# Capacity control of air coils for heating and cooling: transfer functions, drive power and system design.

1 INTRODUCTION

There is an abundance of liquid-to-air coils used as air heaters or air coolers for air conditioning, refrigeration etc. They are sometimes designed for variable air flow, as in VAV air-conditioning systems, and sometimes with constant air flow as in supermarket display cabinets. In other applications, such as hydronic radiators for heating or chilled beams for cooling, the air-side relies on natural convection for heat transfer.

The design capacity of an air coil is rarely used and must therefore be reduced accordingly, traditionally by means of on-off operation or control valves. Shunt groups are often used with constant liquid flow in the coil irrespective of demand. Traditional control will result in excessive pressure drop due to high flows and the need for balancing valves and control valves with authority. The result is excessive drive power to the pumps, which is further aggravated by the traditionally low efficiency of small pumps.

In this paper we will look at alternative ways of control of the coil capacity and how these affect the transfer function (controlled variable/controlling variable), pressure drop and pumping power of the system. The aim of the discussion is to show the advantages of direct flow control by means of decentralized pumps (1), (7). The advantages are quantified by an example with fan-coil units for heating and cooling as analyzed in a report by Fahlen (5).

2 CAPACITY CONTROL OF AIR COILS

The thermal capacity of an air coil can be calculated by means of the [epsilon]-NTU method and specified inlet air and liquid (brine) temperatures to the coil ([t.sub.a1] and [t.sub.b1). The general expression will be:

[Q.sub.a] = [epsilon] * [C.sub.min] * ([t.sub.b1] - [t.sub.a1]) [W] (1)

2.1 Control principles

Applying logarithmic differentiation (2) to equation (1), the sensitivity of the thermal capacity to changes in the respective parameters can be estimated,

[[[DELTA][Q.sub.a]]/[Q.sub.a]] = [[[DELTA][epsilon]]/[epsilon]] + [[[DELTA][C.sub.min]]/[C.sub.min]] + [[[DELTA]([t.sub.b1] - [t.sub.a1])]/([t.sub.b1] - [t.sub.a1])] [-] (2)

where [epsilon] = [epsilon](R, [N.sub.tu]) is the effectiveness of the coil, R = [C.sub.min]/[C.sub.max] is the ratio of the minimum and maximum heat capacity flow rates, and [N.sub.tu] = U * A/[C.sub.min]. In high-flow systems [C.sub.a] [less than or equal to] [C.sub.b], and then [C.sub.min] = [V.sub.a] * [[rho].sub.a] * [c.sub.p,a]. This type of sensitivity analysis is helpful in formulating linearized transfer functions in control system design. It is straightforward also to differentiate [epsilon] = [epsilon]([C.sub.a], [C.sub.b]). Equation (2) also indicates the main possibilities of capacity control:

* Primary side (liquid) supply temperature: [t.sub.b,s] (with no mixing arrangement, [t.sub.b1] = [t.sub.b,s]; see 3.1 and 3.2)

* Primary side (liquid) inlet temperature: [t.sub.b1] (with mixing, [t.sub.b1] = ([V.sub.cv]/[V.sub.ac]) * [t.sub.b,s] + (1-[V.sub.cv]/[V.sub.ac]) * [t.sub.b2]

* Primary side (liquid) flow rate: [C.sub.b] =[V.sub.b] * [[rho].sub.b] * [c.sub.p,b] (t.sub.b1] = [t.sub.b,s] = constant)

* Secondary side (air) flow rate: [C.sub.a] = [V.sub.a] * [[rho].sub.a] * [c.sub.p,a] (t.sub.a1] = constant)

2.2 Designations and assumptions

The discussion presumes a liquid-to-air coil for heating or cooling of air. This means that thermal capacity is positive when the air temperature is raised and negative when reduced. Also, losses from the coil are neglected, i.e. [Q.sub.a] = -[Q.sub.b]. Index "a" is used for air and "b" for the liquid as many of the original applications described by Fahlen (3), (4), (6) were on refrigeration with a single-phase brine (= b). In the analysis of transfer functions, Fahlen (3) has motivated the use of arithmetic instead of logarithmic mean temperature differences ([theta] is used for temperature difference between two media and [DELTA]t for temperature change of one medium). Figure 1 illustrates the system of designations.

[FIGURE 1 OMITTED]

Mean temperature differences:

Possible simplifications

[[theta].sub.m] [approximately equal to] [[theta].sub.gm] if [[theta].sub.1] < 5 * [[theta].sub.2] and

[[theta].sub.m] [approximately equal to] [[theta].sub.am] if [[theta].sub.1] < 3 * [[theta].sub.2] with

[[theta].sub.am] = ([[theta].sub.1] + [[theta].sub.2])/2,

[[theta].sub.gm] = [square root of ([[theta].sub.1] * [[theta].sub.2])] and

[[theta].sub.lm] = ([[theta].sub.1] - [[theta].sub.2])/ln([[theta].sub.1]/[[theta].sub.2])

To compare alternative solutions Fahlen (3) has introduced a non-dimensional controlling variable and a non-dimensional controlled variable (the controlled variable is usually a temperature or capacity):

Controlling variable: [x.sub.b] = [[actual value (x %)]/[design value (100 %)]]; Controlled variable [y.sub.a] = [[actual capacity (y %)]/[design capacity (100 %)]];

Fahlen has also introduced a base load for the pump consisting of the pressure drop of the basic pipe work and the coil. This is the minimum pressure drop of the system used as a benchmark. Additional pressure drops for control purposes will increase the drive power in relation to the benchmark.

3 TRANSFER FUNCTIONS OF AIR COILS

Due to varying loads etc. the demand situation will differ between the various rooms of a building. The supply temperature should be modulated to make the deciding terminal unit operate at its design flow. All other units will then operate at reduced capacity with the given supply temperature. Variable capacity can be achieved by means of temperature control (supply or mixing) and/or flow control as indicated in 2.1.

3.1 Supply temperature control

From an efficiency perspective, supply temperature should never be higher than required for heating or lower than required for cooling. In heating systems, it is common to have an outdoor temperature related feed-forward control. This, however, is less common in cooling applications where a fixed temperature, required only at the design load, is often used. From equation (2) we see that the third term provides the relative change of capacity as a function of the relative change in temperature difference. Hence, according to Fahlen (3), the transfer function becomes:

[y.sub.a] = [x.sub.b] [-] with [x.sub.b] = [([t.sub.b,s] - [t.sub.a1])/[([t.sub.b,s] - [t.sub.a1])d]] [-] and [y.sub.a] = [[Q.sub.a]/[Q.sub.a,d]] [-] (1)

Supply temperature control is linear and direct. It should always be included as a primary control method of heating as well as cooling coils. Also, there is no pressure drop penalty involved in supply temperature control.

3.2 Primary, liquid-side inlet temperature control

Figure 2 illustrates the principle of controlling capacity by means of a variable liquid-side inlet temperature to the coil. This can be achieved by the illustrated arrangement, i.e. a three-way valve and constant supply flow, or by using a two-way valve and variable supply flow. Truschel and Fahlen (3) provide expressions for how capacity varies with the flow rate through the control valve CV. As the coil flow is constant, the brine-side temperature efficiency [[eta].sub.b] will be a constant from the coil design. The supply temperature ([t.sub.b,s]) and liquid flow through the coil are also constant.

[FIGURE 2 OMITTED]

Variable inlet temperature

Transfer function:

[y.sub.a] = [[x.sub.b]/([[eta].sub.b] * (1 - [x.sub.b]) + [x.sub.b])]

Pumping power:

[[W.sub.tot]/[w.sub.base,d]] = [1/[1-[beta]]] * (1 - [beta] * [[[DELTA][p.sub.ac]]/[[DELTA][p.sub.base]]]+[[[DELTA][p.sub.bv1]]/[[DELTA][p.sub.base]]]+ (1 - [beta]) * [[[DELTA][p.sub.bv2]]/[[DELTA][p.sub.base]]])

The transfer function shows that although capacity is a linear function of the inlet temperature, it is certainly not a linear function of flow rate. However, the control valve flow is the controlling variable and the lower [[eta].sub.b] is the more nonlinear will be the transfer function. The pumping power of this particular configuration will remain constant irrespective of the thermal heat transfer and it will be much larger than the required base power (pipe system plus coil). Assuming a control authority [beta] = 0.5, we see that the drive power at the design condition will be typically more than twice the base value (depending on how much balancing that is required).

3.3 Primary, liquid-side flow rate control

Fahlen (3) has derived a relation between the controlling variable [x.sub.b] (liquid flow ratio) and the controlled variable [y.sub.a] (coil capacity ratio). The transfer function [y.sub.a] = [y.sub.a] ([x.sub.b]) can be shaped to conform to desired characteristics by means of the overall size, the ratio between the liquid and air side heat transfer capacity and the air and liquid flow rates according to:

[y.sub.a] = [[F(x.sub.b) * (1 + [K.sub.d5] + [K.sub.d6]) * [x.sub.b]]/[[x.sub.b] + F([x.sub.b]) * ([K.sub.d5] * [x.sub.b] + [K.sub.d6])]] [-] with F([x.sub.b]) = [x.sub.b.sup.m] * [[1 + [K.sub.d1]]/[1 + [K.sub.d1] * [x.sub.b.sup.m]]] and [x.sub.b] = [[v.sub.b]/[v.sub.b.d]] [-] (4)

The design constants [K.sub.d1] to [K.sub.d6] in the relations above are given by (m is the flow related heat transfer exponent):

Design heat transfer capacity

Ratio of heat transfer capacity on the liquid side to that of the air side: [K.sub.d1] = [[[alpha].sub.b,d] * [A.sub.b,d]/[[alpha].sub.a,d] * [A.sub.a,d]] [-];

Design air flow rate (a = air)

Air temperature change, i.e. required flow rate, at the design condition: [K.sub.d2] = [DELTA][t.sub.a,d] [K];

Design liquid flow rate (b = brine)

Liquid temperature change, i.e. required flow rate, at the design condition: [K.sub.d3] = [DELTA][t.sub.b,d] [K];

Design heat exchanger size

Mean temperature difference, i.e. required heat exchanger size, at the design condition: [K.sub.d4] = [[theta].sub.am,d] [K];

Facilitating constants

Constants introduced to simplify the relation for [y.sub.a]: [K.sub.d5] = [[K.sub.d2]/[K.sub.d4]] [-] and [K.sub.d6] = [[K.sub.d3]/[K.sub.d4]] [-].

Flow control can be realized by means of on-off operation (not discussed in this paper) or continuous variation using either a) a central pump (CP) and valve control (see figure 3) or b) a decentralized pump (DP) with continuous VSD control (see figure 4). Continuous flow control will have a nonlinear transfer function. However, just as in the case with inlet temperature control, it is possible to tune the characteristics by means of suitable coil design. With a), see figure 3, the current practice is to operate the pump P1 with VSD control of the supply pressure. If this is kept constant at the design level, then pumping power will decrease linearly with flow rate but faster in relation to a capacity reduction (c.f figure 5). With [beta] = 0.5 the design pump power will be more than twice the benchmark value.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

(a) Flow control using a two-way valve

Transfer function:

[y.sub.a] = [[F(x.sub.b) * (1 + [K.sub.d5] + [K.sub.d6]) * [x.sub.b]]/[[x.sub.b] + F([x.sub.b]) * ([K.sub.d5] * [x.sub.b] + [K.sub.d6])]]

Pumping power (constant pressure control):

[[W.sub.tot]/[W.sub.base,d]] = [x.sub.b] * (1/[1 - [beta]]) * (1 + [([[DELTA][p.sub.bv]]/[[DELTA][p.sub.base]]).sub.d])

In the case of b), decentralized pump control (DP-VSD), see figure 4, pumping power will equal the benchmark value at the design capacity. With reduced flow and reduced capacity, power will drop very quickly (c.f. figure 5). Note that no balancing or control valves are required in this case.

(b) Flow control using a decentralized VSD pump

Transfer function:

[y.sub.a] = [[F(x.sub.b) * (1 + [K.sub.d5] + [K.sub.d6]) * [x.sub.b]]/[[x.sub.b] + F([x.sub.b]) * ([K.sub.d5] * [x.sub.b] + [K.sub.d6])]]

Pumping power:

[[W.sub.tot]/[W.sub.base,d]] = [x.sub.b.sup.n+1] 0 < n < 2 depending on flow regime

(laminar, transitional or turbulent flow)

3.4 Summary y

Irrespective of whether capacity is controlled by means of inlet temperature or coil flow rate, the transfer function will be nonlinear in terms of the controlling valve or pump flow rate. However, the shape of the transfer function may be tuned by means of heat exchanger design. In particular, designing for laminar liquid flow (6) will drastically change the flow related heat transfer and pressure drop.

Figure 5 shows the substantial difference in pumping power between the DP-VSD solution and the standard CP-VSD solutions (two or three-way valves). At reduced capacity, the power requirement of the DP-VSD system drops very quickly. The CP-VSD with two-way valve also drops quickly if the CP pressure can be reduced to a minimum level for the entire system. If, as in the standard case, the pressure has to be maintained in case one room in the building requires full capacity, then the drop is only linear.

[FIGURE 5 OMITTED]

4 CENTRALIZED VS DECENTRALIZED PUMPS - EXAMPLE WITH FAN COILS

Fahlen (5) has exemplified the many advantages of decentralized pump operation in a report that analyzes a hydronic fan-coil system. The report compares a central pump (CP) with constant-speed drive (CSD), a central pump (CP) with variable-speed drive (VSD), and a solution that uses decentralized pumps (DP) with variable-speed drive (VSD). Figures 6 (a) and (b) illustrate the basic designs. The new solution in (b), DP-VSD, can be controlled by first changing the supply temperature until the first DP-unit operates at full speed. The remaining DP-units will then operate at reduced speed. This provides intrinsic feed-forward control that automatically considers both internal and external loads. Fahlen (5) provides comparisons for CP-CSD, CP-VSD with a design flow velocity of maximum 1 m/s and CP-VSD with a design flow velocity of maximum 0.6 m/s. The latter alternative is by far the most advantageous for the CP alternative but even so we shall soon see that this results in much higher pumping power than the DP solution.

[FIGURE 6 OMITTED]

To compare the relative pumping power of a specific situation with that of the benchmark design value Fahlen (5) has introduced [r.sub.p] = ratio of current value (p%) and the design value (100%) of the fan coil. The following figures indicate the effect of design flow velocity (1 or 0.6 m/s) and system solution (CP or DP): 1 m/s, CP-VSD: [r.sub.100] = 9.57, [r.sub.50] = 4.79, [r.sub.20] = 1.92; 0.6 m/s, CP-VSD: [r.sub.100] = 4.68, [r.sub.50] = 2.32, [r.sub.20] = 0.93; and 0.6 m/s, DP-VSD: [r.sub.100] = 2.31, [r.sub.50] = 0.32, [r.sub.20] = 0.02.

4.1 Centralized pump with variable-speed drive (CP-VSD)

Figure 7 shows that as capacity is reduced from 100 to 50% the pressure drop in the control valve goes from being more than half the total to represent virtually all of the pressure drop and consequently nearly all the pumping power.

[FIGURE 7 OMITTED]

4.2 Decentralized pump with variable-speed drive (DP-VSD)

Figure 8 indicates the substantial reduction in pressure drop and pumping power accomplished by the decentralized pump solution. Note that this is the worst coil of the system. With a DP system all other coils will require a smaller pumping head whereas in a CP system balancing valves will make all branches have the same head requirement.

[FIGURE 8 OMITTED]

5 DISCUSSION AND CONCLUSIONS

In order to be energy efficient, an HVAC system must provide the required function with low use of heating, cooling and electricity, i.e. the system must be heat efficient as well as electricity efficient. To be heat efficient, a system must be able to match actual demand for heat supply or heat removal at room level. This requires a high degree of local control in order to achieve a supply with the right quality (temperature), the right quantity (heating or cooling capacity) at the right time and in the right place. To be electricity efficient, a system must transfer and deliver the required thermal energy with low use of electricity. This requires heat exchanger designs with low pressure drop, e.g. optimized for laminar flow, and distribution systems with low pressure drop.

This paper has shown that there are substantial advantages lurching in designs with decentralized, locally controlled flow generators. The simple examples of this report indicate potential savings on electric drive power in the range 30 - 50% at full capacity, 40 - 80% at half capacity and 50 - 95% at 20% capacity. We have indicated that it is possible to design controllable hydronic heating and cooling systems without introducing distribution pressure drops from balancing and control valves. This can be achieved by means of new system concepts and new technologies for motors and motor drives. Energy efficiency, however, is not the only potential advantage in using decentralized flow generators. Below is a summary (5) of the most important positive aspects:

* Reduction of the use of heat and electricity

* Control authority with no extra pressure drop

* Feed-forward temperature control inherently available for outdoor as well as indoor generated load variations

* Improved matching between supply and heating or cooling demand in relation to the local, actual demand

* Increased number of degrees of freedom in system design

* Flexibility during retrofits and reconstruction (no need for rebalancing when units are added or subtracted)

* Fewer components and much reduced need for adjustment during commissioning

* Central adjustment of local flows (possibility of intrinsic flow measurement with no measuring pressure drop)

* Integrated heat-allocation measurement a future possibility

* Simple disaggregated performance statistics by means of the Building Automation System (delivery at room level is controlled by one, local unit and may be monitored; this is a useful feature both for a disaggregated information on weak positions in the building, provides information for future upgrading according to EPBD, and the need for service and maintenance)

6 ACKNOWLEDGMENTS

We acknowledge the support of Gothenburg Energy, the Swedish Energy Agency, Formas-Bic and all companies participating in our project on Efficiency of building related pump and fan operation (1).

8 NOMENCLATURE

A = area [[m.sup.2]]

C = heat capacity flow rate [[W.sub.th]/K]; C = [c.sub.p] * M

N = number (of transfer units) [-]

Q = power, thermal [[W.sub.th]]

T = thermodynamic temperature [K]

W = power, mechanical or electric [W]

[beta] = valve authority [-]: [beta] = [DELTA][p.sub.cv]/[DELTA][p.sub.tot]

[epsilon] = efficiency, thermal power [W/[W.sub.max]]

[rho] = density [kg/[m.sup.3]]

c = specific heat capacity [J/(kg * K)]

M = mass flow rate [kg/s]; M = [rho] * V

p = pressure [Pa]

t = celsius temperature [[degrees]C]

V = volume flow rate [[m.sup.3]/s]

[alpha] = coefficient of heat transfer [W/([m.sup.2] * K)]

[DELTA] = difference, change (e.g. pressure, temp.)

[eta] = efficiency, temperature [K/[K.sub.max]]

[tau] = time [s] or [h]

8.1 Subscripts

a = air

b = (brine) liquid

cv = control valve

p = pump

1 = inlet of coil

ac = air-coil

bv = balancing valve

d = design condition

tu = transfer units

2 = outlet of coil

8.2 Abbreviations

BV = balancing valve

CP = centralized pump

DP = decentralized pump

NTU = number of transfer units

CHV = check valve

CV = control valve

EPBD = Energy Performance of Buildings Directive

VSD = variable speed drive

9 REFERENCES

(1.) Fahlen, P, 2004. Efficiency of building related pump and fan operation - Applications, system solutions, motor technology and control. Research application: STEM, (Building Services Engineering.) Gothenburg, Sweden.

(2.) Fahlen, P, Voll, H, Naumov, J, 2004. Efficiency of pump operation in hydronic heating and cooling systems. Energy for Buildings, Vilnius, Lithuania, 2004-10-07-08. (Rehva.)

(3.) Fahlen, P, 2007. Capacity control of air coils in systems for heating and cooling - Transfer functions and drive power to pumps and fans. R2007:01, (Building Services Engineering.) Gothenburg.

(4.) Fahlen, P, Markusson, C, Haglund Stignor, C, 2007. Capacity control of liquid-cooled air-coolers. 22nd International Congress of Refrigeration, Beijing, China, 2007-08-21-26. (International Institute of Refrigeration.)

(5.) Fahlen, P, 2009. Capacity control of hydronic fan-coil units - Reduction of pump work. R2009:02, (Building Services Engineering.) Gothenburg.

(6.) Haglund Stignor, C, Fahlen, P, Sunden, B, 2007. Design of different types of secondary loop cooling systems in supermarkets - Comparison of energy use and costs. 22nd International Congress of Refrigeration, Beijing.

(7.) Paarporn, S, 2000. Local pumping system. ASHRAE Journal, vol. 42, no. 9, 2007-02, pp. 25-30. (ASHRAE.).

Per Fahlen

Member ASHRAE

Caroline Markusson

Research Student

Per Fahlen is a professor in the Department of Energy and Environment, Chalmers University of Technology, Gothenburg, Sweden.

Caroline Markusson is a research student in the same department.
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Author: Printer friendly Cite/link Email Feedback Fahlen, Per; Markusson, Caroline ASHRAE Transactions Report 4EUSW Jan 1, 2011 3492 Rehab of DDC at corporate National Training Center- three years of therapy. Proportional-Integral-Plus (PIP) control system for individual thermal zones in a small ventilated space. Engineering design Heating Heating equipment Transfer functions