Total pressure

The total pressure (also total pressure ) describes the pressure that occurs in a flowing medium when the flow velocity isentropically (lossless) is or would be reduced to almost a standstill; this is often the case with a stagnation point flow. The pitot tube is an effective means of measuring the total pressure . Since the pressure is easiest to measure at stagnation points at which the medium is at rest, the total pressure is also referred to as rest pressure or stagnation pressure . ${\displaystyle p_{\mathrm {t} }}$

Different components contribute to the total pressure for incompressible and for compressible fluids.

Incompressible case (liquid)

In the case of an incompressible fluid , the total pressure is the sum of the static pressure that an observer moving with the flow would feel and the dynamic dynamic pressure : ${\displaystyle p}$ ${\displaystyle p_{\mathrm {dyn} }}$

${\displaystyle {\begin{aligned}p_{\mathrm {t} }&=p+p_{\mathrm {dyn} }\\&=p+{\frac {\rho }{2}}\cdot v^{2}\end{aligned}}}$

With

• the density of the fluid ${\displaystyle \rho }$
• the flow velocity . ${\displaystyle v}$

Through the above The relationship can be used to determine the flow velocity from the measurement of static pressure and total pressure, as can be done with a Prandtl probe .

Compressible case (gas)

In the case of a compressible fluid, there is a relationship between density, pressure and temperature, which in connection with a flow leads to the laws of gas dynamics.

Assuming that internal friction (friction between the gas molecules ) and external friction (friction between the gas and the pipe wall) are negligible and the height of the flow path is constant, the total pressure remains constant over the entire flow path, while the static pressure is dependent can vary from the flow rate.

Classification and demarcation

The simplified energy approach according to Bernoulli via the pressure change within a flow only applies exactly to liquids.

In the case of a gas flow, on the other hand, the static pressure also changes the density and temperature of the medium. This is reflected in the internal and kinetic energy of the medium or the flow and must therefore be taken into account in the energy approach .

The extended power equation (energy per time) according to Bernoulli along the flow path of an ideal gas flowing without friction is thus:

${\displaystyle P=p_{\text{ruhe}}\cdot {\dot {V}}=p_{\text{s}}\cdot {\dot {V}}+{\dot {m}}\cdot c_{\text{v}}\cdot T+{\frac {1}{2}}\cdot {\dot {m}}\cdot {w^{2}}+{\dot {m}}\cdot g\cdot z={\text{konst}}\qquad (1)}$

The individual summands stand for the following proportions of the performance : ${\displaystyle P}$

${\displaystyle p_{\text{s}}\cdot {\dot {V}}}$	= Proportion of mechanical energy (pressure × volume = force × displacement)
${\displaystyle {\dot {m}}\cdot c_{\text{v}}\cdot T}$	= Share of the amount of heat transported or the thermal energy contained in the gas
${\displaystyle {\frac {1}{2}}\cdot {\dot {m}}\cdot {w^{2}}}$	= Share of the kinetic energy of the moving gas mass
${\displaystyle {\dot {m}}\cdot g\cdot z}$	= Share of the potential energy (positional energy) of the gas

Here are:

${\displaystyle p_{\text{s}}}$	= static pressure (absolute)
${\displaystyle {\dot {V}}}$	= Volume flow (= volume per time)
${\displaystyle {\dot {m}}}$	= Mass flow (= mass per time)
${\displaystyle c_{\text{v}}}$	= specific heat capacity of the gas at constant volume
${\displaystyle T}$	= Temperature of the gas (absolute)
${\displaystyle w}$	= Flow velocity
${\displaystyle g}$	= Acceleration due to gravity
${\displaystyle z}$	= geodetic height

Assuming that the proportion of potential energy is negligible (flow is horizontal, i.e. the density of gases is usually much lower than that of liquids), the last term of equation (1) can be removed. With the basic equations ${\displaystyle \Delta z=0}$

${\displaystyle {\dot {V}}={\dot {m}}/\rho }$

${\displaystyle \rho =p_{\text{s}}/(R\cdot T)}$

${\displaystyle R=c_{\text{p}}-c_{\text{v}}}$

With

• ${\displaystyle \rho }$= Density of the gas
• ${\displaystyle R}$= Gas constant
• ${\displaystyle c_{\text{p}}}$ = specific heat capacity of the gas at constant pressure

follows from the rest of the equation:

${\displaystyle {\begin{aligned}\Rightarrow {\frac {P}{\dot {V}}}=p_{\text{ruhe}}&=\underbrace {p_{\text{s}}+\rho \cdot c_{\text{v}}\cdot T} _{\rho \cdot c_{\text{p}}\cdot T}+{\frac {1}{2}}\cdot \rho \cdot w^{2}={\text{konst}}\qquad {\text{(2)}}\end{aligned}}}$

Influence of speed

Speed ​​dependence of the temperature

Speed ​​dependence of the pressure

each with an air flow ${\displaystyle T_{\text{t}}=20{\text{°C}}}$

Equation (2) states that with a steady , heat-insulated flow of an ideal gas, the temperature of the gas decreases with increasing flow velocity and vice versa, provided that there is no energy exchange with the environment. When the flow speed increases, part of the undirected microscopic molecular speed (temperature) is transformed into a directed macroscopic speed.

It is similar with the pressure. Here, too, the static pressure decreases with increasing flow velocity, since the molecular velocity (and thus the momentum exchange ) perpendicular to the wall is reduced in favor of the flow velocity.

At a flow speed of 0 m / s, static pressure and total pressure are identical, as are static temperature and resting temperature.

calculation

Since the total pressure can only be measured at a point at which the flow velocity is negligibly small (hence the designation rest pressure or stagnation pressure), it usually has to be calculated. For an ideal gas, this can be done as follows:

${\displaystyle p_{\text{t,x}}=p_{\text{x}}\cdot \left[{\frac {1}{2}}+{\sqrt {{\frac {\kappa -1}{2\cdot \kappa }}\cdot R_{\text{i}}\cdot T_{\text{t,x}}\cdot \left({\frac {\dot {m}}{A_{\text{x}}\cdot p_{\text{x}}}}\right)^{2}+{\frac {1}{4}}}}\ \right]^{\frac {\kappa }{\kappa -1}}\qquad (3)}$

Here are:

${\displaystyle p_{\text{t,x}}}$	= Total pressure at the point  ${\displaystyle x}$
${\displaystyle p_{\text{x}}}$	= static pressure at the point  ${\displaystyle x}$
${\displaystyle \kappa }$	= Isentropic exponent (for example, in air. ) ${\displaystyle \kappa =1{,}4}$
${\displaystyle R_{\text{i}}}$	= individual gas constant
${\displaystyle T_{\text{t,x}}}$	= Resting temperature at the point  ${\displaystyle x}$
${\displaystyle {\dot {m}}}$	= Mass flow
${\displaystyle A_{\text{x}}}$	= Cross-sectional area of ​​the flow path at the point  . ${\displaystyle x}$

Since with an ideal gas the idle temperature along a heat-insulated flow path, in which the flow velocity is negligible, remains constant (this also applies to the pressure reduction due to friction / throttling), it can usually be recorded at any point along the flow path. Whether the flow can be regarded as thermally insulated depends not only on the thermal conductivity of the environment but also on how large the temperature differences between the gas flow and the environment are and how large the ratio of surface area to mass flow is.

Equation (3) also assumes that the flow profile of the gas flow has a rectangular velocity distribution . This is almost the case with turbulent flow (which occurs at higher speeds and where there are significant differences between the total pressure and the static pressure). However, according to the static pressure for any, i. H. At least partially laminar flow profile is sought, so numerical methods must be used, since equation (3) obviously cannot be converted according to the static pressure.

Outflow process from a container and diffuser

During the discharge process from a container through a well-rounded nozzle (almost frictionless discharge with usually negligible heat transfer ), the total pressure, which corresponds to the static pressure in the neck part of the nozzle, corresponds to the internal pressure of the container.

The static pressure at the diffuser inlet can also be largely reduced to the total pressure via a diffuser with a small widening angle, as long as the flow velocity at the diffuser inlet is below the associated speed of sound .

application

In technical practice, it often makes more sense to work with the “correct” total pressure instead of the static pressure that is usually easy and directly measurable. Unfortunately, this is often overlooked and leads to further questions and problems. Two examples from the field of flow measurement and determination of flow parameters on pneumatic components should clarify this fact.

example 1

If the inlet pressure is measured on a flow-optimized diffuser with a smaller pressure measuring tube (tube on which the static pressure of the flow can be tapped at right angles to the direction of flow) and the back pressure with a larger pressure measuring tube due to the different connection cross-sections, the paradox occurs that the back pressure is greater than the pre-pressure (as long as the speed of sound in the diffuser is not exceeded). So the medium seems to flow from the lower pressure to the higher static pressure.

However, if the flow is viewed from the point of view of the pressure at rest, the pressure at rest in the direction of flow will never increase, but at best remain almost constant.

Example 2

A common flow rate  parameter on pneumatic components is the qnN according to VDI 3290. With the qnN, the correlating flow is measured with a static inlet or inlet pressure of 6 bar and a static outlet or outlet pressure of 5 bar. If the qnN is determined on a valve terminal (unit with a common supply connection and several individually switchable single valves), the effect occurs that the quotient of the flow rate divided by the number of switched (identical) valves does not remain constant (or even decreases, as one might be due to an internal limitation), but increases progressively with each further switched valve. The cause of this effect is that the static inlet pressure is kept constant by active readjustment , but the invisible static pressure increases steadily and leads to an increased back pressure at the last valve slices, which results in an increased flow.

literature

• Association of German mechanical and plant engineering : VDMA standard sheet 24 575: Flow measurement of pneumatic components. Application of ISO 6358: 1989 taking into account the influence of the flow rate .
• Werner Wunderlich, Erwin Bürk, Wolfgang Gauchel: Measurement in fluid technology / flow measurement. Special features in pneumatics . In: O + P. Fluid technology for mechanical and plant engineering , 2010, April, ISSN  0341-2660

Individual evidence

1. ^ TU Graz: pipe flow, profile flow. (PDF; 792 kB) Archived from the original on January 29, 2016 ; Retrieved July 26, 2013 .