LU marks In lecture notes we considered the Semenov model for spontaneous combustion de dr =f (0) f (0) - exp [0] - 0 where o is essentially the temperature difference between the combustible material and its surroundings, " is scaled time and , v 2 0, is known as the Semenov number. In this question we investigate a simplification of the Semenov model in which 1 ( 0 ) - 4 ( 1 + 0 + 2 ) - 0 . (a) Explain how the simplified model can be obtained from the exact model. (b) Is it possible for the temperature (0) in the simplified model to become negative? 3 (c) Show that the steady-state solutions of the simplified model satisfy the equation 1 =g (0), where you must identify the function g (8).30 / 59 4.4 Spontaneous Combustion 4.4.6 Analysing the scaled equation We have obtain the following model for spontaneous combustion de dry = exp 1 + 60 - 0. We make the approximation that 1+ 0 = 1. 8 Under this approximation the model becomes do dT= = 6e - 0. (9) The steady-states of equation 9 satisfy the equation 6 = de-. 30 / 594.4.6 Analysing the scaled equation (page two) Question 5.5 By sketching the graph 6 = 0e-0 deduce that There are no steady-state solutions if 6 > e-1. There is one steady-state solution if 6 = e-1. There are two steady-state solutions, 651 and 6$2 with 651 ocr: the system is supercritical. . No steady-state solutions: figure 4 (iii). . Temperature increases without bound. . Physically: spontaneous combustion, or ignition, occurs. . 6 0 provided 2R4.4.7 Conditions for spontaneous combustion? (page 7) Question 5.10 From the definition of the Semenov number EVQAp 6 = hST2R identify the properties of the material and the storage conditions that promote the risk of spontaneous combustion. Solution Following parameters increase the Semenov number and consequently the risk of spontaneous combustion: . high volume to surface area (V/S). . material undergoing highly exothermic reaction (Q). . highly reactive material (A) . high density (p). . low heat-transfer coefficient (h). . high ambient temperature (72) (provided that 0