5.5 Affinity Laws & Parallel/Series Pump/Fan System Curves

5.5 Affinity Laws & Parallel/Series Pump/Fan System Curves

The Affinity Laws (homologous scaling relationships) govern the performance of geometrically similar turbomachinery—including centrifugal pumps, fans, and blowers—under variations in rotational speed, impeller diameter, and fluid density. On the NCEES PE Mechanical exam, affinity law calculations are routinely paired with system resistance curves, Variable Frequency Drive (VFD) energy evaluations, and parallel/series pump combinations.

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1. The Affinity Laws for Constant Impeller Diameter (Variable Speed $N$)

When a pump or fan changes rotational speed ($N$, in $\text{RPM}$) while maintaining a fixed impeller diameter and constant fluid density:

+-----------------------------------------------------------------------------+
| THE ROTATIONAL SPEED AFFINITY LAWS (Constant Diameter D)                   |
+-----------------------------------------------------------------------------+
| 1. Volumetric Flow Rate scales linearly with speed:                         |
|    Q_2 / Q_1 = N_2 / N_1                                                    |
|                                                                             |
| 2. Total Head / Static Pressure scales with speed squared:                  |
|    H_2 / H_1 = (N_2 / N_1)^2       or       P_2 / P_1 = (N_2 / N_1)^2       |
|                                                                             |
| 3. Brake Horsepower (Power) scales with speed CUBED:                        |
|    BHP_2 / BHP_1 = (N_2 / N_1)^3   or       P_2 / P_1 = (N_2 / N_1)^3       |
+-----------------------------------------------------------------------------+

The Cubic Power Relationship & VFD Energy Savings

The cubic power law ($\text{BHP} \propto N^3$) demonstrates the tremendous energy savings of Variable Frequency Drives (VFDs). If building cooling load drops and a secondary chilled water pump speed is reduced by $20%$ ($N_2 / N_1 = 0.80$):

$\text{Flow Rate:} \quad Q_2 = 0.80 \times Q_1 \quad (20\% \text{ reduction})$

$\text{Head:} \quad H_2 = (0.80)^2 \times H_1 = 0.64 \times H_1 \quad (36\% \text{ reduction})$

$\text{Power:} \quad \text{BHP}_2 = (0.80)^3 \times \text{BHP}_1 = 0.512 \times \text{BHP}_1 \quad (48.8\% \text{ energy reduction!})$

VFD vs. Throttling Valve: Reducing flow to $80%$ by closing a discharge triple-duty balancing valve keeps the pump running at full $100%$ speed, riding the pump curve up to higher head and consuming approximately $88% - 92%$ power. Slowing the pump with a VFD drops power consumption to $51.2%$, saving over $40%$ more energy.

2. Impeller Trimming Laws (Constant Speed $N$)

When a pump impeller is physically machined (trimmed) on a lathe to reduce its outer diameter ($D_1 \rightarrow D_2$) at constant rotational speed (valid for moderate trims up to $15% - 20%$ of original diameter):

$\frac{Q_2}{Q_1} = \frac{D_2}{D_1}, \quad \frac{H_2}{H_1} = \left( \frac{D_2}{D_1} \right)^2, \quad \frac{\text{BHP}_2}{\text{BHP}_1} = \left( \frac{D_2}{D_1} \right)^3$

Combined Speed and Diameter Scaling

If both rotational speed and impeller diameter vary simultaneously:

$\frac{Q_2}{Q_1} = \left( \frac{N_2}{N_1} \right) \left( \frac{D_2}{D_1} \right)$

$\frac{H_2}{H_1} = \left( \frac{N_2}{N_1} \right)^2 \left( \frac{D_2}{D_1} \right)^2$

$\frac{\text{BHP}_2}{\text{BHP}_1} = \left( \frac{N_2}{N_1} \right)^3 \left( \frac{D_2}{D_1} \right)^3$

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3. Gas Density Variations in Fans & Air Handling Systems

Unlike incompressible hydronic water circuits, air handling systems operate across varying air densities due to elevation (altitude) and temperature.

Fan Performance at Variable Air Density (Constant RPM & Duct Geometry)

1. Volumetric Airflow ($Q$, in $\text{CFM}$) is INDEPENDENT of density: A fan is a constant-volume machine. At fixed RPM, it displaces the exact same geometric volume of air regardless of density: $Q_2 = Q_1 \quad (\text{CFM}_2 = \text{CFM}_1)$
2. Mass Flow Rate ($\dot{m}$) scales directly with density: $\dot{m}_2 = \dot{m}_1 \left( \frac{\rho_2}{\rho_1} \right)$
3. Static & Total Pressure ($P$, in $\text{in. w.g.}$) scale directly with density: $P_2 = P_1 \left( \frac{\rho_2}{\rho_1} \right)$
4. Fan Brake Horsepower ($\text{BHP}$) scales directly with density: $\text{BHP}_2 = \text{BHP}_1 \left( \frac{\rho_2}{\rho_1} \right)$

+-----------------------------------------------------------------------------+
| AIR DENSITY DERATING SUMMARY (Denver 5,000 ft vs Sea Level)                |
+-----------------------------------------------------------------------------+
| Parameter                     Sea Level (rho = 0.075)   Denver (rho = 0.0624) |
| Volumetric Flow (CFM)         10,000 CFM                10,000 CFM (No Change)|
| Total Pressure (in. w.g.)     2.50 in. w.g.             2.08 in. w.g. (-16.8%)|
| Fan Power (BHP)               5.00 hp                   4.16 hp       (-16.8%)|
| Thermal Sensible Heat (q_s)   1.08 * CFM * Delta_T      0.90 * CFM * Delta_T  |
+-----------------------------------------------------------------------------+

4. Parallel vs. Series Pump Configurations

When multiple pumps are piped together, their combined characteristic curve is constructed by summing individual pump curves according to hydraulic topology.

+-----------------------------------------------------------------------------------------+
| PARALLEL VS. SERIES PUMP COMBINATIONS                                                   |
+-----------------------------------------------------------------------------------------+
| PARALLEL CONFIGURATION (Common Headers):                                                |
|                                                                                         |
|       +---[ Pump A ]---+                                                                |
|   --->|                |---> Combined Flow: Q_total = Q_A + Q_B at identical Head H     |
|       +---[ Pump B ]---+                                                                |
|                                                                                         |
|   - Curve Construction: Add flows horizontally (Q_total = 2 * Q_single at each H)       |
|   - Purpose: Increase volumetric flow rate in low-resistance / flat system curves       |
|                                                                                         |
| SERIES CONFIGURATION (Booster in Line):                                                 |
|                                                                                         |
|   --->[ Pump A ]---->[ Pump B ]---> Combined Head: H_total = H_A + H_B at flow Q        |
|                                                                                         |
|   - Curve Construction: Add heads vertically (H_total = 2 * H_single at each Q)         |
|   - Purpose: Increase total pressure / head in high-resistance / steep system curves    |
+-----------------------------------------------------------------------------------------+

The Parallel Pump Diminishing Returns Phenomenon

A common engineering misconception is that adding an identical second pump in parallel doubles the total system flow rate ($Q_{\text{parallel}} = 2 \times Q_{\text{single}}$). This is false in real piping networks!

  Head (ft)
    ^
    |                                  / System Curve: H = C * Q^2 (Steep Parabola)
    |                                 /
    |                 Combined       /
    |                 Parallel      /  <--- Operating Point 2 (Two Pumps: Q_total)
    |                 Curve        /       Q_total ~ 1.25x to 1.40x Q_single
    |      Single     ========+===/=========
    |      Pump      /        |
    |      Curve    /         |
    |     ========+/==========+==================
    |    /        |           |
    |   /         | Operating Point 1 (Single Pump: Q_single)
    +---+---------+-----------+-----------------------------> Flow Q (GPM)
        0      Q_single    Q_total  (NOT 2 * Q_single!)

Because the system friction curve rises quadratically ($H_{\text{sys}} \propto Q^2$), operating two identical pumps in parallel increases system flow by only $25% \text{ to } 40%$ (typically $Q_{\text{parallel}} \approx 1.25 \text{ to } 1.40 \times Q_{\text{single}}$), while each individual pump delivers less than its single-pump design flow ($Q_{\text{each}} = Q_{\text{total}} / 2$) at a higher head.

5. Worked Engineering Examples

Worked Example 1: Fan Affinity Laws & Speed Reduction

A supply air fan currently operates at $N_1 = 1750\text{ RPM}$, delivering $Q_1 = 12,000\text{ CFM}$ against a total static pressure of $P_1 = 2.50\text{ in. w.g.}$ with a motor shaft power of $\text{BHP}_1 = 7.50\text{ hp}$. A building balance requires reducing the airflow to $Q_2 = 9,600\text{ CFM}$. Assuming constant duct configuration and standard air density:

1. What is the required new fan speed ($N_2$)?
2. What is the new fan static pressure ($P_2$)?
3. What is the new Brake Horsepower ($\text{BHP}_2$)?

Step 1: Determine speed ratio from flow affinity law: $\frac{N_2}{N_1} = \frac{Q_2}{Q_1} = \frac{9,600\text{ CFM}}{12,000\text{ CFM}} = 0.80$

$N_2 = 0.80 \times 1750\text{ RPM} = 1400\text{ RPM}$

Step 2: Calculate new static pressure using the squared law: $P_2 = P_1 \left( \frac{N_2}{N_1} \right)^2 = 2.50\text{ in. w.g.} \times (0.80)^2 = 2.50 \times 0.64 = 1.60\text{ in. w.g.}$

Step 3: Calculate new Brake Horsepower using the cubic law: $\text{BHP}_2 = \text{BHP}_1 \left( \frac{N_2}{N_1} \right)^3 = 7.50\text{ hp} \times (0.80)^3 = 7.50 \times 0.512 = 3.84\text{ hp}$

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Worked Example 2: Mathematical Parallel Pump Operating Point

A closed chilled water loop has a system curve defined by $H_{\text{sys}} = 0.00024 Q^2$ (where $H$ is in feet and $Q$ is in $\text{GPM}$). A single pump has a characteristic curve $H_p = 80 - 0.00008 Q_p^2$.

1. Find the operating point $(Q_1, H_1)$ with one pump running.
2. Find the total system operating point $(Q_{\text{total}}, H_{\text{total}})$ when two identical pumps operate in parallel.

Step 1: Single Pump Operating Point: $H_p = H_{\text{sys}} \implies 80 - 0.00008 Q_1^2 = 0.00024 Q_1^2$

$80 = 0.00032 Q_1^2 \implies Q_1^2 = \frac{80}{0.00032} = 250,000$

$Q_1 = \sqrt{250,000} = 500\text{ GPM}$

$H_1 = 0.00024 \times (500)^2 = 60.0\text{ ft}$

Step 2: Two Pumps in Parallel: In parallel, each pump delivers half the total flow: $Q_p = Q_{\text{total}} / 2$.

$H_{p,\text{parallel}} = 80 - 0.00008 \left( \frac{Q_{\text{total}}}{2} \right)^2 = 80 - 0.00002 Q_{\text{total}}^2$

Equating parallel pump head to system head: $80 - 0.00002 Q_{\text{total}}^2 = 0.00024 Q_{\text{total}}^2$

$80 = 0.00026 Q_{\text{total}}^2 \implies Q_{\text{total}}^2 = \frac{80}{0.00026} = 307,692$

$Q_{\text{total}} = \sqrt{307,692} = 554.7\text{ GPM} \approx 555\text{ GPM}$

$H_{\text{total}} = 0.00024 \times (554.7)^2 = 73.8\text{ ft}$

$\text{Flow Increase Ratio} = \frac{554.7\text{ GPM}}{500\text{ GPM}} = 1.109 \quad (\text{Only an } 11\% \text{ increase in flow, NOT } 100\%!)$

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6. NCEES Reference Handbook Navigation & Exam Tips

• Affinity Laws Location: In the Reference Handbook under Fluid Mechanics $\rightarrow$ Turbomachinery $\rightarrow$ Affinity Laws.
• Static Head Affinity Trap: The Affinity Laws ($H \propto N^2$) scale the pump curve itself, but they do NOT represent the operating point trajectory if the system has static head ($H_{\text{static}} \ne 0$). If static head exists, the operating point must be calculated by finding the intersection of the scaled pump curve and the system curve ($H = H_{\text{static}} + C Q^2$).