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Measuring heat usage in radiator

16 Jun 2015  | Ben Smith

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Sound waves in a fluid propagate more quickly when traveling in the downstream direction (that is, with the direction of flow) than when traveling in the upstream direction (that is, against the flow) (figure 2). We can use this fact to our advantage.

Figure 2: Measuring the velocity of a fluid by differential time-of-flight.

In figure 2 the system is configured to measure time of flight. Time of flight is defined as the time that elapses from the instant a burst of ultrasonic sound is launched from one transducer (transducer A in the diagram) to the instant that the burst is received in another transducer (transducer B). The time of flight will be dependent on the physical properties of the medium and the velocity of flow. The faster the fluid flows in the burst direction, the less the time of flight will be.

When a burst of ultrasonic sound is launched from transducer B and received in transducer A, the situation is reversed. Now, the velocity of the fluid is working against the sound burst. The faster the fluid flows, the greater the time of flight will be.

This is how you measure flow velocity. First, launch a burst in the downstream direction and count the time that it takes to be received. Now launch a burst in the upstream direction and count the time that takes to be received. If the two times are equal, then the medium (the water) is stationary. But if the upstream count is greater than the downstream count, the difference gives you a very precise indication of the velocity of the medium. Now factor in the pipe diameter and apply a conversion constant, and you can report the flow rate in your units of choice: gallons per minute, liters per second, or cubic meters per hour.

Doing the math
The velocity of propagation of sound in any medium under stated conditions of temperature and pressure is generally considered to be a constant, C0. For a particular path length, L, the time of flight is given by:

In water, the velocity of propagation of sound is about 1,497m/s at room temperature. If the spool body has a path length of about 10cm, the time of flight will be about 67µs.

This formula applies only if the medium is stationary. If the medium is moving in a direction parallel to, and in the same direction as, the pulse path (downstream propagation), the propagation velocity is increased and the propagation time is reduced:

Similarly, if the medium is moving in a direction parallel to, and in the opposite direction from, the pulse path (upstream propagation), the propagation velocity is decreased and the propagation time is increased:

Now, if we take a measurement in the upstream direction and a second measurement in the downstream direction, and then calculate the difference, we obtain:

We can find a common denominator on the right side of the equation:

But the velocity of sound in the medium is much greater than the velocity of the medium, so we can eliminate the v2 term. We can also cancel the C0 term in the numerator:

Solving this equation for v we obtain:

If C0 is assumed to be about 1,497m/s at room temperature, and if we assume a path length of 10cm, then the fluid velocity would be the time difference multiplied by about 11.2 × 106m/s. Conversely, a velocity of 1m/s would provide a time difference of 89.2ns.

We must now resolve a new difficulty: how to accurately and with sufficient resolution measure a time difference of less than 100ns?

There are a number of techniques to accurately measure time intervals that do not involve running an oscillator at microwave frequencies. For example, the MAX35101 Time-to-Digital Converter measures time differences with an accuracy of better than 20ps and a resolution of about 4ps.

Since the fluid velocity equation strongly depends on the velocity of sound in water, and since the velocity of sound in water is strongly dependent on temperature, it is necessary to measure the temperature and adjust C0 accordingly. That leads us to the second difficulty to be resolved: how to accurately and inexpensively measure the temperature of the flowing water?

Actually, measuring temperature is not that difficult. Many suppliers provide solid-state devices that deliver a temperature reading with reasonably good accuracy. But if the meter is already measuring time differences with a high accuracy, can that same facility be used to measure temperature too?

Measure temperature by measuring time

When considering how to measure temperature in an industrial setting, two technologies stand out: thermocouples and resistive temperature detectors (RTDs). There is a place for each in temperature detection, but a decision on an implementation usually comes down to one criterion. If the application needs to measure very hot (greater than 600°C) temperatures, a thermocouple is a better choice. In virtually every other situation, however, the RTD is a better choice...and it is our choice here.

An RTD is typically a fine coil of platinum wire or a thin film of platinum metal on a ceramic or other inert base. As the temperature increases, the resistance of the conductor increases. If the temperature range is relatively narrow, the change in resistance is a simple quadratic function with temperature.

Above 0°C, the resistance of a platinum RTD is given by:

For typical RTD sensors, A has a value of about 3.9083 × 103/°C and B has a value of about -0.5775 × 106/°C2. At 90°C (typical inlet temperature for a water-sourced radiative heat system), a 1,000Ω RTD will exhibit a resistance of 1,347.07Ω. If the outlet temperature is room temperature, the 1kΩ RTD will exhibit a resistance of 1,097.35Ω, for a difference of 249.72Ω. This resistance range is easily measured.

One method of determining temperature given the resistance of the RTD is to solve the quadratic equation given above for T and plug in RT. The problem with this supposedly "simple" solution is accurately measuring the resistance of the RTD. Remember that the device in the heat meter measures time with a high degree of accuracy; it does not measure resistance.

Fortunately, there is a simple way to convert resistance to time: allow a capacitor to discharge through the resistance and count the amount of time it takes to discharge to a known voltage level. Figure 3 shows how it works:

Figure 3: Temperature-to-time conversion circuit.

In figure 3, Q1 turns on to precharge the capacitor prior to the measurement interval. When the capacitor is completely charged, then Q1 turns off and Q2 turns on to begin discharging the capacitor. The time measurement logic simultaneously begins counting time. When the voltage on the capacitor falls below the comparator's threshold level, the time measurement logic stops counting and reports the count. Then Q2 is turned off and Q1 is turned on to precharge the capacitor again (figure 4). Since resistance is proportional to temperature and a higher resistance corresponds to a longer period to discharge the capacitor, a greater count will correspond to a higher temperature.

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