# Piping Geometry Factor

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Piping geometry factor, abbreviated as Fp, a dimensionless number, is the pressure and velocity changes caused by fittings such as bends, expanders, reducers, tees, and Y's if directly conected to the valve.  It is used when analyzing the flow of fluids through pipes, pipelines, or ducts with non-standard geometries, such as bends, elbows, valves, fittings, and other obstructions.  The piping geometry factor quantifies how the irregularities or features in the piping system affect the resistance to flow and the pressure drop.  It is particularly relevant in engineering and design processes to determine the efficiency and performance of a fluid transport system.

### Liquid Control Valve Sizing Index

The value of the piping geometry factor can vary significantly depending on the specific geometry and the type of flow (laminar or turbulent).  It is often determined empirically through experiments or obtained from tables or charts provided in fluid mechanics references.

In practice, engineers use various empirical correlations and equations to estimate the piping geometry factor for different flow situations and pipe configurations.  These equations take into account factors like the type of fitting, the Reynolds number, and other flow parameters.  The piping geometry factor is necessary for calculating pressure drops, flow rates, and energy losses in piping systems.  It helps engineers and designers optimize piping layouts, select appropriate pipe sizes, and minimize energy consumption in fluid transport systems.

## Step 1  -  Piping Geometry Factor

### Piping Geometry Factor Formula

$$F_p \;=\; 1 \;/\; \sqrt{1 + ( \Sigma K \;/\; 0.00214 ) \; ( C_v \;/\; D_v^2 )^2 }$$

### Piping Geometry Factor - Solve for Fp

 algebraic sum, K flow coefficient, Cv nominal valve size, Dv

Symbol English Metric
$$F_p$$ = piping geometry factor $$dimensionless$$
$$\Sigma K$$ = algebraic sum $$dimensionless$$
$$C_v$$ = flow coefficient $$dimensionless$$
$$D_v$$ = nominal valve size $$in$$ $$mm$$

### algebraic sum Formula

$$\Sigma K$$  (the algebraic sum of the velocity head loss coefficient for all the fittings that are attached to the valve)

$$\Sigma K \;=\; K_1 + K_2$$

$$\Sigma K \;=\; K_1 + K_2 + K_{B1} + K_{B2}$$

### Piping Geometry Factor K - Solve for K

 resistance coeff of upstream ftg, K1 resistance coeff of downstream ftg, K2 inlet Bernoulli coefficient, KB1 outlet Bernoulli coefficient, KB2

Symbol English Metric
$$K_1$$ = resistance coefficient of upstream fittings $$dimensionless$$
$$K_2$$ = resistance coefficient of downstream fittings $$dimensionless$$
$$K_{B1}$$ = inlet Bernoulli coefficient $$dimensionless$$
$$K_{B2}$$ = outlet Bernoulli coefficient $$dimensionless$$

### algebraic sum Plug-ins Formulas

$$\Sigma K$$  (the algebraic sum of the velocity head loss coefficient for all the fittings that are attached to the valve)

$$K_1 \;=\; 0.5 \; ( 1 - ( D_v^2 \;/\; d_1^2 ) \;)^2$$     (inlet expander / reducer)

$$K_2 \;=\; 1.0 \; ( 1 - ( D_v^2 \;/\; d_2^2 )\;)^2$$     (outlet expander / reducer)

$$K_1 + K_2 \;=\; 1.5 \; ( 1 - ( D_v^2 \;/\; d^2 ) \;)^2$$     (for a valve installed between identical expander / reducer)

$$K_{B1} \;=\; 1 - ( D_v \;/\; d_1 )^4$$     (inlet Bernoulli coefficient)

$$K_{B2} \;=\; 1 - ( D_v \;/\; d_2 )^4$$     (outlet Bernoulli coefficient)

$$K_{B1}$$ or $$K_{B2}$$     (are only used when the diameter of the piping approaching the valve is different from the diameter of the piping leaving the valve)

### Solve for K1

 nominal valve size, Dv upstream pipe inside diameter, d1

### Solve for K2

 nominal valve size, Dv downstream pipe inside diameter, d2

### Solve for K1 + K2

 nominal valve size, Dv pipe inside diameter, d

### Solve for KB1

 nominal valve size, Dv upstream pipe inside diameter, d1

### Solve for KB2

 nominal valve size, Dv downstream pipe inside diameter, d2
Symbol English Metric
$$K_1$$ = resistance coefficient of upstream fittings $$dimensionless$$
$$K_2$$ = resistance coefficient of downstream fittings $$dimensionless$$
$$K_{B1}$$ = inlet Bernoulli coefficient $$dimensionless$$
$$K_{B2}$$ = outlet Bernoulli coefficient $$dimensionless$$
$$D_v$$ = nominal valve size $$in$$ $$mm$$
$$d$$ = pipe inside diameter $$in$$ $$mm$$
$$d_1$$ = pipe inside diameter, upstream $$in$$ $$mm$$
$$d_2$$ = pipe inside diameter, downstream $$in$$ $$mm$$

### Liquid Critical Pressure Recovery Factor Formula

Measure the efficiency in maintaining pressure control when dealing with fluids near their critical pressure.

$$N_2$$ is a constant value of 890 when d, D is inch

$$N_2$$ is a constant value of 0.00241 when d, D is mm

$$F_{lp} \;=\; [ \; ( K_1 + K_{B1} \;/\; N_2 ) \; ( C_v \;/\; D_v^2 )^2 + ( 1 \;/\; F_l^2 ) \; ] ^{- \frac{1}{ 2}}$$

### Solve for Flp

 resistance of upstream fittings, K1 inlet Bernoulli coefficient, KB1 flow coefficient, Cv nominal pipe size, Dv control valve 100% open, Fl

Symbol English Metric
$$F_{lp}$$ = liquid critical pressure recovery factor $$dimensionless$$
$$K_1$$ = resistance coefficient of upstream fittings $$dimensionless$$
$$K_{B1}$$ = inlet Bernoulli coefficient $$dimensionless$$
$$N_2$$ = constant (value = 890) $$in$$ $$mm$$
$$C_v$$ = flow coefficient $$dimensionless$$
$$D_v$$ = nominal valve size $$in$$ $$mm$$
$$F_l$$ = control valve, $$C_v$$ at 100% open $$dimensionless$$

## Step 2  -  Pressure Required for Sizing

### liquid critical pressure ratio Formula

It's the ratio of the pressure drop across the valve to the critical pressure of the liquid.  This becomes relevant when dealing with liquids, particularly near or at their critical point.

$$F_f \;=\; 0.96 - 0.28 \; \sqrt{ P_v \;/\; P_c }$$

### Solve for Ff

 vapor pressure at inlet temp, Pv critical pressure at inlet temp, Pc

Symbol English Metric
$$F_f$$ = liquid critical pressure ratio $$dimensionless$$
$$P_v$$ = vapor pressure of fluid at inlet temperature $$lbf \;/\; in^2$$  $$Pa$$
$$P_c$$ = critical pressure of fluid at inlet temperature $$lbf \;/\; in^2$$  $$Pa$$

### valve pressure choked Formula

If  $$\Delta P_{valve} \le \Delta P_{choked}$$ , then  $$\Delta P \;=\; \Delta P_{sizing}$$

The choked valve pressure is the pressure at which the flow through the valve becomes choked.  It is the upstream pressure at which the velocity of the fluid reaches the speed of sound, and any further reduction in pressure does not lead to an increase in flow rate.

$$\Delta P_{choked} \;=\; ( F_lp \;/\; F_p )^2 \; ( P_1 - F_f \; P_v )$$

### Solve for ΔP

 liquid critical press recov factor, Flp piping geometry factor, Fp valve inlet pressure, P1 liquid critical pressure ratio, Ff vapor press of fluid at inlet temp, Pv
Symbol English Metric
$$\Delta P_{choked}$$ = valve pressure choked  $$lbf \;/\; in^2$$ $$Pa$$
$$F_{lp}$$ = liquid critical pressure recovery factor $$dimensionless$$
$$F_p$$ = piping geometry factor $$dimensionless$$
$$P_1$$ = valve inlet pressure $$lbf \;/\; in^2$$  $$Pa$$
$$F_f$$ = liquid critical pressure ratio $$lbf \;/\; in^2$$  $$Pa$$
$$P_v$$ = vapor pressure of fluid at inlet temperature $$lbf \;/\; in^2$$  $$Pa$$

## Step 3  -  Control Valve $$C_v$$

### conversion between Cv / Kv

• $$C_v \;=\; 1.16 \; K_v$$
• $$K_v \;=\; 0.862 \; C_v$$

### Control Valve $$C_v$$ Formula

Once $$C_v$$ is calculated, check to see if $$C_v$$ is within the limits of the selected control valve.  If not, the next size of control valve should be selected and the calculations repeated.

$$N_1$$ is a constant value of 1.00 when Q is gpm and P is psig

$$N_1$$ is a constant value of 0.865 when Q is $$Nm^3 /h$$ and P is bar

$$N_1$$ is a constant value of 0.0865 when Q is $$Nm^3 /h$$ and P is kPa

$$C_v \;=\; Q \;/\; ( N_1 \; F_p \; \sqrt{ \Delta P_{choked} \;/\; ( \rho_s \;/\; \rho_r ) } \;\; )$$

### Solve for Cv

 volumetric flow rate, Q piping geometry factor, Fp valve pressure choked, ΔP substance density, ρs reference density, ρr
Symbol English Metric
$$C_v$$ = control valve  $$gpm$$
$$Q$$ = volumetric flow rate $$ft^3 \;/\; sec$$  $$m^3 \;/\; s$$
$$N_1$$ = constant (value = 1.0)  $$gpm$$  -
$$F_p$$ = piping geometry factor $$dimensionless$$
$$\Delta P_{choked}$$ = valve pressure choked $$lbf \;/\; in^2$$  $$Pa$$
$$\frac{\rho_s}{\rho_r}$$ = relative density of fluid at inlet temperature $$dimensionless$$
$$\rho_s$$   (Greek symbol rho) = substance density $$lbm \;/\; ft^3$$ $$kg \;/\; m^3$$
$$\rho_r$$   (Greek symbol rho) = reference density $$lbm \;/\; ft^3$$ $$kg \;/\; m^3$$