## What are units and dimensions?

Dimensions are the basic concepts of the measurement of quantities. Units are the means of expressing the dimensions. For instance, an area has a dimension of L2 and a unit of m2.

## Fundamental and secondary quantities

The fundamental quantities are independent and invariable to the changes in other quantities. There are seven fundamental quantities- length, mass, time, temperature, electric current, luminous intensity, and amount of matter.

Secondary quantities are the quantities that are derived from or using the fundamental quantities or any other secondary quantity or both fundamental and secondary quantities. For example, the volume is a secondary quantity. The dimension of volume is L3, which is derived using the fundamental quantity Length (L). Density is also a secondary quantity. The dimension of density is ML-3, which is derived by using a fundamental quantity mass (M) and a secondary quantity Volume (L3).

## Units and dimensions in chemical engineering calculations

The units and dimensions are the necessary concepts in the calculation of chemical processes. The basic principle of any chemical engineering calculation is that the equation must be dimensionally consistent. It means while performing various chemical engineering calculations, the net dimensions of the quantities on the left-hand side of any equation must be equal to the net dimensions of the right-hand side of the equation. For instance, if an equation states, A + B = X . Y - Z, then the net dimensions of A + B must be equal to the net dimensions of X . Y - Z for the equation to be valid.
For the quantities to add or subtract in an equation, the dimensions of each quantity must be alike, whereas, for the quantities to multiply or divide in an equation, the dimensions of each quantity may be alike or different. Hence, in the above equation, quantities A and B must have the same dimensions. The dimensions of X and Y may be alike or different, but the net dimensions of X.Y should be equal to the net dimension of Z.
The units and dimensions are useful for the chemical process analysis. The units and dimensions also play a significant role in the dimensional and non-dimensional (dimensionless) numbers. A dimensional number has certain dimensions, either fundamental or secondary.

### Dimensional numbers

Dimensional numbers are the set of variables that have some specified dimensions. For example, Length has a dimension L. Hence, length is a dimensional number.

### Dimensionless numbers

Dimensionless numbers are the set of variables without net dimensions of net units. There are two types of constants- dimensional constants and dimensionless constants. Dimensional constants have dimensions, such as the gravitational constant g. The dimensionless constants do not have dimensions, such as the Reynolds number. The common dimensionless quantities used in chemical engineering are- Reynold's number, Prandtl number, Nusselt number, Sherwood number, Froude's number, Grashof number, Mach number, Schmidt number, and so on.

 Sr. No. Dimensionless numbers Formula 1 Reynold's number $\frac{\rho vD}{\mu }$ 2 Prandtl number $\frac{\nu }{\alpha }$ 3 Nusselt number $\frac{h.L}{{k}_{f}}$ 4 Sherwood number $\frac{{k}_{m}}{D/L}$ 5 Froude's number $\frac{v}{\sqrt{g.D}}$ 6 Grashof number $\frac{g.\beta .{\rho }^{2}.\left({T}_{s}-{T}_{f}\right).{L}^{3}}{{\mu }^{2}}$ 7 Mach number $\frac{v}{c}$ 8 Schmidt number $\frac{\mu }{\rho .D}$

The units and dimensions are useful for the material balance in a chemical process.

### Material balance

The concept of material balance is based on the law of conservation of mass. The material balance includes the calculation of the quantities of materials that enter or leave a system. As per the theory of material balance, the quantity of material that enters a system shall be equal to the quantity of material that leaves a system. Generally, a chemical reaction takes place in two types of systems - an open system and a closed system. In an open system, any material can cross the system boundary and enter it or exit from it, whereas, in a closed system, any material can't enter or exit the system. The reactions need to be written properly by considering all the losses and verified using the basic principle, that is the equation should be dimensionally consistent.

## Context and Applications

The units and dimensions play a significant role in every chemical reaction that takes place in a laboratory, chemical plant, and so on. Hence, it is necessary to study and properly apply the concept of units and dimensions. The units and dimensions are useful for the students undergoing the following graduate courses-

• Bachelors of Engineering or Technology (Chemical)
• Masters of Technology (Chemical, Petroleum, Oil)

## Practice Problems

Q1. Which of the following are defined as the basic concepts of the measurement of quantities?

a) Units

b) Dimensions

c) Reactions

d) Stream diagrams

Explanation– Dimensions are defined as the basic concepts of the measurement of quantities.

Q2. Which of the following are defined as the means of expressing the dimensions?

a) Units

b) Dimensionless constants

c) Dimensional constants

d) Material Balance

Explanation– Units are defined as the means of expressing the dimensions.

Q3. Which of the following is the set of variables without net dimensions or net units?

a) Dimensionless numbers

b) Dimensional numbers

c) Net numbers

d) Material constants

Explanation– Dimensionless numbers are the set of variables without net dimensions or net units.

Q4. On which law is the concept of material balance based?

a) Law of conservation of momentum

b) Newton's law of cooling

c) Law of conservation of mass

d) Henry's Law

Explanation– The concept of material balance is based on the law of conservation of mass.

Q5. Which of the following is a type of system in which a chemical reaction can take place?

a) Spacial system

b) Open system

c) Vacuum system

d) Exterior system

Explanation– Open system is a type of system in which a chemical reaction can take place.

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### Units and Dimensions

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