确定度 (K/s) m_sample, # 样品质量 (kg) delta_m, # 质量仪器误差 (kg) V_heat, delta_V, # 加热电压及仪器误差 (V) F, delta_F, # 面积及仪器误差 (m²) r, delta_r, # 电阻及仪器误差 (Ω) R_thickness, delta_R, # 厚度及仪器误差 (m) delta_uV_inst, # 电压表仪器误差 (uV) S_sensitivity, # 热电偶灵敏度 (mV/K, 预留接口，暂不计入不确定度) ): """ 计算导热系数 λ 和比热容 c 的合成不确定度。 λ = V²·R / (4·F·r·ΔT) u_r(λ) = √[ (2·u_V/V)² + (u_R/R)² + (u_F/F)² + (u_r/r)² + (u_ΔT/ΔT)² ] u(ΔU): 稳态区间多点的 A 类 + 仪器 B 类合成; u_r(ΔT) ≈ u(ΔU)/|ΔU_mean| c = V² / (2·r·m·|κ|) u_r(c) = √[ (2·u_V/V)² + (u_r/r)² + (u_m/m)² + (u_κ/|κ|)² ] u_κ 来自冷却曲线线性拟合 """ sqrt3 = Decimal(str(math.sqrt(3))) lam = result["lambda"] q_c = result["q_c"] # --- u(ΔU): A 类 + B 类 --- steady_delta_uV = result["steady_delta_uV"] delta_U_mean, u_delta_U, u_a_dU, u_b_dU, _ = calculate_stats( steady_delta_uV, delta_uV_inst ) # u_r(ΔT) ≈ u(ΔU) / |ΔU_mean|，忽略 S 的不确定度 ur_deltaT = (u_delta_U / abs(delta_U_mean)) if delta_U_mean != 0 else Decimal("0") # --- 单次测量的 B 类相对不确定度 --- ur_V = (delta_V / sqrt3) / V_heat ur_F = (delta_F / sqrt3) / F ur_r = (delta_r / sqrt3) / r ur_R = (delta_R / sqrt3) / R_thickness ur_m = (delta_m / sqrt3) / m_sample # --- u_r(λ) --- ur_lam_sq = (Decimal("2") * ur_V)**2 + ur_R**2 + ur_F**2 + ur_r**2 + ur_deltaT**2 ur_lam = Decimal(str(math.sqrt(float(ur_lam_sq)))) u_lam = lam * ur_lam # --- u_r(c) --- ur_cool = (u_dTdtau / dT_dtau_cool) if dT_dtau_cool != 0 else Decimal("0") ur_c_sq = (Decimal("2") * ur_V)**2 + ur_r**2 + ur_m**2 + ur_cool**2 ur_c = Decimal(str(math.sqrt(float(ur_c_sq)))) c_val = (Decimal("2") * q_c * F / (m_sample * dT_dtau_cool)) if dT_dtau_cool != 0 else Decimal("0") u_c = c_val * ur_c return { "u_lam": u_lam, "ur_lam": ur_lam, "u_c": u_c, "ur_c": ur_c, "delta_U_mean": delta_U_mean, "u_delta_U": u_delta_U, "u_a_dU": u_a_dU, "u_b_dU": u_b_dU, "ur_deltaT": ur_deltaT, "ur_V": ur_V, "ur_F": ur_F, "ur_r": ur_r, "ur_R": ur_R, "ur_m": ur_m, "ur_cool": ur_cool, } # ────────────────────────────────────────────── # 输出函数 # ────────────────────────────────────────────── def print_data_table(result, S): """打印加热阶段数据表格""" n = len(result["time_min"]) print(f"\n{'时间(min)':>10} {'中心面(uV)':>12} {'加热面(uV)':>12} {'ΔU(uV)':>10} {'ΔT(K)':>10}") print("-" * 60) for i in range(n): t = result["time_min"][i] uc = result["center_uV"][i] uh = result["heating_uV"][i] du = result["delta_uV"][i] dt = result["delta_T"][i] marker = " ← 稳态起始" if t == result["ss_start_min"] else "" print(f"{t:>10} {uc:>12} {uh:>12} {du:>10} {float(dt):>10.2f}{marker}") def print_full_result(result, c_val, dT_dtau_cool, u_result): """打印含不确定度的完整计算结果""" lam = result["lambda"] lam_final, u_lam_final = scientific_round(lam, u_result["u_lam"]) c_final, u_c_final = scientific_round(c_val, u_result["u_c"]) W = 58 print(f"\n{'='*W}") print(f" {result['material']} — 计算结果") print(f"{'='*W}") print(f" 稳态起始时间 = 第 {result['ss_start_min']} 分钟") print(f" 稳态温差 ΔT = {float(result['delta_T_steady']):.4f} K") print(f" 加热趋近速率 dT/dτ = {float(result['dT_dtau']):.6f} K/s") print(f" 冷却速率 |κ| = |dT/dτ| = {float(dT_dtau_cool):.6f} K/s") print(f" 热流密度 q_c = {float(result['q_c']):.4f} W/m²") print(f"-" * W) print(f" 稳态 ΔU (均值) = {float(u_result['delta_U_mean']):.3f} uV") print(f" u_A(ΔU) = {float(u_result['u_a_dU']):.4f} uV") print(f" u_B(ΔU) = {float(u_result['u_b_dU']):.4f} uV") print(f" u(ΔU) = {float(u_result['u_delta_U']):.4f} uV") print(f"-" * W) print(f" 不确定度贡献 (相对):") print(f" u_r(V) = {float(u_result['ur_V'])*100:6.3f}% (λ & c 共用)") print(f" u_r(F) = {float(u_result['ur_F'])*100:6.3f}% (λ)") print(f" u_r(r) = {float(u_result['ur_r'])*100:6.3f}% (λ & c 共用)") print(f" u_r(R) = {float(u_result['ur_R'])*100:6.3f}% (λ)") print(f" u_r(ΔT) = {float(u_result['ur_deltaT'])*100:6.3f}% (λ)") print(f" u_r(m) = {float(u_result['ur_m'])*100:6.3f}% (c)") print(f" u_r(κ) = {float(u_result['ur_cool'])*100:6.3f}% (c)") print(f"-" * W) print(f" u_r(λ) = {float(u_result['ur_lam'])*100:.2f}% → u(λ) = {float(u_result['u_lam']):.6f} W/(m·K)") print(f" u_r(c) = {float(u_result['ur_c'])*100:.2f}% → u(c) = {float(u_result['u_c']):.2f} J/(kg·K)") print(f"{'='*W}") print(f" ★ 导热系数 λ = ({lam_final} ± {u_lam_final}) W/(m·K)") print(f" ★ 比热容 c = ({c_final} ± {u_c_final}) J/(kg·K)") print(f"{'='*W}") # ────────────────────────────────────────────── # 主程序 # ────────────────────────────────────────────── def main(): # --- 1. 实验参数设置 --- W = 58 print("=" * W) print(" 导热系数 & 比热容计算 (稳态平板法 — 一维无限大平板)") print("=" * W) # 默认实验参数 V_HEAT_DEFAULT = "18.00" # 加热电压 (V) F_DEFAULT = "0.006885" # 有效加热面积 = 0.85 × 0.09 × 0.09 (m²) R_RESISTANCE_DEFAULT = "110" # 加热电阻 (Ω) R_THICKNESS_DEFAULT = "0.01" # 样品厚度 (m) S_SENSITIVITY_DEFAULT = "0.04" # 热电偶灵敏度 (mV/K) N_POINTS_DEFAULT = 18 # 加热段数据点数 N_COOL_DEFAULT = 5 # 冷却段数据点数 print(f"\n--- 实验参数 (按 Enter 使用默认值) ---") V_heat = Decimal(input(f"加热电压 V (V, 默认 {V_HEAT_DEFAULT}): ").strip() or V_HEAT_DEFAULT) 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()