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Report on scaling methodology

Scaling of Gas Distribution, Mixing Processes

Introduction

Scales and Scaling

The purpose of scaling usually is to inter- or extrapolate quantities a along a scale or a reference quantity A = a_r . Scales may be basic quantities like meter m , kilogram kg , or second s , then they are frequently referred to as units. However, in the following scales are rather natural or even composed quantities of several constants which are characteristic for the observed phenomenon. In any case scales allow to introduce dimensionless quantities denominated wit a star in the following: $$ a = A a^\ast $$

The actual law how-to scale might be found by observation and intuition. These observations provide an empirical "law" which relates an observed, dependent quantity to the varied, independent quantity. Thus predictions for certain relevant quantities for non tested domains might be provided. If a phenomenon scales, then the dependent quantity correlates to the independent quantity and the pairs of values plotted in a double log diagram should will lie on a straight line.

The chances for success, i.e. getting a more universal relation, are usually furthered if the dependent and independent quantities are transformed with a sub-set of scales to dimensionless quantities.

Scale factors

The scale factor \hat{a} relates the two corresponding scales in the model A^' = {a_r}^' and in the prototype A = a_r : $$ \hat{a} = A^'/A $$ If there is a homologous scaling corresponding values of this quantity are scaled from the model to the prototype with this scale factor. Of course the scaling factor for the different quantities like length (classical size scale), time, temperature, electrical current, force or mass might differ. These scale factors are called primary scale factors as they described the scaling of the basic units, whereas all derived quantities are scaled by secondary scale factors. For instance, the secondary scale factor for the velocities \hat{v} can be easily derived by dividing the scale factor for the lengths \hat{l} by the scale factor for the time \hat{t} : $$ \hat{v} =\hat{l} / \hat{t} $$

In many engineering applications observations are made with a size-scale model, which is a geometrical model of the prototype. For some applications not all of the dimensions of the model have to be scaled equally, instead the known laws require special rules to preserve a consistent scaling of forces, for instance. One example is the scale modelling of airfoils, sometimes labelled as parametric scaling.

Similitude, Similarity, Model, Prototype

Similitude and similarity are synonyms in this context and define the possibility to transfer observations from a model to a prototype.

The methods, described in the following, allow to get a conceptional picture of a complicated phenomena, which might be not directly accessible in a prototype. Reasons which make a direct study impossible are the real size of the prototype, too big or too small to handle the prototype easily, the general possibility to conduct experiments in our usual environment (e.g. gravity) and simply the implied costs of the prototype which might be lost or damaged in an experiment. Another motivation might be to exclude unwanted qualities of a scenario to be studied, e.g. to use a non-combustible gas for studying the dispersion of a burning gas like hydrogen.

Similarity methods include the Parameter Approach, the Law and the Equation Approach. The Parameter Approach requires experience, the Law Approach the identification of basic laws involved, and the Equation Approach needs a valid theoretical model for the whole coupled phenomena, usually defined by a set of differential equations.

Dimensional Analysis

The dimensional analysis provides a methodology to identify suitable combinations of parameters influencing the problem on the basis of the fundamental dimensions of a problem and some pure mathematical recipes. With the help of dimensional analysis the minimum number of independent dimensionless parameters, including the observed and varied quantity might be identified in a unique way. Based on this set of dimensionless parameters other observations might be easily translated in this usually unique systems of variables. However, this methodology requires a sound understanding of the relevant physical phenomena and a proper pre-selection of the parameters entering the mathematical scheme.

All methods allow to derive the dimensionless parameters and in some cases even the relative importance of single effects. For similarity all dimensionless parameters have to be the same in the model and in the prototype.

Relaxation

At least for complex phenomena a perfect scaling, similarity respectively, with identical dimensionless parameters in the model and prototype cannot be fulfilled or only fulfilled if model and prototype are identical, what would make the whole effort obsolete. Then one has to relax the model requirements. One way is to identify less important influences or partial phenomena and to omit their modelling. Alternatively a temporal or spatial separation might help, when certain phenomena prevail in a certain time span or area only.

The following chapters first introduce the established methodology and then show applications to a few hydrogen safety relevant dispersion problems, the SBEPV3 Garage problem and a horizontal buoyant jet.

Methodologies

In the following some methodologies related to similarity and dimensional analysis will be introduced. We start with methodologies, like empirical identification of scales or identification of relevant parameters, the Parameter Approach. Basic models describing the phenomena under consideration are not required for this method. Instead a mathematically formal procedure can be applied to reduce the number of quantities to a minimum number of dimensionless quantities.

The methods described in the following sections, the Law and Equation Approach, need some basic physical models or even a complete mathematical frameworks for the relevant phenomena. So, provided there is a good model - usually a set of differential equations - we can derive dimensionless parameters and even identify the relative importance of a single effect for a scenario. These methods do not require a broad understanding of the physics but simply assume that the problem is modelled correctly with these equations and that the analyst is able to identify the right representative quantities or scales.

All methods yield dimensionless parameters or \Pi-numbers which have to be the same in the model and in the prototype to guarantee similarity: $$ {\Pi_i}^' = \Pi_i $$ From these conditions scale factors for basic quantities, like time, length, forces etc, can be derived, which can instruct the analyst in designing a model, model experiment respectively.

Parameter Approach

Sufficient experience and/or good intuition allow the analyst identifying the relevant quantities, named parameters here, of the scenario. Some of the parameters will be dimensionless from the very beginning, but most of the parameters will involve dimensions (called units in the introduction). Equipped wit an even better experience the analyst might be able to combine several parameters with dimensions to a smaller set of dimensionless parameters, which allow to transfer even other observations in this parameters domain more easily.

A more systematic approach for the latter process is provided by the Dimensional Analysis.

Dimensional Analysis

The actual physical relationship between observed and relevant independent dimensional quantities parameters characterising a problem should not depend on the choice of the units of measurement. This means that this relation can always be formulated as a relation between dimensionless quantities. The reformulation with dimensionless quantities implies a reduction of the numbers of parameters as in the original set of dimensional parameters the units of measurements were additional hidden (parasitic) quantities. Some dimensional parameters take the role of the units of measurement.

The universal and simplified formulation of the basic relations is exactly the attractive capability of the dimensional analysis. The mathematical implementation of this concept is the Buckingham Pi-theorem.

Pi-Theorem

All mechanical problems can be described with quantities having units composed of the units of the basic set of quantities length L, mass M, and time T. In some cases the replacement of the mass M by the force F might be advantageous. However, this will not change the approach in principle.

The engineering SI units for these 3 basic units are $$ [L]=m $$ $$ [M]=kg $$ $$ [T]=s $$ Together with the unit for the electrical quantity current {I=A$}, this forms the so-called mk(g)sA-system of basic units. Alternative basic units could be cm, g, and s like used for the physical c(m)gs-system. In the following we will use exclusively the mksA-system, also recommended in the international standards (SI).

All units of the relevant quantities including the observed quantity may be expressed by products of these basic units. Example for these dimensional expressions are: $$ [v]=[L]^1[M]^0[T]^{-1}=m^1kg^0s^{-1} $$ $$ [F]=[L]^1[M]^1[T]^{-2}=m^1kg^1s^{-2} (= N) $$ $$ [p]=[L]^1[M]^{-1}[T]^{-2}=m^{-1}kg^1s^{-2}(= Pa) $$

Step 1

Before the actual dimensional analysis based on the Buckingham Pi-theorem, a sound selection of the relevant quantities has to be done. We separate the observed key parameter and remove all dimensionless parameters from this set for the further consideration. This will yield the set of dimensional relevant parameters A_1, A_2,...., A_n.

Exactly this initial step requires a good physical background knowledge with an insight in the relevant physical phenomena. Thereby this methodology is more knowledge or experience based than the other similarity methods described later.

Now, we assume that there is a functional relationship for the observed dimensional quantity A: $$A=f(A_1, A_2,...., A_n)$$.

Step 2

Set up a matrix consisting of 3 rows for the basic units and n+1 columns for all dimensional quantities in the above relation.

 $$A$$$$A_1$$$$A_2$$...$$A_n$$
m$$a_1^0$$$$a_1^1$$$$a_1^2$$...$$a_1^n$$
kg$$a_2^0$$$$a_2^1$$$$a_2^2$$...$$a_2^n$$
s$$a_3^0$$$$a_3^1$$$$a_3^2$$...$$a_3^n$$

Where the matrix a_m^n contains the exponents in the dimensional expressions introduced above.

Step 3

Check the rank of the matrix a_m^n. In most cases it should be 3. In special cases, like static or 0-dimensional problems, there might be a reduced rank. Practically speaking this means that one (or more) line(s) of the matrix above drop out. The actual rank determines the new dimension m.

Step 4

Determine a suitable reference system consisting of m basic dimensional quantities, columns of the matrix a_m^n respectively. Exchange columns so that the reference system consisting of the m columns reside on the left hand side of the matrix. Use linear combinations of the rows or a Gaussian substitution to make the left hand side mxm sub-matrix a unity matrix or a upper triangular sub-matrix at least.

Step 5

The homogeneous system $$ a_m^n b_n = 0$$ provides definitions for (n+1)-m dimensionless quantities \Pi_i. To this end for each dimensionless quantity the remaining (n+1)-m elements of the solution vector have to be fixed. The following transposed b vectors reduce the degree of freedoms such that the linear system may be solved and from each solution a dimensionless parameters is derived: $$ (b_0,b_1,b_2;1,0,0,...,0); $$ $$ (b_0,b_1,b_2;0,1,0,...,0); $$ $$ (b_0,b_1,b_2;0,0,1,...,0); $$ ... $$ (b_0,b_1,b_2;0,0,0,...,1); $$

Step 6

Solve the system for each of the reduced unknown b vector above. The result for each system is a dimensionless quantity $$ \Pi_i = a_i * a_0^{b_{0,i}}* a_1^{b_{1,i}}* a_2^{b_{2,i}}; i = m..n+1 $$ In some cases these dimensionless parameters might look unknown. However, with some experience one can fix the right hand side vectors in Step 5, such that more conventional dimensionless parameters are directly delivered. If not this is managed Step 7

Step 7

From suitable products of the derived dimensionless parameters including those from the initial list. known dimensionless \Pi-numbers may be produces.

Law Approach

For the Law Approach one has to identify the relevant single effects, influences respectively, and their basic "constitutional" laws. The single effects are in most cases contributions to the force acting on the system or energies entering the overall energy balance. However, in this approach these effects are not balanced like in the Equation Approach, where the actual balancing equations and implicitly the interrelations of all effects are considered. Therefore the Law Approach might be seen as a methodology in between the Parameter Approach and the Equation Approach.

Step 1

Decide what are the important effects. In a gas mixing process this will be inertia, friction, buoyancy and diffusion.

Step 2

Give for each term identified in Step 1 the law defining the related forces or energies. For inertia this would be the Newton's law which allows to determine the inertial forces as the product of mass m and acceleration a : $$ F_{inertia} = m a $$

Step 3

Now we replace the terms in the laws provided in step 2 by suitable representative quantities. By this replacement we formally also replace "equality" by a "correspondence". So for the example above this will give and "estimate" for the representative inertial force $$ F_{r,inertia} \simeq m_r l_r / t_r^2 = M L / T^2 $$ where capital letter denominate the representative quantities (here representative mass M , length L and time T . Or - as one should not have force and mass terms simultaneously - one could introduce some material properties here: $$ F_{r,inertia} \simeq \rho L^4 / T^2 = \rho L^2 V^2 $$ This could be also the preferred form for any problem in the field of continuum flow.

Step 4

Now we relate the term on the right hand side, representing the intertial forces in this case, to a fictive external force F . This yields a dimensionless \Pi -number. $$ \Pi_{inertia} = {{ \rho L^2 V^2 } \over F } $$

These dimensionless \Pi -numbers resemble these numbers one could have found alternatively via the parameter approach. Like in Step 7 of the Dimensional Analysis combinations of these \Pi -numbers - in most cases ratios of them - will eliminate the artificially introduced fictive external force and will yield the more conventional \Pi -numbers.

For example, in step 2 we might have additionally introduced for the frictional forces in a fluid $$ F_{friction} = \mu {{ \partial v } \over { \partial y }} A $$ According to the described procedure, this will give $$ \Pi_{friction} = {{ \mu V L } \over F } $$ The ratio of the two \Pi -numbers is the well known Reynolds-number Re $$ {{\Pi_{inertia}} \over {\Pi_{friction}}} = {{ L V } \over { \mu / \rho } } = Re $$

The law approach provides a reasonable compromise between the difficult selection of parameters, formal mathematical dimension analytical approach and the too general Equation approach. Still one has to chose the right laws relevant for this phenomena.

The application in the second part of this report will concentrate on this methodology.

Equation Approach

The Equation Approach presumes that there is a valid mathemetical model interconnecting all relevant phenomena under consideration. For non-reactive compressible multi-component flows this would be the set of Navier-Stokes-Equations including the mass, momentum, energy and species conservation.

$$ { { \partial \rho } \over { \partial t } } + \vec{ \nabla } \cdot ( \rho \vec{ v } ) = 0 $$ $$ { { \partial \rho \vec{ v } } \over { \partial t } } + \vec{ \nabla } \cdot ( \rho \vec{ v } \otimes \vec{ v } + p I ) = \rho \vec{ g } + \vec{ \nabla } \cdot \tau $$ $$ { { \partial \rho e_t } \over { \partial t } } + \vec{ \nabla } \cdot ( \rho \vec{ v } h_t ) = \vec{ \nabla } \cdot ( - \vec{ q } ) + \vec{ \nabla } \cdot ( \tau \vec{ v } ) + \rho \vec{ g } \cdot \vec{ v } $$ $$ { { \partial \rho Y_i } \over { \partial t } } + \vec{ \nabla } \cdot ( \rho ( \vec{ v } + {\vec{ v }}_{d,i} ) Y_i ) = 0 $$

This is a set of partial differential equations not easily solved for real problems.

Even in studying the character of these equations it helps to reformulate these equations in a dimensionless form. Additional motivation for doing this is the knowledge that the set of used units shall not affect the validity of these basic conservation laws and that, secondly, nature knows only little about human created units like "meter", "second" and "kilogram".

Instead we choose natural scales for the length, time, etc, provided by the problem itself.

$$ l = L l^*; t = T t^*; \rho = \rho_r \rho^* p = P p^* $$

As a derived quantity the velocities should be scaled by: $$ v = V v^* = {L \over T} v^* $$

The scales or representative quantities are indicated by capital letters or by the subscript r. Inserting this in the Navier-Stokes equations gives

$$ { { \partial \rho^* } \over { \partial t^* } } + \vec{ \nabla^* } \cdot ( \rho^* \vec{ v^* } ) = 0 $$ $$ { { \partial \rho^* \vec{ v^* } } \over { \partial t^* } } + \vec{ \nabla^* } \cdot ( \rho^* \vec{ v^* } \otimes \vec{ v^* } + {P \over \rho_r V^2 } p^* I ) = {L g \over V^2 } \rho^* {\vec{ e }}_g + {\mu_r \over \rho_r L V } \Delta^* v^* $$ $$ {c_{v,r} \over c_{p,r} } { { \partial \rho^* e_t^* } \over { \partial t^* } } + \vec{ \nabla^* } \cdot ( \rho^* \vec{ v^* } h_t^* ) = - { \lambda_r \over \rho_r c_{p,r} L V } \Delta T + { \mu_r V^2 \over V L \rho_r c_{p,r} T_r } \Delta + \rho \vec{ g } \cdot \vec{ v } $$ $$ { { \partial \rho Y_i } \over { \partial t } } + \vec{ \nabla } \cdot ( \rho ( \vec{ v } + {\vec{ v }}_{d,i} ) Y_i ) = 0 $$

Application to the "Garage" Scenario

The above methodology will be applied to the garage scenario. In the HySafe internal project InsHyde releases in closed spaces have been studied. These studies were supported by experimental and numerical simulations of releases in garage-like geometries. In cooperation typical hydrogen sources were identified. So typical releases are in the order of 1 g H2/s. Assuming an orifice of 20 mm in diameter the gas velocity u at the exit would reach 38 m/s in average.

Attach:setupGarage.jpg Δ | Figure 1. Setup of the garage scenario to be analysed

The initial mixing tests in the INERIS gallery facility applied hydrogen directly. The INERIS facility is described in detail in the documents of the HySafe SBEP-V3 description http://www.hysafe.net/SBEPV3.

The actual experiments are characterised by two time domains:

  1. the relatively short injection period from t_0 = 0s to t_1 (240 s typically)
  2. the subsequent diffusion period from t_1 to the end of the measurement which could be several hours

In the following chapter we will use the law approach to derive the relevant dimensionless quantities involved. The equation approach as explained in xyz? will be used to check the completeness (usually this step is not required). As the analysis of these dimensionless parameters will prove that perfect similarity may not be provided we will relax the problem by dividing the whole process in the two domains in the following sub-chapters. The initial vertical jet injection and the subsequent diffusion phase - as introduced above will be considered separately. For the first we will rely on models available from literature. For the latter we will apply a dimensional analysis.

The release process when relatively light gases are injected at low Mach speed in the atmosphere is governed by the forces of inertia, buoyancy and viscosity. Hence the following laws apply:

Law for interial forces $$ F_{inertia} \simeq \rho L^2 V^2 \Rightarrow \Pi_i = { F \over {\rho L^2 V^2}} $$ Law for gravitational forces $$ F_{gravitation} \simeq \Delta \rho L^3 g \Rightarrow \Pi_g = { F \over {\Delta \rho g L^3 }} $$ Law for viscous forces $$ \tau \simeq \mu { V \over L } \Rightarrow F_{viscous} \simeq \mu L V \Rightarrow \Pi_v = { F \over {\mu L V }} $$

As we don't have any external characteristic force F, only ratios of the above ratios will be of interest. In these ratios the fictional force F drops out and we get two well known \Pi-Numbers from the above three laws: The Reynolds number Re $$ Re = { \Pi_v \over \Pi_i } = { L V \over \nu } $$ and the Richardson number Ri $$ Ri = { \Pi_i \over \Pi_g } = { {\Delta \rho g L } \over \rho V^2 } $$ It is easy to see that the Richardson number is closely related to the Froude number Fr introduced in the Equation Approach description: $$ Ri = { {\Delta \rho} \over \rho } { 1 \over Fr^2 } = { 1 \over Fr^{\star} } $$ Where Fr^{\star} defines the densimetric Froude number

Release Phase

The release process when relatively light gases are injected at low Mach speed in the atmosphere is governed by the forces of inertia, buoyancy and viscosity. Hence the following laws apply:

Law for interial forces $$ F_{r,inertia} \simeq \rho L^2 V^2 \Rightarrow \Pi_i = { {\rho L^2 V^2} \over F } $$ Law for gravitational forces $$ F_{r,gravitation} \simeq \Delta \rho L^3 g \Rightarrow \Pi_g = { {\Delta \rho g L^3 } \over F} $$ Law for viscous forces $$ \tau \simeq \mu { V \over L } \Rightarrow F_{r,viscous} \simeq \mu L V \Rightarrow \Pi_v = { {\mu L V } \over F } $$

As we don't have any external characteristic force F, only ratios of the above ratios will be of interest. In these ratios the fictional force F drops out and we get two well known \Pi-Numbers from the above three laws: The Reynolds number Re $$ Re = { \Pi_i \over \Pi_v } = { {L V } \over {\mu / \rho} } = { {L V } \over \nu } $$ and the Richardson number Ri $$ Ri = { \Pi_i \over \Pi_g } = { {\Delta \rho g L } \over \rho V^2 } $$ It is easy to see that the Richardson number is closely related to the Froude number Fr introduced in the Equation Approach description: $$ Ri = { {\Delta \rho} \over \rho } { 1 \over Fr^2 } = { 1 \over Fr^{\star} } $$ Where Fr^{\star} defines the densimetric Froude number.

In the model of the described experiments the flammable hydrogen is replaced by the inert gas helium. Helium is usually selected as its properties are close to hydrogen . However, for similarity the relevant dimensionless parameters, here the Re and Ri have to be identical, what can be expressed in the following scale factor requirements:

From Re^' = Re it follows $$ { { \hat{l} \hat{v} } \over \hat{\nu} } = 1 $$ and from Ri^' = Ri it follows $$ { { \hat{\Delta \rho} \hat{l} } \over {\hat{\rho} \hat{v}^2} } = 1 $$ where it has been assumed that also the experiments will be made on earth ( \hat{g} = 1 ) and use the same external gas, namely air ( \hat{\rho} = 1 ).

Because of the selected model gas the remaining two material scale factors are also fixed: $$ \hat{\nu} = {{\nu_{He}} \over {\nu_{H2}}} \approx 1.095 $$ $$ \hat{\Delta \rho} = {{\rho_{air}-\rho_{He}} \over {\rho_{air}-\rho_{H2}}} \approx 0.927 $$

The above requirements may be reformulated to yield the remaining velocity and size scale: $$ \hat{v} = \sqrt[3]{ {\hat{\nu} \hat{\Delta \rho}} \over {\hat{\rho}}} \approx 0.8 $$ $$ \hat{l} = \sqrt[3]{ {\hat{\rho} \hat{\nu}^2} \over \hat{\Delta \rho} } \approx 1.37 $$

So for similarity the velocities in the helium jet should be reduced by 20% and the geometry of the helium experiments should be enlarged by 37%, roughly a third.

With the same size in the model experiment and the prototype only velocities may be scaled only after further relaxation. Assuming that the buoyancy effects are stronger than the frictional effects the Ri -number provides the following orientation $$ \hat{v} = \sqrt{ \hat{\Delta \rho} \over \hat{\rho}} \approx 0.68 $$ Under these conditions the characteristic helium release velocities should be reduced by a third. Violating this will make a comparison of the release phases and all subsequent processes difficult.

Scaling of vertical jet

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)5.04.49.35
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 -6C14.4513.31.95 x 10-30.2330.265 <= z <= 0.382z>= 1.43
Sensor$$ z/L_{Mo}$$Region$$ C_{Bouss} $$ ( vol % )C ( vol % )Co ( % )
164.79BJ20.11 (22.21 for BP)16.94 (18.4 for BP)16.5
156.94BP1210.8-
149.09BP7.677.168.04
1310.39BP6.155.826.52

Horizontal buoyant jet

For the numerical simulation of turbulent buoyant jets, normally there are two methods available: integral methods and simulation by CFD.

Integral methods are based on the basic laws of conservation of mass, momentum, energy, and species concentration. To integrate the partial differential equations, empirical profile shapes (i.e. Gaussian distribution) for velocity, temperature and concentration are assumed. The entrainment of ambient gas into the turbulent buoyant jet is also needed to be assumed to close the equation system [5-8]. Pantokratoras[9] modified the integral Fan-Brooks model[10] to calculate the horizontal penetration of inclined thermal buoyant water jets, and the modified model predictions are in good agreement with the trajectory measurements.

Jirka[11-12] formulated an integral model, named CorJet, for turbulent buoyant jets (round and planar) in unbounded stratified flows, i.e., the pure jet, the pure plume, the pure wake. Based on the Boussinesq approximation, CorJet integral model appears to provide a sound, accurate and reliable representation of complex buoyant jet physics.

However integral methods are difficult to extend to more complex turbulent buoyant flows which are essentially three dimensional and often obstacles and walls interact with the flow. It is very difficult to prescribe the profile shapes and to relate the entrainment rate to all of the local parameters.

Therefore CFD methods become more popular which do not make assumptions for the profile and entrainment but yield them as a part of the solution. However, if not applying DNS, the turbulence model in the engineering CFD code is one of the most significant factors to affect the simulations of turbulent buoyant jets and plumes. Most of the turbulence models were developed and tested for non-buoyant flows. Turbulence models including buoyant effects [13] were reviewed by Hossain and Rodi[14]. In recent studies models with buoyancy related modification were evaluated, assessing in particular buoyancy effect on the production and dissipation of the turbulent kinetic energy in buoyant plumes[15-17]. The standard models were suspected to seriously underpredict the spreading rate of vertical buoyant jets and to overpredict the entrainment of horizontal flows.

Comparing to the vertical buoyant jet, much fewer data and calculations for the study of horizontal turbulent buoyant jets can be found in the open literatures[18-23]. Almost all the models and calculations available are based on Boussinesq approximation, which the environmental and atmospheric engineers are more interested in. What we are concerning is hydrogen, helium, or steam released into air, which involves a large density difference between the jet and ambient, making the Boussinesq approximation invalid.

In the table below, integral models of horizontal buoyant round and plane jets are introduced. The system of ordinary differential equations can be solved by 4th order Runga-Kutta method to obtain the horizontal buoyant jet trajectory, the velocity, the density, and the tracer concentration. For small density difference cases, the Boussinesq integral model can obtain reasonable results as the CorJet model shows in the table. For hydrogen safety analysis, the large density difference will violate the Boussinesq approximation.

 Horizontal buoyant plane jetHorizontal buoyant round jet
Diagram of horizontal buoyant jet into uniform ambientAttach:D68_Fig1.jpg ΔAttach:D68_Fig2.jpg Δ
Volume flux$$ q_0 = U_0 b_0 $$(specific flux)$$Q_0 = U_0 A_0 {\rm{ = }}U_0 \pi r_0^2 $$
Momentum flux (in kinematic units)$$m_0 = U_0^2 b_0 $$ (specific flux)$$M_0 = U_0^2 A_0 = U_0^2 \pi r_0^2 $$
Buoyancy flux(in kinematic units)$$j_{\rm{0}} = U_0 \left( {{{(\rho _a - \rho _0 )g} \over {\rho _{\rm{0}} }}} \right)b_0 $$ (specific flux)$$J_0 = U_0 \left( {{{(\rho _a - \rho _0 )g} \over {\rho _0 }}} \right)\pi r_0^2 $$
Jet/plume transition length scale$$L_M = m_0 /j_0^{2/3} $$$$L_M =M_0^{3/4}/J_0^{1/2} $$
Froude number$$Fs = {{U_0 } \over {\sqrt {g'b_0 } }} = {{U_0 } \over {\sqrt {\left( {{{(\rho _a - \rho _0 )g} \over {\rho _0 }}} \right)b_0 } }}$$$$Fs = {{U_0 } \over {\sqrt {g'r_0 } }} = {{U_0 } \over {\sqrt {\left( {{{(\rho _a - \rho _0 )g} \over {\rho _0 }}} \right)r_0 } }}$$
AssumptionsThe fluids are incompressible; The flow is fully turbulent. Molecular transport can be neglected in comparison with turbulent transport which means there is no Reynolds number dependence; The profiles of velocity, buoyancy, and concentration are similar at all cross sections normal to the jet trajectory; longitudinal turbulent transport is small compared with latitudinal convective transport.
Velocity profile$$u = u_s e^{ - n^2 /b^2 } $$$$u = u_s e^{ - r^2 /b^2 } $$
Density deficiency profile$${{\rho _a - \rho } \over {\rho _a }} = \left( {{{\rho _a - \rho _s } \over {\rho _a }}} \right)e^{ - n^2 /(\lambda b)^2 } $$$${{\rho _a - \rho } \over {\rho _a }} = \left( {{{\rho _a - \rho _s } \over {\rho _a }}} \right)e^{ - r^2 /(\lambda b)^2 } $$
Tracer concentration profile$$c = c_s e^{ - n^2 /(\lambda b)^2 } $$$$c = c_s e^{ - r^2 /(\lambda b)^2 } $$
entrainment$$E = 2\alpha \rho _a u_s $$$$E = 2\pi \alpha b\rho _a u_s $$
Entrainment coefficient$$\alpha = \alpha _j \exp \left[ {\ln \left( {{{\alpha _p } \over {\alpha _j }}} \right)\left( {{{Ri_{j - p} } \over {Ri_p }}} \right)^{3/2} } \right]$$$$\alpha = \alpha _j \exp \left[ {\ln \left( {{{\alpha _p } \over {\alpha _j }}} \right)\left( {{{Ri_{j - p} } \over {Ri_p }}} \right)^2 } \right]$$
Jet entrainment coefficient$$\alpha _j = 0.052 \pm 0.003$$$$\alpha _j = 0.0535 \pm 0.0025$$
Plume entrainment coefficient$$\alpha _p = 0.102$$$$\alpha _{\rm{p}} = 0.0833 \pm 0.0042$$
Width ratio$$\lambda = 1.35$$$$\lambda = 1.19$$
conservation equations$$\int_{ - \infty }^\infty {{{\partial (\rho u)} \over {\partial s}}dn} = 2\alpha \rho _a u_s $$ $$\int_{ - \infty }^\infty {{{\partial (\rho uu\cos \theta )} \over {\partial s}}dn} = 0 $$ $$\int_{ - \infty }^\infty {{{\partial (\rho uu\sin \theta )} \over {\partial s}}dn} = g \int_{ - \infty }^\infty {(\rho _a - \rho )dn} $$ $$\int_{ - \infty }^\infty {{{\partial (cu)} \over {\partial s}}} dn = 0 $$$$\int_0^\infty {\int_0^{2\pi } {{{\partial (\rho u)} \over {\partial s}}r} drd\varphi } = 2\pi \alpha b\rho _a u_s $$ $$\int_0^\infty {\int_0^{2\pi } {{{\partial (\rho uu\cos \theta )} \over {\partial s}}r} drd\varphi } = 0 $$ $$\int_0^\infty {\int_0^{2\pi } {{{\partial (\rho uu\sin \theta )} \over {\partial s}}r} drd\varphi } = \int_0^\infty {\int_0^{2\pi } {(\rho _a - \rho )grdr} } d\varphi $$ $$\int_0^\infty {\int_0^{2\pi } {{{\partial (cu)} \over {\partial s}}r} drd\varphi } = 0 $$

Boussinesq integral model for horizontal buoyant plane jet.

Hydrogen-Helium Similarity

Buoyancy effect in the mean flow

To investigate on the relation between helium and hydrogen buoyancy we need to consider the non-dimensional form of the buoyancy term in the vertical momentum equation. If we introduce L and U as the reference length and velocity scales, then this term takes the form of a local Richardson number, expressing the ratio between buoyancy and inertial forces: $$ Ri_{local}={{(\rho_a-\rho) g L} \over {\rho U^2}} $$ In this analysis we shall use the Boussinesq approximation (mass fraction f<<1), which implies replacing the mixture density with the density of ambient air in all the terms of the mean flow conservation equations except for the density difference in the buoyancy term. Additionally by introducing the local molar concentration C the above buoyancy term becomes proportional to a global Richardson number Ri : $$ Ri_{local}= C Ri $$ where $$ Ri= {{(\rho_a-\rho_g) g L} \over {\rho_a U^2}} $$

The relation between helium and hydrogen buoyancy can now be assessed using this global Ri, assuming same velocity and length scales: $$ {{Ri_{He}}\over{Ri_{H2}}}= {{\omega_{He}-1} \over {\omega_{H2}-1}} {{\omega_{H2}} \over {\omega_{He}}} \approx 0.92 $$

The proximity of the ratio of the Richardson numbers to 1.0 justifies the use of He instead of H2 for molar concentration measurements in regions of the flow were the Boussinesq approximation is valid. This result can be compared to Agranat et al., 2004 Invalid BibTex Entry!, who derived the ratio of the Richardson numbers to be 0.47 (compared to 0.92 here), which is strongly different to the present result and certainly does not justify performing hydrogen dispersion tests using helium. The reason for this discrepancy is that Agranat et al. defined their Richardson number by dividing with the density of the gas and not of the air as has been done in the present analysis. Use of the density of the gas could be justified only where the mixture is very rich, e.g. very close to the source. In these regions of course the Boussinesq approximation is not valid.

Buoyancy effect in the turbulent kinetic energy

Buoyancy affects the mean flow directly and indirectly. The direct influence is through the buoyancy term in the vertical momentum equation as shown in the above paragraph. The indirect influence is through the buoyant production term of the turbulent kinetic energy (TKE) equation. With the eddy viscosity approach and the introduction of the molar concentration this term is: $$ G=-\rho w g = {\nu_t \over Sc_t} {{\partial \rho} \over {\partial z}} g = - {\nu_t \over Sc_t} {{\partial C} \over {\partial z}} g (\rho_a - \rho_g) $$

By introducing, L and U as reference length and velocity scales and under the Boussinesq approximation the non-dimensional form of the buoyant production term becomes proportional to the Richardson number and inversely proportional to the turbulent Schmidt number: $$ {{G} \over {\rho_a U^3 / L}} = - {\nu_t \over {U L} } {{\partial C} \over {\partial z}} {Ri \over Sc_t} $$

This result shows that the effect of the released substance on the buoyant production term can be investigated using the ratios of Richardson and turbulent Schmidt numbers. The Richardson ratios were close to 1.0 as shown in the above paragraph. The turbulent Schmidt number is usually taken as 0.7 and independent of the substance. Therefore, under the Boussinsq approximation the effect on the buoyant TKE production term of replacing hydrogen with helium in hydrogen dispersion tests is considered to be small. This is not valid for regions of the flow were the Boussinesq approximation does not hold.

Axial concentration in axi-symmetric jets

Using the definitions and correlations introduced above for axi-symmetric vertical jets we can compare the axial molar concentrations for helium versus hydrogen releases. It can be easily shown that the helium to hydrogen Froude ratios are: $$ {{F_{He}}\over{F_{H2}}} = {{\omega_{H2}-1}\over{\omega_{He}-1}} \approx 2.2 $$ Then, one can obtain the helium to hydrogen mass fraction ratios as follows: In the non-buoyant jet region: $$ \left( {{f_{He}}\over{f_{H2}}} \right)_{NBJ} = \left( {{\omega_{H2}}\over{\omega_{He}}} \right)^{1 \over 2} \approx 1.4 $$ In the intermediate buoyant jet region: $$ \left( {{f_{He}}\over{f_{H2}}} \right)_{BJ} = \left( {{\omega_{H2}-1}\over{\omega_{He}-1}} \right)^{1 \over 8} \left( {{\omega_{H2}}\over{\omega_{He}}} \right)^{1 \over 2} \approx 1.6 $$ In the buoyant plume region: $$ \left( {{f_{He}}\over{f_{H2}}} \right)_{BP} = \left( {{\omega_{H2}-1}\over{\omega_{He}-1}} \right)^{1 \over 3} \left( {{\omega_{H2}}\over{\omega_{He}}} \right)^{2 \over 3} \approx 2.1 $$

Under the Boussinesq approximation, the helium to hydrogen molar fraction ratios become: $$ \left( {{C_{He}}\over {C_{H2}}} \right) = \left( {{f_{He}} \over {f_{H2}}} \right) \left( {{\omega_{H2}}\over{\omega_{He}}} \right) $$

Then for the non buoyant jet region: $$ \left( {{C_{He}}\over {C_{H2}}} \right)_{NBJ} \approx 0.71 $$ In the intermediate buoyant jet region: $$ \left( {{C_{He}}\over {C_{H2}}} \right)_{BJ} \approx 0.81 $$ In the buoyant plume region: $$ \left( {{C_{He}}\over {C_{H2}}} \right)_{BP} \approx 1.03 $$ The higher the proximity of these ratios to 1.0 to more justified is the use of He instead of H2 for molar concentration measurements.

After Release Phase

Again the garage experiment is characterised by the two time domains. As a complete scaling of the overall experiment is difficult and - in fact - not necessary, we decided to analyse this experiment in two steps. For the first phase the upward directed jet has been compared to the empirical scaling in the literature. For the second phase we want to know who the injected gas behaves in the nominally closed room.

After switching off turbulence generating sources the turbulence connected with the contained gases will decay. Most experiments on decaying turbulence have been performed in wind tunnels up to now, where the distance from the grid gives the decay time t, if the mean velocity is known. In the garage experiment the mean velocity after the release phase is zero. So either one could measure the fluctuations in the gas layer close to the roof with its high hydrogen content or one could refer to scaling considerations like in [Lohse] etc.

Scaling of diffusion

One has to differ two regimes in the diffusion process of gases. The basic diffusion is the molecular diffusion which is solely based on the stochastic motion and interference of the molecules. This process might be supported by micro-vortices generally associated with turbulence in the gas. This turbulent diffusion is typically orders of magnitude more efficient.

Diffusion may transport mass, impulse and heat. For each transport there is a coefficient characterising the corresponding transport properties for the given thermodynamic state. Disregarding the actual transported quantity the 2nd Fick's law allows to identify the units of the diffusion coefficients $$ [D] = [\nu] = [a] = {m^2 \over s} $$ Evidently ratios of these coefficients deliver dimensionless parameters characterising the gas itself: the Prandtl number $$ Pr = {\nu \over a} $$ relates the impulse transport to the heat transport and the Schmidt number $$ Sc = {\nu \over D} $$ the impulse transport to the mass transport. As the transport properties mainly depend on the degrees of freedom of the gas species, the related dimensionless numbers nearly the same for two-atomic gases (like H2, O2 and N2) for instance.

After switching off turbulence generating sources the turbulence connected with the contained gases will decay. Most experiments on decaying turbulence have been performed in wind tunnels up to now, where the distance from the grid gives the decay time t, if the mean velocity is known. In the garage experiment the mean velocity after the release phase is zero.

Applying the Pi-theorem to the momentum decay in a turbulent gas layer we will receive the following matrix:

 $$\epsilon$$$$u$$$$L$$$$ t_\ell $$$$\nu$$
m$$ 2$$$$ 1$$$$ 1$$$$ 0$$$$ 2$$
kg$$ 0$$$$ 0$$$$ 0$$$$ 0$$$$ 0$$
s$$-3$$$$-1$$$$ 0$$$$ 1$$$$-1$$

It is evident that the problem, the rank of the dimension matrix respectively, is reduced as the dynamic row represented by the kg is zero. So there must be 5-2=3 dimensionless parameters for this problem. The experienced reader easily identifies the Reynolds number, the Taylor-Reynolds respectively, $$ Re = {{u L} \over \nu} $$

$$ Re_\lambda = {{u \lambda} \over \nu} = {{u^2} \over {{{\partial u} \over {\partial x}} \nu}} $$

the dimensionless decay time $$ \Pi_t = {{t_\ell} \over \tau} ={{t_\ell \nu} \over {L^2}} $$ and - maybe more difficult to see immediately - the ratio of dissipation times in a characteristic time step and of the specific kinetic energy of the flow field: $$ \Pi_e = {{\epsilon L} \over {u^3}} $$

As for this problem the Re_\lambda depends implicitly on the time of the observation Re_\lambda, the whole might be summarised to $$ \Pi_e = f(Re_\lambda(t)) $$

In Fig.2 we can identify the turbulence lifetime t_\ell where the Reynolds number starts to drop close to t / \tau \approx 10^{-1} where Re_\lambda \approx 16.

Attach:FigLohseReOverT.jpg Δ | Figure 1. Time dependence of the Taylor-Reynolds number for freely decaying turbulence

From further considerations and from experience one may derive that also for the dimensionless decay time t_\ell / \tau there should be for high Reynolds numbers a Reynolds number independent limit, which is according to [Lohse]: $$ \Pi_{t, max} = 0.18 $$ For the garage benchmark this would yield approximately one minute after the release end for the turbulence lifetime.

Again, these findings are based on the assumptions of free and isotropic turbulence. Deviations for the garage example are the wall effects of the roof and the influence of the gravitation.

After the turbulent regime only the molecular diffusion remains.

Conclusions

Scaling and similarity provide valuable tools for planning of gas mixing experiments and for extrapolating the experimental findings. With some experience the analyst is able to differ important and negligible influences. To this end he has to identify the relevant parameters, processes or mathematical models in one initial step. However, without experience this initial choice might be wrong or incomplete and lead to wrong conclusions. Under these conditions even the methods importing a certain knowledge, like the equations approach, will not improve the situation considerably.

In complex settings the requirement for scaling can be so strong, such that similarity becomes identity and therefore obsolete. In these cases spatial or temporal decomposition of the problem can relax the requirements as demonstrated for the garage example. A general conclusion for the material scaling in gas mixing experiments is, that the replacement of hydrogen by the model fluid helium violates the similarity conditions only minor. With a slight geometrical scaling (slightly larger experiment with helium) the similarity will be even improved. The application to the evaporation of a LH2 pool is recommended for an exercise.

In a future extension of this report the scaling found in the literature for reactive flows, including fires and gas explosions, will be summarised. The methodology will be applied to some standard benchmark cases where hydrogen fire or explosions have been modeled.

Basic properties of H2,He,air

In the previous chapters refrence was made to some essential properties of hydrogen, helium, frequently used as a safe model fluid, and air into which both gases are dispersed.

propertyH2Heair
density (at NTP 1 atm, 20C) 0.0835 kg/m^3 0.166 kg/m^3 1.21 kg/m^3
kinematic viscosity 1.05 10^{-4} m^2/s 1.15 10^{-4} m^2/s 1.51 10^{-4} m^2/s
diffusion coefficient 6.1 10^{-5} m^2/s 5.7 10^{-5} m^2/s ... m^2/s
diffusion coefficient in air 7.5 10^{-5} m^2/s ... 10^{-5} m^2/s ... m^2/s
sonic speed (at 1 atm, 20C) 1306 m/s 1005 m/s 300 m/s
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