# Flow

## Flow

## Critical Flow

### Critical Flow FORMULA

\( Fr = \frac {v } { \sqrt {g \;\cdot\; h} } \) \( Froude \; number \;=\; \frac { velocity } { \sqrt { \; gravitational \; acceleration \;\;x\;\; depth \; of \; flow \; relative \; to \; the \; channel \; bottom } } \)

Where:

\(Fr\) = Froude number

\(v \) = velocity of the flow

\(g \) = gravitational acceleration

\(h \) = depth of flow relative to the channel bottom

## Curb gutter Flow Rate

### Curb gutter Flow Rate Formula

\( Q = \frac { 0.56 } { n } P^{5/3} m^{1/2} q^{8/3}\) \( gutter \; flow \; rate \;=\; \frac { 0.56 } { Manning's \; roughness \; coefficient } \; wetted \; perimeter^{5/3} \;\;x\;\; slope^{1/2} \;\;x\;\; flow \; width^{8/3}\)

Where:

\( Q \) = gutter flow rate

\( n \) = Manning's roughness coefficient

\( P \) = wetted perimeter (roadway cross slope)

\( m \) = longitudinal slope

\( q \) = flow width

Solve for:

\(n = \frac { 0.56 } { Q } P^{5/3} m^{1/2} Q^{8/3} \)

\(P = \frac { Qn } { 0.56 m^{1/2} Q^{8/3} } ^{3/5} \)

\(m = \frac { Qn } { 0.56 P^{5/3} Q^{8/3} } ^2 \)

\(Q = \frac { Qn } { 0.56 P^{5/3} m^{1/2} } ^{3/8} \)

## Flow Coefficient

Flow coefficient can be described as the volume (in US gallons) of water at 60°F that will flow per minute through a valve with a pressure drop of 1 psi across the valve. This gives us a method to compare flow capabilities of different valves. The flow coefficient allows us to determine what size valve is required for a given application.

Flow Coefficient is primarily used when sizing control valves. However, it can be used to characterize other types of valves such as ball valves and butterfly valves.

### Flow Coefficient Formula

\(C_v = Q \sqrt {\frac{SG} {\Delta p} }\) \( flow \; coefficient \;=\; flow \; rate \; capacity \sqrt {\frac{ specific \; gravity } { difference \; in \; pressure } }\)

Where:

\(C_v\) = flow coefficient

\(Q\) = flow rate capacity

\(SG\) = specific gravity of fluid (water at 60°F = 1.0000)

\(\Delta p\) = pressure differential

## Flow Rate

Flow rate ( \(Q\) ) measure the amount of fluid that flows in a given time past a specific point.

### Flow Rate formula

\(Q = \frac { \Delta V } { \Delta t } \) \(flow \; rate \;=\; \frac { difference \; in \; volume } { difference \; in \; time } \)

Where:

\(Q\) = flow rate

\(\Delta t\) = time differential

\(\Delta V\) = volume differential

## Flow Regime

### Bubble Flow

Bubble flow occurs when liquid occupies the bulk of the cross-section and vapor flows in the form of bubbles along the top of the pipe. During this phase the vapor and liquid velocities are about the same.

### Plug Flow

The bubbles in the pipe eventually coalesce as the vapor rate increases. Plug flow is similar to slug flow but liquid is the continuous phase along the bottom of the pipe.

### Stratified Flow

As the vapor rate continues to increase, the plugs join and become a continuous phase. In fully stratified flow, vapor flows along the top of the pipe and liquid flows along the bottom. The interface between the two phases is relatively smooth and the flow area occupied by each phase remains constant. In uphill flow, stratified flow rarely occurs with wavy flow being favored.

### Wavy Flow

As the vapor rate increases even more in the pipe during stratified flow, the top most layer begins to form waves. As the vapor rate increases, the amplitude of the waves increases. Wavy flow can occur when flow moves uphill, in horizontal pipe or in downhill flow. Most often, it is found in piping that is horizontal but it may occur in uphill flow. When the flow is moving downhill, wavy flow can still occur but the amplitude of the waves is not as great.

### Slug Flow

Slug flow occurs when the speed of the vapor phase pushes the waves from the wavy flow regime onto each other. The waves grow until the liquid waves touch the top of the pipe and form frothy slugs. The velocity of frothy slugs and the "slugs" of vapor is faster than the liquid velocity. Slug flow is found in lower velocities in piping running uphill than in horizontal piping. When running downhill, it takes higher vapor rates to establish slug flow than in horizontal pipe.

Because the slugs are moving faster than the average liquid velocity, care should be taken to avoid slug flow around fittings. Severe water hammer may occur when changing flow direction when slug flow is occurring.

### Annular Flow

Annular flow is a two-phase flow regime where the liquid forms a film of varying thickness along the wall of the pipe and the vapor phase flows at a higher speed down the middle of the pipe. The interface between the vapor and liquid phase is not entirely well defines. Part of the liquid is sheared off from the film by the vapor and is carried along in the core as entrained droplets. At the same time, turbulent eddies in the vapor deposit droplets on the liquid film. Due to the different forces on the fluid, the thickness of the liquid film is not constant across the cross section of the pipe. The effects of gravity can cause the thickness of the fluid film towards the bottom of the annulus to be bigger than the top. Downstream of bends, most of the liquid will be at the outer wall.

### Spray Flow

Spray Flow is also known as mist or dispersed flow occurs when two-phase flow where the liquid phase is the dispersed phase and exists in the form of many droplets, while the gas phase is the continuous phase. This occurs when the velocity of the vapor tears the liquid film away from the wall and is carried by the vapor as entrained droplets.

Sprays are formed for industrial, commercial, agricultural, and power generation purposes by injection of a liquid stream into a gaseous environment. In addition, sprays can form naturally in a falling or splashing liquid.

## Laminar Flow

Laminar flow generally happens when dealing with low Reynolds Numbers in pipes. This could be due to low velocities, large diameters or high viscosities. Laminar flow can be modeled as a series of liquid cylinders in the pipe, where the innermost parts flow the fastest, and the cylinder touching the pipe isn't moving at all.

Shear stress in laminar flow is independent of the density - ρ, and the shear stress depends almost only on the viscosity - μ.

## Orifice Flow Rate

Orifice flow rate is the amount of fluid that flows in a given time.

### Orifice Flow Rate Formula

\(Q = C_d A_o \sqrt { 2Gh } \) \( flow \; rate \;=\; discharge \; coefficient \;\;x\;\; orifice \; area \; \sqrt { \; 2 \;\;x\;\; gravitational \; constant \;\;x\;\; center \; line \; head } \)

Where:

\(Q\) = flow rate

\(C_d\) = discharge coefficient

\(A_o\) = orifice area

\(G\) = gravitational constant

\(h\) = center line head

Solve for:

\(G = \frac { \left ( \frac { Q } { C_d A_o } \right) ^2 } { 2H } \)

\(H = \frac { \left ( \frac { Q } { C_d A_o } \right) ^2 } { 2G } \)

## Slug Flow

Slug flow occurs when the speed of the vapor phase pushes the waves from the wavy flow regime onto each other. The waves grow until the liquid waves touch the top of the pipe and form frothy slugs. The velocity of frothy slugs and the "slugs" of vapor is faster than the liquid velocity. Slug flow is found in lower velocities in piping running uphill than in horizontal piping. When running downhill, it takes higher vapor rates to establish slug flow than in horizontal pipe.

Because the slugs are moving faster than the average liquid velocity, care should be taken to avoid slug flow around fittings. Severe water hammer may occur when changing flow direction when slug flow is occurring.

## Stratified Flow

As the vapor rate continues to increase, the plugs join and become a continuous phase. In fully stratified flow, vapor flows along the top of the pipe and liquid flows along the bottom. The interface between the two phases is relatively smooth and the flow area occupied by each phase remains constant. In uphill flow, stratified flow rarely occurs with wavy flow being favored.

## Venturi Tube Flow Rate

### Venturi Tube Flow Rate Formula

\(Q = C_v A \sqrt {2 g \Delta h} \) \( volumetric \; flow \; rate \;=\; flow \; coefficient \;\;x\;\; cross \; section \; area \; \sqrt { \; 2 \;\;x\;\; gravitational \; acceleration \;\;x\;\; head \; loss } \)

\(Q = C_v A \sqrt { \frac { 2 \Delta P} {\rho} } \) \( volumetric \; flow \; rate \;=\; flow \; coefficient \;\;x\;\; cross \; section \; area \; \sqrt { \; \frac { \; 2 \;\;x\;\; pressure \; loss } { density } } \)

Where:

\(Q\) = volumetric flow rate / flow rate

\(C_v\) = flow coefficient

\(A\) = cross section area

\(\Delta P\) = pressure loss

\(\rho\) (Greek symbol rho) = density

\(g\) = gravitational acceleration

\(\Delta h\) = head loss

## Volumetric Flow Rate

Volumetric flow rate ( \(Q\) or \(V\) ) (also called flow rate) is the amount of fluid that flows in a given time past a specific point.

### Volumetric Flow Rate formula

\(Q = A v \) \( volumetric \; flow \; rate \;=\; cross section \; area \;\;x\;\; flow \; velocity \)

\(Q = \frac {V} {t} \) \( volumetric \; flow \; rate \;=\; \frac { volume } { time } \)

\(Q = k i A \) \( volumetric \; flow \; rate \;=\; hydraulic \; conductivity \;\;x\;\; hydraulic \; gradient \;\;x\;\; cross section \; area \)

Where:

\(Q\) = volumetric flow rate

\(A\) = cross section area

\(i\) = hydraulic gradient

\(k \) = hydraulic conductivity

\(t\) = time

\(v\) = flow velocity

\(V\) = area volume

## Wavy Flow

As the vapor rate increases even more in the pipe during stratified flow, the top most layer begins to form waves. As the vapor rate increases, the amplitude of the waves increases. Wavy flow can occur when flow moves uphill, in horizontal pipe or in downhill flow. Most often, it is found in piping that is horizontal but it may occur in uphill flow. When the flow is moving downhill, wavy flow can still occur but the amplitude of the waves is not as great.

Tags: Equations for Flow Rate