Gas Injection Study
7.2 Analysis of Injector Flow Path
by Martellucci and Rie (1971), reported that air mass injection at low mass flow rates was destabilizing, but at higher mass flow rates the injection was stabilizing.
Schneider (2010) states of that result: “The downstream movement of transition for higher blowing rates is very surprising and must be viewed skeptically unless it can be supported by additional information. It seems possible that there is some error in the inferences from the surface impact pressures.” Notably, none of the reviewed studies involved carbon dioxide injection. The present study examines the impact of carbon dioxide gas injection on high enthalpy boundary layers both computationally and experimentally.
0 20 40 60 80 100 120 140 160 0
1 2 3 4
CO 2 mass flow [g/s] R2 = 0.9992
c1 = 3.01e−05 (95% CI: 1.6e−05 to 4.4e−05) c2 = 0.0197 (95% CI: 0.0174 to 0.0221) Intercept = 0.119 (95% CI: 0.03 to 0.21) Needle valve open
Needle valve 2/9 turns closed Needle valve 5/9 turns closed Needle valve 7/9 turns closed Quadraticfit
0 20 40 60 80 100 120 140 160
−10 0 10
EM1 Output [LPM]
Residual [%]
Figure 7.3: Sensirion EM1 (air) inline thermal mass flow meter calibrated against a rotameter for use in measuring CO2 flows. The quadratic fit coefficients for the calibration are indicated as c1 and c2.
injected through the porous media into the boundary layer. The reported mass flow rates were therefore too high by more than an order of magnitude. To mitigate this error in any future work, the EM1 mass flow meter was selected for its small size, which would allow it to be placed directly into the flow path inside the cone, near the injector plenum. Should further injection experiments be carried out, this is recommended.
However, as will be shown below, the pressure drop in the flow path between the run tank and the injector plenum is minimal for the relevant conditions, and the true mass flow rates through the injector section for the experiments in Section 7.4 may therefore be found based upon the ex post facto porous media calibration, using the measured run tank pressure and the calculated pressure at the boundary layer edge, Pe, to find the pressure differential upon which the mass flow rate depends.
Figure 7.4 is a schematic of the injector flow path. It will be shown that for relevant conditions, the run tank pressure P1 is essentially identical to the plenum pressure P1, which implies that the correct mass flow rates may be determined from
L = 4.05 m ID = 0.01016 m Vrt= 0.0189 m3
Injector
A = 0.003466 m2 t = 0.0016 m P’1
P2 P1 ID = 0.0100 m
Figure 7.4: Schematic (not to scale) diagram of the injector flow path showing pipe length, on/off ball valve with an opening time of ∼30 ms, sharp-edged entrance at the run tank, and sharp-edged entrance exit into the plenum behind the porous metal injector, which injects into the test section (a total of four junctions and three 90◦ turns between the ball valve and the plenum are omitted for clarity, but are included in the pipe analysis: see Table 7.1).
a steady-state calibration on the pressure drop from P1 to P2. First, assume a CO2 mass flow rate of 0.5 g/s from the maximum tested run tank pressure P1 = 172 kPa.
As it happens, this is about 25% higher than the maximum mass flow rate observed in the experiments, and therefore would be expected to exceed the maximum pressure drop for any of the conditions treated below. Flow velocity at the outlet of the tank, U1, can be calculated from the tank temperature, pressure, and the outlet area:
U1= mRT˙ 1 P1Apipe
For CO2 at the conditions described, U1 = 2.01 m/s. The Reynolds number based on pipe diameter D is defined as:
ReD = ρU1D μ
For the present pipe internal diameter D = 0.01016 m, ReD ≈ 5000, so the flow is expected to be turbulent. For turbulent pipe flow, from Anderson (1990) equation 3.97, the effects of shear stress can be expressed in terms of a friction coefficientf, and for a calorically perfect gas the variation of the mean cross-sectional flow properties between two locations in the pipe, x1 and x2, can be recast in terms of the Mach
numbers M1 and M2 at each location to find the friction equation:
x2
x1
4fdx
D =
− 1
γM2 − γ+ 1 2γ ln
M2 1 + γ−21M2
M2
M1
(7.1)
Taking x=L∗ as the distance where M = 1, 4 ¯f L∗
D =
1−M2
γM2 + γ+ 1 2γ ln
(γ+ 1)M2
2 + (γ−1)M2 (7.2)
Here, ¯f is an approximate average (or constant) value for the friction factor. For a Swagelok stainless steel tube, the equivalent roughness ε is about 0.0015 mm. There- fore, from ε/D = 1.48×10−4 and ReD, the friction factor f = 0.031 may be found from a Moody diagram, and is used as ¯f.
L = 4.05 m ID = 0.01016 m
L
eL
2*
L
1*
1 2 *
Figure 7.5: Schematic (not to scale) diagram of the physical pipe length L and total equivalent length of the fittings Le, with notional sonic pipe lengths from stations 1 and 2 indicated as L∗1 and L∗2. Finding the conditions at which sonic velocity is reached at station * for the two pipe lengths L∗1 and L∗2 permits the calculation of conditions at station 2.
The so-called minor losses due to pipe system components in the flow path may be represented, for approximately known conditions, as an equivalent lengthening of the tube by a quantity Le. Numerical values for the loss coefficients of various fittings must be determined empirically; the coefficients used here are from Moran et al. (2003) and tabulated along with their computed equivalent lengths in Table 7.1.
The integral on the left hand side of Equation (7.1) can be expressed with L∗1 and
Component Quantity Loss Coeff. KL Equiv. Length Le (each) [m]
90◦ bend 3 0.30 0.0798
Fully open ball valve 1 0.05 0.0133
Union, threaded 4 0.08 0.0213
Entrance, sharp-edged 1 0.50 0.1330
Plenum exit, sharp-edged 1 1.00 0.2660
Total Le= 0.737 m Table 7.1: Loss coefficients for injection system components from Moran et al. (2003), converted into equivalent pipe lengths.
L∗2 taken as the length of tube necessary for the flow to reach sonic velocity from the conditions at station 1 and station 2, respectively (see Figure 7.5).
4 ¯f(L+Le)
D = 4 ¯f L∗1
D − 4 ¯f L∗2
D (7.3)
For a pipe of the given diameter, length, components, and assumed constant friction coefficient, we have:
f¯(L+Le)
D = 17.7
Equations (7.2) and (7.3) can be solved iteratively to find M2, which is in turn used with M1 to calculate the change in pressure and temperature from adiabatic relations:
T2
T1 = 2 + (γ−1)M21 2 + (γ−1)M22
P2 P1 = M1
M2
2 + (γ−1)M21 2 + (γ−1)M22 Mach number M1 can then be found from:
M1= U1
√γRT
For a CO2 mass flow rate of 0.5 g/s and run tank pressure P1 = 172 kPa, M1 = 0.00745. The solution of Equations (7.2) and (7.3) is found by iteration to be
M2 = 0.007469. Therefore, P1/P1 = 0.9974 and T1/T1 = 0.99999996, so there is a maximum pressure drop of less than 0.3% in the injection system, and virtually no temperature change. Therefore, the conditions in the injector plenum are assumed to be the conditions in the run tank. The resulting corrected mass flow rates are recorded as part of Table 7.4 in Section 7.4.