Mechanical energy balance

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Mechanical energy balance
Piping system example.png
Main subject
General macroscopic balance
Related Pages
Important Concepts

Variables used on this page

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Definition Variable Additional Information
Shaft Work Mechanical shaft work done on the surroundings. This will be positive when work is done on the system or negative when the system does work on its surroundings. When there is no shaft work for example, pumps or turbines,
Compression work The reversible conversion of mechanical energy into internal energy due to fluid compression. For an incompressible fluid
Viscous dissipation The irreversible conversion of mechanical energy into internal energy due to viscous flow. For a frictionless fluid
Flow velocity Velocity is a vector. Average velocity in a single component system often reported as a scalar and is . Often the brackets are not used for average velocity. In multi-component systems, both a mass average velocity vector and a molar average velocity vector can be defined .
Density Mass per unit volume
Pressure
Viscosity Also known as dynamic viscosity
Mass flow rate or more generally


Unit normal The unit normal is orthogonal (or normal, or perpendicular) to a differential surface of area dA.
Kinetic energy correction factor For fully-developed laminar flow in pipes, the kinetic energy correction factor is 0.5. For fully-developed turbulent pipe flow, the correction factor ranges from 0.9 and 0.96.
Shaft Work 'd'-Form Term in the 'd'-form of the mechanical energy balance. This is obtained by dividing the by the mass flow rate.
Viscous Dissipation 'd'-Form Term in the 'd'-form of the mechanical energy balance. This is obtained by dividing the by the mass flow rate.
Acceleration of Gravity The acceleration of gravity is or . When using English units the gravitational conversion factor, , will be useful.
Distance in z direction Distance in z direction or high based on a reference point.
Velocity Velocity relative to a system boundary at a point.

Introduction

The concept of mechanical energy is based on the the classical understanding that potential energy can be completely converted to kinetic energy and vice versa in the absence of friction or non-conservative forces. The development of the mechanical energy balance for a flowing system is quite involved. Here, we just give the final result:

            (1)

where is the rate of reversible conversion of mechanical energy into internal energy due to compression of the fluid and is defined as

            (2)

This term is positive if the fluid is under compression and negative if the fluid is under expansion; it is zero if the fluid is incompressible. The viscous loss term is the irreversible conversion of mechanical energy into internal energy due to viscous dissipation.

            (3)

Its precise form depends on the constitutive equation to describe viscosity. For instance, for an incompressible Newtonian, fluid in cylindrical coordinates:

            (4)

which is the sum of the square of the velocity gradients multiplied by the viscosity, which is always a positive value. If the fluid is inviscid (no viscosity) or has no velocity gradients, the viscous loss term will also be zero. The viscous loss terms can be determined if the flow profile is completely known (i.e. by solving the Navier-Stokes equations). However, this is only feasible for laminar flow. Most flows in practice are turbulent, and we do not know the exact flow profile. Hence, the values of such terms come from experiments, and we will use correlations to determine the viscous loss.

Viscous losses

In fluid flow, friction between adjacent fluid elements travelling at different velocities (or directions) causes a permanent loss of mechanical energy (measured either as pressure or head). If we consider laminar flow, each stream line has a different velocity (ranging from a maximum velocity at the centerline to zero velocity at the pipe wall). At the boundary of each stream line, therefore, friction is generated due to the different velocities. This is similar to rubbing your hands together. The friction generates heat which increases the temperature of the fluid and reduces the mechanical energy. In turbulent flow, the presence of eddies and vortexes further adds to friction and causes an additional decrease in available head. For many systems, however, the velocity and pressure profiles cannot be easily calculated. Rather, we take several sets of experimental data and develop empirical correlations between dimensionless variables in geometrically similar systems.

The viscous loss is divided into two parts: major losses defined with a friction factor and minor losses defined with a loss coefficient

            (5)

Major losses occur due to fluid flow through straight sections of pipe and minor losses occur due to changes in flow direction due to pipe fittings, bends, etc. The terms "major" and "minor" are historical terms and have nothing to due with the magnitude. Minor losses can be much greater than major losses. In the above equation, the subscript denotes different sections of pipe diameter with length and the subscript refers to the number of pipe fittings, bends, etc, each of which have their unique value for the loss coefficient.

Mechanical energy balance for control volumes with one inlet and one outlet

For control volumes with one inlet and one outlet, the mechanical energy balance can be written as

            (6)

where the outlet is denoted with subscript 2 and the inlet is denoted with subscript 1. If the fluid is incompressible, the density is constant and , hence the above equation simplifies to

            (7)

In the case of compressible fluids, if the density, pressure, and velocity of the fluid do not vary over the cross-section of the control volume, Equation (2) can be directly integrated as

            (8)

and Equation (6) can be expressed for a compressible fluid

            (9)

Note that (8) breaks down if there are significant changes in density, pressure, and/or velocity over the cross-section, and hence care must be made that this assumption is satisfied.

Bernoulli equation

At steady-state, if the flow has no viscous losses and there is no shaft work, the mechanical energy balance with one inlet and one outlet reduces to the well-known Bernoulli equation

            (10)

Equation of hydrostatic pressure

In the absence of flow, the steady-state mechanical energy reduces to the equation of hydrostatic pressure

            (11)


See also