## 2 Heat Transfer Theory of Linear Heat Sources in Water Media
### Derivation of Heat Transfer Formula for Linear Heat Sources  
When the heating rod of an active heating and temperature measurement system is placed in water, its heat transfer model is idealized as a slender linear heat source situated in an infinitely large liquid environment. At any given moment, the temperature within the linear heat source is uniformly distributed. Moreover, during the prolonged heat generation process, the temperature of the liquid environment surrounding the linear heat source remains unaffected by external influences (external temperatures are considered constant) [9]. The non-steady-state heat conduction partial differential equation with internal heat sources is applicable to this problem, namely as follows:  
\[\frac{\partial T}{\partial t}=a\
abla^{2}T+\frac{\Phi^{\prime}}{\rho c} \tag{1}\]  
Since the internal thermal resistance of the object can be neglected and temperature is independent of spatial coordinates, the second-order derivative term of temperature is zero. Therefore, the above equation can be simplified to the following:  
\[\frac{\partial T}{\partial t}=\frac{\Phi^{\prime}}{\rho c} \tag{2}\]  
Here, \(\Phi^{\prime}\) is considered a generalized heat source. The heat exchange boundary here is not a calculated boundary, so the heat exchanged on the interface should be converted into a volume heat source for the entire object:  
\[Ah(T-T_{\infty})+\Phi^{\prime}V=Q \tag{3}\]  
The [PERSON]'s Law of Cooling equation describes the temperature change in an object under free convection conditions and is commonly used to describe the heat exchange process between a gas or liquid and its surrounding medium. According to the [PERSON]'s Law of Cooling and the law of conservation of energy [10],  
\[\rho cV\frac{\partial T}{\partial t}=Q-Ah(T-T_{\infty}) \tag{4}\]  
In the equation, \(\rho\) is the density of the linear heat source per unit length; \(c\) is the specific heat capacity of the linear heat source; \(A\) is the surface area of the linear heat source per unit length; \(h\) is the convective heat transfer coefficient of the linear heat source material; \(T\) is the surface temperature of the object; \(T_{\infty}\) is the ambient temperature. To solve Equation (4), the following could be obtained:  
\[\frac{\partial T}{Q-Ah(T-T_{\infty})}=\frac{\partial t}{\rho cV} \tag{5}\]By introducing the excess temperature \(\theta=T-T_{\infty}\), the following could be obtained:  
\[\frac{\partial\theta}{Q-Ah\theta}=\frac{\partial t}{\rho cV} \tag{6}\]  
Integrating with respect to \(t\), the following could be obtained:  
\[\int_{\theta_{0}}^{\theta}\frac{d\theta}{Q-Ah\theta}=-\int_{0}^{t}\frac{1}{ \rho cV}dt \tag{7}\]  
If heating is stopped at time \(t_{1}\), then \(Q=0\), and the following could be obtained:  
\[\rho cV\frac{\partial T}{\partial t}=-Ah(T-T_{\infty})=-Ah\theta \tag{8}\]  
In the equation, \(T_{0}\) is the initial temperature of the linear heat source surface and \(V\) is the volume of the linear heat source in contact with seawater per unit length. The initial condition expressed in terms of excess temperature is as follows:  
\[\theta(0)=T-T_{\infty}=\theta_{0} \tag{9}\]  
To solve Equation (9), the following could be obtained:  
\[\frac{d\theta}{\theta}=\frac{Ah}{\rho cV}dt \tag{10}\]  
Integrating with respect to \(t\) from \(t_{1}\) to \(t\), the following could be obtained:  
\[\frac{\partial T}{\partial t}=\frac{\Phi^{\prime}}{\rho c} \tag{2}\]0  
\[\frac{\partial T}{\partial t}=\frac{\Phi^{\prime}}{\rho c} \tag{2}\]1  
\[\frac{\partial T}{\partial t}=\frac{\Phi^{\prime}}{\rho c} \tag{2}\]2  
Based on Equations (9) and (11), the analytical solution for the temperature variation in a linear heat source at a measurement point in a water medium with respect to time is derived as follows:  
\[T(t)-T_{\infty}=\left\{ \begin{array}{l}\frac{Q}{Ah}(1-\frac{|Q-Ah(T_{0}-T_{ \infty})|}{Ah}e^{-\frac{Ah}{\rho cV}t}),0<t<t_{1}\\ (T_{0}-T_{\infty})e^{-\frac{hA}{\rho cV}(t-t_{1})},t\geq t_{1}\end{array} \right. \tag{12}\]