CFD simulation of fume spread in rotational symmetry

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(1)Bror Persson and Ludovic Romanov*). CFD Simulation of Foam Spread in Rotational Symmetry SP AR 2004:03 Fire Technology Borås 2004. *) Ecole Nationale Supérieure en Informatique Automatique Mécanique Energétique et Electronique, Valenciennes, France.

(2) 2. Abstract In a recently conducted EU project, FOAMSPEX, theoretical models were developed describing the spread of foam on burning liquid surfaces in cylindrical pools. The friction data used in these models were based on cold flow experiments with foam spread in a half symmetrical pool with radius 10 m. These friction data are not supposed to be generally valid for arbitrary pool sizes. In order to extend the applicability, and to improve the predictive capability of the approximate models, a scaling law for the friction coefficient with respect to pool size and inlet radius has been derived. Drainage effects are disregarded in the analysis. The scaling law introduced here is based on results from CFD simulations of foam spread on liquid surfaces in cylindrical pools. The simulations were carried out by making use of the CFD program FLUENT. In this CFD program it is not possible to model the nonNewtonian behaviour of the foam, instead the foam has been modelled as a highly viscous fluid with Newtonian properties. Still it is expected that the main features of the friction problem can be described in this way. The CFD simulations indicate that the original model exaggerates the friction for large pools. This means that the times required to cover a large pool in cold flow will be shorter than previous estimates imply. For the case with fire the reduction in the frictional resistance will have less importance because the friction is then only a minor part of the total restraining forces against foam spread.. Key words: Fire fighting foam, foam spread modelling, CFD-simulations. SP Sveriges Provnings- och Forskningsinstitut SP Arbetsrapport 2004:03 ISSN 0284-5172 Borås 2004. SP Swedish National Testing and Research Institute SP Technical Note 2004:03 Postal address: Box 857, SE-501 15 BORÅS, Sweden Telephone: +46 33 16 50 00 Telex: 36252 Testing S Telefax: +46 33 13 55 02 E-mail: info@sp.se.

(3) 3. Contents Abstract. 2. Contents. 3. 1. Introduction. 5. 2 2.1 2.2 2.3. Theory Governing equations Friction model Analytical solution. 6 6 7 9. 3 3.1 3.2 3.3. CFD calculations Description of the Volume of Fluid (VOF) model Comparison with experiments in a half symmetry pool with water Determination of friction coefficients for arbitrary pool sizes. 10 10 10 11. 4. Improved approximate solution. 14. 5. Application to large scale tanks. 15. 6. Concluding remarks. 17. 7. References. 18.

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(5) 5. 1. Introduction. In the foam spread models described by Persson et al [1] the frictional resistance between the foam and the liquid is described by a model based on approximate solutions of the boundary layer flow in the liquid. The friction model is expressed as a friction coefficient times a functional relationship involving the inlet velocity of the foam and the instantaneous extension of the foam blanket. This implies that the friction coefficient will be dependent upon both the inlet radius and the pool size. The CFD calculations performed are not expected to simulate all aspects of the foam spread problem but will give sufficient information of the main features of the viscous flow as far as the viscous resistance is concerned. The calculations do not give the friction coefficients explicitly but rather indicates the relative variation of the friction coefficients when varying the pool size and the inlet radius..

(6) 6. 2. Theory. A brief summary of the governing equations of foam spread, the model describing the viscous friction, and the approximate analytical solution will be given here. For more details, see [1].. 2.1. Governing equations. Consider the spread of foam on a liquid fuel surface in a circular pool. The foam is assumed to be gently applied at an inlet radius r = R0 with a constant volume flow rate V& (m3/s) and advancing symmetrically in radial direction, see Figure 1. The instantaneous thickness of the spreading foam layer is denoted by h(r , t ) , the thickness at r = R0 is h0 , and the instantaneous location of the foam front is given by r = R(t ) . It is assumed that the viscosity of the foam is several orders of magnitude larger than the viscosity of water and ordinary hydrocarbon fuels. This means that the velocity gradients in the foam will be small compared to the gradients in the fluid and consequently the velocity in the foam layer can to a good approximation be assumed to be constant across the thickness. The local velocity of the foam is denoted by u i.e. u = u (r , t ) , and the inlet velocity by u 0 . A more detailed discussion of the governing equations given here can be found in Persson et al. [1].. Figure 1. Schematic of a foam layer spreading on a liquid surface. The foam is applied from the left at an inlet radius R0 with a constant volume flow rate, V& , and a friction force, τ f , opposes the spread. The foam is exposed to a radiation flux, q& r , causing an evaporation mass loss, m& vr , and a radiation-induced drainage, m& dr , in addition to the ordinary drainage, m& d .. Assuming that drainage effects can be disregarded, which implies that the density of the foam is constant, the continuity equation can be expressed. ∂ ∂ (rh) + (ruh) = 0 ∂t ∂r. When inertia forces are neglected the equation of motion takes the form. (1).

(7) 7. ( ). rτ f g 1 ∂ rh 2 = − 1 −  ρ 2  S  ∂r. (2). Here the left hand side expresses the driving force due to the gradient of the hydrostatic pressure across the foam layer, the right hand side expresses the friction force due to the shear stress between foam and liquid. Furthermore, g denotes the gravitational constant, S is the ratio between the density of the liquid and the density of the foam ρ . This parameter is called the expansion ratio and is normally used as one quality parameter of the foam. In this analysis both the foam density and the expansion ratio are assumed constant. Equations (1) and (2) constitute the set of equations describing the foam spread in a circular pool when drainage effects can be neglected. At the inlet ( r = R0 ) the density of the fresh foam is known, while the layer thickness h0 and the inlet velocity u 0 are connected through the equation. 2πR0 u 0 h0 = V&. (3). In the subsequent analysis the density of the foam will be denoted without subscript. An additional boundary condition is provided by the closure at the front of the foam layer, i.e.. h( R, t ) = 0. 2.2. (4). Friction model. When foam is applied at the centre of a circular pool the resistance to the foam spread will mainly emanate from the shear forces between the foam and the liquid. The flow set up in the pool can be considered as a backward boundary layer flow, i.e. a flow where the boundary layer thickness is increasing from the inlet towards the advancing front. At the foam front the flow field in the liquid cannot be described in terms of a boundary layer flow, therefore the resistance in the model is treated as a mean over the entire foam area rather than as a local entity. The shear stress τ f appearing in the momentum equation will be estimated by making use of solutions for boundary layer flow in the liquid. It is assumed that the viscosity of the foam is much larger than that of the liquid, which means that the shear deformation of the foam can be neglected. In addition it is assumed that the liquid layer beneath the foam is deep enough to be considered as semi-infinite. Furthermore, the flow is assumed to be in a quasi steady state meaning that the time variation in the velocity field is slow and can be neglected.. An integral balance for the momentum change of the boundary layer in a liquid of infinite depth can be written. ρl. d δ 2 ∫ ru ⋅ dy = rτ f dr 0. (5). where ρ l is the density of the liquid, r is a co-ordinate in radial direction (inlet at r = R0 ), δ is the thickness of the boundary layer, y is a co-ordinate perpendicular to.

(8) 8. the liquid surface (positive downwards), and τ f is the shear stress at the interface between the foam and the liquid. By assuming a second order polynomial for the velocity profile, the resulting equation for the boundary layer thickness δ can be expressed. ρl. (. ). 10 µ l rU d rU 2δ = dr δ. (6). after introducing the shear stress defined by. 2µ lU  ∂u   = δ  ∂y  y = 0. τ f = − µ l . (7). Here µ l denotes the dynamic viscosity of the liquid and U is the velocity of the foam layer. The notation U has been introduced for the velocity to indicate that it is a mean velocity rather than a local velocity dependent upon both the radial co-ordinate r and the time t . To simplify the calculations, the velocity is assumed constant. Upper and lower bounds for the velocity are u 0 (the inlet velocity) and R& (the front velocity), i.e. u 0 > U > R& . After applying the boundary condition δ = 0 at r = R0 , and assuming U constant the solution of equation (6) takes the form. δ =. 20 µ l r 3ρ lU. (8). Inserting (8) into (7) yields. τf =. µ lU 3 3 ⋅ ρl 5 ρl r. (9). A mean value of the shear stress over the total foam area ( τ f ) can be defined by integrating equation (9). The resulting expression indicates that it would be conceivable to write the shear stress as 3. τ f =k⋅. u0 2. (10). g ( R). where k is a constant and where g ( R ) is a function of the extension. Comparing the analytical solution based on equation (10) with experimental results indicates a best fit of 1. the data when assuming g ( R ) = R 4 , see [1]. It is to be noted that the friction coefficient k is not a material constant but linked to different variables through equation (10). This means that the friction coefficient may show some dependence upon length scale when extrapolating the results to very large pools..

(9) 9. When calculating foam spread on other liquids than water the friction coefficient k can be corrected according to [1]. kl =. ρl µl ⋅ kw ρwµ w. (11). Here suffix “ l ” refers to the actual liquid and “ w ” to water. In the cold flow experiments with water it was found that k w = 0.25 for AFFF-AR foam and k w = 0.42 for FP foam. As ρ w µ w ≈ 1 for water the corrected friction coefficient can accordingly be expressed (suffix for liquid dropped for k ). k = 0.25 ρ l µ l for AFFF-AR foam k = 0.42 ρ l µ l for FP foam. 2.3. (12a) (12b). Analytical solution. A simple approximate analytical solution of equations (1) – (2) subjected to the boundary conditions (3) and (4) has been derived [1]. The expression for the extension and the inlet thickness can be written 1.  ρ 4c 4 R06V& 8  31 14  R = 1.038 ⋅ t 31 ; R >> R0 4   k  . (13a). 1.  k 8V& 15  31 3  h0 = 0.559 ⋅ t 31  ρ 8c8 R12  0  . (13b). Here ρ denotes the density of the expanded foam (constant) and the parameter c is defined as.  1 c = g 1 −   S. (14). In the derivation of these approximate solutions use has been made of the assumption that the 1. function g (R ) appearing in equation (10) can be expressed as g ( R ) = R 4 and that the extension R is large compared to the inlet radius i.e. R >> R0 . It is noted that the approximate analytical solution depends upon the inlet radius, both explicitly in the equation and implicitly through the friction coefficient and the inlet velocity..

(10) 10. 3. CFD calculations. In the CFD calculations the foam has been modelled as a highly viscous Newtonian liquid. A description of the program used, FLUENT, can be found in [2]. The FLUENT program does not have the option of a true multi-phase flow, which would be the proper description of foam spread on a liquid surface. Instead a model called the Volume of Fluids (VOF) has been used.. 3.1. Description of the Volume of Fluid (VOF) model. In this model the different phases or fluids in the problem is described by means of a volume fraction α n where n denotes the n th fluid. The VOF model can model two or more immiscible fluids by solving a set of momentum equations and tracking the volume fraction of each of the fluids throughout the domain. In each control volume in the computational domain, the volume fractions of all phases sum up to unity. The fields for all variables and properties are shared by the phases and represent volume-averaged values. Thus the variables and properties in any given cell are either purely representative of one of the phases, or representative of a mixture of the phases, depending upon the volume fraction values. The following three conditions for the volume fraction of the n th phase are possible:. αn = 0 αn = 1 0< α n <1. the cell is empty (of the n th fluid) the cell is full (of the n th fluid) the cell contains the interface between the fluids. Based on the local value of α n , the appropriate properties and variables will be assigned to each control volume within the domain. More details about the VOF method can be found in [2].. 3.2. Comparison with experiments in a half symmetry pool with water. In Figures 2 and 3 below a comparison is shown between the calculated extension and the extension obtained in the ½-symmetry tests with foam spread on water as reported in [1]. The following input data has been used in the calculations: Pool radius. R p = 10 m. Foam density. ρ f = ρ = 100 kg/m3. Foam viscosity. µ f = 0.1 Ns/m. Liquid density. ρ w = 1000 kg/m3 (water) µ w = 0.001 Ns/m (water). Liquid viscosity.

(11) 11. AF101_CFD-Exp. AF107_CFD-Exp 10 R (m). R (m). 10. 8. 6. 8. 6. Rcalc Rexp. 4. Rcalc Rexp. 4. 2. 2. 0. 0 0. 20. 40. 60. 80. 100. Figure 2. 0. 10. 20. 30. 40. 50. 60. t (s). t (s). Comparison between experiments and calculated results for AFFF-AR foam. Volume flow rates of expanded foam are 0.038 m3/s (left) and 0.094 m3/s (right). CFD results are for a Newtonian fluid with density 100 kg/m3 and viscosity 0.1 Ns/m, spreading on water with a viscosity of 0.001 Ns/m. Inlet radius 1 m.. FP105_CFD-Exp. R (m). 10. 8. 6. 4. Rcalc Rexp. 2. 0 0. 10. 20. 30. 40. 50. t (s). Figure 3. Comparison between experiments and calculated results for FP foam. Volume flow rate of expanded foam is 0.145 m3/s. CFD results are for a Newtonian fluid with density 100 kg/m3 and viscosity 0.1 Ns/m, spreading on water with a viscosity of 0.001 Ns/m. Inlet radius 1 m.. From Figures 2 and 3 it is seen that the agreement is acceptable showing that the main features of the foam spread is well described and that the results support the assumption that the main restraining force is due to the viscous friction between the foam and the liquid. The calculations of the half symmetry case are performed just to verify that the main features of the foam-spread process can be modelled by CFD. No attempt has been made to adjust the “foam” viscosity in the calculations to make a best fit or to explain the difference between the foam-spread properties of the AFFF-AR and the FP foam appearing in the experiments.. 3.3. Determination of friction coefficients for arbitrary pool sizes. Calculations have been performed for a number of combinations of pool radius, inlet radius and expanded volume flow rates. The computed time history of the extension has then been compared with the analytical solution, equation (13a), to determine the best value of the friction coefficient giving a full cover of the pool area for the same time as in the CFD calculations. Figure 4 summarises the results..

(12) 12. k/R0**1.5/Vp**0.055. 0.08 0.07 0.06 y = 12.649 * x^(-1.912) R= 0.98951 0.05 0.04 0.03 0.02 0.01 0 10. 20. 30. 40. 50. 60. Rp (m). Figure 4. Calculated friction coefficients for different inlet and pool radii (R0 and Rp in m) and for various volume flow rates (Vp m3/s).. The correlation summarising the CFD results in Figure 4 can be expressed. k=. 12.65 ⋅ R01.5 ⋅ V& 0.055 R1p.91. (15). The friction coefficient according to equation (15) cannot be directly used for calculating the foam spread as it is not derived from results for real expanded foam but for a highly viscous Newtonian liquid (with a density close to a real expanded foam). However, it is assumed that the correlation according to equation (15) gives a sufficiently accurate functional dependence between the friction coefficient, the pool radius, the inlet radius and the volume flow rate. To make the results more generally applicable the correlation must be modified to agree with the experimental results obtained for true foams. On that account equation (15) is normalised with the experimental friction coefficient k10 obtained in the experiments (the suffix 10 indicates that the experiments were performed in a pool with radius 10 m). A characteristic value of the expanded foam flow in the experiments was V&10 = 0.12 m3/s while the inlet radius was 1 m. Thus, equation (15) can be rewritten.  10 k = R01.5   Rp k10 . 1.91.    . 0.055  V&     0.12 . (16a). or. R01.5V& 0.055 k = 72.34 k10 R1p.91. ρl µl. (16b). where the correction for arbitrary liquids (fuel) according to equation (11) has been introduced..

(13) 13. Numerical values for the friction coefficient k10 for water have been given in section 2.2. Equation (16b) together with the appropriate friction coefficient k10 can now be introduced together with equation (13a) to calculate the foam spread for arbitrary pool sizes..

(14) 14. 4. Improved approximate solution. By combining equations (13a) and (16b) it is possible to formulate a modified version of the theoretical solution that is independent of the inlet radius. The modified version takes the form.  ρ 2c 2   R = kp  ρl µl   . 0.0856. 1. 3. ⋅ V& 3 ⋅ t 5. (17). where the constant k p takes the values. k p = 0.64 for AFFF-AR foam and k p = 0.59 for FP foam. It is to be noted that the present formulation does not include the effect of the drainage. This means that the predicted times for coverage of a pool with a certain size will be underestimated..

(15) 15. 5. Application to large scale tanks. Results based on equation (13a) for large-scale tanks were reported in [1]. To get an indication of the effect of the improved friction data a series of calculations are presented here. Cetral inlet - T ank diameter 24 m. Central inlet - Tank diameter 80 m 40 Foam front, R (m). Foam front, R (m). 12. 10. 8. 6. 35 30 25 20. Orig solution (R0=1) Orig solution (R0=2) Improved solution. Orig solution (R0=1) Orig solution (R0=2) Improved solution. 15. 4. 10 2. 5. 0. 0 0. 10. 20. 30. 40. 50. 60. 70. 0. 50. 100. 150. 200. 250. Figure 5. FP - water - exp.ratio 3. 14. 14 Application rate (L/m2/min). Application rate (L/m2/min). 350. Foam front as function of time for a tank with diameter 24 m (left) and diameter 80 m (right). Application rate 4.1 L/m2/min and expansion ratio 3.. AF - water - exp. ratio 3. 1 min 2 min 5 min 10 min. 12 10 8. 12 1 min 2 min 5 min 10 min. 10 8. 6. 6. 4. 4. 2. 2. 0. 0 0. 20. 40. 60. 80. 100. 120. 0. 20. 40. 60. AF - water - exp.ratio 7. 100. 120. FP - water - exp. ratio 7 Application rate (L/m2/min). 14 12 1 min 2 min 5 min 10 min. 10. 80. Diameter (m). Diameter (m). Application rate (L/m2/min). 300. Time (s). Time (s). 8. 14 12. 8. 6. 6. 4. 4. 2. 2. 0. 1 min 2 min 5 min 10 min. 10. 0 0. 20. 40. 60. 80. 100. Diameter (m). Figure 6. 120. 0. 20. 40. 60. 80. 100. 120. Diameter (m). Application rate as function of tank diameter in order to cover the surface in 1, 2, 5 and 10 minutes according to equation (17). Assumptions: one central inlet, water as “fuel”, no drainage, expansion ratio 3 and 7..

(16) 16. AF - diesel oil - exp. ratio 3. FP - diesel oil - exp. ratio 3 14 Application rate (L/m2/min). Application rate (L/m2/min). 14 12 1 min 2 min 5 min 10 min. 10 8. 12. 8. 6. 6. 4. 4. 2. 2. 0. 1 min 2 min 5 min 10 min. 10. 0 0. 20. 40. 60. 80. 100. 120. 0. 20. 40. Diameter (m). AF - diesel oil - exp. ratio 7. 80. 100. 120. 100. 120. FP - diesel oil - exp. ratio 7 14 Application rate (L/m2/min). 14 Application rate (L/m2/min). 60. Diameter (m). 12 1 min 2 min 5 min 10 min. 10 8. 12. 8. 6. 6. 4. 4. 2. 2. 0. 1 min 2 min 5 min 10 min. 10. 0 0. 20. 40. 60. 80. 100. Diameter (m). Figure 7. 120. 0. 20. 40. 60. 80. Diameter (m). Application rate as function of tank diameter in order to cover the surface in 1, 2, 5 and 10 minutes according to equation (17). Assumptions: one central inlet, diesel oil, no drainage, expansion ratio 3 and 7.. From Figure 5 it is seen that the improved friction model gives results that are not differing substantially from the original model for moderate pool sizes, say up to approximately 30 m. For very large pools the situation is different, here the improved model indicates a considerable reduction in the friction compared to the original formulation. However, it must be kept in mind that the influence of drainage becomes much more pronounced for larger pools. This tends to decrease the speed of the foam i.e. increases the spreading time. Thus, the results for large pools without drainage must not be considered as predictions of the true spreading times..

(17) 17. 6. Concluding remarks. The present CFD simulations indicate that the original approximate viscous friction model reported in [1] does not scale the results from the small-scale experiments (pool radius 10 m and inlet radius 1 m) sufficiently well as the pool size increases. The reason is mainly connected with the inlet radius used in the experiments. In the original friction model the velocity of the foam at the inlet was used as a characteristic velocity for determining friction coefficients from the experimental results. Thus, the friction data are implicitly dependent upon the inlet radius. As the pool size increases the inlet radius becomes less important compared to the outer radius, which is clear from equation (15). At times when the foam front is much larger than the inlet radius one would expect the dependence of the inlet radius to be negligible. This is also the case as is seen from equation (17) where the dependence upon the inlet radius has disappeared. In this respect the formulation of the approximate theoretical solution as displayed in equation (17) is superior compared to the original formulation according to equation (13a). The CFD results reported are based upon the same experiments as the original formulation of the friction model. Thus, the CFD simulations agree with the original model for moderate pool sizes, say up to a diameter of 30 m. For larger pool sizes the CFD results indicate a reduced friction, which will have consequences for the time required to completely cover a pool, especially in cold flow situations. The original model will overestimate the time compared to the CFD results. However, for the case with fire the present modification means less. The influence of the viscous friction is small compared to other restraining effects against foam spread on a burning surface. For foam spread on a burning surface the dominating mechanisms governing the spreading time is rather drainage, evaporation of water and destruction of foam at the foam front..

(18) 18. 7. References. [1]. Persson B., Lönnermark A., Persson H., Mulligan D., Lancia A., and Demichaela M., ”FOAMSPEX: Large Scale Foam Application – Modelling of Foam Spread and Extinguishment”, SP Report 2001:13, 2002.. [2]. FLUENT 5 User´s Guide, Fluent Incorporated, July 1998..

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