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Aerodynamic Networks

Aerodynamic networks are made of a concatenation of fluid elements filled with a compressible medium which can be considered as an ideal gas. An ideal gas satisfies

| (74)

where p is the pressure, is the density, is the specific gas constant and is the absolute temperature. A fluid element (see section 4.1.17) consists of three nodes: in the end nodes the temperature and pressure are the unknowns, in the middle node the mass flow is unknown. The end nodes play the role of crossing points in the network, whereas the middle nodes represent the flow within one element. To determine these unknowns, three types of equations are available: conservation of mass and conservation of energy in the end nodes and conservation of momentum in the middle node. Right now, only stationary flow is considered.

The stationary form of the conservation of mass for compressible fluids is expressed by:

| (75)

where the velocity vector. Integration over all elements connected to end node i yields:

| (76)

where is the mass flow from node i to node j or vice versa. In the above equation is always positive.

The conservation of momentum or element equations are specific for each type of fluid section attributed to the element and are discussed in Section 4.2 on fluid sections. For an element with end nodes i,j it is generally of the form (for positive , where p is the total pressure and is the total temperature), although more complex relationships exist. Notice in particular that the temperature pops up in this equation (this is not the case for hydraulic networks).

The conservation of energy in stationary form requires ([16]):

| (77)

where is the external heat flux, is the body flux per unit of mass and is the body force per unit of mass. is the total enthalphy satisfying:

| (78)

where is the specific heat at constant pressure and is the absolute temperature (in Kelvin). This latter formula only applies if is considered to be independent of the temperature. This is largely true for a lot of industrial applications. In this connection the reader be reminded of the definition of total temperature and total pressure (also called stagnation temperature and stagnation pressure, respectively):

| (79)

and

| (80)

where . and are also called the static temperature and static pressure, respectively.

If the end nodes of the elements are considered to be large chambers, the velocity is zero. In that case, the total quantities reduce to the static ones, and integration of the energy equation over all elements belonging to end node yields [13]:

| (81)

where is the convection coefficient with the walls. Notice that, although this is not really correct, a slight temperature dependence of is provided for. If one assumes that all flow entering a node must also leave it and taking for both the value corresponding to the mean temperature value of the entering flow, one arrives at:

| (82)

where .

The calculation of aerodynamic networks is triggered by the *HEAT TRANSFER keyword card. Indeed, such a network frequently produces convective boundary conditions for solid mechanics heat transfer calculations. However, network calculations can also be performed on their own.

A particularly delicate issue in networks is the number of boundary conditions which is necessary to get a unique solution. To avoid ending up with more or less equations than unknowns, the following rules should be obeyed:

- The pressure and temperature should be known in those nodes where mass flow is entering the network. Since it is not always clear whether at a specific location mass flow is entering or leaving, it is advisable (though not necessary) to prescribe the pressure and temperature at all external connections, i.e in the nodes connected to dummy fluid elements.
- A node where the pressure is prescribed should be connected to a dummy fluid element. For instance, if you have a closed circuit add an extra dummy fluid element to the node in which you prescribe the pressure.

Next: Hydraulic Networks Up: Types of analysis Previous: Diffusion mass transfer in Contents