Conservation of Mass – Definitions, Caclulations and Examples

The first law of thermodynamics states the conservation of energy principle in nature. In different kinds of systems, we use the conservation of energy principle. Also, there is a natural law about the conservation of mass. Similarly, it is about the balanca of input masses and output masses inside and outside the system. We will delve into the conservation of mass topic here. You can find information on these subjects;

• What is the conservation of mass principle?
• Mass flow rate and volume flow rate phenomenon.
• Conservation of mass with calculations.
• Mass balance in engineering systems.
• Application examples of this principle.
• FAQs about conservation of mass.

What is the Conservation of Mass?

The general explanation of the conservation of mass principle is just like the energy principle. The input masses and output masses are the same for different systems. So, there is a balance between the masses inside a system. And also, we use a control volume approach while we are dealing with the conservation of mass in engineering systems.

In different kinds of applications, we need to apply the conservation of mass principle. Because it is one of the general principles in nature. We need to take special care of this phenomenon in our calculations. So, we will use different kinds of calculations in the conservation of mass.

Mass Flow Rates and Volume Flow Rates

In the conservation of mass principle, mass flow rate and volume flow rates are very important. Also, we call them to flow rates. You know that we use flow rates in different kinds of applications. These flow rates are generally on a mass basis and volume basis.

We calculate the mass flow rate in a fluid flow with this formula;

In this formula;

• m is the mass flow rate which has the unit of kg/s or lb/s in English units. 
• Also, the q is the density of the flowing fluid that has the unit of kg/m3 or lb/ft3.
• Vavg is the average velocity of the flowing fluid which has the unit of m/s or ft/s.
• A is the area of the cross-section of mass flow. The unit is m2 or ft2.

As you see above, with the increasing area, flow velocity, and area, the mass flow rate increases.

Also, we can use the volume flow rate notion. We can calculate the volume flow rate with this formula;

This is also another form of flow rate. If you deal with the volumes of masses instead of masses, you can use the volume flow rate. Furthermore, the only difference in the volume flow rate, there is no density in the calculation.

Also, the unit of the volume flow rate is m3/s or ft3/s.

These two values are very important in the conservation of mass calculations.

Example About Mass and Volume Flow Rates

Example 1: There is a fluid flow through a pipe section. The diameter of the pipe section is 5 centimeters. Also, the velocity of the fluid is 5 m/s inside this pipe. If we consider the density of the fluid as 2250 kg/m3. The pipe section instantly shrinks to 2-centimeter diameters. Find;

• Mass and volume flow rate of this fluid flow.
• The velocity of fluid flow in pipe 2.

Solution;

The calculation of mass and volume flow rates is very simple. Flow rates are the same in two sections of the pipe. This is because of the conservation of mass principle. If we put the values inside the mass flow rate and volume flow rate formulas;

• mass flow rate = density*Area*velocity;

= pi*(0.05^2)*2250*5;

Mass Flow Rate = 88.31 kg/s.

• volume flow rate = Area*velocity

= pi*(0.05^2)*5;

Volume Flow Rate = 0.039 m3/s.

• Also, to find the velocity of the fluid in the shrunk section, we need to use the conservation of mass principle;

Input Volume Flow = Output Volume Flow

We can use the volume flow because the densities are the same in each section. Because it is an incompressible fluid.

pi*(0.05^2)*5 = pi*(0.02^2)*velocity

If we take the velocity from here;

Velocity of fluid flow in the second section = 31.25 m/s.

Conservation of Mass Principle

The importance of the conservation of mass principle emerges here. So, we can use the conservation of mass principle in different kinds of applications. We can build the conservation of mass principle with the mass flow rates.

In the engineering approach, we need to specify a control volume. The control volume is our system. So, there are input and output masses inside this control volume. We need to take special care of this system.

Because of the conservation of mass principle, the input mass must be equal to the output mass. If think about this principle for fluid flow systems, we can build this arrangement;

According to this equation, if the input mass flow is bigger than the output mass flow, the total mass of the system increases. Otherwise, it decreases. Also if the input and output mass flows are the same, there is no total change in the mass flow.

Also, we can write this conservation of mass principle like this;

We just wrote the same equation by writing the formula inside the mass flows. So, an average input and output velocities and input and output areas are very important.

Example to Understand the Conservation of Mass Principle

Example 2: Consider that we have a control volume that there is a mass flow inside it. The input velocity of the fluid inside the control volume is 2 m/s. Also, the diameter of the input pipe is 5 centimeters. Also, the output velocity of the fluid from another pipe is 1 m/s. The diameter of the output pipe is 3 centimeters. The density of the fluid is 2250 kg/m3.

• Calculate the total mass and volume flow rates into the control volume.
• Calculate the total fluid mass after 60 seconds inside the control volume.

Solution; 

First of all, we need to use the conservation of mass principle to calculate the total mass and volume flow rates inside the control volume. So;

Input mass flow – Output mass flow = Total mass flow

density*(area of inlet)*(inlet velocity) – density*(area of outlet)*(outlet velocity) = total mass flow

2250*(0.05^2)*2 – 2250*(0.03^2)*1 = 92.25 kg/s.

So, if we want to find the total volume flow rate, we just need to divide this value by density;

92.25/2250 = 0.041 m3/s.

Because the input mass flow is bigger than the output mass flow, there is a positive change of mass inside the control volume according to the conservation of mass principle. So, if we multiply with seconds, we can find the total mass of fluid inside the control volume;

Total mass = (total mass flow)*(time)

= 92.25*60;

Total mass after 60 seconds in the control volume = 5535 kg.

Effects of Different Types of Flows on Conservation of Mass Principle

In fluid mechanics, we are dealing with different kinds of fluid flows. Also, we use these different kinds of fluid flows in engineering systems. So, while we are designing our systems, we need to consider the different types of flows.

Conservation of Mass for One, Two and Three Dimensional Flows

In the analysis of different engineering systems, the number of dimensions is very important. Because it affects the complexity of our calculations.

There are no one and two-dimensional flows in nature technically. They are the general considerations and assumptions for different engineering systems.

For one-dimensional fluid flows, there is no meaning of area because the fluid flow is linear. Also for the two-dimensional fluid flow, the area value is linear. We can use lengths for the area. For three-dimensional flow, it is as it is.

Steady and Unsteady Fluid Flows

This is also a very important consideration. In most practical applications, we consider fluid flows as steady flows. And in the conservation of mass principle, we consider the fluid flows as steady flows.

Also, if we need to deal with the conservation of mass calculations of unsteady flows, we need to calculate the time-averaged values. This is an approach to considering unsteady fluid flows as steady fluid flows.

Laminar and Turbulent Fluid Flows

Laminar and turbulent natures of flows do not affect the calculations of conservation of mass. We are dealing with the flow velocity and densities of fluids. We are not dealing with the flow regime or type of fluid in these calculations.

Effects of Compressibility and Incompressibility of Fluids

This is a very important effect on the conservation of mass principle. Because, if the fluid is compressible, the density of the fluid changes. As you know from above, density is a very important factor in the mass flow. If the density of fluid changes, the total mass flow changes.

If there is a difference in the densities of fluid flows, there will be changes in the mass flow rates. So, the conservation of mass will be affected by this situation.

Above all, most engineering systems are incompressible systems. And the density in incompressible systems is not changing. So, we can use the volume rate calculations for these systems.

Internal and External Flows

This is also another classification of fluid flows. But there is no effect of internal and external flows in the conservation of mass calculations. We are dealing with the speed of the flow, the area o the flow cross-section, and the density of the fluid flow.

So, internal and external flows are not important in the calculations of mass flow and volume flow rates.

Inviscid and Viscous Flows on Conservation of Mass

In general, we assume the fluid flows in different aspects. If we think about the inviscid and viscous flows in general, there are no effects of these types of flows in mass flow calculations. It is about the regime of the flow.

There are inviscid and viscous flow regions of different fluid flows. But in general, we are not considering the inviscid and viscous flow regions in these calculations. We are not dealing with these aspects.

General Relativity on Mass and Energy

Also, there is a relation between the eenergy and the mass. This energy is the total energy of the mass. From this relation, we can understand that there is convertibility between mass and energy. The conversion takes place with this famous equation;

In this famous equation of Einstein,

• E is the total energy of a mass.
• m is the total mass of the object.
• c is the speed of light in space.

So, there is a strict relationship between mass and energy.

Application Areas of Conservation of Mass

Actually, in all the systems that have mass transfer, we use the conservation of mass equation. We can give different kinds of applications for it.

Heat Exchangers

Heat exchangers are very important devices that we use in building heating applications. In heat exchangers, there is a heat transfer between the fluid and the burning fuel. And this fluid transfers the heat to other sides of the buildings.

So in heat exchanger systems, there is an inlet flow and outlet flow of fluid. And if we think of the heat exchanger as a control volume, the inlet and the outlet mass flows are the same. We are making our heat transfer calculations according to the mass flow rates.

Pumps and Turbines

Pumps and turbines are also very important applications for the mass flow of fluids. They are giving energy or extracting energy from flowing fluids. So, there are inlet fluids and outlet fluids in pump and turbine systems.

Also, if we think of the pump and turbines as control volumes, the mass flow of the fluid inlet and fluid outlet are the same. So, there is a conservation of mass in these systems.

Car Engines

In car engines, there are lots of kinds of systems that we use the conservation of mass principle. For example in the pistons, there is a fuel and air mixture inlet, and chemical reaction and outlet of the burnt fuels.

There is no net mass accumulation in the pistons and cylinders. So, the inlet mass flow must equal the outlet mass flow in car engines.

As you understand from this example, the conservation of mass principle is also valid for chemical reactions.

Conclusion

So, the conservation of mass is a very important law of nature. Like the conservation of energy, we use this law in lots of kinds of systems. This phenomenon is generally related to mass flow and volume flow rates. We generally build the conservation of mass by using the mass and volume flow equations.

Also, there are effects of types of fluid flows on this phenomenon. We need to consider the types and regimes of flows while we are building our equations.

And the conversion between mass and energy according to general relativity is also worth mentioning in this topic.

In different practical applications, we use the conservation of mass equation. For example in heat exchangers, car engines, and other kinds of applications, we build mass equations.

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FAQs About Conservation of Mass