Chilled Water Systems for Cooling Applications

Chilled water systems represent the primary cooling distribution method for medium to large commercial, institutional, and industrial facilities. Unlike direct expansion refrigeration systems that distribute refrigerant, chilled water systems use water as a secondary heat transfer medium, providing superior flexibility in zoning, capacity modulation, and central plant optimization. This analysis examines the fundamental physics, system configurations, component selection methodologies, and energy optimization strategies governing chilled water system design.

Fundamental System Physics

Heat Transfer Fundamentals

The cooling capacity delivered by a chilled water system derives from sensible heat transfer between the chilled water and the conditioned space air. The fundamental energy balance equation governs system sizing:

$$ $$\dot{Q} = \dot{m} \cdot c_p \cdot \Delta T = \rho \cdot \dot{V} \cdot c_p \cdot \Delta T$$ $$

Where:

$\dot{Q}$ = cooling capacity (kW or tons)
$\dot{m}$ = mass flow rate (kg/s or lb/hr)
$c_p$ = specific heat of water = 4.186 kJ/(kg·K) or 1.0 Btu/(lb·°F)
$\Delta T$ = supply-return temperature difference (°C or °F)
$\rho$ = water density = 1000 kg/m³ or 8.33 lb/gal
$\dot{V}$ = volumetric flow rate (m³/s, L/s, or gpm)

For water systems using standard units:

$$ $$\dot{Q}_{tons} = \frac{\dot{V}_{gpm} \cdot \Delta T_F}{24}$$ $$

$$ $$\dot{Q}_{kW} = \frac{\dot{V}_{L/s} \cdot \Delta T_C}{14.3}$$ $$

Design Delta-T Selection

The temperature difference between supply and return water fundamentally impacts system economics. Larger delta-T values reduce flow rate requirements, enabling smaller pipes, pumps, and reduced pumping energy, but may compromise heat transfer effectiveness at terminal units.

Standard design delta-T values:

10°F (5.6°C): Traditional design, conservative approach
12-14°F (6.7-7.8°C): Modern practice balancing equipment performance and pumping energy
16-20°F (8.9-11.1°C): High delta-T systems requiring careful terminal unit selection

Flow rate calculation for a given delta-T:

$$ $$\dot{V}_{gpm} = \frac{\dot{Q}_{tons} \cdot 24}{\Delta T_F}$$ $$

Example: 500-ton cooling load with 12°F delta-T requires:

$$ $$\dot{V} = \frac{500 \times 24}{12} = 1000 \text{ gpm}$$ $$

Delta-T Syndrome

Delta-T degradation represents a persistent operational challenge where actual system delta-T falls below design values, causing excessive flow rates and pump energy consumption. Primary causes include:

Coil bypass flow: Improperly balanced terminal units allowing water to bypass coils
Oversized pumps: Excessive flow overcoming control valve authority
Low load conditions: Reduced heat transfer at partial load with constant flow
Chiller control: Supply temperature reset reducing delta-T
Piping design: Short-circuiting or unbalanced distribution

The energy impact scales significantly:

$$ $$\text{Power}_{pump} \propto \dot{V}^3$$ $$

A 20% increase in flow rate from delta-T degradation (12°F design to 10°F actual) increases pump energy by:

$$ $$\left(\frac{1.2}{1.0}\right)^3 = 1.728 \text{ (73% increase)}$$ $$

System Configurations

Primary-Secondary (Decoupled) Systems

Primary-secondary pumping decouples chiller flow (primary loop) from distribution flow (secondary loop) using a common pipe or decoupler. This configuration allows chillers to operate at optimal flow rates independent of building load variations.


graph TD
    CH1[Chiller 1] -->|Primary Pump 1| CP[Common Pipe]
    CH2[Chiller 2] -->|Primary Pump 2| CP
    CH3[Chiller 3] -->|Primary Pump 3| CP
    CP -->|Secondary Pump 1| LOAD1[Building Load 1]
    CP -->|Secondary Pump 2| LOAD2[Building Load 2]
    CP -->|Secondary Pump 3| LOAD3[Building Load 3]
    LOAD1 --> CP
    LOAD2 --> CP
    LOAD3 --> CP

Common Pipe Sizing

The common pipe must maintain low pressure drop to prevent primary-secondary interaction. Velocity should not exceed 2-4 ft/s (0.6-1.2 m/s):

$$ $$v = \frac{\dot{V}}{A} = \frac{4\dot{V}}{\pi D^2}$$ $$

Where:

$v$ = velocity (ft/s)
$\dot{V}$ = flow rate (ft³/s, converted from gpm: gpm/448.8)
$D$ = pipe diameter (ft)

Common pipe length should be minimized (typically 10-20 pipe diameters) to reduce mixing and maintain temperature separation.

Flow Relationships

Three operating conditions exist:

Primary flow > Secondary flow: Excess chilled water returns through common pipe to chillers (rare)
Primary flow = Secondary flow: Ideal balanced condition
Primary flow < Secondary flow: Return water bypasses through common pipe, mixing with supply (typical at partial load)

The mixing temperature when secondary exceeds primary:

$$ $$T_{supply,mixed} = \frac{\dot{V}_{pri} \cdot T_{pri} + (\dot{V}_{sec} - \dot{V}_{pri}) \cdot T_{return}}{\dot{V}_{sec}}$$ $$

Variable Primary Flow (VPF)

Variable primary flow eliminates the secondary loop, using variable speed drives on primary pumps to modulate flow based on building demand. This configuration reduces pump count, initial cost, and pumping energy but requires careful chiller minimum flow protection.


graph TD
    VFD1[VFD Primary Pump 1] --> CH1[Chiller 1]
    VFD2[VFD Primary Pump 2] --> CH2[Chiller 2]
    VFD3[VFD Primary Pump 3] --> CH3[Chiller 3]
    CH1 --> DIST[Distribution System]
    CH2 --> DIST
    CH3 --> DIST
    DIST --> BPV[Bypass Valve]
    BPV --> CH1
    BPV --> CH2
    BPV --> CH3

Minimum Flow Protection

Chillers require minimum flow to prevent evaporator freeze-up and ensure adequate heat transfer. Protection strategies include:

Bypass valve: Opens when system flow drops below chiller minimum
Pump minimum speed: VFD minimum frequency (typically 30-40 Hz)
Chiller staging: Shut down chillers to maintain flow through operating units

Bypass valve sizing equation:

$$ $$C_v = \frac{\dot{V}_{gpm}}{\sqrt{\frac{\Delta P_{psi}}{SG}}}$$ $$

Where:

$C_v$ = valve flow coefficient
$\dot{V}_{gpm}$ = design flow rate through bypass
$\Delta P_{psi}$ = differential pressure across valve (psi)
$SG$ = specific gravity of fluid (1.0 for water)

Distributed Pumping

Distributed pumping places pumps at terminal units or zones rather than centrally. This approach optimizes pressure delivery, reduces pipe sizing, and enables independent zone operation but increases equipment count and maintenance complexity.

Applications include:

High-rise buildings with significant elevation changes
Campus systems with distributed buildings
Retrofit installations adapting existing infrastructure

Pump head calculation for distributed systems:

$$ $$H_{total} = H_{friction} + H_{elevation} + H_{equipment} + H_{control}$$ $$

Chiller Types and Performance

Chiller Classification

Chilled water systems employ various chiller technologies, each with distinct performance characteristics:

| Chiller Type | Compressor | Capacity Range | Efficiency (kW/ton) | Part-Load Performance | Applications | |--------------|------------|----------------|---------------------|----------------------|--------------| | Air-Cooled Scroll | Scroll | 20-160 tons | 1.0-1.2 | Moderate | Small commercial, no cooling tower | | Air-Cooled Screw | Screw | 150-500 tons | 0.85-1.05 | Good | Medium commercial, water scarcity | | Water-Cooled Screw | Screw | 150-750 tons | 0.55-0.70 | Very Good | Industrial, process cooling | | Water-Cooled Centrifugal | Centrifugal | 200-10,000+ tons | 0.45-0.60 | Excellent | Large commercial, central plants | | Magnetic Bearing Centrifugal | Centrifugal | 150-2,000 tons | 0.40-0.55 | Excellent | High-efficiency applications | | Absorption | Heat-driven | 100-1,500 tons | 15-20 lb steam/ton | Good | Waste heat recovery, CHP |

Chiller Performance Curves

Chiller efficiency varies with operating conditions, primarily affected by:

Entering condenser water temperature (ECWT) or entering air temperature
Leaving chilled water temperature (LCWT)
Part-load ratio (PLR)

The integrated part-load value (IPLV) metric accounts for part-load operation:

$$ $$\text{IPLV} = 0.01A + 0.42B + 0.45C + 0.12D$$ $$

Where A, B, C, D represent efficiency at 100%, 75%, 50%, and 25% load respectively.

Performance correction for non-standard conditions:

$$ $$\text{kW/ton}_{actual} = \text{kW/ton}_{rated} \times f_{ECWT} \times f_{LCWT} \times f_{PLR}$$ $$

Typical centrifugal chiller part-load curve shows improving efficiency from 100% to 60-70% load:

$$ $$\text{kW/ton} = a + b \cdot PLR + c \cdot PLR^2 + d \cdot PLR^3$$ $$

Representative coefficients for variable-speed centrifugal:

a = 0.15
b = 0.20
c = 0.30
d = 0.35

Chiller Staging and Sequencing

Multiple chiller plants require optimization algorithms to minimize energy consumption. The fundamental approach stages chillers to maximize efficiency:

Equal loading: Distribute load evenly across operating chillers
Incremental efficiency: Stage next chiller when incremental efficiency improves
Manufacturer curves: Use actual performance data for staging decisions

Staging decision criterion:

$$ $$\text{Stage on when: } \frac{\text{kW}_{n+1}}{\text{Tons}_{n+1}} < \frac{\text{kW}_n}{\text{Tons}_n}$$ $$

Where subscript $n$ represents current operating chillers and $n+1$ represents adding one additional chiller.

Pump Selection and Analysis

Pump Head Calculation

Total pump head consists of several components:

$$ $$H_{total} = H_{friction} + H_{static} + H_{equipment} + H_{control}$$ $$

Friction Losses

Pipe friction follows the Darcy-Weisbach equation:

$$ $$\Delta P_f = f \cdot \frac{L}{D} \cdot \frac{\rho v^2}{2}$$ $$

Where:

$f$ = Darcy friction factor (dimensionless)
$L$ = pipe length (ft or m)
$D$ = pipe diameter (ft or m)
$\rho$ = fluid density (lb/ft³ or kg/m³)
$v$ = fluid velocity (ft/s or m/s)

Converting to head in feet of water:

$$ $$H_f = \frac{\Delta P_f}{\rho g} = f \cdot \frac{L}{D} \cdot \frac{v^2}{2g}$$ $$

For turbulent flow in commercial steel pipe, the Colebrook equation determines friction factor:

$$ $$\frac{1}{\sqrt{f}} = -2\log_{10}\left(\frac{\epsilon/D}{3.7} + \frac{2.51}{Re\sqrt{f}}\right)$$ $$

Where:

$\epsilon$ = pipe roughness (ft), typically 0.00015 ft for steel
$Re$ = Reynolds number = $\frac{\rho v D}{\mu}$
$\mu$ = dynamic viscosity

Hazen-Williams equation provides simplified friction calculation:

$$ $$H_f = \frac{4.52 \cdot L \cdot \dot{V}^{1.852}}{C^{1.852} \cdot D^{4.87}}$$ $$

Where:

$H_f$ = friction head loss (ft per 100 ft)
$\dot{V}$ = flow rate (gpm)
$C$ = Hazen-Williams coefficient (120 for steel, 150 for copper)
$D$ = inside diameter (inches)

Equipment Losses

Component pressure drops from manufacturer data:

| Component | Typical Pressure Drop | |-----------|----------------------| | Chiller evaporator | 10-20 ft | | Air handler coil | 8-15 ft | | Fan coil unit | 5-10 ft | | Control valve (wide open) | 3-5 ft | | Strainer | 2-5 ft | | Heat exchanger | 8-15 ft |

Control Valve Authority

Control valve authority ensures adequate control range:

$$ $$\beta = \frac{\Delta P_{valve}}{\Delta P_{valve} + \Delta P_{coil}}$$ $$

Target authority: 0.25-0.50 for acceptable control. Lower authority causes poor control and delta-T degradation.

Design valve pressure drop:

$$ $$\Delta P_{valve} = \frac{\beta}{1-\beta} \cdot \Delta P_{coil}$$ $$

For $\beta$ = 0.33 and coil drop of 10 ft:

$$ $$\Delta P_{valve} = \frac{0.33}{1-0.33} \cdot 10 = 4.9 \text{ ft}$$ $$

Pump Power and Efficiency

Pump power consumption derives from hydraulic principles:

$$ $$P_{hydraulic} = \frac{\rho g \dot{V} H}{1000}$$ $$

Where:

$P_{hydraulic}$ = hydraulic power (kW)
$\rho$ = water density (kg/m³)
$g$ = gravitational acceleration (9.81 m/s²)
$\dot{V}$ = flow rate (m³/s)
$H$ = total head (m)

In imperial units:

$$ $$P_{hp} = \frac{\dot{V}_{gpm} \cdot H_{ft} \cdot SG}{3960}$$ $$

Actual motor power accounts for pump and motor efficiency:

$$ $$P_{motor} = \frac{P_{hydraulic}}{\eta_{pump} \cdot \eta_{motor}}$$ $$

Typical efficiencies:

Pump: 70-85% (larger pumps more efficient)
Motor: 90-95% (premium efficiency motors)
VFD: 95-97%

Variable Speed Pumping

Variable frequency drives modulate pump speed to match system demand, following affinity laws:

$$ $$\frac{\dot{V}_2}{\dot{V}_1} = \frac{N_2}{N_1}$$ $$

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

$$ $$\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^3$$ $$

Where:

$\dot{V}$ = flow rate
$H$ = pump head
$P$ = power
$N$ = rotational speed
Subscripts 1 and 2 represent initial and final conditions

Energy savings from variable speed operation scale dramatically with the cubic relationship. Reducing speed to 75% (75% flow) reduces power to:

$$ $$P_2 = P_1 \times 0.75^3 = 0.42 P_1 \text{ (58% energy reduction)}$$ $$

This relationship drives the economic justification for VFD implementation on chilled water pumps.

Differential Pressure Control

Control Strategies

Differential pressure (DP) sensors provide feedback for VFD pump speed control. Sensor location and setpoint strategy critically impact energy performance:

1. Fixed DP Setpoint at End of System

Traditional approach maintaining constant differential pressure at the critical (most remote) circuit:

$$ $$\Delta P_{setpoint} = \Delta P_{coil} + \Delta P_{valve} + \Delta P_{piping,critical}$$ $$

Advantages:

Guaranteed pressure at all loads
Simple control logic

Disadvantages:

Excessive pressure at partial load when remote loads are inactive
Higher pumping energy

2. Reset DP Setpoint

Reduces differential pressure setpoint when control valves approach full open:

$$ $$\Delta P_{reset} = \Delta P_{max} - \left(\Delta P_{max} - \Delta P_{min}\right) \times \frac{N_{open}}{N_{total}}$$ $$

Where:

$N_{open}$ = number of valves >90% open
$N_{total}$ = total number of control valves

Typically implemented as:

Reduce DP setpoint when all valves <90% open
Increase DP setpoint when any valve >95% open
Rate limit changes to prevent instability

Energy savings: 20-40% pump energy reduction compared to fixed DP.

3. Flow-Based Control

Directly controls pump to deliver required system flow based on load:

$$ $$\dot{V}_{system} = \sum_{i=1}^{n} \dot{V}_{terminal,i}$$ $$

Requires flow meters on terminal circuits or calculated flow from control valve positions.

Sensor Placement

DP sensor location significantly affects control stability and energy performance:

At pump discharge: Measures only pump-generated pressure, simple but provides no load feedback
Two-thirds into distribution: Balances coverage of building zones
Critical circuit: Ensures most remote load receives adequate pressure
Multiple sensors: Use highest signal from multiple locations

Bypass Valves and Decoupling

Bypass Valve Functionality

Bypass valves (also called differential pressure bypass valves) protect chillers from low flow conditions in variable flow systems. The valve opens when system pressure exceeds setpoint, diverting flow from supply to return.

Bypass valve sizing methodology:

Determine maximum bypass flow requirement:

$$ $$\dot{V}_{bypass,max} = \dot{V}_{chiller,min} - \dot{V}_{system,min}$$ $$

Calculate available differential pressure:

$$ $$\Delta P_{available} = \Delta P_{pump,max} - \Delta P_{chiller}$$ $$

Size valve coefficient:

$$ $$C_v = \frac{\dot{V}_{bypass,max}}{\sqrt{\Delta P_{available}}}$$ $$

Example: 1000 gpm chiller with 400 gpm minimum, 200 gpm minimum system flow, 60 psi pump pressure, 15 psi chiller drop:

$$ $$\dot{V}_{bypass} = 400 - 200 = 200 \text{ gpm}$$ $$

$$ $$\Delta P = 60 - 15 = 45 \text{ psi}$$ $$

$$ $$C_v = \frac{200}{\sqrt{45}} = 29.8$$ $$

Select next larger standard valve size.

Decoupler Design

In primary-secondary systems, the decoupler (common pipe) hydraulically separates primary and secondary loops. Proper design requires:

Low pressure drop: <0.5 ft to prevent interaction
Short length: 10-20 diameters to minimize mixing
Bridge connections: Close spacing between supply and return connections

Decoupler pressure drop verification:

$$ $$\Delta P_{decoupler} = f \cdot \frac{L}{D} \cdot \frac{v^2}{2g} < 0.5 \text{ ft}$$ $$

Temperature mixing when secondary exceeds primary flow:

$$ $$T_{mixed} = \frac{\dot{V}_{pri} \cdot T_{LCHWT} + \dot{V}_{bypass} \cdot T_{RCHWT}}{\dot{V}_{sec}}$$ $$

Where:

$\dot{V}{bypass} = \dot{V}{sec} - \dot{V}_{pri}$
$T_{LCHWT}$ = leaving chilled water temperature from chillers
$T_{RCHWT}$ = return chilled water temperature from building

Waterside Economizer Integration

Free Cooling Principles

Waterside economizers utilize low ambient conditions to provide cooling without chiller operation. When outdoor wet-bulb temperature drops sufficiently below required supply water temperature, a heat exchanger or direct cooling tower connection provides “free cooling.”

Available free cooling capacity:

$$ $$\dot{Q}_{free} = \dot{m} \cdot c_p \cdot (T_{RCHWT} - T_{LCHWT,target})$$ $$

Subject to constraint:

$$ $$T_{LCHWT,target} \geq T_{WB,ambient} + \Delta T_{approach} + \Delta T_{HX}$$ $$

Where:

$T_{WB,ambient}$ = ambient wet-bulb temperature
$\Delta T_{approach}$ = cooling tower approach (typically 5-7°F)
$\Delta T_{HX}$ = heat exchanger temperature difference (if used, 2-5°F)

Configuration Types

1. Plate-Frame Heat Exchanger

Indirect connection between condenser water and chilled water loops through heat exchanger:


graph LR
    CT[Cooling Tower] --> CWP[Condenser Water Pump]
    CWP --> HX[Plate-Frame HX]
    HX --> CT
    CHW[Chilled Water Return] --> HX
    HX --> CHWSUP[Chilled Water Supply]

Heat exchanger effectiveness:

$$ $$\varepsilon = \frac{T_{CWS,in} - T_{CWS,out}}{T_{CWS,in} - T_{CHW,in}}$$ $$

Typical effectiveness: 0.60-0.85 for counterflow plate-frame exchangers.

Required heat exchanger capacity:

$$ $$UA = \frac{\dot{Q}}{\Delta T_{LMTD}}$$ $$

Where log-mean temperature difference:

$$ $$\Delta T_{LMTD} = \frac{(T_{CWS,in} - T_{CHW,out}) - (T_{CWS,out} - T_{CHW,in})}{\ln\left(\frac{T_{CWS,in} - T_{CHW,out}}{T_{CWS,out} - T_{CHW,in}}\right)}$$ $$

2. Integrated Waterside Economizer

Direct connection using cooling tower to chill building supply water through integrated chiller-economizer:

Advantages:

No separate heat exchanger
Lower first cost
Reduced maintenance

Disadvantages:

Water treatment criticality (tower water in building loop)
Limited to integrated chiller designs

Control Sequences

Economizer operation modes:

100% free cooling: Chillers off, economizer provides full load
Partial free cooling: Economizer pre-cools, chillers supplement
Chiller-only: Ambient too warm for economizer benefit

Mode selection logic:

$$ $$\text{If } T_{WB} + \Delta T_{approach} + \Delta T_{HX} < T_{CHWST,target} - 2°F \text{: Enable economizer}$$ $$

Energy savings calculation:

$$ $$\text{Annual Savings} = \sum_{h=1}^{8760} \left(\text{kW}_{chiller,h} - \text{kW}_{economizer,h}\right) \times \text{hours}$$ $$

Where economizer energy includes:

Condenser water pump
Cooling tower fan
Heat exchanger pump (if applicable)

Typical economizer energy: 0.05-0.15 kW/ton versus 0.50-0.70 kW/ton for chiller operation.

Thermal Storage Systems

Storage Fundamentals

Thermal energy storage (TES) shifts cooling production to off-peak periods, reducing demand charges and capitalizing on lower nighttime efficiency. The fundamental storage equation:

$$ $$E_{stored} = m \cdot c_p \cdot \Delta T = \rho \cdot V \cdot c_p \cdot \Delta T$$ $$

Where:

$E_{stored}$ = stored energy (kWh or ton-hours)
$m$ = mass of storage medium (kg or lb)
$V$ = volume (m³ or gal)
$\Delta T$ = temperature difference between charged and discharged states

Converting to ton-hours:

$$ $$\text{Ton-hours} = \frac{V_{gal} \times 8.33 \times \Delta T_F}{12,000}$$ $$

Storage Technologies

| Technology | Storage Medium | Temperature | Energy Density | Efficiency | Applications | |------------|----------------|-------------|----------------|------------|--------------| | Chilled Water | Water | 42-50°F | 5-8 Btu/gal-°F | 90-95% | Large systems, retrofit | | Ice (full storage) | Ice | 14-25°F | 144 Btu/lb | 85-90% | Limited space, high density | | Ice (partial storage) | Ice/water | 25-40°F | 100-120 Btu/lb | 87-92% | Demand reduction, retrofit | | Phase Change Material | Eutectic salts | 47°F | 60-80 Btu/lb | 88-93% | Moderate density, stable temp |

Storage Strategies

Full Storage

Chillers operate only during off-peak periods, storage meets entire on-peak load:

$$ $$V_{storage} = \frac{\dot{Q}_{peak} \times t_{discharge}}{\rho \cdot c_p \cdot \Delta T}$$ $$

Where:

$\dot{Q}_{peak}$ = peak cooling load
$t_{discharge}$ = on-peak duration (hours)

Example: 500-ton peak load, 12-hour on-peak period, 16°F delta-T:

$$ $$V = \frac{500 \times 12 \times 12,000}{8.33 \times 16} = 539,568 \text{ gallons}$$ $$

Partial Storage (Load Leveling)

Chillers operate continuously, storage supplements during peak periods:

$$ $$\dot{Q}_{chiller} = \frac{\int_0^{24} \dot{Q}_{load}(t) dt}{24}$$ $$

Storage size based on area between load profile and chiller capacity:

$$ $$E_{storage} = \int_{t_1}^{t_2} \left(\dot{Q}_{load}(t) - \dot{Q}_{chiller}\right) dt$$ $$

Partial storage reduces chiller size and storage volume compared to full storage while still providing demand reduction.

Demand Limiting

Storage discharges only during utility demand peaks to minimize demand charges:

$$ $$\text{Savings}_{\text{demand}} = (\dot{Q}_{reduced} - \dot{Q}_{with storage}) \times \text{Rate}_{\text{\$/kW}} \times 12 \text{ months}$$ $$

Storage System Hydraulics

Thermal storage requires careful piping design to maintain thermal stratification and prevent mixing.

Series configuration (charge-discharge flow path through storage):


graph LR
    CH[Chiller] --> STG[Storage Tank]
    STG --> LOAD[Building Load]
    LOAD --> CH

Parallel configuration (storage in separate loop):


graph TD
    CH[Chiller] --> DIV{Diverting Valve}
    DIV --> LOAD[Building Load]
    DIV --> STG[Storage Tank]
    STG --> MIX{Mixing Valve}
    LOAD --> MIX
    MIX --> CH

Diffuser design for stratification:

$$ $$Fr = \frac{v_{inlet}}{\sqrt{g \cdot H \cdot \frac{\Delta \rho}{\rho}}} < 1$$ $$

Where:

$Fr$ = Froude number (dimensionless)
$v_{inlet}$ = inlet velocity (m/s)
$g$ = gravitational acceleration (9.81 m/s²)
$H$ = tank height (m)
$\Delta \rho / \rho$ = relative density difference

Froude number <1 indicates stable stratification.

System Diversity and Load Profiles

Diversity Factors

Chilled water systems serving multiple zones rarely experience simultaneous peak loads. Diversity factors account for this non-coincident loading:

$$ $$\text{Diversity Factor} = \frac{\sum \dot{Q}_{zone,peak}}{\dot{Q}_{system,peak}}$$ $$

Typical diversity factors:

| Building Type | Diversity Factor | Description | |--------------|------------------|-------------| | Single zone | 1.00 | No diversity | | Multiple floors, similar use | 0.85-0.95 | Time-based diversity | | Mixed-use building | 0.70-0.85 | Usage pattern diversity | | Campus system | 0.60-0.80 | Building diversity | | District cooling | 0.50-0.70 | Large-scale diversity |

Applying diversity:

$$ $$\dot{Q}_{plant} = \frac{\sum \dot{Q}_{zone,peak}}{\text{DF}} \times \text{SF}$$ $$

Where:

$\text{DF}$ = diversity factor
$\text{SF}$ = safety factor (1.10-1.20)

Load Duration Curves

Load duration curves plot cooling load against hours of occurrence, enabling energy analysis and equipment sizing optimization:

$$ $$\text{Annual Energy} = \sum_{i=1}^{n} \dot{Q}_i \times \Delta t_i$$ $$

Key metrics from load duration analysis:

Peak load: Maximum instantaneous demand
Equivalent full-load hours: Total annual energy divided by peak load
Part-load distribution: Percentage of hours at various load levels

Equivalent full-load hours:

$$ $$\text{EFLH} = \frac{\sum_{h=1}^{8760} \dot{Q}_h}{\dot{Q}_{peak}}$$ $$

Typical commercial buildings: 1,500-3,000 EFLH/year.

Bin Analysis

Bin analysis correlates building load with outdoor temperature for energy modeling:

$$ $$E_{annual} = \sum_{i=1}^{n_{bins}} \dot{Q}(T_{bin,i}) \times h_{bin,i} \times \frac{\text{kW}}{\text{ton}}(T_{bin,i})$$ $$

Where:

$T_{bin,i}$ = representative temperature for bin $i$
$h_{bin,i}$ = hours in bin $i$
$\dot{Q}(T)$ = building load at temperature $T$
$\text{kW/ton}(T)$ = chiller efficiency at temperature $T$

Condenser Water Systems

Cooling Tower Fundamentals

Cooling towers reject heat from chiller condensers through evaporative cooling. The governing heat and mass transfer relationship:

$$ $$\dot{Q}_{tower} = \dot{m}_w \cdot c_p \cdot (T_{CWR} - T_{CWS}) = \dot{m}_a \cdot (h_{out} - h_{in})$$ $$

Where:

$\dot{m}_w$ = water mass flow rate
$T_{CWR}$ = condenser water return temperature
$T_{CWS}$ = condenser water supply temperature
$\dot{m}_a$ = air mass flow rate
$h$ = air enthalpy

Tower performance characterized by approach and range:

$$ $$\text{Approach} = T_{CWS} - T_{WB}$$ $$

$$ $$\text{Range} = T_{CWR} - T_{CWS}$$ $$

Where $T_{WB}$ = ambient wet-bulb temperature.

Typical design values:

Approach: 5-7°F (tighter approach requires larger tower)
Range: 10-15°F (matches chiller condenser delta-T)

Tower effectiveness:

$$ $$\varepsilon = \frac{\text{Range}}{\text{Range} + \text{Approach}} = \frac{T_{CWR} - T_{CWS}}{T_{CWR} - T_{WB}}$$ $$

Tower Selection and Sizing

Tower thermal capacity rated in terms of heat rejection:

$$ $$\dot{Q}_{tower} = \dot{Q}_{cooling} \times \left(1 + \frac{1}{\text{COP}}\right)$$ $$

For 500-ton chiller at 0.60 kW/ton (COP = 5.86):

$$ $$\dot{Q}_{tower} = 500 \times \left(1 + \frac{1}{5.86}\right) = 585 \text{ tons}$$ $$

Tower performance varies with operating conditions according to manufacturer curves, typically expressed as:

$$ $$\frac{\dot{Q}_{actual}}{\dot{Q}_{rated}} = f\left(\frac{\dot{V}_w}{\dot{V}_{w,rated}}, \frac{\dot{V}_a}{\dot{V}_{a,rated}}, T_{WB}, T_{CWR}\right)$$ $$

Variable Speed Tower Fans

Tower fan power follows cubic relationship with speed (affinity laws):

$$ $$P_{fan} = P_{rated} \times \left(\frac{N}{N_{rated}}\right)^3$$ $$

Fan speed modulates to maintain condenser water supply temperature setpoint. As ambient conditions cool, fan speed reduces, providing substantial energy savings.

Annual tower fan energy:

$$ $$E_{fan} = \sum_{h=1}^{8760} P_{fan,h} \times 1 \text{ hour}$$ $$

Where $P_{fan,h}$ varies with load and ambient conditions each hour.

Condenser Water Temperature Optimization

Lower condenser water temperature improves chiller efficiency but increases tower fan energy. The optimal setpoint balances these competing effects:

$$ $$P_{total} = P_{chiller}(T_{CWS}) + P_{tower}(T_{CWS})$$ $$

Minimize total power:

$$ $$\frac{dP_{total}}{dT_{CWS}} = 0$$ $$

Typical optimal range: 65-75°F depending on load and ambient conditions.

Chiller power sensitivity to condenser water temperature:

$$ $$\frac{d(\text{kW/ton})}{dT_{CWS}} \approx 0.01 \text{ to } 0.02 \text{ kW/ton per °F}$$ $$

For 500-ton chiller at 0.60 kW/ton base:

$$ $$\Delta P = 500 \times 0.015 \times \Delta T_{CWS} \text{ kW}$$ $$

A 5°F reduction saves approximately 37.5 kW of chiller power.

Evaporative Loss and Makeup Water

Cooling towers lose water through evaporation, drift, and blowdown:

Evaporation rate:

$$ $$\dot{V}_{evap} = \frac{\dot{Q}_{tower}}{h_{fg} \times \rho_w}$$ $$

Simplified:

$$ $$\dot{V}_{evap,gpm} \approx 0.001 \times \dot{V}_{CW,gpm} \times \Delta T_F$$ $$

Where $\dot{V}_{CW}$ = condenser water flow rate.

Blowdown maintains water quality by limiting concentration:

$$ $$\dot{V}_{blowdown} = \frac{\dot{V}_{evap}}{\text{COC} - 1}$$ $$

Where $\text{COC}$ = cycles of concentration (typically 3-6).

Total makeup water:

$$ $$\dot{V}_{makeup} = \dot{V}_{evap} + \dot{V}_{blowdown} + \dot{V}_{drift}$$ $$

Drift typically 0.001-0.01% of circulation rate for modern drift eliminators.

Water Treatment and Corrosion Control

Water Quality Requirements

Chilled water and condenser water systems require chemical treatment to prevent:

Corrosion of metallic components
Scale formation from mineral precipitation
Biological growth (algae, bacteria, fungi)
Fouling from suspended solids

Critical water quality parameters:

| Parameter | Chilled Water | Condenser Water | Impact | |-----------|---------------|-----------------|--------| | pH | 7.5-9.0 | 6.5-8.5 | Corrosion, scale | | Hardness (ppm CaCO₃) | <50 | <400 | Scale formation | | Alkalinity (ppm CaCO₃) | 50-200 | 50-400 | pH buffering, scale | | Chlorides (ppm) | <100 | <500 | Corrosion | | Sulfates (ppm) | <100 | <500 | Corrosion, scale | | Total Dissolved Solids | <500 | <1500 | Scale, corrosion | | Biological count (CFU/mL) | <1000 | <10,000 | Biofouling |

Corrosion Mechanisms

Electrochemical corrosion occurs when dissimilar metals contact in an electrolyte (water):

$$ $$\text{Anode: } M \rightarrow M^{n+} + ne^-$$ $$

$$ $$\text{Cathode: } O_2 + 2H_2O + 4e^- \rightarrow 4OH^-$$ $$

Corrosion rate follows Faraday’s law:

$$ $$\text{Corrosion Rate} = \frac{I \cdot M}{n \cdot F \cdot A \cdot \rho}$$ $$

Where:

$I$ = corrosion current (A)
$M$ = molecular weight (g/mol)
$n$ = electrons transferred
$F$ = Faraday constant (96,485 C/mol)
$A$ = surface area (cm²)
$\rho$ = density (g/cm³)

Typical corrosion rates expressed as mils per year (mpy):

Excellent: <2 mpy
Acceptable: 2-5 mpy
Poor: >5 mpy

Scale Formation

Scale forms when mineral solubility limits are exceeded. Calcium carbonate represents the most common scale:

$$ $$Ca^{2+} + 2HCO_3^- \rightarrow CaCO_3 \downarrow + H_2O + CO_2$$ $$

Langelier Saturation Index (LSI) predicts scaling tendency:

$$ $$\text{LSI} = \text{pH}_{actual} - \text{pH}_s$$ $$

Where $\text{pH}_s$ = pH at calcium carbonate saturation.

Interpretation:

LSI > 0: Supersaturated, scale-forming
LSI = 0: Saturated, neutral
LSI < 0: Undersaturated, corrosive

Target range: -0.5 to +0.5 for balanced condition.

Ryznar Stability Index provides alternate assessment:

$$ $$\text{RSI} = 2\text{pH}_s - \text{pH}_{actual}$$ $$

Target RSI: 6.5-7.0 for stable conditions.

Treatment Strategies

Closed Loop Systems (Chilled Water)

Closed systems require one-time treatment with long-term inhibitors:

Corrosion inhibitors: Sodium nitrite, sodium molybdate (200-1000 ppm)
pH control: Sodium hydroxide or potassium hydroxide (pH 8.5-9.5)
Biocides: Initial slug dose, periodic maintenance

Treatment concentration:

$$ $$\text{Dose}_{lbs} = \frac{V_{gal} \times \text{ppm}_{target}}{120,000}$$ $$

Inhibitor depletion rate typically 10-20% per year requires monitoring and makeup.

Open Loop Systems (Condenser Water)

Open systems continuously lose water through evaporation, requiring ongoing treatment:

Scale inhibitors: Phosphonates, polymers (5-20 ppm)
Corrosion inhibitors: Azoles, phosphates (2-10 ppm)
Biocides: Oxidizing (chlorine, bromine) and non-oxidizing
pH control: Acid feed for alkalinity management

Blowdown control maintains concentration:

$$ $$\text{COC} = \frac{\text{[Conductivity]}_{CW}}{\text{[Conductivity]}_{makeup}}$$ $$

Chemical feed rate:

$$ $$\dot{V}_{chemical} = \frac{\dot{V}_{makeup} \times \text{ppm}_{target}}{10^6 \times \rho_{chemical}}$$ $$

Filtration

Side-stream filtration removes suspended solids:

Filter flow rate typically 1-10% of system flow:

$$ $$\dot{V}_{filter} = 0.01 \text{ to } 0.10 \times \dot{V}_{system}$$ $$

Particle removal efficiency:

| Filter Type | Particle Size | Efficiency | Pressure Drop | |-------------|---------------|------------|---------------| | Strainer | >500 μm | 80-90% | 2-5 psi | | Sand filter | 20-100 μm | 90-95% | 5-10 psi | | Cartridge filter | 5-50 μm | 95-99% | 5-15 psi | | Bag filter | 1-25 μm | 90-98% | 3-8 psi |

Energy Optimization Strategies

System-Level Optimization

Chilled water system energy consumption includes:

$$ $$E_{total} = E_{chiller} + E_{CHW\_pump} + E_{CW\_pump} + E_{tower} + E_{aux}$$ $$

Optimization targets each component while maintaining comfort and reliability.

Chilled Water Temperature Reset

Raising chilled water supply temperature during low-load periods reduces chiller lift and improves efficiency:

$$ $$\text{COP}_{Carnot} = \frac{T_{evap}}{T_{cond} - T_{evap}}$$ $$

Practical chiller efficiency improves approximately:

$$ $$\Delta(\text{kW/ton}) \approx -0.01 \text{ to } -0.015 \text{ per °F increase in CHWST}$$ $$

Reset schedule based on outdoor air temperature or load:

$$ $$T_{CHWST} = T_{min} + (T_{max} - T_{min}) \times \left(1 - \frac{\dot{Q}_{current}}{\dot{Q}_{design}}\right)$$ $$

Typical range: 42°F at design load to 50-55°F at minimum load.

Constraints:

Maintain adequate dehumidification (dewpoint control)
Ensure terminal unit capacity at elevated temperature
Monitor valve positions (reset limit when valves saturate)

Condenser Water Temperature Optimization

Discussed previously, balances chiller efficiency gains against tower fan energy:

$$ $$T_{CWS,optimal} = f(\dot{Q}_{load}, T_{WB}, \text{Chiller curves}, \text{Tower curves})$$ $$

Requires real-time optimization or pre-computed lookup tables.

Pump Speed Optimization

Variable speed pumps reduce energy consumption following cubic relationship:

Annual pump energy savings from variable speed:

$$ $$\text{Savings} = \sum_{h=1}^{8760} P_{constant,h} - P_{variable,h}$$ $$

Typical savings: 30-60% versus constant speed operation.

Control optimization:

Differential pressure reset (described previously)
Flow-based control matching demand
Predictive control based on load forecasts

Chiller Plant Optimization

Multi-chiller plants require sequencing and load distribution optimization:

Objective function minimizes total plant power:

$$ $$\min P_{total} = \sum_{i=1}^{n} P_{chiller,i}(\dot{Q}_i) + P_{pumps}(\dot{V}) + P_{towers}(T_{CWS})$$ $$

Subject to constraints:

$\sum_{i=1}^{n} \dot{Q}i = \dot{Q}{load}$ (meet load requirement)
$\dot{Q}_{min,i} \leq \dot{Q}i \leq \dot{Q}{max,i}$ (chiller capacity limits)
$\dot{V}_{min,i} \leq \dot{V}_i$ (minimum chiller flow)

Solution methods:

Lagrange multipliers: Analytical solution for differentiable curves
Dynamic programming: Discrete optimization
Real-time optimization: Use building automation system and chiller plant meter data

Equal marginal efficiency criterion:

$$ $$\frac{dP_1}{d\dot{Q}_1} = \frac{dP_2}{d\dot{Q}_2} = \cdots = \frac{dP_n}{d\dot{Q}_n}$$ $$

Heat Recovery

Chiller condenser heat recovery captures rejected heat for simultaneous heating loads:

Available heat:

$$ $$\dot{Q}_{cond} = \dot{Q}_{evap} + P_{compressor}$$ $$

Condenser heat available at 85-105°F suitable for:

Domestic hot water preheating
Space heating during swing seasons
Pool heating
Process heating

Heat recovery effectiveness:

$$ $$\varepsilon_{HR} = \frac{\dot{Q}_{recovered}}{\dot{Q}_{cond,available}}$$ $$

Economic analysis:

$$ $$\text{Savings}_{annual} = \frac{\dot{Q}_{recovered} \times \text{hours}}{\eta_{boiler}} \times \text{Cost}_{fuel}$$ $$

Where $\eta_{boiler}$ represents efficiency of displaced heating system.

Thermal Zoning and Diversity

Optimize chilled water distribution by thermal zones:

Core zones: Constant cooling loads year-round
Perimeter zones: Variable loads based on solar and envelope
Process loads: Independent of weather
Ventilation loads: Dependent on outdoor air conditions

Separate distribution enables:

Temperature optimization by zone characteristics
Reduced mixing losses
Improved control and delta-T

Load aggregation diversity:

$$ $$\dot{Q}_{plant} = \sqrt{\left(\sum \dot{Q}_{sensible}\right)^2 + \left(\sum \dot{Q}_{latent}\right)^2}$$ $$

This vector sum acknowledges non-coincident peak conditions.

Practical Design Example

System Design Specification

Building: 200,000 ft² office building, 10 stories Location: Chicago, IL Cooling load: 800 tons peak (4.0 tons/1000 ft²) Hours of operation: 6 AM - 8 PM weekdays, 8 AM - 4 PM Saturday Design delta-T: 14°F (42°F supply, 56°F return)

Load Profile Analysis

Monthly peak loads:

| Month | Peak Load (tons) | Load Factor | Equivalent Full-Load Hours | |-------|------------------|-------------|---------------------------| | January | 240 | 0.30 | 80 | | February | 280 | 0.35 | 100 | | March | 400 | 0.50 | 150 | | April | 520 | 0.65 | 200 | | May | 640 | 0.80 | 240 | | June | 760 | 0.95 | 280 | | July | 800 | 1.00 | 310 | | August | 800 | 1.00 | 300 | | September | 680 | 0.85 | 260 | | October | 480 | 0.60 | 180 | | November | 320 | 0.40 | 120 | | December | 240 | 0.30 | 80 | | **Annual** | **800** | **0.65** | **2,300** |

Chiller Selection

Select three chillers for capacity, efficiency, and redundancy:

Configuration: Three 350-ton water-cooled centrifugal chillers with variable speed drives

Capacity analysis:

Total installed: 1,050 tons (131% of peak)
N+1 redundancy: 700 tons (87.5% of peak with one chiller down)
Optimal loading: 267 tons per chiller at peak (76% of chiller capacity)

Performance specifications:

| Load Point | Efficiency (kW/ton) | Total Plant (kW) | |------------|---------------------|------------------| | 100% (800 tons) | 0.52 | 416 | | 75% (600 tons) | 0.48 | 288 | | 50% (400 tons) | 0.46 | 184 | | 25% (200 tons) | 0.50 | 100 |

IPLV calculation:

$$ $$\text{IPLV} = 0.01(0.52) + 0.42(0.48) + 0.45(0.46) + 0.12(0.50) = 0.474 \text{ kW/ton}$$ $$

Chilled Water Pumping

Configuration: Variable primary flow with three dedicated chiller pumps

Flow rate per chiller:

$$ $$\dot{V} = \frac{350 \times 24}{14} = 600 \text{ gpm}$$ $$

Head calculation:

Pipe friction: 35 ft
Chiller evaporator: 15 ft
Coil pressure drop: 12 ft average
Control valve: 5 ft
Fittings and accessories: 8 ft
Total: 75 ft

Pump power at design:

$$ $$P_{hp} = \frac{600 \times 75}{3960 \times 0.78 \times 0.94} = 15.6 \text{ hp}$$ $$

Select 20 hp motor with VFD.

Annual pumping energy (simplified):

$$ $$E_{pump} = 3 \times 15.6 \times 0.746 \times 2,300 \times 0.65^3 = 22,150 \text{ kWh}$$ $$

Where $0.65^3$ represents average speed cubed based on load factor.

Condenser Water System

Cooling tower selection: Three 410-ton induced-draft counterflow towers with VFD fans

Heat rejection per tower:

$$ $$\dot{Q}_{tower} = 350 \times 1.17 = 410 \text{ tons}$$ $$

(117% of chiller capacity based on 0.52 kW/ton)

Design conditions:

Entering water: 95°F
Leaving water: 85°F
Range: 10°F
Wet-bulb: 78°F
Approach: 7°F

Condenser water flow:

$$ $$\dot{V}_{CW} = \frac{410 \times 24}{10} = 984 \text{ gpm} \approx 1000 \text{ gpm per tower}$$ $$

Condenser water pump head:

Pipe friction: 25 ft
Chiller condenser: 20 ft
Tower: 15 ft
Control valve: 5 ft
Fittings: 10 ft
Total: 75 ft

Pump power per unit:

$$ $$P_{hp} = \frac{1000 \times 75}{3960 \times 0.78 \times 0.94} = 26.0 \text{ hp}$$ $$

Select 30 hp motor with VFD.

Tower fan power: 15 hp per tower

Annual condenser water system energy:

$$ $$E_{CW} = 3 \times (26.0 + 15.0) \times 0.746 \times 2,300 \times 0.65^2 = 70,800 \text{ kWh}$$ $$

Total System Energy

Annual energy summary:

| Component | Annual Energy (kWh) | Percentage | |-----------|---------------------|------------| | Chillers | 872,000 | 87.0% | | CHW pumps | 22,150 | 2.2% | | CW pumps | 56,400 | 5.6% | | Tower fans | 51,800 | 5.2% | | **Total** | **1,002,350** | **100%** |

System efficiency:

$$ $$\text{kW/ton}_{system} = \frac{1,002,350}{800 \times 2,300} = 0.544 \text{ kW/ton}$$ $$

This represents excellent performance for a water-cooled centrifugal plant with optimized auxiliary equipment.

Economic Analysis

Capital cost estimate:

Chillers (3 × $180,000): $540,000
Cooling towers (3 × $45,000): $135,000
CHW pumps (3 × $15,000): $45,000
CW pumps (3 × $18,000): $54,000
Piping and valves: $250,000
Controls and instrumentation: $80,000
Installation labor: $200,000
Total: $1,304,000

Operating cost ($0.10/kWh):

$$ $$\text{Cost}_{annual} = 1,002,350 \times 0.10 = \$100,235$$ $$

Comparison to air-cooled system:

Air-cooled efficiency: 1.05 kW/ton
Air-cooled annual energy: 1,932,000 kWh
Annual savings: 929,650 kWh × $0.10 = $92,965
Simple payback: $304,000 incremental cost / $92,965 = 3.3 years

Conclusion

Chilled water system design requires integrated analysis of thermodynamic performance, fluid mechanics, heat transfer, and energy economics. Optimal configurations balance first cost, energy efficiency, reliability, and operational flexibility. Key design principles include:

Delta-T preservation: Maintain design temperature difference through proper coil selection, balancing, and control valve authority
Variable flow implementation: Utilize VFDs on pumps and tower fans to capture substantial energy savings at part-load conditions
Chiller plant optimization: Select equipment sizes and implement controls to optimize total plant efficiency across the load spectrum
Water quality management: Implement comprehensive treatment programs to prevent corrosion, scaling, and biological growth
System diversity utilization: Account for non-coincident loads to rightsize equipment and reduce capital and operating costs
Performance monitoring: Meter and track key parameters to verify design performance and identify optimization opportunities

The analytical methods presented enable rigorous system design based on fundamental physics and proven engineering principles. Proper application of these methodologies yields chilled water systems that deliver reliable, efficient cooling throughout their 20-30 year service life.