Karra Model 

The Karra Model is a mathematical formula used to predict how a vibrating screen separates particles into:

• Oversize (won’t pass the screen)

• Undersize (passes the screen)

Engineers use it because it is:

✔ simple
✔ realistic
✔ based on screen geometry
✔ works well for industrial screens

1. Why Do We Need the Karra Model?

Every screening process asks one question:

“If I feed this mixture of particle sizes to the screen, how much will go through?”

But screening is NEVER perfect.

Some small particles don’t pass.
Some large particles sneak through.

So we need a model that predicts:

• Probability of passing

• Effect of particle size

• Effect of screen area

• Effect of open area

• Effect of screen inclination

Karra’s Model does exactly this.

Core Idea Behind Karra Model

Karra said:

Every particle has a probability of passing through a screen opening.
The closer the particle size is to the aperture size, the lower the probability.
The smaller the particle compared to the aperture, the higher the probability.

The Famous Karra Curve (S-Curve)

Karra expresses the passing probability using an S-shaped curve:

• Near 0% pass for particles MUCH bigger than the aperture

• Near 100% pass for particles MUCH smaller

• Smooth transition in between

4. The Basic Formula (Easy Form)

Karra’s passing probability:

$$P = \frac{1}{1+\left(\frac{d}{d_{50}}\right)^m}$$

Where:

• P = % passing through the screen

• d = particle size

• d₅₀ = size at which 50% particles pass

• m = slope parameter (controls steepness of the S-curve)

Meaning:

• When d = d₅₀, P = 50%

• When d ≪ d₅₀, P → 100%

• When d ≫ d₅₀, P → 0%

What Affects d₅₀ and m in the Karra Model?

Karra gave formulas for d₅₀ and m that depend on:

Screen aperture size (A)

Bigger aperture → easier to pass → larger d₅₀

Screen inclination

Higher inclination → poorer screening → bigger d₅₀

Screen area

More area → more chance to pass → smaller d₅₀

Material characteristics

Shape, density, moisture, and near-size content all affect m.

Most Important Interpretation

Karra model does not tell you how fast the material moves.
It does not model time.
It only predicts the probability of passing, given the screen conditions.

Screen capacity models (e.g., Mogensen, Bokan, Plitt) handle flowrate.
Karra focuses ONLY on accuracy of separation.

Example Case:

• Screen aperture = 5 mm

• d₅₀ predicted = 3.2 mm

• m = 4

• Particle size = 4 mm

Now calculate probability of passing:

$$P = \frac{1}{1+\left(\frac{4}{3.2}\right)^4}$$

Compute:

$$P = \frac{1}{1 + 2.44} = 0.29$$

Karra S-Curve (d50 = 3.2 mm, m = 4)

Summary

$$\frac{\mathrm{}}{}$$

Concept	Meaning
Karra Model	Predicts passing probability in screens
Curve Type	S-shaped probability curve
Key Equation	$P = 1 / [1 + (d/d_{50})^m]$
d₅₀	particle size with 50% passing
m	slope (sharpness) of curve
Big particles	low probability
Small particles	high probability
Uses	Screening efficiency prediction