Practical Models for Thermodynamic Measurement of Pump Efficiency: Derivation and Validation

As a fluid machine widely used in industrial and agricultural applications, the accurate assessment of pump operating efficiency is important for reducing energy consumption and maintaining efficient system operation. In hydraulic systems, pump pressure, flow rate, and power are closely coupled with operating conditions and constitute fundamental parameters for performance regulation and energy utilization [1]. Accurate determination of pump efficiency is therefore a prerequisite for identifying energy-saving opportunities. At present, there are mainly two methods for testing pump efficiency: the hydraulic method (traditional method) and the thermodynamic method. The main difference between the two methods is that the hydraulic method needs to directly measure the pump flow rate, and then calculate the pump efficiency by using a formula, and the prediction accuracy of the pump efficiency depends on the accuracy of the flow measurement; the thermodynamic method, on the contrary, does not need to directly measure the pump flow rate, and can directly measure the pump efficiency, and then back-calculate the pump flow rate from the shaft power of the pump, thereby obtaining the performance parameters of the pump [2, 3].

The theory of the thermodynamic method was proposed as early as the 1960s [4-6], but constrained by the extreme requirement for the measurement accuracy of temperature difference (millikelvin level), its large-scale engineering application was not gradually carried out until the last ten years or so. Thorne and Neal [7] published a study on the independent evaluation of the thermodynamic method, summarized the practical guidelines and the temperature probe calibration procedures of the direct thermodynamic method, and identified the consistency of the method and the problems to be studied through a series of field comparison tests; the Korean scholar Bae et al. [8] verified the consistency between the thermodynamic method and the classical hydraulic method through precise comparison experiments, confirming the reliability of this method for pump performance assessment; Clifford [9] systematically developed a long-term high-precision differential thermometer set for industrial environments, and achieved a measurement accuracy of 0.29 mK in a throttling calorimeter; Ye et al. [10] proposed an optimized thermodynamic efficiency measurement method adapted to pump-turbines, and completed comparison tests at several pumped storage power stations; Patil [11] applied the thermodynamic theory to other liquid machines; Li et al. [12] introduced entropy production analysis into the loss identification of hydraulic machines for the first time, and subsequent studies have widely applied it to centrifugal pumps, pump-turbines, multi-stage pumps and other equipment. The above studies are mostly limited to specific equipment or laboratory comparisons and therefore provide limited guidance for standardized field assessment across different operating environments. More generally, engineering energy systems require clearly defined and comparable performance indicators if technical performance is to be evaluated consistently across operating conditions [13]. In view of these limitations, it is necessary to establish a thermodynamic-method-based pump efficiency measurement framework that considers both measurement accuracy and engineering practicability, so as to fill the gap between the sophisticated theory and the complex industrial field, and to provide a strong technical guarantee for the energy saving, consumption reduction and safe operation of pump systems.

3.1 Thermodynamic temperature-difference method

According to the thermodynamic steady flow energy equation, the pump efficiency calculation formula (1) can be changed into:

$\eta=\frac{h_{2 s}-h_1}{\left(h_2-h_1\right)+\Delta e^{\prime}}$                  (2)

where,

η — efficiency of the pump;

h₂ₛ — specific enthalpy at the pump outlet under the ideal state (isentropic process);

h₁ — specific enthalpy at the pump inlet in the actual non-isentropic compression process;

h₂ — specific enthalpy at the pump outlet in the actual non-isentropic compression process;

$\Delta e^{\prime}$ — the part of the heat energy converted from various losses that is not absorbed by the liquid.

According to the thermodynamic theory, the calculation formula of the enthalpy change under the ideal state (isentropic process) is as shown in Eq. (3), and the calculation formula of the enthalpy change under the actual non-isentropic state is as shown in Eq. (4).

$h_{2 s}-h_1=\bar{v}\left(p_2-p_1\right)$                      (3)

where,

$\bar{v}$ — mean specific volume of water;

$p_2-p_1$ — pressure difference between the inlet and outlet of the pump.

$h_2-h_1=\bar{a}\left(p_2-p_1\right)+\bar{C}_p\left(T_2-T_1\right)$                        (4)

where,

$\bar{a}$ — coefficient of compressibility of water, which is called the isentropic factor in the international standard;

$\bar{C}_p$ — mean specific heat capacity of water;

$T_2-T_1$ — temperature difference between the inlet and outlet of the pump.

Substituting Eqs. (3) and (4) into Eq. (2), the basic mathematical model for testing the pump efficiency by the thermodynamic method can be obtained as follows:

$\eta=\frac{\bar{v}\left(p_2-p_1\right)}{\bar{a}\left(p_2-p_1\right)+\bar{C}_p\left(T_2-T_1\right)+\Delta e^{\prime}}$                     (5)

According to different objects and needs, different simplifications and calculations can be carried out on Eq. (5). Dividing the numerator and the denominator of Eq. (5) by $\bar{v}\left(p_2-p_1\right)$ at the same time, Eq. (6) is obtained.

$\eta=\frac{1}{\frac{\bar{a}}{\bar{v}}+\frac{\bar{C}_p\left(T_2-T_1\right)}{\bar{v}\left(p_2-p_1\right)}+\frac{\Delta e^{\prime}}{\bar{v}\left(p_2-p_1\right)}}$                     (6)

Because the pump head $H=\frac{\bar{v}\left(p_2-p_1\right)}{g}$, Eq. (6) can be changed into:

$\eta=\frac{1}{\frac{\bar{a}}{\bar{v}}+\frac{\bar{C}_p\left(T_2-T_1\right)+\Delta e^{\prime}}{g H}}$                 (7)

Under different pressures and different temperatures, the pump efficiency can all be calculated according to the above equation. Taking the case of 1 standard atmospheric pressure and a water temperature of 20 °C as an example, it is found from the tables that: $\bar{C}_p=4186 \mathrm{~J} /(\mathrm{kg} \cdot \mathrm{K})$, $\bar{v}=0.001002 \mathrm{~m}^3 / \mathrm{kg}$,  $\bar{a}=9.39 \times 10^{-4} \mathrm{~m}^3 / \mathrm{kg}$, $g=9.81 \mathrm{~m} / \mathrm{s}^2$. Substituting these into Eq. (7) gives:

$\eta=\frac{1}{0.937+\frac{426.7 \Delta T}{H}+\frac{\Delta e^{\prime}}{\mathrm{g} H}}$           (8)

where,

$\Delta T$ — temperature difference between the outlet and inlet of the pump;

H — head of the pump.

$\Delta e^{\prime}$ is mainly the bearing and shaft seal friction losses and the heat dissipation loss of the pump casing, and the proportion of the losses is usually less than $1 \%$. In general engineering applications, item $\Delta e^{\prime}$ can be neglected, and then the above equation can be simplified as:

$\eta=\frac{1}{0.937+\frac{426.7 \Delta T}{H}}$                       (9)

3.2 Thermodynamic pressure-difference method

In order to solve the problem of the difficult measurement of the small temperature difference between the inlet and outlet of the pump, some researchers have adopted the partial expansion method and the zero temperature-difference method [20]. These two methods do not directly measure the small temperature difference between the inlet and outlet of the pump, but use the pressure loss caused by the pressure reducing device installed in the pressure measuring pipeline, which corresponds to the temperature rise, for measurement. Specifically, a sampling water flow is extracted at the pump inlet, and an expansion valve is set between the sampling pipeline and the remaining measuring components, so that the temperature of the water flow rises to the pump outlet temperature after passing through the expansion valve. In this way, in the denominator of the pump efficiency calculation Eq. (5), item $\bar{C}_p\left(T_2-T_1\right)$ becomes zero, and the pressure at this time $p_{11}$ is measured by a pressure sensor with sufficient accuracy, and item $\bar{a}\left(p_2-p_1\right)$ in the denominator of Eq. (5) becomes $\bar{a}\left(p_2-p_{11}\right)$, so that the efficiency value can be determined, and its expression is as follows:

$\eta=\frac{\bar{v}\left(p_2-p_1\right)}{\bar{a}\left(p_2-p_{11}\right)+\Delta e^{\prime}}$                   (10)

The partial expansion method and the zero temperature-difference method can carry out the measurement and achieve a very high accuracy under the condition that the temperature difference between the inlet and outlet of the pump is very small, but the actual measurement process is relatively complicated, and the measuring pipeline cannot work in an environment lower than the atmospheric pressure, so there are few engineering applications. However, according to the design concept of the zero temperature-difference method, the temperature difference $\Delta T$ in Eq. (8) can be expressed in the form of pressure and pressure difference, namely:

$\eta=\frac{1}{0.937+\frac{426.7}{H} \mu\left(\frac{p_1}{p_a}\right)^n\left(\frac{\Delta p}{p_2}\right)^m}$                      (11)

where,

$p_1$ — absolute pressure at the inlet of the pump;

$p_2$ — absolute pressure at the outlet of the pump;

$\Delta p$ — pressure difference between the outlet and inlet of the pump;

$p_a$ — atmospheric pressure;

$\mu$ — coefficient, taken as 1.01;

m, n — exponents.

According to Karassik et al. [21], m = 0.75-1, n = 1.75-2.

Using pressure measurement instead of temperature difference measurement not only avoids the technical and cost problems of small temperature difference measurement, but also has a small time-lag coefficient and a fast response of pressure measurement, which is beneficial to the on-line testing of pump efficiency. The thermodynamic pressure-difference method test system for pump efficiency is simpler, and the test process is more convenient, and it is a simple and practical water efficiency test model.

In order to comprehensively evaluate the reliability of the thermodynamic pressure-difference method (Eq. (11)) in practical engineering application, based on the internationally used ISO Guide to the Expression of Uncertainty in Measurement (GUM) framework, a quantitative analysis is carried out on the uncertainty of the final pump efficiency caused by the measurement uncertainty of the core measurement parameters in the model (the inlet absolute pressure $p_1$, the outlet absolute pressure $p_2$ and the atmospheric pressure $p_a$).

5.1 Measurement model and uncertainty propagation law

According to the uncertainty propagation law of ISO GUM, when the input quantities are independent of each other, the combined standard uncertainty $u_c(\eta)$ of input $\eta$ is:

$\begin{aligned} & u_c(\eta)=\sqrt{\left(\frac{\partial \eta}{\partial p_1} u\left(p_1\right)\right)^2+\left(\frac{\partial \eta}{\partial p_2} u\left(p_2\right)\right)^2+\left(\frac{\partial \eta}{\partial p_a} u\left(p_a\right)\right)^2}\end{aligned}$             (14)

Let the sensitivity factor $K=1-0.937 \eta$, and the sensitivity coefficients of the input quantities are derived:

(1) Sensitivity coefficient of the inlet absolute pressure $p_1$:

$c_{p 1}=-\eta K\left(\frac{n}{p_1}-\frac{m}{p_2-p_1}\right)$                   (15)

(2) Sensitivity coefficient of the outlet absolute pressure $p_2$:

$c_{p 2}=-\eta K m\left(\frac{1}{p_2-p_1}-\frac{1}{p_2}\right)$                    (16)

(3) Sensitivity coefficient of the atmospheric pressure $p_a$:

$c_{p a}=\eta K n \frac{1}{p_a}$                  (17)

5.2 Calculation of a typical operating condition example

The test data of Case 1 at the optimum operating point (flow rate Q = 51.8 m³/h) are selected for substitution and checking calculation.

Operating condition parameters: the pump efficiency by the thermodynamic temperature-difference method $\eta=56.11 \%$, the pump head $\mathrm{H}=125.2 \mathrm{~m}$, the pump inlet absolute pressure $p_1=46.3  \mathrm{kPa}$, the pump outlet absolute pressure $p_2=1263.3$ kPa , and the atmospheric pressure $p_a=101.2  \mathrm{kPa}$.

Model parameters: m = 0.75, n = 1.75, and the sensitivity factor is calculated as K = 0.474.

Substituting the above data into the uncertainty propagation formula, the calculation results are shown in Table 6.

Table 6. Summary table of the uncertainty components of the pump efficiency measurement

Taking the coverage factor k = 2 (corresponding to a confidence probability of about 95%), it can be obtained that the expanded uncertainty is 1.83%.

From the analysis of the above calculation results, it can be known that:

(1) The inlet pressure $p_1$ is the absolute dominant factor of the measurement uncertainty. It can be seen from the variance contribution rate that the uncertainty of $p_1$ contributes 99.79% of the total variance. It is suggested that in the measurement by the pressure-difference method, a high-precision (>0.05% FS) pressure sensor must be used for the inlet absolute pressure $p_1$, and this is the core way to improve the accuracy of the efficiency measurement.

(2) The uncertainty of the outlet pressure $p_2$ can be neglected.

(3) The influence of the atmospheric pressure $p_a$ is at a relatively low level. Since the reference value of $p_a$ is large, its variance contribution is only 0.20%, and the use of a conventional digital barometer can meet the engineering requirements.

(4) Mutual confirmation with the results of the residual analysis.

The ISO GUM analysis shows that, under the instrument accuracy of 0.1% FS, the theoretical combined standard uncertainty of the model is about 0.915%. Combined with the residual diagnosis results in Table 5, it can be seen that the residuals are completely within the coverage range of the theoretical uncertainty (k = 1), and are mainly caused by the small measurement error of the inlet pressure. This verifies, from the metrological point of view, the reliability and the accuracy margin of Eq. (11) as an engineering application model, and it fully meets the engineering accuracy requirements of the on-site pump efficiency test.

It should be noted that in the model of this paper, both the density and the specific heat capacity are taken as the constants corresponding to 20 ℃. Within the range of 20 ± 0.23 ℃, the change rate of the density of water is about 2.3 × 10⁻⁴ kg·m⁻³·℃⁻¹, and the change rate of the specific heat capacity is about 1 × 10⁻³ kJ·kg⁻¹·℃⁻¹. The relative error of the density introduced thereby does not exceed 5 × 10⁻⁵, and the relative error of the specific heat capacity does not exceed 2 × 10⁻⁴. Through error propagation, the combined influence of the two on the heat exchange quantity obtained by the pressure-difference method does not exceed 0.07%, which is smaller than the relative error of the temperature difference corresponding to the calibration uncertainty of the temperature sensor itself. Therefore, under the operating conditions studied in this paper, neglecting the continuous correction of the physical properties with temperature will not have a substantial influence on the main conclusions.