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I. Damage Assessment Based on Number of Affected Layers
The mass of damaged paper can be estimated by determining the total surface area of the affected layers and multiplying it by the basis weight. The calculation is as follows:
Damaged mass (kg) = Basis weight (kg/m²) × Width (m) × π × Roll diameter (m) × Number of damaged layers
Where:
- Basis weight is converted from g/m² to kg/m² (e.g., 250 g/m² = 0.250 kg/m²);
- Roll width and diameter are expressed in meters;
- π is the mathematical constant approximated as 3.1416.
Example:
A paper roll with a basis weight of 250 g/m² (0.250 kg/m²), width of 1374 mm (1.374 m), and diameter of 1100 mm (1.100 m) exhibits damage across 20 layers upon unpacking.
Calculation: 0.250 × 1.374 × 3.1416 × 1.100 × 20 = 23.7 kg (rounded to one decimal place).
II. Damage Assessment Based on Radial Depth of Surface Defects
When damage manifests as a localized indentation or cavity on the outer surface, the damaged mass may be estimated using geometric approximation of the annular segment removed. Two complementary methods are provided:
Method A - Empirical Approximation Formula
Damage ratio (%) = [4T(D − T)] / [(D + d)(D − d)] × 100%
where:
- T = depth of damage (cm),
- D = outer diameter of the roll (cm),
- d = inner diameter (core tube diameter) (cm).
Example:
A roll with D = 92 cm, d = 10 cm, and T = 3 cm yields:
[4 × 3 × (92 − 3)] / [(92 + 10) × (92 − 10)] × 100% ≈ 12.77%.
Given a total roll mass of 380 kg, the damaged mass is: 380 × 0.1277 = 48.5 kg.
Method B - Geometric Cross-Sectional Ratio (Core-Ignoring Approximation)
Damage ratio (%) = {[(D − 2T)/D]² − 1} × 100%
Using the same parameters:
[(92 − 6)/92]² − 1 = (0.9348)² − 1 ≈ −0.1262 → absolute value 12.62% (corrected for sign convention).
Thus, damaged mass = 380 × 0.1262 = 48.0 kg.
Note: This method assumes a uniform core diameter and omits explicit core dimension input; it is suitable for preliminary estimation when core data are unavailable.
III. Damage Assessment for Central End-Face Perforations
For circular defects located concentrically on the end face (i.e., not at the periphery but within the cross-sectional plane), the damaged area corresponds to an annular region bounded by inner and outer radii relative to the roll's center.
Damage ratio (%) = [(r₁² − r₂²) / R²] × 100%
where:
- R = outer radius of the roll (cm),
- r₁ = outer radius of the damaged zone (cm),
- r₂ = inner radius of the damaged zone (cm).
Example:
A roll with outer diameter D = 84.5 cm → R = 42.25 cm. A central end-face defect has r₁ = 37.5 cm and r₂ = 34.8 cm.
Damage ratio = [(37.5² − 34.8²) / 42.25²] × 100% = [(1406.25 − 1211.04) / 1785.06] × 100% ≈ 10.94%.
For a 380 kg roll, damaged mass = 380 × 0.1094 = 41.6 kg.
All calculations assume homogeneous paper density and uniform winding tension. Field verification and calibration against empirical sampling are recommended for high-accuracy loss assessment.