Minimization of fan energy consumption in ventilation and (or) air-conditioning systems as an optimization calculus of variations

The article presents an optimization calculus of variations of fan energy consumption in ventilation and (or) air-conditioning systems. It defines an air flow rate function that depends on the time of operation in the defined room size, starting conditions and the function of hazardous substances emission rate in the room. The differential form of air flow rate dependence on density of hazardous substances allows to establish a connection between air pollution in the room and a fan air flow rate, i.e. fan energy consumption. Creating a fan energy model experiment in the room in different conditions allows to minimize energy consumption to 5–30% depending on existing conditions.


Introduction
Minimization of energy consumption plays an important role in air ventilation and air-conditioning. This is also related to the minimization of electrical energy consumption required for electrical engine of fans that create air intake to establish the conditions for ventilation and air conditioning of rooms.
Traditionally the rate of consumed air flow V (m 3 /s) in ventilation systems is calculated as an equation [1][2][3][4]: where Ks (kg/s)capacity of emitted hazardous substances in the room with a size of P (m 3 ), sz (kg/m 3 )density of hazardous substances in the outdoor air intake, smax (kg/m 3 )maximum allowable density of hazardous substances in ventilated air.
If in the equation (1) we exchange Ks  Qs (J/s) = excessive heat in the room, s  I (or s  ct) (J/m 3 ), where Idensity of air enthalpy,  air density, cspecific heat capacity, t -temperature of the ventilated air, an equation of the consumed air volume that depends on indoor and outdoor temperature [4,5] is established.
Equations (1) and (1') have been developed based on the operating conditions of the fan [1], responsible for constant air flow V (m 3 /s) and constant emission rate of hazardous substances in the room Ks, due to which the density of hazardous substances in the ventilated air does not exceed the maximum allowed density smax (kg/m 3 ).
Equations (1) and (1') present the solution to the difference equation of hazardous substances density balance in the room air [1] for a time period Equations (1) and (1') can be used for calculation of ventilation and air-conditioning systems under assumption that a fan pumps the air flow V (m 3 /s) on a regular basis. Contemporary electronic control systems can regulate fan operation mode in time. Consequently, to minimize the energy consumption in ventilation and (or) air-conditioning systems it is important to define a task related to the optimization of variable conditions of fan operation (optimal time functions of fan operation) that provides acceptable conditions in the ventilated room depending on the density of emitted hazardous substances.
The purpose of the article is to explore the variable operation modes of fans in the room that consume minimum energy under the given conditions.

Optimization problem of fan energy consumption minimization
Electrical capacity of the fan W is proportional to excessive pressure p of the fan and its air flow consumption rate V [6].
On the other hand, for fans in the operation mode excessive pressure is considered as proportional to the square of air flow consumption rate 2 pV  for simple ventilation systems and to ,2 n p V n  for the network systems [4].
Therefore, the capacity of the fan equals where kcoefficient of proportionality that depends on the air density and temperature as well as other aspects of the fan. If we assume that the fan operates for a particular time period [0, T], then the energy E of the fan equals 1 00 ( , ) TT n The equation (5) presents the energy consumed for room ventilation. The equation (5) is an objective function in the optimization problem. The problem of ensuring required where energy (5) is a function of air flow V (m 3 /s) consumed by ventilation system and system operation time .
As a result, optimization problem of energy consumption minimization can be presented as the following: with the following limitations Minimization problem (6) is a calculus of variations problem [7,8], the idea of which is to find such a function of consumed airflow V() with dependence on time , that can show the minimum of equation (5) under the condition (7), that states that the density of hazardous substances does not exceed the maximum allowed density in the room with the capacity P (m 3 ).

Energy consumption of a fan with a constant airflow
It can be assumed that the system has a fan that operates in different modes and intakes the air flow that depends on time: Energy E0 consumed during fan operation mode that provides a constant air inflow V during the time is defined as a target value. Since V = const, then: If equation (1) is used in (8), dependence of energy consumption on density of hazardous substances is defined as: The equation (9) provides several obvious conclusions. First, energy E0 consumed for ventilation is proportional to the emission rate of hazardous substances in the room Ks to the (1 + n) degree. Secondly, energy consumption is growing to the (1 + n) degree if the density of hazardous substances sz in the outdoor air intake is too high. Thirdly, energy consumption is growing proportionally to the operating time of the fan .

Optimization of energy consumption of a fan with a variable airflow
Possibilities of minimization of energy consumption by means of fan operation mode modification in time are presented in this section. Since in the difference model (2) density of hazardous substances in the room s = s(V, ) has a simple dependence on time  and air intake V, calculus of variations problem (6) can be reduced to one-function problem s = s(). For this purpose, variable V = V() in the equation (6) is excluded after the transformation of the equation (2): Differential equation (10)  The minimization problem consists in finding among all the potential curves s() such curves that show the minimum of the function (11), calculated based on the dependence for V().
To solve the calculus of variations problem (11) one first needs to solve the Euler-Poisson equation [ Or as follows: The equation (15) allows to find dependence s(), which presents the optimal (minimal) energy consumption. By defining the dependence s(), air flow intake and energy consumption can be calculated based on equations (10) and (11).

Model experiment of air flow intake and energy consumption
The equation (15)

3.1
In this section the assumption Ks = const is examined. Ks = const means that the emission rate of hazardous substances is constant and does not depend on time ( The equation (17) is solved in the following way:  Numerical model experiment for the abovementioned data and starting conditions shows the following result: Е1 = 1.647·10 9 k (J), where Е1/Е0 = 0.707 = 70.7%. Consequently, the use of optimal function V() allows to decrease fan energy consumption by approximately 30%. Fig. 2 shows dependencies V() and s(). Since the operation mode s() = smax, s() = 1.1sz has been chosen, the 60% of fan operation is directed to keep s = s(), and for the 40% of the remaining time the fan increases its capacity to decrease s to the level of s() = 1.1sz.
Operating mode for different s(), s() is calculated in a similar way.

3.2
In this section assumption Based on this equation, optimal function V() is calculated from equation (10). Correlation between Е1/Е0 equals 94.2%.

Conclusions
The problem of the fan energy consumption minimalization has been approached as a calculus of variations problem. This approach has allowed to identify air flow rate function V() depending on the time and defined room size, initial conditions and the function of hazardous substances emission rate in the room Ks().
The differential form (10) of air flow rate dependence V() on hazardous substances density s() has allowed to establish a connection between air pollution in the room and a fan air flow rate and, consequently, a fan energy consumption.
The model experiment of a fan energy in the room in different operation modes has shown the possibility to minimize energy consumption by 5-30% depending on operating conditions.