INTRODUCTION

'Losses' are permanent conversions of mechanical energy to thermal energy in a flowing stream of fluid. Loss coefficients or K factors are ratios of mechanical energy differences (irreversibilities) across a component in a conduit system to kinetic energy at a specified section, either upstream or downstream. K factors are a convenient means of attributing to a given component the ratio of the irreversibilities it will create within system of conduits to the kinetic energy at a fixed section.

Miller seems to have been one of the first analysts to break with the tradition of blind acceptance of the use of incompressible loss coefficients for compressible fluid calculations. For this, he is to be commended. We will attempt an analysis of Miller's methods before proceeding to what we believe will be a more fruitful analysis of the work of Benedict et al.

THREE MAJOR CONSIDERATIONS

There are three major considerations in the simulation or estimation of fluid flow in networks. These considerations must be followed by specific actions. The first consideration 1) is the examination of the possibility of choking with gases and flashing liquids. The second 2) is the estimation of mechanical energy losses in straight, uniform section conduits. The third 3) is in the estimation of mechanical energy losses across fittings, perhaps of different entry and exit sections. Included in this last consideration is what occurs when flows combine or diverge.

1) The possibility of choked flow must be examined by applying one of the choked flow criteria: -dP/dU|_choke = G/g_c or -dP/dv|_choke = G²/g_c. These criteria are developed in Flow of Industrial Fluids. The choked flow criteria can be examined with the aid of the incremental method or the analytic method.

2) The correlations for straight conduit usually involve the Moody or the Fanning friction factors. The Moody friction factor is numerically four times the Fanning factor and the equations are adjusted accordingly. We use the Moody coefficient exclusively. Friction factors for straight, uniform cross section pipe are generally well established but they become less reliable at the approach to 'sonic' (choked) velocity and beyond with compressible fluids.

3) For fittings, the original K factors (proportion of identifiable kinetic energy converted to thermal energy by an impediment to flow) are well established for incompressible flows (liquids). The same factors tend to be used even for compressible flows for want of experimental data and because early research suggested there was no difference between factors for incompressible and compressible fluids. We will try to explain the problem that exists because of this assumption and to suggest solutions. The works of Miller and of Benedict et al offer help in dealing with compressible fluids.

The general so-called head loss equation, ADIABATIC CONSTRAINT

The general so-called head loss equation is derived from the differential form of the Bernoulli equation with irreversibilities. This equation was written under the adiabatic constraint. There is no dq term.

[1]

Equation [1] is set up as a functional relationship equated to zero. Each term represents differential energy per unit mass, including the last term, called by some the 'lost work'. The Greek deltas represent differential quantities that are not 'exact' – the path (relation of variables as they change from one state to another) has to be known before the integration can be performed. 'Exact' differentials are those such as dz and UdU whose integrated values depend only upon the limits of integration, 1 and 2. All of the integrals represent energy changes between sections numbered one and two. The first term on the right is the shaft work transmitted to or from the fluid. The second term is the differential change in static, pressure energy. The third term is the differential change in kinetic energy. Alpha, in the third term, is the velocity profile correction; here, assumed constant between sections. The fourth term is the differential change in potential energy involved in elevation change. The fifth term represents the so-called losses (the difference in mechanical energy between the two locations).

Actually, there are no energy losses: the mechanical energy is simply transformed into thermal energy and a temperature difference results. The word 'losses' might be justified if it were defined in terms of mechanical energy conversions.

SIMPLIFYING ASSUMPTIONS

There is an adiabatic constraint (no Ôq term). If no prime mover exists in the segment of conduit being considered, the first term on the right in equation [1] is zero. If the velocity profile is fully developed, alpha often is considered equal to one, but this assumption should be checked. If a gas is involved, and if the elevation changes are reasonably small, the fourth term is zero.

The above assumptions allow the formulation of the constrained 'head loss' equation. Again, the word 'loss' would be better replaced by the phrase 'conversion of mechanical to thermal energy'.

[2]

If the density is reasonably constant, the following integrated equation results:

[3]

DARCY-WEISBACH EQUATION

The empirical Darcy-Weisbach equation for constant cross sections with constant density fluids (liquids) is given in [4] (with constant density, the velocities are constant when the section does not change).

[4]

The mechanical energy conversion to thermal energy between two locations of a constant section conduit can be measured by the pressure drop divided by the density with constant density fluids (liquids). The Moody friction factor, f_M, can be estimated by dividing the mechanical energy losses or irreversibilities by the length to diameter ratio and by the kinetic energy. Correlations can be made among the variables L, D, U, P and r.

The empirical Darcy-Weisbach equation, [4], is a constrained version of the more general head loss equation, [1]. It is constrained to conduits of constant section and fluids of constant density. It relates changes in mechanical energy or pressure drop divided by density to the length to diameter ratio, the kinetic energy and an empirically determined coefficient. It is not applicable without modification to other cases.

LOSS COEFFICIENTS, K FACTORS

Irreversibilities due to the presence of a component in a conduit system are considered separately from those due to conduits. The use of the K factor is applied strictly to components, not conduit. Conduits are treated by the use of the Darcy-Weisbach equation and the 'friction' factor. An exception to this statement does occur when some people use a K factor correlation for pipe.

Problem 1: The word 'loss' is almost never fully defined. This means that it is often interpreted erroneously as a loss of pressure. This misconception is exacerbated by the phenomenon of pressure recovery.

Solution 1: Understand the problem. The word 'loss' is applied to the change in mechanical energy due to the presence of a component in a conduit system: the component can be a valve, bend or fitting. 'Losses' involve conversions from mechanical energy to thermal energy (internal energy plus heat energy flow). The conversions, although referred to the component, may occur upstream and, especially, downstream from it. There is no real loss of energy. Energy is conserved.

A loss coefficient involves the ratio between the mechanical energy conversion to thermal energy and the kinetic energy at a section. The purpose of the coefficient is to allow a simple number to be applied as a multiplying factor to a known kinetic energy at a known section

in order to obtain the change in mechanical energy attributable to a component.

It should be borne in mind that the change may occur far downstream and even a little upstream of the component introducing the disturbance.

Problem 2: The locations of the pressure and velocity measurements are not necessarily coincidental with the upstream and downstream faces of the component.

Solution 2: Measure the pressure and velocity at a convenient location sufficiently far from the component that the velocity profile is re-established. This technique effectively permits estimation of pressure and average velocity when neither the metering device nor the component are present. For the downstream measurement, make sure that recovery is complete from the profile disturbances due to the component. This usually means a distance of 30 to 50 pipe diameters of separation of the flow measuring element from the component, downstream. Project the measured numbers upstream and downstream.

Problem 3: Even if the points of measurement of flowing velocities and pressures could be made coincidental with the upstream and downstream faces of the component, the conversion of one form of energy to another takes place over 30 to 50 pipe diameters downstream of the component and about one pipe diameter upstream as the turbulence created by the presence of the component dies out.

Solution 3: Same as solution 2.

Loss coefficients or K factors apply to fittings or components of conduit systems. They are sometimes applied to conduit of constant section, but they were not intended for this use when they were conceived.

ORIGINAL DEFINITION OF A LOSS COEFFICIENT

The original loss coefficient applied only to incompressible fluids (liquids).

[5]

The purpose of the original loss coefficient was to give a multiplying factor to the average kinetic energy at a specific section in order to estimate changes in mechanical energy without having to measure them. The coefficient must be associated with a known section. The subscript, x, is to ensure that the correct velocity is used. Note that the velocity may change with even incompressible fluids at steady-state if the sectional area changes. The value of K must be established by experiment and, once established, may be used to establish conversions (losses) for similar components. The value is specific to the particular component and sometimes to the flow rate and to the location at which the velocity is measured or to which it is projected (to calculate kinetic energy).

Note that it is not always possible to measure velocity close to the entrance to or the exit from a fitting and that distance from these sections will involve additional irreversibilities that sometimes can be accommodated by projecting the measured velocities upstream or downstream.

MILLER'S COEFFICIENTS

Miller makes use of the concept of 'total' pressure in deriving his method. For incompressible fluids, Miller's K factors are identical to the traditional ones. Miller's K factors for compressible fluids have not been established as frequently as those for incompressible fluids, unfortunately. The total pressure concept probably was suggested by the total temperature and total enthalpy concepts. However, there is a difference. Under adiabatic conditions, the total enthalpy and the total temperature are constant along a conduit, but the total pressure is not, because of recovery and irreversibilities. Note that Benedict et al also use the concept of total pressure, but their correlation was made to mechanical energy and not to kinetic energy.

Total pressure is defined as the sum of the actual, flowing pressure and the product of density and kinetic energy. If the latter product is examined it will be seen to have units of pressure. The product of density and kinetic energy is also known as the 'velocity pressure'.

[6]

The same equation can be applied to the downstream section if the subscript 1 becomes 2. The quantity P_T2 is less than P_T1.

INCOMPRESSIBLE FLUIDS

For an incompressible fluid flowing through a component of a horizontal conduit with a changing section, the mechanical energy converted to thermal energy as a fraction of the kinetic energy at section 1 is given by the first equation of the set [7].

[7]

By multiplying the expression by the constant density, we have changed the units to those of pressure. We can re-arrange the last expression of the set [7].

[8]

From the second from last equation, above, the difference in total pressures across a component equals the loss coefficient times the constant density and the kinetic energy at a section. Stated differently, the difference in total pressures is proportional to the density times the kinetic energy at a section for a constant density fluid. The proportionality coefficient is the K factor. For liquids, density is considered constant, independent of flow; for gases and vapours, it is not.

From the definition of total pressure and velocity pressure, for incompressible fluid flow, we can develop the relationships of [9].

[9]

R₁ and R₂ are defined as the ratios of the flowing pressures to the total pressures at their respective locations. The ratios P_vi/P_Ti are ratios of velocity pressure to total pressure.

From the difference of the total pressures upstream and downstream given in the set [8], substitution of [9] gives [10].

[10]

The ratio of the downstream to upstream total pressures is equal to one minus the K factor (related to upstream kinetic energy) times one minus the ratio of the upstream actual to total pressures. Remember that, in the case of gases and vapours, the K factor is not constant.

Note that all the terms of equation [10] are dimensionless and that the ratio of total pressures (downstream to upstream) may be found as a function of upstream terms only.

The first equation of the set [10] can be manipulated.

[11]

Miller defines the ratio of the flowing upstream pressure to the total upstream pressure as R₁ and the ratio of flowing downstream pressure to total downstream pressure as R₂ (equation set [9]).

MILLER'S K FACTORS COMPARED TO THE ORIGINAL K FACTORS – INCOMPRESSIBLE FLUIDS

For an incompressible fluid, the density is constant and the two formulae must produce the same value of K. If the first formula of the set [11] can be resolved to the standard form, we will have proved the equivalence.

[12]

The equivalence is proved for incompressible fluids.

COMPRESSIBLE FLUIDS

We will now investigate Miller's method for compressible fluids. Equation [6], for total pressure, still applies and it can be used for downstream conditions if the subscript is changed to 2. Equation [8] becomes the development in set [13].

[13]

The product of the two ratios in the first term of the last of equations [13] is developed in [14].

[14]

The last ratio in equation [14] is the ratio of the mechanical energies across a component. This ratio could be thought of as an efficiency since it represents the fractional mechanical energy unconverted to thermal energy after the fluid has passed the component. One minus this ratio (the last equation of the set [13]) represents the fractional conversion of mechanical energy to thermal energy. The fractional conversion of mechanical energy to thermal energy is seen to equal the ratio of the velocity pressure to the total pressure.

If equation [14] is inserted into the last one of [13], it can be seen that the Miller K factor for a compressible fluid represents the ratio between the fractional conversion of mechanical energy to thermal energy and the velocity pressure as a fraction of total pressure.

How is the Miller loss coefficient measured in compressible flow? From equation [13], we can write equation [15].

[15]

The last expression in [15] is similar to equation [12], the incompressible fluid equation with correct downstream density but it tells us that we have to measure pressure, density and velocity upstream and downstream of the component, so that K₁ for gases is not numerically the same as K₁ for liquids. The two sets of values should be projected from the points of measurement to the upstream and downstream faces of the component. A projection method will be described later in this paper.

The original loss coefficient, K, is the ratio of the change in mechanical energy across a component to the kinetic energy, usually at the inlet to the component but sometimes at the outlet from it. It is a measure of the inefficiency of the component or of the fractional conversion of mechanical to thermal energy.

Equation set [15] does not give the influence of changing flow rates on the K factor. It simply tells how the K factor is measured. In a pipe tee, for instance, the K factors could be measured for different flow ratios, branch to discharge from run, and correlated with these ratios as Miller suggests. It is evident that K factors established for incompressible flows should only be used with circumspection for compressible flows and that K factors that ignore the effect of flow ratios in combining or dividing components leave something to be desired. Most engineering companies have their own rules of thumb that allow them to add safety factors to their methods of computation.

The author of this paper has formed the opinion that the cavalier treatment of the difference between incompressible and compressible K factors in the general literature on fluid flow is a source of many engineering errors. This treatment can be added to the lack of clarity of description and to the use of such terms as 'head' and 'feet (or meters) of fluid' and 'fluid friction' contributing to the long list of factors that cause confusion.

RATIO OF INCOMPRESSIBLE AND COMPRESSIBLE K FACTORS

We will establish the ratio between the incompressible and the compressible K factors so that the two may be compared on the same basis. Note that although one can imagine two hypothetical fluids, one incompressible, the other compressible, to have the same conditions at section 1, they will differ at section 2.

[16]

THE USE OF THE EXPANSION COEFFICIENT, Y, FOR VALVES

In the last equation of [16], the ratio of densities suggests the use of the inverse of the expansion factor, Y squared.

[16-a]

The coefficient, Y, is used to correct for different densities across a component. For valves, the curves of Y versus x (pressure drop divided by upstream pressure) are essentially straight lines terminating at a value of x equal to x_T. The value of x_T is best determined from the choked flow criterion, not from Y_T equals 0.667. See the paper Driskell, ISA Handbook of Control Valves. Driskell states that, for valves, the linear relationship of Y to x has been confirmed experimentally to a tolerance of about 2%.

The slopes of the Y versus x curves for valves vary with the type of valve, being steep for an angle venturi valve and less steep for a single seat globe valve.

Driskell states that the slope is also influenced by the ratio of specific heats of the gas involved. Air is taken as the standard compressible fluid and a correction factor is applied as follows in [17].

[17]

Note that the slope of the expansion factor, Y, is a function of one variable, x, the ratio of the pressure drop across a component to the upstream pressure. Essentially, it is linear with this ratio. The remaining terms are constants for a given fluid.

The slope of the expansion factor equation for valves is minus 1/(3F_kx_T). The factors in the denominator are non-dimensional as are x and Y. The coefficient, Y, is a multiplying factor that gives the influence of the expansion on mass flow through a component of any sort – contraction or enlargement.

The factor x_T depends on the type of valve or fitting. The value of F_k will depend upon the gas mixture and a molar average can be used for gas mixtures.

The expansion factor, Y, will vary from one, when there is no flow, to a fractional value, which Driskell claims to be 0.667 when flow is choked. The author of this paper believes the choked value to vary with choked pressure as was pointed out in the book Flow of Industrial Fluids.

By substituting equation [16a] and [17]into [16], it will be seen that the ratios of incompressible to compressible K factors vary with x, and, therefore, with flow. As the incompressible K factors are constant, the compressible ones must vary with flow.

Table 1 is extracted from the ISA Handbook of Control Valves.

TABLE 1, REFERENCE DATA FOR GASES AND STEAM
	SP. GRAVITY G	SP. HEATS RATIO k	FACTOR Fk
Acetylene	0.897	1.28	0.914
Air	1.000	1.40	1.00
Ammonia	0.587	1.29	0.921
Argon	1.377	1.67	1.19
Carbon Dioxide	1.516	1.28	0.914
Carbon Monoxide	0.965	1.41	1.01
Ethylene	0.967	1.22	0.871
Helium	0.138	1.66	1.19
Hydrogen Chloride	1.256	1.40	1.00
Hydrogen	0.0695	1.40	1.00
Methane	0.553	1.26	0.900
Methyl Chloride	1.738	1.20	0.857
Nitrogen	0.966	1.40	1
Nitric Oxide	1.034	1.40	1.00
Nitrous Oxide	1.518	1.26	0.900
Oxygen	1.103	1.40	1.00
Sulphur Dioxide	2.208	1.25	0.893
Steam (dry saturated) P1
0-80		1.32	0.94
80-245		1.30	0.93
245-475		1.29	0.92
475-800		1.27	0.91
800-1050		1.26	0.90
1050-1250		1.25	0.89
1250-1400		1.23	0.88

UTILITY OF MILLER'S METHOD

To emphasise the fact that Miller's K factors for compressible fluid flow are different from the original ones for incompressible fluids, we will add an upper case M as a subscript. Equations [18] may be derived directly from the first equation of the set [12].

[18]

To establish or to use Miller's loss coefficients, one must know the projected values P_, r and U to the upstream and downstream faces of the component of interest. The following sketch shows the placement of orifice plates used to measure velocity at the upstream tap location of each plate. The pressures are also measured at the same locations. The densities can be measured or estimated from pressure and temperature measurements.

Algorithm of Balzhiser, Samuels and Eliassen

We will use the algorithm of Balzhiser, Samuels and Eliassen described in Appendix IV of Flow of Industrial Fluids to project the values of pressure and density to the faces of the component. The distances from the faces to the upstream tap locations of the plates will be set at 50 pipe diameters to ensure that the velocity profile has become normal in each case. The average projected velocities at the cross sections of the faces may be calculated from G/r_i. We will use the ideal gas approximation to prove the utility of the method. The reader is invited to use a more exact approximation such as the R-K EOS.

The two simultaneous equations involved come from a first law, energy balance and a second law, force balance. The energy balance uses the ideal gas approximation. The force balance uses the Peter Paige equation. Both equations are given in the set [19].

[19]

The constant B is equal to gamma/(gamma – 1). The specific heat at constant pressure, c_P, is taken as 0.24 Btu/lb_m-F for air. The conversion factor from Btu's to ft-lb_f, J, is 778.16.

We assume steady-state flow and constant but unequal upstream and downstream sections.

In the example, we will use data from Crane in order to check the results. Assume ideal air flowing adiabatically in a 2 inch, schedule 40 pipe upstream and a 4 inch schedule 40 pipe downstream of a component. The nominal pressure and temperature are 100 psig and 60 F measured at the upstream tap of the upstream orifice plate. The nominal density is 0.596 lb_m/ft³.

If we take the mass flow rate as 1.655 lb_m/s (corresponding to 166.6 ft³/minute from Crane), we find that G₁ is 1.655/0.0233 or 71.025 lb_m/s-ft² and G₂ is 1.655/0.0884 or 18.722 lb_m/s-ft². The friction factors at complete turbulence will be taken as 0.019 for the 2 inch pipe and 0.017 for the four inch pipe from Crane. The downstream pressure, P₂, will be 90 psig.

The two simultaneous equations shown in [19] must have their suffixes changed for the two situations. For the two inch pipe, upstream, the suffix 2 becomes U . This is shown in the set [20]. For the 4 inch pipe, downstream, the suffix 1 becomes D. This is shown in set [22]. The partial differentials also change for the upstream pipe. Those for the downstream pipe remain the same. This is due to the forward calculation upstream and the backwards calculation downstream. We will give all formulae once more. The word 'forward' refers to calculation in the direction of flow. 'Backwards' refers to the opposite.

FORMULAE FOR UPSTREAM CALCULATIONS

[20]

[21]

FORMULAE FOR downSTREAM CALCULATIONS

[22]

[23]

CHOICE OF METERING DEVICES

Orifice plates are chosen for the metering devices as they inherently are velocity measuring devices. The component for which we wish to establish the Miller coefficient is an arbitrary one. We will use a ten psi difference between the upstream taps of each orifice plate and estimate the resulting pressures, densities and velocities at the component faces. This assumes that the remaining pressure drop occurs across the component for the given flow rate. The upstream pressure, P₁, would be controlled upstream and the downstream pressure, P₂, would be due to a controlled back pressure. The mass flow rate could be controlled by the component (a needle valve, for example).

SIMULATION INPUT AND RESULTS

The input to the simulation is given below.

The results of the simulation are given below.

Note that the pressure difference across the component is 9.03 psi whereas that across the entire assembly is 10 psi. The Miller K factor for the above example can be found as follows:

COMPARISON OF MILLER'S COMPRESSIBLE K FACTORS WITH INCOMPRESSIBLE ONES

To find the real incompressible loss coefficient, we must have experimental data on the component. However, for purposes of comparison, we can approximate the incompressible loss factor by assuming an incompressible fluid and that the velocities in the pipes will be related by beta squared. We will assume the same upstream and downstream pressures but unchanging densities.

These assumptions mean that the numbers plugged into the above equation change giving the following result:

The comparison, although based on an assumption, serves to show that incompressible loss coefficients are not equal to Miller's compressible coefficients and that the particular coefficients must always be used in their intended equations.

CONCLUSIONS AND SUMMARY

This paper discussed the concept of 'loss' coefficients for a component of a conduit system. The loss coefficient or K factor is a convenient means of estimating changes in mechanical energy using the K factor and the calculated kinetic energy at a specified location.

The paper also presented the Darcy-Weisbach equation and introduced the Moody 'friction' factor, both are used for estimating irreversibilities in straight, uniform section conduit. An extensive discussion on Miller's coefficients for piping components followed and it was shown that there was a difference between coefficients established for incompressible fluids and those for compressible ones.

This study has served to establish that Miller's methodology is not simple. If the methods of Benedict et al prove to be valid, there is probably no reason to use Miller's methods.

One important conclusion that results from Miller's work is that the use of incompressible loss coefficients for compressible fluids is not justifiable without well established safety factors.

Miller's coefficients should be obtained from data supported by reliable experiments and they should be used only with Miller's formulae. The conventional coefficients should be used for incompressible fluids and with circumspection for compressible fluids.