Combining Equations Reinforcement Questions

1.

Combine the two equations below to make the inner pipe diameter (di) the subject:

begin mathsize 14px style ϕ space equals space fraction numerator 16 over denominator R e end fraction end style

begin mathsize 14px style R e space equals space fraction numerator rho u d subscript i over denominator mu end fraction end style

Where:

  • ρ = fluid density

  • u = fluid velocity

  • di = pipe inner diameter

  • μ = fluid viscosity

  • size 14px ϕ = friction factor

Let’s start with the friction factor equation:

begin mathsize 14px style ϕ space equals space fraction numerator 16 over denominator R e end fraction end style

Then, let’s rearrange to set Re as the equation subject:

begin mathsize 14px style R e space equals space 16 over ϕ end style

Now, we can substitute in the equation for Re in place of the term:

begin mathsize 14px style fraction numerator rho u d subscript i over denominator mu end fraction space equals space 16 over ϕ end style

Then, we multiply both sides of the equation by the fluid viscosity (μ):

begin mathsize 14px style rho u d subscript i space end subscript equals space fraction numerator 16 mu over denominator ϕ end fraction end style

Finally, to rearrange the equation to isolate di we divide both sides of the equation by the fluid density (ρ) and velocity (u):

begin mathsize 14px style d subscript i space end subscript equals space fraction numerator 16 mu over denominator ϕ rho u end fraction end style

2.

Combine and rearrange the two equations below to make u subscript 80 m m end subscript the subject.

begin mathsize 14px style Q subscript 100 m m space end subscript equals space Q subscript 80 m m end subscript end style

begin mathsize 14px style Q space equals space A u end style 

Let’s start with what we know:

begin mathsize 14px style Q subscript 100 m m end subscript space equals space Q subscript 80 m m end subscript end style

begin mathsize 14px style Q space equals space A u end style

Since begin mathsize 14px style Q space equals space A u end style and begin mathsize 14px style Q subscript 100 m m end subscript space equals space Q subscript 80 m m end subscript end style then begin mathsize 14px style A subscript 100 m m space end subscript space u subscript 100 m m end subscript end style  begin mathsize 14px style equals space A subscript 80 m m space end subscript space u subscript 80 m m end subscript end style

To make begin mathsize 14px style u subscript 80 m m end subscript end style  the subject we just need to divide both sides of the equation by begin mathsize 14px style A subscript 80 m m end subscript end style :

begin mathsize 14px style u subscript 80 m m end subscript space equals space fraction numerator A subscript 100 m m space end subscript u subscript 100 m m end subscript over denominator A subscript 80 m m end subscript end fraction end style

Next, we add some dimension to the lesson by introducing units of measurement.

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