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This one, I hope, is pretty simple. I'm working on a school project that's looking at the feasibility of a methane fueled rocket motor. Right now I'm looking at regenerative cooling for the nozzle itself and found the following paper that seemed to give one of the better descriptions on the design and function of regeneratively cooled nozzles that I've found so far in my literature review:

https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19670004203.pdf

To be clear, this paper's on a nuclear thermal rocket, which is so cool, but I'm interested specifically in the heat transfer aspect of the propellant-cooled nozzle. I was going to try to code up a quick version of this study, validate it against the results, and use it as a foundation upon which to build a more specific model for my methane-fueled rocket motor.

Unfortunately, I'm having a bit of trouble understanding the following convective heat transfer coefficient relation published in the paper (page 11 in the pdf above). To be clear, this is the coefficient which allows calculation of how much heat energy is transferred from the hot-gas flow inside the nozzle to the cold-gas flow in the coolant tubes surrounding the nozzle:

$h_C = \frac{0.0208}{d^{0.2}}(\frac{\dot{m}_C \rho_{CF}}{A_C \rho_{CS} \mu_{CF}})^{0.8} K_{CF} Pr_{CF}^{0.4} c_2 c_3$

Where $d$ is the hydraulic diameter of the coolant tubes running axially along the rocket nozzle, $\dot{m}_C$ is the mass flow rate of the coolant, $\rho$ denotes densities, $\mu$ is the dynamic (absolute) viscosity of the coolant/propellant, $A_C$ is the total cross sectional area of all the coolant tubes in the rocket nozzle, $K$ refers to the thermal conductivity of the propellant/coolant, $Pr$ is the Prandtl number, and $c_2$ / $c_3$ are correction coefficients that aren't super relevant to my upcoming question. The subscripts $C$ refer to bulk coolant/propellant properties, $CF$ to properties of the coolant/propellant "film" (averaged boundary layer conditions), and $CS$ refers to what the paper calls "static" properties, which I assume to mean stagnation properties.

The question I've got is this: how would one go about determining "film" properties (those labelled with the $CF$ subscript)?

The following source, along with my general understanding of fluid mechanics indicates that there is not much variance of pressure over the boundary layer for turbulent gaseous flow (which I believe would be a reasonable first order approximation of the conditions cited by the paper within the original nozzle-design report which states that the maximum expected Mach number within the coolant tubes is 0.5).

https://aip-scitation-org.mines.idm.oclc.org/doi/abs/10.1063/1.1762413

Any suggestions on how I could evaluate $\rho_{CF}$, $K_{CF}$, $\mu_{CF}$, and $Pr_{CF}$? I'm currently assuming that there's not much difference between bulk and film properties, but I doubt the original author of the NASA study would have included those film terms if they were not significant. For the record, I'm most curious about the $\rho_{CF}$ and possibly $\mu_{CF}$. The Prandtl number and the thermal conductivity of the fluid I'm reasonably comfortable assuming as constant to start out with.

Thanks! -Dave

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First of all, let me point out that it looks like the thermal conductivity of the gas is 0.0208, and that the quantity $\frac{hd}{k}$ represents the dimensionless Nussult number for heat transfer. The expression in parenthesis raised to the 0.8 power combined with the diameter d also now raised to the 0.8 power represents the dimensionless Reynolds number raised to the 0.8 power. The equation then takes the standard form of correlation developed for many heat transfer applications: $$Nu=KRe^{0.8}Pr^{0.4}$$

Now for what conditions to evaluate the physical properties. Bird, Stewart, and Lightfoot make specific recommendations on the evaluation of the properties: "Usually this is done by means of an empiricism-namely, evaluating the physical properties at some appropriate average temperature. Throughout this chapter, unless explicitly stated otherwise, it is understood that all physical properties are to be calculated at the film temperature $T_f$ defined as follows: .... b. For submerged objects with uniform surface temperature $T_0$ in a stream of liquid (fluid) approaching with uniform temperature $T_{\infty}$,$$T_f=\frac{1}{2}(T_0+T_{\infty})$$"

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  • $\begingroup$ Hi Chet, thanks for the feedback, very helpful, particularly your comment regarding the form of heat transfer correlations using $Nu$, $K$, $Re$, and $Pr$. Also I read you loud and clear on the assumed relation for film temp. Presumably it'd work just fine for metrics like film density and film viscosity as well. I believe that $\rho_{\inf}$ and $\mu_{\inf}$ are likely to be easy to evaluate. Any thoughts on how to evaluate density and dynamic viscosity at the wall? $\endgroup$
    – D. Hodge
    Jun 10, 2020 at 18:57
  • $\begingroup$ At the wall, you use the wall temperature. For a gas at relatively low pressure, viscosity is a function only of temperature. For the density at the wall you use the pressure and temperature in the ideal gas law. At higher pressures, you use the law of corresponding states to get the density and the viscosity. For viscosity, a corresponding states correlation is given in Bird et al. $\endgroup$ Jun 10, 2020 at 20:12

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