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 Engineering, Indian Institute of Science, Bangalore. India. dmp_mech@yahoo.com. 1, Department of Aerospace Engineering, Indian Institute of Science, Bangalore. India. tss@aero.iisc.ernet.in 1,3 Department of Aerospace Engineering, Indian Institute of Science, Bangalore. India. jshankar@aero.iisc.ernet.in Abstract Fragmentation behavior of coal particle subjected to detonation wave generated by plasma was studied theoretically. Temperature of the shocked gas was calculated by ankine-hugoniot condition. Detonation wave is initiated by plasma cartridge at the one end of a detonation tube. Coal particles were subjected to a temperature shock whose magnitude depends on the Mach number of the detonation wave. Coal particle is assumed suspended. Heat transfer process in the coal particle was carried out mathematically using non dimensional formulation. The temperature shock was found to generate thermal stresses and strains that fragmented the coal particle. Stresses and strains were calculated at various times and radii. A thermal stress model is used to determine boundaries for coal fragmentation. Failure occurs when the thermal stress developed due to failure theory exceeds the ultimate strength of coal. Maximum Principal Strain theory was used to generate plots to predict time, temperature and location of fracture. A model for coal particle fragmentation when subjected to a detonation wave is proposed. Keywords Coal Particle, Detonation Wave, Spontaneous Fragmentation
Nomenclature r adius of coal particle T Temperature of coal particle in Kelvin k Thermal conductivity in W m -1 K -1 t Time in seconds Dimensionless radius T i Initial temperature T Surrounding temperature a Outer radius of coal particle C p Specific heat of coal J kg -1 K -1 θ Dimensionless time α Thermal Diffusivity in m s -1 τ Dimensionless Temperature ξ Function of only μ Function of θ only. λ Separation constant x Arbitrary variable ν Poisson s ratio E Young s modulus in N m - J, Y Bessel s functions s Laplace transform variable σ Stress in Pa M Mach number p Order of pole u arbitrary function β Coefficient of thermal expansion in K -1 Subscripts r radial t Tangential n Normal n1 Maximum principal stress n Minimum principal stress m oot number 1 Introduction Complete coal combustion composed of three major stages: thermal heating, devolatilization, defragmentation and burning. Defragmentation can be of two types: (i) defragmentation resultant of internal pressure developed due to devolatilization, (ii) defragmentation resultant of stresses developed due to thermal heating. The combustion of clouds of coal particle in shock-heated mixtures of oxygen and nitrogen was studied by Nettleton et al. [1]. The shock tube method was described to combust the coal particles. It was found that coal particles burnt in less than ms. Mathematical modeling of heat treatment and combustion process carried out by Enkhzhargal et al.[]. The thermal stresses developed in coal particles were studied using asymptotic method. The induced stresses were compared with ultimate tensile strength. It was concluded that particles of size 10 mm and large collapse at medium temperature. Mechanical submodel based on Weibull theory in fluidized bed combustion was proposed by Chirone et al. [3]. Three submodels namely: thermal submodel, volatile transport submodel, and volatile
release submodel were presented. Experiments were conducted on large coal particles. It was concluded that coal particle break in hemispherical shape and parallel to its bedding plane. Stresses induced due to devolatilization were studied by No et al. [4]. Fragmentation behaviour of coal particle due to Knudsen Pressure wave in non slagging cyclone combustor was studied in this work. A model was proposed in which central region was loosely fragmented. The fragmentation behaviour of large coal particle in drop tube furnace was studied by P. Dacombe et al. [5]. Various parameters like temperature of furnace, carbon content, moisture content, volatile matter etc influencing the fragmentation process were reported. The work presented here reports analytical and numerical studies on thermal fragmentation of coal when subjected to a detonation wave in a detonation combustor. Direct initiation of detonation is brought about by an electric plasma cartridge at one end of a detonation tube. Particle is assumed spherical shape. The governing differential equations and boundary conditions are nondimensionalized and solved by two different techniques. Analytical solution of the heat transfer process in the coal particle has been obtained. Thermal stresses and strains induced were calculated. Fragmentation behavior is predicted by maximum principal strain theory. Three dimensional plots are presented showing time and location for probable fragmentation of coal particle. A thermal stress model for coal particle fragmentation when subjected to a detonation wave is proposed. Governing Equations and Boundary Conditions Unsteady heat conduction equation with spherical symmetry is [6]. 1 T ( r, t) T ( r, t) kr C p r r r t (1) and the boundary conditions are: T ( a, t) T () where T is the skin temperature induced by the detonation wave. And T r0 0 r (3) The initial condition is: T(r,0)=T i (4) Assuming constant thermal conductivity Eq. 1 simplifies to T T 1 T r r r t (5) Introducing dimensionless variables T ( r, t) T r t (, ),, Ti T a a (6) Eq. 5 becomes (, ) (, ) (, ) (7) with boundary and initial condition and, (1, ) 0 (8) 0 0 (9)
(, 0) 1 (10) 3 Analytical Solutions 3.1 Solution by Separation of Variable Setting (, ) ( ) ( ) (11) where ξ is function of only and μ is function of only. Eq. 7 becomes ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) (1) Dividing by ξ μ 1 ( ) ( ) 1 ( ) (13) From which we obtain 0 (14) And ( ) C1e (15) The solution of Eq. 14 is 1 ( ) CJ1/( ) C3Y1/ ( ) (16) Substituting Eq. 15 and Eq. 16 in Eq. 11 and using Bessel s identity [7] [8] gives e (, ) C sin( ) C 3 cos( ) (17) Applying boundary condition of Eq. 9 gives e C sin( ) C 3 cos( ) 0 (18) After simplifying and substituting =0, we get C 3 =0. This gives the solution C (, ) sin( ) e (19) Applying the boundary condition of Eq. 8 gives C (1, ) sin( ) e 0 (0) C 0 sin( ) e 1 (1) Either C = 0 or sin 0. C = 0 is a trivial solution. Therefore, we have sin 0 o,,,, n resulting in C (, ) sin( n) e n1 () Applying initial condition of Eq. 10
Therefore (,0) sin( ) e 1 n1 n1 C C sin( n ) 1 Multiplying both the sides by sin( ) (where m m ) and integrating, m 1 1 sin( ) d C sin( )sin( ) d m n m 0 0 n1 Using orthogonality concept,. H. S. = 0 if m n and. H. S. 0 if m = n 1 sin( 0 m) d C m 1 sin ( ) d 0 n1 After simplification, cos( m ) ( 1) m C m m m Where, m = 1,,.m. Thus the final solution is m T ( r, t) T ( 1) m (, ) sin( m ) e T T i Where sin m 0 is the eigen condition and m m (m = 1,,.) are the eigen values. 3. Solution by Laplace Transform Taking Laplace Transform of Eqs. 7 to 10, (, ) (, ) (, ) And the boundary and initial conditions becomes (1, ) 0 (1, S) 0 And m m1 d (, s) 0 0 0 0 d (, 0) 1 (, 0) 1 From Eq. 9 d d s (, s) (,0) d d This is Bessel s differential equation and its solution after simplification is C sinh( ) 1 s (, s) s Applying boundary conditions of Eq. 30 m (3) (4) (5) (6) (7) (8) (9) (30) (31) (3) (33) (34)
C sinh( ) 1 s (1, s) 0 s (35) Therefore 1 C s sinh( s ) (36) The complete transformed solution is sinh( s) 1 (, s) ssinh s s (37) The final solution as a function of and can be found by taking inverse Laplace Transform of Eq. 37. sinh( s) 1 (, ) 1 (, s) 1 1 s sinh s s (38) The inverse of the above equation can be found by Bromwich s Contour Integral method [9]. The method can be given as If, f ( t) F( s) (39) 0 Then 1 tz f ( t) F( z) e dz Sum of esidues i f ( z) (40) And esidues are given by, p1 1 d [ f ( z); z j ] lim ( z z ) ( ) p 1 j f z ( p 1)! zz j dz (41) Using the Bromwich's Contour Integral method and simplifying, the inverse of the Eq. 40 will give the final solution as below, m T ( r, t) T ( 1) m (, ) sin( m ) e Ti T m1 m (4) 4 Calculation of Thermal Stresses developed It is necessary to calculate thermal stresses to understand fracture pattern of coal particle. 4.1 adial Stresses adial stress [9] is given by E 1 a 1 r r Tr dr Tr dr 3 0 3 (1 ) a r 0 After substituting the value of temperature T from Eq. 4 and simplifying, m m 4 E( Ti T ) ( 1) e cos m 1 cos( m) sin( m) r 3 (1 ) m1 m m m m (43) (44) 4. Tangential Stresses Tangential stress [10] is given by
E a 1 r t Tr dr Tr dr T 3 0 3 (1 ) a r 0 (45) After substituting the value of temperature T from Eq. 4 and simplifying, m m E( Ti T ) ( 1) e cos m sin( m ) 1 cos( m) sin( m) t 3 (1 ) m1 m m m m (46) 5. Calculation and Fragmentation Analysis Numerical values of analytical solution were obtained by writing C language code. The 3D graphs were plotted for temperature, developed resultant stress and maximum strain against and time. Typical values for various parameters of Bituminous coal were taken from reference [] [4]. The initial coal temperature was taken as T i = 300 K. Three typical size of coal particles were taken as 50 μm, 100 μm, and 150 μm. The Mach numbers of shock waves were taken as 3, 5, and 7. For simplicity, spherically symmetric coal particle is considered. Hence stress components consist of radial stress component σ r and tangential stress component σ t. Stresses and strains developed due to Maximum Principal Strain theory were calculated. Ultimate strength was considered as limiting criteria of fracture for the theory. In the Figs. from 4 to 6, solid lines are for 50 μm, dashed lines for 100 μm, and dotted lines for 150 μm size coal particles. The state of stresses can be expressed as shown in Fig. 1. The ultimate strength plane is shown as black colored horizontal plane in Figs. from 5 to 7. Figure 1. Stress state Figure. Coal particle temperature distributions at detonation Mach no. 3 Figure 3. Coal particle temperature distributions at detonation Mach no. 5
Figure 4. Coal particle temperature distributions at detonation Mach no. 7 5.1 Maximum Principal Strain Theory This theory was developed by Saint Venant [11]. 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. Mathematically, it is expressed as, n1 n u (47) (a) 50 μm (b) 100 μm (c) 150 μm Figure 5. Developed thermal stress distribution at detonation Mach no. 3 (a) 50 μm (b) 100 μm (c) 150 μm Figure 6. Developed thermal stress distribution at detonation Mach no. 5
(a) 50 μm (b) 100 μm (c) 150 μm Figure 7. Developed thermal stress distribution at detonation Mach no. 7 esultant stresses as given by Eq. 47 were compared with ultimate strength as shown in Figs. from 5 to 7. Fig. 5 suggest that 50 μm size coal particle reaches highest stress at 1.38 ms and =0.1, 100 μm coal reaches highest at.76 ms and =0.1, and 150 μm reaches highest at two locations. At ms it reaches highest at =0.95 and at 8.8 ms it reaches highest at =0.1. Time and location where stress reaches maximum value for Mach number 3 along with Mach numbers 5 and 7 and for all sizes of coal particle are noted down in the Table 1. 5. Maximum Strain Maximum strain is given by E E n1 n max (48) (a) 50 μm (b) 100 μm (c) 150 μm Figure 8. Developed thermal stain distribution at detonation Mach no. 3 (a) 50 μm (b) 100 μm (c) 150 μm Figure 9. Developed thermal stain distribution at detonation Mach no. 5
(a) 50 μm (b) 100 μm (c) 150 μm Figure 10. Developed thermal stain distribution at detonation Mach no. 7 Solution to Eq. 48 in the time range 0 to 8.8 milliseconds was obtained. The detailed observations where strain reaches highest value for Mach numbers 3, 5 and 7 and for all sizes of coal particle are noted down in the Table 1 below. Table 1. Time and location where developed thermal stress and strain reaches peak value Mach Size in Number μm Exposure time in ms Location 3 1.38 0.1 5 50 1.38 0.1 7 All the times Throughout 3.76 0.1 5 100 4.41 and 0.85 0.1 and 0.95 7 All the time Throughout 3 and 8.8 0.95 and 0.1 5 150 All the time Throughout 7 All the time Throughout 6 Conclusions 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 qualitatively different from conventional coal combustion. There is also possible scope of reconciling proposed fragmentation model with other failure theories. eferences [1] Nettleton, M. A.; Stirling,., The combustion of clouds of coal particles in shock-heated mixtures of oxygen and nitrogen. Proc. oy. Soc. Lond. A 1971, 3, 07-1. [] Enkhzhargal Kh.; Salomatov, V. V., Mathematical Modeling of the Heat Treatment and Combustion of a Coal Particle I. Heating Stage, J. of Engg. Phy. and Thermophysics 010, 83, 891-901. [3] Chirone,.; Massimilla, L., Primary Fragmantation of Coal in Fluidized Bed Combustion, Twenty-
Second Symposium (International) on Combustion/The Combustion Institute 1988, 67-77. [4] No, S. Y.; Syred N., Thermal Stress and Pressure Effects on Coal Particle Fragmentation and Burning Behaviour in a Cyclone Combustor, J. of Inst. of Energy 1990, 63,195-0. [5] Dacombe, P.; Pourkashanian, M.; Williams, A.; Yap, L.,. Combustion-induced fragmentation behavior of isolated coal particles Fuel 1999, 78, 1847-1857. [6] Carslaw H S. Conduction of Heat in Solids (nd ed.), Oxford Press, 1959. [7] Kreyszig E, Advanced Engineering Mathematics (8th ed.), John Wiley, 006. [8] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (8th ed.), Dover Publications. [9] Dean G. Duffy Transform Methods for Solving Partial Differential Equation (nd ed.), Chapman and Hall, 004. [10] Timoshenko, S. P.; Goodier, J. N., Theory of Elasticity (3rd ed.),mcgraw Hill Book Company, 1970. [11] Shigley, J. E., Mechanical Engineering Design, Tata McGraw Hill Book Company, 000.