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Differential Temperature Transducer
An Iterative Method of Evaluating the Signal

Picture of a Delta-T Transducer

The Differential Temperature Transducer has been used for a number of years to measure the change in temperature of closed fluid-streams.

The instrument is based on the thermopile, which is inherently a differential device; even when used as an absolute-temperature detector, a thermocouple or thermopile does so only by virtue of a reference temperature made available to it. One junction (the cold junction) serves as the "reference" for the other junction (the hot junction). Where "absolute" temperature is being measured, a standard reference cold-junction of 0°C (ice bath temperature) is used.

A difficulty, however, is that thermocouple characteristics are nonlinear. Figure 1 shows a typical E(mV) vs T(temperature) curve. The National Bureau of Standards (now NIST) has published tables of thermocouple characteristics which define the nonlinearities, and various linearizing schemes are used to translate the millivolt thermocouple signal into an accurate temperature reading.

Figure 1, Typical E(mV) vs T(temperature curve.

In the Differential Temperature Transducer, another difficulty arises: the reference temperature is not a fixed value. If the low-temperature side is the "reference," then the sensitivity (mV/°C) can vary considerably, depending on the value of that low-temperature point. Another way to describe the problem is by recognizing that, depending on where on the thermocouple characteristic curve one is operating, the conversion factor (slope) for signal strength to differential temperature is a variable; the curve of E (output) vs Temperature (T) is not linear.

Some information must be available, in addition to the DeltaT signal itself, to define the correct sensitivity (S):

S = dE/dT (1)

Note that S, the Seebeck coefficient for the particular thermocouple materials, is a function of both ΔT and Tch,where Tch is a characteristic temperature, such as Tlow, Thigh, or Tavg.

The temperature difference to be determined (ΔT) is:

ΔT=ΔE/S   (2)

where ΔE is the measured signal, in millivolts, and S is an average, or effective, sensitivity over the range Tlow to Thigh.

Equations (1) and (2) indicate the problem: S, in general, is a stronger function of Tch than of ΔT.

THE ITERATIVE METHOD

To implement the linearization method herein, we assume an approximate value of Tlow (or better yet, a measured value). Figure 1 depicts a thermocouple characteristic, taken from published data.

We wish to linearize the curve between two points, Tlow and Thigh. On Figure 1, Tlow is point 1; Thigh is point 2. The millivolt signal to be linearized (ΔE) corresponds to Thigh - Tlow.

The sensitivity, the slope of the characteristic curve, is evaluated at Tlow by drawing a tangent to the curve at the point. Let us call this dE/dT1. Using this sensitivity at point 1 and ΔE, the measured signal, we compute the first approximation (ΔTi) to ΔT:

ΔT<sub>i</sub> = ΔE / ( dE/dT<sub>i</sub>)	 (3)

Clearly, ΔTi exceeds ΔTtrue because of the increasing slope of the characteristic curve with increasing temperature. If, then, a new characteristic temperature is computed, an average between Tlow and an apparent Thigh, using ΔTi, is:

T<sub>avi</sub> = T<sub>low</sub> + ΔT<sub>i</sub>/2  		(4)

Point 3 on the curve corresponds to Tavi and, if a new tangent is drawn at point 3, a still better value of effective sensitivity is found:

ΔT<sub>ii</sub> = ΔE / (dE/dT<sub>i</sub> ) 		(5)

The result of the computation is re-inserted into (4), giving yet another (and better) point of the curve. With each iteration, the slope (sensitivity) lies closer and closer to the chord 1-2, whose slope is the correct value.

This procedure is nothing more than a graphical version of the familiar Newton-Raphson iteration procedure used to solve for zeroes of a polynomial.

Conveniently, the NBS has provided curve-fitting information in the form of coefficients of an 8th-order polynomial that closely approximates the thermocouple characteristic curve for copper-Constantan (type T) over a range of 0° to 400°C (see Table 1):

E = a<sub>1</sub>T + a<sub>2</sub>T<super>2</super> + ... a<sub>8</sub>T<super>8</super>		(6)

It is a simple matter to differentiate (6) to arrive at a polynomial expression for dE/dT:

dE/dT = a<sub>1</sub> + 2a<sub>2</sub>T T ... 8a<sub>8</sub>T<super>7</super>  (7)

The following simple algorithm can quickly, within one or two iterations (depending on the accuracy required and the magnitude of ΔT), yield a satisfactory result:

dE/dT(T<sub>1</sub>) = a<sub>1</sub> + 2a<sub>2</sub>T<sub>2</sub> + ... 8a<sub>8</sub>T<sub>1</sub>   (8)  

ΔT<sub>1</sub> = ΔE / (dE/dT)(T<sub>1</sub>)         (9) 

T<sub>avi</sub> = T<sub>1</sub> + ΔT<sub>1</sub>/2    (10)

For the first pass, let T1 = Tlow; then return to (8) with Tavi until ΔT converges to a stable value.

Note: The Delta-T Differential Temperature Transducer's thermopile consists of 10 pairs of junctions. Hence the sensitivity coefficients (a1 ... a8) used in the computation exceed the published values in NBS 125 by a factor of 10.

RESULTS

For a typical case where Tlow = 20°C is erroneously estimated at 25°C, then the error in ΔT is found to be 0.1°C after two iterations. At that portion of the curve, for small errors in Tlow, the error in ΔT is given by: ΔT<sub>error</sub> = 0.02 T<sub>low error</sub>      (11)

The more accurate the knowledge of the reference, or low temperature, the more accurate will be the calculated ΔT that we measure. Fortunately, the results are not very sensitive to this estimate.

If the value of Tlow is known to vary considerably, it might be convenient to provide a simple, inexpensive temperature probe at the fluid inlet to the Differential Temperature Transducer to measure the actual temperature. The Delta-T transducer can be provided with such probe incorporated within the housing if desired.

TYPICAL CALCULATION

Img of Typical Calculation for ΔT