8 As part of an industrial process a single pump causes the intake of a liquid chemical to the bottom end of a tube, draws it up the tube and then discharges it through a nozzle at the top end of the tube. The tube is straight and narrow, 35 m long and inclined at an angle of \(26 ^ { \circ }\) to the horizontal. The chemical arrives at the intake at the bottom end of the tube with a speed of \(6.2 \mathrm {~ms} ^ { - 1 }\). At the top end of the tube the chemical is discharged horizontally with a speed of \(14.3 \mathrm {~ms} ^ { - 1 }\) (see diagram). In total, the pump discharges 1500 kg of chemical through the nozzle each hour.
\includegraphics[max width=\textwidth, alt={}, center]{98053e88-1aec-4b0d-ae5f-ece4ad340266-5_405_1175_685_242} In order to model the changes to the mechanical energy of the chemical during the entire process of intake, drawing and discharge, the following modelling assumptions are made.

• At any instant the total resistance to the motion of all the liquid in the tube is 40 N .
• All other resistances to motion are ignored.
• The liquid in the tube moves at a constant speed of \(6.2 \mathrm {~ms} ^ { - 1 }\).
  1. State one other modelling assumption which is required to model the changes to the mechanical energy of the liquid with the given information.
  2. Determine the power at which the pump is working, according to the model.

When the power at which the pump is working is measured it is in fact found to be 450 W .

1. Find the difference between the total amount of energy output by the pump each hour and the total amount of mechanical energy gained by the chemical each hour.
2. Give one reason why the model underestimates the power of the engine.