How do I know air cleaners even work?

Because they have been studied, and found to be very effective at removing aerosols from the air: 

• Classrooms:   “…our study shows that box fan air cleaners can serve as an effective low-cost alternative for mitigating airborne transmission risks in poorly ventilated spaces.”
• Hospitals (preprint):  “A single bed hospital room equipped with an excellent ventilation rate (∼ 14 air changes per hour) can clear the aerosols in 20 minutes. However, with the addition of two air cleaners, the clearance time became 3 times faster (in 6 minutes and 30 seconds).”  
  • “What? How can there be an 14 ACH ventilation rate, but only 3 ACH to clear the aerosols? 14 ACH is every 4 mins?”
  • Answer: “An ACH is the rate to remove 63% of the air (not 100%). It requires 5x this time (5 x – 1/ACH (ie lifetime / residence time) to clear 99% (ie remaining = 0.37 ^ 5 < 0.01) of the air. Think of a drop of red dye in a sink - easy to remove the first 50% really long to remove it all."
• Hospital (published): “With 2 small domestic air cleaners in a single patient room of a hospital ward, 99% of aerosols could be cleared within 5.5 minutes.”
• A systemic review found:” Portable HEPA filter systems have been shown to be another effective tool in reducing viral load.”
• CDC MMWR: “Portable HEPA air cleaners offer a simple means to increase the filtration of aerosol particles from a room without modifying the existing building ventilation system…”
• FORD engineers did a study on a single filter version, called the Scrappy Filtration Kit..

The following Federal Agencies recommend them (along with many state public health agencies):

• CDC directly recommends them: “Use portable high-efficiency particulate air (HEPA) fan/filtration systems to enhance air cleaning (especially in higher risk areas such as a nurse’s office or areas frequently inhabited by people with a higher likelihood of having COVID-19 and/or an increased risk of getting COVID-19).”
• ASHRAE recommends them
• EPA recommends them.
• The Corsi/Rosenthal Box as discussed in the recent “Delta Schools Airborne” Webinar.
• Media: NPR, This Old House,

If these air cleaners are so great, why haven’t any large organizations rolled them out?

They have!

At least 20,000 by the Ontario, Canada government – in classrooms, gyms, libraries with no mechanical ventilation.  As approved by their Chief Medical Officer of Health.

Toronto School District itself.

New York City, but there’s a note of caution there.  Experts have indicated that NYC did not make the best choice.