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Flow in an engine cylinder needs to account for combustion of fuel. Efficient combus- tion requires oxygen. Hence, modeling such flows requires an equation for the oxygen concentration field. Arbitrary Eulerian CV unit outward diffusive normal flux fluid velocity n v Differential surface patch of area Δ. Besides being transported advectively by the motion of the air, oxygen can also enter or leave the control surface by the process of diffusion. The direction and magnitude of the diffusive transport of oxygen per unit area at any point on the control surface is given by a vector field denoted as j. This "diffusive flux” vector has the units of kg of oxygen transported by diffusion per sq. m per second. The rate of consumption of oxygen around any point due to combustion per unit volume is denoted as G (units: kg-of-oxygen consumed per m’per sec). Assume that this rate G is a known function of oxygen concentration. The verbal equation describing the conservation of oxygen is: Rate of accumu- Net rate of oxy- Net rate of oxy- Net consumption lation of oxygen = gen into CV by + gen into CV by - of oxygen due to in CV advection diffusion combustion The field c(r, t) denotes the concentration of oxygen i.e. its density in kg of oxygen per m? [4] [3] [5] (a) Use Reynolds' Transport Theorem to express the left-hand side of the verbal equation above as an integral over the Eulerian control volume. (b) Express the advective term on the right-hand side of the verbal equation as an integral over the Eulerian control surface. (c) The rate of consumption of oxygen around any point due to combustion per unit volume is denoted as G (units: kg-of-oxygen consumed per m’per sec). In general, this rate G may vary with position and time. How will you express the reaction term on the right-hand side of the verbal equation in terms of G? (d) Besides being transported advectively by the motion of the air, oxygen can also enter or leave the control surface by the process of diffusion. The direction and magnitude of the diffusive transport of oxygen per unit area at any point on the control surface is given by a vector field denoted as j. This “diffusive flux” vector has the units of kg of oxygen transported by diffusion per sq. m per second. How would you mathematically express as an integral the rate of net gain of oxygen by the CV due to diffusion i.e. the diffusion term on the right-hand side of the verbal equation? (5)

C! (e) Using an arbitrary Eulerian control volume (CV) within a flowing air whose [6] velocity field is v, derive the following PDE for the concentration field, c: Dc ac +v. Vc=-cv.v-V.j-G. Dt at Develop first the integral form of the conservation equation starting from the accumulation term. Then reduce the integral form to the PDE above. (f) We face a closure problem and need to relate j to the concentration field to [6] achieve a closed set of equations if this equation is to be solved along with the INSEs. We know that chemicals diffuse from regions of higher concentration to lower concentration. Further, we expect stronger diffusion when the spatial gra- dient of the concentration is larger. Based on this idea suggest the simplest pos- sible "law" relating j to the concentration field c to express this idea. Substitute this equation in the PDE above to derive the final advection-diffusion-reaction equation for chemical species such as oxygen. (g) In the absence of diffusion and reaction, what would cause the concentration of [6] oxygen in a fluid particle to change as it carried by the flow? Total for Question 2: 35