14.7 Viscosity and turbulence  (Page 4/14)

Using flow rate: air conditioning systems

An air conditioning system is being designed to supply air at a gauge pressure of 0.054 Pa at a temperature of $20\text{°}\text{C}.$ The air is sent through an insulated, round conduit with a diameter of 18.00 cm. The conduit is 20-meters long and is open to a room at atmospheric pressure 101.30 kPa. The room has a length of 12 meters, a width of 6 meters, and a height of 3 meters. (a) What is the volume flow rate through the pipe, assuming laminar flow? (b) Estimate the length of time to completely replace the air in the room. (c) The builders decide to save money by using a conduit with a diameter of 9.00 cm. What is the new flow rate?

Strategy

Assuming laminar flow, Poiseuille’s law states that

We need to compare the artery radius before and after the flow rate reduction. Note that we are given the diameter of the conduit, so we must divide by two to get the radius.

Solution

1. Assuming a constant pressure difference and using the viscosity $\eta =0.0181\text{mPa}\cdot \text{s}$ ,
   
   $Q=\frac{\left(0.054\text{Pa}\right)\left(3.14\right){\left(0.09\text{m}\right)}^4}{8\left(0.0181\times {10}^{\mathrm{-3}}\text{Pa}\cdot \text{s}\right)\left(20\text{m}\right)}=3.84\times {10}^{\mathrm{-3}}\frac{{\text{m}}^3}{\text{s}}.$
2. Assuming constant flow $Q=\frac{dV}{dt}\approx \frac{\text{Δ}V}{\text{Δ}t}$
   
   $\text{Δ}t=\frac{\text{Δ}V}{Q}=\frac{\left(12\text{m}\right)\left(6\text{m}\right)\left(3\text{m}\right)}{3.84\times {10}^{\mathrm{-3}}\frac{{\text{m}}^3}{\text{s}}}=5.63\times {10}^4\text{s}=15.63\text{hr}.$
3. Using laminar flow, Poiseuille’s law yields
   
   $Q=\frac{\left(0.054\text{Pa}\right)\left(3.14\right){\left(0.045\text{m}\right)}^4}{8\left(0.0181\times {10}^{\mathrm{-3}}\text{Pa}\cdot \text{s}\right)\left(20\text{m}\right)}=2.40\times {10}^{\mathrm{-4}}\frac{{\text{m}}^3}{\text{s}}.$
   
   Thus, the radius of the conduit decreases by half reduces the flow rate to 6.25% of the original value.

Significance

In general, assuming laminar flow, decreasing the radius has a more dramatic effect than changing the length. If the length is increased and all other variables remain constant, the flow rate is decreased:

Doubling the length cuts the flow rate to one-half the original flow rate.

If the radius is decreased and all other variables remain constant, the volume flow rate decreases by a much larger factor.

Cutting the radius in half decreases the flow rate to one-sixteenth the original flow rate.

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