Mass And Volume Flow Rate: Formula And Equation
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Table of Contents
- Mass Flow Rate
- Mass Flow Rate Equation
- Volume Flow Rate Equation
- Mass Flow Rate to Velocity
- Continuity Equation Fluids
- Lesson Summary
- FAQs
- Activities
Fluid Flow and the Continuity Equation
Fluids can be liquids or gases. Liquids can not be compressed, but gases can. For liquids, the mass flow of the liquid or the volume flow has to be constant. Let's work some practice problems dealing with the continuity equation. Solve the problems and then check your answers under the solutions section.
Practice Problems
1. 100 cm3 of water flow through a 2 cm2 cross-sectional area pipe every minute. What is the volume flow rate in cubic cm per second?
2. A pipe has an initial cross-sectional area of 2 cm2 that expands into a 5 cm2 area. Initially, the velocity of the water through the smaller cross-sectional area of the pipe is 20 cm/s. What is the volume flow rate?
3. In number 2, determine the velocity of the water through the larger cross-sectional area section of the pipe.
Solutions
1. 100 cm3 / 60 seconds = 1.67 cm3 / s
2. (cross-sectional area)(velocity) = (2)(20) = 40 cm3 / s
3. We will use the continuity equation to determine the velocity of the water through the smaller cross-sectional area pipe.
A1 v1 = A2 v2
(2)(20) = 5v2
40 = 5v2
v2 = 8 cm/s
What is mass flow rate in fluid mechanics?
In fluid mechanics, the mass flow rate is defined as the ratio of the change in the mass of a flowing fluid with the change in time.
How to calculate mass flow rate?
To calculate mass flow rate, divide the change in mass with the change in time: dm/dt = (m_2 - m_1) = (t_2 - t_1).
If the volumetric flow rate was given, the mass flow can be found by multiplying the density of the fluid with the volumetric flow rate: Q * density.
What is the formula for volume flow rate?
The volume flow is the change in volume per change in time: dV/dt = (V_2 - V_1)/(t_2 - t_1).
Another formula that is widely used is the following:
Q = v * A,
Where v is the velocity of the fluid and A is the cross-section area the fluid is passing through.
If the mass flow rate was given, the volume flow rate can be sought by dividing the mass flow rate by density.
Create an accountTable of Contents
- Mass Flow Rate
- Mass Flow Rate Equation
- Volume Flow Rate Equation
- Mass Flow Rate to Velocity
- Continuity Equation Fluids
- Lesson Summary
Mass Flow Rate
Mass flow rate is defined as the mass of matter that passes through an area per unit of time. It is a measure of the rate of the movement of passing fluids (liquids and gases) through a defined area. Mass flow rate is a very important variable, which is commonly seen in engineering and fluid mechanics. Its standard SI unit is kilogram per second (kg/s).
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- 4:30 Flow Rate
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Mass Flow Rate Equation
The mass flow rate equation can be easily derived by simply reviewing the definition of mass flow rate and recalling its standard unit. The units of mass flow rate are kg/s, mass per time, which indicates that mass flow rate is the ratio of the change in a fluid's mass to the change in time. The following is the mass flow rate formula:
{eq}\dot{m}=dm/dt {/eq}
Where {eq}dm {/eq} is the change in mass, {eq}dt {/eq} is the change in the time, and {eq}\dot{m} {/eq} is the mass flow rate. The dot on top of m is used to differentiate between regular mass, m, and the mass flow rate.
Mass Flow Rate to Volume Flow Rate
The volume flow rate often referred to as the volumetric flow rate, is the volume of a fluid passing a cross-sectional area per time.
Both mass and volume flow rates are related to one another the same way mass and volume are related to each other. In a sense, the mass flow rate is the measure of the amount of fluid flowing through, say, a pipe, while the volumetric flow rate is the measure of 3D space occupied by the fluid that's passing through a pipe.
The ratio of mass and volume yields density, using this relationship is the key to converting the mass flow rate to volumetric flow rate:
{eq}Q=\frac{\dot{m}}{\rho } {/eq}
Converting volumetric flow rate to mass flow rate:
{eq}\dot{m}=Q*\rho {/eq}
Note: Q is the volumetric flow rate
How to Calculate Mass Flow Rate?
The provided examples will show how to calculate mass flow rate:
- Example 1: Water is poured into a tank whose capacity is 80 {eq}m^3 {/eq}, filling it to the brim. What is the mass flow rate if the amount of time to empty the tank in 2 hours? (Density of water is equal to 1000 {eq}kg/m^3 {/eq}).
{eq}m = v * \rho = 80 * 1000 = 80000 kg {/eq}
{eq}\dot{m}=m/t = 80000/2 = 40000 \frac{kg}{hr}*\frac{1hr}{3600s}=11.1\frac{kg}{s} {/eq}
- Example 2: When 400 g of water is drained from a cylinder, the remaining amount of water in the cylinder is 200 g. Find the mass flow rate if the time in this process was 60 seconds.
{eq}\dot{m}=dm/dt = (m_2 - m_1)/(t_2 - t_1) = (200 - 400)/(60 - 0)=-3.33\frac{kg}{s} {/eq}
Water's mass flow rate is (-)3.33 kg/s. The negative sign indicates that the water is exiting the system. A positive mass flow rate means that it's entering the system.
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Volume Flow Rate Equation
The volumetric flow rate equation that relates to the flow velocity is:
{eq}Q = v * A {/eq}
Where Q is the volumetric flow rate, v is the fluid velocity, and A is the area of the cross-section the fluid is passing. It can also be defined as the ratio of the change in the volume of the fluid with change in time {eq}Q = dV/dt {/eq}.
The standard units of Q is {eq}[m^3/s] {/eq} cubic meter per seconds. If the value of the mass flow rate is given, the volumetric flow rate can be computed by simply dividing the mass flow rate by the density of the fluid, as shown in the previous section.
The provided examples should help in computing the volumetric flow rate.
Example 1
Find the mass flow rate of a liquid that is flowing at a velocity of 9.2 {eq}m/s {/eq} through a circular pipe whose inner radius is 2 cm. (Liquid density is 940 {eq}kg/m^3 {/eq}.)
The area of the pipe: {eq}A = \pi * (r)^2 = \pi * (2cm)^2 = 12.56cm^2 * \frac{(1 m)^2}{(100cm)2} = 0.001256 m^2 {/eq}
The volumetric flowrate: {eq}Q = v * A = 9.2*0.00125 = 0.0115 m^3/s {/eq}
The mass flowrate: {eq}\dot{m}=Q*\rho=940 kg/m^3 * 0.0115 m^3/s = 10.81 kg/s {/eq}
The mass flow rate is equal to 10.81 {eq}kg/s {/eq}.
Example 2
A liquid with a density of 999 {eq}kg/m^3 {/eq} is flowing through a circular pipe whose inner diameter is 5 cm. The velocity of the liquid is 10.5 {eq}m/s {/eq}. What is the mass flow rate of the liquid?
Radius of the pipe: {eq}r = d/2 = 5/2 = 2.5 cm {/eq}
The area of the pipe: {eq}A =\pi * (r)^2 = \pi * (2.5)^2 = 19.63 cm^2 * \frac{(1 m)^2}{(100cm)2} = 0.00196 m^2 {/eq}
Volumetric flowrate: {eq}Q = v * A = 10.5*0.00196 = 0.0206 m^3/s {/eq}
Mass flowrate: {eq}\dot{m}=Q*\rho=999 kg/m^3 * 0.0206 m^3/s = 20.6 kg/s {/eq}
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Mass Flow Rate to Velocity
The velocity of the fluid is related to the mass flow rate through the volumetric flow rate equation. Since:
{eq}\dot{m}=Q*\rho {/eq}
and {eq}Q = v * A {/eq}
then {eq}\dot{m}=(v * A)*\rho {/eq}.
The provided example will show how the velocity can be calculated from a given mass flow rate:
Example: Velocity from Mass Flow Rate
Water flows through a circular pipe whose radius is 1.5 cm at a flowrate of {eq}0.056 m^3/s {/eq}. What is the velocity of the flowing water? ({eq}\rho = 998 kg/m^3 {/eq})
Area of pipe: {eq}A = \pi * (r)^2 = \pi * (1.5)^2 = 7.06 cm^2 * \frac{(1 m)^2}{(100cm)2} = 0.000706 m^2 {/eq}
Mass flow rate: {eq}\dot{m}=Q* \rho = 0.056*998 = 55.88 kg/s {/eq}.
Velocity of water: {eq}v = \dot{m}/(\rho * A)= 55.88/(998*0.000706) = 79.3m/s {/eq}
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Continuity Equation Fluids
It is a given fact that neither mass nor energy can be created nor destroyed, and it is the same case for fluids passing through pipelines. The amount entering the pipe is equal to the amount exiting it, which is the principle of mass conservation. In this case, this is called the continuity equation. Before proceeding any further in discussing this equation, it should be noted that such fluids, as mentioned above, are ideal fluids, and they fall under the following assumptions:
- Fluids are incompressible, which is true for liquids but not for gases, but for simplicity's sake, it is assumed that all fluids are incompressible.
- The resistance fluids exhibit again flow is assumed to be negligible, that is, fluids are non-viscous.
- Fluid particles flow smoothly and steadily with no fluctuations. In other words, the fluid is laminar.
 |
The assumption about fluids being incompressible simplifies things such that the volumetric flow rate at any given point in a pipe segment is the same as the rest of the points.
Continuity Equation Derivation
The continuity equation derivation is made simple by recalling that the volumetric flow rate is the volume of fluid passing through a certain point in a pipe segment per unit time {eq}Q = V/t {/eq}, this can be used as a starting point for the derivation.
 |
The continuity equation states that the amount of fluid in point x in a pipe must be the same in point y through a pipe. Two cross-sections, 1 and 2, are highlighted in the image stating the fact that the mass of fluid is conserved and is related to the volume through density; it can be said that:
{eq}\dot{m_1}=\dot{m_2} {/eq}
{eq}Q_1 \rho = Q_2 \rho \rightarrow Q_1 = Q_2 {/eq} (density gets canceled since it's the same value on both the right and left-hand sides of the equation.)
The flow of fluid through segment 1: {eq}Q_1 = V_1/t {/eq}
Cylinder volume is {eq}V = A * L {/eq}, so {eq}Q_1 = (A_1 * L_1)/t {/eq}. Velocity is equal to length (displacement) divided by time {eq}v = L / t {/eq}, which makes:
{eq}Q_1 = A_1 \frac{L_1}{t} = A_1 * v_1 {/eq}.
Repeating the same process in segment two yields {eq}Q_2 = A_2 * v_2 {/eq}. The volume of fluid flowing through section 1 is equal to the volume of fluid flowing through section 2, thus:
{eq}Q_1 = Q_2 \rightarrow A_1 * v_1 = A_2 * v_2 {/eq}
The pipe segment expanded, as shown in the picture. For the fluid to conserve its mass in motion as dictated by the principle of conservation, the velocity of the fluid approaching segment 2 will decrease, since the area of that cross-section has increased. The continuity equation shows that a decrease in the cross-section area is coupled with the increase in the fluid velocity such that the amount flowing in point x is the same as in point y.
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Lesson Summary
Mass flow rate is an important variable in engineering and fluid mechanics, it is the amount of fluid flowing per unit time, and its SI unit is kg/s. The mass flow rate formula is:
{eq}\dot{m}=dm/dt {/eq}
The volumetric flow rate refers to the 3D space occupied by a fluid flowing through a cross-section per time. It is the ratio of the change in the volume of a flowing fluid to the change in time. This is related to the mass flow rate the same way regular mass and volume are related; their relationship with density can be used to convert between mass and volume flow rates. The following equations are used to find the volume flow rate:
- {eq}Q = v * A {/eq}
- {eq}Q = dV/dt {/eq}
- {eq}Q = \frac{\dot{m}}{\rho} {/eq}
The continuity equation is based on the conservation of mass principle where the amount of fluid flowing in must be the same as the amount flowing out. Fluids that follow the continuity equation are:
- Laminar flow smoothly and steadily.
- Non-viscous exhibit no resistance to flow.
- Incompressible are described as ideal fluids.
The equation of continuity states that if the cross-section of a pipe decreases, the velocity of the fluid increases, given the flow rate before and after the narrowing of the pipe is the same. As shown in the formula {eq}Q_1 = Q_2 \rightarrow A_1 * v_1 = A_2 * v_2 {/eq}. The speed of fluid changes according to the cross-section of the segment in a way that the amount of fluid flowing between each point is uniform. For example, the speed of fluid passing through a narrow segment increases.
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Video Transcript
Fluids in Motion
There's no doubt, fluids like to move! Just imagine a roaring river or a summer breeze blowing, and you can understand what I mean. Fluid dynamics is a tricky subject because there are different kinds of ways that fluids can move, and we don't even fully understand exactly how they do some of the amazing things they do. Because of this, we're going to talk about fluids in a very specific sense, making a few assumptions for this lesson.
First, we are going to assume that the fluid is incompressible, meaning that its density can't be changed. Try to condense 8 ounces of water into a 4 ounce container, and you'll see what this means!
Second, we're going to assume that the flow of the fluid is laminar, meaning that it is a steady, constant flow that doesn't change with time. Think of water running through a straight, narrow creek instead of pooling and twisting in small nooks and crannies along the bank.
Finally, we're going to assume that the fluid is non-viscous, meaning that there is no resistance to flow. Viscosity is the resistance to flow, so we would say that honey is more viscous than water because water flows much more easily than honey. But for this lesson, our fluid experiences no resistance.
The Equation of Continuity
OK, now that we've gotten that out of the way, let's look at how fluids move.
You know that you can make the water in a garden hose come out faster if you partially block the opening. This is because the same amount of water has to travel through that smaller opening as the larger opening.
This is true of any type of fluid and any type of tube or pipe - toothpaste coming out of its tube, blood flowing through your arteries, and water through pipes. The moving fluid can't be stored in the tube or pipe - it must travel through. And, the same volume of fluid that goes in must come out.
But when we make the area in the tube or pipe smaller, like we did with the garden hose, the fluid speeds up because the same volume of water has to go through a smaller area than before. This relationship between the area inside the pipe (the pipe's internal diameter) and the velocity of the fluid is expressed in the equation of continuity, written as v1 A1 = v2 A2 . Here, v is the velocity of the fluid, and A is the area that fluid travels through.
Because this is an equation, it means that the product of either side has to equal the product of the other. So if the area on the either side decreases, it means that the velocity on the same side of the equation has to increase accordingly.
It sounds simple enough, but let's work through an example to see how the speed changes depending on the area the fluid is traveling through.
Say that you have some fluid flowing through a pipe. At one end, the pipe has an internal diameter of 10.0 cm. But down the line at a second point, the internal diameter of the pipe is only 5.0 cm. The initial speed of the fluid moving through the pipe is 5.0 m/s, but we want to know what the speed is at the second point where the pipe is narrower. And here's where we can use the equation of continuity to help us figure it out.
First, we need to rearrange our equation to get v2, the velocity of the fluid at the second location, alone on one side. Next, we need to do some quick conversions to make sure we're working with the correct units. Since our velocity is in meters per second, we need to change our pipe diameters to meters as well. This gives us 0.10 m for the first point and 0.05 m for the second.
Finally, since we're working with the area of a circle, the pipe, our area values will use the radii of the openings instead of the diameters. This gives us π*(0.05 m)^2 for A1, and π*(0.025 m)^2 for A2.
Now all that's left to do is plug in our known values and solve! When we do, we find that v2 equals 20 m/s. That's quite an increase! Can you see how the equation of continuity shows how the speed of the fluid is faster in narrower areas than wider ones? The same amount of fluid has to pass through, so it goes through faster to make up for the smaller area.
Flow Rate
We can also understand fluid dynamics by calculating the flow rate of a fluid, which is the rate at which a volume of fluid flows through a tube. This is different from the speed - the flow rate is the time frame in which an amount of fluid flows, whereas the speed is simply how fast the fluid flows.
In equation form, flow rate is represented as:
Q = ΔV / Δt
where Q is the volume flow rate, V is the volume of fluid, and t is the time in seconds. The Greek symbol Δ means change in, so we read this as: the volume flow rate equals the change in volume over the change in time.
Let's try an example with this equation. Say you have a garden hose that fills a 5.0 liter bucket in 10 seconds, and we want to know the flow rate at which the water comes out of the end of the hose.
Using our flow rate equation, we simply plug in our values and solve the equation. When we do so, we get: Q = 5.0 L / 10 s, which means that Q = 0.5 L/s.
Both the equation of continuity and this flow rate equation show us how the flow rate itself is constant at any point in the tube. One variable doesn't change without affecting the other, so as the diameter of the tube decreases, the speed must increase to make sure that the flow rate stays the same. If the flow rate is 0.5 L/s at one point, it will still be 0.5 L/s at the second point. In order to keep that flow rate constant, the fluid must travel faster to move the same amount of volume in a given time interval.
Lesson Summary
Fluids are dynamic. They like to move, but this movement is not always well understood. We can, however, describe fluid motion in terms of an ideal fluid when we make some assumptions. Assuming that a fluid is incompressible, has laminar flow, and is non-viscous, we can describe how fluids move through tubes and pipes.
Specifically, fluids move so that the same volume of fluid that goes into the tube must come out. This means that the fluid speeds up as it passes through narrower areas. This is described with the equation of continuity. This equation, written as v1 A1 = v2 A2 helps us understand how when the area A decreases, the velocity v must increase to keep the equation equal.
We can also describe a fluid's flow rate, which is the rate at which a volume of fluid flows through a tube. Expressed in equation form, flow rate is: Q = ΔV / Δt, where Q is the flow rate, ΔV is the change in volume, and Δt is the change in time.
Together, the flow rate equation and the continuity equation tell us that this flow rate is constant at all points in a tube. This is because the amount of volume entering the tube must be the same as the amount of volume leaving it, so in order to compensate for narrower spaces, the fluid must speed up to push on through.
Learning Outcomes
Once you've completed this lesson, you should be able to:
- Describe three characteristics of an ideal fluid
- Identify the equation of continuity and the flow rate equation
- Explain how these two equations describe the flow of fluids through tubes
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