Dear pandaba,

Calculating the acceleration of a particle in an airflow is done along Newton’s laws.

The acceleration force is calculated with the general resistance formula:

Force = resistance factor * *gas density * vrel^2 * π/4*^2

The formula Force = mass * acceleration

gives

acceleration = Force/mass.

Further:

Velocity2 = veolocity1 + acceleration * d(time)

Traveled distance = (velocity2 + velocity1)/2 , when d(time) is taken small enough.

This approach has to be done with numeric integration, as the acceleration is changing with the particle velocity and the (expanding) gas velocity.

In the situation that the particle acceleration equals the particle deceleration, due to collisions and friction, the flow has reached a quasi-stationary condition.

Due to the always expanding gas, the quasi stationary velocity constantly increases along the pipeline, until the pipe diameter is changed.

In vertical flow upwards, the acceleration is reduced by gravity.

In vertical flow downwards, the acceleration is increased by gravity.

In horizontal flow, gravity is theoretically not influencing the acceleration, because in this situation the gravity is perpendicular to the particle flow direction.

Have a nice day

Teus ■

Originally Posted by pandaba

Hi,

To find the acceleration length in vertical and horizontal gas solid flow , which formula to be used. Is it same for horizontal and vertical flow. I also want to find the corresponding pressure drop . Some one please help me in theory and related correlations to predict these values.

Thanks

Formula for calculating the acceleration length is :

L/D = [e**3.32] [d/D]**0.953 [Meu]**-0.0912 [Rho/Rhop]**-0.924

Where, L = acceleration length, meters

d= Particle Diameter, mm

D= Pipe Inside Diameter, mm

Meu= Solids to Air Ratio

Rho= Air Density, Kg/m3

Rhop= Particle Density, Kg/m3

For pressure drop due to acceleration, please refer to my article: Theory and Design of Pneumatic Conveying Systems.

Regards,

Amrit Agarwal

Pneumatic Conveying Consulting

polypcc@aol.com

Attachments

acceleration length 2 (DOC)

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Originally Posted by pandaba

Thanks Amit,



One thing to ask "whether the acceleration length is same for both horizontal and vertical flow". I think it will be smaller for vertical case as explained by Teus.

Dear Pandaba,

If the vertical section of the line is after the horizontal section, solids entering the vertical section have already been accelerated to a higher than the saltation velocity. This is the preferred line layout.

Regards,

Amrit Agarwal

Pneumatic Conveying Consulting ■

While traveling through the bend, the particle is decelerated by friction along the outer bend wall.

After the bend, the lost kinetic energy is compensated by reacceleration after the bend.

There is certainly an (re)-acceleration path after the bend.

The vertical acceleration is definitely influenced by the direction of the gravity and also is the reached quasi stationary particle velocity, where the acceleration energy equals the deceleration energy caused by friction and collisions.

BR

Teus ■

Dear Pandaba,

The term "Acceleration Length" is generally used to determine the length of horizontal pipe that is required after the solids pick-up point to accelerate the solids from zero velocity to a sufficiently high conveying velocity that prevents solids from plugging up the first bend in the conveying line.

The loss of conveying velocity in the bends, from the bend inlet to the bend outlet, depends upon many factors including bend shape and design, conveying conditions such as solids to air ratio, and physical properties of the solids. It can vary from 10% to 50%. Therefore, the selected acceleration length must provide for this loss in the first bend.

Because this loss of velocity in the bends is difficult to calculate precisely, sufficient length of horizontal pipe is provided after each bend so that solids recover their lost velocity before entering another bend. Pressure drop due to this velocity recovery is generally accounted for by equating each bend to certain equivalent number of pipe diameters.

Regards,

Amrit Agarwal

Pneumatic Conveying Consulting ■

Good day,

I calculated the velocity gradient for a horizontal to vertical bend and a horizontal bend.

The pipe section after the bends was divided in approx. 15 sections of 0.1 m and calculated in a time domain of 0.0001 second.

The product is cement and the pipe diameter is 200 mm.

The calculated velocities and pressure drop before and after the bend are represented in the attached files.

In the bend, the particle loses velocity, due to friction along the outer bend wall.

As a result of this particle velocity loss, the remaining cross section for the conveying gas decreases, resulting in a higher gas velocity.

After the bend, the particle velocity increases by the re-acceleration and the resulting gas velocity decreases again.

The sharp pressure drop after the bend is caused by the velocity loss in the bend plus a small pressure drop for the conveying losses over the acceleration length after the bend.

From the curves, the “equivalent” length is calculated as:

For the horizontal to vertical bend:

Pressure drop per meter = 0.095 bar/m

Pressure drop over acceleration length = 0.03 bar

“Equivalent” length for the bend based on pressure drop = 0.03 / 0.095 = 0.316 m

Related to a pipe diameter of 0.2 m, the “equivalent” length is 0.316/.2 = 1.6 times the pipe diameter.

For the horizontal to horizontal bend:

Pressure drop per meter = 0.075 bar/m

Pressure drop over acceleration length = 0.05 bar

“Equivalent” length for the bend based on pressure drop = 0.05 / 0.075 = 0.667 m

Related to a pipe diameter of 0.2 m, the “equivalent” length is 0.667/.2 = 3.33 times the pipe diameter.

The “equivalent” length for a bend varies with the orientation of the bend and I assume that this “equivalent” length also varies with product and chosen velocities.

I prefer the direct calculation above the guessed “equivalent” length method.

Take care

Teus

Attachments

bendacclengthhor_hor (PDF)

bendacclengthhor_vert (PDF)

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