Air Receiver - Doubt on SCFM & CFM

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Air compressor is feeding an air drier to bring the moisture contents down to -40degC. Dried air is then stored in an air receiver before it is distributed to dried air users. An air receiver is designed to provides sufficient dried air to users in the events of air compressor or air drier trip.

A receiver with volume of V₀ is charged with dried air (MW_air) at P₀ & T₀. A constant air demands of Q_S scfm shall be supplied for a period of t minutes prior to plant shutdown before it internal pressure reach P₁ . Following is the schematic of a typical air receiver with it control valve.

As per many articles, the following equation has been common used to estimate the Air Receiver size:

V₀ = Q x P_a / (P₁ - P₂) x t

where,
V₀ = Receiver volume (cu ft)
t = time for receiver to go from maximum pressure (P₁) to minimum pressure (P₂), (minutes)
Q = Free air flow (scfm)
P_a = atmospheric pressure (psia)
P₁ = maximum pressure (psig)
P₂ = minimum pressure (psig)

Source :
i) Compressed Air Receivers, Engineering Toolbox
ii) Air Receiver Volume Calculations For Short Term Demands, Aeris Corporation
iii) Sizing the Air Receiver, Air Technologies
iv) Sizing Air Receiver, Kaeser Compressors

If we carefully examine the unit used for Free air flow, Q, source (i)-(iii) are using SCFM. However, source (iv) is using CFM (or ACFM). There was a discussion if the SCFM or CFM shall be used.

An effort has been put in to derive the equations in order to clear this doubt.

Earlier post "Relate Normal to Actual Volumetric Flow" has derived a simple relationship.

Q₂/Q₁= z₂/z₁ x P₁/P₂ x T₂/T₁ [Eq.1]

1 & 2 being the condition 1 and 2.

From Universal Gas Law,

P₀V₀ = z₀.(M₀/MW).R.T₀ [Eq.2]

At any time interval-i,
Mass of air in receiver,

M_i = (P_i.V_i.MW)/(z_i.R.T_i) [Eq.3]

Density of air,

M_i/V_i = (P_i.MW)/(z_i.R.T_i) [Eq.4]

At i=0,
initial condition,

M₀ = P₀/(z₀.T₀).(V₀.MW/R) [Eq.5-0a]

At i=1,
Air being used for a period of dt₁, there is no physical volume change, V₀ =V₁=V₂ =V₃...

M₁ = P₁/(z₁.T₁).(V₀.MW/R) [Eq.5-1a]

Mass change,

==> dM₁ = M₀ - M₁

dM₁ = [P₀/(z₀.T₀) - P₁/(z₁.T₁)].(V₀.MW/R) [Eq.5-1b]

Mass change base on flow Q₁

From [Eq. 1], make first condition as actual volumetric flow (1) and second condition as Standard conditions,

==> Q₁/Q_s= z₁/z_s x P_s/P₁ x T₁/T_s
==> Q₁ = Q_s x z₁/z_s x P_s/P₁ x T₁/T_s

Pressure drop from P₀ to P₁ for a period dt₁,

Mass of air being removed from air receiver,

==> dM₁ = Q₁ x M₁/V₁ x dt₁
==> dM₁ = Q_s x z₁/z_s x P_s/P₁ x T₁/T_s x (P₁.MW)/(z₁.R.T₁) x dt₁

dM₁ = Q_s x P_s/T_s x MW/(z_s.R) x dt₁ [Eq.5-1c]

At i=2,

M₂ = P₂/(z₂.T₂).(V₀.MW/R) [Eq.5-2a]
dM₂ = [P₁/(z₁.T₁) - P₂/(z₂.T₂)].(V₀.MW/R) [Eq.5-2b]
dM₂ = Q_s x P_s/T_s x MW/(z_s.R) x dt₂ [Eq.5-1c]

.

.
.

At i=n,

M_n = P_n/(z_n.T_n).(V₀.MW/R) [Eq.5-na]
dM_n = [P_n-1/(z_n-1.T_n-1) - P_n/(z_n.T_n)].(V₀.MW/R) [Eq.5-nb]
dM_n = Q_s x P_s/T_s x MW/(z_s.R) x dt_n [Eq.5-nc]

At i=F,

M_F = P_F/(z_F.T_F).(V₀.MW/R) [Eq.5-Fa]
dM_F = [P_F-1/(z_F-1.T_F-1) - P_F/(z_F.T_F)].(V₀.MW/R) [Eq.5-Fb]
dM_F = Q_s x P_s/T_s x MW/(z_s.R) x dt_F [Eq.5-Fc]

From i=0 to i=F,
With [Eq.5-xb]
==> dM_0-F = [P₀/(z₀.T₀) - P_F/(z_F.T_F)].(V₀.MW/R) [Eq.6]

With [Eq.5-xc],
==> dM_0-F = Q_s x P_s/T_s x MW/(z_s.R) x dt₁
+ Q_s x P_s/T_s x MW/(z_s.R) x dt₂
+ Q_s x P_s/T_s x MW/(z_s.R) x dt_3
.
.
.
+ Q_s x P_s/T_s x MW/(z_s.R) x dt_F
==> dM_0-F = Q_s x P_s/T_s x MW/(z_s.R) x [dt₁+dt₂+dt₃+...+dt_F]
==> dM_0-F = Q_s x P_s/T_s x MW/(z_s.R) x dt_Total [Eq.7]

[Eq.6]=[Eq.7],
==> [P₀/(z₀.T₀) - P_F/(z_F.T_F)].(V₀.MW/R) = Q_s x P_s/T_s x MW/(z_s.R) x dt_Total
==> [P₀/(z₀.T₀) - P_F/(z_F.T_F)].(V₀) = Q_s x P_s/(T_s.z_s) x dt_Total

As physical volume has no change, V₀ =V₁
Assume infinite heat transfer with ambient which lead to isothermal operation, T₀ =T_F
Assume minimal changes in z₀, z₀=z_F=z_s=1 at ideal condition,

V₀ = Q_s x P_s / (P₀ - P_F).(T₀ / T_s) x dt_Total

If T_s =T₀:

V₀ = Q_s x P_s / (P₀ - P_F) x dt_Total [Eq.8]

From [Eq.1],
==> Q₀ x P₀ / T₀ = Q_s x P_s / T_s
==> Q₀ x P₀ = Q_s x P_s [Eq.9]


From [Eq.7] and [Eq.8],
==> V₀ = Q_s x P_s / (P₀ - P_F) x dt_Total

V₀ = Q₀ x P₀ / (P₀ - P_F) x dt_Total [Eq.10]

Concluding remarks
To size a receiver, two equations have been derived base on flow unit SCFM and CFM. Both equations can be used to size an air receiver. However, correct unit shall be used for dedicated equation. Equation [8] shall be used with SCFM while [Eq.10] shall be used with CFM.