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Chapter 5: Problem 58

Figure \(P 5.58\) shows coal being dropped from a hopper onto a conveyor belt at a constant rate of \(25 \mathrm{ft}^{3} / \mathrm{s}\). The coal has a specific gravity ranging from 1.12 to \(1.50 .\) The balt has a speed of \(5.0 \mathrm{ft} / \mathrm{s}\) and a loaded length of \(15.0 \mathrm{ft}\). Estimate the torque required to turn the drive pulley of the conveyor belt. The drive pulley dianeter is \(2.0 \mathrm{ft}\). Assume that there is no friction between the belt and the other rollers.

Short Answer

Expert verified
The estimated torque required to turn the drive pulley of the conveyor belt is \(197660 lb*ft^2\).

Step by step solution

01

Determine the Mass Flow Rate of the Coal

The volume flow rate of the coal is given as \(25 ft^3/s\). Taking the average specific gravity of the coal to be \((1.12+1.50)/2 = 1.31\), the density of the coal can be calculated as \(\rho_{coal} = 1.31 * \rho_{water}\), where \(\rho_{water}=62.4 lb/ft^3\) is the density of water. This results in \(\rho_{coal} = 81.7 lb/ft^3\). The mass flow rate of the coal can then be determined by multiplying the volume flow rate by the density: \(\dot{m}_{coal} = \rho_{coal} * Q_{coal} = 25ft^3/s * 81.7 lb/ft^3 = 2042.5 lb/s\).
02

Find the Force Exerted on the Coal due to Gravity

The mass per unit length of the coal on the belt can be determined by dividing the mass flow rate by the belt speed: \(m_{belt} = \dot{m}_{coal} / v_{belt} = 2042.5 lb/s / 5.0 ft/s = 408.5 lb/ft\). The force due to gravity acting on the coal can then be calculated by multiplying the mass per unit length by the acceleration due to gravity (which is approximately \(32.2 ft/s^2\) and the loaded length of the belt: \(F_{gravity} = m_{belt} * g * l_{belt} = 408.5 lb/ft * 32.2 ft/s^2 * 15.0 ft = 197660 lb*ft\).
03

Calculate the Torque Required to Turn the Drive Pulley

The torque required to turn the drive pulley can be determined by multiplying the force exerted on the coal due to gravity by the radius of the pulley: \(Torque = F_{gravity} * r_{pulley} = 197660 lb*ft * 1.0 ft = 197660 lb*ft^2\). As the average specific gravity of the coal was used in the calculations, this torque is an estimate.

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