Question

An important design consideration in the two-phase pipe flow of solid-liquid mixtures is the terminal settling velocity below which the flow becomes unstable and eventually the pipe becomes clogged. On the basis of extended transportation tests, the terminal settling velocity of a solid particle in the rest water is given by VL = FL2gD(S − 1)−−−−−−−−−−√ , where FL is an experimental coefficient, g the gravitational acceleration, D the pipe diameter, and S the specific gravity of solid particle. What is the dimension of FL? Is this equation dimensionally homogeneous?

Answer 1

FL is an experimental coefficient in the equation VL = FL^2gD(S-1)^(-1/2), and its dimension depends on the specific experiments. The equation is not dimensionally homogeneous because the units of each term do not cancel out to give [L/T].

Step-by-step explanation:

The question is asking for the dimension of FL in the equation VL = FL2gD(S-1)−1/2, and whether the equation is dimensionally homogeneous. The dimension of FL can be determined by looking at the units on both sides of the equation. The equation is dimensionally homogeneous if the units on both sides of the equation are the same. To determine this, we need to consider the units of each variable in the equation.

VL represents the terminal settling velocity, which has units of length per time (L/T). FL is an experimental coefficient, so its dimension is unknown and will depend on the specific experiments. For the equation to be dimensionally homogeneous, the dimension of FL needs to be [L/T].

To determine whether the equation is dimensionally homogeneous, we need to consider the units of each variable in the equation. Let's break down the units of each term:

• VL: [L/T]
• FL: [?]
• g: [L/T2]
• D: [L]
• S: [dimensionless]

As we can see, the units of each term in the equation do not cancel out to give [L/T]. Therefore, the equation is not dimensionally homogeneous.