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Chapter 7: Problem 7

A mixing basin in a sewage filtration plant is stirred by a mechanical agitator with a power input \(\dot{W} \doteq F \cdot L / T\). Other parameters describing the performance of the mixing process are the fluid absolute viscosity \(\mu \doteq F \cdot T / L^{2},\) the basin volume \(V \doteq L^{3}\) and the velocity gradient \(G \doteq 1 / T\). Determine the form of the dimensionless relationship.

Short Answer

Expert verified
The dimensionless relationship between the parameters of the mechanical agitator in the sewage filtration plant can be represented as: \(\mu / (\dot{W} \cdot V \cdot G) = K\), where K is a dimensionless constant.

Step by step solution

01

Analyzing the given terms

We are given four parameters: power input \(\dot{W}\), absolute viscosity \(\mu\), basin volume \(V\), and velocity gradient \(G\). Their forms are as follows: - \(\dot{W} \doteq F \cdot L / T\) - \(\mu \doteq F \cdot T / L^{2}\)- \(V \doteq L^{3}\)- \(G \doteq 1 / T\) Where F represents force, L represents length and T represents time.
02

Establishing a dimensionless relationship

To establish a dimensionless relationship, we need to form a group that consists of one of the dependent parameters (on the left hand side) and other non-repeating variables (on the right hand side). Let's consider the dependent parameter as \(\mu\) and other variables as \(\dot{W}\), \(V\) and \(G\). We will then have a relationship formed as: \(\mu / (\dot{W} \cdot V \cdot G) = K\) ,where K is a dimensionless constant.
03

Confirming the dimensionless relationship

After substitution of variables with their respective forms, we get \(F \cdot T / L^{2} \div {(F \cdot L / T) \cdot L^{3} \cdot (1 / T)} = K\(1 \div L^{2}\)Considering the dimensions of the above equation, the left hand side and right hand side both have dimensions L^{-2} which confirms the dimensionless relationship between the parameters.

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Most popular questions from this chapter

(See The Wide World of Fluids article titled "Ice Engineering." Section \(7.9 .3 .)\) A model study is to be developed to determine the force exerted on bridge piers due to floating chuaks of ice in a river. The piers of interest have square cross sections. Assume that the force, \(R\), is a function of the pier width, \(b\), the thickness of the ice, \(d\), the velocity of the ice, \(V\), the acceleration of gravity, \(g,\) the density of the ice, \(\rho_{i},\) and a measure of the strength of the ice, \(E_{i},\) where \(E_{i}\) has the dimensions \(F L^{-2}\) (a) Based on these variables determine a suitable set of dimensionless variables for this problem. (b) The prototype conditions of interest include an ice thickness of 12 in. and an ice velocity of \(6 \mathrm{ft} / \mathrm{s}\). What model ice thickness and velocity would be required if the length scale is to be \(1 / 10 ?(\mathrm{c})\) If the model and prototype ice have the same density, can the model ice have the same strength properties as that of the prototype ice? Explain.

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