Energy Efficiency Optimization of Pum** Stations and Fan Stations

Abstract

Many occasions and fields have requirements for transporting liquids and gases, so pumps and fans are widely used in chemical industry, pharmaceuticals, power plants, steel mills, papermaking, oil refining, water conservancy, water supply, drainage and irrigation, sewage, secondary water supply, hospitals, office buildings, shop** malls, hotels, hotels and other occasions. The total electricity consumption of pumps and fans accounts for 30–35% of the world's total electricity consumption.

Many occasions and fields have requirements for transporting liquids and gases, so pumps and fans are widely used in chemical industry, pharmaceuticals, power plants, steel mills, papermaking, oil refining, water conservancy, water supply, drainage and irrigation, sewage, secondary water supply, hospitals, office buildings, shop** malls, hotels, hotels and other occasions (Fig. 23.1). The total electricity consumption of pumps and fans accounts for 30–35% of the world's total electricity consumption.

In the field of energy saving for pumps and fans, speed control devices have developed rapidly, especially frequency conversion speed control technology has become relatively mature. At present, many internationally renowned companies, such as Siemens, ABB, Schneider, Fuji, Toshiba, Sanken, General Electric, AB, etc., have launched water pump fan type inverters. In terms of energy-saving calculation of water pumps and fans, many multinational companies have also launched some tools, such as ABB’s “PumpSave” and “FanSave” software, and Siemens’ “SinaSave” software. However, there are still many problems in terms of versatility, accuracy and energy-saving thoroughness.

The energy-saving transformation of some pum** stations and fan stations is far from the expected results, which not only damages the reputation of energy-saving equipment manufacturers, but also makes users dissatisfied. Such cases include large multinationals and listed companies. Some pum** stations and fan stations have increased power consumption after energy-saving renovations. The reason is unknown, and the two sides started arguing. Since there are many ways to save energy, and each has its own advantages and disadvantages, some users simply think that they can use the equipment of the company that promises the largest energy-saving ratio. Turning technical issues such as energy-saving ratio into business issues that can be negotiated. Many strange phenomena have occurred. The promised power saving ratio has broken the law of conservation of energy, and the two parties are still negotiating seriously.

Fig. 23.1

Application of pump and fan

There are different opinions on energy-saving methods, and the vast number of energy-saving workers feel clouded. Some experts’ articles are well written, but the conclusion is only correct for the water pump fan station they experimented with. If generalized, there will be problems. Some energy-saving methods are very good, but they are only suitable for specific working conditions. The following problems are what you have to face:

Where are inverters suitable for application?

Is there no potential for energy saving if the inverter is already used?

In fact, many special cases can also illustrate some general trends:

• (1) When the frequency converter works at 50 Hz, it is better to run directly at power frequency to save power. Using methods such as series resistance speed regulation and liquid viscosity speed regulation, the efficiency is close to 100% when running close to the rated speed, which is more energy-saving than a frequency converter;

• (2) When multiple pumps are running, the variable frequency pump that only maintains pressure but does not output water wastes 100% of energy. Turning off the pump at this time will not affect the output pressure and flow of the pump station;

• (3) Even if the frequency converter is used and has a good power-saving effect, if there is no optimal operation method, it may still cause a large waste of electric energy.

23.1 The Total Energy Consumption of a Fan Station

The energy consumed by a fan is

$$W={k}_{8}\frac{Q{P}_{0}}{\eta (Q)}$$

(23.1)

where W is the energy consumed by the fan, k₈ is a constant, Q is the flow rate of the fan, and η is the energy efficiency of the fan, P₀ is the total pressure provided by the fan.

A fan station has m fans, and the total pressure P₀ provided by each fan is equal. The total energy consumption of this fan station is

$${W}_{t}={\sum }_{i=1}^{m}{W}_{i}={k}_{8}{P}_{0}{\sum }_{i=1}^{m}{Q}_{i}\frac{1}{\eta ({Q}_{i})}$$

(23.2)

where Q_i is the output flow of the i-th fan, η_i is the energy efficiency of the i-th fan, and W_i is the energy consumed by the i-th fan. W_t is the total energy consumption of the fan station.

23.2 Total Energy Consumption of a Pum** Station

The energy consumed by a pump is

$$W=\frac{Q{H}_{0}}{367\eta (Q)}$$

(23.3)

where W is the energy consumed by the pump, Q is the flow rate of the pump, and η is the energy efficiency of the pump, H₀ is the full head provided by the pump.

A pum** station has m pumps, and the full head H₀ provided by each pump is equal. The total energy consumption of this pum** station is

$${W}_{t}=\sum_{i=1}^{m}{W}_{i}=\frac{{H}_{0}}{367}{\sum }_{i=1}^{m}{Q}_{i}\frac{1}{{\eta }_{i}({Q}_{i})}$$

(23.4)

where Q_i is the output flow of the i-th pump, η_i is the energy efficiency of the i-th pump, and W_i is the energy consumed by the i-th pump, W_t is the total energy consumption of the pum** station.

23.3 Unification of Energy Efficiency Optimization for Pum** Stations and Fan Stations

Comparing the energy consumption expressions of the water pump station and the fan station, the changing parts are the same. Only one type of energy efficiency optimization method needs to be solved to generalize to the other type. Below we only analyze the energy efficiency optimization of pum** stations.

For a pum** station that outputs a constant full head H0, the efficiency of the i-th centrifugal variable speed pump is shown in Fig. 23.2.

Fig. 23.2

Centrifugal pump efficiency function

In Fig. 23.2, Q is flow (m³/hour), and η is the efficiency, Q_ie is the flow rate at the maximum efficiency η_ie, Q_im is the maximum flow rate of the pump.

$$Q\ge 0$$

(23.5)

$${\eta }_{i}\left(Q\right)\ge 0$$

$${\eta }_{i}\left(0\right)=0$$

$${\eta }_{i}^{{\prime}{\prime}}(Q)<0$$

Approximately, we have

$${\eta }_{i}(Q)={\sum }_{i=0}^{\infty }{a}_{i}{Q}^{i}=Q{\sum }_{i=1}^{\infty }({a}_{i}{Q}^{i-1})$$

(23.6)

$${\sum }_{i=1}^{\infty }\left({a}_{i}{Q}^{i-1}\right)>0$$

$$ a_{0} = 0 $$

The minimum energy consumption of the pum** station is

$$\begin{array}{c}{minW}_{t}=min\left(\frac{{H}_{0}}{367}{\sum }_{i=1}^{m}{Q}_{i}\frac{1}{{\eta }_{i}({Q}_{i})}\right)\\ \begin{array}{c} s.t. \sum_{i=1}^{m}{Q}_{i}={Q}_{t} >0\\ {Q}_{imx}\ge {W}_{i}>0\end{array}\end{array}$$

(23.7)

where Q_t is the total flow rate of the pum** station.

23.4 Typical Single Closed-Loop PID Control Method

A typical single closed-loop control of a pum** station is shown in Fig. 23.3.

Fig. 23.3

Typical single closed-loop control block diagram

In Fig. 23.3, the typical PID control method is to measure the actual value H'₀ to be controlled, compare H'₀ with the set value H₀, and feed the error back to the PID. The PID changes a manipulated variable to control the frequency converter and adjust the speed of the motor and pump. When H’₀ is less than H₀, PID increases the pump speed. If H’₀ is still less than H₀ when the pump speed reaches the maximum speed, PLC or DCS will turn on another pump. When H’₀ is greater than H₀, PID reduces the pump speed. If H’₀ is still greater than H₀ when the pump speed reaches the minimum speed, PLC or DCS will shut down a pump.

This control method does not consider the energy consumption of the pump station, so it cannot achieve minimum power consumption.

23.5 Optimal Control of Pum** Stations Composed of Identical Energy Efficiency Pumps

A pum** station has n water pumps of the identical model running. The pum** station is used to transport water at a constant full head H₀. The total flow rate is Q_t. The flow rate of the i-th water pump is Q_i. Q_i is greater than zero. The speed of all pumps is variable, and the full head of the i-th pump remains at H₀, we have

$${Q}_{i}>0$$

(23.8)

$${\sum }_{i=1}^{n}{Q}_{i}={Q}_{t}$$

η (Q_i) is the i-th pump efficiency at the point (Q_i, H₀), η_t is the total efficiency of the pum** station, W_t is the total power consumption of the pum** station.

$${W}_{t}=\frac{{H}_{0}}{367}{\sum }_{i=1}^{n}{Q}_{i}\frac{1}{\eta ({Q}_{i})}$$

(23.9)

Based on the discussion in the previous chapters, we know that the optimal control method is to keep

$${Q}_{1}={Q}_{2}=...={Q}_{n}=\frac{{Q}_{t}}{n}$$

(23.10)

This is “Same pump, Same load”, it is equivalent to “Same pump, Same speed”, “Same pump, Same frequency” and “Same pump, Same efficiency”.

The minimum value of the total power consumption is

$${\mathit{minW}}_{t}={W}_{0}\frac{1}{\eta (\frac{{Q}_{t}}{n})}$$

(23.11)

The ideal work P₀ is

$${W}_{0}=\frac{{Q}_{t}{H}_{0}}{367}$$

(23.12)

The overall optimal efficiency is

$$ {\text{max}}\eta_{{\text{t}}} \left( {{\text{Q}}_{{\text{t}}} } \right) = \eta \left( {\frac{{{\text{Q}}_{{\text{t}}} }}{{\text{n}}}} \right) $$

(23.13)

23.6 Pum** Station Composed of Pumps of Similar Efficiency

We define the load rate γ as

$$\gamma =\frac{Q}{{Q}_{e}}$$

(23.14)

We call η_N(γ) as the normalized efficiency function of a pump. The normalized efficiency function η_N(γ) has a shape shown in Fig. 23.4.

Fig. 23.4

The normalized efficiency function η_N (γ)

In Fig. 23.4, γ is a variable and η_N is the efficiency. η_N and η have the following relationship.

$$\eta (Q)=\eta (\gamma {Q}_{e})={\eta }_{N}(\gamma )$$

(23.15)

If the normalized efficiency functions of two different pumps are identical, we have

$${\eta }_{N1}(\gamma )={\eta }_{N2}(\gamma )$$

(23.16)

We refer to them as pumps with similar efficiencies. A pum** station composed of pumps with similar efficiencies is called a similar efficiency pum** station.

Assume that γ_i is the load rate of the i-th pump, and its form is

$${\gamma }_{i}=\frac{{Q}_{i}}{{Q}_{ie}}$$

(23.17)

For a pum** station consisting of n pumps with similar efficiencies, the total power consumption is of the form

$${\text{W}}_{t}=\frac{{H}_{0}}{367}{\sum }_{i=1}^{n}\frac{{Q}_{i}}{{\eta }_{i}({Q}_{i})}=\frac{{H}_{0}{Q}_{t}}{367}{\sum }_{i=1}^{n}(\frac{1}{{Q}_{t}}\frac{{\gamma }_{i}{Q}_{ie}}{{\eta }_{N}\left({\gamma }_{i}\right)})={\text{W}}_{0}\frac{1}{{\eta }_{t}}$$

(23.18)

where W₀ is the ideal work, η_t is the overall energy efficiency of the pum** station.

$$ {\text{W}}_{{0}} = \frac{{{\text{H}}_{0} {\text{Q}}_{{\text{t}}} }}{367} $$

(23.19)

$$ \eta_{t} = {1 \mathord{\left/ {\vphantom {1 {\sum_{i = 1}^{n} \left( {\frac{1}{{Q_{t} }}\frac{{\gamma_{i} Q_{ie} }}{{\eta_{N} \left( {\gamma_{i} } \right)}}} \right)}}} \right. \kern-0pt} {\sum_{i = 1}^{n} \left( {\frac{1}{{Q_{t} }}\frac{{\gamma_{i} Q_{ie} }}{{\eta_{N} \left( {\gamma_{i} } \right)}}} \right)}} $$

23.7 Energy Efficiency Optimization of a Similar Efficiency Pum** Station

When the flow rate of each running pump is greater than zero, consider the minimization of the total power consumption

$${\mathit{minW}}_{t}$$

(23.20)

$$s.t.{\gamma }_{i}>0,i=\text{1,2},\dots n$$

$${\sum }_{i=1}^{n}{\gamma }_{i}{Q}_{ie}={Q}_{t}$$

We consider three kinds of situation.

• (1) n = 2

There are two variables.

We have

$${\gamma }_{1}{Q}_{1e}+{\gamma }_{2}{Q}_{2e}={Q}_{t}$$

(23.21)

$${\gamma }_{1}>0$$

$${\gamma }_{2}>0$$

The objective function P_t is expressed as

$${W}_{t}=\frac{{H}_{0}}{367}(\frac{{\gamma }_{1}{Q}_{1e}}{{\eta }_{N}({\gamma }_{1})}+\frac{{\gamma }_{2}{Q}_{2e}}{{\eta }_{N}({\gamma }_{2})})$$

(23.22)

The optimal condition is

$${{W}_{t}}{\prime}({\gamma }_{1})=0$$

(23.23)

We have

$$\begin{gathered} \frac{{Q_{{1e}} \eta _{N} \left( {\gamma _{1} } \right) - \gamma _{1} Q_{{1e}} \eta _{N}^{\prime } \left( {\gamma _{1} } \right)}}{{\eta _{N}^{2} \left( {\gamma _{1} } \right)}} \hfill \\ + \,\frac{{ - Q_{{1e}} \eta _{N} \left( {\frac{{Q_{t} - \gamma _{1} Q_{{1e}} }}{{Q_{{2e}} }}} \right) + \left( {Q_{t} - \gamma _{1} Q_{{1e}} } \right)\frac{{Q_{{1e}} }}{{Q_{{2e}} }}\eta _{N}^{\prime } \left( {\frac{{Q_{t} - \gamma _{1} Q_{{1e}} }}{{Q_{{2e}} }}} \right)}}{{\eta _{N}^{2} \left( {\frac{{Q_{t} - \gamma _{1} Q_{{1e}} }}{{Q_{{2e}} }}} \right)}} = 0 \hfill \\ \end{gathered}$$

(23.24)

It is easy to see that

$${\gamma }_{1}={\gamma }_{2}=\frac{{Q}_{t}}{{Q}_{1e}+{Q}_{2e}}$$

(23.25)

is an optimal point.

The minimum value of the total power consumption is

$${\mathit{minW}}_{t}=\frac{{H}_{0}{Q}_{t}}{367}\frac{1}{{\eta }_{N}(\frac{{Q}_{t}}{{Q}_{1e}+{Q}_{2e}})}=\frac{{W}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{Q}_{1e}+{Q}_{2e}})}$$

(23.26)

The maximum energy efficiency of the pum** station is

$$\text{max}{\eta }_{t2}={\eta }_{N}\left(\frac{{Q}_{t}}{{Q}_{1e}+{Q}_{2e}}\right)={\eta }_{N}\left(\frac{{Q}_{t}}{\sum_{i=1}^{2}{Q}_{ie}}\right)$$

(23.27)

• (2) n = 3

There are three variables.

Based on known conditions, we have

$${\gamma }_{1}{Q}_{1e}+{\gamma }_{2}{Q}_{2e}+{\gamma }_{3}{Q}_{3e}={Q}_{t}$$

(23.28)

$${\gamma }_{1}>0$$

$${\gamma }_{2}>0$$

$${\gamma }_{3}>0$$

W_t expression becomes

$${W}_{t}=\frac{{H}_{0}}{367}(\frac{{\gamma }_{1}{Q}_{1e}}{{\eta }_{N}({\gamma }_{1})}+\frac{{\gamma }_{2}{Q}_{2e}}{{\eta }_{N}({\gamma }_{2})}+\frac{{\gamma }_{3}{Q}_{3e}}{{\eta }_{N}({\gamma }_{3})})$$

(23.29)

Assume that γ₁ is a fixed optimal point and only γ₂ and γ₃ are variables. We have

$${\gamma }_{2}{Q}_{2e}+{\gamma }_{3}{Q}_{3e}={Q}_{t}-{\gamma }_{1}{Q}_{1e}=cons\mathit{tan}t$$

(23.30)

Based on the above conclusions at n = 2, the optimal point is at

$${\gamma }_{2}={\gamma }_{3}$$

(23.31)

Assume that γ₂ is a fixed optimal point and only γ₁ and γ₃ are variables. We have

$${\gamma }_{1}{Q}_{1e}+{\gamma }_{3}{Q}_{3e}={Q}_{t}-{\gamma }_{2}{Q}_{2e}=cons\mathit{tan}t$$

(23.32)

According to the conclusion at n = 2, the optimal point is at

$${\gamma }_{1}={\gamma }_{3}$$

(23.33)

Similarly, assume that γ₃ is a fixed optimal point and only γ₁ and γ₂ are variables. We have

$${\gamma }_{1}{Q}_{1e}+{\gamma }_{2}{Q}_{2e}={Q}_{t}-{\gamma }_{3}{Q}_{3e}=cons\mathit{tan}t$$

(23.34)

According to the conclusion at n = 2, the optimal point is at

$${\gamma }_{1}={\gamma }_{2}$$

(23.35)

So that we have the optimal points

$${\gamma }_{1}={\gamma }_{2}={\gamma }_{3}=\frac{{Q}_{t}}{{\sum }_{i=1}^{3}{Q}_{ie}}$$

(23.36)

The minimum value of the total power consumption is

$${\mathit{minP}}_{t}=\frac{{P}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{3}{Q}_{ie}})}$$

(23.37)

The maximum overall energy efficiency of the pum** station is

$$\text{max}{\eta }_{t3}={\eta }_{N}\left(\frac{{Q}_{t}}{\sum_{i=1}^{3}{Q}_{ie}}\right)$$

(23.38)

• (3) n = k

There are k variables.

The above conclusion can be extended to the situation at n = k, the optimal point is

$${\gamma }_{1}={\gamma }_{2}=...={\gamma }_{k}=\frac{{Q}_{t}}{{\sum }_{i=1}^{k}{Q}_{ie}}$$

(23.39)

This is “Same load rate, Same efficiency “.

The minimum value of the total power consumption is

$${\mathit{minW}}_{t}=\frac{{P}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{k}{Q}_{ie}})}$$

(23.40)

The maximum overall energy efficiency of the pum** station is

$$\text{max}{\eta }_{tk}={\eta }_{N}\left(\frac{{Q}_{t}}{\sum_{i=1}^{k}{Q}_{ie}}\right)$$

(23.41)

23.8 Optimal Number of Running Pumps

For pum** stations composed of the identical pumps, if n is the optimal, there must be

$$\frac{{P}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n-1}{Q}_{ie}})}\ge \frac{{P}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n}{Q}_{ie}})}\le \frac{{P}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n+1}{Q}_{ie}})}$$

(23.42)

where

$${\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n-1}{Q}_{ie}})\le {\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n}{Q}_{ie}})\ge {\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n+1}{Q}_{ie}})$$

(23.43)

For pum** stations composed of the pumps with similar energy efficiency, if n is the optimal, we have

$$ \frac{{P_{0} }}{{\eta_{N} \left( {\frac{{Q_{t} }}{{\sum_{i = 1}^{n1} Q_{ie} }}} \right)}} \ge \frac{{P_{0} }}{{\eta_{N} \left( {\frac{{Q_{t} }}{{\sum_{i = 1}^{n} Q_{ie} }}} \right)}} $$

(23.44)

where

$$ \eta_{N} \left( {\frac{{Q_{t} }}{{\sum_{i = 1}^{n1} Q_{ie} }}} \right) \le \eta_{N} \left( {\frac{{Q_{t} }}{{\sum_{i = 1}^{n} Q_{ie} }}} \right) $$

(23.45)

n1 is any combination other than this n combination.

23.9 Optimal Switch of the Pum** Station with the Identical Model Pumps

Assuming that n is optimal, the total power consumption is the smallest

$${\mathit{minW}}_{t}=\frac{{W}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n}{Q}_{ie}})}$$

(23.46)

$$n\le m$$

$${\eta }_{N}\left(\frac{{Q}_{t}}{{\sum }_{i=1}^{n-1}{Q}_{ie}}\right)\le {\eta }_{N}\left(\frac{{Q}_{t}}{{\sum }_{i=1}^{n}{Q}_{ie}}\right)\ge {\eta }_{N}\left(\frac{{Q}_{t}}{{\sum }_{i=1}^{n+1}{Q}_{ie}}\right)$$

$$\gamma ({Q}_{t},n)=\frac{{Q}_{t}}{{\sum }_{i=1}^{n}{Q}_{ie}}$$

When Q_t changes, we consider three situations.

• (1) The γ(Q_t, n) is less than 1.

η_e is the maximum efficiency, and 1 is the load rate at η_e, as shown in Fig. 23.5.

Fig. 23.5

η_N (γ) curve of the pum** station with the identical pumps when γ(Q_t, n) is less than 1

In Fig. 23.5, we have

$$ \max \left( {\eta_{N} \left( {Q_{t} ,n - 1} \right),\eta_{N} \left( {Q_{t} ,n} \right),\eta_{N} \left( {Q_{t} ,n + 1} \right)} \right) = \eta_{N} \left( {Q_{t} ,n} \right) $$

(23.47)

When Q_t increases, the load rate γ(Q_t, n) increases, η_N(Q_t, n) and η_N(Q_t, n + 1) increase also. However when γ(Q_t, n) > 1, Q_t continues to increase, η_N(Q_t, n + 1) still increases, η_N(Q_t, n) decreases. When η_N(Q_t, n) < η_N(Q_t, n + 1), n + 1 is the optimal, we should increase the number of running pumps from n to n + 1, leading to

$$ \max \left( {\eta_{N} \left( {Q_{t} ,n - 1} \right),\eta_{N} \left( {Q_{t} ,n} \right),\eta_{N} \left( {Q_{t} ,n + 1} \right)} \right) = \eta_{N} \left( {Q_{t} ,n + 1} \right) $$

(23.48)

The optimal switch point is

$$ \eta_{N} (Q_{t} ,n) = \eta_{N} (Q_{t} ,n + 1) $$

(23.49)

When Q_t decreases, the load rate γ(Q_t, n) decreases, η_N(Q_t, n) decreases also, however η_N(Q_t, n-1) increases. When η_N(Q_t, n) < η_N(Q_t, n-1), n-1 is the optimal, we should decreases the number of running pumps from n to n-1, which yields,

$$ \max \left( {\eta_{N} \left( {Q_{t} ,n - 1} \right),\eta_{N} \left( {Q_{t} ,n} \right),\eta_{N} \left( {Q_{t} ,n + 1} \right)} \right) = \eta_{N} \left( {Q_{t} ,n - 1} \right) $$

(23.50)

The optimal switch point is

$$ \eta_{N} (Q_{t} ,n) = \eta_{N} (Q_{t} ,n - 1) $$

(23.51)

• (2) The γ(Q_t, n) is greater than 1, as shown in Fig. 23.6.

  Fig. 23.6

  

  η_N (γ) curve of the pum** station with the identical pumps when γ(Q_t, n) is greater than 1

In Fig. 23.6, we have

$$ \max \left( {\eta_{N} \left( {Q_{t} ,n - 1} \right),\eta_{N} \left( {Q_{t} ,n} \right),\eta_{N} \left( {Q_{t} ,n + 1} \right)} \right) = \eta_{N} \left( {Q_{t} ,n} \right) $$

(23.52)

When Q_t increases, the load rate γ (Q_t, n) increases, and η_N(Q_t, n + 1) increases also, however η_N(Q_t, n) decreases. When η_N( Q_t, n) < η_N(Q_t, n + 1), n + 1 is the optimal, we should increase the number of running pumps from n to n + 1, and we have

$$ \max \left( {\eta_{N} \left( {Q_{t} ,n - 1} \right),\eta_{N} \left( {Q_{t} ,n} \right),\eta_{N} \left( {Q_{t} ,n + 1} \right)} \right) = \eta_{N} \left( {Q_{t} ,n + 1} \right) $$

(23.53)

The optimal switch point is

$$ \eta_{N} \left( {Q_{t} ,n} \right) = \eta_{N} \left( {Q_{t} ,n + 1} \right) $$

(23.54)

When Q_t decreases, the load rate γ(Q_t, n) decreases, η_N(Q_t, n) and η_N(Q_t, n-1) both increase. When γ(Q_t, n) < 1, Q_t continues to decrease, η_N(Q_t, n-1) still increases, however η_N(Q_t, n) decreases. When η_N(Q_t, n) < η_N(Q_t,n-1), n-1 is the optimal, we should decreases the number of running pumps from n to n-1, and we have

$$ \max \left( {\eta_{N} \left( {Q_{t} ,n - 1} \right),\eta_{N} \left( {Q_{t} ,n} \right),\eta_{N} \left( {Q_{t} ,n + 1} \right)} \right) = \eta_{N} \left( {Q_{t} ,n - 1} \right) $$

(23.55)

The optimal switch point is

$$ \eta_{N} (Q_{t} ,n) = \eta_{N} (Q_{t} ,n - 1) $$

(23.56)

• (3) The γ(Q_t, n) is equal to 1

When Q_t increases, the load rate γ(Q_t, n) increases, η_N(Q_t, n + 1) increase also, however η_N(Q_t, n) decreases. When η_N(Q_t, n) < η_N(Q_t, n + 1), n + 1 is the optimal, we should increase the number of running pumps from n to n + 1, which leads to,

$$ \max \left( {\eta_{N} \left( {Q_{t} ,n - 1} \right),\eta_{N} \left( {Q_{t} ,n} \right),\eta_{N} \left( {Q_{t} ,n + 1} \right)} \right) = \eta_{N} \left( {Q_{t} ,n + 1} \right) $$

(23.57)

The optimal switch point is

$$ \eta_{N} (Q_{t} ,n) = \eta_{N} (Q_{t} ,n + 1) $$

(23.58)

When Q_t decreases, the load rate γ(Q_t, n) decreases, η_N(Q_t, n) decreases also, however η_N(Q_t, n-1) increases. When η_N(Q_t, n) < η_N(Q_t, n-1), n-1 is the optimal, we should decreases the number of running pumps from n to n-1, and we have

$$ \max \left( {\eta_{N} \left( {Q_{t} ,n - 1} \right),\eta_{N} \left( {Q_{t} ,n} \right),\eta_{N} \left( {Q_{t} ,n + 1} \right)} \right) = \eta_{N} \left( {Q_{t} ,n - 1} \right) $$

(23.59)

The optimal switch point is

$$ \eta_{N} (Q_{t} ,n) = \eta_{N} (Q_{t} ,n - 1) $$

(23.60)

This is “Same efficiency Switching”.

23.10 Optimal Switch of a Similar Efficiency Pum** Station

Assuming that n is optimal, the total power consumption is the smallest and the overall operating energy efficiency is the highest.

$${\mathit{minW}}_{t}=\frac{{W}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n}{Q}_{ie}})}$$

(23.61)

$$n\le m$$

$${\eta }_{N}\left(\frac{{Q}_{t}}{{\sum }_{i=1}^{n1}{Q}_{ie}}\right)\le {\eta }_{N}\left(\frac{{Q}_{t}}{{\sum }_{i=1}^{n}{Q}_{ie}}\right)$$

$$\gamma ({Q}_{t},n)=\frac{{Q}_{t}}{{\sum }_{i=1}^{n}{Q}_{ie}}$$

n1 is any combination other than this n combination.

When Q_t changes, we consider three situations.

• (1) The γ(Q_t, n) is less than 1

η_e is the maximum efficiency, and 1 is the load rate at η_e, as shown in Fig. 23.7.

Fig. 23.7

η_N (γ) curve of the similar efficiency pum** station when γ(Q_t, n) is less than 1

In Fig. 23.7, we have

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n\right)$$

(23.62)

When Q_t increases, the load rate γ(Q_t, n) increases, η_N(Q_t, n) and η_N(Q_t, n1) increase also. However when γ(Q_t, n) > 1, Q_t continues to increase, η_N(Q_t, n1) still increases,η_N(Q_t, n) decreases. When η_N(Q_t, n) < η_N(Q_t, n1), n1 is the optimal, we should change the number of running pumps from n to n1, which leads to,

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n1\right)$$

(23.63)

The optimal switch point is

$${\eta }_{N}({Q}_{t},n)={\eta }_{N}({Q}_{t},n1)$$

(23.64)

When Q_t decreases, the load rate γ(Q_t, n) decreases, η_N(Q_t, n) decreases also, however η_N(Q_t, n2) increases. When η_N(Q_t, n) < η_N(Q_t, n2), n-1 is the optimal, we should decrease the number of running pumps from n to n2, and we have

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n2\right)$$

(23.65)

The optimal switch point is

$${\eta }_{N}({Q}_{t},n)={\eta }_{N}({Q}_{t},n2)$$

(23.66)

• (2) The γ(Q_t, n) is greater than 1, as shown in Fig. 23.8

  Fig. 23.8

  

  η_N (γ) curve of the similar efficiency pum** station when γ(Q_t, n) is greater than 1

In Fig. 23.8, we have

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n\right)$$

(23.67)

When Q_t increases, the load rate γ(Q_t, n) increases, η_N(Q_t, n + 1) increases also, however η_N(Q_t, n) decreases. When η_N( Q_t, n) < η_N(Q_t, n1), n1 is the optimal, we should change the number of running pumps from n to n1, and we have

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n1\right)$$

(23.68)

The optimal switch point is

$${\eta }_{N}\left({Q}_{t},n\right)={\eta }_{N}\left({Q}_{t},n1\right)$$

(23.69)

When Q_t decreases, the load rate γ(Q_t, n) decreases, η_N(Q_t, n) and η_N(Q_t, n2) both increase. When γ(Q_t, n) < 1, Q_t continues to decrease, η_N(Q_t, n2) still increases, however η_N(Q_t, n) decreases. When η_N(Q_t, n) < η_N(Q_t,n2), n2 is the optimal, we should change the number of running pumps from n to n-1, leading to

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n2\right)$$

(23.70)

The optimal switch point is

$${\eta }_{N}({Q}_{t},n)={\eta }_{N}({Q}_{t},n2)$$

(23.71)

• (3) The γ(Q_t, n) is equal to 1

When Q_t increases, the load rate γ(Q_t, n) increases, η_N(Q_t, n1) increase also, however η_N(Q_t, n) decreases. When η_N(Q_t, n) < η_N(Q_t, n1), n1 is the optimal, we should change the number of running pumps from n to n1, which yields,

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n1\right)$$

(23.72)

The optimal switch point is

$${\eta }_{N}({Q}_{t},n)={\eta }_{N}({Q}_{t},n1)$$

(23.73)

When Q_t decreases, the load rate γ(Q_t, n) decreases, η_N(Q_t, n) decreases also, however η_N(Q_t, n2) increases. When η_N(Q_t, n) < η_N(Q_t, n2), n2 is the optimal, we should change the number of running pumps from n to n2, and we have

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n2\right)$$

(23.74)

The optimal switch point is

$${\eta }_{N}({Q}_{t},n)={\eta }_{N}({Q}_{t},n2)$$

(23.75)

This is “Same efficiency Switching”.

23.11 Pressure Measurement of Pump and Fan

Take the water supply pipe network as an example, as shown in Fig. 23.9. If the resistance loss in the pipe network is not considered and there is no external energy input, according to the law of conservation of energy, the energy of any two cross sections 1–1 and 2–2 in the pipe network should be equal, that is, the pressure energy, potential energy, and kinetic energy of the liquid on the two cross sections should be equal.

Fig. 23.9

Bernoulli's equation

In Fig. 23.9, p1, v₁, h₁, and s₁ respectively represent the pressure (pa, N/m²), water velocity (m/s), height (m), and area (m²) of the 1–1 cross section, while p₂, v₂, h₂, and s₂ respectively represent the pressure, water velocity, height, and area of the 2–2 cross section. The pressure F₁ (N) of cross section 1–1 and the pressure F₂ of cross section 2–2 are calculated as follow

$${F}_{1}={p}_{1}\times {s}_{1}$$

(23.76)

$${F}_{2}={p}_{2}\times {s}_{2}$$

Calculate the volumetric flow rate Q as follow

$$Q={s}_{1}\times {v}_{1}={s}_{2}\times {v}_{2}$$

(23.77)

where Q is volumetric flow rate, m³/s.

Calculate the mass flow rate as follow

$$m=\gamma \times Q$$

(23.78)

where m is the mass of the volumetric flow rate Q, and m is expressed in kg, γ is the density of the fluid in kg/m³. Unit time Δt = 1 s, the pressure energy at cross sections 1–1 and 2–2 in the pipeline causes changes in potential energy and kinetic energy as follow

$$\left({p}_{1}\times {s}_{1}\right)\times {v}_{1}\times \Delta t-\left({p}_{2}\times {s}_{2}\right)\times {v}_{2}\times \Delta t=\left(mg{h}_{2}-mg{h}_{1}\right)+\left(\frac{m{{v}_{2}}^{2}}{2}-\frac{m{{v}_{1}}^{2}}{2}\right)$$

(23.79)

where h₁ and h₂ are heights, m. Both sides of Eq. (23.79) are divided by mg, and the terms with the same base standard on both sides are merged. The relationship between pressure energy, potential energy, and kinetic energy per unit weight is obtained as follows:

$$ \frac{{p_{1} }}{{\gamma g}} + h_{1} + \frac{{v_{1} ^{2} }}{{2g}} = \frac{{p_{2} }}{{\gamma g}} + h_{2} + \frac{{v_{2} ^{2} }}{{2g}} $$

(23.80)

where h₁ and h₂ are the positional water heads, p₁ / (γ g) and p₂/ (γ g) is the pressure head (also known as static pressure), and v₁²/(2 g) and v₂²/(2 g) are the kinetic head (also known as dynamic pressure). Equation (23.80) is the famous Bernoulli equation. It represents the relationship between pressure energy, potential energy, and kinetic energy that can be converted into each other without external work and without resistance loss.

In the case of h₁ = h₂, an increase in kinetic energy will lead to a decrease in pressure energy. Similarly, a decrease in kinetic energy will lead to an increase in pressure energy. For example, if the pipeline becomes thicker, the flow rate will decrease, the dynamic pressure will decrease, and the static pressure will increase. If the pipeline becomes thinner, the flow rate will increase, the dynamic pressure will increase, and the static pressure will decrease. A fire water gun is an example of increasing dynamic pressure.

The total head H provided by the water pump (m, calculated from the inlet water surface) is converted into pressure energy, potential energy, and kinetic energy in the pipeline network without considering losses, leading to

$$H=\frac{{p}_{1}}{\gamma g}+{h}_{1}+\frac{{{v}_{1}}^{2}}{2g}=\frac{{p}_{2}}{\gamma g}+{h}_{2}+\frac{{{v}_{2}}^{2}}{2g}$$

(23.81)

It must be noted that the concept of head in the pump characteristic curve and pipeline network characteristic curve includes three parts: pressure energy, potential energy, and kinetic energy. When implementing energy-saving control for actual pum** stations, we are concerned about the operating parameters from the pump inlet to the pump outlet. We need to have a comprehensive understanding of the pressure, flow rate, and liquid level situation in this section, as shown in Fig. 23.10.

Fig. 23.10

Measurement of parameters in a pump station

In Fig. 23.10, h0, h1, h2, and h3 represent the liquid level height of the water tank, the height from the pump inlet pressure gauge to the pump shaft, the height from the pump outlet pressure gauge to the pump shaft, and the height from the pipe network pressure gauge to the pump shaft. P₁, P₂, and P₃ represent the water pump inlet pressure, water pump outlet pressure, and pipe network pressure, respectively. F represents the flow rate of the pump station.

When the liquid level h₀ of the water tank is higher than the center of the water pump shaft, if the inlet pipeline of the water pump is shorter, the resistance loss is not significant, and P₁ is positive pressure. When the liquid level of the water tank is lower than the center of the water pump shaft, or if the inlet pipeline of the pump is longer, the resistance loss is significant, resulting in P₁ being negative pressure. P₁ is static pressure.

When using a pressure gauge to measure static pressure, the actual pressure value at the measure point is the displayed value on the pressure gauge, plus the additional pressure added by the height of the pressure measuring element on the pressure gauge. Therefore, the actual static pressure at the inlet of the water pump is equal to P₁/(γg) + h₁, the actual static pressure at the outlet of the water pump is equal to P₂/(γg) + h₂. The actual static pressure of the pipeline network is equal to P₃/(γg) + h₃. Everyone must pay attention that the height of the pressure measuring element on the pressure gauge and the height of the pressure gauge display are sometimes not the same, unless they are integrated. The electrical signals measured by the pressure measuring element can be transmitted to far places to display.

Pressure p (pa), height h (m) and density γ (kg/m³), their conversion relationship is as follows:

$$h=\frac{p}{\gamma g}$$

(23.82)

Due to density γ different, the same pressure value results in different heights. For water, the density γ = 1000 kg/m³, 1Mpa pressure value is equivalent to a height of 102 m water, 1 kg/cm² pressure corresponds to a height of 10 m water, 1 standard atmospheric pressure (ATM) corresponds to a height of 10.33 m water, and 1 bar corresponds to a height of 10.2 m water. For mercury liquids, the same pressure is much lower when converted to a height of mercury. On the contrary, for oil with lower density, it is higher.

$$\begin{gathered} {\text{1 }}\left( {{\text{kg m}}} \right) = {\text{9}}.{\text{8 }}\left( {\text{J}} \right) \hfill \\ {\text{1 }}\left( {{\text{kW h}}} \right) = {\text{367}},{\text{17}}0\,\,({\text{kg m}}) \hfill \\ \end{gathered}$$

(23.83)

The flow rate of the pump is divided by the cross-sectional area of the pipeline to obtain the average flow velocity in the pipeline, and then the dynamic pressure is calculated. The total head of a pump is equal to the total pressure at the pump outlet (static pressure + dynamic pressure + h₂) minus the total pressure at the pump inlet (static pressure + dynamic pressure + h₁). When the diameter difference between the inlet and outlet of the pump is not significant, it can be approximated that the inlet and outlet dynamic pressures are equal. The total head of the pump is approximately equal to the static pressure at the outlet minus the static pressure at the inlet. In order to reduce resistance loss, the flow rate of pipelines designed according to specifications is generally not too high, so the proportion of dynamic pressure v₂²/(2 g) is not too large and can sometimes be ignored.

For the fan system, the dynamic pressure value already accounts for a significant proportion of the total pressure and cannot be ignored anymore, due to the density γ of the gas very small, the pressure formed by the installation height h of the pressure gauge is γgh, the numerical value is very small and can be ignored, so only static pressure and dynamic pressure are considered in the fan system.

The static pressure in the fan system can be directly measured using a pressure gauge or pressure transmitter, as shown in Fig. 23.11.

Fig. 23.11

Measurement of static pressure in fan systems

The measurement of dynamic pressure in a fan system is usually obtained by subtracting the static pressure from the total pressure. The total pressure and static pressure can be directly measured using a pitot tube, as shown in Fig. 23.12.

Fig. 23.12

Static pressure and total pressure

In Fig. 23.12, point A is located at the center of the pipeline, directly facing the direction of the incoming air. The pressure tap at point A measures the total pressure in the pipeline, while point B is located at the center of the pipeline. The pressure tap at point B is perpendicular to the direction of the incoming air, and measures the static pressure in the pipeline. The dynamic pressure is equal to the total pressure minus static pressure. The total pressure and static pressure are simultaneously fed into the two pressure taps of the differential pressure transmitter to directly measure the dynamic pressure.

For ventilation fans, the pressure is not high, and sometimes the static pressure h₁, full pressure h, and dynamic pressure h₂ can be directly measured on site using a U-shaped tube, as shown in Figs. 23.13, 23.14, and 23.15.

Fig. 23.13

Measuring static pressure with a U-tube

Fig. 23.14

Measure full pressure using a U-tube

Fig. 23.15

Using a U-shaped tube to measure dynamic pressure

Use pressure sensors to measure the pressure in pipelines and containers.

23.12 Practical Case of Energy Saving in a Pum** Station

The above theory and two theorems have been successfully applied to nearly a thousand pum** stations in the fields of building secondary water supply, urban water supply, steel, petrochemical, pharmaceutical and other fields.

A pum** station with a daily water supply of 400,000 tons is shown in Fig. 23.16. It has three speed-regulating centrifugal pumps of the identical model.

Fig. 23.16

The water delivery pump station of the water plant

The rated head of the pump is 39 (m), the rated flow is 4500 (m³/h), the rated speed n₀ is 980 (r/min). The flow-efficiency curve of the pump at the rated speed is shown in Fig. 23.17.

Fig. 23.17

Flow-efficiency curve Q-η

The flow-head curve H(Q) of the pump at the rated speed n₀ is shown in Fig. 23.18.

Fig. 23.18

Flow-Head Curve Q-H

The pum** station pressurizes the water to a fixed height H₀, H₀ is equal to 20 (m).

Based on the above conclusions, “Same pump, Same frequency” and “Same efficiency Switching”, referred to as “4S” technology, there is a curve as shown in Fig. 23.19.

Fig. 23.19

Optimal running curve

The optimal switching point Q₁₂ of 1 pump and 2 pumps is at Q_t = 4050(m3/hour), which leads to

$$Q_{{12}} = 4050\,({\text{m}}^{3} /{\text{h}})$$

(23.84)

When the total flow Q_t is less than Q₁₂, use 1 water pump and keep Q₁ = Q_t; when the total flow Q_t is greater than Q₁₂, use 2 water pumps and keep Q₁ = Q₂ = 0.5Q_t.

The optimal switching point Q₂₃ of 2 water pumps and 3 water pumps is at Q_t = 7254(m3/hour), and we have

$${Q}_{23}=7254({m}^{3}/h)$$

(23.85)

When the total flow Q_t is less than Q₂₃, use 2 water pumps and keep Q₁ = Q₂ = 0.5Q_t; when the total flow Q_t is greater than Q₂₃, use 3 water pumps and keep Q₁ = Q₂ = Q₃ = Q_t/3.

For example, if Q_t = 7100 (m3/hour), optimal control is achieved with 2 pumps, and the optimal control method is to keep

$${Q}_{1}={Q}_{2}=\frac{{Q}_{t}}{2}=3550({m}^{3}/h)$$

(23.86)

The minimum total power consumption is

$${\mathit{minP}}_{t}=\frac{{Q}_{t}{H}_{0}}{367{\eta }_{1}(\frac{{Q}_{t}}{2})}=437.465(kW)$$

(23.87)

Similarly, when Q_t = 11,700 (m3/hour), 3 pumps are used to achieve optimal control, and the optimal control method is to keep

$${Q}_{1}={Q}_{2}={Q}_{3}=\frac{{Q}_{t}}{3}=3900({m}^{3}/h)$$

(23.88)

The minimum total power consumption is

$${\mathit{minP}}_{t}=\frac{{Q}_{t}{H}_{0}}{367{\eta }_{1}(\frac{{Q}_{t}}{3})}=738.124(kW)$$

(23.89)

This case was successfully applied in May 1999.

23.13 Judgment Method for Optimal Efficiency Operation of Pum** Stations

Judgment method: For a pum** station composed of the identical type of pump, measure the operating efficiency curve within the full flow range when H = 0.5H_e. H_e is the rated head of the pump. If the pum** station operates with optimal energy efficiency, the curve has no jump point and the switching points are between n and n + 1 or n and n-1, as shown in Fig. 23.20; if the pump station operating efficiency curve jumps up and down, it is definitely not optimal.

Fig. 23.20

Yao curve

We call the optimal operating efficiency curve of H = 0.5H_e as the Yao curve.

The optimal operating efficiency curve when the operating head of the pum** station satisfies 0 < H < 0.7 h is in line with the above rules and can also be used as a Yao curve.

23.14 Stability of Pumps and Fans at Rated Speed

For some pumps and fans with low specific speed, their Q-H curves often appear in hump shape, while for pumps and fans with high specific speed, their Q-H curves often appear in saddle shape, as shown in Figs. 23.21 and 23.22.

Fig. 23.21

Q-H curves with a hump

Fig. 23.22

Q-H curves with a saddle

When such a pump or fan is connected to the pipe network, there may be more than one operating point, resulting in unstable operation.

Take the hump-shaped Q-H characteristic curve of a pump as an example, as shown in Fig. 23.23.

Fig. 23.23

Working status of pumps and fan with hump in Q-H curve

In Fig. 23.23, curve 1 is the Q-H curve of the pump at the rated speed ne, and curve 2 is the characteristic curve of the water supply network. The head of the pump reaches the highest point H_m at point B in the figure, and the pump flow rate corresponding to the highest head H_m is Q_m. For such a pump and pipe network, there are two operating points, A and C. If the system works at point A, due to interference, the working point A moves to the left. The head of the pump on the left side of point A is less than the head required by the pipe network, the flow rate decreases, and the working point continues to move to the left until the flow rate of the pump drops to zero. If the disturbance causes the working point A to move to the right, the head of the pump on the right side of A will be higher than the pressure required by the pipe network, the flow rate of the pump will increase. The working point will continue to move to the right until it reaches point C, and the system will be stable down, so the working point A is unstable, and it cannot return to the original working position after being disturbed.

Let's take a look at the working conditions of point C. Suppose that after the disturbance occurs, the working point C moves to the left. The head of the pump on the left side of point C is higher than the head required by the pipe network, so the flow increases, and the working point moves to the right again, return to point C. If the interference causes the working point C to move to the right, the lift of the pump on the right side of point C will be lower than the pressure required by the pipe network. The flow rate of the pump will decrease, the working point will move to the left, and return to point C, the system automatically stabilized, so the working point C is stable.

In fact, corresponding to the pump shown in Fig. 23.23, the working area on the left side of Q_m is an unstable area, and the working area on the right side of Q_m is a stable area. When the system works in an unstable region, the system is very likely to become unstable.

For the pump whose Q-H characteristic curve is a saddle shape, the analysis of its stability is the same as the above process.

Next, we take a type of pump with a container (water storage, oil storage, etc.) at the outlet as an example to analyze the instability of pump operation. Assume that the operating system of the pump is as shown in Fig. 23.24, and the Q-H characteristic curve of the pump is in the shape of a hump.

Fig. 23.24

A water tank in the pipeline

In Fig. 23.24, the water pump supplies water to the water tank, and the water flowing out of the water tank is supplied to the user. For the water pump, the difference between the liquid level of the water tank and the liquid level of the inlet pool is both the geodesic height Hg of the pipe network and the water outlet pressure of the water pump. It must be greater than H_g, and the pump can send out water, the characteristic equation of the pipe network is:

$$H={H}_{g}+K\times {Q}^{2}$$

(23.90)

If the water tank is not far from the water pump, the second part (K × Q²) is smaller, and the characteristic curve of the pipe network is relatively flat. As shown in Fig. 23.25, when the water tank is very close to the water pump, the pipe network characteristic curve is approximately a horizontal straight line.

Fig. 23.25

Working status of a pump with hump in Q-H curve

In Fig. 23.25, Curve 1 is the Q-H characteristic curve of the water pump, and Curve 2, Curve 3 and Curve 4 are the characteristic curves of the pipe network at different tank liquid levels.

When the liquid level of the water tank is H_g = H1, the characteristic curve of the pipe network is curve 2. At this time, the user’s water consumption is Q_d, the user’s water consumption is greater than Q_m, and the working point is stable at point D. When the user’s water consumption increases, the water tank’s liquid level will decrease, the characteristic curve of the pipe network will move downward, and the flow rate of the pump will increase. The network can realize self-stabilizing operation.

When the water consumption of the user is Q_m, the liquid level of the water tank reaches the highest point, and the characteristic curve of the pipe network moves up to the highest point, which is curve 4. At this time, if the water consumption of the user decreases, the characteristic of the pipe network moves from point B to the left, and the pump characteristic curve is separated from the characteristic curve of the pipe network. The head of the pump is less than the pressure of the water in the water tank, and the water flows back. Because there is a check valve in front of the pump, the water cannot flow back, and the flow rate of the pump drops to zero. As the user uses water, the liquid level of the water tank gradually decreases. When the liquid level of the water tank drops below the closing head H_c of the pump, the head of the pump is greater than the pressure of the water tank, and the pump starts to discharge water. Since the water consumption of the user is small, the pump’s flow rate is larger than the user's water consumption, the liquid level of the water tank rises, and the characteristic curve of the pipe network moves up. The pump will oscillate back and forth in the unstable range from H_c to H_m. The amplitude of the oscillation depends on the size of the user's flow rate. When the distance is small, the oscillation period will be very short.

For such a system, only when the user's water consumption is greater than Q_m, the system can achieve stable water supply, and the liquid level of the water tank will be stable at a certain level.

For the fan system, there is usually no check valve at the outlet of the fan, so when the shock occurs, the gas backflow phenomenon will appear, which is called the surge phenomenon. The division of the unstable area of ​​the fan is the same as the analysis process of the above-mentioned water pump.

The pipe network characteristic equation of the gas path is generally:

$$P={K}_{1}\times {Q}^{2}$$

(23.91)

The characteristic curve of the pipe network is shown in Fig. 23.26.

Fig. 23.26

Q-P characteristic curve of gas path

In Fig. 23.26, the curve is the Q-H characteristic curve of the fan. For the fan system, in most cases, as long as there is wind pressure, there will be flow in the pipeline, so the shape of the pipe network characteristics of the air supply system is basically a quadratic curve passing through the origin P_g = 0.

In fact, there are also some fan systems, only when the wind pressure is greater than a certain value, there is flow in the pipeline. For such pipelines, P_g ≠ 0, the pipe network characteristic curve is not a quadratic parabola passing through the origin, so what kind of pipeline has this characteristic? For example, the pipe network with a water tank filter in the air circuit, and the pipe network with a constant pressure opening valve, that is the case. Take the pipe network with a water tank filter in the air circuit as an example, as shown in Fig. 23.27.

Fig. 23.27

A water tank in the fan pipeline

The shape of the characteristic curve of the pipe network is shown in Fig. 23.28. This type of pipe has flow rate only when the pressure is greater than P_g.

Fig. 23.28

The characteristic curve of the pipe network with water tank

Next, we use an example of a type of fan with a large container at the outlet to analyze the unstable operation of the fan. Assume that the fan system is shown in Fig. 23.29, and the Q-H characteristic curve of the fan in the figure is a hump shape.

Fig. 23.29

A large container or a thick pipe network in the fan pipeline

In Fig. 23.29, the fan supplies air to the user through a large container (thicker pipeline). Because the large container has the function of storing gas, the pressure of the pipe network is greater than the outlet pressure of the fan when the fan is powered off for a short time. If the fan outlet valve is not closed, air from the large container will flow back to the fan.

The operation analysis of the fan’s flow-pressure characteristic curve and the pipe network characteristic curve is shown in Fig. 23.30.

Fig. 23.30

Working status of fan with hump in Q-P curve

In Fig. 23.30, curve 1 is the Q-P characteristic curve of the fan. When the gas consumption of the user is Qc, the characteristic curve of the pipe network is curve 2. Since the gas consumption of the user is greater than Q_m, the working point is stable at point C. When the air volume increases, the pressure of the pipe network decreases and the flow rate of the fan increases. When the air consumption of the user decreases, the pressure of the pipe network increases and the flow rate of the fan decreases. The fan and pipe network can realize self-stabilizing operation.

When the gas consumption of the user is Q_m, the output pressure of the fan reaches the highest. At this time, if the gas consumption of the user decreases and the pressure of the pipe network increases again, the output pressure of the fan will be lower than that in the pipe network (also a large container). The large container supplies gas to the user, and the gas in the large container also flows back to the fan. The fan has a negative flow rate, and the operation curve enters the left side of the P coordinate axis. With the gas consumption, the pressure in the large container drops rapidly. When the pressure in the large container drops below the closing head P_c of the fan, the pressure of the fan is greater than the pressure in the large container, and the fan starts to supply air to the outside. Due to the small air consumption of the user, the flow rate of the fan is larger than the user's air consumption, and the pressure in the large container rises, the fan will oscillate back and forth in the unstable range from P_c to P_m, and the oscillation range depends on the size of the user's flow. When the distance between P_c and P_m is small, the oscillation cycle will be accelerated. For such a system, only when the user's gas consumption is greater than Q_m, the system can achieve stable gas supply.

The stability analysis of the fan and pipe network curve of the saddle-shaped Q-P is shown in Fig. 23.31.

Fig. 23.31

Pipeline behavior with saddle Q-P curve

In Fig. 23.31, curve 1 is the Q-P characteristic curve of the fan. When the gas consumption of the pipe network is at point Q_a, the pipe network curve is curve 2, and the operating point is A. The system can run stably. The analysis process is as before. When the pipe network When the gas consumption of the network is at Q_b, the working point is B, and the pipe network curve is curve 3. There will be two intersection points B and E between the pipe network curve and the Q-P characteristic curve. When the gas consumption of the pipe network is less than Q_b such as curve as shown in 4, the pipe network curve of the system and the Q-P characteristic curve of the fan will have three intersection points C, D and F. When the gas flow rate of the pipe network is less than Q_b, the system will easily enter an unstable operation state, and the analysis process is omitted.

When the pipe network curve intersects with the Q-P characteristic curve of the fan, as shown in Fig. 23.32, the system is likely to enter an unstable operation state, and the analysis process is omitted.

Fig. 23.32

Q-P curve with saddle and pipe network curve

In Fig. 23.32, curve 1 is the Q-P characteristic curve of the fan, and curve 2 is the pipe network curve.

When the pump or fan does not adjust the speed, the stability judgment and stability guarantee measures are summarized as follows:

• (1) When the flow head Q-H characteristic curve of the pump or the flow pressure Q-P characteristic curve of the fan are in the shape of a hump or a saddle, such a pump or fan may have unstable operation. It is the most fundamental factor affecting the stable operation. If possible, choose a pump or fan with a flow head (pressure) characteristic curve without hump and saddle shape.

• (2) If it is not possible to choose a pump or fan without hump and saddle shape on the flow head (pressure) characteristic curve, select the operating point to make the system run in the stable area on the right side of the highest head (highest pressure) point, and avoid working at the highest head (highest pressure) point to the unstable area on the left. We can change the characteristic curve of the pipe network, change the vane angle of the inlet deflector to change the flow head (pressure) characteristic curve of the pump or fan, or use a bypass valve or other measures to increase the flow rate.

23.15 Stability Criteria After Speed Adjustment of Pumps and Fans

For pumps or fans with hump and saddle-shaped characteristic curves of flow head (pressure), the problem that one frequency corresponds to two flow points will also appear after the implementation of frequency conversion speed regulation, so if no targeted measures are taken, there will also be problems Control instability, as described in the automatic control theory in a large number of pages, stability is the primary problem of the control system.

For the pump or fan with hump and saddle shape on the flow-head (pressure) characteristic curve, the system is set to automatically control according to the constant pressure H_s, as shown in Fig. 23.33.

Fig. 23.33

Stability of constant pressure speed regulation

In Fig. 23.33, curve 1 is the pipe network characteristics required for constant pressure, and the required constant pressure value is Hs. Curve 2 is the connecting curve of the highest point of the Q-H characteristic curve after speed regulation. Curve 2 intersects the constant pressure line 1 at Q_m, and curves 3, 4, 5, and 6 are the Q-H characteristic curves of the pump or fan at four speeds of n1, n2, n3, and n4 respectively.

It can be seen from Fig. 23.33 that when the water consumption Q of the pipe network is greater than Q_m, the larger the flow rate of the pipe network (Q3 to Q4), the higher the speed of the pump or fan (n3 to n2). From the principle of automatic control, this is a normal feedback control system, and general controllers or regulators can control this system very well. When the flow rate Q of the pipe network is less than Q_m, the flow rate of the pipe network is smaller (Q2 to Q1), and the higher the required speed (n3 to n2). This adjustment method is opposite to the above normal situation, the automatic control system has the problem of control law uncertainty. Q_m is the critical unstable point of the system, and the right side of Q_m is the stable operation area, the left side of Q_m is the unstable operation area.

Assume that the system automatically controls pressure according to the characteristics of the pipe network under different flow rates, as shown in Fig. 23.34.

Fig. 23.34

Stability after speed change

In Fig. 23.34, curve 1 is the characteristic curve of the pipe network, and curves 3, 4, and 5 are the Q-H characteristic curves of the pump or fan at three speeds of n1, n2, and n3.

It can be seen from Fig. 23.34 that the characteristics of the pipe network are tangent to curve 5 at Q_m. When the water consumption Q of the pipe network is greater than Q_m, the greater the flow rate of the pipe network (Q2 to Q3), the higher the speed of the pump or fan (n2 to n1). From the principle of automatic control, this is a normal feedback control system, and general controllers or regulators can control this system very well. When the flow rate Q of the pipe network is less than Q_m, the pipe network has smaller flow rate, the higher the required speed. This adjustment method is contrary to the above normal situation. The automatic control system has control uncertainty problems. Q_m is the critical unstable point of the system, and the right side of Q_m is the stable operation area. The left side of Q_m is the unstable operation area.

The occurrence of these unstable phenomena is mainly due to the fact that the flow-head (pressure) characteristic curve of the pump or fan is not like the normal flow-head (pressure) characteristic curve, and a hump appears.

Although the flow head (pressure) characteristic curve of some water pump fans has no hump, its top changes gently in a wide range. In such a system, a small change in the speed in the low flow area will cause a large change in flow, which will also give Control stability brings problems. At this time, it is necessary to improve the accuracy of control output, reduce the range of output change, and not use sensitive speed signals as control targets. Use the flow signal to limit the range of frequency changes.

Stability criteria (Yao’s stability criteria): Assuming that Q_m and P_m are the flow rate and head corresponding to the highest hump point of the pump or fan flow head curve, and H_sv is the full head (or full pressure) set by the process, and the pump or fan is centrifugal, the necessary condition for the stable operation of the pump or fan after speed regulation is

$$Q>{Q}_{m}\times \sqrt{\frac{{H}_{SV}}{{H}_{m}}}$$

(23.92)

Note: This oscillation is not an oscillation caused by the mechanical resonance frequency of the pump (for example, a long-axis water pump), but a controlled oscillation caused by the shape of the pump characteristic curve.

23.16 Overload Problems of Pumps and Fans

For the pump or fan with low specific speed, the flow rate-shaft power Q-N characteristic curve, the greater the flow rate, the greater the shaft power. If the outlet pressure is very low or the outlet is damaged, the flow rate will be too large, and the motor driving the pump or fan is prone to overload and heat generation, exceeding the rated power of the matched motor, the motor is overheated, the alarm trips, and it burns out, as shown in Fig. 23.35.

Fig. 23.35

Q-N flow rate-shaft power characteristic curve (1)

In Fig. 23.35, when the flow rate is greater than Q_m, the shaft power of the pump or fan is greater than N_m, and the output power of the motor driving the pump or fan is close to the rated power. If the flow rate is larger, the motor will be overloaded. To avoid overload, we need to use current transmitter monitors the running current of the motor, and when it is found that the running current is too large, the valve is turned off a little. To ensure that the flow rate of the pump or fan will not be too large.

For the pump or fan with high specific speed, the flow rate-shaft power Q-N characteristic curve, the smaller the flow rate, the greater the shaft power. So for this kind of pump or fan, if a valve is used to control the outlet flow rate, it must be noted that if the valve opening is too small, the flow rate is too small or zero, which will cause the shaft power of the pump or fan to rise too fast, exceeding the rated power of the motor, and the motor will overheat, and the alarm will trip, as shown in Fig. 23.36.

Fig. 23.36

Q-N flow rate-shaft power characteristic curve (2)

In Fig. 23.36, when the flow rate is less than Q_m, the shaft power of the pump or fan is greater than N_m, and the output power of the motor driving the pump or fan is close to the rated power. If the flow rate decreases further, the motor will be overloaded. In order to avoid the overload problem, measures need to be taken to ensure that the flow rate control valve of the pump or fan is not closed or closed too small, and a current transmitter is used to monitor the operating current of the motor. When the operating current is found to be too large, the valve is opened a little.

23.17 Mechanical Resonance Problem of Pumps and Fans

For oscillations caused by the mechanical resonance frequency of pumps and fans (such as long-axis pumps), dangerous high and low speed limit zones should be set to avoid operating in this range.

For example, for the ACS510 inverter, set the codes 2501 (CRIT SPEED SEL), 2502 (CRIT SPEED LO), and 2503 (CRIT SPEED HI).

23.18 Other Issues that Need Attention

Most water supply systems consist of multiple water pumps connected in parallel. A small amount of water drip** from the pump shaft can ensure heat dissipation at the shaft end. The pressure at the center of a centrifugal pump is low, and air will be sucked into the pump through the shaft end. Air accumulation in the pump will cause the pump to be unable to supply water, or the flow rate of the pump will be reduced. Gradually reduce until no water flows out. This can easily happen when the water level in the pum** station's inlet pool is lower than the height of the pump shaft. This should attract enough attention. At this point, beginners don’t need to panic. Note the minimum fluid level in the inlet tank.

23.19 Conclusion

There is no need to establish an accurate mathematical model of a pum** station, based on the characteristics of the energy efficiency function, this chapter presents a constrained, nonlinear, integer-real-number hybrid energy efficiency optimization method for pum** stations.

This optimization method includes two theorems: optimal load distribution theorem and optimal switching theorem.

Optimal load distribution theorem: The optimal load distribution method of a pum** station is to keep the operating energy efficiency of each operating pump equal, i.e., Yao Theorem 1.

$${\eta }_{\text{N}}\left({\gamma }_{1}\right)={\eta }_{\text{N}}\left({\gamma }_{2}\right)=\dots ={\eta }_{N}\left({\gamma }_{n}\right)$$

(23.93)

Optimal switching theorem: The optimal switching point for the number of operating pumps is at the point of equal efficiency or at the maximum output point of the pumps, i.e., Yao Theorem 2.

$${\eta }_{\text{N}}\left(\frac{{Q}_{t}}{{\sum }_{i=1}^{n}{Q}_{ie}}\right)={\eta }_{\text{N}}\left(\frac{{Q}_{t}}{{\sum }_{i=1}^{k}{Q}_{ie}}\right) or \frac{{Q}_{t}}{{\sum }_{i=1}^{n}{Q}_{ie}}={\gamma }_{1m}$$

(23.94)

We call above optimization method as the Quantum Optimization Method of Pum** Stations.

We call above theory as the Energy Efficiency Predictive Theory of Pum** Stations.