According to the idea of momentum balance in Bernoulli's equation, the mechanical balance for the flow through a pipe is written as:
$\dfrac{p_1}{\gamma}+z_1+\dfrac{v_1^2}{2g}=\dfrac{p_2}{\gamma}+z_2+\dfrac{v_2^2}{2g}$ Eq. (01)
Therefore, it is easy to express the (total) mechanical energy of the fluid at any point, along the pipe, as,
$H=\dfrac{p}{\gamma}+z+\dfrac{v^2}{2g}$ Eq. (02)
On the other hand, the quantity of transferred energy by a pump to the fluid is precisely the difference between the energy of the fluid at the pump's discharge port and its energy at the suction port. This is,
$H_B=H_D-H_A$ Eq. (03)
Also, if Eqs. (02-03) are combined an expression for $H_B$ in terms of fluid velocity, pressure, fluid and geometrical properties can be obtained. This is as follows,
$H_B=\dfrac{p_D}{\gamma}+z_D+\dfrac{v_D^2}{2g} - \left( \dfrac{p_A}{\gamma}+z_A+\dfrac{v_A^2}{2g} \right)$whcih can be reorganized as,
$H_B=\dfrac{p_D-p_A}{\gamma}+\left(z_D-z_A\right)+\dfrac{v_D^2-V_A^2}{2g}$ Eq. (04)
and in terms of the flow rate $Q$ Eq. (04) becomes,
$H_B=\dfrac{p_D-p_A}{\gamma}+\left(z_D-z_A\right)+\dfrac{Q^2}{2g}\left( \dfrac{1}{A_D^2} -\dfrac{1}{A_A^2}\right)$ Eq. (05)
where $A$ is the pipe cross section. If, for any reason pressure transducers are not located at the suction and discharge ports but a little far (below or above) some correction would need to be made into Eq. (05).
Also, the hydraulic power $N_H$ transferred by the pump to the fluid is usually expressed as,
$N_H=\gamma Q H_P$ Eq. (06)
The power from the electric motor
The power from the electric motor can be estimated with,
$N_M=M\omega$ Eq. (07)
where $M$ is the torque defined in terms of the force $F$ and the radius $r$ as,
$M=Fr$ Eq. (08)
Also, the angular velocity $\omega$ is defined in terms of the turns $n$ as,
$\omega=\dfrac{\pi}{30}n$ Eq. (09)
In this way, the power $N_M$ in EQ. (07) can be rewritten as,
$N_M=\dfrac{\pi}{30}FRn$ Eq. (10)
so that the mechanical power can be estimated from measurable parameters in the equipment. $F$ is the torque exerted and $r$ the length of the shaft.
About the efficiency
Efficiency in important to have an idea of how well a pump is working or to determined possible faults. Efficiency could depen on factors such as,
- mechanical wearing,
- mechanical overload,
- changes in the fluid viscosity.
The mechanical power comes from the electricity supplied to the motor which converts into mechanical energy by making the shaft to spin which in turn makes the fluid to move. The capacity of the electric motor to convert electrical energy into mechanical is not perfect and may range between 70% to 95% or more according to the motor design and manufacturer. Let us called this efficiency $\eta_i$.
Also, the mechanical energy taken by the shaft is not totally transferred to the fluid as motion. Again, there is certain efficiency, less than 100%, for this. Factor affecting this efficiency can be friction and heat losses. Let us call this efficiency $\eta_M$.
Then, what would be the global efficincy of a pump? It necessarily would be a ratio of the power, mechanical and electrical, since not all electrical power is converted into mechanical. This is,
$\eta=\dfrac{N_H}{N_M}$ Eq. (11)
Dimensional analysis
There are common physical forces involved in the operation of a pump that lead to dimensional groups. These can be obtain for the flow rate $Q$, the head or mechanical energy $H$ and for the mechanical power $N$. These are,
$Q^*=\dfrac{Q}{n^2D^3}$ Eq. (12)
$H^*=\dfrac{Hg}{n^2D^2}$ Eq. (13)
$N^*=\dfrac{N}{n^3D^5}$ Eq. (14)
where $D$ is the impeller diameter. From Eqs. (12-14) scale-up and similarity can be implemented for pumps of different size so that measurements and extra-work can be saved.
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| Fig. 01 Ejemplos de curvas características de una bomba |
The curves
Therefore, if you want to plot the data of a pump you need to know the impeller diameter and instruments to measure the pressure and flow rate. An schematic of the instruments and is position is briefly presented in Fig. 02.
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| Fig. 02 Instruments for pump measurements |
So that once you measure the pressure and flow rate for different impeller speeds and torque ya should be able to use the equations above to make the plots displayed in Fig. 01.
Any question? Write in the comments and I shall try to help.
Other stuff of interest
- LE01 - AC and DC voltage measurement and continuity test
- The pros and cons of bulb thermometers
- Some examples of temperature instruments
- Minor losses - Formulas
- What is a process variable?
- What are the most important process variables?
- Time dependence of process variables
- A list of process variables
Ildebrando.
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