6629aa12-89dc-4544-9971-6ab8b17aadbe 31 4 5 false false 0 decimal
&[DATE] &[TIME] - &[FILENAME]
&[PAGENUM] / &[COUNT]
 0.99.7610 0.99.7610

Derive HVAC Constants using SMATH

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

 <p fontName="Arial">Water</p>

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 1.9 Reciprocal 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 @ 60F https://webbook.nist.gov/chemistry/fluid/

c.pHIw 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

Volumetric Heat Constant @ HI Temp

VHC.HIw ρ.HIw c.pHIw * : 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)

 BTU hr / min * gal Δ°F * / 500.8643

Water Volumetric Heat Constant used in empirical formulas @ HI Temp Generic Text Book Equation uses 500

VHC.HIw BTU hr / min * gal Δ°F * / 500.8643

Technically this is not to "scientific "standards....
Relative density, or specific gravity,is the ratio of the density (mass of a unit volume) of a substance to the density of a given reference material. Specific gravity for liquids is nearly always measured with respect to water [at its densest (at 4 °C or 39.2 °F)] The term "relative density" is often preferred in scientific usage.
]]> 0.9994 BTU hr / min * gal Δ°F * / 500.5518
As can be seen, the differenctce, for water, is nominal
]]>

The empirical equation in textbooks is:

500 BTU hr / min * gal Δ°F * / * Q * gal min / * ΔT * Δ°F * Q ΔT * BTU hr / * 500 <p fontName="Arial">Air</p>

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

2017 ASHRAE Fundamentals 1.4 Reciprocal for air @ 68F

ρ.AMCAa 1 13.2977 / lbm ft 3 ^ / * : lbm ft 3 ^ / 0.0752

AMCA 210 3.2.8 CAGI and ISO use this too

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.pAMCAa .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

Volumetric Heat Constant @ AMCA Temp

VHC.AMCAa ρ.AMCAa c.pAMCAa * : BTU ft 3 ^ Δ°F * / 0.0181

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

BTU hr / min * ft 3 ^ Δ°F * / 1.0852

Air Volumetric Heat Constant used in empirical formulas @ AMCA Temp Generic Text Book Equation uses 1.08 or 1.085

VHC.AMCAa BTU hr / min * ft 3 ^ Δ°F * / 1.0852

The empirical equation in textbooks is:

1.085 BTU hr / min * ft 3 ^ Δ°F * / * Q * ft 3 ^ min / * ΔT * Δ°F * Q ΔT * BTU hr / * 1.085
Relative density, or specific gravity,is the ratio of the density (mass of a unit volume) of a substance to the density of a given reference material. For gases, the reference is air at room temperature (20 °C or 68 °F). For Air, ADR is the ratio that is used as it accounts for pressure and temperature. NOTE, CAGI and ISO use 0% relative humidity (RH); whereas AMCA uses 36% RH.
]]> <p fontName="Arial">Velocity Pressure</p>

Fan Pressure and Velocity Relationship

See Aaon "Value in Air"

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

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

 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

 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 ^ * ≡

Units are slightly off. Lets pull out the velocity component units, convert to seconds, and expand the squared

ΔP ρ lbm ft 3 ^ / * 5.1986 lbm ft 2 ^ / inWC / * / 2 32.174 * ft s 2 ^ / * / Vel 2 ^ ( * ft min / ( 2 ^ * : ρ Vel 2 ^ ( * 2 32.174 * 5.1986 * / inWC * 0.0003

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

 s 2 ^ min 2 ^ / 3600

Notice the 1 in lieu of .0003 from above! All variables are now strings as unts are canecelled out

ΔP ρ Vel 2 ^ ( * 2 32.174 * 5.1986 * 3600 * / inWC * 1

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

 1097.3925

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

Generic equation has been seen as 1096, 1096.7, or 1097. Typically due to round off error e.g. 5.2 in lieu of 5.1986

VP 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

 4000.321 4001.753 ft 3 ^ lbm / 13.2977

Eliminate units and combine numeric numbers

Eliminate units and combine numeric numbers

 3.6466 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

 4001.7515 4009.3235

Empirical formulas @ AMCA Temp Generic Text Book Equation uses 4005.

VP.AMCAa Vel 1097.3925 3.6466 * / ( 2 ^ ≡ ( Vel 4001.7515 / ( 2 ^ inWC * ≡

Empirical formulas @ traditional 70F Temp Generic Text Book Equation uses 4005. Typically due to round off error e.g.1096, 1096.7,or 1097

VP.70 Vel 1097.3925 3.6535 * / ( 2 ^ ≡ ( Vel 4009.3235 / ( 2 ^ inWC * ≡ 4004.236 4006.7934 4007.8895