Mechanical energy balance equation

Isothermal Gas Flow in Pipes and Channels Isothermal compressible flow is often encountered in long transport lines, where there is sufficient heat transfer to maintain constant temperature. Velocities and Mach numbers are usually small, yet compressibihty effects are important when the total pressure drop is a large fraction of the absolute pressure. For an ideal gas with p = pM. JKT, integration of the differential form of the momentum or mechanical energy balance equations, assuming a constant fric tion factor/over a length L of a channel of constant cross section and hydraulic diameter D, yields,... 

The modifications are substituted into the mechanical energy balance (Equation 4-1) to determine u, the average discharge velocity from the leak ... 

Because there is no pressure change and no pump or shaft work, the mechanical energy balance (Equation 2-28) reduces to... 

The mechanical energy balance (Equation 4-1) also applies to adiabatic flows. For this case it is more conveniently written in the form... 

For liquids stored at their saturation vapor pressure, P = Ps, and Equation 4-91 is no longer valid. A much more detailed approach is required. Consider a fluid that is initially quiescent and is accelerated through the leak. Assume that kinetic energy is dominant and that potential energy effects are negligible. Then, from a mechanical energy balance (Equation 4-1), and realizing that the specific volume (with units of volume/mass) v = 1/p, we can write... 

Flow through spring-type reliefs is approximated as flow through an orifice. An equation representing this flow is derived from the mechanical energy balance (Equation 4-1). The result is similiar to Equation 4-6, except that the pressure is represented by a pressure difference across the spring relief ... 

When elevation head and work transfer are neglected, the mechanical energy balance equation (6.13) with the friction term of Eq. (6.18) become... 

Equation 2.9-10 is the total differential energy balance, and it contains both thermal and mechanical energies. It is useful to separate the two. We can do this by taking the dot product of the equation of motion with the velocity vector v to get the mechanical energy balance equation ... 

Integration of Eq. 2.9-11 leads to the macroscopic mechanical energy balance equation, the steady-state version of which is the famous Bernoulli equation. Next we subtract Eq. 2.9-11 from Eq. 2.9-10 to obtain the differential thermal energy-balance... 

Mechanical Energy Balance Equation. Assuming no heat transfer to or from the surroundings (q = 0) and that frictional losses end up being dissipated as heat, the Steady Flow Energy Equation for a constant density fluid becomes ... 

The Mechanical Energy Balance Equation (14), is the key equation when calculating pressure changes through piping systems. See Section 3.10 for examples of the use of the Mechanical Energy Balance Equation in such calculations. 

Equation (16) is similar to (14), the Mechanical Energy Balance Equation, but with gAhf and wx both equal to zero. 

The Mechanical Energy Balance Equation (14) for a constant density fluid is ... 

The Mechanical Energy Balance Equation (14), may be used to determine the fluid pressure at the inlet, to the pump (see Section 3.10). This must exceed the liquid s vapour pressure by the NPSH requirement of the pump, otherwise the pipework should be redesigned or a different pump with a smaller NPSH requirement installed. This explains why pumps are normally located at the lowest part of the pipework (maximum head) and close to the source (minimum inlet, pipework pressure drop). 

The absolute pressure (or head) of the liquid at the pump inlet, the suction side, must always be greater than the fluid vapour pressure by the NPSH. Let us use the Mechanical Energy Balance Equation (14), to find the pressure at the pump inlet. 

The second term in the mechanical-energy balance. Equation 5.1, is the change in potential energy and requires no comment. The third term is "pressme work" and its evaluation depends on whether the fluid is compressible or incompressible. Because the increase in pressure across the fan is small, we treat the flow as essentially incompressible. Thus, the fluid density may be removed from the integral sign and the mechanical energy balance becomes... 

Starting with the open system balance equation, derive the steady-state mechanical energy balance equation (Equation 7.7-2) for an incompressible fluid and simplify the equation further to derive the Bernoulli equation. List all the assumptions made in the derivation of the latter equation. 

MECHANICAL-ENERGY BALANCE. Equation (4.29) may be written over a short length of conduit in the following differential form ... 

Steady-state flow is never achieved with foamed fluids rather, the flow is dynamic. Foams flow dynamically because the pressure, which is continually changing, affects the viscosity, flow rate, and density of the foam at any given interval in the tubular. This problem can be accounted for by numerically integrating the mechanical energy balance equation from bottomhole to surface conditions. 

Finally, we substitute Eq. (2.7-26) into (2.7-10) and 1/p for V, to obtain the overall mechanical-energy-balance equation ... 

To evaluate the upstream pressure pi we use the mechanical-energy balance equation (2.7-28) assuming no frictional losses and turbulent flow. (This can be checked by calculating the Reynolds number.) This equation then becomes, for a = 1.0,... 

Applying the mechanical-energy-balance equation (2.7-28) to points 1 and 2,... 

This is the mechanical-energy loss due to skin friction for the pipe in N m/kg of fluid and is part of the F term for frictional losses in the mechanical-energy-balance equation (2.7-28). This term (Pi—Pz)/ for skin friction loss is different from the (p, — Pz) term, owing to velocity head or potential head changes in Eq. (2.7-28). That part of F which arises from friction within the channel itself by laminar or turbulent flow is discussed in Sections 2.10B and in 2. IOC. The part of friction loss due to fittings (valves, elbows, etc.), bends, and the like, which sometimes constitute a large part of the friction, is discussed in Section 2.10F. Note that if Eq. (2.7-28) is applied to steady flow in a straight, horizontal tube, we obtain (pi — Pz)/p = F. 

Frictional losses in mechanical-energy-balance equation. The frictional losses from the friction in the straight pipe (Fanning friction), enlargement losses, contraction losses, and losses in fittings and valves are all incorporated in the F term of Eq. (2.7-28) for the mechanical-energy balance, so that... 

---