• bivariate population balance;
  • ethanol-fueled spray combustors;
  • liquid biofuel performance;
  • liquid propellant chemical rockets;
  • evaporation-controlled combustion intensity;
  • multiphase chemical reaction engineering


We present a bivariate population balance-based formulation of the performance of well-mixed adiabatic combustors fed by ethanol (EtOH)-containing sprays of prescribed droplet size distribution (DSD) and composition. Our historically interesting example is the fuel-cooled V-2 chemical rocket—using 75 wt % EtOH + H2O solution, and oxidizer O2(L). Of special interest are the predicted combustion “intensity” (GW/m3) and efficiency (EtOH fraction vaporized) at each ratio of combustor mean residence time to feed-droplet characteristic vaporization time. Our formulation exploits a quasi-steady, gas-diffusion-controlled individual droplet evaporation rate law, and the method-of-characteristics to solve the associated first-order population balance partial differential equation governing the joint distribution function n(m1, m2) of the fuel spray exiting such a chamber, where m1 = EtOH mass/droplet, and m2 = H2O mass/droplet. Besides the combustor efficiency and intensity, this bivariate distribution function enables predictions of corresponding unconditional DSD, and the joint distribution function(diam., droplet temperature)—perhaps measurable. Our numerically exact formulation/results also provide valuable test cases for convenient approximate methods (bivariate moment and spectral/weighted residual) to predict these “correlated” bivariate distribution functions in more complex situations. © 2011 American Institute of Chemical Engineers AIChE J, 2011