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re: psych chart challenge: true or false?
29 aug 2002
gary r. lloyd  wrote:

>>>assuming a thermostat setpoint between 70f and 80f, in order to
>>>achieve 50% rh @ setpoint temperature, the wet bulb temperature
>>>leaving the evaporator coil must drop to at least 20f below the
>>>thermostat setpoint temperature.

>>the approximate vapor pressure of 70 f air at 50% rh
>>pa = 0.5exp(17.863-9621/(460+70)) = 0.3742 "hg, and
>>twb(r) = 9621/(22.47-ln(460+70+100pa-twb(r)) = 518 r,
>>after some iteration, a la bowen, ie 58 f, so it's false. 
 
>i have no idea what any of this means, and it seems to contradict my
>psych chart. :)

the first part

  >>the approximate vapor pressure of 70 f air at 50% rh
  >>pa = 0.5exp(17.863-9621/(460+70)) = 0.3742 "hg, and

is a clausius-clapeyron approximation to find the vapor pressure of 70 f
air at 50% rh. handy if you have a $5.99 casio fx-260 scientific calculator
but no psych tables... "exp" is "e to the x", the inverse of the natural
log ("ln") function. this is a simpler version of equation 4 on page 6.11
of the 1993 ashrae hof. 

table 3 of the 1993 ashrae hof says 70 f air at 100% rh has a vapor
pressure of 0.73964 "hg, and 50% of that is 0.36982. let's use that.

the second part

  >>twb(r) = 9621/(22.47-ln(460+70+100pa-twb(r)) = 518 r,
  >>after some iteration, a la bowen, ie 58 f, so it's false. 

is a way to find the approx. wet bulb temp with a calculator vs tables
and charts using bowen's 1926 equation. it says the ratio of evaporative 
to convective heat loss for a pond at temp tw (f) in ta air with vapor
pressure pa is 100(pw-pa)/(tw-ta), independent of windspeed, where pw
and pa are vapor pressures in "hg. at the wet bulb temp, this ratio is
minus 1, ie 100(pw-pa) = ta-tw, which (combined with the clausius approx)
leads to this equation for twb(r), in absolute (rankine) temperature: 

  >>twb(r) = 9621/(22.47-ln(460+70+100pa-twb(r)) =... 518 r,

    twb(r) = 9621/(22.47-ln(566.98-twb(r)). 

start with a guess: say twb = 60 f, ie 520 r. plug it in on the right:

    twb(r) = 9621/(22.47-ln(566.98-520) = 516.7

and we get 516.7 r (56.7 f). plug that in again on the right, and

    twb(r) = 9621/(22.47-ln(566.98-516.7) = 518.6.

repeating a few more times makes 517.5, 518.1, 517.8, and 518.0, so
the wet bulb temp is approximately 518-460 = 58 f.

it looks like the hof psych chart says 70 f air at 50% rh has twb = 59.4 f.
the hof has some more complicated iterative formulas to find this.

nick




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