## 3 Data Processing and Uncertainty Analysis  
### Data Processing  
The efficiency, thermal resistance, effectiveness, and heat transfer coefficient (h) were calculated, and the results were compared to evaluate performance improvements. Air flow rates were utilized to determine the heat transfer within the ducts. The air mass flow rate (_in_) was calculated using Equation (1):  
\[\dot{m}=\rho\cdot V_{avg}\cdot A \tag{1}\]  
where \(\rho\) is density (kg\(\cdot\)m\({}^{-3}\)), \(V_{avg}\) is the average air velocity (m\(\cdot\)s\({}^{-1}\)), and \(A\) is the cross-sectional area of the duct (m\({}^{2}\)). \(\rho\) and dynamic viscosity (\(\mu\)) (kg\(\cdot\)m\({}^{-1}\cdot\)s\({}^{-1}\)) are calculated using Equations (2) and (3) [49]:  
\[\rho=351.99\ T^{-1}+344.84\ T^{-2} \tag{2}\]  
\[\mu=\frac{\left(1.459\times 10^{-6}T^{1.5}\right)}{\left(110.77+T\right)} \tag{3}\]  
The nanofluid utilized in this study exhibits pseudo-plastic behavior due to the inclusion of CMC in the base fluid, which alters its rheological characteristics [36; 50; 51]. Therefore, the apparent viscosity (\(\eta\)) of the CMC/water solution is determined using the power-law model expressed as [33]:  
\[\eta=m\dot{\gamma}^{\eta-1} \tag{4}\]  
Figure 7: General view of the experimental setup. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test Case No.** & **Heating Power (W)** & \
\begin{tabular}{c} **Hot Air Velocity** \\ **(m/s)** \\ \end{tabular} &
\begin{tabular}{c} **Cold Air Velocity** \\ **(m/s)** \\ \end{tabular} \\ \hline
1 & 1000 & 0.555 & 0.437 \\
2 & 1000 & 0.555 & 0.751 \\
3 & 1000 & 0.985 & 0.437 \\
4 & 1000 & 0.985 & 0.751 \\
5 & 2000 & 0.555 & 0.437 \\
6 & 2000 & 0.555 & 0.751 \\
7 & 2000 & 0.985 & 0.437 \\
8 & 2000 & 0.985 & 0.751 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Operating conditions for all experiments. where \(\dot{\gamma}\) is the shear rate, m is the consistency index, and n is the power-law index. The apparent viscosity was estimated using the power-law model parameters reported in the literature for similar CMC-based nanofluids with metal oxide nanoparticles [33]. In the power-law model, a value of \(n=1\) corresponds to a Newtonian fluid, while \(0<n<1\) is a shear-thinning (pseudo-plastic) fluid, and \(n>1\) represents a shear-thickening (dilatant) fluid. Experimental studies show that a CMC-water solution with a concentration of 0.0-0.4% can be approximated as a pseudo-plastic fluid for power-law index 0 < n < 1. In this study, CMC-water with a concentration of 0.4% was selected as the base fluid. This solution exhibits non-Newtonian pseudo-plastic behavior with a flow behavior index of n = 0.76 and m = 0.0785 [52]. This concentration level has been widely used in the literature to induce non-Newtonian behavior in nanofluid systems [36; 50]. To estimate the viscosity of nanofluids, the [PERSON] model provides a widely accepted extension of the classical [PERSON] equation. According to [PERSON]'s formulation, the effective viscosity of a nanofluid (\(\mu_{nf}\)) is related to the viscosity of the base fluid (\(\mu_{f}\)) and the particle volume fraction (\(\varnothing\)) as follows [53]:  
\[\mu_{nf}=\mu_{f}\cdot\frac{1}{(1-\varnothing)^{25}} \tag{5}\]  
The thermal conductivity of the nanofluid (\(k_{nf}\)) is calculated using Equation (6) [52]. \[k_{nf}=k_{f}\cdot\left[\frac{k_{p}\ +2\ k_{f}+2\varnothing\ \left(k_{p}\ -k_{f} \right)}{k_{p}+2k_{f}-\varnothing\ \left(k_{p}\ -k_{f}\right)}\right] \tag{6}\]  
where \(k_{f}\) and \(k_{p}\) are the thermal conductivities of the base fluid and nanoparticles, respectively. The CMC-water solution with 0.4% CMC concentration was considered to have a thermal conductivity of \(k_{f}=0.613\,\mathrm{W/mK}\)[52]. The amount of heat transferred from the hot air to the working fluid is determined using Equation (7):  
\[Q_{e}=\dot{m}_{h}\cdot c_{p}\cdot(T_{int,\ t}-T_{out,\ h}) \tag{7}\]  
where \(c_{p}\) is the specific heat of air (k)\(\cdot\)kg\({}^{-1}\).K\({}^{-1}\)).