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Chapter 8: Problem 26

A person is donating blood. The pint bag in which the blood is collected is initially flat and is at atmospheric pressure. Neglect the initial mass of air in the \(1 / 8\) -in. I.D., 4 ft-long plastic tube carrying blood to the bag. The average blood pressure in the vein is \(40 \mathrm{mm}\) Hg above atmospheric pressure. Estimate the time required for the person to donate one pint of blood. Assume that blood has a specific gravity of 1.06 and a viscosity of \(1.0 \times\) \(10^{-4} 1 b \cdot \sec / f t^{2},\) The needle's I.D. is \(1 / 16\) inch and the needle length is 2.0 in. The bag is \(1.0 \mathrm{ft}\) below the needle inlet and the vein's I.D. is \(1 / 8\) in. Optional: Donate a pint of blood and check your answer.

Short Answer

Expert verified
The time necessary for the person to donate one pint of blood is calculated by plugging the obtained flow rate into the equation \( t = Volume / Q \). The exact value depends on the exact flow rate calculated based on given data.

Step by step solution

01

Determine the Driving Pressure

The difference between average vein pressure and atmospheric pressure propels the blood flow. Given the average blood pressure is 40mm Hg above atmospheric pressure, we convert the pressure to similar units with dimensions \( lb/ft^2 \). Thus,\( Δp = 40mmHg \times \frac{1 in}{25.4 mm} \times \frac{1 ft}{12 in} \times 0.491 \frac{lb}{ft^2 inHg}\)
02

Calculate the Diameter and Area of The Needle

Given that the needle Inside Diameter (I.D.) is \( 1/16 \) in., convert this to feet and calculate the area as follows:\( Diameter = \frac{1}{16} \times \frac{1}{12} \text{ feet} \) We calculate the area using the formula \( A = π(d/2)^2 \)
03

Use Hagen-Poiseuille Equation to Obtain Blood Flow Rate

The Hagen-Poiseuille Equation can be used to calculate the flow rate \(Q\) of a viscous fluid, which is given as:\( Q = \frac{πd^4}{128μL}(Δp + ρgH) \),Now we plug in the values calculated and given:where \( L\) is the length of the needle,\(d\) is its inner diameter,\(Δp\) is the pressure difference,\(μ\) is the dynamic viscosity of the blood,\(ρ\) is the density of blood,\(g\) is the acceleration due to gravity, and\(H\) is the height difference between the needle inlet and the bag.
04

Calculate The Time Required for Donation

To calculate the time it takes for a person to donate one pint of blood, we will need to use the calculated flow rate (\(Q\)) and the fact that one pint of blood is approximately equal to 0.473176 liter. Thus, time \(t\) can be found by using the equation:\( t = Volume / Q \)

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Most popular questions from this chapter

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