Patadiya Dharmeshkumar Makanlal. Aeroacoustics and Plasma Dynamics Lab, Department of Aerospace Engineering, Indian Institute of Science, Bangalore

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1 Study of and Induced Primary Fragmentation and of Coal Particles Subjected to Plasma Initiated Detonation using Weibull Theory Patadiya Dharmeshkumar Makanlal Aeroacoustics and Plasma Dynamics Lab, Department of Aerospace Engineering, Indian Institute of Science, Bangalore December, 24

2 2 3 Overview

3 Detonation combustion of coal holds promise of higher system and combustion efficiencies. Primary fragmentation of coal in the combustion process is known to substantially impact efficiency. Coal is likely to remain an important energy source and hence improvements in coal combustion technology will have major impact. Coal combustion is composed of four major stages heating Devolatilization Defragmentation due to thermal and/ volatilization Burning The work presented here reports analytical and numerical on thermal and volatilization induced fragmentation of coal when subjected to a detonation wave.

4 Present Work A detonation wave is generated using a plasma cartridge at one end of a detonation tube. Response of coal particle to detonation waves of Mach numbers 3 to 8 are studied. and volatilization induced primary fragmentation under detonation condition is aim of this work. induced due to combined thermal and volatilization effects are calculated. Convective and radiative boundary condition is used. code was written and the solution obtained is compared with analytical solution. Particle is assumed fractured when maximum principal stress exceeds ultimate strength.

5 Research overview I started with developing HTM of simple constant temperature boundary condition using analytical methods. I predicted particle fracture time and location due to thermal stress only using failure criteria suggested by various failure theories. Applied this model to solve more realistic boundary conditions; convective and radiative boundary conditions. After successfully developing HTM analytically I moved on to develop numerical model for solution of more complex problem. I developed CFD code to solve convective and radiative boundary condition and compared it with analytical solutions. It matched with minimum error. After developing HTM numerically I developed VM of volatile matter.

6 Research overview I linked HTM and VM with SMM and obtained principal stresses developed in coal particle. Here it ends the linking of all models and gives DSM. After successfully developing DSM I developed FCM using Weibull s WLT. This model is capable of giving fragmentation time, fragmentation location, temperature at the time of fragmentation, volatilization matter present at fracture, flow of volatile at fracture, pressure at fracture. This ends complete PFM which includes HTM, VM, SMM, DSM and FCM. After developing PFM for single coal particle I developed SM which is capable of giving me average, Standard Deviation, PDF, CDF of the mixture of different size coal particles.

7 for Simple Model Unsteady heat conduction equation with spherical symmetry is. ( ) 2 T (r, t) T (r, t) r 2 kr = ρc p () r r t And the boundary conditions are: T (r o, t) = T (2) where T is the skin temperature induced by the detonation wave. And T r = (3) r= The initial condition is: T (r, ) = T i (4)

8 Analytical Solution Solution can be obtained by two different analytical techniques Solution from Seperation of Variable technique τ(r, θ) = T (r, t) T T i T = 2( ) m sin(λ m R)e λ2mθ (5) λ m R m= where sin λ m = is the eigen condition and λ m = mπ (m =, 2,...)are the eigenvalues. And solution from Laplace Transform technique τ(r, θ) = T (r, t) T T i T = 2( ) m sin(λ m R)e λ2 m θ (6) λ m R m= Where eigen condition is sinh( z)=. The eigen values are the poles of the function f (z), which are, z = λ 2 m = m 2 π 2

9 Dharmeshkumar, profile in Kelvin Time in Seconds Dimensionless radius R Time in Seconds (a) M = 3 (b) M = 5 in Kelvin Time in Seconds in Kelvin Dimensionless radius R Dimensionless radius R (c) M = 7 Figure: Coal particle temperature distribution

10 Principal Stress Theory This theory was suggested by Rankine. The theory is based on calculating maximum principal stresses induced in the coal particle and comparing it with ultimate strength. σ n,2 = σ (σr ) r 2 2 ± + σ 2 2 t σ uu N (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal stress distribution at detonation Mach no. 3

11 Principal Stress Theory (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal stress distribution at detonation Mach no. 5

12 Principal Stress Theory (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal stress distribution at detonation Mach no. 7

13 Principal Theory This theory was developed by Saint Venant. According to this theory, the failure or yielding occurs at a point in the member when the maximum principal strain in the bi-axial stress system reaches the limiting value of strain. σ n ν σ n2 = σ uu N (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal stress distribution at the detonation Mach no. 3

14 Developed Stress (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal stress distribution at the detonation Mach no. 5

15 Developed Stress (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal stress distribution at the detonation Mach no. 7

16 Principal strain is given by ɛ max = σ n E νσ n2 E (a) 5 µ (b) µ (c) 5 µ Figure: strain distribution at the detonation Mach no. 3

17 Principal (a) 5 µ (b) µ (c) 5 µ Figure: strain distribution at the detonation Mach no. 5

18 Principal (a) 5 µ (b) µ (c) 5 µ Figure: strain distribution at the detonation Mach no. 7

19 Shear Stress Theory This theory was suggested by Guest and Tresca. The theory is based on calculating maximum shear stresses induced in the coal particle and comparing it with ultimate strength. τ max = (σr 2 ) 2 + σ 2 t = σ uu 2 N (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal shear stress distribution at detonation Mach no. 3

20 Shear Stress Theory (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal shear stress distribution at detonation Mach no. 5

21 Shear Stress Theory (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal shear stress distribution at detonation Mach no. 7

22 Distortion Energy Theory This theory was suggested by Hencky and Von Mises. According to this theory, the failure or yielding occurs at a point in a member when the distortion strain energy per unit volume, also called as shear strain energy, in bi-axial stress system reaches the limiting distortion energy per unit volume. σ 2 n + σ2 n2 σ nσ n2 = σ uu N (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal stress distribution at detonation Mach no. 3

23 Distortion Energy Theory (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal stress distribution at detonation Mach no. 5

24 Distortion Energy Theory (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal stress distribution at detonation Mach no. 7

25 Energy Theory This theory was developed by Haigh. According to the theory, the failure or yielding occurs at a point in member when the strain energy per unit volume in bi-axial stress system reaches the limiting strain energy per unit volume. σ 2 n + σ2 n2 2νσ nσ n2 σ uu N (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal stress distribution at detonation Mach no. 3

26 Energy Theory (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal stress distribution at detonation Mach no. 5

27 Energy Theory (a) 5 µ (b) µ (c) 5 µ Figure: Developed thermal stress distribution at detonation Mach no. 7

28 on failure theory (a) Rankine s Theory (b) Saint Venant s Theory (d) Henckey s Theory (e) Haigh s Theory (c) Guest s Theory Figure: Comparision of failure theories for 5 µm size coal particle at M = 3

29 Observations Coal particles subjected to a detonation wave experience highly stressed and strained inner and outer regions. Three different regimes emerge in coal particle based on the different particle sizes since the coal particle is subjected to temperature shock. The largest particles observed exploded into smaller fragments as break up develops throughout the coal particle. The medium particles observed fragmented in the outer region and left over surviving fraction of same particles are then fragment in the interior. The smallest particles observed being fragmented in the interior. As the Mack number increases the entire process rapidly speeds up. This observations suggest that coal particle under the effect of detonation wave is highly stressed and strained. This suggests that detonation combustion of coal is quali-

30 Table: Time and location where developed thermal stress and strain reaches peak value Mach No. Size µm Exposure time 3 secs Location All the time Throughout and.85. and.95 7 All the time Throughout 3 2 and and. 5 5 All the time Throughout 7 All the time Throughout

31 : HTM and SMM heat transfer equation is ( ) 2 T (r, t) T (r, t) r 2 kr = ρc p r r t Radial and Tangential stresses are given by σ r = 2βE [ r ( ν) r 3 Tr 2 dr r ] r 3 Tr 2 dr σ t = βe [ 2 r ( ν) r 3 Tr 3 dr + r ] r 3 Tr 2 dr T

32 for : VM and SMM governing volatilization are ( ) V t = k Ea exp RT Where Total radial stress is given by r (Nr 2 ) = r 2 ρ c V M vol t r 2 porepɛ N = 8µτ t R g T (V V ) n p r σ r,tot = p p s + σ r

33 Conditions behind Detonation Rankine Hugoniot relations used to calculate conditions behind shock wave T 2 = [2γM2 (γ )][(γ )M 2 + 2] T (γ + ) 2 M 2 p 2 = 2γM2 (γ ) p (γ + )

34 Boundary and Initial Conditions Boundary conditions are: k T r = h(t T ) + σ b ɛ b (T 4 T ) 4 r=r And initial conditions are: T r = r= p r=r = p s T (r, ) = T init p t= = p s = p init

35 Principal : DSM Principal stresses are the eigenvalues of the stress tensor τ rr τ rθ τ rφ σ = τ θr τ θθ τ θφ τ φr τ φθ τ φφ

36 Weibull s Weakest Link Fracture Criteria: FCM Sequence of events or objects depend on the support of the whole. The whole is only as reliable as the weakest member or link. The basic assumption for the model is that all materials contain inhomogeneities which are distributed at random. When the defects become the fracture origin, it is found that failure is triggered by the largest defect present or, in other words, weakest element present. It is possible to indicate a definite probability of the rupture occurring at a given stress or at given time. Three parameter Weibull s failure probability is given by P f = e ( σ σu σo And fracture criteria is set to ) mdv R f = R i for P f,i = max i P f,i and P n f, Pn f,i P f,b

37 in Kelvin Dimensionless Radius Profile.5 Time x 3 3

38 Comparison Between Analytical and Solutions in K M=3, Without Radiation,time=.38 ms M=3, With Radiation,time=.38 ms M=3,Analytical Without Radiation,time=.38 ms Dimensionless Radius

39 Radial Stress Radial Stress in N/m Dimensionless Radius Radial Stress Profile.5 Time x 3

40 Tangential Stress Tangential Stress in N/m Dimensionless Radius Tangential Stress Profile.5 Time x 3

41 Principal Principal Stress (σ ) in N/m x 3.5 Time Principal Stress Profile (a) σ.6.8 Dimensionless Radius Minimum Principal Stress (σ 3 ) in N/m Intermediate Principal Stress (σ 2 ) in N/m Minimum Principal Stress Profile.6 Dimensionless Radius.8 (c) σ 3.5 Dimensionless Radius.5.5 Time Intermediate Principal Stress Profile.5 (b) σ 2 Figure: Principal stresses at M = x Time x 3

42 Plots Dharmeshkumar, Dimensionless Radius 3.5 x Dimensionless Radius Time (a) Volatile Generation x Time (b) Flow of Volatile Pressure Profile 5 x 2 5 x Volatile Matter Pressure in N/m2 Molar Flow 5 x 4 x Molar Flow of Volatile in kmol/m2 sec Volatile Matter Generated in kg/kg of coal Volatile Matter Generated Dimensionless Radius. 3.5 x Time (c) Volatile Pressure Figure: Volatile matter plots at M = 4

43 Effect of Radial Stress Radial stress comparision with and without volatilizatoin at constant CTE Without With Dimensionless Radius Figure: Comparision of σ r with and without volatilization at M=8

44 Effect of Local Failure P f Local failure probability P f x Dimensionless Radius.4 Time Figure: Local failure probability at M = 7

45 Effect of Overall Failure P f,o vmax=.3 vmax= Fracture Time in seconds x 3 Figure: Comparison of failure probability for different initial volatile matter content at M = 4

46 Fracture Data Mach No Detonation Fracture Time Fracture Fracture Location e e e e e e

47 solution obtained is compared with analytical solution and it matches with minimum error. There is no temperature difference of the numerical solution with and without radiation and hence can be concluded that effect of radiation is negligible. does not have significant effect on radial stress for the particle size studied here. Particle takes longer time to fragment in the interior when subjected to lower Mach number (M=3). It takes shorter time for particle to fragment in the surface region when subjected to higher Mach number (M > 3).

48 Fragmentation Table: Typical particle size distribution Particle size in µm Percentage to % 5 to % 9 to % 2 to % > % 4 m long detonation tube constructed Travel time for M = 3 is 383 µs M = 8 is µs. Desirable if particle fractures before detonation travel time. Travel Time = 4 3 γrt = 4 3 =.3 83 s

49 Statistical equations PDF for normal distribution is given by and CDF is f (x) = σ (x µ) 2 2π e 2σ 2 φ(x) = σ e (x µ)2 2σ 2 2π dx

50 Time statistics Table: Statistical data derived from fracture time for mixture of coal particle subjected to detonation waves of Mach nu=6 Radius Fracture Time Avg. Time Variance PDF CDF 5 5.7e e-5.993e e e e-5.993e e e e-5.993e e e e-5.993e e e e-5.993e e e e-5.993e e e e-5.993e e e e-5.993e e e e-5.993e e e e-5.993e e e e-5.993e e e e-5.993e e

51 PDF and CDF for time Density Function PDF plot for fracture time Fracture Time in seconds x 4 CDF plot for fracture time Fracture Time in seconds x 4 (a) PDF (b) CDF Figure: PDF and CDF plots for fracture time Cumulative Density Function

52 Volatile matter statistics Table: Volatile matter generation statistics size Volatile at Fracture Average Volatile Volatile Variance PDF CDF e e e e e e e e e e e e e e e- 7.57e e e e e e e e e e e e e e e e- 8.4e e e e e e e e e e e e- 3.38e e e e e

53 PDF and CDF for volatile matter generation Density Function x PDF Plot for Volatile Matter Generation at Fracture Volatile Matter Generated at Fracture in kg/kg x of coal (a) PDF Cumulative Density Function CDF plot for Volatile Matter Generation at Fracture Volatile Matter Generated at Fracture in kg/kg x of coal (b) CDF Figure: PDF and CDF plots of volatile matter generated at fracture for M = 6

54 atile statistics Table: Volatile matter flow statistics size Flow of Volatile at Fracture Average Flow Flow Variance PDF CDF e e-6.449e e e e-6.449e e e e-6.449e e e e-6.449e e e e-6.449e e e e-6.449e e e e-6.449e e e e-6.449e e e e-6.449e e e e-6.449e e e e-6.449e e e e-6.449e e

55 PDF and CDF for flow of volatile generation Density Function 4 x PDF Plot for Volatile Matter Flow at Fracture Flow of Volatile Matter at Fracture in kmol/m x s 5 (a) PDF Cumulative Density Function CDF plot for Volatile Matter Flow at Fracture Flow of Volatile Matter at Fracture in kmol/m x 2 s 5 (b) CDF Figure: PDF and CDF plots of volatile matter flow at fracture for M = 6

56 based on this thesis Journals Patadiya D. M., Jaisankar S., Sheshadri T. S., Detonation Initiated Disintegration of Coal Particle Due to Energy Theory, Journal of Coal Science and Engineering pp , Dec Jaisankar S., Patadiya D. M., Sheshadri T. S. Shock Wave Induced Fragmentation of Coal Particles Under review in Fuel. 3 Patadiya D. M., Jaisankar S., Sheshadri T. S., Application of Weibull theory in studying fragmentation statistics of coal particle mixture when subjected to detonation wave Under preparation. Conferences and Symposia Patadiya D. M., Jaisankar S., Sheshadri T. S., Application of Principal Theory for Study of Coal Particle Disintegration when Subjected to Detonation Wave, ICCS&T 23, pp , Oct Patadiya D. M., Jaisankar S., Sheshadri T. S., Computational Model for and Induced Spontaneous Fragmentation of Coal Particle Proc. of the Second Intl. Conference on Advances in Mechanical and Robotics Engineering - AMRE 24, pp , Oct Patadiya D. M., Jaisankar S., Sheshadri T. S., Model for and Induced Spontaneous Fragmentation of Coal Particle Under review in Frontier of Computational Physics 25

57 Thank You!

Application of Maximum Principal Strain Theory for Study of Coal Particle Disintegration when Subjected to Detonation Wave

Application of Maximum Principal Strain Theory for Study of Coal Particle Disintegration when Subjected to Detonation Wave Patadiya D. M. 1,1, Sheshadri T. S. 1,, Jaishankar S 1,3. 1,1 Department of Aerospace