Derivation of HVAC equation

[bagofmilk]

First of all, what is the name of this equation:
ΔT = SP_total / (2.7*η_f)

This is used in HVAC applications to determine the total temperature rise across the supply fan.

Where:
ΔT = change in temperature (F)
SP = Total Static Pressure (inH2O)
η_f = fan efficiency (decimal form)


And where is this derived from? Can someone provide a link or specifics?
I started off with the first law:
Assume:
m_dot(1) = m_dot(2) = m_dot
Δz = 0

ΔU = 0 = Q - W + m_dot(Δh + 0.5(ΔV²))

Δh = c_pΔT
m_dot = ρAV

This seemed to get me nowhere...so I went to Bernoulli's (neglect altitude change)

(P₁/ρ) + 0.5(V₁²) = (P₂/ρ) + 0.5(V₂²)

ΔP = 0.5ρ(ΔV²)

but I see nowhere where temperature is involved...im lost.

 

[chicopee]

Is the temperature rise that of the fan motor or of the air being displaced? and from which reference did you get that first equation?

 

[dfaunce]

Hi Chicopee,
Page 5, equation 12

http://www.learningincontext.com/PiC-Web/pdfs/AirSys.pdf

 

[bagofmilk]

Yes. Thank you dfaunce. Do you know where that equation came from, even that document doesn't really describe where it came from.

 

[DRWeig]

bagofmilk,

Did you look into applying the ideal gas law at the end of your original post derivation? That should get you to temperature...

Best to you,

Goober Dave

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[EnergyProfessional]

It looks like to calculate temperature rise over a fan. If you use all metric units you inlcude heat capacity of air, and the energy of the pressure rise and you would have flowrate twie (whcih cancels out). The equation you quoted is the IP equivalent and the "2.7" includes all the unit conversions etc. and gets rid of some units... this isn't really scientific and also explains why IP equations kind of calculate something... but don't really show what they actually do.

 

[bagofmilk]

Hi bagofmilk,
I actually just experienced this at work. You had it right by following the 1st Law, but you must also neglect Kinetic Energy as the Velocities are essentially the same. With that you will be left with Work equal to mass flow times change in enthalpy. Then remember that mass floe rate is (density*Area*Velocity) and that enthalpy is (cp*change in temperature).

Density is 0.075, cp is 0.24 and the rest is converting to inH2O

 

[trashcanman]

15 yard penalty for illegal use of units! Loss of down.

You cannot get Deg. F by dividing static pressure by a pure number (fan efficiency). Temperature rise (english units) = BTUH/(cfm*1.08).
Remember Heat rate = mass flow x Heat Capacity x Change in temperature.
Then, change in temp = Heat rate/(mass flow x heat capacity).

 

[DRWeig]

With that you will be left with Work equal to mass flow times change in enthalpy

Don't forget that it's heat of compression plus paddlewheel work being done (fan inefficiency). Throw it together, consolidate the constants, and you're almost there.

Best to you,

Goober Dave

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