Work Energy Principle equation

The work-energy principle states that the pressure drop between two junctions must be the same for all paths between the two junctions. The work-energy equation is used to calculate the pressures in the network.

Important: The work energy equation is solved at each element, simultaneously with the continuity equation at each node, and the temperature equation at each node, to satisfy the two principles of network solutions.

The work-energy principle equation needs to satisfy the same pressure loss for all the paths between the two junctions.

The pressure drop, or , can be expressed in general terms as a function of the volumetric flow rate, Q, for any branch of elements off a junction node as:

where and are variables that are dependent on the empirically determined pressure-drop flow-rate relationship used, and N refers to the order of equation used to describe the pressure drop.

This equation is valid both for cases where pressure is added to, or lost from, a network. When pressure is added to a network, for example when pumps are used, the pressure is considered to be a negative pressure loss.

Note: In addition to the pressure drop

between two junctions, the following also contribute to pressure loss and must be accounted for in the pressure loss calculations:

Frictional head loss due to fluid shear at the channel wall. This frictional head loss is always present throughout the length of the channel, and is solved in the Coolant Flow analysis, using the Darcy-Weisbach equation.
Minor losses, caused by local disruptions of the fluid stream such as valves, bends, tee junctions and the like.
Pumps, which add pressure and are considered to be negative head loss.

The work-energy principle equation needs to represent the pressure drop relationship in every element in the network.

Solving the general work energy principle equation for each element

Referring to the coolant flow computational domain, in general the pressure drop relationship for any of the branch elements j of node i can be expressed as:

where is the pressure drop flow rate relationship derived from the work energy principle above, is the connectivity operator, (+) or (-), is the mass flow rate, and is a function of the density of the fluid in the network. For incompressible fluids, such as water, this is a constant.

Note: The general work energy principle equation is further modified, using the Darcy-Weisbach equation, to take into account frictional head loss, minor losses, and negative head loss from the pump.