Estimations for Gas-Side Wall Temperature Within Rocket Engines

Question

I'm in the process of reading Modern Engineering for Design of Liquid Propellant Rocket Engines and am stuck on this one example given in chapter 4 on the calculations for the cooling system around a thrust chamber. Here is a direct quote from the textbook:

Since there is no solid deposit on the chamber walls, an average gas-side wall temperature of $1500^{\circ}\text{ R}$ is assumed, and a $T_{wg}/(T_c)_{\text{ns}}$ value of $1500/5740$ or $0.26$ is used to determine the $\sigma$ values from Fig. 4-28.

For a little bit of context, the problem asks to calculate the overall gas-side thermal conductance value $h_g$ for an engine that uses $\text{LOX}$ and $\text{LH}_2$ by using Bartz equation

$$h_g=\frac {0.026}{D_t^{0.2}}\left(\frac {\mu^{0.2}C_p}{\operatorname{Pr}^{0.6}}\right)_{\text{ns}}\left(\frac {(p_c)_{\text{ns}}g}{c^*}\right)^{0.8}\left(\frac {D_t}R\right)^{0.1}\left(\frac {A_t}A\right)^{0.9}\sigma$$

Where $\sigma$ is another complicated expression that makes use of the $T_{wg}/(T_c)_{\text{ns}}$ ratio mentioned in the excerpt above. And the variable $T_{wg}$ is the temperature of the side of the wall around the combustion chamber that interfaces with the hot combustion gases.

I'm having a hard time digesting the part where the authors just assumed that the side of the wall that interfaces with the hot combustion gases would be $1500^{\circ}\text{ R}$. What made them choose that value?

Does anybody know of any resources or equations that would allow one to predict the value of $T_{wg}$ or roughly approximate it instead of using sheer guess-work? The authors mention it as if they pulled the number out of thin air.

Answer 1

It's definitely not clear or stated explicitly, but I think that the 1500 deg R is a design goal, i.e. the cooling for the chamber will be designed such that the wall temp doesn't exceed approximately that value.

Some support for this idea is given further on in the book on page 111 (this is the 2nd edition, available at NTRS) where it says:

To avoid the "hot shortness" or low-ductility properties of Inconel X in the range 1200 deg - 1400 deg F, the mean temperature of the tube wall must be kept under 1000 deg F (or 1460 deg R).