The problem requires calculating the total wet pick-up based on the bone-dry weight of the warp sheet, the target add-on percentage, and the concentration of the size paste.

Understanding Key Terms

• Bone-dry warp sheet weight: The weight of the warp sheet material without any moisture. Given as 12 kg.
• Target add-on (%): The percentage of the size solids applied relative to the bone-dry weight of the warp sheet. Given as 12%.
• Concentration (%, w/w): The percentage of actual size solids present in the total size paste mixture. Given as 12%.
• Wet pick-up (kg): The total weight of the size paste (including water) applied to the warp sheet. This is what needs to be calculated.

Calculating Add-on Weight (Size Solids)

First, determine the weight of the size solids that need to be applied. This is calculated using the target add-on percentage and the bone-dry weight.

Let $W_{dry}$ be the bone-dry weight of the warp sheet.

Let $AddOn_{\%}$ be the target add-on percentage.

Weight of Size Solids ($W_{solids}$) = $W_{dry} \times \frac{AddOn_{\%}}{100}$

Substituting the given values:

$W_{solids} = 12 \text{ kg} \times \frac{12}{100} = 12 \text{ kg} \times 0.12 = 1.44 \text{ kg}$

Calculating Total Wet Pick-up (Size Paste)

The calculated weight ($1.44$ kg) represents the mass of the *size solids*. The size paste itself is only 12% solids by weight. The total wet pick-up is the total weight of the paste required to achieve this $1.44$ kg of solids.

Let $W_{paste}$ be the total weight of the wet size paste (wet pick-up).

Let $C_{\%}$ be the concentration of size solids in the paste.

The relationship is: $W_{solids} = W_{paste} \times \frac{C_{\%}}{100}$

Rearranging to find $W_{paste}$:

$W_{paste} = \frac{W_{solids}}{\frac{C_{\%}}{100}} = W_{solids} \times \frac{100}{C_{\%}}$

Substituting the values:

$W_{paste} = \frac{1.44 \text{ kg}}{\frac{12}{100}} = \frac{1.44 \text{ kg}}{0.12} = 12 \text{ kg}$

Therefore, the total wet pick-up is 12 kg.