The rigid model assumes that the pipeline is not deformable and the liquid is incompressible; therefore, system flow-control operations affect only the inertial and frictional aspects of transient flow. Given these considerations, it can be demonstrated using the continuity equation that any system flow-control operations results in instantaneous flow changes throughout the system, and that the liquid travels as a single mass inside the pipeline, causing a mass oscillation. If liquid density and pipe cross section are constant, the instantaneous velocity is the same in all sections.

These rigidity assumptions result in an easy-to-solve ordinary differential equation; however, its application is limited to the analysis of surge. Newton's second law of motion is sufficient to determine the dynamic hydraulic of a rigid water body during the mass oscillation:

dH = f (L/D)(V|V|/2g) + (L/g) (dV/dt)

If a steady-state flow condition is established-that is, if dV/dt = 0-then this Equation equation simplifies to the Darcy-Weisbach formula for computation of head loss over the length of the pipeline. However, if a steady-state flow condition is not established because of flow control operations, then three unknowns need to be determined: H ₁ (t) (the left-hand head), H ₂ (t) (the right-hand head), and V(t) (the instantaneous flow velocity in the conduit). To determine these unknowns, the engineer must know the boundary conditions at both ends of the pipeline.

Using the fundamental rigid-model equation, the hydraulic grade line can be established for each instant. The slope of this line indicates the head loss between the two ends of the pipeline, which is also the head necessary to overcome frictional losses and inertial forces in the pipeline. For the case of flow reduction caused by a valve closure (dQ/dt < 0), the slope is reduced. If a valve is opened, the slope increases, potentially allowing vacuum conditions to occur. The change in slope is directly proportional to the flow change. Generally, the maximum transient head envelope calculated by rigid water column theory (RWCT) is a straight line, as shown in the following figure.

Figure 14-4: Static and Steady HGL versus Rigid and Elastic Transient Head Envelopes

The rigid model has limited applications in hydraulic transient analysis because the resulting equations do not accurately model pressure waves caused by rapid flow-control operations. The rigid model applies to slower surge or mass oscillation transients, as defined in Wave Propagation and Characteristic Time. HAMMER only utilizes rigid column theory under certain conditions (see Extended CAV Method).