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Here Elastic energy U = j fdx Elastic coenergy U' = fxdf f Fig. 1-26. Force-displacement relationship for a spring. we show the displacement-force relationship to be nonlinear, which is the general case. For the region where the curve is linear, the analytic expres sion is written, f=Kx (Newtons) (1-35) where K is the spring constant. Over this linear region, the spring satisfies Hooke's law. Physical springs satisfy this relationship for forces below the elastic limit of the material. Deviations from this linear relation may occur for a number of reasons including, temperature effects, rotation of the spring ends, hysteresis, and others.

Attention is called to the analogy of this laminar flow law with Ohm's law for the electric circuit because the flow (current) is directly proportional to the head (voltage). ) (1-56) Note that laminar resistance is constant, and is directly analogous to electrical resistance. However, laminar flow is not often encountered in industrial practice. 6), A = area of restriction, m2, g = acceleration due to gravity, m/sec2, h = head of liquid, m. Unlike Eq. (1-55) for laminar flow, this expression shows a square law flow relation between Q and h.

For turbulent flow through pipes, orifices and valves, the steady-flow energy equation (the first law of thermodynamics) for adiabatic flow of ideal gases is, w = KAYV2g(Pl-p2)lp (kg/sec) where w = gas flow rate, kg/sec, A = area of restriction, Y = rational expansion factor = y = specific heat ratio for gases, p = gas density, kg/m3, K = a flow coefficient, p = pressure, kg/m2. p\^V—zr (1-62) 44 Modeling of System Elements Pi nr p2 Fig. 1-42. Fluid (gas) resistance and capacitance. The turbulent gas flow resistance is defined as, R= (sec/m2) %' (1-63) It is not possible to easily calculate R from Eq.