BIO. Artery Blockage. A medical technician is trying to determine what percentage of a patient's artery is blocked by plaque. To do this, she measures the blood pressure just before the region of blockage and finds that it is 1.20×10 4 Pa, while in the region of blockage it is 1.15×10 4 Pa. Furthermore, she knows that blood flowing through the normal artery just before the point of blockage is traveling at 30.0 cm/s, and the specific gravity of this patient's blood is 1.06. What percentage of the cross-sectional area of the patient's artery is blocked by the plaque?

Hey everyone. So this problem is working with fluid flow. Let's see what they're giving us and see what they're asking. So we know that there is a pipe in that pipe. Some amount of that pipe is blocked by scaling. The team measures the pressure at two different places. The first place, it measures a pressure of 2.6 times to the fifth pascal's and then the second pressure at the narrow region is 2.5 times 10 to the fifth pascal's. They use flow rates and determine that the flow at the kind of open area, the area before the scaling is 4m/s. And they have determined that the fluid has a specific gravity of .896. We are asked to calculate the percentage of the pipes area that is occupied by the scales. So the first thing that we need to do here is recognize that when we're working with the same fluid flowing along a pipe in two different areas, that's most likely the Bernoulli's equation type problem. So let's recall that equation first, See what we have and see what we need to find. So Bernoulli's equation is P one plus one half roe V one squared equals P two plus one half row V two squared. They don't tell us anything about height differentials or anything like that. So we can kind of ignore the height term of that. Bernoulli's equation. We also need to recall our continuity equation which is given as a one V one equals a two V two for flow at two different points in the same pipe and then they do talk about here, specific gravity. So let's just recall the specific gravity is the density of the fluid over the density of water, where the density of water is a given is a constant. Sorry, it's a 1000 kg per meters cubed. And so from there we can calculate the density of this fluid Is 897. kg Per meter. Cute! Alright, let's go back into the problem and see what else they tell us. So, P one is given to us as 2.6 times 10 to the 5th Pascal's. P two is 2. times 10 to the 5th Pascal's and V one. This is that region before the scaling Is given as four m/s. Okay, so um From there we can rearrange the Bernoulli equation to solve for V because in the end right there asking about area, so we're going to want to solve the continuity equation as our last step. So we know P 12. times 10 to the fifth pascal's plus one half. We now know our density of fluid. 897.6 kilograms per meters cubed. Times are V one which is four m per second. That term is squared Equals P two. That was also given two times 10 to the 5th Pascal's plus one half. Again, density. We already solved for times V two squared. So we're gonna plug all of that into our calculator and Soul for V two, B two is 11.8 m/s. So now we're gonna use that to solve in first to solve our area using our continuity equation. So A one B one equals a two B two. We actually don't have either area, but they're asking us for a percentage of area that's occupied by scales. So let's just solve a one in terms of A two and go from there. Okay, so A one equals a two times V two over V one. We know V two and V one now. So that comes out to a two equals 2.95, A one equals 2.95 times a two. So another way of looking at the percentage of the area that is blocked Is to say it's 1 - the percentage of the total area of the scales. So that's gonna be a two over 81. We already know that a one is 2.95 times a two. So we are going to use that and that comes out to one minus 10.339, That's .661. They're asking us for percent Multiply that by 100 and that's .661%. So if we look at our potential options, that aligns with B, The percentage of the pipes area that is occupied by the scales is 66.1%. That's all for this problem. We'll see you in the next video.

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Problem 42

Textbook Question

BIO. Artery Blockage. A medical technician is trying to determine what percentage of a patient's artery is blocked by plaque. To do this, she measures the blood pressure just before the region of blockage and finds that it is 1.20×10⁴ Pa, while in the region of blockage it is 1.15×10⁴ Pa. Furthermore, she knows that blood flowing through the normal artery just before the point of blockage is traveling at 30.0 cm/s, and the specific gravity of this patient's blood is 1.06. What percentage of the cross-sectional area of the patient's artery is blocked by the plaque?

Verified step by step guidance

First, understand that the problem involves fluid dynamics, specifically the application of Bernoulli's equation and the continuity equation to determine the blockage in an artery.

Convert the velocity of blood flow from cm/s to m/s for consistency in units. Since 1 cm = 0.01 m, the velocity is 0.30 m/s.

Use Bernoulli's equation, which states that the sum of the pressure energy, kinetic energy per unit volume, and potential energy per unit volume is constant along a streamline. The equation is:

P + \frac{1}{2} ρ v^{2} + ρ g h = constant

, where

P

is the pressure,

ρ

is the density,

v

is the velocity,

g

is the acceleration due to gravity, and

h

is the height. Assume height changes are negligible.

Calculate the density of blood using its specific gravity. Specific gravity is the ratio of the density of a substance to the density of water. Since the specific gravity is 1.06, the density of blood is

ρ = 1.06 \times 1000 = 1060

kg/m³.

Apply the continuity equation, which states that the product of cross-sectional area and velocity is constant for incompressible flow:

A_{1} v_{1} = A_{2} v_{2}

. Use Bernoulli's equation to find the velocity in the blocked region and then solve for the change in cross-sectional area to find the percentage of blockage.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Bernoulli's Principle

Bernoulli's Principle states that in a fluid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or potential energy. This principle is crucial for understanding how changes in blood pressure and velocity relate to the narrowing of an artery due to plaque. It helps explain the pressure difference observed before and within the blockage.

Continuity Equation

The Continuity Equation in fluid dynamics asserts that the mass flow rate must remain constant from one cross-section of a pipe to another. For incompressible fluids like blood, this means that the product of cross-sectional area and velocity is constant. This concept is essential for determining how the velocity of blood changes as the artery narrows due to plaque buildup.

Specific Gravity

Specific gravity is the ratio of the density of a substance to the density of a reference substance, typically water for liquids. In this context, the specific gravity of blood (1.06) is used to calculate its density, which is necessary for applying Bernoulli's equation and understanding the dynamics of blood flow through the artery. It provides a basis for comparing the blood's properties to those of water.

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