Beyond the affinity laws

Engineered Systems, August, 2004 by Tumin Chan

The affinity laws are frequently cited for energy savings calculations in VSD applications with pumps and fans, and they are just as frequently misused in an attempt to simplify those calculations. The most flagrant misuse of this methodology is to apply the cubic function directly on the kW formulation. Dig in to make sure you avoid overestimating or underperforming.

The affinity laws apply to centrifugal fans as they do to centrifugal pumps. They are based on the following principles of fluid dynamics:

1. Q (flow) [alpha] RPM (speed of the fan or pump): in mathematical terms [Q.sub.1] / [Q.sub.2] = RP[M.sub.1] / RP[M.sub.2];

2. P (pressure) [alpha] [Q.sup.2] (flow): in mathematical terms [P.sub.1] / [P.sub.2] = [([Q.sub.1/3] / [Q.sub.2]).sup.2]; and

3. W (power) [alpha] Q (flow): in mathematical terms [W.sub.1] / [W.sub.2] = [([Q.sub.1] / [Q.sub.2]).sup.3].

The terms Q(flow) and RPM(speed of the fan or pump) in the above expressions, by virtue of Equation 1, are interchangeable. It is important to note that the above affinity laws as stated are only valid in theory where the flow [Q.sub.1] has simply been changed to Q: at the design stage. In this situation, the designer can determine the corresponding revised [W.sub.2] required delivering the revised flow [Q.sub.2]. The designer can then resize the pump or fan accordingly.

Many engineers recognize the fact that the affinity laws are for theoretical ideal situations; in order to correct for this shortcoming in dealing with a real situation, every engineer brings his own adjustment factor to modify the affinity laws. The above cubic exponent in Equation 3 is modified and substituted by an exponent 2.x, where 0 < x < 9, to be conservative in their energy calculation estimates. There lies the dilemma to the owner, who has commissioned the evaluation of installing a VSD on a pump, for example. There is no consensus on a fixed value of 2.x, as 2.x is characterized by each engineer as "based on experience." This characterization leaves plenty of room for interpretation, as the amount of experience of each engineer is different.

In a hypothetical situation where three different engineers have submitted calculations for the exact same study, and assuming that all three engineers are working on the same exact load profile, all three results will be different because all three engineers would have used three different adjustments of 2.x. None of them could claim with confidence that their results are the correct ones and all of them would be wrong, in this, case what does an owner do since the differences can be quite significant, for example, between 2.1, 2.4, and 2.7 exponents?

ACTUAL DYNAMIC CONDITIONS

In an actual situation, the design flow and the fan/pump have already been determined, and the goal is to vary the flow Q in proportion to the HVAC loads; this situation is dynamic and it requires at least one controlling parameter such that the fan/pump will deliver the required amount of flow to meet the HVAC end use demand. There are numerous methods available in defining the control parameters; however, the discussion of the selection of the control parameters is beyond the scope of this article. Suffice it to say that the preferred and most common control parameter has been the pressure in the circuit (air or water).

Typically, an HVAC circuit is comprised of multiple branches: the flow to each of the branches is regulated by the opening or closing of a damper (air system) or a two-way valve (water system). As the flow is varied, the total pressure of the circuit varies accordingly as dictated by the affinity laws. The fan or pump can respond more dynamically to a change in pressure in the circuit than to a change in temperature of the returning fluid in the circuit. For this reason, an artificial minimum pressure is maintained in a closed loop circuit.

This minimum pressure is expressed as a percentage of the peak design. Per ASHRAE Standard 90.1-2001, this minimum setpoint must not be greater than 1/3 of total design pressure (see [section] 6.3.3.2.2 or [section] 6.3.3.2.3). In an open loop hydraulic system, the pump must overcome a certain amount of lift and thus the typical "square" pressure/flow curve is shifted upward by the amount of required lift. As a consequence, the second law (2) of the above affinity laws is modified in the following manner.

1. [P.sub.1] / [P.sub.2] = a b* [([Q.sub.1] / [Q.sub.2]).sup.2], where (0< a < 1 and a b = 1).

In a simplified mathematical term, substituting [P.sub.1] / [P.sub.2] = y and [Q.sub.1] / [Q.sub.2] = x, the above expression can be modified such that:

2. y = a b * [x.sup.2].

This relationship is illustrated in Figure 1 and is comparable to the theoretical ideal relationship between the pressure and flow.

[FIGURE 1 OMITTED]

Another expression for the power W is: [Wa.sup.1] = (Q * P) / (C * [eta]) where C is a unit conversion constant, and [eta] is the efficiency of the fan/pump at the given Q. For two different flow situations, [Q.sub.1] and [Q.sub.2], the corresponding power [W.sub.1] and [W.sub.2] can be calculated in the following manner in equations (3) and (4) respectively.