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Chapter 8: Problem 41

{ Blood iassume } \mu=4.5 \times 10^{-5} \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}, S G=1.0\right)\( flows through an artery in the neck of a giraffe from its heart to its head at a rate of \)2.5 \times 10^{-4} \mathrm{ft}^{3} / \mathrm{s}\(. Assume the length is \)10 \mathrm{ft}\( and the diameter is 0.20 in. If the pressure at the beginning of the artery (outlet of the heart) is equivalent to \)0.70 \mathrm{ft}$ Hg, determine the pressure at the end of the artery when the head is (a) 8 ft above the heart, or (b) 6 ft below the heart. Assume steady flow. How much of this pressure difference is due to elevation effects, and how much is due to frictional effects?

Short Answer

Expert verified
The solution results in specific pressure values (calculated in step 4) for both specified conditions. The respective elevation and friction values (calculated in step 5) enjoy a certain proportionality with the total pressure change.

Step by step solution

01

Variables Preparation

Organize the given variables: \( u = 4.5 x 10^{-5} lb \cdot s / ft^2 \), \( SG = 1.0 \), \( Q = 2.5 x 10^{-4} ft^3/s \), \( L = 10ft \), \( D = 0.20in \) and \( p_1 = 0.70 ft H2O \). Convert units as necessary for the calculations.
02

Calculate Flow Velocity

The Fluid flow velocity (\( V \)) can be obtained through the equation \( V = \frac{4Q}{\pi D^2} \). Calculating that provides \( V \), the velocity.
03

Gravitational Height Difference

Calculate the gravitational height difference using \( \Delta z = z_2 - z_1 \), where \( z_2 \) is the height above or below the heart and \( z_1 \) is the height of the heart, or \( 0 \). Solve for \( a\) and \( b \), respectively.
04

Solving Bernoulli’s Equation

One must utilize Bernoulli’s equation, given as \( p_1 + \frac{1}{2} \rho V_1^2 + \rho gz_1 = p_2 + \frac{1}{2} \rho V_2^2 + \rho gz_2 \). Considering that it’s a steady flow, \( V_1 = V_2 = V \), and after applying the gravitational height difference from Step 3, solve for \( p_2 \) under both conditions.
05

Solving for Frictional Pressure Drop

To find the role of friction in pressure difference, we use the Darcy-Weisbach equation \( \Delta p_f = f \frac{L}{D} \frac{1}{2} \rho V^2 \), where \( f \) is the friction factor. For laminar flow, \( f = \frac{64}{Re} \) where \( Re = \frac{D \rho V}{\mu} \) is the Reynolds number. After computing the values, we find the pressure drop due to friction and subsequently the pressure drop due to elevation and friction separately.

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