Anemometer: Difference between revisions

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Revision as of 15:22, 28 November 2011

Description

An anemometer, a flow sensing device, consists 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 Fig.1 below) from which the fluid velocity can be determined.

The physical model can be expressed by the convection-diffusion partial differential equation[4]:

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

where $ρ$ denotes the mass density, $c$ is the specific heat, $κ$ is the thermal conductivity, $v$ is the fluid velocity, $T$ is the temperature and $\dot{q}$ 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 meshing and discretizing by the finite element method. The order of the system is $n = 29008$ .

Example with 1 parameter:

The $n$ dimensional ODE system has the following transfer function $G (p) = c ((s E - A_{v 0} - p (A_{v 1} - A_{v 0}))^{- 1} b)$

with the fluid velocity $p (= v) \in [0, 1]$ as single parameter. Here $E$ is the heat capacitance matrix, and Failed to parse (unknown function "\jmat"): {\displaystyle \jmat M_{d}} incorporates the heat dissipation and thus contains the convection and conduction terms. $b$ 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. $A$ are the stiffness matrices with $_v 0$ for pure diffusion, $_v 1$ diffusion and convection. Thus, for obtaining pure convection you have to compute $A_{v 1} - A_{v 0}$ .

@@ Line 27: / Line 27: @@
 The solid model has been generated and meshed in ANSYS.
 Triangular PLANE55 elements have been used for meshing and discretizing by the finite element
-method. The order of the system is $n = 29008$.\\
+method. The order of the system is <math>n = 29008</math>.
-\vspace{0.5cm}
-\noindent
+Example with 1 parameter:
-Example with 1 parameter:\\
-The $n$ dimensional ODE system has the following transfer function
+The <math>n</math> dimensional ODE system has the following transfer function
-\[
+<math>
 G(p) = c((sE - A_{v0}- p(A_{v1} - A_{v0}))^{-1}b)
-\]
+</math>
-with the fluid velocity $p(=v) \in [0, 1]$ as single parameter.
-Here $E$ is the heat capacitance matrix,
+with the fluid velocity <math>p(=v) \in [0, 1]</math> as single parameter.
-% and $\jmat M_{d}$ incorporates the heat dissipation and thus contains the convection and conduction
+Here <math>E</math> is the heat capacitance matrix,
-%terms.
+and <math>\jmat M_{d}</math> incorporates the heat dissipation and thus contains the convection and conduction
-$b$ 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. $A$ are the stiffness matrices with $\_v0$ for pure diffusion, $\_v1$ diffusion and convection. Thus, for obtaining pure convection you have to compute $A_{v1} - A_{v0}$.\\
+terms. <math>b</math> is the load vector which is derived from separating the spatial and temporal variables in <math>\dot{q}</math> and the FEM discretization w.r.t. the spatial variables. <math>A</math> are the stiffness matrices with <math>\_v0</math> for pure diffusion, <math>\_v1</math> diffusion and convection. Thus, for obtaining pure convection you have to compute <math>A_{v1} - A_{v0}</math>.
-\vspace{0.5cm}