tailieunhanh - Aerodynamics for engineering students - part 3

phương trình Giao thông vận tải cho các chất gây ô nhiễm trong lĩnh vực hai dimensionalflow Trong một trong nhiều ứng dụng kỹ thuật là quan tâm đến việc vận chuyển chất gây ô nhiễm một dòng chảy chất lỏng. Các chất gây ô nhiễm có thể là bất cứ thứ gì từ một hóa chất gây ô nhiễm môi trường hạt vật chất. | 102 Aerodynamics for Engineering Students 3 Transport equation for contaminant in two-dimensional flow field In many engineering applications one is interested in the transport of a contaminant by the fluid flow. The contaminant could be anything from a polluting chemical to particulate matter. To derive the governing equation one needs to recognize that provided that the contaminant is not being created within the flow field then the mass of contaminant is conserved. The contaminant matter can be transported by two distinct physical mechanisms namely convection and molecular diffusion. Let c be the concentration of contaminant . mass per unit volume of fluid then the rate of transport of contamination per unit area is given by dx dyj where i and j are the unit vectors in the X and y dừections respectively and T is the diffusion coefficient units m2Ịs the same as kinematic viscosity . Note that diffusion transports the contaminant down the concentration gradient . the transport is from a higher to a lower concentration hence the minus sign. It is analogous to thermal conduction. a Consider an infinitesimal rectangular control volume. Assume that no contaminant is produced within the control volume and that the contaminant is sufficiently dilute to leave the fluid flow unchanged. By considering a mass balance for the control volume show that the transport equation for a contaminant in a twodimensional flow field is given by de de dC r c C n dt dx dy dx2 dy2 J b Why is it necessary to assume a dilute suspension of contaminant What form would the transport equation take if this assumption were not made Finally how could the equation be modified to take account of the contaminant being produced by a chemical reaction at the rate of mc per unit volume. 4 Euler equations for axisymmetric flow a for the flow field and coordinate system of Ex. 1 show that the Euler equations inviscid momentum equations take the form rdu du du dp p PSr- dt dr dzJ dr fdw . dw dw dp ue