# Control of space pressurization for sealed or tight rooms.

INTRODUCTION

Laboratory ventilation systems typically apply space pressurization to limit the spread of air contaminants from the laboratory room to surrounding areas. In most facilities, this is a straight-forward, routine process. In certain special facilities, the rooms are built to very low leakage specifications. This makes the pressurization control system very sensitive, requiring special control techniques. For a number of reasons, there is a trend toward tighter room envelopes in laboratories. making this a good time to analyze the lab pressurization problem from start to finish.

The aim of this analysis is to identify the factors that affect performance of room pressure control systems, and to show the effects of varying mechanical parameters. Control system analysis methods based on linear or linearized models can provide that kind of understanding. This does not mean the non-linearity of the system is unimportant or negligible; it means there is value in looking past it to gain the benefits of analysis.

MECHANICAL MODEL

The mechanical phenomena that characterize the system are modeled with basic equations from fluid mechanics. The equations are combined into a system and parameters are assigned.

Figure 1 represents the basic mechanical system. To keep the analysis manageable, there is one mechanical supply flow and one mechanical exhaust. Air exchange with surrounding spaces is represented by one infiltrating (or exfiltrating) air flow to one ambient pressure.

[FIGURE 1 OMITTED]

Room air pressure

The pressure within the lab room is affected by the air flow rates in and out of the room. That relationship is approximated by the ideal gas law. Taking the derivative against time, and assuming a constant room air temperature relates the rate of pressure change to the air flow rates.

[P.sub.R] = mRT/V

[P.sup.R] = mRT/V

It is convenient to discuss the system in terms of the various volumetric flow rates the make up the net mass flow rate into the room. To derive the room pressure equations, we neglect variations in air density through the system. This simplifying assumption is clearly not true. The air in the supply duct is denser than air in the room or the exhaust duct. The changes in room pressure that we study are directly related to density variations. However, the objective of the analysis does not depend on including density in the calculations. With the assumption in effect, the pressure rate is:

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The parameter [P.sub.0] is defined as the absolute room pressure at the nominal operating point.

Air flow to and from the room

Supply: The supply system is represented by a high pressure at a reference point in the supply duct, pushing air through a variable flow resistance into the room. Pressure losses along the supply duct upstream and downstream of the supply damper are lumped into one variable resistance parameter [(C.sub.s]) and included in the non-linear function [[(D.sub.s](X.sub.s])) that represents air flow resistance vs. damper stroke.

[Q.sub.S] = ([P.sub.1] - [P.sub.R] [).sup.n] [C.sub.S] [D.sub.S]( [X .sub.S])

The flow resistance parameter is a discharge coefficient in the orifice flow equation that models the supply air flow. It models mechanical sizing effects. The non-linear function takes values between 0 (perfectly sealed damper) and 1.0 (fully open damper.) The non-linearity makes it possible to represent characteristics of any type of damper and authority effects.

The pressure exponent (n) in the flow equation is taken as 0.5, meaning the flow is proportional to the square root of the pressure difference.

A real supply system serves more than one terminal. The individual terminals affect each other by changing the pressure drop in common sections of the duct work. The usual practice of controlling the pressure at a point in the duct system tends to reduce the effect, but it does not eliminate it. It is possible to extend this model to study multiple rooms and their effects but that is not within the scope of this paper.

Exhaust: The exhaust system is modeled the same way. Here, room pressure is the high side of the orifice and the suction pressure is in the exhaust duct.

[Q.sub.E] = ([P.sub.R]-[P.sub.2]).sup.n] [C.sub.E] [[D.sub.E]([X.sub.E])

Infiltration: Air exchange with surrounding spaces is modeled with the orifice flow equation. The parameter ([C.sub.I]) sizes the air leakage path between the room and adjacent spaces. The exponent on the pressure difference is taken here as 0.5 but values closer to 1.0 can be selected to represent more nearly laminar flow.

[Q.sub.I] = ([P.sub.I]-[P.sub.R] ).sup.n] [C.sub.I]

Combined Mechanical Model

Equations for the room air pressure, and the air flows in and out are combined to represent a non-linear system. For purposes of this analysis, we consider the inputs to be the duct pressure and damper position for supply and exhaust, and the pressure in the adjacent space. From these we can calculate the three air flows and the room pressure. Figure 2 illustrates the structure of the model in block diagram form.

[FIGURE 2 OMITTED]

CONTROL SYSTEM

The previous section presents a model that can calculate room pressure and air flows from the input values of ambinet pressure and damper positions. In this section we model the other half of the system so we can calculate damper positions from flows and pressures. This enables us to analyze the closed loop. Simple models for the end devices (sensors and actuators) are combined with models of the control algorithms. Laplace Transform based transfer functions describe the dynamics of the control system components.

The damper motors and filtered air flow sensors are each modeled as first order lags. The damper motor model relates positions (x) to controller output values. (y) The flow sensor model relates measured flow values [(Q.sub.M]) to actual flow values.(Q)

[Q.sub.EM] = [FQ.sub.E] = [Q.sub.E]/( [[tau].sub.F]S + 1)

[Q.sub.SM] = [FQ.sub.S] = [Q.sub.S]/( [[tau].sub.F]S + 1)

[X.sub.E] = [MY.sub.E] = [Y.sub.E]/( [[tau].sub.M]S + 1)

[X.sub.S] = [MY.sub.S] = [Y.sub.S]/( [[tau].sub.M]S + 1)

The mechanical model and the control system model fit together. When they are combined, we can analyze closed loop behavior.

Flow Offset Control

The first control algorithm modeled is the standard flow offset control. PID calculations operate dampers to drive the supply and exhaust flows to their setpoints. The supply flow setpoint is less than the measured exhaust flow by a fixed offset.

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[FIGURE 3 OMITTED]

Pressure Feedback

Pressure feedback is another common control strategy for pressurized rooms. The diagram shows the closed loop with the same mechanical model elements connected to a different control system model. The pressure control loop in the top half of the diagram operates the supply air damper. A flow control loop operates the exhaust damper.

As with the flow tracking system, we manipulate the blocks to eliminate the interconnecting loops before analyzing the system.

LINEARIZED SYSTEM FOR ANALYSIS

To enable control system analysis, this model is linearized at an operating point. The non-linear air flow equations are differentiated against each input and approximated by the first term of a Taylor expansion. To keep the problem simple, the duct pressures are taken as constants; the room pressure, the ambient pressure and the damper positions are taken as variable inputs to the air flow equations. The supply flow equation is linearized as follows:

[Q.sub.S] =[ ([P.sub.1] -[P.sub.R]).sup.n] [C.sub.S] [D.sub.S]( [X.sub.S])

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[Q.sub.S] [approximately equal to] [ ([P.sub.1] -[P.sub.RO]).sup.n] [C.sub.S] [D.sub.S]( [X.sub.SO] )+ n[ ([P.sub.1] -[P.sub.RO]).sup.n-1] [C.sub.S] [D.sub.S]( [X.sub.SO]) ([P.sub.RO] -[P.sub.R]) + [ ([P.sub.1] -[P.sub.RO]).sup.n] [C.sub.S] [N.sub.S]( [X.sub.S] - [X.sub.SO])

The three terms in the expression are the supply flow at nominal operating point, the deviation in supply due to changes in room pressure and the deviation in supply flow due to changes in supply damper position. The value, N that multiplies the deviation in damper position is the local slope of the non-linear damper curve. The damper position, X and the air flow resistance factor D(X) both range from zero to one. The local slope N may take any positive value. It is greater than 1.0 on the steep part of the damper curve, and less on a shallow part of the curve. For analysis purposes, it is not necessary that we identify the slope at any particular point, or determine a particular installed damper curve, as long as we check an appropriate range of values for N, covering steep, average and shallow characteristics.

[FIGURE 4 OMITTED]

For convenience, the linearized air flow equation is rewritten with a more intuitive set of sizing parameters. The discharge coefficient is replaced with values that represent the air flow at the nominal pressure drop. The nominal supply flow is defined as the supply flow, with the damper fully open (D(x)=1) and the room pressure at the nominal operating point.

[Q.sub.SNOM] = [C.sub.S] [([P.sub.1] - [P.sub.RO]).sup.n]

[C.sub.S] = [Q.sub.SNOM]/ [([P.sub.1] - [P.sub.RO]).sup.n]

When this expression is substituted for the discharge coefficient, all the parameters in the linearized air flow equation are values that describe the desired operating point.

[Q.sub.S] [approximately equal to] [Q.sub.SO] + [Q.sub.SNOM] [-n D([X.sub.SO])/ ([P.sub.1] -[P.sub.RO]) ([P.sub.1] -[P.sub.RO]) + [N.sub.S] ( [X.sub.S] - [X.sub.SO])]

[Q.sub.S] [approximately equal to] [Q.sub.S] - [R.sub.S] ([P.sub.1] -[P.sub.RO]) + [Q.sub.SNOM] [N.sub.S] ( [X.sub.S] - [X.sub.SO])

These are the same three terms, but expressed in a handier form for further analysis. The factor, R expresses the effect of room air pressure on supply flow. The factors, QSNOM and NS, represent air terminal sizing and local slope of the damper curve. Together they express the effect of a damper movement.

The same mathematical steps apply to the exhaust flow.

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[Q.sub.S] [approximately equal to] [Q.sub.EO] + [R.sub.E] ([P.sub.R] -[P.sub.RO]) + [Q.sub.SNOM] [N.sub.E] ( [X.sub.E] - [X.sub.EO])

The infiltrating air flow equation is simpler because there is no damper input. The factor, L, expresses the effect of room air pressure on the infiltrating air flow. It describes the leakage characteristics of the room.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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[FIGURE 5 OMITTED]

The block diagram of the linear model illustrates the relationship between the leakage parameter (L) and the mechanical flow parameters (Rs and Re) that characterize the ducted flows. Each is expressed as a ratio of flow to pressure. In a room with typical leakage characteristics, the L parameter is much larger than values for Rs and Re. This means a change in room pressure is made up almost entirely by changes in the infiltrating flow, with very little effect on the supply and exhaust flows. If L is small, the infiltration does not compensate so readily for changes, and pressure changes have more effect on the mechanical flows.

This representation also illustrates all the feedback paths in the system, not just the ones associated with the PID controllers. It is the view required to analyze the effects of parameters and groups of parameters on the system dynamics. For example, we can investigate the effect of the leakage parameter in comparison to the room flow parameters, Re and Rs.

The linearized flow equations can be combined with the pressure feedback system for the same purpose. This diagram is a little simpler, with fewer interlocking feedback loops.

[FIGURE 6 OMITTED]

CONCLUSION

These complete feedback models can be used to analyze the dynamics of the pressurization control systems and to develop rules for effective design and tuning. Similar models can be developed for other pressurization control systems.

ACKNOWLEDGEMENTS

The author thanks Joseph Coogan and Benjamin Jeong for their assistance in preparing this manuscript.

NOMENCLATURE

[C.sub.E] Discharge coefficient for exhaust flow

[C.sub.I] Discharge coefficient for infiltrating air flow

[C.sub.S] Discharge coefficient for supply flow

[D.sub.E] Function expressing exhaust damper curve

[D.sub.S] Function expressing supply damper curve

F Transfer function representing air flow sensing equipment

[G.sub.E] Transfer function representing exhaust flow controller

[G.sub.S] Transfer function representing supply flow controller

[K.sub.E] Proportional gain for exhaust flow control

[K.sub.S] Proportional gain for supply flow control

L Factor representing effect of room pressure on infiltrating flow

m Mass of air in room

M Transfer function representing damper motor

n Exponent in orifice flow equation

[N.sub.E] Factor representing local slope of non-linear exhaust damper curve

[N.sub.S] Factor representing local slope of non-linear supply damper curve

[P.sub.1] Air pressure in the supply duct

[P.sub.2] Air pressure in the exhaust duct

[P.sub.I] Air pressure in space adjacent to room, at the source of infiltrating air

[P.sub.I0] Air pressure driving the infiltration at the operating point

[P.sub.R] Air pressure in room

[P.sub.R0] Room pressure value at the analyzed operating point

Q Air flow

[Q.sub.E] Exhaust air flow

[Q.sub.EM] Measured value of exhaust flow

[Q.sub.ES] Setpoint for exhaust flow control

[Q.sub.ENOM] Exhaust flow at nominal pressures and fully open damper

[Q.sub.I] Infiltrating air flow

[Q.sub.INOM] Infiltrating flow at the nominal pressures

[Q.sub.S] Supply air flow

[Q.sub.SS] Setpoint for supply flow control

[Q.sub.SM] Measured value of supply flow

[Q.sub.SNOW] Supply flow at nominal pressures and fully open damper

R Gas constant

[R.sub.E] Factor representing effect of room pressure on exhaust flow

[R.sub.s] Factor representing effect of room pressure on supply flow

S Laplace Transform operator

T Temperature of air in room

V Volume of air in room

[X.sub.E] Exhaust damper position

[X.sub.E0] Exhaust damper position at the analyzed operating point

[X.sub.S] Supply damper position

[X.sub.S0] Supply damper position at the analyzed operating point

[Y.sub.E] Output value of the exhaust flow controller

[Y.sub.S] Output value of the supply flow controller

P Density of air in room

[[tau].sub.ED] Time constant associated with derivative in exhaust flow controller

[[tau].sub.EI] Time constant associated with integrator in exhaust flow controller

[[tau].sub.F] Time constant related to flow sensing filter

[[tau].sub.M] Time constant associated with damper motor

[[tau].sub.SD] Time constant associated with derivative in supply flow controller

[[tau].sub.SI] Time constant associated with integrator in supply flow controller

James Coogan

Member

James Coogan is a Senior Principal Engineer, at Siemens Building Technologies, Buffalo Grove, IL, USA.
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