Chapter 2, Problem 2.13P

To determine

The volume occupied by $1000\text{lb}$ of refuse following baling with the same waste as it would be in a garbage can.

Answer to Problem 2.13P

The loose volume is $4.77{\text{yd}}^{\text{3}}$ .

The baled volume is $1.2{\text{yd}}^3$ .

Explanation of Solution

Given data:

Weight of the refuse is $1000\text{lb}$ .

Loose density for miscellaneous waste is $300\text{lb}/{\text{yd}}^{\text{3}}$ .

Baled density for miscellaneous waste is $1000\text{ld}/{\text{yd}}^{\text{3}}$ .

Concept used:

Refer Table 2-5 and P2-13 for the composition, loose density and baled density of the refuse.

Component	Percent by weight	Loose density$\left(\text{lb}/{\text{yd}}^{\text{3}}\right)$	Baled density$\left(\text{lb}/{\text{yd}}^{\text{3}}\right)$
Newsprint	$21$	$20-55$	$720-1000$
Office paper	$15$	$400$	$700-750$
Cardboard	$8$	$350$	$1000-1200$
Glass	$15$	$500-700$	$1800-2700$
HDPE	$3$	$15$	$65$
PETE	$5$	$30-40$	$400-500$
Yard waste	$23$	$24$	$400-500$
Aluminum can	$4$	$50-74$	$250$
Miscellaneous	$11$	$300$	$1000$

Calculation:

Calculate the loose density.

  ${\rho }_{\text{l}}=\left[\begin{array}{c}21\%\text{of}20\text{lb}/{\text{yd}}^{\text{3}}+15\%\text{of}400\text{lb}/{\text{yd}}^{\text{3}}\\ +8\%\text{of}350\text{lb}/{\text{yd}}^{\text{3}}+15\%\text{of}500\text{lb}/{\text{yd}}^{\text{3}}\\ +3\%\text{of}15\text{lb}/{\text{yd}}^{\text{3}}+5\%\text{of}30\text{lb}/{\text{yd}}^{\text{3}}\\ +23\%\text{of}24\text{lb}/{\text{yd}}^{\text{3}}+4\%\text{of}50\text{lb}/{\text{yd}}^{\text{3}}\\ +11\%\text{of}300\text{lb}/{\text{yd}}^{\text{3}}\end{array}\right]$

  $=\left[\begin{array}{l}\frac{21}{100}\times 20\text{lb}/{\text{yd}}^{\text{3}}+\frac{15}{100}\times 400\text{lb}/{\text{yd}}^{\text{3}}\\ +\frac{8}{100}\times 350\text{lb}/{\text{yd}}^{\text{3}}+\frac{15}{100}\times 500\text{lb}/{\text{yd}}^{\text{3}}\\ +\frac{3}{100}\times 15\text{lb}/{\text{yd}}^{\text{3}}+\frac{5}{100}\times 30\text{lb}/{\text{yd}}^{\text{3}}\\ +\frac{23}{100}\times 24\text{lb}/{\text{yd}}^{\text{3}}+\frac{4}{100}\times 50\text{lb}/{\text{yd}}^{\text{3}}\\ +\frac{11}{100}\times 300\text{lb}/{\text{yd}}^{\text{3}}\end{array}\right]$

  $=\left[\begin{array}{c}4.2\text{lb}/{\text{yd}}^{\text{3}}+60\text{lb}/{\text{yd}}^{\text{3}}+28\text{lb}/{\text{yd}}^{\text{3}}\\ +75\text{lb}/{\text{yd}}^{\text{3}}+0.45\text{lb}/{\text{yd}}^{\text{3}}+1.5\text{lb}/{\text{yd}}^{\text{3}}\\ +5.52\text{lb}/{\text{yd}}^{\text{3}}+2\text{lb}/{\text{yd}}^{\text{3}}+33\text{lb}/{\text{yd}}^{\text{3}}\end{array}\right]$

  $=209.67\text{lb}/{\text{yd}}^{\text{3}}$

Here, the loose density is ${\rho }_{\text{l}}$ .

Calculate the baled density.

  ${\rho }_{\text{b}}=\left[\begin{array}{c}21\%\text{of}720\text{lb}/{\text{yd}}^{\text{3}}+15\%\text{of}700\text{lb}/{\text{yd}}^{\text{3}}\\ +8\%\text{of}1000\text{lb}/{\text{yd}}^{\text{3}}+15\%\text{of}1800\text{lb}/{\text{yd}}^{\text{3}}\\ +3\%\text{of}65\text{lb}/{\text{yd}}^{\text{3}}+5\%\text{of}400\text{lb}/{\text{yd}}^{\text{3}}\\ +23\%\text{of}400\text{lb}/{\text{yd}}^{\text{3}}+4\%\text{of}250\text{lb}/{\text{yd}}^{\text{3}}\\ +11\%\text{of}1000\text{lb}/{\text{yd}}^{\text{3}}\end{array}\right]$

  $=\left[\begin{array}{l}\frac{21}{100}\times 720\text{lb}/{\text{yd}}^{\text{3}}+\frac{15}{100}\times 700\text{lb}/{\text{yd}}^{\text{3}}\\ +\frac{8}{100}\times 1000\text{lb}/{\text{yd}}^{\text{3}}+\frac{15}{100}\times 1800\text{lb}/{\text{yd}}^{\text{3}}\\ +\frac{3}{100}\times 65\text{lb}/{\text{yd}}^{\text{3}}+\frac{5}{100}\times 400\text{lb}/{\text{yd}}^{\text{3}}\\ +\frac{23}{100}\times 400\text{lb}/{\text{yd}}^{\text{3}}+\frac{4}{100}\times 250\text{lb}/{\text{yd}}^{\text{3}}\\ +\frac{11}{100}\times 1000\text{lb}/{\text{yd}}^{\text{3}}\end{array}\right]$

  $=\left[\begin{array}{c}151.2\text{lb}/{\text{yd}}^{\text{3}}+105\text{lb}/{\text{yd}}^{\text{3}}+80\text{lb}/{\text{yd}}^{\text{3}}\\ +270\text{lb}/{\text{yd}}^{\text{3}}+1.95\text{lb}/{\text{yd}}^{\text{3}}+20\text{lb}/{\text{yd}}^{\text{3}}\\ +92\text{lb}/{\text{yd}}^{\text{3}}+10\text{lb}/{\text{yd}}^{\text{3}}+110\text{lb}/{\text{yd}}^{\text{3}}\end{array}\right]$

  $=840.15\text{lb}/{\text{yd}}^{\text{3}}$

Here, the baled density is ${\rho }_{\text{b}}$ .

Write the expression to calculate the loose volume.

  $V_{\text{l}}=\frac{W}{{\rho }_{\text{l}}}$   ..... (I)

Here, the loose volume is $V_{\text{l}}$ and the weight of the refuse $W$ .

Substitute $1000\text{lb}$ for $W$ and $209.67\text{lb}/{\text{yd}}^{\text{3}}$ for ${\rho }_{\text{l}}$ in Equation (I).

  $\begin{array}{c}{V}_{\text{l}}=\frac{1000\text{lb}}{209.67\text{lb}/{\text{yd}}^{\text{3}}}\\ =4.77{\text{yd}}^{\text{3}}\end{array}$

Write the expression to calculate the baled volume.

  $V_{\text{b}}=\frac{W}{{\rho }_{\text{b}}}$  ..... (II)

Substitute $1000\text{lb}$ for $W$ and $840.15\text{lb}/{\text{yd}}^{\text{3}}$ for ${\rho }_{\text{b}}$ in Equation (II).

  $\begin{array}{c}{V}_{\text{b}}=\frac{1000\text{lb}}{840.15\text{lb}/{\text{yd}}^{\text{3}}}\\ =1.2{\text{yd}}^{\text{3}}\end{array}$

Conclusion:

Thus, the loose volume is $4.77{\text{yd}}^{\text{3}}$ .

The baled volume is $1.2{\text{yd}}^3$ .