Jtdylktuy

Consider the finite volume element in a cylindrical coordinate system shown in the figure below whose faces are bounded by the radii $r=bar{r}_{i-1}$ and $r=bar{r}_{i}$.

The issue is how to correctly determine the nodal centre of this element. The conventional approach appears to be to use the geometric mid-point radius as the nodal location such that

$$
r_i = frac{1}{2}(bar{r}_{i-1}+bar{r}_{i-1})
hspace{2cm}text{[Eq. 1]}
$$

I believe that a better definition can be determined as follows. Let us describe the variation of some physical entity $phi(r)$ over the interval $bar{r}_{i-1}le rle bar{r}_{i}$ about some point $r_i$ in a first order Taylor expansion as

$$
phi(r)=phi(r_i) + frac{partialphi}{partial r}Bigg|_{r=r_i}(r-r_i) + O(|r-r_i|^2)
hspace{2cm}text{[Eq. 2]}
$$

Now, let us propose that this nodal location should be such that the value of the property $phi(r)$ is representative of the entire element such that its volumetric integral can be expressed as

$$
Vphi(r_i)=int_{r=bar{r}_{i-1}}^{r=bar{r}_{i}}phi(r)rdr
hspace{2cm}text{[Eq. 3]}
$$

where

$$
V=int_{r=bar{r}_{i-1}}^{r=bar{r}_{i}}rdr=frac{1}{2}(bar{r}_{i}^2-bar{r}_{i-1}^2)
hspace{2cm}text{[Eq. 4]}
$$

Substituting [Eq. 2] in [Eq. 3] results in

$$
int_{r=bar{r}_{i-1}}^{r=bar{r}_{i}}(r-r_i)rdr =0
hspace{2cm}text{[Eq. 5]}
$$

from which the nodal centre $r_i$ is

$$
r_i=
frac{2}{3}
frac
{bar{r}_{i}^3-bar{r}_{i-1}^3}
{bar{r}_{i}^2-bar{r}_{i-1}^2}
hspace{2cm}text{[Eq. 6]}
$$

which I will note, is also different from the neutral radius (volume bisector) i.e.,

$$
r_i=
sqrtfrac
{bar{r}_{i}^2+bar{r}_{i-1}^2}{2}
hspace{2cm}text{[Eq. 7]}
$$

Questions:

1) I couldn’t find anything along the lines of Eq. (6) in the literature. I don’t even know what to formally name this radius. Could someone point me to a reference utilising the same or a similar approach.

2) Do you agree with the rationale that I used in deriving [Eq. 6]. If not, why?

The differences between the radius estimates are minuscule and largely negligible as seen in the table below Both nodal centre estimates were implemented in a nodal analysis. The differences in the nodal temperatures are less than 1%, but greater than the % difference between the nodal centre estimates themselves.