An impact press consists of a piston that is accelerated rapidly by pumping high pressure gas into the cylinder. The high-speed piston slams into the material to be flattened. Derive the equations needed to perform the design and analysis of this system according to the following suggested procedure and assumptions. Assume that the gas enters at a constant and known flow rate Ṁ₁ at constant and known state 1 (T₁, P₁), and that the inlet gas is mixed instantaneously with the fluid inside, which at any instant is at T, P, which vary with time. We will neglect the kinetic and potential energy of the gas, but must consider the kinetic and potential energies of the piston. We will treat the piston as frictionless, and assume that the pressure on the workpiece side of the piston is constant at P₀, the atmospheric pressure. We will assume that heat transfer is negligible, and that the internal energy of the piston does not change. Denote the piston mass by M_p, the piston area by A_p, and the downward piston velocity by ν_p (be sure not to confuse p with the gas volume ν_p = A_p(L − z)). Write the rate-basis energy and mass balances at time t for the control volume above the piston; this will give you two differential equations involving the unknowns.

Then, write the rate-basis energy balance on the piston, which will give you another differential equation (the same as would be obtained from Newton’s law). Using the ideal gas equations, these three balances and the definition ν_p = −dz/dt give you a coupled set of four ordinary differential equations for the variables z, ν_p, T, and M. Express all of the terms in these equations as a function of these four variables and known quantities, such as Ṁ₁, T₁, A_p, L, M_p, P₀, the acceleration of gravity g, and constants for the gas. These are the equations that would be solved (analytically or numerically) to design the press.