e available for a particular valve, provided by the manufacturer in tabulated format.
For example, the graph of q (with Δp = 4 psi) as a function of x and the graph of C as a function of x, are shown in Below Figures (a) and (b) respectively, for the flow characteristics of our hypothetical valve, depicted in above Figure.
The different values of C obtained as the valve travels through its full range (or stroke as it is also called) from closed to open, is generally expressed in the following format, by valve manufacturers:
C = CV×f(x) …Equation 1
where f(x) is called the inherent valve characteristic of a particular valve. For example, from the graph of C, in Above Figure (b), we could express C as:
C = 40x
and we conclude therefore from Equation 1, that CV = 40 and f(x) = x.
The function f(x) varies from 0 (valve closed with x equal to 0) to 1 (valve fully open with x equal to 1). The inherent valve characteristic f(x), is intimately linked to the flow rate q through a valve, as we can see if we replace C with CVf(x) (as per Equation 1), to rewrite the valve Equation in the form:
It is indeed clear from above Equation that if we keep Δp constant (for a given liquid, G is constant as well), q and f(x) will have exactly the same shape.
A valve for which f(x) = x, is called a linear valve because the flow rate will change linearly with valve opening x. Depending however on the valve design, f(x) may also reflect quick opening or equal percentage (slow opening) characteristics, as shown in Below Figure.
The flow behaviour of a valve is dictated by the manner in which the flow areas A1 and A2 changes with valve position and therefore by the style and design of the valve trimming and in particular the design of the valve seat and closure member (plug).
The quick opening flow characteristic provides for maximum change in flow rate at low valve travels with a nearly linear relationship. Additional increases in valve travel gives sharply reduced changes in flow rate.
The linear flow characteristic curve allows the flow rate to be directly proportional to the valve travel (Δq/Δx equals a constant) or in terms of the inherent valve characteristic, f(x) = x.
An equal percentage valve starts initially with a slow increase in flow rate with valve position which dramatically increases as the valve opens more.
The term equal percentage for a slow opening characteristic curve may at first be confused with the description of a linear characteristic curve.
However, for an equal percentage valve, Δq/Δx at any stage, is proportional to the flow rate q at that moment. This is in contrast with a linear characteristic for which Δq/Δx is constant.
That Δq/Δx is proportional to q, may be rephrased as Δq/q is proportional to Δx. This means the percentage change Δq with respect to the current flow rate q (that is (Δq/q)×100), is equal at every valve travel position x for the same change in valve travel Δx, hence the term ‘equal percentage’.
The inherent valve characteristic for an equal percentage valve is exponential in nature and is normally given by valve manufacturers in the form
where R is a constant for the valve.
The exponential behaviour of an equal percentage valve is explored further in Below Example
An equal percentage valve delivers 8 gpm of water when the valve is 50% open (x=0.5). When the valve is 60% open (x=0.6), the flow rate increases to 16 gpm. Estimate the flow rate through the valve when it is 70% open (x=0.7).
Assume that the pressure drop across the valve remains constant.
When the valve opens from 50% to 60%, the flow rate changes from 8 gpm to 16 gpm. This means that Δq is 8 gpm. when Δx is 0.1 and q = 8.
Therefore Δq/q = 8/8 = 1 and the percentage change is 100% (Δq/q×100) at x = 0,5.
For an equal percentage valve, the percentage change in flow rate when the valve opens from 50% to 60%, (Δx = 0.1) must be equal to the percentage change in flow rate when the valve opens from 60% to 70% (the same Δx of 0.1).
Therefore Δq/q at x = 0.6 must also be 1 (or 100%) for Δx = 0,1. But q is 16 gpm. when the valve is 60% open and we