# Mass Flow Rate: Calculation and Measurement

Mass flow rate is the measure of the amount of mass passing through a cross-sectional area per unit of time. It is typically expressed in units such as kilograms per second (kg/s) and is commonly used in fluid dynamics to quantify the rate at which a fluid substance is moving through a system.

In this article, we will discuss the calculation of mass flow rate, the effect of compressibility on mass flow rate, mass flowmeters, and practical applications in different industries.

## What is Mass Flow Rate

In fluid mechanics, mass flow rate is the quantity of mass passing through a given surface per unit of time. To illustrate this, consider the diagram below showing a fluid flowing through a circular pipe.

Based on the conservation of mass, mass is neither created nor destroyed. Hence, the mass of fluid entering the pipe is the same amount of fluid exiting the pipe. It follows that, at any plane perpendicular to the center line of the tube, the same amount of mass passes through.

The amount of mass passing through this plane is called the mass flow rate, which can be calculated as the product of the fluid density, the cross-sectional flow area, and the velocity of the fluid. This is shown in the following equation:

• = mass flow rate [kg/s]
• = fluid density [kg/m3]
• = cross-sectional flow area [m2]
• = fluid velocity [m/s]

If the volumetric flow rate is known, then it is simply equal to the product of the volumetric flow rate and the fluid density:

Where:

• Q = volumetric flow rate [m3/s]

Knowing the mass flow rate is important in ensuring the correct operation of various mechanical and thermodynamic processes that involves fluid flow. For example, engineers need to account the mass flow rate in determining the power output of engines, the efficiency of heat exchangers, and for proper chemical reaction management in process industries.

## Effect of Compressibility on Mass Flow Rate

Based on the mass flow rate equation above, it seems that increasing the velocity of a moving fluid could indefinitely increase the mass flow rate for a given cross-sectional flow area. However, in real fluids, the density changes as velocity increases due to the effects of compressibility. Therefore, in order to accurately determine the mass flow rate in higher velocities, we need to consider this phenomenon.

For an ideal compressible gas, the mass flow rate can be calculated using the following equation:

Where:

• pt = total pressure [Pa]
• Tt = total temperature [K]
• γ = specific heat ratio [unitless]
• R = gas constant [8.314 J/mol-K]
• M = Mach number [unitless]

In the equation above, the specific heat ratio, also known as the adiabatic index, is the ratio of the heat capacity of the fluid at constant pressure to its heat capacity at constant volume. Its value varies depending on the type of gas, with common values around 1.4 for diatomic gases like air.

The Mach number is a dimensionless parameter defined as the ratio of the fluid velocity to the speed of sound.

The compressible mass flow rate equation reveals that increasing the area, increasing the total pressure, or decreasing the total temperature will boost the mass flow through a tube. However, the impact of increasing velocity, or the Mach number, is more intricate.

If we were to hold the area, total pressure, and temperature constant, and plot the mass flow rate against the Mach number, we would observe that the mass flow rate reaches a maximum value when the Mach number equals one. This maximum flow limit is known as the choking of the flow. Substituting M = 1 into the equation above allows us to determine the value of the choked mass flow rate as follows:

## Mass Flowmeters

Mass flowmeters are instruments designed for the precise measurement of the mass rate of flow passing through a pipe or conduit. Unlike volume flowmeters, these devices measure the mass flow directly, providing a useful metric for many industrial applications where the density of the flowing medium can vary with temperature and pressure.

There are two types of mass flowmeters: Thermal and Coriolis.

### Thermal Mass Flowmeters

Thermal mass flowmeters operate on the principle that the amount of heat carried away by a flowing fluid is directly proportional to its mass flow rate. These flowmeters inject a known quantity of heat into the fluid and then measure the temperature change downstream.

### Coriolis Mass Flowmeters

Another common type is the Coriolis flowmeter, which utilizes the Coriolis effect. As the fluid flows through a vibrating tube, the Coriolis forces induce a phase shift that can be measured and from which the mass flow rate is derived.

To ensure precise measurements, installation parameters must be rigorously followed. This includes proper alignment, avoiding areas of turbulence caused by bends or valves, and maintaining a straight run of pipe both upstream and downstream of the flowmeter.

Regular calibration of mass flowmeters is critical to maintain accuracy. Factors such as changes in fluid characteristics or wear and tear can affect performance over time.

## Practical Applications

### Chemical Process Industry

In the chemical process industry, determining the mass flow rate is important in the design of chemical reactors to ensure reaction efficiency and safety. It is especially critical in processes like the continuous stirred-tank reactor (CSTR) system, where maintaining correct reactant inflow and product outflow is key to achieving desired conversion rates.

### Thermal Power Plants

In thermal power plants, the mass flow rate of steam through turbines determines the energy output. Maintaining optimal flow rates of steam can improve the efficiency of heat exchange and electricity generation processes.

### Aerospace Engineering

In aerospace engineering, jet propulsion and the design of intake systems of aircraft engines heavily rely on the accurate calculation of mass flow rate. It informs the thrust produced by the engine.

### HVAC Systems

In heating, ventilation, and air conditioning systems, calculating and controlling mass flow rate is vital to ensure the right distribution of temperature-controlled air throughout buildings.

### Automotive Industry

In the automotive industry, proper mass flow rate calculation of fuel and air into the combustion chamber is essential for engine performance optimization. It influences the stoichiometry of the fuel mixture and subsequently, the engine’s power and emission levels.

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