## 2 Experimental Measurements
### Numerical Procedure  
Numerical modeling was carried out using the finite element method with ANSYS Fluent (2023 R2) software and a discrete solver. The nanofluid was regarded as Newtonian due to the low concentration of metallic nanoparticles. The flow was treated as incompressible, and a turbulent flow regime was identified at the applied flow rate.  
The Navier-Stokes equations are solved numerically in conjunction with the energy and continuity equations. In the flow field, TPMS is modeled as a solid structure, which adds complexity to the model because of the closed-end channels present. Consequently, it is necessary to direct the flow toward the lower pressure region. The flow velocity decreases after high flow rates within the structure. This irregular pattern allows longer liquid circulation time, enabling better heat removal. However, the complexity affects equation convergence. To overcome this challenge, the fluid equations are first solved by assuming that the flow is initially zero. Once convergence is achieved, the model is re-evaluated using the flow field results as initial conditions. The updated mathematical model incorporates the governing equations for fluid flow and heat transfer, resulting in a more efficient and precise convergence process. The convergence threshold is defined by the reduction of residuals associated with velocity, pressure, and temperature fields to values less than \(10^{-6}\), ensuring a high level of accuracy in the numerical solution. Although this process may extend the computational time, the enhanced mesh structure significantly improves the accuracy of the solution. The formulations utilized in this analysis are as follows [28; 29; 30]:  
\(x\)-direction  
\[\rho_{f}\bigg{[}u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} +w\frac{\partial u}{\partial z}\bigg{]}=-\frac{\partial p}{\partial x}+\mu_{f }\bigg{[}\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y ^{2}}+\frac{\partial^{2}u}{\partial z^{2}}\bigg{]} \tag{7}\]  
\(y\)-direction  
\[\rho_{f}\bigg{[}u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y }+w\frac{\partial v}{\partial z}\bigg{]}=-\frac{\partial p}{\partial y}+\mu_{ f}\bigg{[}\frac{\partial^{2}v}{\partial x^{2}}+\frac{\partial^{2}v}{\partial y ^{2}}+\frac{\partial^{2}v}{\partial z^{2}}\bigg{]} \tag{8}\]  
\(z\)-direction  
\[\rho_{f}\bigg{[}u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y }+w\frac{\partial w}{\partial z}\bigg{]}=-\frac{\partial p}{\partial z}+\mu_{ f}\bigg{[}\frac{\partial^{2}w}{\partial x^{2}}+\frac{\partial^{2}w}{ \partial y^{2}}+\frac{\partial^{2}w}{\partial z^{2}}\bigg{]} \tag{9}\]  
Energy Formulation  
\[\rho_{f}C\rho\bigg{[}u\frac{\partial T}{\partial x}+v\frac{\partial T}{ \partial y}+w\frac{\partial T}{\partial z}\bigg{]}=k_{f}\bigg{[}\frac{ \partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{ \partial^{2}T}{\partial z^{2}}\bigg{]} \tag{10}\]  
In the model, \(u\), \(v\), and \(w\) represent the velocity components in the \(x\), \(y\), and \(z\) directions, respectively. These variables characterize the fluid motion in three-dimensional space and are essential for describing the flow behavior within the TPMS structure. The effect of gravity has been neglected for the TPMS gyroid structure under forced convection conditions. This approach is considered appropriate for simplifying the model and reducing the computational time. Under forced convection conditions, the motion of the flow is primarily determined by an externally applied force; thus, the effect of gravity is thought to play a secondary role.