**3. Assessment of the prediction performance of the existing heat transfer correlations**

### **3.1 Assessment method**

As seen in **Table 3**, the heat transfer correlations for SCW have been proposed in different years and might be developed on different basis of experimental data. As a result, the applicability of each correlation might be different. As reported by Pioro et al. [8] and Lei et al. [2], there exist distinct discrepancies between the results predicted by different correlations. It is necessary to quantitatively evaluate the prediction performance of the existing correlations.

In this part, the prediction performance of the existing heat transfer correlations are quantitatively estimated by introducing four parameters, i.e., *σ*<sup>1</sup> (mean relative deviation, MRD), *σ*<sup>2</sup> (mean absolute deviation, MAD), *σ*<sup>3</sup> (standard deviation, SD), and *ρ*xy (correlation coefficient between the predicted values and experimental values), as defined by Eq. (1) through Eq. (5).

$$\sigma\_1 = \sum\_{i=1}^{n} e\_i / N \tag{1}$$

$$\sigma\_2 = \sum\_{i=1}^{n} |e\_i| / N \tag{2}$$

$$
\sigma\_3 = \sqrt{\sum\_{i=1}^{n} (e\_i - \sigma\_1)/(N-1)} \tag{3}
$$

where *e*<sup>i</sup> is

$$e\_i = \left[ \mathbf{Nu\_{cal}} \text{-N} \mathbf{u\_{exp}} \right] / \text{Nu\_{exp}} \tag{4}$$

$$\rho\_{xy} = \frac{\text{Cov}(X, Y)}{\sqrt{D(X)}\sqrt{D(Y)}}\tag{5}$$

where *D*(*X*) refers to the variance of the experimental data *X*, *D*(*Y*) refers to the variance of the calculated results Y, and *Cov*(*X*, *Y*) is the covariation of *X* and *Y* [9]. The closer the *ρ*xy is to 1.0, the better the correlation is [9].

*Heat Transfer Correlations for Supercritical Water in Vertically Upward Tubes DOI: http://dx.doi.org/10.5772/intechopen.89580*
