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# ScalingOfVerticalJet

Jets resulting from leaking pressure equipment are a key safety issue. Much work for developing simplified relations to predict the "size", the entrainment, the mixing behavior and other details of these jets has been done.

The simplest case is a non-buoyant sub-sonic jet, like air in air from a low-pressure reservoir. Complexity increases with increasing pressure leading to under-expanded jets and changing the involved components, like releasing hydrogen in air, which introduces buoyancy forces in a gravity field. Even more complex are jets including phase changes or reactions like in a ignited jet fire.

In the following section we will detail the problem of releasing hydrogen vertically from a reservoir with undercritical pressure ratio. This resembles the source of the garage problems described before.

### Non-Boussinesq similarity in subsonic axi-symmetric jets

#### Flow structure

According to Panchapakesan and Lumley (1993) the flow field of axi-symmetric (vertical) jets can be divided into there regions, based on the relative importance of buoyancy: the non-buoyant jet region (NBJ), the buoyant jet region (BJ) and the buoyant plume region (BP). Chen and Rodi (1980) (ChenCJ:1980) after reviewing the available experimental data suggested that the NBJ region exists for values of the non dimensional length while the buoyant plume region for values of , where the non-dimensional length is defined as the ratio of the axial distance from the jet origin (denoted below by subscript 0) divided by the Morton length scale L_{Mo}: $$z_1 = {z \over L_{Mo}}$$ with $$L_{Mo}=F^{1 \over 2} \omega^{1 \over 2} d$$ with the density ratio \omega , densimetric Froude number F and Richardson number Ri defined as follows $$\omega={\rho_a \over \rho_0}; F={{\rho_0 U^2_0} \over {(\rho_a-\rho_0 )gd} } ; Ri={1 \over F}$$

#### Axial concentration correlations

Non-Boussinesq empirical similarity relations are found in the literature and summarised below for the mass fraction f for the three jet regions:

• non-buoyant jet (inertia dominated) region,
• buoyant jet region and
• in the far distance from the jet origin the buoyant plume region
 non buoyant jets $$z/L_{Mo} \le 0.5$$ $$\omega q_{NBJ} = A_{NBJ}\left( {{z \over d}} \right)^{ - 1} \omega ^{{1 \over 2}}$$ buoyant jets $$0.5 < z/L_{Mo} \le 5$$ $$\omega q_{BJ} = A_{BJ}\left( {{z \over d}} \right)^{ - {5 \over 4}} F^{{1 \over 8}} \omega ^{{7 \over 16}}$$ plumes $$5 < z/L_{Mo}$$ $$\omega q_{BP} = A_{BP}\left( {{z \over d}} \right)^{ - {5 \over 3}} \left( \omega F \right)^{{1 \over 3}}$$

The constants as suggested in the different references are summarised in the following table

 Source $$A_{NBJ}$$ $$A_{BJ}$$ $$A_{BP}$$ Dai et al 10.73 Papanicolaou and List (1987) 9.45 Chen and Rodi (1980) 5.0 4.4 9.35 Shabbir and George (1992) 8.0 George et al. (1977) 7.75 Cleaver at el. (1994) 6.04 Paranjpe (2004) 4.4 Ogino et al. (1980) 4.8

Note: The constants A for buoyant plumes are related to the constants reported in Dai et al. (1994) by factor of (\pi/4)^(2/3) .

To find the molar fraction C the following relation can be used: $$C={{ \omega f } \over { 1 + f (\omega -1) }} = {{\rho_a-\rho} \over {\rho_a-\rho_0}}$$

Under the Boussinesq approximation (f<<1) , the relationship between molar concentration and mass fraction becomes: $$C_{Bouss} \approx \omega f$$

### Evaluation of correlations based on INERIS-6C

Table 2 below shows the NBJ, BJ and BP regions for the conditions of experiment INERIS-6C, see Lacome et al. (2007) and Venetsanos et al. (2007) . Table 3 shows that sensors 13 to 15 lie inside the BP region while sensor 16 close to the boundary between BJ and BP. The experimental concentrations in the plume region are observed to be higher compared to the Chen and Rodi (1980) correlations. Larger mean concentrations (and narrower plumes) were also reported in the measurements by Dai et al. (1994). For sensor 16 on the other hand the experimental concentration is lower compared to the buoyant jet correlation of Paranjpe (2004). Finally it can be observed that the difference between the calculated molar concentrations with and without the Boussinesq approach decreases with increasing distance from the source.

 Test $$\omega$$ $$F$$ $$Ri$$ Morton length (m) NBJ region (m) BP region (m) INERIS -6C 14.4 513.3 1.95 x 10-3 0.233 0.265 <= z <= 0.382 z>= 1.43
 Sensor $$z/L_{Mo}$$ Region $$C_{Bouss}$$ ( vol % ) C ( vol % ) Co ( % ) 16 4.79 BJ 20.11 (22.21 for BP) 16.94 (18.4 for BP) 16.5 15 6.94 BP 12 10.8 - 14 9.09 BP 7.67 7.16 8.04 13 10.39 BP 6.15 5.82 6.52