11.4 Transforming distribution functions  11. Special topics  11.2 Using units in

11.3 Nomograms

With the current availability of computers, there has been a decline in reliance on graphic methods for the solution of equations, and the material in this section is mostly unknown to the current generation of scientists. In engineering, graphic presentations have long been important tools, and they still serve important roles in atmospheric science (e.g., in thermodynamic diagrams). This optional section on nomograms is included here partly because this material is seldom seen and yet is a useful technique for the rapid, although approximate, solution of many equations. The author often uses nomograms in the course of airborne research, where it is difficult to solve equations with the speed needed to make flight decisions.

A nomogram is a graphic representation of the solution of an equation, usually involving more than two variables, where the solution can be read by using a straightedge to connect points representing two variables and then determine the value of a third. The equation A+B=C can be used to illustrate the geometrical basis for nomograms. Because (A+B)/2=C/2, the average of A and B is (1/2)C, and the solution can be represented by three lines as in Fig. 11.1 where the scale relationship used for the variable C is half that used for A and B.

 

Figure 11.1:  Nomogram representing the solution of the equation A+B=C.

 

The diagram becomes more useful and less trivial if A, B, and C are functions of other variables, e.g., as in the relationship A(x)+B(y)=C(z), in which case the axes can be labeled according to the values of x, y, and z that give the corresponding values of A, B, and C. Product forms like A(x)B(y)=C(z) can be constructed by using logarithmic scales, because $\log(A)+\log(B)=\log(C)$.

It is also possible to construct nomograms representing relationships among more than three variables. For example, A+B+C+D=E can be represented by an initial nomogram to give the sum A+B, followed by another that uses the resulting point from the first as input to give (A+B)+C, followed by a fourth to complete the sum. Another technique often used in such sequences is to incorporate an ordinary graph as a way of representing one relationship or to change units or scales, as in the example that follows.

 

Figure 11.2:  Nomogram representing the relationship among temperature (T), dewpoint (T_D), and isobaric wet-bulb temperature (T_WB) as a function of pressure. The circled points connected by the straight line show corresponding values for a pressure of 800 mb.

 

Figure 11.2 shows an example. In this case, the nomogram was constructed to represent the wet-bulb equation (cf. Example 9.1):

$$T_{wb} = T + {{L_V}\over{C_p}}(r-r_s(T_{wb}))$$ (11.8)

 

where L_v is the latent heat of vaporization, C_p the specific heat of air at constant pressure, r the water vapor mixing ratio, and r_s(T_wb) the saturation mixing ratio at the wet-bulb temperature. The mixing ratio is a function of the dewpoint temperature T_d and the pressure p:

$$r = \epsilon {{e_s(T_d)}\over{(p-e_s(T_d)}}$$ (11.9)

 

where $\epsilon$ is the ratio of molecular weights of water and air and e_s(T_d) is the saturation vapor pressure at the temperature T_d. The wet-bulb equation can then be rewritten as

$$T_{wb} + {{L_v}\over{C_p}}r_s(T_{wb}) = T + {{L_v}\over{C_p}} r_s(T_d) \ .$$ (11.10)

 

The left side of this equation is the quantity plotted on the left side of the nomogram, and the right side is the last term in this equation; these two terms are labeled by the appropriate values of T_wband T_d, respectively, and in both cases a set of lines is plotted to show the pressure dependence of the terms.

For a more extensive discussion of nomograms, including those with more complicated geometry, see Menzel (1960), pp. 141..

   
 11.4 Transforming distribution functions  11. Special topics  11.2 Using units in