EE proj pump-valve combination

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ECET 380-Numerical Methods-Project 1 – Week 1

Table of Contents Introduction .................................................................................................................................................. 1

The Pump ...................................................................................................................................................... 1

The Pump Curve ........................................................................................................................................ 2

Pumps in series ......................................................................................................................................... 3

Pumps in parallel ....................................................................................................................................... 3

The system curve .......................................................................................................................................... 4

Where does the pump operate? ................................................................................................................... 4

The Project Itself ........................................................................................................................................... 5

What you need to do ................................................................................................................................ 6

Tips for solving this project ........................................................................................................................... 6

What should be in your report ..................................................................................................................... 6

Introduction

This week’s project concerns itself with a pump-valve combination, in which a pump discharges to a valve, which then discharges back to a pump. For ECE students, think of a pump as a current dependent voltage source (the voltage is dependent on the current), and the valve is a resistor. Later we will show how the system obeys a mechanical version of Kirchoff’s Voltage Law. ECE students have actually seen this kind of a system: In ECEL 302 you worked with BJTs, which is basically an electrical version of a pump.

The Pump The purpose of a pump is to increase the pressure of a flow so it can flow. In a closed system, the purpose of a pump is to provide enough flow energy such that it overcomes system losses. It is very common to characterize pumps in terms of head gain. The units of head gain is in meters.

The head gain of a pump is often characterized by the following equation:

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𝐻𝐻 = 𝐻𝐻0𝑙𝑙𝑙𝑙�𝑒𝑒 + 𝐴𝐴 𝐹𝐹 𝐹𝐹0

+ 𝐵𝐵 � 𝐹𝐹 𝐹𝐹0 � 2

+ 𝐶𝐶 � 𝐹𝐹 𝐹𝐹0 � 3

�

Where

H = pump head gain (m)

H0 = Maximum head (also known as shutoff head – no flow rate exists) (m)

F = volumetric flow rate (m3/s)

F0 = maximum volumetric flow rate (m3/s)

A, B, C are coefficients

In our model, the pump will have a head of H0 at F=0, and a head of 0 at F=F0. This allows us to place one of the coefficients in term of the other coefficients. Using the limit there is no head at the maximum flow rate, we start our derivation:

0 = 𝐻𝐻0𝑙𝑙𝑙𝑙(𝑒𝑒 + 𝐴𝐴 + 𝐵𝐵 + 𝐶𝐶)

In order for the equation to be satisfied, we see that

𝑒𝑒 + 𝐴𝐴 + 𝐵𝐵 + 𝐶𝐶 = 1

Solving for C, we have

𝐶𝐶 = 1 − 𝑒𝑒 − 𝐴𝐴 − 𝐵𝐵

The Pump Curve

The pump curve displays the head gain of the pump as a function of the pump capacity (here, it is volumetric flow rate). Figure 1 displays a typical pump performance curve. Note this is the head gain for a single pump.

Figure 1 Example pump performance curve for single pump

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Pumps in series

When two pumps are in series, the total head gain of the pumps is the sum of the head gain from the individual pumps for the same flow rate. This is the same as having two voltage sources in series: the total voltage gain is additive. Figure 2 displays an example performance for a single pump and two pumps in series.

Figure 2 Example pump performance curve for single and series operation

Pumps in parallel

When pumps are in parallel, the flow rate entering each pump is the same (assuming identical pumps). Additionally, the pump head gain for each pump is the same. This is the same as identical voltage sources in parallel: The current is evenly distributed, while the voltage across the voltage sources are the same. Thus, for the same head, the flow rates are additive. Figure 3 displays an example of two pumps in parallel.

Figure 3 Example performance curve for pumps in parallel

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The system curve

Now that we have a basic idea of how pumps operate, we need to know how the system performance behaves. Typical parts of a system are valves and piping. These components provide a head loss. An electrical analogy of system parts are resistors. Each component has their basic equation that characterizes the head loss. However, in this project, we will focus only on a valve, which has a heal loss characteristic of

𝐻𝐻𝐿𝐿 = 𝐾𝐾 𝑣𝑣2

2𝑔𝑔

Where

K = head loss coefficient

v = fluid velocity (m/s)

g = gravitational acceleration (m/s2)

While K is unitless, we really don’t want to use the velocity, but rather the volumetric flow rate. If at any one time the valve is not moving, we assume a constant cross sectional area. This allows us to rewrite the head loss equation as

𝐻𝐻𝐿𝐿 = 𝐾𝐾′ �̇�𝑉2

2𝑔𝑔

Where �̇�𝑉 is the volumetric flow rate (m3/s). Note that K’ is NOT unitless.

Where does the pump operate?

In a closed system, a centrifugal pump (this is the pump this project is using) will operate at the point where the pump head gain equals the system head loss. In other words, the net head gain/loss in a closed system is zero. This is analogous to Kirchoff’s Voltage Law: The voltage sources will provide enough voltage such that the sum of the voltage drops in a loop equals the sum of the voltage gains in a loop. Figures 2-3 indicate the pump flow rate for single, series, and parallel operation (points 1 and 3 in each of these figures).

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The Project Itself

In this project, you will be determining the operating points of a single pump, two identical pumps in series, and two identical pumps in parallel for a given system. Note in this system we only focus on the valve head loss equation above, and assume no other losses in the loop. The following characteristics used in this project are:

Parameter Variable Value Shutoff Head (m) H0 50

Maximum Flow Rate (m3/s) F0 100 Pump Parameter A 0 Pump Parameter B -2 Pump Parameter C 1-e-A-

B Valve Head Loss Parameter K’ 0.2 Gravitational Acceleration

(m/s2) G 9.8

Note each of the pump parameters above is for a single pump.

For this project, you need to display your results in a graph format shown below:

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Note the blue lines are the pump performance curves (single, series, parallel), and the red line is the system curve. The black dots indicate the pump operating points for each configuration.

What you need to do Note you need to solve the operating points using two different approaches: the bisection and false-position methods. So you should have two different plots for your report, each in the format shown above. The solutions should have a relative error of 0.1%.

Tips for solving this project

1. Use anonymous (inline) functions in MATLAB, or lambda functions in Python. 2. You are needing to find the intersection of the pump characteristic and system curves.

This means you need to find the solution to ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑔𝑔𝑒𝑒𝑔𝑔𝑙𝑙 − ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = 0 3. If you are an ECE student, think of this in terms of voltage sources and resistors

What should be in your report

The following sections should be in your report:

1. Introduction 2. Theory of pump/system operation (note: Do NOT go into detail in this section, but

rather a basic introduction to the theory, such as noted above) 3. Theory of bisection and false-position methods 4. System configuration and parameters 5. Results, but graphical and numerical 6. Discussion 7. Appendix that displays your code you wrote