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About Dampers : Bernoulli’s Equation

Bernoulli’s Equation

In order to derive a leakage and pressure drop formula it is necessary to start with Bernoulli's Equation of Energy Balance.

 

 

Where:

P1

= Upstream Pressure, PSIA

r1

= Upstream Gas Density, Lb/Ft

V1

= Upstream Velocity, Ft/Sec

Z1

= Upstream Altitude, Ft

g

= Gravity, 32.2 Ft/Sec2

P2

= Downstream Pressure, PSIA

r2

= Downstream Gas Density, Lb/Ft

V2

= Downstream Velocity, Ft/Sec

Z2

= Downstream Altitude, Ft

Since there is no change in altitude between the upstream and downstream sides of the damper, Z1 = Z2 and drops out of the equation.  Normally gas is considered to be a compressible fluid, but if DP is less than .1P1, the gas may be considered an incompressible fluid.  Assuming the downstream pressure to be atmospheric and that all pressures in the equation are in terms of absolute pressure, a check may be made to insure that standards are within the criteria specified.           

DP = P1A - P2A = .1P1A

(P1 + 14.7) - (P2 + 14.7) = .1 (P1 + 14.7)

(P1 + 14.7) - 14.7 = .1 (P1 + 14.7)

P1  = .1 (P1 + 14.7)

P1  = .1 P1  + 1.47

.9 P1  = 1.47

P1 = 1.47/.9 = 1.6333 PSIG

= 45.2 in. WC

Almost all of our work is less than 45.2 in. WC and therefore, it can be safe to assume the gas to be incompressible.  Once the gas is assumed incompressible, the density remains the same between the upstream and downstream sides, (r1 = r2 = r).  Dropping the Z’s and combining the r’s, Bernoulli’s Equation becomes:

Rearranging generates:

Because this equation describes flow through an orifice, the downstream velocity (V2) will be much larger than the upstream velocity (V1).  In this case V22 - V12 = V22 and, therefore, V1 can be assumed to equal zero.  Further reducing of the equation yields:

    and then  

If  DP = P1 - P2, then:

Where:

DP

= Differential Pressure, PSIA

r

= Gas Density, Lb/Ft

g

= Gravity, 32.2 Ft/Sec2A

V2

= Downstream Pressure, PSI

This equation gives the velocity of a gas through an orifice.

A common form of the velocity formula in leakage calculation is as follows:                                               

Where:                

V is given in ft./min.

DP is given in WG

r is given in lb./ft.3

The number 1097 is gravity constant and the conversion of other variables to a more common unit as follows: 

 

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