Episode 04. Flow Cofficients and Characteristics in Valves
Summary
There are many types of valves, but the common factor we need to check is whether the valve can pass the desired amount of fluid. This episode introduces flow and resistance coefficients, which help determine how much flow a valve can handle. Finally, we discuss valve flow characteristics and their different types.
1. Introduction
As mentioned in the previous episode, valves have various functions, such as turning flow on/off or regulating it. However, one thing all valves have is that fluid passes through them. In other words, regardless of its function, the valve must be able to accommodate the required flow rate. To ensure the required amount of fluid flows through, we must select a valve that can handle that flow rate. So, how do we know how much fluid a particular valve can allow through? In this episode, we look at flow coefficients and discuss the valve’s (especially the control valve) flow characteristics using the flow coefficients.
(Figure 4.1)
2. Definition of flow & resistance coefficients
The flow coefficient indicates how much fluid can flow. It depends on the units used, so let’s look at the two commonly used coefficients: \({\small\displaystyle C_v}\) and \({\small\displaystyle K_v}\).
The \({\small\displaystyle C_v}\) represents the flow capacity (gallons per minute) of 60°F water that can flow through an opening with a 1psi pressure drop.
\({\small\displaystyle Q[\text{gal/min}],\,\Delta P[\text{psi}]}\)\( {\small\displaystyle\begin{align} C_v \quad&=\quad Q\cdot\sqrt{\frac{S_g}{\Delta P}} \tag{Eq 4.1} \end{align} } \)
\({\small\displaystyle K_v}\) is the flow coefficient expressed in mixed SI units. It shows the flow capacity (cubic meters per hour) of water between 5°C and 40°C that can flow through an opening with a 1bar pressure drop.
\({\small\displaystyle Q[\text{m}^3/\text{hr}],\,\Delta P[\text{bar}]}\)\( {\small\displaystyle\begin{align} K_v \quad&=\quad Q\cdot\sqrt{\frac{S_g}{\Delta P}} \tag{Eq 4.2} \end{align} } \)
Valves also have a resistance coefficient. We can express the resistance coefficient using the friction factor as follows.
\( {\small\displaystyle\begin{align} \xi \quad=\quad f_v\frac{L}{D} \tag{Eq 4.3} \end{align} } \)
The flow coefficient and resistance coefficient are related to each other. We can determine their relationship using the Darcy-Weisbach equation mentioned in Episode 01.
\( {\small\displaystyle\begin{align} \frac{\Delta P}{\rho g} \quad&=\quad \frac{v^2}{2g}\cdot\frac{fL}{D} \tag{Eq 1.3} \end{align} } \)
Converting units and applying (Eq 1.3), we can derive the following relationships between the flow and resistance coefficients. The units for each item are as follows: \({\small\displaystyle d[\text{mm}],\,D[\text{m}],\,g[\text{m}/\text{s}^2],\,L[\text{m}],\,v[\text{m/s}],\,\rho[\text{kg}/\text{m}^3]}\)
\( {\small\displaystyle\begin{align} C_v \quad&=\quad 0.0462\frac{d^2}{\sqrt{\xi}} \tag{Eq 4.4} \\[15pt] K_v \quad&=\quad 0.040\frac{d^2}{\sqrt{\xi}} \tag{Eq 4.5} \\[15pt] K_v \quad&=\quad 0.865C_v \tag{Eq 4.6} \end{align} } \)
(Eq 4.4) to (Eq 4.6) represent the relationships between the coefficients, and we can find the process of deriving these equations in (Table 4.1).
| \( {\displaystyle \Delta P } \) | \( {\displaystyle \frac{\rho v^2\xi}{2} } \) | \( {\displaystyle \text{Pa} } \) | \( {\displaystyle 1.45\times 10^{-4}\cdot\frac{\rho v^2\xi}{2} } \) | \( {\displaystyle \text{psi} } \) | \( {\displaystyle 10^{-5}\cdot\frac{\rho v^2\xi}{2} } \) | \( {\displaystyle \text{bar} } \) |
| \( {\displaystyle Q } \) | \( {\displaystyle \frac{\pi D^2v}{4} } \) | \( {\displaystyle \text{m}^3/\text{s} } \) | \( {\displaystyle 15850.3\cdot\frac{\pi D^2v}{4} } \) | \( {\displaystyle \text{gal/min} } \) | \( {\displaystyle 3600\cdot\frac{\pi D^2v}{4} } \) | \( {\displaystyle \text{m}^3/\text{hr} } \) |
| \( {\displaystyle C_v } \) | \( {\displaystyle \begin{align} Q\times\sqrt{\frac{S_g}{\Delta P}} \,&=\, 12448.8 D^2v\times\sqrt{\frac{2(\rho/\rho_w)}{1.45\times 10^{-4}\rho v^2\xi}} \,=\, 12448.8D^2v\times\frac{1}{v}\sqrt{\frac{13.80}{\xi}},\quad \rho_w\,=\,1000\,\text{kg}/\text{m}^3 \\[10pt] \,&=\,46233.8\times\frac{D^2[\text{m}^2]}{\sqrt{\xi}} \,=\, 46233.8\times\frac{10^{-6}d^2[\text{mm}^2]}{\sqrt{\xi}} \,=\, 0.0462\times\frac{d^2}{\sqrt{\xi}} \end{align} } \) | |||||
| \( {\displaystyle K_v } \) | \( {\displaystyle\begin{align} Q\times\sqrt{\frac{S_g}{\Delta P}} \,&=\, 2827.4D^2v\times\sqrt{\frac{2(\rho/\rho_w)}{10^{-5}\rho v^2\xi}} \,=\, 2827.4D^2v\times\frac{1}{v}\sqrt{\frac{200}{\xi}},\quad \rho_w\,=\,1000\,\text{kg}/\text{m}^3 \\[10pt] \,&=\, 39986\times\frac{D^2[\text{m}]}{\sqrt{\xi}} \,=\, 39986\times\frac{10^{-6}d^2[\text{mm}]}{\sqrt{\xi}} \,=\, 0.040\times\frac{d^2}{\sqrt{\xi}} \end{align} } \) | |||||
3. Flow characteristics and rangeability
In control valves, the opening position can vary during operation. So, when selecting a control valve, we need to consider how the flow rate changes with the valve’s opening position. The relationship between the valve opening and the flow coefficient is known as the flow characteristic. Flow characteristics are different when a valve is in an inherent state (inherent flow characteristics) and when it is installed in a system (installed flow characteristics). Let’s discuss these in more detail.
Rotary valves, such as ball and butterfly valves, are difficult to modify closure elements. This makes it difficult to change the inherent flow characteristics. As a result, they require specially designed closure elements. However, in globe valves, we can adjust the flow characteristics based on the type of trim.
Inherent flow characteristics
(Figure 4.2)
We can divide the flow characteristics into three main types. Let’s take a look at (Figure 4.2).
- Quick opening flow characteristic
The quick opening flow characteristic is that the flow passing through from the initial opening is large. Due to this characteristic, it is often used when on/off functionality is required.
- Linear flow characteristic
The linear flow characteristic is that the flow rate changes uniformly with the valve opening. Generally, it provides a greater flow rate than the equal percentage type across the entire valve opening range.
- Equal percentage flow characteristic
The equal percentage flow characteristic is that the flow rate increases proportionally with the valve opening. It offers higher rangeability compared to other types. We will discuss this in more detail below.
Installed flow characteristics
Flow characteristics can vary within a system due to pressure losses in the piping. Let’s explore how a valve’s inherent flow characteristics change when the valve valve is installed within a system. Consider a pumping system where a control valve regulates the flow rate. The downstream pressure of the valve is maintained at 2barg. The pump’s performance curve and the pressure drop within the piping are depicted in (Figure4.3).
Based on (Figure 4.3) and (Eq 4.2), we can summarize the pressure information and flow coefficient for each flow rate. Where the fluid is water between 5°C and 40°C, and we ignore the effects of the fitting before and after the valve.
(Figure 4.3)
| \( {\displaystyle Q\,[\text{m}^3/\text{hr}] } \) | \( {\displaystyle P_{\text{discharge}}\,[\text{barg}] } \) | \( {\displaystyle \Delta P_{\text{piping}}\,[\text{bar}] } \) | \( {\displaystyle \Delta P_{\text{valve}}\,[\text{bar}] } \) | \( {\displaystyle \text{required}\,K_v } \) | \( {\displaystyle \begin{gather} \text{percent of} \\ \text{maximum}\,K_v \end{gather} } \) |
|---|---|---|---|---|---|
| \( {\displaystyle 30 } \) | \( {\displaystyle 8.6 } \) | \( {\displaystyle 1 } \) | \( {\displaystyle 5.6 } \) | \( {\displaystyle 12.68 } \) | \( {\displaystyle 8.45 } \) |
| \( {\displaystyle 60 } \) | \( {\displaystyle 8.3 } \) | \( {\displaystyle 1.2 } \) | \( {\displaystyle 5.1 } \) | \( {\displaystyle 26.57 } \) | \( {\displaystyle 17.71 } \) |
| \( {\displaystyle 90 } \) | \( {\displaystyle 8 } \) | \( {\displaystyle 1.5 } \) | \( {\displaystyle 4.5 } \) | \( {\displaystyle 42.43 } \) | \( {\displaystyle 28.29 } \) |
| \( {\displaystyle 120 } \) | \( {\displaystyle 7.2 } \) | \( {\displaystyle 2.1 } \) | \( {\displaystyle 3.1 } \) | \( {\displaystyle 68.16 } \) | \( {\displaystyle 45.44 } \) |
| \( {\displaystyle 150 } \) | \( {\displaystyle 6 } \) | \( {\displaystyle 3 } \) | \( {\displaystyle 1 } \) | \( {\displaystyle 150 } \) | \( {\displaystyle 100 } \) |
If each valve’s inherent flow characteristics follow (Figure 4.2), we can plot the installed flow characteristics using (Table 4.2). (Figure 4.4) shows the linear flow characteristic, while (Figure 4.5) shows the equal percentage flow characteristic.
(Figure 4.4)
(Figure 4.5)
We now know that flow characteristics change within a system. Therefore, we need to consider the system and determine which flow characteristic is the most suitable.
Rangeability
Along with flow characteristics, another factor to consider is “Rangeability.” Adjusting the valve opening from 0% to 100% doesn’t necessarily mean it can control the flow across the entire range from zero to full flow. Rangeability refers to the ratio of maximum to minimum flow rate that a valve can control. High rangeability means the valve can control a wide range of flow rates. So, it allows us to accommodate a variety of conditions. Usually, when a valve has high rangeability, its sensitivity is lower when closing. However, in the opposite case, the sensitivity increases.
Nomenclature
\({\small\displaystyle C_v}\) : flow coefficient
\({\small\displaystyle D}\) : diameter
\({\small\displaystyle f_v}\) : friction factor of valves
\({\small\displaystyle K_v}\) : flow coefficient in mixed SI
\({\small\displaystyle L}\) : length
\({\small\displaystyle \Delta P}\) : pressure drop
\({\small\displaystyle Q}\) : volumetric flow rate
\({\small\displaystyle S_g}\) : specific gravity
\({\small\displaystyle v}\) : velocity
\({\small\displaystyle \rho}\) : density
\({\small\displaystyle \rho_w}\) : density of water
\({\small\displaystyle \xi}\) : resistance coefficient