A simple rate-of-climb indicator (see Fig. 2C.7) under the proper circumstances the simple apparatus shown in the figure can be used to measure the rate of climb of an airplane. The gauge pressure inside the Bourdon element is taken as proportional to the rate of climb. For the purposes of this problem the apparatus may be assumed to have the following properties: (i) the capillary tube (of radius R and length L, with R

(a) Develop an expression for the change of air pressure with altitude, neglecting temperature changes, and considering air to be an ideal gas of constant composition.

(b) By making a mass balance over the gauge, develop an approximate relation between gauge pressure p, – Po and rate of climb vz for a long continued constant-rate climb. Neglect change of air viscosity, and assume changes in air density to be small.

(c) Develop an approximate expression for the “relaxation time” trel of the indicator that is, the time required for the gauge pressure to drop to 1/e of its initial value when the external pressure is suddenly changed from zero (relative to the interior of the gauge) to some different constant value, and maintained indefinitely at this new value.

(d) Discuss the usefulness of this type of indicator for small aircraft.

(e) Justify the plus and minus signs in the figure.