thermodynamic and fluid-dynamic analysis of
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Thermodynamic and fluid-
dynamic analysis of
stages
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Classification of these methods depend on the type of
hypothesis formulated to analyze the machine flow rate. Onthe most general level we may distinguish between:
Monodimensional methods. This term indicates a group of
models deriving from application of the hypothesis of
monodimensional flow in the stage.
Non-viscous methods. This refers to numerical techniques
based on flow analysis in the individual components of the
stage in the approximation of non-viscous flow.
Viscous methods. These methods are based on flow
analysis conducted through numerical integration of the flowviscous equations.
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MonodimensionalMethods
Assumes that the fluid conditions are uniform over
certain flow cross-sections.
These cross-sections are taken before and after the
impeller as well as at inlet and exit of the entire machine. A specific operating condition is assumed, defined by the
following parameters ,assumed to be known:
P00= total inlet pressure
T00= total inlet temperaturem =mass flow rate
N=impeller speed of rotation
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Analysis of Impeller Inlet Section
The flow between stage inlet section and impeller inlet
section can usually be considered isentropic.
The conditions in section 1 can be evaluated by applying thecontinuity and momentum equations.
Determination of the quantities relevant to the streamline
passing in proximity to the blade tip is particularly important
since it is here that the highest relative Mach numbers are
found.
The meridian component of the absolute velocity Cm1 can
be determined by:
Where
r1= specific volume &CD
=blockage factor due to presence of theblades.
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The tangential component of the absolute velocity Cq1
depends on whether or not inlet guide vanes are utilized. In
the absence of vanes, we will have Cq1=0.
And also:
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Analysis of Impeller Discharge Section
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Monodimensional Analysis of Diffusers
Assume the inlet condition to diffuser is the exit from
impeller.
The most important diffuser performance parameter is
the pressure recovery coefficient Cp defined by theequation:
This parameter is utilized to quantify the amount kinetic
energy transferred to the fluid by the impeller, whichconverting into potential energy.
The diffusors most frequently utilized in centrifugal
stages can be classified under two headings: free vortex
and bladed.
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The approach most frequently used in analyzing free
vortex diffusors hypothesizes a succession of
monodimensional condition sections with r=constant lying
between impeller discharge section and diffusor dischargesection.
The fluid-dynamic balance equations relevant to this
representation, inclusive of the friction terms deriving from
the presence of side walls, can be integrated numericallystarting from known conditions in the discharge section.
This procedure can be used to evaluate the fluid-dynamic
state on discharge from the diffusor and the consequentperformance of the component.
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The best-known of these
correlations refers to
experiments conducted by
Runstadler on diffusors of
bidimensional geometry with
straight walls diverging on a
single plane. It shows that
the recovery coefficient
depends on a number ofgeometric and aerodynamic
parameters, such as the
length/width ratio L/w, throat
section, and divergence
angle 2.
With the bladed diffusor, the approach most commonly
employed for evaluating the performance of this component
consists of experimental correlations.
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Disadvantages of The monodimensional
methods
1. impossibility of obtaining an accurate representation of
the fluid-dynamic field at all machine points.
2. impossibility of diagramming the detailed geometry of
the components and its influence on the fluid-dynamic
characteristics.
3. need to introduce empirical data in the form of various
experimental correlations.
to overcome at least some of these limitations ,we need
for analysis methods capable of resolving, throughnumerical calculation procedures.
Because of the complexity and expense of using
viscous models, attention was initially focused on
models based on the hypothesis of non-viscous.
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Non-viscous Numerical Methods
The non-viscous methods can be divided into
four categories:
Bidimensional solutions relevant to streamline
surfaces lying in the hub-to-shroud direction
Bidimensional solutions relevant to streamline
surfaces lying in the blade-to-blade direction
Quasi-three-dimensional solutionsThree-dimensional solutions
In each of these categories the methods can be
classified to streamline curvature methods and
partial derivative methods.
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The streamline curvature methods are based on
the integration of ordinary differential equations
of the first order: these describe the momentum
balance along directions defined by the so-called quasi-normals to the streamlines.
The partial derivative methods are based on the
integration of differential equations with the
partial derivatives which describe the balance of
mass, that of quantity of motion and that of
energy at a point in the calculation domain.
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Bidimensional Solutions Relevant to Streamline
Surfaces in the Hub-to-shroud Direction.
These methods are based on representation of the
conditions existing on a hypothetical mean streamline
surface, extending in the hub-to-shroud direction within
the area lying between two adjacent blades.
A typical calculation code for this category, utilizing the
streamline curvature approach, is based on integration of
the momentum balance equations & mass balance
equations.
The codes based on the partial derivations approach
frequently utilize Wus formulation
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Bidimensional Solutions Relevant to
Streamlines in the Blade-to-blade Direction
These methods are based on the representation of conditionsin hypothetical streamlines consisting of surfaces between
two contiguous blades.
Many of these methods employ the streamline curvature
formulation. The most widely used approach utilizing finite differences
methods, frequently based on the formulation proposed by
Stanitz.
Have ability for evaluating the pressure and velocitydistribution, and consequently to predict the behavior of the
boundary layers in the real machine.
can be utilized as constituent elements of quasi-three-
dimensional or three-dimensional procedures.
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Quasi-three-dimensional and Three-
dimensional Solutions
A frequently used technique consists of producing quasi-
three-dimensional representations , obtained by
combining two bidimensional solutions relevant to
Streamline Surfaces in the Hub-to-shroud & Blade-to-
blade Direction .
Using of three-dimensional methods make an actual
representation. This model developed by Hirsch, Lacor,
and Warzee which utilizes a finite-element procedure.
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Viscous Methods The term viscous methods indicates a family of calculation
codes based on procedures of numerical integration of theviscous, compressible, and three-dimensional equations of
motion.
the system formed of the complete Navier-Stokes equations
in non-stationary form, the laws of fluid, and the equations
that specify the dependency of viscosity and thermal
conductivity on other variables.
In the case of laminar flow, a numerical simulation based on
this methods give accurate description & do not have another
information based on empirical data. It is possible to simulate a turbulent flow through integration of
the Navier-Stokes equations in non-stationary form.