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Chapter 11: Problem 55

A patient recovering from surgery is being given fluid intravenously. The fluid has a density of \(1030 \mathrm{kg} / \mathrm{m}^{3},\) and \(9.5 \times 10^{-4} \mathrm{m}^{3}\) of it flows into the patient every six hours. Find the mass flow rate in \(\mathrm{kg} / \mathrm{s}\).

Short Answer

Expert verified
The mass flow rate is approximately \(4.53 \times 10^{-5}\, \mathrm{kg/s}\).

Step by step solution

01

Understand the given parameters

We need to determine the mass flow rate of fluid in \(\mathrm{kg}/\mathrm{s}\). We are given: density of the fluid, \(\rho = 1030\, \mathrm{kg}/\mathrm{m}^3\), and the volume that flows in every 6 hours, \(V = 9.5 \times 10^{-4}\, \mathrm{m}^3\).
02

Convert time to seconds

Since the mass flow rate has to be determined in \(\mathrm{kg}/\mathrm{s}\), we need to convert the time from hours to seconds: \(6\, \text{hours} = 6 \times 3600 = 21600\, \text{seconds}\).
03

Calculate the mass of the fluid

The mass \(m\) of the fluid can be calculated using the formula: \(m = \rho \cdot V\). Therefore, \(m = 1030\, \mathrm{kg}/\mathrm{m}^3 \times 9.5 \times 10^{-4}\, \mathrm{m}^3 = 0.9785\, \mathrm{kg}\).
04

Calculate the mass flow rate

The mass flow rate \(\dot{m}\) is given by the formula \(\dot{m} = \frac{m}{t}\), where \(t\) is time in seconds. Substituting the known values: \[ \dot{m} = \frac{0.9785\, \mathrm{kg}}{21600\, \mathrm{s}} \approx 4.53 \times 10^{-5}\, \mathrm{kg/s} \]
05

Review the results for units and accuracy

Ensure that the calculated mass flow rate has the correct units. The mass \(0.9785\, \mathrm{kg}\) processed over \(21600\, \mathrm{s}\) results in \(\dot{m} \approx 4.53 \times 10^{-5}\, \mathrm{kg/s}\). This confirms the setup and operations are correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Density
In fluid dynamics, understanding the concept of density is crucial. Density is defined as the mass per unit volume of a substance. It is commonly denoted by the Greek letter \(\rho\) and expressed in units of \(\mathrm{kg/m}^3\). In our exercise, the fluid has a density of \(1030 \, \mathrm{kg/m}^3\). This means each cubic meter of the fluid weighs 1030 kilograms. Knowing the density is vital when calculating the mass of the fluid, especially when given the volume.
  • Density is significant because it helps to determine how much mass is contained within a given volume.
  • It allows for the conversion between mass and volume, which is essential for calculating flow rates.
When dealing with problems involving fluids, always double-check the density value as it can vary significantly depending on temperature and pressure conditions.
Mass
Mass is a measure of the amount of matter in an object or fluid, typically measured in kilograms (kg). In the context of fluid flow, mass can be calculated when density and volume are known. The relationship between mass, density, and volume is expressed by the formula:\[ m = \rho \cdot V \]where \(m\) is mass, \(\rho\) is density, and \(V\) is volume. In the step-by-step solution, this equation is used to compute the mass of the fluid that flows into the patient.
  • Mass is a fundamental property that indicates the amount of substance available.
  • In fluid dynamics, knowing the mass flowing through points of a system helps determine the behavior of the fluid.
The mass of fluid that flowed, as computed, is \(0.9785 \, \mathrm{kg}\), which was arrived at by multiplying the given density by the specified volume.
Volume
Volume is the amount of space that a substance occupies, measured in cubic meters (\(\mathrm{m}^3\)). In our scenario, the given volume of fluid is \(9.5 \times 10^{-4} \, \mathrm{m}^3\). This is the amount of fluid that flows into the patient every six hours.
  • Volume is used with density to calculate mass, crucial for determining flow rates.
  • Understanding how flow volumes relate to changes in system conditions (like pressure and temperature) is essential.
The accurate measurement of volume is important in medical applications like intravenous therapy to ensure proper dosage.
Fluid Dynamics
Fluid dynamics is the study of fluids (liquids and gasses) in motion. It is a branch of physics that plays a crucial role in various fields, from engineering to medicine. In our exercise, fluid dynamics principles are used to understand how fluid flows into a patient's body at a specific rate.
  • The mass flow rate, a key concept here, provides the rate at which mass travels through a given surface.
  • Fluid dynamics considers properties like viscosity and density, influencing flow behavior.
In medical contexts, understanding these principles ensures that patients receive the correct amount of nutrients and medications intravenously. The calculated mass flow rate of about \(4.53 \times 10^{-5} \, \mathrm{kg/s}\) demonstrates how much fluid is administered per second. This precise control is crucial in clinical settings.

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Most popular questions from this chapter

A full can of black cherry soda has a mass of \(0.416 \mathrm{kg}\). It contains \(3.54 \times 10^{-4} \mathrm{m}^{3}\) of liquid. Assuming that the soda has the same density as water, find the volume of aluminum used to make the can.

If a scuba diver descends too quickly into the sea, the internal pressure on each eardrum remains at atmospheric pressure, while the external pressure increases due to the increased water depth. At sufficient depths, the difference between the external and internal pressures can rupture an eardrum. Eardrums can rupture when the pressure difference is as little as \(35 \mathrm{kPa} .\) What is the depth at which this pressure difference could occur? The density of seawater is \(1025 \mathrm{kg} / \mathrm{m}^{3}\).

The aorta carries blood away from the heart at a speed of about \(40 \mathrm{cm} / \mathrm{s}\) and has a radius of approximately \(1.1 \mathrm{cm} .\) The aorta branches eventually into a large number of tiny capillaries that distribute the blood to the various body organs. In a capillary, the blood speed is approximately \(0.07 \mathrm{cm} / \mathrm{s},\) and the radius is about \(6 \times 10^{-4} \mathrm{cm} .\) Treat the blood as an in compressible fluid, and use these data to determine the approximate number of capillaries in the human body.

Two identical containers are open at the top and are connected at the bottom via a tube of negligible volume and a valve that is closed. Both containers are filled initially to the same height of \(1.00 \mathrm{m},\) one with water, the other with mercury, as the drawing indicates. The valve is then opened. Water and mercury are immiscible. Determine the fluid level in the left container when equilibrium is reestablished.

A hydrometer is a device used to measure the density of a liquid. It is a cylindrical tube weighted at one end, so that it floats with the heavier end downward. The tube is contained inside a large "medicine dropper," into which the liquid is drawn using the squeeze bulb (see the drawing). For use with your car, marks are put on the tube so that the level at which it floats indicates whether the liquid is battery acid (more dense) or antifreeze (less dense). The hydrometer has a weight of \(W=5.88 \times 10^{-2} \mathrm{N}\) and a cross-sectional area of \(A=7.85 \times 10^{-5} \mathrm{m}^{2}\) How far from the bottom of the tube should the mark be put that denotes (a) battery acid \(\left(\rho=1280 \mathrm{kg} / \mathrm{m}^{3}\right)\) and (b) antifreeze \(\left(\rho=1073 \mathrm{kg} / \mathrm{m}^{3}\right) ?\)
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