<content> Water </content>

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Derive HVAC Constants using SMATH

Q ṁ c.p ΔT * * ≡

This is the bounding equation that is used to derive the empirical equations for air and water

<content> Water </content>

Assuming standard conditons per the Hydraulic institute (HI)

HI 3.1-3.5 (2015)3.2.38.4 Temp.HI 60 °F * :

2017 ASHRAE Fundamentals 1.9Reciprocal for water @ 60F ρ.HIw 1 .01603 / lbm ft 3^/ * : lbm ft 3^/ 62.383

ṁ is a mass flow RATE. US, Gallons per minute GPM

Heat capacity of water at constant pressure @ 60F

Specific Heat of Water @ 60Fhttps://webbook.nist.gov/chemistry/fluid/ c.pw 1.001 BTU lbm Δ°F * / * :

ASME STEAM TABLES...

BTU is defined as the amount of work needed to raise the temperature of one pound of water
by one degree Fahrenheit

Water Volumetric Heat Constant @ HI Temp Water.vhcHI ρ.HIw c.pw * : BTU ft 3^Δ°F * / 62.4454

The heat transfer equation incorporates a flow rate, water is GPM or gal/min which converts to ft^3 per minute which introduces a time factor which is picked up in the heating value as heat flow rate. As flow rates in IP systems we need to multiply by 1
* Flow RATE multiply units of 60 min/hr and
* CONVERT gallons to ft^3 (.13368)

Water.vhcHI Water.vhcHI 60 * min hr / * : BTU hr / min * gal Δ°F * / 500.8643

Water Volumetric Heat Constant  used inempirical formulas @ HI TempGeneric  Text Book Equation uses 500 Water.vhcHI BTU hr / min * gal Δ°F * / 500.8643

The empirical equation in textbooks is:

Why is this not showing 500? 500 BTU hr / min * gal Δ°F * / * Q * gal min / * ΔT * Δ°F * BTU hr / 673418623255747000 Q * ΔT * 1346837246511500 /

BTU hr / 500 flow * ΔT * ≡

500 BTU hr / min * gal Δ°F * / * Q * gal min / * ΔT * Δ°F * #

500 BTU hr / min * gal Δ°F * / * Q * gal min / * ΔT * Δ°F * Q ΔT * BTU hr / * 500

Assuming standard conditons per the Air Movement and Control Association (AMCA)

2017 ASHRAE Fundamentals 1.4Reciprocal for air @ 68F ρ.AMCAa 1 13.2977 / lbm ft 3^/ * : lbm ft 3^/ 0.0752

AMCA 210 3.2.8 Temp.AMCA 68 °F * :

ṁ is a mass flow RATE. US, Cubic Feet per second CFM

Heat capacity of air at constant pressure @ 68F

2017 ASHRAE F 30.65 - Refrigerant 729 (Air) Properties of Gas at 14.696 psia (one standard atmosphere)F18.15 Eq 8 c.pa .2405 BTU lbm Δ°F * / * :

BTU is defined as the amount of work needed to raise the temperature of one pound of water
by one degree Fahrenheit

Air Volumetric Heat Constant @ AMCA Temp Air.vhcAMCA ρ.AMCAa c.pa * : BTU ft 3^Δ°F * / 0.0181

Air.vhcSTD Air.vhcAMCA 60 * min hr / * : BTU hr / min * ft 3^Δ°F * / 1.0852

Air Volumetric Heat Constant  used inempirical formulas @ AMCA TempGeneric  Text Book Equation uses 1.08 or 1.085 Air.vhcAMCA BTU hr / min * ft 3^Δ°F * / 1.0852

The empirical equation in textbooks is:

Why is this not showing 1.085? 1.085 BTU hr / min * gal Δ°F * / * Q * ft 3^min / * ΔT * Δ°F * BTU hr / 1366427540114100 Q * ΔT * 168354655813937 /

BTU hr / 1.085 flow * ΔT * ≡

<content> Velocity Pressure </content>

Fan Pressure and Velocity Relationship

See Aaon value in air

The relationship between pressure and velocity of the air stream can be shown using
Bernoulli’s equation where:

ΔP ρ Vel 2^(* 2 g * / ≡

let's look at units, ignoring numbers, using vel. units of ft/min...

lbm ft 3^/ ft min / (2^* ft s 2^/ / lbm in 2^/ 0

You'll notice this is similar to PSI, but PSI is lbf.
HVAC calculations often express pressure in terms of inches of water gauge (inwg). The
previous equation must be further developed so that the solution is given in inches of water
gauge assuming standard conditions

ρ.HIw ft 12 inWC * / * lbm ft 2^/ inWC / 5.1986

Assume 1 foot height column: NOTE inch unit
and creating a new unit from length; essentially multiply by 1

Check unit cancelation ΔP ρ lbm ft 3^/ * 5.1986 lbm ft 2^/ inWC / * / 2 g.e * / Vel 2^(* ft min / (2^* ≡

Why is this not showing a numeric value? ΔP ρ lbm ft 3^/ * 5.1986 lbm ft 2^/ inWC / * / 2 32.174 * ft s 2^/ * / Vel 2^(* ft min / (2^* : 25566638599935 ρ * inWC * Vel 2^* 30789142158182200000 /

ΔP ρ 5.2 / Vel 2^(* 2 32.174 * / ≡

In order to cancel out units by hand, convert s^2 to min^2 since IP measures in FPM. This is multiplied in the denominator

60 s min / * (2^s 2^min 2^/ 3600

ΔP ρ Vel 2^(* 2 32.174 * 5.1986 * 3600 * / inWC * ≡

As the units cancel, we will pay attention only to the numeric values, in the denominator

Units cancel out appropriately above.  Lets look at the denominator 2 32.174 * 5.1986 * 3600 * 1204270.2461

This is an unweildly number. The equation has a numerator number that is squared. let's combine these two items

ΔP ρ Vel 2^(* 1204270.2461 / ≡

1204270.2461 sqrt 1097.3925

Take the square root of the denominator and bring it under Vel^2

Generic equation has been seen as 1096 or 1097.Typically due to round off error e.g. 5.2 in lieu of 5.1986 ΔP ρ Vel 1097.3925 / (2^* inWC * ≡

This is where we will "deviate" a bit. We will do 2 calculations to show the differences
1) AMCA calculates air density at 68F
2) HVAC typically uses 70F

10972^ρ.AMCAa ft 3^lbm / * / sqrt 4000.321

1204270.2461 ρ.AMCAa ft 3^lbm / * / sqrt 4001.753

υ.AMCAa 1 ρ.AMCAa / : ft 3^lbm / 13.2977

υ.70F 13.3482 ft 3^lbm / * :

Eliminate units and combine numeric numbers

υ.AMCAa υ.AMCAa lbm ft 3^/ * sqrt : 3.6466

υ.70F υ.70F lbm ft 3^/ * sqrt : 3.6535

Although NOT an unweildly number, we want to combine values. The equation has a number that is squared. Let's combine these two items

Take the square root of the denominator and bring it under Vel^2

1097.3925 3.6466 * 4001.7515

1097.3925 3.6535 * 4009.3235

Empirical formulas @ AMCA TempGeneric  Text Book Equation uses 4005. ΔP.AMCA Vel 1097.3925 3.6466 * / (2^≡ (Vel 4001.7515 / (2^inWC * ≡

Empirical formulas @ traditional 70F TempGeneric  Text Book Equation uses 4005.Typically due to round off error e.g.1096 or 1097 ΔP Vel 1097.3925 3.6535 * / (2^≡ (Vel 4009.3235 / (2^inWC * ≡

1096 3.6535 * 4004.236

1097 3.6535 * 4007.8895