### Wednesday, August 13, 2008

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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 V

_{0}is charged with dried air (MW

_{air}) at P

_{0}& T

_{0}. 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 P

_{1}. 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:

V

_{0}= Q x P_{a}/ (P_{1}- P_{2}) x twhere,

V

_{0}= Receiver volume (cu ft)

t = time for receiver to go from maximum pressure (P

_{1}) to minimum pressure (P

_{2}), (minutes)

Q = Free air flow (scfm)

P

_{a}= atmospheric pressure (psia)

P

_{1}= maximum pressure (psig)

P

_{2}= 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

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.

Q

_{2}/Q_{1}= z_{2}/z_{1}x P_{1}/P_{2}x T_{2}/T_{1}[Eq.1]1 & 2 being the condition 1 and 2.

From Universal Gas Law,

P

_{0}V_{0}= z_{0}.(M_{0}/MW).R.T_{0}[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,

M

_{0}= P_{0}/(z_{0}.T_{0}).(V_{0}.MW/R) [Eq.5-0a]At i=1,

Air being used for a period of dt

_{1}, there is no physical volume change, V

_{0}=V

_{1}=V

_{2}=V

_{3}...

M

_{1}= P

_{1}/(z

_{1}.T

_{1}).(V

_{0}.MW/R) [Eq.5-1a]

Mass change,

==> dM

_{1}= M_{0}- M_{1}dM

_{1}= [P_{0}/(z_{0}.T_{0}) - P_{1}/(z_{1}.T_{1})].(V_{0}.MW/R) [Eq.5-1b]Mass change base on flow Q

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

==> Q

Pressure drop from P

Mass of air being removed from air receiver,

==> dM

==> dM

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

==> Q

_{1}/Q_{s}= z_{1}/z_{s}x P_{s}/P_{1}x T_{1}/T_{s}==> Q_{1}_{ }= Q_{s}x z_{1}/z_{s}x P_{s}/P_{1}x T_{1}/T_{s}Pressure drop from P

_{0}to P_{1}for a period dt_{1},Mass of air being removed from air receiver,

==> dM

_{1}= Q_{1}x M_{1}/V_{1}x dt_{1}==> dM

_{1}= Q_{s}x z_{1}/z_{s}x P_{s}/P_{1}x T_{1}/T_{s}x (P_{1}.MW)/(z_{1}.R.T_{1}) x dt_{1}dM

_{1}= Q_{s}x P_{s}/T_{s}x MW/(z_{s}.R) x dt_{1}[Eq.5-1c]M

dM

dM

_{2}= P_{2}/(z_{2}.T_{2}).(V_{0}.MW/R) [Eq.5-2a]dM

_{2}= [P_{1}/(z_{1}.T_{1}) - P_{2}/(z_{2}.T_{2})].(V_{0}.MW/R) [Eq.5-2b]dM

_{2}= Q_{s}x P_{s}/T_{s}x MW/(z_{s}.R) x dt_{2}[Eq.5-1c].

.

.

At i=n,.

M

dM

dM

_{n}= P_{n}/(z_{n}.T_{n}).(V_{0}.MW/R) [Eq.5-na]dM

_{n}= [P_{n-1}/(z_{n-1}.T_{n-1}) - P_{n}/(z_{n}.T_{n})].(V_{0}.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

dM

dM

_{F}= P_{F}/(z_{F}.T_{F}).(V_{0}.MW/R) [Eq.5-Fa]dM

_{F}= [P_{F-1}/(z_{F-1}.T_{F-1}) - P_{F}/(z_{F}.T_{F})].(V_{0}.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}= [P

_{0}/(z

_{0}.T

_{0}) - P

_{F}/(z

_{F}.T

_{F})].(V

_{0}.MW/R) [Eq.6]

With [Eq.5-xc],

==> dM

_{0-F}= Q

_{s}x P

_{s}/T

_{s}x MW/(z

_{s}.R) x dt

_{1}

+ Q

_{s}x P

_{s}/T

_{s}x MW/(z

_{s}.R) x dt

_{2}+ 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 [dt

_{1}+dt

_{2}+dt

_{3}+...+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],

==> [P

_{0}/(z

_{0}.T

_{0}) - P

_{F}/(z

_{F}.T

_{F})].(V

_{0}.MW/R) = Q

_{s}x P

_{s}/T

_{s}x MW/(z

_{s}.R) x dt

_{Total}==> [P

_{0}/(z

_{0}.T

_{0}) - P

_{F}/(z

_{F}.T

_{F})].(V

_{0}) = Q

_{s}x P

_{s}/(T

_{s}.z

_{s}) x dt

_{Total}

As physical volume has no change, V

_{0}=V

_{1}

Assume infinite heat transfer with ambient which lead to isothermal operation, T

_{0}=T

_{F}

Assume minimal changes in z

_{0}, z

_{0}=z

_{F}=z

_{s}=1 at ideal condition,

V

_{0}= Q_{s}x P_{s}/ (P_{0}- P_{F}).(T_{0}/ T_{s}) x dt_{Total}If T

_{s}=T

_{0}:

V

_{0}= Q_{s}x P_{s}/ (P_{0}- P_{F}) x dt_{Total}[Eq.8]From [Eq.1],

==> Q

_{0}

_{ }x P

_{0}/ T

_{0}= Q

_{s}x P

_{s }/ T

_{s}

==> Q

_{0}

_{ }x P

_{0}= Q

_{s}x P

_{s}[Eq.9]

From [Eq.7] and [Eq.8],

==> V

_{0}= Q

_{s}x P

_{s}/ (P

_{0}- P

_{F}) x dt

_{Total}

V

_{0}= Q_{0}_{ }x P_{0}/ (P_{0}- 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.

Labels: Air

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