Payload Bay Venting — theory

The pressure inside a vented payload bay during ascent. Based on C. P. Hoult, Payload Bay Venting in Flight (rev. 2), 2012.

Open the calculatorOriginal memo (PDF)

Overview

As a rocket climbs, the air outside thins, but the air in a payload bay can only leave through its vents. Until it does, the bay is pressurised against the outside and the shell is in tension; in extreme cases the shell fails, as part of the payload shroud of the first Space General Astrobee 1500 did early in its first-stage burn. The lag also corrupts any measurement of atmospheric pressure made inside the bay. Vents should make the bay pressure follow the outside closely.

Bay pressure

The analysis is simple but captures the main effects. The vents are cylindrical (throat and exit areas equal) and all exit at the same pressure, so their areas simply add; three or more equally spaced vents keep angle of attack from biasing the bay. The flow through them is adiabatic and isentropic, the bay air is stagnant and expands isentropically from its lift-off state p_0,T_0, and air is a perfect gas.

The vent mass flow is \dot m=-\rho_Ta_TM_TAC_N, with throat conditions from isentropic flow out of the bay, and the bay loses mass at V\,d\rho_C/dt with \rho_C=\rho_0(p_C/p_0)^{1/\gamma}. Matching the two:

\frac{dp_C}{dt}=-\gamma\sqrt{\gamma RT_0}\,\frac{A}{V}\,C_NM_T\left[1+\frac{\gamma-1}{2}M_T^2\right]^{-\frac{\gamma+1}{2(\gamma-1)}}p_0^{-\frac{\gamma-1}{2\gamma}}\,p_C^{\frac{3\gamma-1}{2\gamma}}

When the outside is at the higher pressure the sign changes and air flows in. The equation is integrated by fourth-order Runge–Kutta along the trajectory, with the time step set from the venting time constant \tau\approx V/(AC_{SON}a_0). Large vents make the equation stiff near equilibrium (the flow goes as \sqrt{\Delta p}); there the bay simply follows the outside and the calculator says so.

The trajectory — height and speed against time — comes from a trajectory simulation; the atmosphere is the standard troposphere for the launch site’s elevation, temperature and latitude.

Vent flow and orifice coefficient

The vent is choked (M_T=1) when the pressure ratio across it exceeds \left(\frac{\gamma+1}{2}\right)^{\gamma/(\gamma-1)}\approx1.893. Below that the throat Mach number is

M_T=\sqrt{\frac{2}{\gamma-1}\left[\left(\frac{p_C}{p_X}\right)^{\frac{\gamma-1}{\gamma}}-1\right]}

(upstream over downstream pressure). The orifice coefficient runs from its incompressible value — about 0.5 for a sharp-edged hole, up to 0.8 with a radiused inner edge — to 0.96–0.98 when choked. Busemann’s hodograph solution for a slit gives the interpolation

C_N=\frac{C_{SON}}{1+\sqrt{1-M_T^2}\left(\frac{C_{SON}}{C_{INC}}-1\right)}

Pressure outside the vents

The vents exit into the flow over the body, at p_X=p_a+\tfrac12\gamma p_aM_\infty^2C_{pX}. Below the critical Mach number of the vent location, the incompressible pressure coefficient (from the Nose Pressure Distribution calculator) is scaled by Kármán–Tsien; above it, a table from a transonic or supersonic method (for example second-order shock-expansion) is interpolated. Vents just behind the nose shoulder see suction, which raises the difference across the shell; the largest difference usually comes near burnout.

Will the door stay on? The allowable pressure difference

A door taped round its edge is pushed outwards by the pressure difference across it. The push on the door is \Delta p\,A_{door}. The tape holds with its peel strength \sigma (force per length of edge) along the edge length L. So the door peels off when (Hoult, “Pressure Strength of Taped-On Doors”, 2011; Espinoza et al., eq. 1)

A_{door}\,\Delta p\ge L\,\sigma\quad\Rightarrow\quad \Delta p_{allow}=\frac{\sigma L}{A_{door}}=\frac{2\sigma(l+w)}{lw}\ \text{(rectangular door)}

Espinoza, Vite, Rios and Hoult tested an 11.88 × 4.25 in door (50.47 in², 32.25 in of edge) taped with three layers of aluminium tape onto a cardboard tube. It failed at about 0.2 psi, a peel strength of about 0.3 lbf/in. They recommend 0.2 lbf/in for sizing vents, and that is the default. Smaller doors, and long narrow ones, hold more. Crenellated edges add edge length without adding area (Hoult).

Verdict (DECISIONS D29): the largest |\Delta p| during the climb against the allowable.

  • SAFE up to the allowable.
  • MARGINAL up to 1.25 times it. With the 0.2 lbf/in design peel, the measured failure point is 1.5 times the allowable.
  • UNSAFE beyond.

You can enter your own allowable instead. The fixes are solved: the smallest vent diameter, by bisection on the full simulation, and the fewest holes of your size.

Verification

With the original field tool’s trajectory (0.01 s steps), atmosphere and inputs, the tests reproduce its bay pressure, vent-exit pressure and integration constant at every checked step, and its largest difference of 0.1437 psi at 5.14 s. The memo’s plot (0.66 psi) belongs to a different case: it is labelled with a vent area of 0.0144 in², not the stored tool’s 0.0124 in². Further tests check the choking limit, the orifice interpolation and the supersonic pressure table.

Notes on the method

  • Choking. The field tool compares the up/downstream pressure ratio (always ≥ 1) with the sonic ratio 0.528, so its vents can never choke. The calculator chokes them at a ratio of 1.893; the difference matters only for very small vents or fast climbs.
  • Inflow. When air flows in, the same equation with the bay conditions is used (as in the memo); this is an approximation.

References

  1. C. P. Hoult, “Payload Bay Venting in Flight (rev. 2),” RS&T, 4 April 2012. View in library
  2. Ames Research Staff, “Equations, Tables and Charts for Compressible Flow,” NACA Report 1135, 1953.
  3. A. Busemann, “Hodographenmethode der Gasdynamik,” ZAMM 12, 1937.
  4. A. Espinoza, B. Vite et al., “Sounding Rocket Allowable Differential Pressure,” AIAA student paper. View in library
  5. C. P. Hoult, “Pressure Strength of Taped-On Doors,” RS&T, 2011. View in library