11.3 Nomograms  11. Special topics  11.1 Introduction

11.2 Using units in equations

At one time, particularly in engineering usage, most equations were expressed in forms that required a particular set of units. An example might be the following form for the equivalent potential temperature:

$$\Theta_e = (T+273.15) \left({{1000}\over{(p-e)}}\right)^{2/7} \exp\left\{{{2.5 r_v}\over{(T+273.15)}}\right\} \ .$$ (11.1)

 

This form gives the equivalent potential temperature in Kelvin if the temperature T is in degrees Celsius, the pressure p and water vapor pressure e are in mb, and the mixing ratio r_v is expressed in g/kg. This form of equation ties the expression to a particular set of units, and often obscures the underlying physical relationships. For example, in this case the factor 2.5 arises from the dependence on the latent heat of vaporization, the specific heat of air at constant pressure, and conversion of units. Because the latent heat varies significantly with temperature, (11.1) only applies to a single temperature and it is not obvious without further explanation how this must be modified for other temperatures. Because of these problems, and the lack of flexibility in units, this form of equation should be avoided.

Instead, current usage favors expressing equations in dimensionless form. For example, the above equation in dimensionless form is

$$\Theta_e = T \left({{p_0}\over{p-e}}\right)^{R_d/C_p} \exp\left\{{{L_vr_v}\over{C_pT}}\right\}$$ (11.2)

 

where R_d  is the gas constant for dry air, L_v is the latent heat of vaporization, C_p the specific heat of air at constant pressure, and T the absolute temperature. Where constants occur, like p₀ in this equation, they are represented by symbols and the value is then given in some system of units, in this case p₀=1000 mb. This form of the equation is dimensionally consistent in any system of units. In this form, arguments like that of the exponential in (11.2) will be dimensionless, and the units used should balance in any correct evaluation of the expression.

A common error, or at least a common imprecision in usage, is to use quantities with dimensions in logarithmic, exponential, or trigonometric functions. For example, the units to be associated with an expression like $\ln(r)$ are unclear, because the value of the expression changes when the units used to express r change. It is preferable to use expressions like $\ln(r/r_0)$ where r₀ is a specified reference radius, because then the expression is independent of the system of units used even when the value of r₀ is expressed in a particular system (e.g., r₀=1$\mu$m). A dimensionally consistent expression can always be found for equations involving logarithms or exponential or trigonometric functions. For precise and clear usage, these forms are preferred, although they are not always adhered to in common usage. If this convention were applied strictly, an expression like

$$\ln(\Theta) = \ln(T) - {{R_d}\over{C_p}}\ln(p) + {\rm constant}$$ (11.3)

 

would be changed to

$$\ln\left({{\Theta}\over{T}}\right) ={{R_d}\over{C_p}}\ln\left({{p}\over{p_0}}\right) \ ,$$ (11.4)

 

because in this form the dimensionally ambiguous terms like $\ln(\Theta)$are avoided. All arguments in logarithmic, exponential, trigonometric, etc., functions should be dimensionless.^11.1

Balancing dimensions in dimensionless equations is a powerful check on the consistency of units, and provides a convenient basis for conversion among units.
 

Example 11.1:

The pattern in the following example, if followed consistently with dimensionless equations, handles all unit conversion problems and in addition provides a valuable check on the consistency of the equation and the values used.^11.2

The problem is to find the equivalent potential temperature using (11.2), for the following conditions: temperature 20, pressure 27 inches of mercury, and dew point 18. In addition, the following values are known: C_p=1005 J kg^-1 K^-1, L_v = 586 cal/g, and the gas constant R_d is 287 J kg^-1 K^-1. At the dewpoint temperature, the saturation vapor pressure is e=20.630 mb. The molecular weights of water (M_w) and of air (M_a) are 18.015 g mol^-1 and 28.964 g mol^-1, respectively.

Units can be converted as needed by multiplying parts of the equation by ratios of units that are unity, such as (100 cm)/(1 m), so as to cancel all units. In this example, that is accomplished as follows:
 
 

• Step 1: Find the mixing ratio from the vapor pressure and the pressure:

$$= {{M_w}\over{M_a}} {{e}\over{(p-e)}}$$  r_v

$$= {{18.015 g\thinspace mol^{-1}}\over{28.964 g\thinspace mol^{-1}}}......\thinspace Hg \times {{1013.25 mb}\over{29.921 in.\thinspace Hg}}- 20.630 mb}}$$

$$= 0.622 \times {{20.630 mb} \over{914.33 mb}}$$

$$= 0.01436 \ {\rm [dimensionless]}$$ (11.5)

 

Note that the pressure ratio 1013.25 mb : 29.921 in. Hg has been used to convert the pressure to mb; both numerator and denominator correspond to one atmosphere, so this ratio is easier to remember and find than the resulting conversion factor of 33.86 mb / (in. Hg).

• Step 2: Evaluate the exponential factor. This should be dimensionless.

$${rl}{{L_vr_v}\over{C_pT}} = {{586 {{cal}\over{g}} \times 0.01436}\over{1005 {{J}\over{kg K}} + 293.15 K}}$$

$$= {{586 {{cal}\over{g}} \times {{4.184 J}\over{cal}} \times 0.01436}\over{1005 {{J}\over{kg K}}\times {{1 kg}\over{1000 g}} \times 293.15 K}} \ \ .$$ (11.6)  =   0.1195

 
• Step 3: Evaluate the expression (11.2):

$${rl}\Theta_e = 293.15 K \left({{1000 mb}\over{27 in.\thinspace Hg \times {{1013.2.......630 mb}}\right)^{{287 J kg^{-1}K^{-1}}\over{1005 J kg^{-1}K^{-1}}} e^{0.1195}$$

$$= 338.9 K \ \ .$$ (11.7)

 

In these examples, the units are carried in the equations and evaluated to obtain the units of the answer. Whenever units fail to cancel, this indicates that a conversion factor is needed (or that incorrect units have been used for one of the quantities in the equation).

Atmospheric science journals and most scientific journals now require use of International System (SI) units. If that system is used consistently, it is not necessary to carry the units in dimensionless equations; units will automatically cancel if SI units are used in a correct nondimensional equation. It is still useful to carry units and check for consistency as in the preceding example.

   
 11.3 Nomograms  11. Special topics  11.1 Introduction