HVAC Controls & Instrumentation

5% of exam

Valve and damper authority and Cv, sensors and actuators, economizer and reset strategies, and PID feedback response.

4 concepts

Exam areas

D. Control Concepts

Control-Valve Sizing: Cv, Authority and Characteristic

Size a modulating valve from its flow coefficient, protect controllability with valve authority, and match an equal-percentage trim to a nonlinear coil for linear control.

A modulating valve is the muscle of every hydronic loop: the controller decides how much heating or cooling a coil should deliver, and the valve translates that decision into a flow rate. Get the valve too big and the first few percent of stroke dumps nearly full flow — the loop becomes an on/off switch that hunts and overshoots. Get the characteristic wrong and a coil that is already nonlinear fights the controller across the whole range. The PE Mechanical exam tests three linked ideas that decide whether a valve actually controls: the flow coefficient

C_v

that sizes it, the authority

\beta

that preserves its shape once installed, and the inherent characteristic chosen to linearize the coil. All three live in the handbook's §11.3 Control Valves (PE Mechanical Reference Handbook — §11 Temperature Controls).

The flow coefficient Cv

The valve flow coefficient is defined as the flow of

60^\circ\text{F}

water, in gpm, that a fully open valve passes at a

1\,\text{psi}

pressure drop. It bundles the valve's geometry into one number you can size against. For any other water flow and drop the relation is a square-root law — flow varies with the square root of the pressure drop — and for fluids other than water you correct with specific gravity

S_g

. Sizing a control valve means picking a

C_v

such that the valve takes a deliberate, healthy pressure drop at design flow; a valve sized to take almost no drop has nothing to control with.

C_v = Q\sqrt{\dfrac{S_g}{\Delta P}} \qquad\Rightarrow\qquad Q = C_v\sqrt{\dfrac{\Delta P}{S_g}}

Inherent characteristic: linear, equal-percentage, quick-opening

The inherent characteristic is the flow-versus-stroke curve measured at constant pressure drop. A linear valve gives flow directly proportional to stem position. A quick-opening valve dumps most of its flow in the first part of the stroke — fine for two-position service, useless for modulation. An equal-percentage valve raises flow by the same percentage of the current flow for each equal increment of stroke, so it is gentle near closed and aggressive near open. That exponential shape is set by the rangeability

R

(the ratio of maximum to minimum controllable flow), and it is the workhorse trim for modulating hydronic coils.

\frac{q}{q_{\max}} = R^{\,(x-1)} \qquad (\text{equal percentage, } x = \text{fractional stroke})

Valve authority: why the installed curve is not the catalog curve

The catalog characteristic assumes constant pressure drop across the valve. In a real circuit, the coil, balancing valve, fittings, and pipe share the pump head with the valve, and as the valve closes its share of the drop rises while the rest of the circuit's drop falls. Authority

\beta

is the ratio of the valve's pressure drop at full flow to the total variable pressure drop of the controlled branch at full flow — the open valve plus everything in series with it. Low authority means the valve takes only a sliver of the branch drop when open, so closing it barely changes flow at first and then collapses flow suddenly: the installed curve bends toward quick-opening and controllability is lost. Designers target

\beta \ge 0.5

(and rarely below

0.25

) precisely to keep the installed curve close to the inherent one.

\beta = \frac{\Delta P_{\text{valve, open}}}{\Delta P_{\text{branch, open}}} = \frac{\Delta P_{\text{valve}}}{\Delta P_{\text{valve}} + \Delta P_{\text{rest}}}

The installed characteristic equation

You can quantify the distortion. If

f = q/q_{\max}

is the inherent flow fraction at a given stroke and

\beta

is the authority, the installed flow fraction

\varphi

follows the relation below. At full stroke

f = 1

and

\varphi = 1

regardless of authority; everywhere else, lower authority pushes

\varphi

above

f

, fattening the low-stroke end of the curve. This is the math behind the rule of thumb: at

\beta \approx 0.5

an equal-percentage valve stays close to equal-percentage once installed (with

R = 50

, half stroke gives

\varphi \approx 0.20

against

f \approx 0.14

), which is what a nonlinear coil needs; as

\beta

falls toward

0.1

the installed curve drifts to linear and then quick-opening.

\varphi = \frac{f}{\sqrt{\beta + (1-\beta)\,f^{2}}}

Matching the valve to the coil

A hot-water or chilled-water coil is itself strongly nonlinear: the first 20–30% of flow delivers more than half the capacity because heat transfer is limited by the air side, so capacity rises steeply at low flow and flattens at high flow. Put a linear valve on such a coil and a small early stroke produces a huge capacity jump — poor control. Put an equal-percentage valve on it and the valve's gentle low-stroke behavior cancels the coil's steep low-flow behavior, yielding capacity nearly proportional to control signal. That cancellation only survives if authority is high enough to keep the installed characteristic equal-percentage rather than quick-opening — which is why sizing

C_v

for a real pressure drop and checking

\beta

are the same design act.

\text{control signal} \;\xrightarrow{\;\text{equal \% valve}\;}\; \text{flow} \;\xrightarrow{\;\text{nonlinear coil}\;}\; \text{capacity} \;\approx\; \text{linear}

Two-way vs three-way and rangeability

Two-way valves throttle flow and create variable-flow systems; pairing them with a variable-speed pump and a differential-pressure setpoint is standard modern practice. Three-way valves (mixing or diverting) keep branch flow roughly constant by bypassing — constant-flow systems, older or specialty applications. Rangeability sets how far down you can turn before control collapses: a valve with

R = 50

controls down to

2\%

of rated flow, but only over the portion of stroke where authority keeps the curve honest. Oversizing throws away rangeability, because design flow then occurs at a small stroke where the valve resolution is coarse and authority is already poor.

Exam strategy

Lead with the square-root law:

C_v = Q\sqrt{S_g/\Delta P}

, and remember it is defined for water at a

1\,\text{psi}

drop, so for water

S_g = 1

and

C_v = Q/\sqrt{\Delta P}

. When asked to size, solve for the

C_v

that gives a design valve drop in the

3

–

10\,\text{psi}

band, then back-check the actual drop at the catalog

C_v

you would pick. For authority questions, write

\beta

as valve-open drop over total branch-open drop and compare to

0.5

; if it is low, the cure is a smaller

C_v

(more valve drop), not a different characteristic. If a problem mentions linear heat transfer or a nonlinear coil, the intended answer is almost always equal-percentage trim. Keep

\Delta P

in psi and

Q

in gpm to stay consistent with the handbook definition.

Key equations

Flow coefficient (general fluid)

C_v = Q\sqrt{\dfrac{S_g}{\Delta P}}

$Q$ in gpm, $\Delta P$ across the valve in psi, $S_g$ = specific gravity (1.0 for water). Defines the valve size.

Flow from Cv

Q = C_v\sqrt{\dfrac{\Delta P}{S_g}}

Square-root law: flow through a fixed opening scales with $\sqrt{\Delta P}$ . Use to back-check actual drop at a chosen $C_v$ .

Cv for water

C_v = \dfrac{Q}{\sqrt{\Delta P}}

$S_g = 1$ simplification. $C_v$ equals the gpm of $60^\circ\text{F}$ water at $1\,\text{psi}$ drop.

Equal-percentage characteristic

\dfrac{q}{q_{\max}} = R^{\,(x-1)}

$x$ = fractional stroke (0–1), $R$ = rangeability (max/min controllable flow). Flow rises by a constant percentage per equal stroke step.

Rangeability

R = \dfrac{q_{\max}}{q_{\min}}

Ratio of maximum to minimum controllable flow at a stated characteristic; larger is better for turndown.

Valve authority

\beta = \dfrac{\Delta P_{\text{valve, open}}}{\Delta P_{\text{branch, open}}}

Open-valve drop divided by total variable branch drop (valve + coil + balancing + fittings) at full flow. Target $\beta \ge 0.5$ .

Installed characteristic

\varphi = \dfrac{f}{\sqrt{\beta + (1-\beta)\,f^{2}}}

Maps inherent flow fraction $f$ to installed flow fraction $\varphi$ given authority $\beta$ . Lower $\beta$ distorts toward quick-opening.

Valve gain

\text{gain} = \dfrac{\Delta q}{\Delta x}

Slope of the installed flow-vs-stroke curve. Good control wants this roughly constant across the operating range.

Worked examples

Sizing a chilled-water coil valve

Problem. A two-way modulating valve must pass $60\,\text{gpm}$ of chilled water to a coil. The designer allots a $6\,\text{psi}$ drop across the wide-open valve. (a) Find the required $C_v$ . (b) If the nearest catalog valve is $C_v = 25$ , what drop will the valve actually take at $60\,\text{gpm}$ ?

Solution. Water, so

S_g = 1

. (a)

C_v = Q/\sqrt{\Delta P} = 60/\sqrt{6} = 60/2.449 = 24.5

. (b) Rearranging the square-root law,

\Delta P = (Q/C_v)^2 = (60/25)^2 = (2.40)^2 = 5.76\,\text{psi}

. The

C_v = 25

valve takes

5.76\,\text{psi}

at design flow — close to the

6\,\text{psi}

allotment, so it is a good selection. Sanity check: a larger

C_v

would take less drop and weaken authority, so rounding up only slightly is correct.

C_v = \frac{60}{\sqrt{6}} = 24.5,\qquad \Delta P_{25} = \left(\frac{60}{25}\right)^2 = 5.76\,\text{psi}

Checking valve authority

Problem. A heating coil branch at design flow has the following pressure drops: coil $4.5\,\text{psi}$ , balancing valve $1.5\,\text{psi}$ , fittings and pipe $1.0\,\text{psi}$ . The wide-open control valve is sized for $7.0\,\text{psi}$ . (a) Compute the valve authority. (b) Is controllability acceptable? (c) If instead the valve were sized for only $2.0\,\text{psi}$ , what authority results?

Solution. Series 'rest of branch' drop:

\Delta P_{\text{rest}} = 4.5 + 1.5 + 1.0 = 7.0\,\text{psi}

. (a)

\beta = \dfrac{\Delta P_{\text{valve}}}{\Delta P_{\text{valve}} + \Delta P_{\text{rest}}} = \dfrac{7.0}{7.0 + 7.0} = 0.500

. (b)

\beta = 0.50

meets the usual

\beta \ge 0.5

target, so the installed characteristic stays close to the inherent equal-percentage shape — acceptable control. (c) With a

2.0\,\text{psi}

valve:

\beta = 2.0/(2.0 + 7.0) = 0.222

. That is below

0.25

, so the installed curve would distort toward quick-opening and control would degrade. Sanity check: bigger valve drop (smaller

C_v

) raises authority — consistent with the trade-off that good control costs pump head.

\beta = \frac{7.0}{7.0+7.0} = 0.500

Installed flow at mid-stroke

Problem. An equal-percentage valve with rangeability $R = 50$ is installed at authority $\beta = 0.50$ . (a) What inherent flow fraction occurs at 50% stroke? (b) What installed flow fraction does that produce? (c) Compare to a poorly sized installation at $\beta = 0.25$ .

Solution. (a) Inherent:

f = R^{(x-1)} = 50^{(0.5-1)} = 50^{-0.5} = 1/\sqrt{50} = 0.141

, i.e.

14.1\%

flow at half stroke — the gentle low-end of equal percentage. (b) Installed at

\beta = 0.50

\varphi = \dfrac{f}{\sqrt{\beta + (1-\beta)f^2}} = \dfrac{0.141}{\sqrt{0.50 + 0.50(0.141)^2}} = \dfrac{0.141}{\sqrt{0.510}} = \dfrac{0.141}{0.714} = 0.198

, i.e.

19.8\%

. (c) At

\beta = 0.25

\varphi = \dfrac{0.141}{\sqrt{0.25 + 0.75(0.141)^2}} = \dfrac{0.141}{\sqrt{0.265}} = \dfrac{0.1414}{0.5148} = 0.275

, i.e.

27.5\%

. Lower authority lifts the half-stroke flow from

14\%

toward

28\%

— the curve is bending toward quick-opening. Sanity check: at full stroke

f = 1

gives

\varphi = 1

for any

\beta

, as required.

\varphi = \frac{0.141}{\sqrt{0.50 + 0.50\,(0.141)^2}} = 0.198

Common pitfalls

•Oversizing the valve so it takes almost no design drop: the first few percent of stroke then passes most of the flow, the loop short-cycles, and rangeability is wasted. Size for a deliberate $3$ – $10\,\text{psi}$ valve drop.
•Confusing inherent and installed characteristics. The catalog curve is at constant $\Delta P$ ; in a circuit the curve distorts by authority. Always check $\beta$ before trusting the published shape.
•Computing authority against the whole pump head instead of the controlled branch's variable drop. Use the open-valve drop over the series valve + coil + balancing + fitting drop of that branch.
•Putting a linear valve on a nonlinear coil and expecting linear capacity. The coil's steep low-flow capacity needs an equal-percentage valve to cancel it.
•Forgetting the $\sqrt{\Delta P}$ law. Flow does not scale linearly with pressure drop; halving flow needs a quarter of the drop at fixed opening.
•Dropping the specific-gravity correction for glycol or other fluids. Use $C_v = Q\sqrt{S_g/\Delta P}$ ; a 30% glycol solution has $S_g \approx 1.03$ – $1.04$ and meaningfully more viscosity effects at high concentration.
•Using a butterfly or standard ball valve for modulation. The handbook (§11.3.3) says butterfly and standard ball valves should be used for two-position (not modulating) service; use a characterized ball valve or a globe valve to modulate.

References

NCEES PE Mechanical Reference Handbook — §11 Temperature Controls (§11.3 Control Valves)
ASHRAE Handbook — Fundamentals, Ch. 7 Fundamentals of Control
ASHRAE Handbook — HVAC Systems and Equipment, Valve characteristics and selection

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