## 2 Thermal Conductivity Methods
### Transient Hot-Wire  
Transient hot-wire (THW) constitutes an intrusive technique that can be used for measuring the thermal conductivity of non-electrically conductive materials, mostly liquids [4]. In case of measuring electrically conductive samples, the wire must be coated with a protective insulating layer [12]. THW is simple in construction and plays an important role in accurately and quickly measuring thermal conductivity of fluids [13]. As mentioned, it is considered as a primary method and was identified as an appropriate technique for obtaining reference data. THW is an absolute method with a working equation and a complete set of corrections reflecting the departure from the ideal model [3]. Toluene, benzene, and water were proposed by the IUPAC as primary standard liquids for the measurement of thermal conductivity, and they were measured with this method with an accuracy of 1% [3]. Many researchers have used this technique to estimate the thermal conductivity of a wide variety of materials such as nanofluids [14; 15; 16], nanocomposites [17], gases [18], and aqueous solutions [19]. The theory of the THW is well-known and can be found elsewhere [20; 21]. The ideal mathematical model is an infinite vertical thin line heat source of zero heat capacity and infinite thermal conductivity immersed in an infinite, stationary, incompressible, and isotropic medium with properties independent of the temperature, and in thermodynamic equilibrium with the wire. When a stepwise constant heat flux per unit length is applied to the wire, it generates pure conductive heat flux that will be transferred from the wire to the immersed material. To obtain the governing equation of the THW method, Equation (1) must be particularized by applying the following initial and boundary conditions:  
\[ \begin{array}{ll}\Delta T(r,t)=0,&t\leq 0,\ \forall r;\\ \lim_{r\to 0}\left(r\frac{\partial T}{\partial r}\right)=-\frac{\dot{q}}{2\pi \lambda},&t\geq 0,\ r=0;\\ \Delta T(r,t)=0,&t\geq 0,\ r\rightarrow\infty\end{array}  \tag{2}\]  
where \(r\), \(t\), and \(\dot{q}\) stand for the radial distance from the centre of the wire, the time, and the constant heat flux per unit length, respectively. Equation (1), particularized with the above initial and boundary conditions, constitutes a standard differential equation and was analytically solved by [PERSON] and [PERSON] [22] as  
\[\Delta T(r,t)=\frac{\dot{q}}{4\pi\lambda}\int_{\frac{\partial T}{\partial t}}^ {\infty}\frac{e^{-u}}{u}du \tag{3}\]in which \(\alpha\) is the thermal diffusivity of the fluid. The last factor of Equation (3) is the so-called exponential integral, which can be expressed as a series expansion [23]:  
\[E_{1}(z)=\int_{z}^{\infty}\frac{\epsilon^{-u}}{u}du=-\gamma-\ln(z)-\sum\limits_{ k=1}^{n}\frac{(-1)}{\mathbb{H}\cdot k}z^{k},\quad z\in\mathbb{C},\ |\arg(z)|<\pi \tag{4}\]  
where \(\gamma\) is the Euler constant. Small values of \(r^{2}/4\alpha t\) fulfil the requirements of Equation (4). At the wire surface, \(r=r_{0}\), the temperature is uniform and equal to the temperature of the conducting medium, and, therefore,  
\[\Delta T(r_{0},t)=\frac{\dot{q}}{4\pi\lambda}\left[\ln\left(\frac{4\alpha t}{ r_{0}2\epsilon^{\gamma}}\right)-\sum\limits_{k=1}^{\infty}\frac{1}{\mathbb{k}k} \left(-\frac{{r_{0}}^{2}}{4\alpha t}\right)^{k}\right] \tag{5}\]  
If the radius of the wire is small enough such that \({r_{0}}^{2}<<4\alpha t\), the sum can be neglected, obtaining the widely spread THW equation [12]:  
\[\Delta T(r_{0},t)=\frac{\dot{q}}{4\pi\lambda}\ln\left(\frac{4\alpha t}{r_{0} ^{2}\epsilon^{\gamma}}\right) \tag{6}\]  
This governing equation constitutes a linear relationship between the temperature rise of the fluid and the natural logarithm of the time. The slope allows one to obtain the thermal conductivity of the immersed medium.