The Transport Equation Colorado State University（科罗拉多州立大学的传输方程）.pdf

The Transport Equation Consider a fluid, flowing with velocity, V, in a thin straight tube whose cross section will be denoted by A. Suppose the fluid contains a contaminant whose concentration at position x at time t will be denoted by u(x,t). Then at time t, the amount of contaminant in a section of the tube between positions, x 1 and x 2 is given by the expression x 2 u x , t A dx amount of contaminant in x ,x at time t 1 2 x 1 Similarly, we can write an expression for the amount of contaminant that flows through a plane located at position, x, during the time interval from t 1 to t2 t2 u x , t A V dt amount of contaminant flowing through a plane at position, x, t 1 during the interval t 1, t2 Then an equation expressing a material balance for the contaminant can be written as follows, x 2 u x , t A dx x 2 u x , t A dx t2 u x , t A V dt t2 u x , t A V dt 2 1 1 2 x 1 x 1 t 1 t 1 i.e., the amount of contaminant in the section x 1,x 2 at time t2 equals the amount of contaminant in the section x 1,x 2 at time t 1, plus the amount of contaminant that flowed through the plane at position x during the time interval t , t minus the amount of 1 1 2 contaminant that flowed through the plane at position x during the time interval t , t . Of 2 1 2 course this equation is based on the assumption that there are no other sources of contaminant in the tube and there is no loss of contaminant through the walls of the tube. Now the fundamental theorem of calculus implies that