Abstract
In a heat source plant, there are many boilers. In the cooling system, there are many chillers. Whether it is a cooling system or a heating system, they are all thermal energy supply systems. There is a minimum energy consumption to provide constant thermal energy.
You have full access to this open access chapter, Download chapter PDF
In a heat source plant, there are many boilers. In the cooling system, there are many chillers. Whether it is a cooling system or a heating system, they are all thermal energy supply systems. There is a minimum energy consumption to provide constant thermal energy.
21.1 Energy Efficiency of a Boiler or a Chiller
The energy efficiency of a boiler or chiller under constant output temperature difference Δt0 (°C) is shown in Fig. 21.1. For the convenience of the following description, we refer to boilers or chillers collectively as equipment or devices.
In Fig. 21.1, Q is the flow rate (tons/hour), η is the efficiency, and Qe is the flow rate at maximum efficiency. We have
Approximately, we have
where
The equipment’s output energy is P(Q), which can be expressed as
where k is a constant related to heat capacity.
21.2 Optimal Control of Heating Systems Composed of Equipment of the Identical Type
A thermal energy supply system has n devices of the identical model. The system is used to supply constant temperature Δt0 water. The total water flow is Qt. The flow of the i-th device is Qi. Qi is greater than zero. The water flow of all devices is variable. We have
η (Qi) is the energy efficiency of the i-th device at point (Qi, Δt0), η is the total energy efficiency of the thermal energy supply system, and Pt is the total energy consumption of the system.
Using the contents of the previous chapters, it is not difficult to prove the following conclusions:
The optimal control method is to keep
The minimum value of total energy consumption is
The ideal work P0 is
The overall optimal efficiency is
21.3 The Energy Efficiency Similarity of Different Equipment
We define the load rate γ as
We call ηN(γ) as the normalized energy efficiency function of an equipment. The Normalized energy efficiency function ηN(γ) has a shape shown in Fig. 21.2.
In Fig. 21.2, γ is the variable and ηN is the efficiency. ηN and η have the following relationship.
If the normalized efficiency functions of two different devices are identical, we have
We call them efficiency-similar devices. A thermal energy supply system containing efficiency-similar devices is called an efficiency-similar heating system.
Let γi be the load rate of the i-th device, and its form is:
For an efficiency-similar heat energy supply system with n-unit running equipment, the total energy consumption has the form
21.4 Optimal Control of an Efficiency-Similar Heating System
When the load rate of each running equipment is greater than zero, consider the minimization of the total energy consumption
We consider three kinds of situation.
-
(1)
n = 2
There are two variables.
We have
The objective function Pt is expressed as
The optimal condition is
We have
It is easy to see that
to be an optimal point.
The minimum value of total energy consumption is
P0 is the ideal work.
-
(2)
n = 3
There are three variables.
Based on known conditions, we have
Pt expression becomes
Assume that γ1 is a fixed optimal point and only γ2 and γ3 are variables. We have
According to the conclusion drawn with n = 2, the optimal point is
Assume that γ2 is a fixed optimal point and only γ1 and γ3 are variables. We have
According to the conclusion at n = 2, the optimal point is at
Similarly, assume that γ3 is fixed optimal point and only γ1 and γ2 are variables. We have
According to the conclusion at n = 2, the optimal point is at
We have the optimal points
The minimum value of total energy consumption is
-
(3)
n = k
There are n variables.
The above conclusion can be extended to the situation at n = k, the optimal point is
The minimum value of total energy consumption is
21.5 Optimal Number of Running Units
If n is the optimal and all equipment are the identical model, there must be
Namely
If n is the optimal and all devices have the identical energy efficiency, there must be
That is
where n1 is the operating number of devices in any combination.
21.6 Optimal Switch of Efficiency-Similar Heating System
Assume that there are M devices in a thermal energy supply system, n devices are running, and n is equal to or less than M. If n is optimal, the total energy consumption is the minimum.
n1 and n2 are the running number of equipment in any combinations.
When Qt changes, we consider three situations.
-
(1)
The γ (Qt, n) is less than 1.
ηe is the maximum efficiency, and 1 is the load rate at ηe, as shown in Fig. 21.3.
In Fig. 21.3, we have
When Qt increases, the load rate γ(Qt,n) increases, and ηN(Qt,n) and ηN(Qt,n1) also increase. However, when γ(Qt,n) > 1, Qt increases, ηN(Qt, n1)) still increases, but ηN(Qt, n) decreases. When ηN(Qt, n) < ηN(Qt, n1), n1 is optimal, we should change the number of running units from n to n1, we have
The optimal switch point is
When Qt decreases, the load rate γ(Qt,n) decreases, and ηN(Qt, n) decreases also. However, ηN(Qt, n2) increases. When ηN(Qt, n) < ηN(Qt, n2), n2 is the optimal, we should change the number of running units from n to n2, we have
The optimal switch point is
-
(2)
The γ(Qt,n) is greater than 1, as shown in Fig. 21.4.
In Fig. 21.4, we have
When Qt increases, the load rate γ (Qt, n) increases, and ηN(Qt, n1) increases also, however ηN(Qt, n) decreases. When ηN(Qt, n)< ηN(Qt, n1), n1 is the optimal, we should change the number of running units from n to n1, we have
The optimal switch point is
When Qt decreases, the load rate γ(Qt, n) decreases, ηN(Qt, n) and ηN(Qt, n2) both increase. When \(\gamma (Q_{t} ,n)\) < 1, Qt decreases, ηN (Qt, n2) still increases, however ηN(Qt, n) decreases. When ηN(Qt, n) < ηN(Qt,n2), n2 is the optimal, we should change the number of running units from n to n2, we have
The optimal switch point is
-
(3)
The \(\gamma (Q_{t} ,n)\) is equal to 1.
When Qt increases, the load rate γ(Qt,n) increases, ηN(Qt, n1) increase also, however ηN(Qt, n) decreases. When ηN(Qt, n) < ηN(Qt, n1), n1 is the optimal, we should change the number of running units from n to n1, we have
The optimal switch point is
When Qt decreases, the load rate γ (Qt, n) decreases, ηN (Qt, n) decreases also, however ηN (Qt, n2) increases. When ηN(Qt, n) < ηN(Qt, n2), n2 is the optimal, we should change the number of running units from n to n2, we have
The optimal switch point is
21.7 Conclusion
The proof of the optimal control and switching method given in this chapter is mainly based on the characteristics of the energy efficiency function, which can be approximated as a concave non-negative function through the origin. The optimal method has the following characteristics:
-
(1)
Includes linear and nonlinear systems,
-
(2)
No systematic mathematical model is required,
-
(3)
High versatility.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Copyright information
© 2024 The Author(s)
About this chapter
Cite this chapter
Yao, F., Yao, Y. (2024). Energy Efficiency Optimization of Heating and Cooling Systems. In: Efficient Energy-Saving Control and Optimization for Multi-Unit Systems. Springer, Singapore. https://doi.org/10.1007/978-981-97-4492-3_21
Download citation
DOI: https://doi.org/10.1007/978-981-97-4492-3_21
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-97-4491-6
Online ISBN: 978-981-97-4492-3
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)