Calculation of Reynolds Numbers Calculate the Reynolds number Re for the following. Assume that the fluid has properties of water: density ρ = 1,000 kg/m^3, viscosity μ = 10^−3 kg/m s, and kinematic viscosity ν = 10^−6 m^2/s. Flow of drugs injected using thumb pressure (about 5 atm) on a 5-ml syrin

Question

Calculation of Reynolds Numbers

Calculate the Reynolds number Re for the following. Assume that the fluid has properties of water: density [latex] ho=1,000 \mathrm{~kg} / \mathrm{m}^{3}[/latex], viscosity [latex]\mu=10^{-3} \mathrm{~kg} / \mathrm{m} \mathrm{s}[/latex], and kinematic viscosity [latex] u=10^{-6} \mathrm{~m}^{2} / \mathrm{s}[/latex].

Flow of drugs injected using thumb pressure (about 5 atm) on a 5-ml syringe through a [latex]\frac{1}{2}[/latex]-inch long, 27-gauge (internal diameter [latex]d[/latex] approximately [latex]300 \mu \mathrm{m}[/latex] ) needle.
Flow of cell culture media at a mean velocity of [latex]10 \mu \mathrm{m} / \mathrm{s}[/latex] through a [latex]d=20[/latex] [latex]\mu \mathrm{m}[/latex] diameter tube machined in a glass microchip, used to culture cells in what is called "continuous perfusion cell culture."
Flow of water at a mean velocity of [latex]80 \mu \mathrm{m} / \mathrm{s}[/latex] through [latex]d=50-\mathrm{nm}[/latex] diameter tubes machined in silicon, used to sort and measure individual DNA molecules.

Accepted Answer

Syringe: Because of the high pressure drop, we anticipate that the flow is probably turbulent, in which case we can use Eqns. (3.33) for the pressure drop and (3.40) for the friction factor in a smooth tube:

[latex]-\Delta p=p_{1}-p_{2}=2 f_{\mathrm{F}} ho u_{\mathrm{m}}^{2} \frac{L}{D}+ ho g \Delta z,[/latex] (3.33)

[latex]f_{\mathrm{F}}=0.0790 \mathrm{Re}^{-1 / 4}[/latex] (3.40)

[latex]-\Delta p=2 f_{\mathrm{F}} ho u_{\mathrm{m}}^{2} \frac{L}{d}, \quad f_{\mathrm{F}}=0.0790 \mathrm{Re}^{-1 / 4}=0.0790\left(\frac{\mu}{ ho \mathrm{u}_{\mathrm{m}} \mathrm{d}} ight)^{1 / 4}.[/latex]

Elimination of the friction factor and rearrangement gives the mean velocity:

[latex]\begin{aligned}u_{\mathrm{m}} &=\frac{(-6.33 \Delta p)^{4 / 7} d^{5 / 7}}{\mu^{1 / 7} ho^{3 / 7} L^{4 / 7}} \&=\frac{\left(6.33 imes 5 imes 10^{5} \mathrm{~Pa} ight)^{4 / 7}\left(300 imes 10^{-6} \mathrm{~m} ight)^{5 / 7}}{\left(10^{-3} \mathrm{~kg} / \mathrm{m} \mathrm{s} ight)^{1 / 7}\left(10^{3} \mathrm{~kg} / \mathrm{m}^{3} ight)^{3 / 7}(0.5 imes 0.0254)^{4 / 7}}=26.6 \mathrm{~m} / \mathrm{s} \\operatorname{Re} &=\frac{u_{\mathrm{m}} d}{ u}=\frac{26.6 imes 300 imes 10^{-6}}{10^{-6}}=7,974,\end{aligned}[/latex]

thus verifying the assumption of turbulent flow.

Cell culture media:

[latex]R e=\frac{u_{\mathrm{m}} d}{ u}=\frac{10 imes 10^{-6} imes 20 imes 10^{-6}}{10^{-6}}=2 imes 10^{-4} .[/latex]

DNA measurement:

[latex]R e=\frac{u_{\mathrm{m}} d}{ u}=\frac{80 imes 10^{-6} imes 50 imes 10^{-9}}{10^{-6}}=4 imes 10^{-6}.[/latex]

Question 12.1: Calculation of Reynolds Numbers Calculate the Reynolds numbe...

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