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Chapter 7: Problem 2

An equation used to evaluate vacuum filtration is $$Q=\frac{\Delta p A^{2}}{\alpha\left(V R w+A R_{f}\right)}$$ Where \(Q \doteq L^{3} / T\) is the filtrate volume flow rate, \(\Delta p \doteq F / L^{2}\) the vacuum pressure differential, \(A \doteq L^{2}\) the filter area, \(\alpha\) the filtrate "viscosity," \(V \doteq L^{3}\) the filtrate volume, \(R \doteq L / F\) the sludge specific resistance, \(w \doteq F / L^{3}\) the weight of dry sludge per unit volume of filtrate, and \(R_{f}\) the specific resistance of the filter medium. What are the dimensions of \(R_{f}\) and and \(\alpha ?\)

Short Answer

Expert verified
The dimensions of \(\alpha\) are \(F T\) (force times time), and the dimensions of \(R_{f}\) are \(L\) (length).

Step by step solution

01

Analyzing and rearranging the equation for \(\alpha\)

From the given equation, we can see that \(\alpha\) is one of the variables in the denominator. To isolate \(\alpha\) we rearrange the formula to this form: \(\alpha=\frac{\Delta pA^{2}}{Q(V R w+A R_{f})}\). This way we have \(\alpha\) on the one side of the equation.
02

Substituting the dimensions for \(\alpha\)

Now, we substitute the dimensions of each symbol into our rearranged equation. \(\Delta p = F/L^2\),\(A = L^2\),\(Q = L^3/T\),\(V = L^3\),\(R = L/F\),\(w = F/L^3\),and \(R_{f} = ?\),So, substituting these dimensions into the equation:\(\alpha = \frac{(F/L^2)(L^2)^2}{(L^3/T)((L^3)(L/F)(F/L^3) + L^2 R_{f})}\)Simplifying this equation, we get that the dimensions of \(\alpha = F T\)
03

Analyzing and rearranging the equation for \(R_{f}\)

Similarly, we rearrange the original equation to isolate \(R_{f}\):\(Q \alpha(V R w+A R_{f}) = \Delta p A^{2}\)\((Q \alpha A - Q \alpha V R w) = \Delta p A^{2}\)So, \(R_{f} = (\Delta p A^{2} - Q \alpha V R w)/(Q \alpha A)\) is the solution for \(R_{f}\).
04

Substituting the dimensions for \(R_{f}\)

Now, again, we substitute the dimensions for each symbol in the rearranged equation:\(\Delta p = F/L^2\),\(A = L^2\),\(Q = L^3/T\),\(V = L^3\),\(R = L/F\),\(w = F/L^3\),and \(\alpha = F T\),Substituting these dimensions:\(R_{f} = ((F/L^2)(L^2)^2 - (L^3/T)(F T)(L^3)(L/F)(F/L^3))/(L^3/T)(F T)(L^2)\)By simplifying this equation, we find the dimensions of \(R_{f} = L\)

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Most popular questions from this chapter

For a certain model study involving a 1: 5 scale model it is known that Froude number similarity must be maintained. The possibility of cavitation is also to be investigated, and it is assumed that the cavitation number must be the same for model and prototype. The prototype fluid is water at \(30^{\circ} \mathrm{C}\), and the model fluid is water at \(70^{\circ} \mathrm{C}\). If the prototype operates at an ambient pressure of \(101 \mathrm{kPa}(\mathrm{abs}),\) what is the required ambient pressure for the model system?

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