To calculate the pressure required to maintain the flow of water through the tube, we can use the Hagen-Poiseuille equation:
ΔP = 8ηLQ / πr^4
Where:
ΔP = Pressure difference
η = Viscosity of water = 0.001 N.s/m^2
L = Length of the tube = 1 km = 1000 m
Q = Flow rate = 10 liters/s = 0.01 m^3/s
r = Radius of the tube = diameter/2 = 10 cm / 2 = 0.05 m
Converting the flow rate to m^3/s:
Q = 10 liters/s = 0.01 m^3/s
Plugging in the values:
ΔP = (8 * 0.001 N.s/m^2 * 1000 m * 0.01 m^3/s) / (π * (0.05 m)^4)
ΔP = (8 * 0.001 * 1000 * 0.01) / (π * 0.000625)
ΔP = 80 / (π * 0.000625)
ΔP = 203403.69 Pa
Therefore, the pressure required to maintain the flow of water at a rate of 10 liters/s through the horizontal tube is approximately 203403.69 Pa.