For two control valves installed in series, the equivalent flow coefficient ( $K_v$ or $C_v$ ) is calculated using the reciprocal of the sum of squares:

\frac{1}{K_{vt}^2} = \frac{1}{K_{v1}^2} + \frac{1}{K_{v2}^2}

Where:

$K_{vt}$ = resulting equivalent $K_v$
$K_{v1}$ = $K_v$ of valve 1
$K_{v2}$ = $K_v$ of valve 2

This arrangement is typically used when a high pressure drop is required or for better rangeability.

Valves in Parallel

For two control valves installed in parallel, the equivalent flow coefficient is the simple sum of their individual coefficients:

K_{vt} = K_{v1} + K_{v2}

This configuration is used to increase the total flow capacity or to provide redundancy in a system.

References

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Mathematical Relationships for Equivalent Kv/Cv

For control valves in series, the equivalent flow coefficient is derived from the inverse of the sum of inverse squares:

\frac{1}{(K_{vt})^2} = \frac{1}{(K_{v1})^2} + \frac{1}{(K_{v2})^2}

Where:

$K_{vt}$ is the resulting total Kv.
$K_{v1}$ and $K_{v2}$ are the Kv values for valve 1 and valve 2, respectively.

For control valves in parallel, the equivalent flow coefficient is a simple sum of the individual coefficients:

K_{vt} = K_{v1} + K_{v2}

Practical Interpretation:

Series: The total restriction to flow increases. The equivalent Kv is always less than the smallest individual Kv in the series path.
Parallel: The total capacity increases. The equivalent Kv is the sum of the individual capacities.

Additional Properties and Practical Considerations

When applying the formulas for equivalent flow coefficients (Kv or Cv), it is useful to understand their underlying assumptions and practical implications for valve sizing and system design.

Series Configuration Properties

Effect on Equivalent Kv: The equivalent Kvt for valves in series is always less than the smallest individual valve's Kv. This is because the series arrangement creates an additive effect on pressure drop, which mathematically results in a smaller combined flow coefficient.
Primary Use Case: Installing control valves in series is a common strategy to manage high differential pressures across a single valve. Splitting the pressure drop can reduce cavitation, flashing, and erosion, improving valve lifespan and control stability.
Assumption: The formula assumes the fluid is incompressible (liquid) and that there are no significant fluid dynamic interactions (like non-uniform flow distribution) between the two valves.

Parallel Configuration Properties

Effect on Equivalent Kv: The equivalent Kvt for valves in parallel is the simple sum of the individual Kv values. This reflects the increased total flow capacity.
Primary Use Case: Parallel installation is used to increase total flow capacity beyond what a single valve can provide, often for operational flexibility or as a backup.
Important Consideration: For accurate control, the valves should have characterized flow curves and precise positioners to ensure stable, proportional splitting of the total flow. Without careful selection and control, one valve may handle a disproportionate share of the flow.

Equivalent Flow Coefficient Calculations

For two control valves installed in series, the equivalent flow coefficient $Kvt$ is calculated using the reciprocal of the sum of squared individual coefficients:

$\frac{1}{(Kvt)^2} = \frac{1}{(Kv_1)^2} + \frac{1}{(Kv_2)^2}$

For two control valves installed in parallel, the equivalent flow coefficient is simply the sum of their individual coefficients:

$Kvt = Kv_1 + Kv_2$

These formulas apply equally to $Kv$ and $Cv$ flow coefficients. The series configuration formula reflects that the total pressure drop increases, reducing the overall flow capacity. The parallel configuration formula indicates that total flow capacity increases additively.

Equivalent Kv for Valve Configurations

The equivalent flow coefficient (Kv or Cv) for control valves in series or parallel configurations is determined by these relationships:

For two valves in series: $\frac{1}{(K_{v,t})^2} = \frac{1}{(K_{v,1})^2} + \frac{1}{(K_{v,2})^2}$

For two valves in parallel: $K_{v,t} = K_{v,1} + K_{v,2}$

Where:

$K_{v,t}$ is the resulting total flow coefficient.
$K_{v,1}$ and $K_{v,2}$ are the flow coefficients of the individual valves.

Quick Reference: Series and Parallel Valve Equations

For control valves in series, the equivalent flow coefficient $Kvt$ is calculated by:

\frac{1}{(Kvt)^2} = \frac{1}{(Kv1)^2} + \frac{1}{(Kv2)^2}

For control valves in parallel, the equivalent flow coefficient $Kvt$ is the sum of the individual coefficients:

Kvt = Kv1 + Kv2

Where:

$Kvt$ = total or equivalent flow coefficient
$Kv1$ , $Kv2$ = flow coefficients of the individual valves

Source: engineeringtoolbox.com