Final answer:
The minimum diameter of the pipe must be around 4.41 mm to ensure a Reynolds number less than 2000. The pressure difference that must be maintained along each kilometer of the pipe is approximately 0.3066 N/m².
Step-by-step explanation:
In order to determine the minimum diameter of the pipe, we need to calculate the Reynolds number using the given flow rate, viscosity, and density of gasoline. Reynolds number (Re) can be calculated using the formula Re = (density x velocity x diameter) / viscosity. Since we want the Reynolds number to be less than 2000, we can rearrange the formula and solve for diameter: diameter = (Re x viscosity) / (density x velocity). Substituting the given values, we get diameter = (2000 x 1.00x10^-3) / (680 x 3.00x10^-2).
Calculating this value, we find that the minimum diameter must be approximately 4.41 x 10^-3 m, or 4.41 mm.
To calculate the pressure difference that must be maintained along each kilometer of the pipe, we can use Bernoulli's equation: P1 + (0.5 x density x velocity1^2) + (density x g x height1) = P2 + (0.5 x density x velocity2^2) + (density x g x height2), where P1 and P2 are the pressures at the start and end points respectively, velocity1 and velocity2 are the velocities at the start and end points respectively, density is the density of gasoline, g is the acceleration due to gravity, height1 and height2 are the heights at the start and end points respectively.
Since the flow rate is constant and the pipe is horizontal, the height difference between the two points is 0 and the equation simplifies to P1 + (0.5 x density x velocity1^2) = P2 + (0.5 x density x velocity2^2). We can rearrange this equation to solve for the pressure difference: ΔP = (0.5 x density x (velocity2^2 - velocity1^2)). Substituting the given values, we get ΔP = (0.5 x 680 x ((3.00x10^-2)^2 - (0))^2).
Calculating this value, we find that the pressure difference that must be maintained along each kilometer of the pipe is approximately 0.3066 N/m².