T-Method Duct Optimization

An advanced duct design optimization technique based on the T-Method is being developed by Tsal, Behls, and Mangel through cooperative research with ASHRAE [Tsal et al., 1986]. The economic analysis of the example from the Duct Design chapter of the ASHRAE Fundamentals Handbook(2001) re152 showed that significant initial or operating cost reductions are obtainable. In addition, the three requirements for optimized designs -- optimum fan selection, pressure balancing, and optimum sectional velocity ratios [Tsal and Adler, 1987] -- are satisfied by the T-method.

The T-Method's advantages over other optimization methods are:

The T-Method has been expanded and found to be capable of optimizing duct systems with air leakage [Tsal et. al., 1998]. The T-Method incorporates the following major procedures:

1)System condensing: Condensing a branched tree system into a single imaginary duct section with identical hydraulic characteristics and the same life-cycle cost as the entire system.

2)Air-handling unit selection: Selecting an optimal fan and establishing optimal system pressure loss.

3) System expansion: Expanding the condensed imaginary duct section into the original system with optimal distribution of pressure selected in step 2.

The T-Method demonstrates that two duct sections connected in series can be condensed into a single imaginary duct section. The imaginary section will have the same: flow, pressure loss sum, and initial cost, as the individual sections. The T-Method shows that K-coefficients, the hydraulics characteristics of each section (which depend on flows, sizes, lengths and fittings) must satisfy the following:

K1-2 = ( K10.833 + K20.833 )1.2

Similarly, two duct sections connected in parallel can be condensed into an imaginary duct section that has the same pressure loss as individual sections, the sum of their flows, and the sum of their initial costs. The relationship between K-coefficients must satisfy the following:

K1-2 = K1 + K2

The T-Method identifies a duct tree system as a sequential number of sections connected in series and in parallel. In a tree system, all individual sections can be condensed into a single imaginary duct section. This process is performed numerically by a recursive procedure that starts from terminals and moves from tee to tee until it hits the root. This movement from tee to tee is the source of the T-method's name.

Most decisions about selecting an air-handling unit will fit into one of the following three categories.

Case 1. The optimum fan pressure is calculated using the classical method where the first derivative is made to equal zero. Once the desired fan pressure and flow are known, the fan can be selected from a catalog.

Case 2. A number of central air-handling units or fans with motors are being considered. The cost of each fan and motor, the total fan pressure, and coefficients of efficiency are known. A comparison is made of life-cycle costs of the system equipped with each fan. The optimum fan is then selected at minimum cost.

Case 3. Fan and motor are pre-selected based on necessary fan pressure.

In the T-method an expansion procedure distributes available fan pressure throughout the system sections. Unlike the condensing procedure, the expansion procedure starts at the root section and continues in the direction of the terminals.

An important advantage of the T-Method is that it can handle constrained optimization processes including non-linearity and integer duct-size rounding. Rounding means selecting a lower or higher nominal duct size. If the lower nominal size is selected, the initial cost decreases, but the pressure loss increases and may cause fan pressure to increase. If the upper nominal size is selected, the initial cost increases but the section pressure loss decreases. This saved pressure means a smaller nominal size can be used in the next sections in the duct network. Therefore, size rounding is also relevant to optimization. The T-Method contains a procedure that predicts the influence of the initial cost of different duct sizes for both a specific duct section and the remaining system. The rounding procedure is efficient but complicates the calculations. For manual calculation, a simplified procedure called the 1/3 boundary procedure is recommended. For this procedure, if a choice is to be made between two commercially available duct sizes where duct "A" is smaller than duct "B," the size difference between A and B is first divided into thirds. Then, if the calculated size is less than A + 1/3, duct A is chosen. If the calculated size is equal to or greater than A + 1/3, duct B is chosen. However, the 1/3 boundary procedure is just a rough approach. If the lower size is selected for a long duct with many local resistances, the pressure loss in the corresponding path may exceed the fan pressure capability. A final advantage of the T-Method shows that it can optimize a duct system with air leakage [Tsal et. al., 1998REFERENCES: Distribution Systems].

All existing analytical and numerical duct design methods except dynamic programming are iterative [Tsal and Adler, 1987]. The T-Method is iterative but relatively simple; it is also able to select the optimum pressure for each system section while incorporating pressure balancing. Many parameters such as C-coefficients for junctions and transitions depend on duct size and are not known at the beginning of the calculation process; they have to be defined during the iterations. The T-Method converges efficiently. Usually, five iterations are sufficient to obtain the optimum solution with a high degree of accuracy.

To optimize a combined supply-return system, the distribution of the pressure losses between the supply and return subsystems must be optimized. The T-Method does this by first condensing each of the subsystems. Next, both condensed sub-roots are interpreted as two sections connected in series and a condensed root section is substituted for them. Then, a fan and motor or central air-handling unit is selected, and the pressure is distributed for the supply and return subsystem as in an expansion procedure. The T-Method can optimize both the supply and return subsystems as one system.

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