T-Method Duct Design

T-Method Optimization Fundamentals

The T-Method represents a mathematical optimization approach to duct design that minimizes total life-cycle cost by systematically balancing initial construction costs against long-term operating expenses. Unlike prescriptive methods such as equal friction or static regain, the T-Method employs calculus-based optimization to determine economically optimal duct sizes for each section of the system.

The method derives its name from the optimization parameter T, which represents the annual cost per unit of total pressure loss. This single parameter encapsulates the economic relationship between fan power consumption and duct construction costs.

Economic Optimization Principle

The T-Method minimizes the total annual cost function:

Total Annual Cost = First Cost × Capital Recovery Factor + Operating Cost

For each duct section:

• First Cost increases with duct size (larger ducts cost more to fabricate and install)
• Operating Cost decreases with duct size (lower velocity reduces pressure loss and fan energy)

The optimal duct size occurs where the derivative of total cost with respect to duct size equals zero, representing the economic balance point between these opposing cost trends.

Mathematical Foundation

The optimization parameter T is calculated as:

T = (e × H × η_m) / (η_f × CRF)

Where:

• e = Energy cost ($/kWh)
• H = Annual operating hours (hours/year)
• η_m = Motor efficiency (decimal)
• η_f = Fan total efficiency (decimal)
• CRF = Capital Recovery Factor (decimal)

The Capital Recovery Factor converts first cost to equivalent annual cost:

CRF = [i(1+i)^n] / [(1+i)^n - 1]

Where:

• i = Interest rate (discount rate)
• n = Analysis period (years)

T-Method Design Procedure

Step 1: Calculate Optimization Parameter T

Determine economic parameters (energy cost, operating hours, interest rate, analysis period) and equipment efficiencies to calculate T. This value remains constant throughout the design.

Step 2: Size Duct Sections Based on Total Pressure

For each duct section, the optimal velocity is:

V_opt = K × (Q / C_1)^(1/3) × (ΔP_t / C_2)^(1/6)

Where:

• K = Optimization constant derived from T
• Q = Airflow rate (cfm)
• C_1, C_2 = Cost coefficients for duct material and energy
• ΔP_t = Available total pressure for the section

Step 3: Apply Total Pressure Budget

Allocate total fan pressure across all sections. The total pressure available for duct losses equals:

ΔP_available = P_fan - P_coils - P_filters - P_terminals - P_fittings

Step 4: Iterative Sizing

Calculate pressure losses for initially sized ducts and compare to allocated pressure budget. Adjust section sizes iteratively until the system balances.

Step 5: Verify Velocity Limits

Check that optimized velocities remain within acoustic and erosion limits. Apply constraints where necessary.

Life-Cycle Cost Components

Economic Parameter Selection

Comparison to Other Methods

T-Method vs Equal Friction:

The equal friction method maintains constant pressure loss per unit length, typically resulting in:

• Higher initial duct cost (oversized in low-flow sections)
• Lower operating cost (reduced total pressure loss)
• No economic optimization basis

T-Method vs Static Regain:

Static regain maintains constant static pressure at each branch takeoff, resulting in:

• Highest initial duct cost (largest ducts to achieve velocity reduction)
• Lowest operating cost (minimized total pressure)
• Optimal for VAV systems but economically inefficient at part load

T-Method Advantages:

• Explicit economic optimization based on actual project costs
• Adapts to varying energy costs and operating schedules
• Produces unique solution for given economic parameters
• Balances first cost and operating cost mathematically

Implementation Considerations

Energy Cost Escalation:

Account for anticipated energy cost increases by using levelized energy cost or adjusting the discount rate. A 3% annual energy escalation over 20 years increases the effective operating cost weight by approximately 30%.

Part-Load Operation:

For VAV systems operating at part load, reduce effective operating hours H by the average diversity factor. A system operating at 60% average flow operates at approximately 35% of full-load pressure, significantly reducing annual energy cost.

Duct Material Cost Variations:

T-Method optimization is sensitive to duct cost coefficients. Rectangular duct costs increase approximately with the 1.6 power of hydraulic diameter, while spiral duct costs increase nearly linearly. Update cost coefficients for regional material and labor rates.

Velocity Constraints:

The unconstrained T-Method may produce velocities outside acceptable ranges:

Apply velocity caps as hard constraints and re-optimize adjacent sections.

Computational Implementation

Modern duct design software implements T-Method through:

1. Economic parameter input (T calculation)
2. System layout definition (sections and connections)
3. Initial sizing using T-Method equations
4. Pressure balance iteration to meet available fan pressure
5. Constraint checking for velocity, aspect ratio, minimum size
6. Final balancing and damper selection

The method requires iterative solution because optimal sizes depend on total pressure distribution, which depends on sizes. Convergence typically occurs within 3-5 iterations.

Sensitivity Analysis

Life-cycle cost optimization is sensitive to economic assumptions:

Perform parametric studies to assess design robustness across reasonable economic scenarios.

Practical Application Limits

The T-Method provides optimal results when:

• Operating schedule is predictable and consistent
• Energy costs are reasonably stable or escalation is estimable
• System operates primarily at design conditions (CAV systems)
• Economic parameters can be determined with confidence

Alternative methods may be preferred when:

• Acoustic performance is paramount (use low-velocity design)
• Future occupancy is uncertain (use conservative equal friction)
• System will operate primarily at part load (static regain for VAV)
• Project economics cannot support optimization analysis

Summary of T-Method Characteristics

The T-Method duct design approach represents the theoretical economic optimum for constant-volume systems with predictable operating profiles. Implementation requires careful determination of economic parameters and iterative calculation, but produces designs that minimize total ownership cost over the analysis period. The method is most valuable for large systems with long operating hours where energy costs dominate life-cycle economics.