Compressed Air System Sizing: Method

Compressed Air System Sizing: Demand, Leaks, Compressors, Receivers, Piping, Moisture, Energy

Scope

The tool follows a plant compressed air system from the tools to the compressor room in the order a designer or an auditor works it: how much free air the plant uses on average and at its peak, how much of the supply is lost to leaks, how much standard air the compressors on the list deliver at the site and how they share the load, how large a receiver carries the peak and how fast it recovers, whether the header and the branches hold the pressure and the velocity within the limits entered, how much water the air carries in and where it condenses, and what the whole system costs to run for a year. It prints results and plain comparisons (a velocity against its limit, a drop against its target, a set point against the pressure required); there is no factor of safety and no margin box, because compressed air is sized to capacity, not to failure. The methods are those of the U.S. Department of Energy's Improving Compressed Air System Performance: A Sourcebook for Industry and its tip sheets, written with the Compressed Air Challenge, of the Unified Facilities Criteria for compressed air (UFC 3-420-02, the successor of Army TM 5-810-4), of Harris's 1917 textbook for the empirical pipe formula, and of the site's own libraries for the Colebrook friction factor and the psychrometric relations.

Conventions and units

Flows are free air at the standard conditions entered, dry (scfm in USCS, L/s in SI); the default standard is 14.696 lbf/in² absolute and 68 °F, the ASME and CAGI reference, and ISO 1217's 14.5 lbf/in² (100 kPa) and 20 °C can be entered instead. Pressures entered are gauge; the barometric pressure follows the standard atmosphere at the entered altitude and is added where an absolute pressure is needed. Temperatures are plain numbers, °F in USCS and °C in SI, and are not converted by the unit-system switch. Lengths, diameters, flows, pressures, volumes, velocities and motor powers go through the site's unit dictionary; the specific power of a compressor is entered as the industry's kW per 100 cfm of rated capacity in both systems, and money is a plain number in the plant's currency.

Demand

Each tool or process is entered with the free air it consumes while it runs, the number of units installed, its duty cycle and the number of units running at the plant's peak, with the pressure it needs at its inlet. The average demand is the sum of units × flow × duty, the peak the sum of the peak units × flow, and the connected load every unit at once. The Sourcebook (Fact Sheet 1) sets the rule the tool follows: the capacity of the system is built on the sum of the averages, the short peaks being met from storage, not on the sum of the maxima. The load factor (average over peak) says how much the demand swings and so how much the control strategy and the storage matter; the diversity factor (peak over connected) says how many of the tools are ever used together. The highest required pressure is the start of the pressure budget.

Leaks

The leak allowance can be entered as a share of the total supply, which the Sourcebook brackets at less than 5 to 10 percent for a well-maintained system and 20 to 30 percent for a poorly maintained one, or estimated from either of its two tests (Fact Sheet 3). With load / unload or start / stop controls and every end use shut off, the compressor cycles on the leaks alone and the leakage as a fraction of its capacity is T / (T + t), the loaded time over the cycle. With other controls, the system is pumped to its working pressure, the compressors stopped, and the time T for the pressure to fall from p₁ to about half of it, p₂, is measured: the leakage of free air is V (p₁ − p₂) / (T p_a) × 1.25, V being the system volume and the 1.25 the Sourcebook's correction for the leakage falling with the pressure during the test. Since leakage through an orifice is choked at any line pressure above about 1 atmosphere gauge, the tool can also turn a count of leaks of an equivalent diameter into a flow with the sonic-orifice equation, ṁ = C_d A p₀ √(γ / (R T₀)) (2/(γ+1))^{(γ+1)/(2(γ−1))}; the DOE tip sheet's leakage table is this flow with C_d = 1 (the tool reproduces it within 3.5 percent), multiplied by 0.97 for a well-rounded and 0.61 for a sharp-edged orifice.

Compressor capacity at the site

A positive-displacement compressor delivers its rated inlet volume whatever the air it draws in; the standard air that volume represents is less at altitude, in warm weather and in humid air. UFC 3-420-02 Appendix A gives the conversion the tool applies, Q_act = Q_std (p_std / (p_act − φ p_vs)) (T_act / T_std): the barometric pressure less the water vapour's share, and the absolute temperatures. The compressors are listed in loading order. At the average supply (demand plus leaks) the first ones run fully loaded as long as the remaining demand exceeds their capacity, the first one not needed in full trims, and the rest are off. The trim compressor's input follows a straight line between the power it draws at zero output and its full-load power, P = P_fl [a + (1 − a) f], with the zero-output fraction a by control type from the Sourcebook's Fact Sheet 5: about 0.2 to 0.3 for a load / unload rotary screw (0.1 to 0.15 for a reciprocating machine), about 0.7 for inlet modulation, 0.1 to 0.15 for a variable-speed drive, zero for start / stop; an entered value overrides it. The Sourcebook's own caution applies: a load / unload compressor realises its unloaded power only when it stays unloaded long enough for its sump to blow down, which takes storage (its Figure 2.5). The full-load input is the entered specific power times the rated capacity, which is how the industry's data sheets state it; the small change of input with altitude is not applied. The system specific power, kW per 100 cfm of standard air delivered at the average supply, is the figure an audit tracks.

Pressure budget

The compressor's load set point must cover the highest point-of-use pressure plus the drops on the way: the branch, the header and the treatment equipment (aftercooler, separator, dryer and filters, usually 3 to 5 lbf/in² for a dryer and more as filters foul). The bottom of the control band, the minimum header pressure, is checked against the point-of-use need plus the piping drops alone, since the treatment sits upstream of the header. Every 2 lbf/in² of set point above the need costs about 1 percent of the compressor energy at full output (Sourcebook Fact Sheet 4), and more through the unregulated uses, so the budget is kept tight and the drops, not the set point, are what to attack.

Receiver

The Sourcebook's receiver equation (Fact Sheet 6) is V = T C p_a / (p₁ − p₂): a receiver of volume V supplies an event of C cfm of free air for T minutes while its pressure falls from p₁ to p₂, p_a being the atmospheric pressure the free air is referred to. With the compressors running during the event C is the net draw, the event demand less the installed capacity, and the same equation turned round gives the time to pump the receiver back up with the surplus over the average supply. The equation is the isothermal ideal-gas mass balance, d(pV)/dt = p_a (S − C): the free air stored between two pressures is V (p₁ − p₂) / p_a, which is why a 500 gal receiver at a 100 lbf/in² differential holds as much as a 5,000 gal receiver at 10 lbf/in², and why a wider control band or a high-pressure offline store does the work of a larger vessel. The tool sizes the receiver for the event from the unload set point to the minimum header pressure, draws the pressure history of the installed receiver through the event and its recovery, and prints the old 1 to 3 gal per cfm rule only for comparison, since the Sourcebook no longer regards it as good practice. Primary storage also lets a load / unload compressor run unloaded long enough to save energy; secondary storage next to a large intermittent user, with a check valve and a metering valve, turns its peak into a small average (Tip Sheets 8 and 9).

Header and branch piping

The main calculation is the isothermal Darcy–Weisbach solution for the compressed gas. The mass flux G = ρ_std Q_std / A is constant along the pipe, and with ρ = p / (R T) the momentum equation p dp = −(f / (2D)) G² R T dx integrates to p₁² − p₂² = f (L + L_eq) G² R T / D, the kinetic-energy term being negligible at these velocities (Harris showed it so in 1917). The friction factor is Colebrook's at the Reynolds number G D / μ, which is the same at every section, computed by the Pipe Flow tool's routine with the roughness of the material chosen; fittings are entered as an equivalent length charged at the pipe's own friction factor; the air is taken at the inlet temperature. For small drops the result equals the incompressible f (L / D) ρ V² / 2 at the inlet density, which the tool prints beside it. Two empirical formulas of long standing are shown for comparison. Harris's (1917, eq. 28a) is f = 0.1025 l v_a² / (3,600 r d^5.31), the loss in lbf/in² for l feet of pipe of d inches inside diameter carrying v_a cfm of free air at a ratio of compression r; it is his 1910 rational formula with the measured coefficients of his Appendix fitted as 0.1025 d^−0.31, and it is the formula behind the loss-per-1,000-ft tables of the CAGI handbook (the tool reproduces the table's 2-in and 3-in rows within about 2 percent). Fritzsche's (1908), as the Engineering ToolBox prints it, is dp = 7.57 q^1.85 L × 10⁴ / (d⁵ p) in kgf/cm² with q in m³/min, L in m and d in mm. Both fall within about 20 percent of the Darcy–Weisbach value on commercial steel pipe at 100 lbf/in² gauge; the Darcy form is the one to trust for other materials, pressures and temperatures. A loop header fed at one point is treated as two legs, each carrying half the flow over half the perimeter. The velocities are compared with the entered limits, the DOE / ORNL guidance being not more than 20 ft/s in a header and 30 ft/s in distribution piping, and the drop with the entered target rate, the usual 1 lbf/in² per 100 ft; UFC 3-420-02 limits the drop from the compressor to the farthest point of use to 5 percent. For each pipe the tool also solves for the inside diameter that just meets the target drop and the one that just meets the velocity limit; the larger, rounded up to a stock size, is the diameter to specify. Each branch starts from the header's far-end pressure.

Moisture and air treatment

The air the compressor takes in carries water vapour at the inlet humidity ratio; the dry-air mass flow is the inlet volume over the specific volume per kilogram of dry air. Compression does not change the humidity ratio but raises the dew point: the same ratio that is well below saturation at atmospheric pressure is saturated at the line pressure at a much higher temperature, because the vapour pressure is the humidity ratio's share of a larger total. At each stage the air keeps the lesser of the ratio it arrives with and the saturation ratio at that stage's pressure and temperature, and the difference condenses: most of the water at the aftercooler and separator, the rest in the dryer down to its pressure dew point. The saturation pressure is Hyland and Wexler's, the humidity ratio, specific volume and dew point ASHRAE's perfect-gas relations, all from the site's psychrometric library. The tool prints the condensate at each stage per day as a mass and a volume (what the drains must pass, with zero-loss traps so that no air is wasted, Tip Sheet 13), the pressure dew point of the air leaving the aftercooler, and both the pressure dew point of the dried air and the atmospheric dew point of the same air, the figure a dew-point meter reads when the sample expands to atmosphere. The dryer's power is entered per 100 cfm from the Sourcebook's figures for each type (0.79 refrigerated, 0.2 deliquescent, 0.8 heat of compression, 2 to 3 heated desiccant, 3 to 4 membrane) and its pressure drop goes into the budget.

Energy and cost

The compressor input at the average supply, from the loading table, plus the dryer, times the operating hours, is the year's energy; times the rate, its cost. This is the Sourcebook's cost equation (Fact Sheet 12) with the part-load factor of the trim compressor's control, and the DOE tip sheet's formula with bhp × 0.746 / motor efficiency when the specific power is built that way. The cost of the leaks charges the leak flow at the system's specific power, the method of Tip Sheet 3, which uses 18 kW per 100 cfm as the typical figure. The saving of a discharge-pressure reduction follows the Sourcebook rule: about 1 percent of the compressor energy for every 2 lbf/in² at full output, plus the response of the unregulated uses, 2 percent × the unregulated share per 2 lbf/in² (its 0.6 to 1.0 percent for 30 to 50 percent unregulated), a total of 1.6 to 2 percent per 2 lbf/in² in the 100 lbf/in² range.

Reading the results

Read the demand table first: the averages size the compressors, the peaks size the storage, and a diversity factor near 1 means the peak is a real simultaneous load, not a coincidence. If the installed standard capacity is short of the average supply at the site, the shortfall is printed and no receiver will cure it; if only the peak exceeds the capacity, the receiver section says how large a vessel carries it and whether the installed one does. The pressure budget shows whether the set point is spent on the tools or on the drops. In the piping section a red figure is a velocity above its limit or a drop above its target, and the two diameters in the last columns say what to specify. The moisture section gives the drains and the dryer their duty and shows why a 38 °F pressure dew point is −5 °F at atmosphere. The energy section ends with the two numbers an audit opens with: what the leaks cost and what 10 lbf/in² less would save. The default case is an illustrative 100 hp two-compressor shop (two 50 hp rotary screws, 250 cfm each, load / unload, 18 kW per 100 cfm) at 100 lbf/in² gauge with about 400 scfm of average supply; its numbers are checked against the DOE tip-sheet and Sourcebook examples, Harris's worked example and a numpy / scipy script, not against a published design of the same plant.

Limits

Steady average loading with one trim compressor and a straight-line part-load model (no sequencing dynamics, no blow-off of centrifugal machines, no manufacturer's part-load curves); the receiver as a lumped isothermal volume with constant event demand and installed capacity; isothermal pipe flow with the Colebrook friction factor and equivalent lengths (no Mach-number effects, no choked branches, no water in the lines); psychrometrics by the perfect-gas relations with the saturation pressure over water (ice below 32 °F is handled by the library, the enhancement factor is not); energy at the average supply with no load-duration profile beyond the hours entered, no demand charges and no heat recovery; piping drops at the peak flow, as a design should be, while the energy is at the average.