Probably the most common method of measuring moderate temperatures of liquids and gases in the chemical and petroleum industries involves a primary element (detector) and a thermowell. The primary element is usually a thermocouple, resistance thermometer, or gas-filled expansion bulb. If a thermocouple is used, it is most apt to be in the form of a "pencil" or sheathed assembly, which provides rigidity and mechanical protection.

The instrument or control engineer is interested in the study of the dynamic behavior of thermowell/primary-element combinations for these reasons:

1. To predict the dynamic behavior quantitatively in order to design a temperature control system quantitatively.

2. To decide on the design of optimum thermowell/primary-element combinations.

To support these interests, we have devised a mathematical second-order model that is simple enough to be used as a practical working tool for design engineers or plant engineers. The details are not reproduced here but are presented elsewhere.9 The model is an improved version of one discussed in Chapters 21 and 22 of reference 2. Calculator programs calculate the two time constants, t„ and r4, and either the step response to ambient temperature changes or frequency response.

Illustrative Examples—Forced Convection

To illustrate the application of the model mentioned above, let us look at several cases.

Example 1: Gas Flow. Let us consider a 1/8-inch OD pencil-type thermocouple in a well 0.405-inch OD by 0.205-inch ID. The service is steam and the base case velocity is 152 ft/sec. The effect of velocity on the two time constants is shown in the following table v = 300 V = 152 V = 50 V = 5.0

ft/sec ft/sec ft/sec ft/sec t„, sec 91.8 92.2 93.4 107

Step-response curves are presented on Figure 11.21. If the thermocouple had been used bare, a single time constant of 1.9 seconds would have been obtained for V = 152 ft/sec. In view of the large annular clearance, there may be some error here in assuming a purely conductive heat-transfer mechanism for the annular fill.

For the base case of 152 ft/sec, let us examine the effect of using different annular fills:

AIR OIL MERCURY

Step responses are given in Figure 11.22. Neglecting the thermal capacitance of oil and mercury as we did may introduce some error here.

Consider next the effect of changing clearance. For the same outside thermowell diameter, for V = 152 ft/sec, for the same thermocouple, and for air in the annular space we obtain:

Annular Clearance 0.040 inch 0.020 inch 0.005 inch t„ sec 92.2 52.4 16.6

time. seconds

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