ASHRAE Journal - June 2009 - 48

• Actuator: A 4 to 20 mA signal moves the actuator through a full stroke of 2.5 in. (63.5 mm) This resulting actuator sensitivity is 2.5 in./(20 – 4) mA = 0.156 in./mA (63.5 mm/(20 – 4) mA = 3.97 mm/mA); • Valve body: A linear valve with a 2.5 in. (63.5 mm) stroke controls flow from 0 to 350 gpm (0 to 22 L/s). Sensitivity is equal to 350 gpm/2.5 in = 140 gpm/in. ([22 L/s]/63.5 mm = 8.8 L/(s · mm); • Process: Assume the heat exchanger has a sensitivity of 0.117°F/gpm (1.03 K/(L · s); • Sensor: A 1,000 W resistance temperature detector (RTD) has a resistance of 1,146.84 to 1,359.70 W over a tem‑ perature range of 100°F to 200°F (38°C to 93°C). The sensitivity is 2.13 W/°F; and • Transmitter: Provides a 4 to 20 mA output signal with an input resistance of 1,146.84 to 1,359.70 W. Transmitter sensitivity equals 0.0752 mA/W. Since the sensitivities are multiplicative, we can determine a process gain as follows: Kp = 0.156 in/mA × 140 gpm/in × 0.117°F/gpm × 2.13 W/°F × 0.0752 mA/W = 0.409 mA/mA This calculation indicates that if the signal to the valve actua‑ tor (process input) changes by 1 mA, there will be a change of 0.409 mA at the output of the transmitter (controller input). If the sensitivities of each of the previously listed components remained constant across the design operating range of the compo‑ nent, then the process would exhibit linear characteristics resulting in constant process gain, a highly desirable quality for a control loop. The previous example was overly simplified by implying the sensitivity of the heat exchanger is constant and by applying a linear valve to the heat exchanger. Heat exchangers are not linear, and we would select an equal percentage valve with the opposite characteristics of the heat exchanger. We must only be concerned about linearizing the two components as a subsystem with the result that the output of the heat exchanger is linear relative to the input to the valve. This is the impetus behind proper control valve selection. Now assume the valve is oversized resulting in the heat‑ex‑ changer‑plus‑valve combination exhibiting variable sensitivity. If one were to assume the rest of the process was linear, exhibiting constant sensitivity, then the product of the two sensitivities results in a variable process gain. In this stated scenario, the result is a system where valve stem lift results in coil heat output as shown in Figure 2. If the system operated normally, between 15% and 70% valve stem lift, the variability in process gain due to the resulting nonlinearity can be determined as follows. Determine the slope of the curve in Figure 2 at each value of stem lift (i.e., 15% and 70%) by drawing a line tangent to each of the two points in question. At 15% stem lift, the slope of a tangent line (valve sensitivity at that point) can be calculated as 40% ÷ 25% = 1.6. At 70% stem lift, valve sensitivity is 13% ÷ 25% = 0.52. As before, if we assume the sensitivity of the rest of the process is constant, then the process gain between these two points of operation varies by a factor of 1.6 ÷ 0.52 » 3. Such variability has the potential to lead to difficulties in controlling the process. 48 ASHRAE Journal 4 to 20 mA Xmtr Controller 4 to 20 mA 100°F to 200°F 160°F 240°F 130°F 350 gpm Figure 1: Simple control of a heat exchanger. 100% 13% Heat Exchanger Output 40% 0 15% 0% 25% 70% 25% Percent Valve Open 100% Figure 2: Determining the gain of a valve-coil subsystem at various operating points. Ideally, one would match system components during the design process so as to minimize this variation in process gain. However, installed characteristics of components often vary from design intent. For example, the inherent characteristic of a control valve assumes a constant pressure drop across the valve. Also, the use of reducers will impact the rated valve Cv. This impact is minimal for globe‑type control valves but can be rather substantial for butterfly and other rotary valves. Both of these factors impact the installed rangeability of the valve, which impacts low‑flow controllability. A valve selected for a heat exchanger may look great on paper, but if the actual installed characteristic varies substantially from design intent, poor controllability due to nonlinear response could pose an issue.1,2 This discussion also applies to VAV boxes, air‑handling unit economizers, etc. If the nonlinearity is not severe, one may be able to tune the control loop despite the nonlinearity. If the nonlinearity is severe, one has limited choices to correct the problem: ashrae.org June 2009

ASHRAE Journal - June 2009

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