Page 278 - The Combined Finite-Discrete Element Method
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EXPANSION OF THE DETONATION GAS      261

           By substituting the expression for density ρ in terms of specific volume v
                                                  1
                                              ρ =                               (8.13)
                                                  v
           into equation (8.12), the gas pressure in terms of specific volume is obtained:

                                                  b−1       b−1
                                  p c   v o 1 + (a/v o  )(v o /v c )
                                     =                                          (8.14)
                                                       b−1
                                  p o   v c    1 + a/v o
           As mentioned before, the solution to the expansion of the detonation gas has been obtained
           in a closed analytical form.
             For a particular explosive, parameters a and b are obtained by an inverse method. For
           instance, in Figure 8.1, the irreversible adiabatic expansion of nitroglycerine detonation
                                                                                  9
           gas is shown. By curve fitting, parameters a and b are estimated a = 10.99e−10 m kg 3
           and b = 4.0.
             The results for pressure as a function of specific volume obtained using equation (8.14)
           compare well with the results given by Johansson. As the specific volume of the gas
           increases, the slope of the log p log v curve decreases. This slope asymptotically
           approaches −1 as the specific volume becomes large. In other words, at very low densities
           the detonation gas approaches the behaviour of an ideal gas. At high density the slope of
           the curve is given by

                             
         
           b−1  (b−1) ln(v o /v c )
                               p c       v o 1 + (a/v o  )e
                           ln      = ln                 b−1                      (8.15)
                               p o       v c     1 + a/v o
                                                       a

                             ln(p c ) =− ln(v c ) + ln 1 +  b−1 e (b−1) ln(v o /v c )
                                                     v o
                                                                b−1
                                     + ln(p o ) + ln(v o ) − ln(1 + a/v o  )     (8.16)

                           1
                                                       Legend
                          0.5                          Isothermal
                                                       Johansson
                           0
                        log p (GPa)  −0.5


                          −1

                         −1.5

                          −2
                         −2.5
                                −3        −2.5      −2        −1.5       −1
                                              log v (m^3/kg)

                   Figure 8.1  Irreversible adiabatic expansion of nitroglycerine detonation gas.
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