Anemometer

Description

Schemantic: 2D-Model-Anemometer

Calculated temperature profile for the Anemometer function

An Anemometer is a flow sensing device, consisting of a heater and temperature sensors before and after the heater, placed either directly in the flow or in its vicinity. They are located on a membrane to minimize heat dissipation through the structure. Without any flow, the heat dissipates symmetrically into the fluid. This symmetry is disturbed if a flow is applied to the fluid, which leads to a convection on the temperature field and therefore to a difference between the temperature sensors (see xx--CrossReference--dft--fig:plots--xx) from which the fluid velocity can be determined.

The physical model can be expressed by the convection-diffusion partial differential equation :

$\rho c \frac{\partial T}{\partial t} = \nabla \cdot (\kappa \nabla T ) - \rho c v \nabla T + \dot q,$

where $\rho$ denotes the mass density, $c$ is the specific heat, $\kappa$ is the thermal conductivity, $v$ is the fluid velocity, $T$ is the temperature, and $\dot q$ is the heat flow into the system caused by the heater.

The solid model has been generated and meshed in ANSYS. Triangular PLANE55 elements have been used for the finite element discretization. The order of the system is $n = 29008$.

Example with 1 parameter:

The $n$ dimensional ODE system has the following transfer function $G(p) = C((sE - A1- p(A2 - A1))^{-1}B)$

with the fluid velocity $p(=v)$ as single parameter. Here $E$ is the heat capacitance matrix, $B$ is the load vector which is derived from separating the spatial and temporal variables in $\dot{q}$ and the FEM discretization w.r.t. the spatial variables. $Ai$ are the stiffness matrices with $i=1$ for pure diffusion and $i=2$ for diffusion and convection. Thus, for obtaining pure convection you have to compute $A2 - A1$.

Example with 3 parameters:

Here, all fluid properties are identified as parameters. Thus, we consider the following transfer function

$G(p_0, p_1,p_2) = C((s \underbrace{(E_s + p_0 E_f)}_{E(p_0)} - \underbrace{( A_{d,s} + p_1 A_{d,f} + p_2 A_c )}_{A(p_1,p_2)} )^{-1}B)$

with parameters $p_0, \, p_1, \, p_2$ which are combinations of the original fluid parameters $\rho, \, c, \, \kappa, \, v: \quad p_0 = \rho c, \, p_1=\kappa,$ and $p_2 =\rho c v,$ see . So far, we have considered the mass density as fixed, i.e. $\rho=1$.

Origin

IMTEK Freiburg, group of Jan Korvink.

Data

Matrices are in the Matrix Market format. All matrices (for the one parameter system and for the three parameter case) can be found and uploaded in Anemometer.tar.gz. The matrix name is used as an extension of the matrix file. The system matrices have been extracted from ANSYS models by means of mor4fem.

For more information about computing the system matrices, the choice of the output, applying the permutation, please look into the readme file.

To test the quality of the reduced order systems, harmonic simulations as well as transient step responses could be computed, see .

Example with 1 parameter:

• .B: load vector
• .E: damping matrix
• .P: permutation matrix
• .A: stiffness matrices (2)

Example with 3 parameters:

• .B: load vector
• .E: damping matrices (2)
• .A: stiffness matrices (5)

References

a) About the Anemometer

b) MOR for non-parametrized Anemometer

c) MOR for parametrized Anemometer

Contact

Ulrike Baur