## Gravity Settling Separation

This article illustrates determination of terminal velocity required for gravity separation. Performing a force balance on liquid droplet settling out of gas stream results in following relationship.

where F_{B} is force due to Buoyancy, F_{D} is
force due to Drag and F_{G} is force due to gravity.

`F`

_{G}= m.g`F`

_{B}= m.ρ_{V}.g/ρ_{L}`F`

_{D}= C_{D}.(ρ_{V}.V_{t}²/2).A_{P}

Assuming spherical liquid droplet with diameter D_{P},
above equation gets reduced to

`F`

_{D}= F_{G}- F_{B}`F`

_{D}= C_{D}.(ρ_{V}.V_{t}²/2).(π.D_{P}²/4)`F`

_{G}- F_{B}= m.g(ρ_{L}- ρ_{V})/ρ_{L}`m = 4/3.(π.D`

_{P}³/8).ρ_{L}

Balancing above equation gives following relation for terminal
velocity V_{t}.

` V`_{t} = [(4gD_{P}/ 3C_{D}).(ρ_{L} - ρ_{V})/ρ_{V} ]^{0.5}

Drag coefficient depends on shape of particle and Reynold's number and can be determined from below graph.

Above curve can be simplified into 3 sections resulting into 3 settling laws.

Stoke's Law

At low Reynold's number < 2, a linear relationship exists
between C_{D} and Re.

`C`

_{D}= 24/Re`Re = D`

_{P}.ρ_{V}.V_{t}/μ_{V}`V`

_{t}= g.D_{P}².(ρ_{L}- ρ_{V})/18μ_{V}

In English units, with D_{P} in feet, V_{t} in
feet/sec, ρ in lb/ft³, g = 32.2 feet/sec² and μ in centipoise,
above equation can be expressed as.

` V`_{t} = 1488.g.D_{P}².(ρ_{L} - ρ_{V})/18μ_{V}

Criterion K is defined to determine the flow regime as following.

` K = D`_{P}[g.ρ_{V}(ρ_{L} - ρ_{V})/μ_{V}²]^{1/3}

Value of K is evaluated for Reynold's number 2 based on Stoke's Law.

`Re = 2`

`V`

_{t}= 2.μ_{V}/(ρ_{V}.D_{P})`K = (36)`

^{1/3}`K = 3.3`

For K < 3.3 Stoke's Law is applicable.

Intermediate Law

For 2 < Re < 500, Intermediate law applies.

`C`

_{D}= 18.5/Re^{0.6}`V`

_{t}= 0.153 g^{0.71}D_{P}^{1.14}(ρ_{L}- ρ_{V})^{0.71}/ ( ρ_{V}^{0.29}. μ_{V}^{0.43})

In English units, with viscosity in centipoise, above equation can be expressed as

` V`_{t} = 3.49 g ^{0.71} D_{P} ^{1.14} (ρ_{L} - ρ_{V})^{0.71}/ ( ρ_{V}^{0.29}. μ_{V}^{0.43})

Newton's Law

Newton's law is applicable for Reynold's number in range of 500 to 200,000.

`C`

_{D}= 0.44`V`

_{t}= 1.74 (g.D_{P}(ρ_{L}- ρ_{V})/ ρ_{V})^{0.5}

Value of K is evaluated for Reynold's number 500 based on Newton's Law.

`Re = 500`

`K = (500/1.74)`

^{2/3}`K = 43.5`

For K in the range of 3.3 to 43.5, Intermediate Law is applicable; for K > 43.5, Newton's Law is applied.

Stoke's Law is applicable for vapor separation from a continuous liquid phase and dispersed liquid separation from a continuous liquid phase. Intermediate Law is typically used for liquid separation from a continuous vapor phase.

Example

Calculate diameter for a vertical vapor liquid separator based
on gravity separation. Gas flowrate is 50,000 lb/h, density is
3.7 lb/ft³ and viscosity is 0.01 cP. Liquid flowrate is 12,000
lb/h, density is 51.5 lb/ft³ and viscosity is 0.42 cP. Assume
liquid particle diameter D_{PL} as 250 micron and vapor
particle diameter D_{PV} as 150 micron.

Terminal velocity is to be calculated for gravity settling of liquid droplet from vapor phase. K Value is calculated to determine the applicable flow regime.

` K = 41.1`

Intermediate Law is applicable for this case. Terminal velocity is calculated based on Intermediate Law.

` V`_{t} = 0.96 feet/sec

Margin of 75% is taken on terminal velocity and vessel diameter is calculated.

` D1 = 2.58 feet`

A second terminal velocity is calculated for disentraining vapor from liquid phase. Typically Stoke's law is applicable for vapor disentrainment. Terminal velocity is calculated based on Stoke's law.

` V`_{t} = 0.07 feet/sec

With margin of 75% on terminal velocity, vessel diameter is calculated.

` D2 = 1.22 feet`

Comparing both diameter values, highest value of 2.58 feet is selected as a minimum diameter required for gravity separation of vapor and liquid.

Resources