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Regenerative cooling#

The cooling study answers an early design question: with these channels and this much coolant, how hot does the wall run, and how much pressure does the coolant lose? It is one-dimensional and steady, and meant for sizing channels. It is not a thermal certification of a chamber.

closeout coolant channels, w × h hot wall, tw and k hot gas at the adiabatic wall temperature, heat flux q in series per station: 1/hg + tw/k + 1/hc Bartz · conduction · Gnielinski
One station of the wall. Heat crosses three resistances in series; the coolant warms as it moves from station to station.

Along the wall#

The wall is the chamber contour from the last engine run, divided into stations. At each station the local area ratio gives a Mach number from the one-dimensional relation with the chamber \(\gamma_f\), subsonic upstream of the throat and supersonic downstream. The gas drives heat into the wall at its recovery temperature,

\[ T_{aw}=T\left(1+r\,\frac{\gamma-1}{2}M^2\right),\qquad r=Pr^{1/3}, \]

which sits just below the chamber temperature everywhere.

Gas side: Bartz#

\[ h_g=\frac{0.026}{D_t^{0.2}}\left(\frac{\mu^{0.2}c_p}{Pr^{0.6}}\right) \left(\frac{p_c}{c^*}\right)^{0.8}\left(\frac{D_t}{R_c}\right)^{0.1} \left(\frac{A_t}{A}\right)^{0.9}\sigma \]

with the property correction

\[ \sigma=\left[\frac12\frac{T_{wg}}{T_c}\left(1+\frac{\gamma-1}{2}M^2\right)+\frac12\right]^{-0.68} \left(1+\frac{\gamma-1}{2}M^2\right)^{-0.12}. \]

\(R_c\) is the throat radius of curvature, \(1.5R_t\) for the EnSim contour. The transport properties are the estimates Bartz recommends when nothing better is known: Eucken's Prandtl number \(Pr=4\gamma/(9\gamma-5)\) and \(\mu=1.184\times10^{-7}M^{0.5}T_c^{0.6}\) Pa·s with \(M\) in g/mol. Both can be supplied directly through the library.

\(\sigma\) depends on the wall temperature, which depends on the heat flux, which depends on \(h_g\). Each station therefore iterates until the gas-side wall temperature changes by less than 0.01 K; it stops with an error after 50 passes rather than returning an unconverged number.

Coolant side: Gnielinski#

Coolant properties are taken at the inlet state. The hydraulic diameter comes from the channel width and height, and

\[ Nu=\frac{(f/8)(Re-1000)Pr}{1+12.7\sqrt{f/8}\,(Pr^{2/3}-1)},\qquad f=(0.79\ln Re-1.64)^{-2} \]

for \(Re\ge3000\). Below 2300 the flow is laminar and \(Nu=3.66\); in between the two are blended linearly. Laminar coolant is almost always a design error: the coefficient drops by an order of magnitude and the wall overheats. The desktop study warns when it happens, and its default channel count is as many channels as fit around the throat at the chosen width and land, which keeps the flow turbulent for engines of any size.

Pressure loss uses the smooth-channel Darcy factor \(f=[-1.8\log_{10}(6.9/Re)]^{-2}\) (Haaland) and \(64/Re\) in laminar flow.

Wall and coolant march#

The three resistances add:

\[ q=\frac{T_{aw}-T_{cool}}{1/h_g+t_w/k+1/h_c}. \]

The gas-side and coolant-side wall temperatures follow from \(q\). Moving to the next station, the coolant absorbs \(q\) over the wetted perimeter and the station length,

\[ \Delta T_{cool}=\frac{q\,\pi D\,\Delta x}{\dot m_c\,c_{p,c}}, \]

and loses \(f\,(\Delta x/D_h)\,\tfrac12\rho v^2\) of pressure. The coolant can enter at the exit and run toward the injector (the usual arrangement) or the other way.

Reading the result#

Melt margin is the difference between the material's melting point and the hottest gas-side wall temperature. Margin to critical temperature compares the coolant with its critical temperature. Both are temperature differences for orientation. They are not safety factors: the wall loses strength long before it melts, and a coolant near its critical point changes its properties far more than the constant-property model assumes.

Not modelled#

Boiling and supercritical property variation, film cooling, rough or ribbed channels, curvature effects on the coolant side, manifold maldistribution, coking, axial conduction in the wall, radiation, and thermal stress.

Sources#

  • D. R. Bartz, "A Simple Equation for Rapid Estimation of Rocket Nozzle Convective Heat Transfer Coefficients", Jet Propulsion 27(1), 1957.
  • V. Gnielinski, "New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow", International Chemical Engineering 16, 1976.
  • T. D. Smith, A Comparison of Techniques for Predicting Local Heat-Transfer Coefficients for Rocket Engines, NASA TM X-71817, 1975.
  • Huzel and Huang, Modern Engineering for Design of Liquid-Propellant Rocket Engines, AIAA, 1992, chapter 4.