Units and Dimensions

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Chapter: Pharmaceutical Engineering: Introduction

The pharmaceutical scientist is familiar with the units (dimensions) of centi-meter (length), gram (mass), and second (time) or the conventional Syste`me Internationale (SI) units of meter, kilogram, and second.

Pharmaceutical Process Engineering

Units and Dimensions

The pharmaceutical scientist is familiar with the units (dimensions) of centi-meter (length), gram (mass), and second (time) or the conventional Syste`me Internationale (SI) units of meter, kilogram, and second. The engineer, in contrast, will express equations and calculations in units that suit quantities he or she is measuring. To reconcile in part this disparity, a brief account of units and dimensions follows.

Mass (M), length (L), time (T), and temperature (°) are four of six funda-mental dimensions, the units of which have been fixed arbitrarily and from which all other units are derived. The fundamental units adopted for this book are the kilogram (kg), meter (m), second (sec), and Kelvin (K). The derived units are frequently self-evident. Examples are area (m2) and velocity (m/sec). Others are derived from established laws of physics. Thus, a unit of force can be obtained from the law that relates force, F, to mass, m, and acceleration, a:

F = kma

where k is a constant. If we choose our unit of force to be unity when the mass and acceleration are each unity, the units are consistent. On this basis, the unit of force is Newton (N). This is the force that will accelerate a kilogram mass at 1 m/sec.

Similarly, a consistent expression of pressure [i.e., force per unit area is Newtons per square meter (N/m2 or Pascal, Pa)]. This expression exemplifies the use of multiples or fractions of the fundamental units to give derived units of practical importance. A second example is dynamic viscosity [M/(L·T)] when the consistent unit kg/(m·sec), which is enormous, is replaced by kg/(m·hr) or even by poise. Basic calculations using these quantities must then include conversion factors.

The relationship between weight and mass causes confusion. A body falling freely due to its weight accelerates at kg·m/sec2 (g varies with height and latitude). Substituting k = 1 in the preceding equation gives W = mg, where W is the weight of the body (in Newtons). The weight of a body has dimensions of force, and the mass of the body is given by

mass(kg) = weight (N) / g(m/sec2)

The weight of a body varies with location; the mass does not. Problems arise when, as in many texts, kilogram is a unit of mass and weight of a kilogram is the unit of force. For example, an equation describing pressure drop in a pipe is

ΔP = 32ulη /d2

when written in consistent units—ΔP as N/m2, viscosity (η) as kg/(m·sec), velocity (u) as m/sec, distance (l) as m, and tube diameter (d) as m. However, if the kilogram force is used (i.e., pressure is measured in kg/m2), the equation must be

ΔP = 32ulη /d2

where g = 9.8 m/sec2. In tests using this convention, the conversion factor g appears in many equations.

The units of mass, length, and time commonly used in engineering heat transfer are kilogram, meter, and second, respectively. Temperature, which is a fourth fundamental unit, is measured in Kelvin (K). The unit of heat is the Joule (J), which is the quantity of heat required to raise the temperature of 1 g of water by 1 K. Therefore, the rate of heat flow, Q, often referred to as the total heat flux, is measured in J/sec. The units of thermal conductivity are J/(m2·sec·K/m). This may also be written as J/(m·sec·K), although this form is less expressive of the meaning of thermal conductivity.

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