Page 46 - Mechanical Engineers Reference Book

Principles of thermodynamics 1/35

which intercept the stream lines at right angles to form an the extensive use of ‘coefficients’ (discharge especially) to
orthogcinal network of characteristic pattern for each flow account for effects which are difficult to model matnema-
field. tically. However, almost ail fluid mechanics equations in
From equations (1.124) and (1.127) it can be seen that common use are based on the conservation of energy. the
conservation of momentum or the fact that the rate of change
(1.130) of momentum may be equated to an applied force, usually a
pressure force. The experimental checking and empirical
amendment to derived formulae is just good engineering
which are the Cauchy-Rieman equations. In such flow fieids practice.
the Laplace equations for d and $ must both be satisfied: This section does not provide rigorous derivations of the
v2+ = 13; and v*$ = o (1.131) various equations quoted. Some appreciation of such deriva-
tions may be required in order to establish the limitations and
Because the fluid is ideal, the Bernoulli equation (1.37) may modifications necessary for the application of the equations,
be readily applied between points in the field, both along and especially to non-standard situations. For this. standard text-
across the stream lines. books on fluid mechanics as listed below should be consulted.

1.5.9.3 Flow patterns Further reading
Examples of simple flow patterns are shown in Figure 1.48 General fluid mechanics
with the equations to their stream functions and velocity Douglas. J. F.. Gasiorek, J. M. and Swaffield, J. A,. Mechanics of
potentials. A source is mathematically a point at which fluid Fluids, third edition, Pitman, London (1986)
appears and flows radially outwards. A sink is a negative Walshaw, A. C. and Jobson, D. A,, Mechanics of Fluids. third
source at which fluid flows radiaily inwards to disappear at a edition, Longman, London (1979)
point (similar to the plug hole in a domestic sink, where, Ireland, J. W., Mechanics of Fluids, Butterworths, London (1971)
however, there is a vortex superposed to produce a spiral Reference tables
Houghton, E. L. and Brock. A. E., Tables for the Compressible
vortex or whirlpool). A vortex is flow in concentric circles with Flow of Dry Air, Edward Arnold, London (1961)
no radial flow. A doublet is the superposilion of a source and a Rogers, G. F. C. and Mayhew, Y. R., Thermodynamic and
sink of equal strength rn, initially a distance 20 apart brought Transport Properties of Fluids, third edition, Blackwell, Oxford
infinitely close together so that the product of their strength (1980)
and the distance between them remains a constant k. k is the Additional computer solutions
strength of the donblet and is equal to 2am. Smith, P. D., BASIC Hydraulics, Butterworths, London (1982)
This is a mathematical concept which is apparently imprac- Boundary layer flow
tical but yields a useful flow pattern. It is often used in Schlicting, H., Boundary Layer Theory, fourth edition,
combination with other simple patterns. McGraw-Hill, New York (1960)
Ideal fluid flow
O‘Neill, M. E. and Chorlton, E’., Ideal and Incompressible Fluid
1.5.9.4 Modelling Dynamics, Ellis Horwood, Chichester (1986)
For students preparing for examinations
Since there can be no flow across a stream line and the fluid is Douglas. J. F., Solutions of Problems in Fluid Mechanics, Parts 1
assume’d inviscid, any stream line in a flow pattern may be and 2, third edition, Pitman, London (1987)
replaced by a solid surface with no effect on the rest of the Brasch, D. J. and Whyman, D., Problems in Fluid Flow, Edward
pattern. A stream line forming a closed contour may be Arnold, London (1986)
replaced by a solid body to model the flow pattern around a
body of the same shape. This provides a method of writing
mathematical models to describe the flow streams around 1.6. Principles of thermodynamics
various shapes.
The limitations of the model must be understood, and the 1.6.1 Introduction
effects of rotation and viscosity particularly considered, when
applying the analysis to real situations. For example, the flow Thermodynamics is concerned with energy transfers in pro-
round a cylinder may be modelled by a combination of doublet cesses. Two modes of transfer are recognized: work (transfer)
and parallel flow. The drag and lift forces calculated by and heat (transfer). From the mechanical engineer’s viewpoint
integrating the resulting pressure forces at the cylinder surface the main interest is the production of shaft power (rate of
appear to be zero. This is obviously not correct. work transfer) by a machine to which the energy is supplied by
In the real situation the viscosity of the fluid produces a heat transfer obtained directly or indirectly from the combu-
boundary layer at the cylinder surface. which, because of stion of a hydrocarbon fuel in air. Before proceeding. the
curvature, separates from the surface to form a wake. The terminology used in thermodynamics must be defined.
presence of the wake disturbs the downstream flow pattern The particular part of the working substance under conside-
2nd the form drag force is a consequence. Viscosity also ration is called the system, and this is separated from the
introduces a surface friction drag. surroundings by a boundary. In the closed or non-flow system
The shedding of vortices from the cylinder surface into the the mass of working substance is constant, but in the open or
wake also produces alternate positive and negative lift forces, flow system there is a mass flow rate across the boundary.
which are not predicted by the ideal flow analysis, although Some processes in reciprocating plant may be considered by
telegraph wires may often be observed vibrating in the wind. non-flow analysis but in steam plant, for example, most are
considered by flow analysis. The state of a system is defined by
the properties (pressure, temperature, etc.). Properties are
1.5.18 Conclusion
normally expressed specifically (Le. per unit mass) to enable
Fluid mechanics is often regarded as an empirical subject charts or tables to be used. The state of simple substances can
which makes use of formulae based only on observed exper- be described by two independent properties, but complex ones
imental results. This misconception is further compounded by such as mixtures need more definition. A change of state is