Mathematical Modelling: Construction and Analysis of a Boiler Temperature Prediction Model | 数学建模：锅炉温度预测模型的建立与分析

📚 Mathematical Modelling: Construction and Analysis of a Boiler Temperature Prediction Model | 数学建模：锅炉温度预测模型的建立与分析

In industrial engineering, the ability to predict the internal temperature of a boiler system is of critical importance for safety, fuel efficiency, and operational stability. This article presents a step-by-step mathematical modelling process, from problem definition and data collection to model construction, parameter estimation, and validation.

在工业工程中，预测锅炉系统内部温度的能力对于安全性、燃料效率以及运行稳定性至关重要。本文将呈现一个循序渐进的数学建模过程，涵盖从问题定义、数据采集到模型构建、参数估计与模型验证的完整流程。

1. Problem Definition and Modelling Objectives | 问题定义与建模目标

Consider an industrial boiler that heats water using a gas burner. The boiler is instrumented to record the water temperature T(t) at regular time intervals, along with the fuel flow rate F(t) and ambient temperature Tₐ. The objective is to construct a mathematical model that accurately predicts future boiler temperature based on current and past observations.

考虑一个使用燃气燃烧器加热水的工业锅炉。锅炉配备仪表，以固定时间间隔记录水温 T(t)、燃料流量 F(t) 以及环境温度 Tₐ。建模目标是构建一个数学模型，能够根据当前及过去的观测数据准确预测锅炉未来的温度。

We assume that the system obeys fundamental physical conservation laws but that internal parameters such as thermal resistance and heat capacity are not directly known. Thus, we adopt a semi-empirical approach: we first derive a physically reasonable model structure, then estimate the unknown parameters from real data.

我们假设系统遵循基本的物理守恒定律，但热阻、热容等内部参数无法直接获知。因此，我们采用半经验方法：首先推导一个物理上合理的模型结构，然后从真实数据中估计未知参数。

Model scope: T(t + Δt) = f(T(t), F(t), Tₐ, parameters)

模型范围：T(t + Δt) = f(T(t), F(t), Tₐ, 参数)

2. Fundamental Physical Principles | 基础物理原理

The heat balance of a boiler can be described by Newton’s law of cooling combined with an input heat source. The rate of change of internal energy in the water equals the heat input from the burner minus the heat lost to the surroundings. Mathematically, for a lumped system:

锅炉的热平衡可以通过牛顿冷却定律与输入热源相结合来描述。水内能的变化率等于燃烧器输入的热量减去向周围环境散失的热量。对集总系统而言，其数学表达为：

m·c·(dT/dt) = Q_input(t) − U·A·(T(t) − Tₐ)

where m is the mass of water, c is the specific heat capacity of water, Q_input(t) is the thermal power supplied by the burner, U is the overall heat transfer coefficient, and A is the effective surface area for heat loss.

其中，m 为水的质量，c 为水的比热容，Q_input(t) 为燃烧器提供的热功率，U 为总传热系数，A 为热损失的有效散热面积。

Since U, A, m, and c may not be known individually, it is advantageous to define two composite parameters: the thermal time constant τ = m·c/(U·A), and the steady-state temperature gain K. These parameters will be estimated directly from data using regression techniques.

由于 U、A、m 与 c 未必能逐一获知，定义两个组合参数将更为有利：热时间常数 τ = m·c/(U·A)，以及稳态温度增益 K。这些参数将直接利用回归技术从数据中估计。

3. Continuous-Time Model Construction | 连续时间模型构建

We now express Q_input(t) as proportional to the fuel flow rate: Q_input(t) = K·F(t)·Δt. Substituting into the heat balance equation and rearranging yields the first-order linear ordinary differential equation:

现在我们将 Q_input(t) 表示为与燃料流量成正比：Q_input(t) = K·F(t)·Δt。将其代入热平衡方程并整理，得到一阶线性常微分方程：

τ · (dT/dt) = K·F(t) − (T(t) − Tₐ)

This equation states that the rate of temperature change is controlled by two competing effects: the heat gain from fuel combustion and the heat loss to the environment. The coefficient τ dictates how quickly the system responds to changes in the fuel input — a large τ implies slow response, typical of systems with large water volume.

该方程表明温度变化率受两个竞争效应控制：燃烧产生的热量增益，以及向环境散失的热量。系数 τ 决定了系统对燃料输入变化的响应速度——τ 越大意味着响应越慢，这是大水量系统的典型特征。

The differential equation has an analytical solution for constant F. Setting the initial temperature as T(0) = T₀, the solution is:

对于恒定 F，该微分方程具有解析解。设初始温度为 T(0) = T₀，其解为：

T(t) = Tₐ + K·F·τ·(1 − e^(−t/τ)) + (T₀ − Tₐ)·e^(−t/τ)

This exponential form shows that the temperature approaches an asymptote Tₐ + K·F·τ as time goes to infinity, and that the gap between the current temperature and the steady-state value diminishes exponentially with rate 1/τ.

该指数形式表明，随着时间趋于无穷，温度将趋近于渐近值 Tₐ + K·F·τ；当前温度与稳态值之间的差距将以速率 1/τ 指数衰减。

4. Discretisation for Real-Time Prediction | 面向实时预测的离散化处理

In practice, sensor data are sampled at discrete time steps. For real-time prediction, it is necessary to convert the continuous differential equation into a discrete-time difference equation. Using a first-order forward Euler approximation with sampling interval h:

实际应用中，传感器数据以离散时间间隔采样。为了实现实时预测，需要将连续微分方程转化为离散时间差分方程。采用采样间隔为 h 的一阶前向欧拉近似：

T(n+1) = T(n) + (h/τ)·[K·F(n) − (T(n) − Tₐ)]

Defining the dimensionless parameter a = h/τ and b = a·K, this becomes an elegant linear recurrence relation:

定义无量纲参数 a = h/τ 以及 b = a·K，上式可转化为简洁的线性递推关系：

T(n+1) = (1 − a)·T(n) + b·F(n) + a·Tₐ

This form is particularly appealing because it expresses the next temperature as a linear combination of the current temperature, the current fuel flow rate, and the ambient temperature. Linear regression can then be applied directly to estimate the coefficients from measured data.

这一形式极具吸引力，因为下一时刻的温度表示为当前温度、当前燃料流量以及环境温度的线性组合。由此可直接应用线性回归，从测量数据中估计各项系数。

5. Parameter Estimation Using Least Squares | 基于最小二乘法的参数估计

With N recorded data points, we can set up a regression problem. Let the observations be denoted as (Tᵢ, Fᵢ, Tₐᵢ). Rewriting the recurrence in measurement form:

设有 N 个记录数据点，我们可以构造一个回归问题。将观测数据记为 (Tᵢ, Fᵢ, Tₐᵢ)。将递推关系改写为含测量的形式：

yᵢ = T(i+1) − T(i) = β₁·T(i) + β₂·F(i) + β₃·Tₐᵢ + εᵢ

where yᵢ is the observed temperature change over one sampling interval, the unknown coefficients are β₁ = −a, β₂ = b, β₃ = a, and εᵢ represents measurement noise and modelling error. The least squares method minimises the sum of squared residuals:

其中 yᵢ 为一个采样间隔内观测到的温度变化量，未知系数为 β₁ = −a、β₂ = b、β₃ = a，εᵢ 代表测量噪声与建模误差。最小二乘法通过最小化残差平方和来实现参数估计：

S(β₁, β₂, β₃) = Σᵢ₌₁ᴺ [yᵢ − (β₁·Tᵢ + β₂·Fᵢ + β₃·Tₐᵢ)]²

The solution is obtained by solving the normal equations ∂S/∂βⱼ = 0. This yields a system of three linear equations that can be solved by matrix inversion. Most scientific computing platforms provide built-in functions for this purpose, making the implementation straightforward.

求解过程通过解正规方程 ∂S/∂βⱼ = 0 来完成，这会产生一个由三个线性方程组成的方程组，可通过矩阵求逆求解。大多数科学计算平台都内置了相应函数，使实现过程变得十分直接。

6. Structured Data Table | 结构化数据表格

The table below provides an illustrative sample of data collected from a test boiler, including temperature, fuel flow rate, ambient temperature, and the calculated temperature change over each sampling interval.

下表给出了从测试锅炉中采集的示例数据，包括温度、燃料流量、环境温度，以及每个采样间隔内计算得到的温度变化量。

Time (s)	T (⁰C)	F (kW)	Tₐ (⁰C)	ΔT (⁰C)
0	20.0	50.0	15.0	0.00
10	22.5	50.0	15.0	2.50
20	24.7	48.0	15.2	2.20
30	26.6	50.0	15.1	1.90
40	28.3	51.0	15.3	1.70
50	29.8	49.0	15.0	1.50
60	31.1	50.0	15.2	1.30
70	32.2	48.0	15.1	1.10

In this sample we see that the temperature increase ΔT per interval gradually decreases despite relatively steady fuel input. This pattern is consistent with the first-order model: the rising water temperature causes greater heat loss to the surroundings, reducing the net rate of energy accumulation.

在该示例中可以观察到，尽管燃料输入保持相对稳定，每个时间间隔内的温度增量 ΔT 却逐步减小。这一趋势与一阶模型的预测相符：水温升高导致向周围环境的散热增加，从而降低了能量的净积累速率。

7. Model Validation and Error Analysis | 模型验证与误差分析

Once the parameters are estimated, we must verify that the model is a faithful representation of the system. A common validation approach is to reserve the last portion of the data — known as the testing set — for comparison against predictions. The fitting portion is called the training set.

参数估计完成后，我们必须验证模型是否忠实反映了系统行为。一种常见的验证方法是将数据末尾的部分保留作为测试集，用于与预测结果进行比较；用于拟合的部分则称为训练集。

Three metrics are particularly useful in assessing model accuracy:

以下三个指标在评估模型精度方面特别有用：

Mean Squared Error (MSE) | 均方误差 (MSE)

MSE = (1/n)·Σ(T_pred − T_actual)². It penalises large deviations more heavily and is the standard metric aligned with least squares estimation.

MSE = (1/n)·Σ(T_pred − T_actual)²。它对较大偏差给予更重的惩罚，是与最小二乘估计相一致的标准指标。
Root Mean Squared Error (RMSE) | 均方根误差 (RMSE)

RMSE = √MSE. Expressing error in the same units as temperature, this is intuitively interpretable.

RMSE = √MSE。误差以与温度相同的单位表示，因而具有直观可解释性。
Coefficient of Determination (R²) | 决定系数 (R²)

R² = 1 − SS_res/SS_tot, where SS_res is the residual sum of squares and SS_tot is the total sum of squares. An R² value close to 1 indicates that the model explains most of the variance in the data.

R² = 1 − SS_res/SS_tot，其中 SS_res 为残差平方和，SS_tot 为总平方和。R² 越接近 1 说明模型解释了数据中的大部分变异。

For our first-order linear model, it is also prudent to examine the residual plot. If residuals display a random pattern around zero, the model is adequate. Systematic patterns would suggest that higher-order dynamics are present and that the model structure needs revision.

对于一阶线性模型，还应检查残差图。若残差在零附近呈随机分布，则模型是合适的；若出现系统性模式，则意味着系统存在更高阶的动力学特征，需要对模型结构进行修正。

8. Prediction and Sensitivity Analysis | 预测与敏感性分析

A well-calibrated model enables two types of prediction: one-step-ahead prediction and multi-step-ahead simulation. In one-step-ahead prediction, we update the model with actual measurements at each time step, thereby resetting any accumulated error. In multi-step simulation, we feed predicted values back into the model, which allows us to forecast far into the future.

经过良好校准的模型可以实现两种类型的预测：单步超前预测和多步超前模拟。在单步超前预测中，每一步都使用实际测量值更新模型，从而消除累积误差。在多步模拟中，预测值被反馈回模型，从而实现对更远未来的预报。

Sensitivity analysis investigates how the prediction error changes when the parameters vary. The temperature prediction T(n+1) is most sensitive to variations in the coefficient (1 − a), which multiplies the current temperature. For a large boiler with slow dynamics, a is small, meaning the current temperature term dominates the next prediction.

敏感性分析研究参数变动时预测误差如何变化。温度预测 T(n+1) 对乘以当前温度的系数 (1 − a) 的变化最为敏感。对于动力学响应缓慢的大型锅炉，a 较小，意味着当前温度项在下一时刻预测值中占据主导地位。

δT(n+1) = T(n)·δ(1 − a) + F(n)·δb + Tₐ·δa

Therefore, accurate estimation of the parameter a is crucial. This can be achieved by ensuring sufficient excitation in the fuel input — if the fuel flow rate remains constant, the data may not reveal enough dynamic information to determine τ precisely.

因此，对参数 a 的精确估计至关重要。这要求燃料输入具有足够的激励变化——如果燃料流量始终恒定，数据将无法揭示足够的动态信息来精确确定 τ。

9. Extended Model: Variable Fuel Rate | 扩展模型：可变燃料速率

In many real scenarios, fuel input is not constant but is adjusted by a control system that reacts to temperature errors. A more sophisticated model incorporates a control law. If a proportional-integral (PI) controller drives the fuel rate, then F(t) becomes a function of the temperature error e(t) = T_setpoint − T(t):

在许多真实场景中，燃料输入并非恒定不变，而是由响应温度偏差的控制系统进行调节。更复杂的模型需要包含控制律。如果比例积分 (PI) 控制器驱动燃料速率，那么 F(t) 将变为温度偏差 e(t) = T_setpoint − T(t) 的函数：

F(t) = Kₚ·e(t) + Kᵢ·∫₀ᵗ e(s)ds

Substituting into the boiler differential equation yields a second-order closed-loop system. The characteristic equation of this system has the general form:

将控制律代入锅炉微分方程，可得到一个二阶闭环系统。该系统的特征方程具有如下一般形式：

s² + 2ζωₙ·s + ωₙ² = 0

The natural frequency ωₙ and damping ratio ζ determine whether the system responds in an oscillatory, critically damped, or overdamped manner. Engineers select Kₚ and Kᵢ to ensure a critically damped response — the fastest response without overshoot.

固有频率 ωₙ 与阻尼比 ζ 共同决定系统表现为振荡响应、临界阻尼响应还是过阻尼响应。工程师通过选择 Kₚ 与 Kᵢ 来实现临界阻尼响应，即在无超调的前提下获得最快的响应速度。

This extension demonstrates how a purely thermal model can be embedded within a broader dynamic system, showing the power of modular mathematical modelling.

这一扩展展示了一个纯热学模型如何嵌入更广泛的动力学系统之中，充分体现了模块化数学建模的强大之处。

10. Conclusion and Practical Recommendations | 结论与实践建议

This article has presented a systematic approach to developing a boiler temperature prediction model. Beginning from physical heat-balance principles, we derived a first-order differential equation, discretised it into a linear recurrence, estimated parameters using least squares, and validated the model’s predictive capability through statistical metrics.

本文展示了建立锅炉温度预测模型的系统化方法。从物理热平衡原理出发，推导了一阶微分方程，将其离散化为线性递推关系，利用最小二乘法估计参数，并通过统计指标验证了模型的预测能力。

Always start from physical principles to establish the model structure. | 始终从物理原理出发建立模型结构。
Use the discrete-domain recursion for implementation as it is computationally efficient and compatible with digital controllers and recorders. | 实现时采用离散域递推形式，其计算效率高，与数字控制器和记录仪兼容。
Actively introduce perturbations in fuel flow during data collection to ensure all model parameters are identifiable. | 在数据采集期间主动对燃料流量施加扰动，以确保所有模型参数均可辨识。
Reserve validation data that is temporally separate from training data to obtain an honest estimate of prediction error. | 保留在时间上与训练数据分开的验证数据，以获得对预测误差的真实估计。
For highly demanding applications, consider nonlinear extensions such as temperature-dependent heat-loss coefficients or multiple-zone compartmentalisation. | 对于要求极高的应用，应考虑非线性扩展，如随温度变化的散热系数或多区域分舱建模。

By following this structured modelling pipeline, engineers can develop reliable predictive tools that support better control decisions, improve energy efficiency, and ensure the safe operation of boiler systems.

遵循这一结构化的建模流程，工程师可以开发出可靠的预测工具，从而为更好的控制决策提供支持，提高能源效率，并确保锅炉系统的安全运行。

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Mathematical Modelling: Construction and Analysis of a Boiler Temperature Prediction Model | 数学建模：锅炉温度预测模型的建立与分析