Question

The power input, P, to a centrifugal pump is a function of the volumetric flow, Q, the impeller diameter, D, the rotational rate, omega, the density, rho, the viscosity, mu, of the fluid, such that: P=f(Q,D,omega,rho,mu). How many dimensionless groups can be formed based on the information provided?

A. 1
B. 2
C. 3
D. 4
E. 5

Answer 1

Using the Buckingham π theorem, which relates the number of dimensionless groups to the number of variables and fundamental dimensions involved in a problem, and given that there are 6 variables across 3 fundamental dimensions, we can form 3 dimensionless groups from the given variables of the power input to a centrifugal pump.

Step-by-step explanation:

To determine the number of dimensionless groups that can be formed from the given variables of the power input to a centrifugal pump, we would typically use the Buckingham π theorem. This theorem relates the number of dimensionless groups to the number of variables and fundamental dimensions involved in a problem.

Given the variables: P (power), Q (volumetric flow rate), D (impeller diameter), ω (rotational rate), ρ (density), and μ (viscosity), we have a total of 6 variables. These variables involve the following fundamental dimensions: M (mass), L (length), T (time), and θ (temperature, affecting viscosity).

According to the theorem, if we have a total of k variables across n fundamental dimensions, the number of dimensionless groups is k - n. Here, the power has dimension ML²T⁻³, volumetric flow rate has dimension L³T⁻¹, the impeller diameter has dimension L, the rotational rate has dimension T⁻¹, density has dimension ML⁻³, and viscosity has dimension ML⁻¹T⁻¹. Thus, we have 6 variables and 3 fundamental dimensions (since μ incorporates temperature but it is not an independent dimension in this context), which gives us 6 - 3 = 3 dimensionless groups.