ASHRAE Journal - December 2020 - 24

TECHNICAL FEATURE

As Robinson6 points out, each effectiveness method
has its own strengths and weaknesses. Range effectiveness is good for reporting extremes of temperature that
might lead to coil freezing or nuisance freeze stat tripping. Statistical effectiveness is good at reporting the
uniformity of the flow for control response. However,
in practice, usually a single sensor is placed in each
airstream, which makes statistical effectiveness more a
tool for design and analysis and not relevant for control
strategies.
Getting the standard deviation of upstream temperature has some inconsistencies in the literature and some
practical difficulties in sample size for a standard deviation. Finaish, et al.,5 defines the upstream standard
deviation as:
SDUS = 0.5(TRA−TOA)

(3)

where TRA is the return air temperature and TOA is the
outdoor air temperature.
This definition differs from the definition in another
paper, where Sauer, et al.,7 defines the upstream standard deviation as:
SDUS =

VRAVOA
(TRA − TOA )
VRA + VOA

(4)

where VRA and VOA are the velocity of the return and outdoor airstreams.
This definition of upstream standard deviation
accounts for different airstream velocities, but was
shown to vary widely when the RA damper area was
changed while holding flow rates and all other variables constant. The derivations of Equation 3 and
Equation 4 were not provided, and the authors of
this paper found no intuitive way to choose between
them.
A simple conservation of energy evaluation neglecting kinetic and potential energy changes yields this
equation:
Q R ArRA cp TRA + Q OA rOAcp TOA = Q MA rMAcp TMA

(5)

Assuming the density, rx, and specific heat, cp, change
is negligible, the equation becomes:
Q RATRA+Q OA TOA = Q MA TMA

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(6)

where Q RA, Q OA and Q MA are the return, outdoor and
mixed air volumetric flow rates, respectively. TRA, TOA
and TMA are the return, outdoor and mixed air temperatures, respectively.
Using the relationship between volumetric flows and
simplifying yields the following:
Q RA + Q AOA = Q MA

(7)

Q RA = (1 - %OA)Q MA

(8)

Q OA = (%OA)Q MA

(9)

TMA = (1 - %OA)TRA + %OA × TOA

(10)

where %OA is the percent of outdoor air.
This is the equation used in an economizer reference
manual8 for finding the mixed air temperature.
Considering TMA as the average theoretical temperature, we can compute a more meaningful upstream
standard deviation for the statistical effectiveness.

∑ (T − T )
n

SDUS =

i =1

n −1

(11)

where the data set consists of TRA and TOA.
Equation 11 will be used in Equation 2 for reporting of
statistical effectiveness in this paper because it does not
skew the numbers based on damper sizes as Equation 4
does, and it provides a weighted average of temperatures based on the percent outdoor air, unlike Equation 3.

Computer-Aided Design (CAD) Model
Preparation and Uncertainties
An air mixing box with dampers facing perpendicular to each other was selected for analysis since this
is the most common configuration. The analysis was
performed in steady state with adiabatic walls. Figure 2
shows the 3D model with dimensions created to match
Configuration 4 tested in Test 7, with the results published in ASHRAE Research Report, RP-1045, " Verifying
Mixed Air Damper Temperature Control and Air Mixing
Characteristics. " 5 To prevent any unrealistic recirculation in the CFD simulation, the inlets and outlet surfaces were extended by 100 in. (2540 mm) and 150 in.
(3810 mm), respectively. There is also some uncertainty
on how the real blades were positioned and their opening angles when performing the test.

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ASHRAE Journal - December 2020

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