Orifices are widely installed in piping systems and hydraulic machinery to produce a regular and reproducible loss of pressure.  Orifices are used to limit flow, or, in branching systems, to balance distributed flow, or to measure flow.


There are several different approaches and equations that are used in literature to estimate the pressure loss through an orifice for a specified flow rate. In many cases the methods do not treat the pressure loss calculation consistently, and common physical properties of the orifice are not used in a standard manner. In some cases, the pressure drop estimate relies on a discharge coefficient, and in other cases it is simply presented as graph plot of pressure drop versus flow rate (or pressure drop derived as a percentage of differential pressure across flow measurement). 


The Pipe Flow Expert software orifice calculations use a more generalized model, which utilizes the broad physical attributes of the orifice to determine the flow characteristics. This generates a K0 value (a k factor) that can be applied in a standard method as follows:


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Where:

H0 = Head Loss (in ft or m of fluid, depending on the units of velocity, ft/s or m/s)

K0 = K factor (for the density and velocity of fluid flow through the orifice diameter)

g   = acceleration due to gravity (in ft/s2 or m/s2)



While this final equation provides a standard method for calculating the pressure drop through different types of orifice, deriving the K0 factor and then applying it to orifice calculations in a complex pipe network (particularly in compressible gas systems where the density and velocity at the orifice inlet varies with pressure changes in the system), is a complex task, which also then relies on the development of further algorithms that allow convergence to balanced solution across the whole model.


The Pipe Flow Expert software handles the complexity of the calculations and makes it easy to perform calculations with orifices in pipe systems. The following sections detail the equations used to derived the orifice K0 value.


Sharp-Edged Orifice in a Straight Pipe 


A sharp-edged orifice in a straight pipe, where the upstream and downstream pipe sizes are the same, as illustrated in Figure 171.

Figure 171  Sharp-Edged Orifice in a Straight Pipe

The loss coefficient (K0) of a sharp-edged orifice in a straight pipe is calculated as follows:



Where the diameter ratio β = d0/d1 and the jet velocity  is given by:





Sharp-Edged Orifice in a Transition Section


A sharp-edged orifice in a transition section, where the upstream and downstream pipe sizes are not the same, is illustrated in Figure 172 and Figure 173.


Figure 172  Sharp-Edged Orifice in a Transition Section (Contraction)



Figure 173  Sharp-Edged Orifice in a Transition Section (Expansion)


The loss coefficient (K0) of a sharp-edged orifice in a transition section is calculated as follows:



Where the diameter ratio β = d0/d1 and the jet velocity  is given by:





Sharp-Edged Orifice Discharging to Atmosphere


A sharp-edged orifice that discharges to atmosphere is illustrated in Figure 174.


Figure 174  Sharp-Edged Orifice Discharging to Atmosphere

For this orifice configuration, the loss coefficient (K0) equation for a sharp-edged orifice in an expanding transition section can be used, however the downstream pipe diameter (d2) is effectively infinite and therefore the equation can be transformed into:


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Round-Edged Orifice in a Straight Pipe 

 


A round-edged orifice in a straight pipe, where the upstream and downstream pipe sizes are the same, is illustrated in Figure 175.

Figure 175  Round-Edged Orifice in a Straight Pipe

The loss coefficient (K0) of a round-edged orifice in a straight pipe is calculated as follows:


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Where the diameter ratio β = d0/d1 and where the jet contraction coefficient  is given by:



In the case of a generously rounded orifice where r/d0 is equal to or greater than 1, the jet contraction ratio  equals 1 and the loss coefficient becomes:





Round-Edged Orifice in a Transition Section


A round-edged orifice in a transition section, where the upstream and downstream pipe sizes are not the same, is illustrated in Figure 176 and Figure 177.


Figure 176  Round-Edged Orifice in a Transition Section (Contraction)


Figure 177  Round-Edged Orifice in a Transition Section (Expansion)


The loss coefficient (K0) of a round-edged orifice in a transition section is calculated as follows:




Where the diameter ratio β = d0/d1 and where the jet contraction coefficient  is given by:



In the case of a generously rounded orifice where r/d0 is equal to or greater than 1, the jet contraction ratio  equals 1 and the loss coefficient becomes:


Round-Edged Orifice Discharging to Atmosphere



A round-edged orifice that discharges to atmosphere is illustrated in Figure 178.


Figure 178  Round-Edged Orifice Discharging to Atmosphere

For this orifice configuration, the loss coefficient (K0) equation for a round-edged orifice in an expanding transition section can be used, however the downstream pipe diameter (d2) is effectively infinite and therefore the equation can be transformed into:



where the jet contraction coefficient  is given by:





Bevel-Edged Orifice in a Straight Pipe 

 


A bevel-edged orifice in a straight pipe, where the upstream and downstream pipe sizes are the same, is illustrated in Figure 179.

Figure 179  Bevel-Edged Orifice in a Straight Pipe

The loss coefficient (K0) of a bevel-edged orifice in a straight pipe is calculated as follows:



Where the diameter ratio β = d0/d1 and where the jet contraction coefficient  is given by:



And where Cb is a function of bevel angle ψ in degrees, and bevel thickness to diameter ratio t/d0 is given by:





Bevel-Edged Orifice in a Transition Section


A bevel-edged orifice in a transition section, where the upstream and downstream pipe sizes are not the same, is illustrated in Figure 180 and Figure 181.


Figure 180  Bevel-Edged Orifice in a Transition Section (Contraction)


Figure 181  Bevel-Edged Orifice in a Transition Section (Expansion)


The loss coefficient (K0) of a bevel-edged orifice in a transition section is calculated as follows:



Where the diameter ratio β = d0/d1 and where the jet contraction coefficient  is given by:



And where Cb is a function of bevel angle ψ in degrees, and bevel thickness to diameter ratio t/d0 is given by:





Bevel-Edged Orifice Discharging to Atmosphere



A bevel-edged orifice that discharges to atmosphere is illustrated in Figure 182.


Figure 182  Bevel-Edged Orifice Discharging to Atmosphere

For this orifice configuration, the loss coefficient (K0) equation for a bevel-edged orifice in an expanding transition section can be used, however the downstream pipe diameter (d2) is effectively infinite and therefore the equation can be transformed into:



Where the diameter ratio β = d0/d1 and where the jet contraction coefficient  is given by:



And where Cb is a function of bevel angle ψ in degrees, and bevel thickness to diameter ratio t/d0 is given by:





Thick-Edged Orifice in a Straight Pipe 

 


A thick-edged orifice in a straight pipe, where the upstream and downstream pipe sizes are the same, is illustrated in Figure 183.

Figure 183  Thick-Edged Orifice in a Straight Pipe

The loss coefficient (K0) of local resistance for thickness t equal to or less than 1.4d of a thick-edged orifice in a straight pipe is calculated as follows:



Where the diameter ratio β = d0/d1 and where the jet contraction coefficient  is given by:



And where Cth is given by:




Thick-Edged Orifice in a Transition Section


A thick-edged orifice in a transition section, where the upstream and downstream pipe sizes are not the same, is illustrated in Figure 184 and Figure 185.


Figure 184  Thick-Edged Orifice in a Transition Section (Contraction)


Figure 185  Thick-Edged Orifice in a Transition Section (Expansion)


The loss coefficient (K0) of a thick-edged orifice in a transition section is calculated as follows:



Where the diameter ratio β = d0/d1 and where the jet contraction coefficient  is given by:



And where Cth is given by:



Thick-Edged Orifice Discharging to Atmosphere



A thick-edged orifice that discharges to atmosphere is illustrated in Figure 186.


Figure 186  Thick-Edged Orifice Discharging to Atmosphere

For this orifice configuration, the loss coefficient (K0) equation for a thick-edged orifice in an expanding transition section can be used, however the downstream pipe diameter (d2) is effectively infinite and therefore the equation can be transformed into:



Where the diameter ratio β = d0/d1 and where the jet contraction coefficient  is given by:



And where Cth is given by: