Beirut explosion: Energy yield from the fireball time evolution in the first 230 milliseconds
Charles Aouad, Wissam Chemissany, Paolo Mazzali, Yehia Temsah, Ali Jahami
NNoname manuscript No. (will be inserted by the editor)
Beirut explosion
Energy yield from the fireball time evolution in the first 230 milliseconds
Charles J.Aouad · Wissam Chemissany · Paolo Mazzali , · YehiaTemsah · Ali Jahami Received: date / Accepted: date
Abstract
The evolution of the fireball resulting fromthe August 2020 Beirut explosion is traced using ama-teur footage taken during the first ∼ ms after thedetonation. 38 frames separated by ∼ ms are extracted from 6 different videos located preciselyon the map. Measurements of the time evolution of theradius R t of the shock wave are traced by the fireball atconsecutive time sequence t . Pixel scales for the videosare calibrated by de-projecting the existing grains si-los building for which accurate drawings are availableand by defining the line of sight incident angles. Theenergy available to drive the shock wave at early timescan be calculated through E = 10 b Kρ o where b is afitted parameter dependant on the relation between R t and t . K is a constant depending on the ratio of specificheats of the atmosphere and ρ is the undisturbed gasdensity. A total energy yield of E ≈ . × Jouleswith a lower bound of 9 . × and an upper boundof ∼ . × or the equivalent of ∼ tons ofT.N.T. is found. Our energy yields are different fromother published studies using the same method. Thiscan present an argument that if the compound thatexploded is fuel rich ammonium nitrate (ANFO), then Charles AouadE-mail: [email protected] Astrophysics Research Institute, Liverpool John MooresUniversity, IC2, Liverpool Science Park, 146 Brownlow Hill,Liverpool L3 5RF, UK · Institute for Quantum Information and Matter, CaliforniaInstitute of Technology, 1200 E California Blvd, Pasadena,CA 91125, USA · Max-Planck Institut fur Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany · Faculty of engineering, Beirut Arab University, Beirut,Lebanon. the actual mass that detonated is less than officiallyclaimed.
Keywords
Explosion physics · Beirut explosion · Fireball · Shock wave · Blast · ammonium nitrate explosion On the 4th of August 2020, an explosion occurred inthe port of Beirut, Lebanon. The explosion happenedafter a fire ignited in warehouse number 12. This tragicevent resulted in massive large scale destruction, severedamages to buildings in an extended radius around thecenter and loss of lives. It was claimed by officials thatthis hangar contained an amount of 2750 tons of am-monium nitrate kept in the port for around 6 years.A few attempts have already been made to evaluate theenergy released by this explosion, with contradicting re-sults.Diaz 2020 measures the evolution of the fireball from26 data points by using publicly available videos toyield a energy of 1 kt of T.N.T. Rigby et al. 2020 usealso available public videos and derive the times of ar-rival of the shock wave at 38 different locations, theyuse empirical relations linking the scaled time of ar-rival with the scaled distance to yield an energy of 0.5kt T.N.T with a higher bound of 1.2 kt T.N.T. Tem-sah, Jahami and Aouad 2020 (personal communication,September, 2010-in preparation) use the structure mod-eling of the existing silos using a 3-dimensional finiteelement non-linear analysis to yield an upper bound of0 . kt T.N.T. Converting the claimed 2750 tons of fuelrich Ammonium Nitrate (ANFO), as officially claimedyields a figure of 1.045 kt T.N.T (taking a 38% con-version factor to T.N.T equivalent (Karlos and Solomos2013)). a r X i v : . [ phy s i c s . pop - ph ] O c t Charles J.Aouad et al. These few and contradicting results create the needfor further investigations in an attempt to evaluate theexact amount of energy released. This may help notonly shed some light on the physical conditions that ledto this tragic event but also study the physical charac-teristics of a rare and such powerful explosion in a purephysical approach.Here we use the theoretical method described byTaylor 1950a; Taylor 1950b and lately used by Diaz2020 for the same event. We extend this investigationby adding more data and by precisely calibrating thescaling.The paper is divided as follows: in section 2, we presentour methodology, in section 3, we present our resultsand observations, leaving the discussion to section 4. Insection 5 we draw our conclusions and discuss possiblelines of future work. ρ ,the pressure p and the velocity u after the passage of ablast wave brought about by a nuclear explosion. Later,he was able to experiment the accuracy of his work af-ter the publication of the photographs of the Trinitynuclear explosion test (Szasz 1984; Mack 1946).In Taylor 1950a, Taylor was able to solve numeri-cally three differential equations, namely an equationof motion, a continuity equation and an equation ofgas state. The boundary conditions given by the Rank-ine Hugoniot equations (Rankine 1870; Hugoniot 1887)have been imposed, followed by the assumption thatthe energy released is to be concentrated in one singlepoint (see Taylor 1950a; Taylor 1950b, for a completemathematical description). This effort has led to themain prediction subject of the current study, in partic-ular, the fifth power of the fireball’s radius (which issupposed to be traced by the shock wave in the earlytimes) R should be proportional to the squared timefrom the explosion t in the form R = t EK − ρ − . (1)Here ρ is the undisturbed gas density, and K is a con-stant that depends on γ the ratio of specific heats ofthe gas and E the part of the energy that has not beenradiated away.This relation can be written as52 log R = alog t + b (2)where a is the slope of this linear relation and is ex-pected to be equal to unity if the observation follows eirut explosion 3 the theoretical prediction. In that case, the energy E can be calculated from Eqs. (1 and 2) via E = 10 b Kρ o (3)where10 b = R t − . (4)the term b can be calculated from a linear fitting to theobserved data Taylor 1950b. Fig. 1 video 2 taken from 1400 m distance. 8 frames sepa-rated by 33.33 ms showing the fireball along with the circlefit. The detonation is assumed to have happened anytime be-tween the first and second frame. Table 1
Videos used to extract frames video rate distance α Reflabel FPS meters ◦ Table 2
Column 1 shows the video label, column 2 showsthe frame rate in frames per second (FPS) for each video.Column 3 shows the distance in meters, column 4 shows theincident line of sight in degrees. The last column shows thevideo reference as taken from social media.
Table 3
Fireball evolution in the first 230 msR pixel scale θ R T videopixels m/pixel meters seconds58 1.4 81.2 0.05 172 1.4 100.80 0.083 182 1.4 114.80 0.116 190 1.4 126 0.149 197 1.4 135.80 0.183 171 1.212 79.236 0.05 289 1.212 99.324 0.083 299 1.212 110.484 0.116 2110 1.212 122.76 0.149 2119 1.212 132.804 0.183 2128 1.212 142.848 0.216 2180 0.334 60.120 0.033 3250 0.334 83.500 0.066 3295 0.334 98.530 0.099 3327 0.334 109.218 0.133 362 1.067 66.154 0.033 470 1.067 74.690 0.050 483 1.067 88.561 0.066 491 1.067 97.097 0.083 497 1.067 103.499 0.100 4105 1.067 112.0.35 0.116 4109 1.067 116.303 0.133 4112 1.067 119.504 0.150 4117 1.067 124.839 0.166 477 0.85 63.725 0.033 5100 0.85 82.760 0.066 5120 0.85 99.312 0.099 5135 0.85 111.726 0.133 5145 0.85 120.002 0.166 5157 0.85 129.933 0.199 5168 0.85 139.864 0.233 5110 0.544 59.840 0.033 6150 0.544 81.600 0.066 6185 0.544 100.640 0.099 6205 0.544 111.520 0.133 6220 0.544 119.680 0.166 6235 0.544 127.840 0.199 6250 0.544 137.088 0.233 6
Table 4
The first column shows the radius of the fireball inpixels; the second column shows the pixel scale calculated foreach video. Column 3 shows the radius in meters. Column4 shows the time from explosion for each frame. Column 5shows the videos labels.
1. The videos are taken at different frame rates (FPS)which are also shown in table 1. The time separationbetween extracted still images is thus limited by thislimitation.2.4 Measuring the time evolution of the fireballWhat we exactly need in this study is to trace the evo-lution of the fireball radius R t as a function of t whichis considered to be the explosion time at time zero.We use the building of the wheat silos to calibratethe pixel scale of the videos by defining the location of Charles J.Aouad et al. Fig. 2
A Google earth map of Beirut showing the location of the 6 videos used in the current report. For each video an incidentline of site is determined along the long facade of the wheat silos building. Video 5 is calibrated using another technique sincethis video does not show the silos building in the frames (see appendix 7.1). The explosion center is shown with black circles. each movie and defining the incident line of sight angletaken with respect to the silos long facade. We use itsaccurate drawings to de-project the main and the sidefacade wherever these two are visible and determine ac-cordingly a pixel scale θ , (in which a pixel correspondsto a physical measurement in meters).We measure the total length of the the building projec-tion in pixels as measured on the CCD; this includesthe side elevation length l ≈ m (including the halfcylinder projection as it defines the border of the pro-jected visible width), and the front elevation length L ≈ . m . The total length corresponding to this pixel di-mensions is given by L sin α + l sin (90 − α ) where α isthe incident angle of sight. The pixel scale θ is thengiven by L sin α + l sin (90 − α ) L total [ pixels ] . This is shown in Fig. 11.The physical value R m will thus be given by : R m = θ × R px (5)here, R m is the radius of the fire ball in meters and R px is the measured radius in pixels. The values of θ are shown in table 3.This procedure does not apply for Video 5 in whichthe silos building is not visible. To calibrate this video, we determine the angular field of view in which a pixelcorresponds to an angle resolution. Using this angularpixel resolution, we can calculate the dimensions of thefireball knowing its distance (630 meters) using basictrigonometry, this is described in the appendix 7.1 Sur-prisingly, we find comparable numbers with differentvideos taken from completely different locations andusing different scaling methods.The separation between consecutive frames changeswith the videos. For most of them (video 1-2-3-4-6), therate is 30 FPS corresponding to time intervals 33.33 ms .For video 5, the rate is 60 FPS corresponding to timeintervals of 16.66 ms .For each frame, we measure the size of the fireballby manually fitting a circle to the luminous edge andconverting the pixels to physical measurements in me-ters using the pixel scale θ derived earlier.It is important to note that some of the videos (namely1,2 and 4) were taken by a shaking hand and thus defin-ing the center of the explosion is executed taking a fixedreference from the picture for which the coordinates ofthe center are corrected for each and every frame. Thelocation of this center of mass doesn’t change horizon- eirut explosion 5 Fig. 3 video 5 taken from a distance of 630 meters fromthe explosion. 9 frames separated by 16.66 ms showing thefireball along with the circle fit. The detonation is assumedto have happened anytime between the first and the secondframe. For pixel calibration method see the appendix7.1 tally, but it changes vertically in the first 66 ms. In fact,the center of the luminous hemisphere rises rapidly dueto buoyancy, and we follow this rise accordingly (Tay-lor 1950b; Bethe et al. 1958). After this initial quickrise, the center stabilises and we identify t by visu-ally checking the frame at which the first bright lightof the explosion is seen. Thus, t can be taken anytimebetween these two frames. We consider an error on de-termining t of 16 ms . (We refer the reader to section2.5 for a complete error analysis review).Using this procedure we can build the evolution of R t with respect to t .The luminous sphere is bounded by a pseudo-sharp edge( ∼ ms ); as this edge starts becoming less sharply de-fined at later stages ( ∼ ms ), we do not extend ourmeasurements further. Our values are shown in table 3.2.5 Error analysisTwo main uncertainties can cause the error on the esti-mation done in the current study, namely the error onmeasuring the radius R and the error in assuming thetime t of the detonation.After all, we are measuring pixels in frames extractedform videos taken by handheld smart phones of limitedframe rates cameras.These errors can be divided into three main cate-gories. 1. Errors in determining the pixel scale, this is mainlydue to uncertainties in defining the precise locationof the camera within at least 1 ◦ and errors in defin-ing the boundaries of the silos border due to noisein the frames and the low resolution for some of thevideos. The combined effect leads to uncertainty inthe pixel scale determination of δθθ ≈ δR pixels R pixels = 3%. Using eq 5 we can write: δR m R m = (cid:115)(cid:18) δθθ (cid:19) + (cid:18) δR pixels R pixels (cid:19) (6)here δR m is the error on the radius of the fireballin meters and δθθ is the error on the pixel scale.Taking the average relative pixel scale δθθ of 5% andan average δR pixels R pixels of 3% leads to a relative error δR m R m = 0 . log R is given by differentiating with respect to R leadingto δ (cid:20)
52 log R m (cid:21) = δR m R m
52 log n ≈ . . (7)3. Error on the assumption of time t of the detona-tion. By checking the frames, it can be seen that thedetonation causes a sudden bright light seen in theframes sequence; thus, the detonation is assumedto happen anytime between these two consecutiveframes (after the frame without bright light and be-fore the frames with the intense light) which can beseen in the first two frames of Figs. 3-7. Videos aretaken at a frame rate of 30 and 60 FPS, so we con-sider an average error on t of 17 ms .The error on the term log t is given by δlog t = 1log n δtt = 0 . δtt . (8)Taking the combined effect of these errors will propa-gate a very large error on the value of the energy E ifthis will be computed only using equation 1 using fewmeasurements of R and t . Our 38 measurements andthe linear fitting will reduce the effect of these errorsand will limit the error to only the one on the fittingparameters, namely the slope a and the intercept b ofequation 2. Charles J.Aouad et al. Fig. 4
For each video the incident angle of sight towards the silos long facade is shown along with the actual picture of thesilos and the de-projection lines. Drawings provided by the Arab center for architecture 2020. For calibration of video 5 referto section 7.1 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 log t [seconds] / l og R [ m e t e r s ] Fire ball radius evolution with time
V1V2V3V4V5V61.031t+6.044
Fig. 5 log R on the Y -axis and log t on the X -axis.The dots represent the fireball radius taken from differentvideos. The best linear fit has the form Y = ax + b where a =1 . . . and b =6 . . . . A clear sharply defined fireball was visible in the first100 ms after the detonation. The boundaries of suchan expanding surface became less sharply pronouncedafter about ∼ ms post-detonation. Then, its bound-ary rapidly faded until it was not clearly defined. Thisis partly due to the rapid cooling caused by the rapiddecrease of over-pressures; partly to the contamination t [seconds] R [ m e t e r s ] Fire ball radius evolution with time
140 [1-exp(-14 t)]
Fig. 6
The red dots are the fireball measurements used inyielding the energy of the explosion, the pink dots are the fire-ball measurements taken at the stopping distance; i.e, whenthe luminous fireball has reached its final radius ∼ m . Apower law of exponent 0.4 which is represented via the blackline is consistent with the theoretical model of Taylor 1950aand represents the shock wave; the bold dashed curve repre-sents a drag theoretical model describing the evolution of thefireball. The two curves depart at ∼ . s , an epoch beyondwhich the fireball does not trace the shock wave anymore. of the atmosphere with the existing dark smokes causedby the first fire and also to the turbulent instabilitiescreated at the boundary of the hot sphere with the sur-rounding atmosphere. Nevertheless, the boundaries ofthis sphere can be reasonably traced, upon examining eirut explosion 7 the extracted frames of all videos, as it expands at thisepoch all the way until the expansion slows down andseems to halt.Right after this, the fireball and the shock wave de-part and the former does not trace the latter any longer( We refer the reader to section 4 for a complete discus-sion ). At this stage, when the temperatures drop, theshock wave causes the formation of the vapour cloud,a.k.a Wilson cloud, clearly spotted in the videos rapidlygrowing, leading a massive damage front ahead of itsboundary. The formation of this cloud is caused by thenegative phase following the passage of the sudden in-crease in pressure, and the adiabatic cooling causingthe atmospheric water vapour to condensate and cre-ate the white appearance. It is important to mentionthat at these epochs the boundary of the shock wave isnot traced by the boundary of the Wilson cloud due tothe delay between the positive pressure and the nega-tive pressure phase; condensation will only happen dur-ing the negative pressure phase. Therefore, using thevapour cloud to trace the shock wave cannot lead tosignificant results.Having 38 measurements of R t exp and t exp , we fit thevalues log R on the Y -axis and the value log t onthe X -axis. The data is consistent with the theoreticalprediction, this fact is remarkable knowing the cumula-tive error margin due to the quality of the data and thelimitations of our procedure. Our best fit to the datais a line of constant gradient of the form log R = a log t + b where a = 1 . . . and b = 6 . . . .Taking K = 0 .
856 as given by Taylor 1950a for a di-atomic gas where γ = 1 . ρ = 1 . kg.m − and byusing Eq. 3, we find the value of E to be 1 . × Joules with a lower bound of 9 . × and an upperbound of ∼ . × or the equivalent of ∼ tons of T.N.T.( here we use a conversion factor be-tween Joules and T.N.T equivalent of 4 . × JoulesT.N.T .This energy yield is a lower estimate. It does not con-sider the amount that has been radiated as heat in theform of electromagnetic radiation, which is consideredinsignificant in a chemical explosion (Bethe et al. 1958),contrary to a nuclear explosion. Moreover, part of thisenergy has been used in demolishing the steel structurehangar in which the explosives where kept, which is alsoassumed to be insignificant.
It can be seen in Fig. 6 that the radius of the shockwave expansion decelerates until it stabilizes at somedistance R m ≈ − m after about ∼ ms . Asmatter of fact, the shock wave at this epoch detachesfrom the fireball and moves ahead of it. This has been observed before (see Taylor 1950b; Mack 1946; Szasz1984; Gordon, Gross, and Perram 2013). At this epochthe shock wave is driven by the remaining energy thathas not been radiated as heat or consumed during theexpansion of the heated air. The epoch during whichthe shock wave can be traced by the fireball is thuslimited.We use a drag model (Gilev and Anisichkin 2006;Gordon, Gross, and Perram 2013) to predict the kine-matics of the fireball. This is given by R f ( t ) = R m [1 − exp ( − κt )] . (9)Here R f ( t ) is the radius of the fireball, R m is the stop-ping distance, a distance at which the fireball expansionbecomes asymptotic, and κ is the drag coefficient. Wefix R m at 140 meters as per our observations and findthe best fit with a value κ = 14.We plot the 2 curves in fig 6. One can notice thatthe drag model curve follows the observed points withinthe range of error. The two curves significantly departat about ∼ − ms , a point after which the as-sumption of the fireball tracing the shock wave doesnot hold any further for the evaluation of the energy.The follow-up of the shock wave beyond this epoch isout of the scope of the current work, however it is worthnoting that such a follow-up is stringently limited andtracing it with these videos is challenging for the fol-lowing reasons: – Tracing the shock wave using the vapour cloud is notconsistent because of the delay in the formation ofthe vapour after the passage of the positive phase ofthe over-pressure. This time delay makes the actuallocation of the pressure front ahead of the vapourcloud. Additionally, pixel scaling set with referenceto the silos building holds only in the vicinity of theexplosion center. However, this cannot be accuratefor extended distances where the dimensional pixelscaling has to be remeasured. – Tracing the shock wave using time of arrival of theshock from frame by frame analysis of the videosmay not be accurate. In fact, the interaction of theshock wave with the dense urban structures causescomplex interactions and diffraction patterns, mak-ing the assumption of an isotropical spherical flowof the shock wave inaccurate (Smith and Rose 2006;Bazhenova, Gvozdeva, and Nettleton 1984; Shi, Hao,and Li 2007). The interaction of the shock wave withthe existing grains silos, for example, has created avisible dark spot within the vapour cloud, signalingthe delay in the propagation of the pressure front
Charles J.Aouad et al. due to the interaction. This can be clearly seen infig 8 in the last panel.Another important point to mention is that the the-oretical work of Taylor 1950a and later his experimen-tal observations (Taylor 1950b) were performed on anuclear explosion. The major assumption is that in anuclear explosion all the energy is released from a sin-gle point in space. This is not the case for a chemicalexplosion.On the contrary, a chemical explosion is more efficientin producing a blast wave than a nuclear explosion ofthe same energy. In the latter, only about half the en-ergy is available to do mechanical work in the form ofa blast wave, while a significant amount ( ∼ R t and t derived by Tay-lor can still be adapted to a blast due to a chemicalexplosion within a range of comparison. This range isprovided in Taylor 1950a, (see page 170-173- Fig. 5)using a comparison between theory and observationaldata taken from experiments. This is given as a functionof the scaled distance log (cid:104) RE − (cid:105) . Here E is the en-ergy yielded from the explosion given in ergs , and R isthe distance in cm . Taylor proposes that the window ofcomparison lies between values of log (cid:104) RE − (cid:105) = − . − . m ) in addition to a total explosive weight of 310 tonsT.N.T as yielded from the current analysis. We find val-ues of log (cid:104) RE − (cid:105) = − . − .
37 for distances be-tween 60 and 100 m . This falls comfortably within thecomparison range. However, we find values of − . ∼ m which lies at the edge of the comparisonrange. It is worth stressing that despite this discrep-ancy with the comparable range, as proposed by Taylor,our data show a remarkable consistency; points taken atdistance of 140 meters still follow the trend as expectedfrom theory. Note that, as previously stated, this dataare severely limited (taken from 6 different videos at 6different locations using different approximate scalingand limited by the resolution). In fact, this indicatesthat the comparison may indeed be extended to largerranges.It is worthwhile to mention that the energy yieldcomputed in this paper falls within the same rangeof values derived by Temsah, Jahami and Aouad 2020 . They model the damage of the silos caused by theblast using a 3D finite element nonlinear model, takinginto consideration the actual reinforcement and mate-rial properties as yielded from lab tests taken from thesite. The silos building consists of three different layerseach composed of 16 concrete cylinders. The first twolayers have been completely destroyed while in the thirdlayer only 2 cylinders got destroyed. In addition, the re-maining silos experienced a permanent lateral displace-ment of 30cm maximum (as surveyed using 3D laserscanning (AMANN 2020)). This provides a way to es-timate the energy yields by running simulations withdifferent energies in order to match the actual damage.They found that, for values of 300 tons T.N.T, thewhole 3 layers are destroyed while a value of 100 tonsT.N.T only the first layer is destroyed. Finally, a yield of200 tons T.N.T reproduces exactly the actual damage(including the 30 cm displacement of the third row).This falls within the same order of magnitude of ourrange, and thereby provides an additional sanity checkon the consistency of our methodology and the resultsreached in the current study. Fig. 7 video 3, taken from a distance of 530 meters from theexplosion. 6 frames separated by 33.33 ms . The fire ball fitsare shown in red circles. Work in preparation, personal communication, Septem-ber 2020.eirut explosion 9
The fireball evolution created by the detonation in Beirutport on the 4th of August 2020 is traced using publiclyavailable videos. The footage are used to extract framesseparated by 16 . − .
33 milliseconds. Pixel calibra-tion is done using the existing silos building and defin-ing accurate line of sight incident angles. The methoddescribed by Taylor 1950a; Taylor 1950b, which relieson a quantitative relation between the fireball radiusand the time from detonation, is used to yield the en-ergy produced by the explosion. Here we draw our mainconclusions :1. The fireball evolution follows the theoretical predic-tion in the sense that its radius R ∝ t . Here R isthe radius of the luminous fireball, and t is the timefrom detonation.2. The fireball expansion reaches an asymptotic limitat about ∼
140 meters; a distance beyond whichthe fireball expansion is brought to a halt. We ar-gue that at this epoch the shock wave departs fromthe fireball and becomes ahead of it. This is con-firmed using a theoretical drag model of the fireballexpansion.3. We’ve found a total energy yield of E = 1 . × Joules– with a lower bound of 9 . × J and anupper bound of ∼ . × J (or the equivalent of ∼ tons of T.N.T). If the detonation of fuelrich ammonium nitrate (ANFO) is the cause behindthis explosion, then the quantity that actually det-onated is 3 times less than the quantity claimed bythe official records should a T.N.T conversion factorof 38% be used.A rapidly expanding vapour cloud has been clearlyseen; it is caused by the negative phase after thepassage of the pressure wave. Future work shall in-clude observational investigations of its kinematicsand possible links to theoretical models. The authors would like to express their thoughts to thevictims of this tragic event and wishes for the recoveryof the wounded. We primarily thank the eyewitnesseswho posted their videos on social media, for withoutthem this article would not have been possible. We arealso thankful to, Alexandra Elkhatib, Philip James andIgor Chilingarian for useful discussions, and RanjeshValavil and Abbas Chamseddine for helps.
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Proceedings of the Royal Society of London. SeriesA. Mathematical and Physical Sciences
In preparation . [throughprivate communication; September 2020]. 2020. λ (in which a pixelcorresponds to an angular measurement) is given by: λ [ ◦ px − ] = 2 β px = tan − . m m px . (10)Here 4.5 m is assumed to be a common length for a carand 132 m and 26 px are the distance to the car andthe pixel size of the car respectively.The radius of the fireball in meters R fm is thus givenby R fm = 630 m × tan φ = 630 m × tan( R fpx λ ) . (11)Here, 630 meters is the distance from the camera tothe explosion, and R fpx is the radius of the fireball inpixels. This is illustrated in Fig. 8.7.2 Frames from videos 1-4-6Frames extracted from different videos located on themap in fig 2. For frame rates refer to table 1. Each eirut explosion 11 Fig. 8 scale calibration for video 5: on the right panel thelength of a passing car is measured in pixels. By determin-ing the distance from the general map and assuming a totallength of the car of 4.5 meters we calculate the angular fieldof view per pixel λ . Using the value of λ and knowing the dis-tance to the explosion site, we can measure the angle φ andconvert the pixel measurements of the fire ball to meters. frame shows the fireball along with the circle fit to de-termine the physical length. The detonation is assumedto have happened anytime between the first and thesecond frame. For pixel calibration method see section2. Fig. 9
Video 1-distance 1146 meters. The dark spot seenwithin the vapour cloud in the last frame is a trace of thediffraction/interaction of the pressure wave with the existingsilos building.
Fig. 10
Video4-distance 666 m