Next: Determining the basic characteristics Up: Aerodynamic and hydraulic networks Previous: Aerodynamic and hydraulic networks Contents

### The variables and the equations

In Sections 4.6.15 and 4.6.16 the governing equations for aerodynamic and hydraulic networks were derived. It was shown that the basic variables for aerodynamic networks are the total temperature, the total pressure and the mass flow. All other variables can be calculated based on these three quantities. This is actually not a unique choice but seems to be best suited for our purposes. For hydraulic networks these reduce to the pressure, temperature and mass flow. This is completely different from the structural unknowns, which are taken to be the temperature and the displacements. Therefore, the degrees of freedom 0 to 3 which are used for structural calculations are redefined for networks according to Table (10)

Table 10: Variables in fluid nodes.
DOF | End node | Middle node
0 | total temperature | -
1 | - | mass flow
2 | total pressure | -
3 | static temperature | -

A distinction is being made between end nodes and middle nodes of fluid elements. Remember that fluid elements consist of two end nodes and one middle node (Section 4.1.18). The mass flow is not necessarily uniquely determined at the end nodes, since more than two branches can come together. Therefore, it is logical to defined the mass flow as unknown in the middle of a fluid element. Similarly, the total temperature or total pressure may not be known within the element, since the exact location of discontinuities (such as enlargements or orifices) is not necessarily known. Consequently, it is advantageous to define the total temperature and total pressure as unknowns in the end nodes. The static temperature is not a basic variable. Once the total temperature, mass flow and total pressure are known, the static temperature can be calculated. It is a derived quantity.

Similar to field nactdof for structural applications a field nactdog is introduced for network applications. It can be viewed as a matrix with 4 rows and as many columns as there are nodes in the model (including structural nodes; this is done to avoid additional pointing work between the local gas node number and the global node number). It indicates whether a specific degree of freedom in a gas node is active: if the entry is nonzero it is active, else it is inactive (which means that the value is known or not applicable because the node is a structural node). The degrees of freedom correspond to the first three rows of Table (10) and are repeated in Table (11) for clarity. Here too, only the first three rows are relevant.

Table 11: Degrees of freedom in fluid nodes.
DOF | End node | Middle node
0 | total temperature | -
1 | - | mass flow
2 | total pressure | -
3 | if 0: label of gas pipe element | -
| or restrictor the node belongs to; |
| if 0: node is is a chamber |

Consequently, if nactdof(2,328) is nonzero, it means that the total pressure in node 328 is an unknown in the system. Actually, the nonzero value represents the number of the degree of freedom attached to the total pressure in node 328. The number of the degree of freedom corresponds with the column number in the resulting set of equations. The fourth row in field nactdog (last row in the Table) is used to determine how the static temperature is to be calculated. If the node belongs to one or two gas pipes or restrictors, the mass flow in one of these elements is used to calculate the static temperature at the node from the total temperature. If the node is a chamber, the static temperature equals the total temperature (velocity is zero). What nactdof is for the degrees of freedom is nacteq for the equations. It is a field of the same size of nactdof but now a nonzero entry indicates that a specific conservation equation applies to the node, cf. Table (12).

Table 12: Conservation equations in fluid nodes.
DOF | End node | Middle node
0 | energy | -
1 | mass | -
2 | - | momentum
3 | if 0: independent node of isothermal | -
| element the node belongs to; |

If nacteq(1,8002) is nonzero, it means that the conservation of mass equation has to be formulated for node 8002. The nonzero value is actually the row number of this equation in the set of equations. If the value is zero, the equation does not apply, e.g. because the mass flow in all adjacent elements is known. The last row in field nacteq (at least for end nodes) is used to account for isothermal conditions. These only apply to gas pipes of type GAS PIPE ISOTHERMAL and exit restrictors preceded by an isothermal gas pipe element. A isothermal element introduces an extra equation specifying that the static temperature in the two end nodes of the pipe is equal. This can be transformed into a nonlinear equation in which the total temperature in one node (the dependent node) is written as a function of the total temperature in the other node and the other variables (total pressure in the nodes, mass flow). To account for this extra equation, the conservation of energy is not expressed for the dependent node (indeed, one can argue that, in order for the static temperatures to be equal an unknown amount of heat has to be introduced in the dependent node. So if nacteq(3,8002)= is nonzero it means that node 8002 is the dependent node in a isothermal relation linking the static nodal temperature to the one of node .

Next: Determining the basic characteristics Up: Aerodynamic and hydraulic networks Previous: Aerodynamic and hydraulic networks Contents