Research on the theoretical selection methods for the flow characteristic of control valves

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Research on the theoretical methods for selecting the flow characteristic of control valves 1. During the design phase of process control systems, correctly selecting the flow characteristic of control valves (hereinafter referred to as selection) is a technical issue that must be addressed; it plays a crucial role in improving the regulation quality of the system. Based on existing literature, the methods for selecting theories are still under investigation; in practice, empirical criteria are mostly used to address this selection issue. However, the empirical guidelines provided in various documents are not all consistent, and some of them do not clearly specify their scope of application. For example, in common heat exchanger outlet temperature control systems, the conclusions regarding the choice between phase-change heat transfer and non-phase-change heat transfer are not always the same. If the selection theory is not understood and empirical guidelines are applied mechanically, it may lead to incorrect selections. Therefore, it is necessary to study the selected theories and methods. Figure 1 Block diagram of the composition of a single-loop feedback control system. As can be seen from Figure 1, once the regulator has been tuned, Kc becomes a constant. The theoretical basis for this choice is Ko = Kv*Kp*Km≈constant (1), where Ko, Kv, and Km are respectively the amplification factors of the system’s generalized object, the control valve, the controlled object, and the sensing transmitter. Analyzing Equation (1) from the perspective of selection: The purpose of using (1) is to compensate for Kp and Km with Kv, and to perform nonlinear correction on Ko. If no disturbances are introduced during system operation, that is, if the operating points of Kp and Km remain unchanged, then no nonlinear correction is required; any type of flow-characteristic control valve will meet the requirements of equation (1). Therefore, in the absence of interference, there is no issue of selection; when studying the theories and methods used for selection, it is necessary to consider interference factors to understand their applicability. (2) The selection process involves determining, based on the variation patterns of Kp and Km under the main disturbances acting on the system, a Kv variation pattern that meets the requirements of equation (1) under such disturbances. Since the variation pattern of Km is determined by the inherent functional relationship between the input and output signals of the detector and transmitter as a whole, it is independent of interference factors. Therefore, the key to solving this selection issue lies in determining and understanding the variation patterns of Kv and Kp under the main disturbing effects on the system. (3) From the perspective of the various existing methods for selection, they all essentially employ qualitative analysis to address the selection issue. Therefore, it is practically useful to understand the patterns of change in Kv for commonly used linear, logarithmic, and quick-opening flow control valves under various disturbances, so as to use this information as a basis for making selections ； Understanding the various common methods for selecting theories, as well as the feasibility and applicability of determining the variation patterns of Kp under interference effects, will provide guidance for designers in choosing the appropriate methods. Based on the above analysis, the issues present in the discussions commonly found in the literature regarding valve selection are mainly as follows: the explanations provided on the flow characteristics of control valves and the value of Kv focus on one particular approach, which is questionable from a selection perspective ； Only the variation patterns of Kv for each valve under ideal operating conditions (i.e., when the pressure drop ΔPv before and after the valve remains constant) are provided for selection; the applicability to disturbances is unclear ； No interference factors were considered to demonstrate the feasibility of determining the variation law of Kv, as well as the suitability of the selected method. Therefore, the selection lacks practical significance and requires revision and supplementation. This paper aims to address the aforementioned issues, and the following section will explore the literature on the flow characteristics of valves and Kv ； Starting from the discussion of the flow characteristics of valves and the four expressions for Kv, and based on the four expressions for Kv of each valve derived using the variables Q/Qmax, Q, C/Cmax, or C, the patterns of change in Kv for linear, logarithmic, and fast-opening flow control valves under the influence of disturbances are determined, along with the applicable ranges for their use ； Demonstrate the feasibility of determining the variation law of Kp under interference, as well as the suitability of the selected method. 2. Explore the literature on the flow characteristics of valves and Kv. 2.1 Analyze the state of relevant variables under interference effects. To facilitate further discussion, it is necessary to first understand the state of these variables under interference effects. The flow equations of the valve at a certain opening degree or fully open are given by equations (2) and (3). Here, g is the acceleration due to gravity, which is a constant value ； ρ—is density; for incompressible fluids, p remains constant ； C—Flow capacity of the valve ； Cmax represents the maximum flow capacity of the pilot-operated control valve; essentially, it is the C value corresponding to the system’s maximum flow rate calculated based on the process parameters. It is the nominal flow capacity Cg of the valve, which has been reasonably rounded and selected by the designers within the valve’s standard product range. Since Cg is a constant value, Cmax is also a constant value. Whether chosen using empirical criteria or theoretical methods, it is usually necessary to identify, among the various disturbances, the one that occurs most frequently and has the greatest impact on the controlled variable y, and this disturbance is then considered as the main disturbance variable of the system; the other minor disturbances are treated as constants. There are various forms of disturbances, which can generally be considered as two types: generalized loads (i.e., disturbances other than Δpv) or Δpv disturbances. Thus, based on equations (2) and (3), it can be seen that: (1) the constant system load is the main disturbance, while secondary disturbances such as Δpv can be regarded as constants ； Qmax is a constant ； Cmax is a constant value ； Q and C are variables. (2) In a constant-value system, Δpv is the main disturbance, while secondary disturbances such as load can be considered constant ； Δpv is the variable ； Cmax is a constant value ； From equation (3), it can be seen that Qmax∝Δpv; in other words, Qmax is a variable ； The state of Q can be analyzed as follows: since the given value X remains constant, and based on the principles of material or energy balance in the process operations as well as the system’s regulation mechanisms, it can be seen that the change in Q caused by the Δpv disturbance is quickly overcome by the system’s adjustments to the valve opening. In the steady state, Q remains at the same value it had before the Δpv disturbance occurred; therefore, Q can still be considered a constant ； C varies with the valve opening, so C is a variable. The above conclusions are cited as given assumptions in the subsequent discussions. 2.2 The discussion format commonly used in the literature: The flow characteristics of a valve are usually defined as a functional relationship between the relative flow rate Q/Qmax of the fluid passing through the valve and the relative stroke L/Lmax of the valve stem. The defining equation is Q/Qmax=f(L/Lmax) (4). The discussion on the flow characteristics of each valve and Kv can be summarized as shown in Table 1. Table 1: Variation patterns of the flow characteristic expression Kv for different valve types. For linear valves, K is a constant, representing the amplification factor of the control valve. For logarithmic valves, Kv∝Q, with values increasing from low to high. For quick-open valves, Kv∝1/Q, with values decreasing from high to low. In the table, K represents the proportionality constant; for linear valves, K=1-1/R; for logarithmic valves, K=InR; and for quick-open valves, K=（1-1/R2）/2 ； R is the ideal adjustable range of the valve, R=Qmax/Qmin=Cmax/Cmin=30. The ideal flow characteristic curves of the various valves as mentioned in the literature are shown in Figure 2. 1: Open quickly ； 2: Straight line ； 3: Logarithmic plot. Figure 2 shows the ideal flow characteristics. By analyzing the definition of valve flow characteristics given in the literature in conjunction with Figure 2, it can be seen that these characteristics are expressed as a functional relationship between two relative dimensionless variables: Q/Qmax and L/Lmax; essentially, this represents the valve’s relative flow characteristics. Therefore, equation (4) should be accurately referred to as the definition equation for the relative flow characteristic of the valve ； Figure 2 should be accurately referred to as the ideal relative flow rate characteristic curves for each valve ； By drawing tangents at the various points on the curve in Figure 2, the changes in their slopes reflect the variations in the relative amplification factor of each valve, expressed as Q/Qmax (denoted as K’VQ) ； The mathematical expression for the tangent is the definition of the valve’s relative amplification factor: K’VQ = d(Q/Qmax) / d(L/Lmax) (5). Therefore, the flow characteristic expression in Table 1 should be accurately referred to as the relative flow characteristic expression of the control valve. As can be seen from Equation (5), this represents the K’VQ expression for each valve (see Table 2); it reflects the variation pattern of K’VQ for each valve, expressed as Q/Qmax, under ideal operating conditions. From Equation (3), it is known that Qmax remains constant under ideal conditions. Therefore, the variation pattern of K’VQ for each valve can be expressed, as shown in Table 1, in the form of the amplification factor K for each valve, expressed using the dimensionless variable Q. Therefore, this is the form of the Kv variation pattern given in Table 1. Therefore, the variation pattern of K given in Table 1 essentially stems from the variation pattern of K'VQ for each valve, under the condition that ideal operating conditions prevail. Therefore, the Kv variation patterns for each valve given in Table 1 are used for selection, but their applicable range is unclear; it is necessary to clarify the scope of applicability to disturbances. (1) The fixed-system load is the main disturbance: Given Δpv, Qmax can be considered a constant. The same conditions that govern the variation patterns of K'VQ or KV for each valve as derived in the above analysis apply here, making it suitable for use. (2) The fixed-value system Δpv is the main interference ； It is known that Δpv and Qmax are variables, which do not meet the conditions under which the variation laws of K'VQ or KV for each valve hold; therefore, Figure 2 is also invalid, and it cannot be used for selection. Therefore, from the perspective of selection, Table 1 should be as shown in Table 2. Table 2: Patterns of K’VQ changes under load disturbances; Patterns of Kv changes under load disturbances. For linear valves, K’VQ = K, meaning it remains constant regardless of changes; Kv also remains constant and is not affected by changes in Q. For logarithmic valves, Kv increases from low to high, with Kv ∝ Q. For quick-opening valves, Kv decreases from high to low