# Experimental Method for Flow Calibration of the Aircraft Liquid Cooling System
## 2 Flow Rate Calibration Scheme  
There are many flow measurement methods [14; 15; 16] proposed interiorly, such as the differential pressure flowmeter method, the volumetric flowmeter method, the float flowmeter method, the blade flowmeter method, the electromagnetic flowmeter method, the vortex flowmeter method and the ultrasonic flowmeter method. Most of these methods need special flowmeters. Many problems will be encountered during the installation of special flowmeters, interfering with the normal operation [17] and affecting the safety check of the liquid cooling system [18]. A variety of flow calibration methods have been proposed based on the physical properties of geometrical characteristics [19; 20], and the most feasible method is the differential pressure flow measurement method. The differential pressure flow measurement principle, is to fix a throttle with an area less than the section area of the pipe in the liquid filled pipe, and the liquid in the pipe will shrink when it passes through the throttle, then the flow velocity will increase and the static pressure will drop at the contraction; finally a certain pressure drop will generate before and after the throttle. As shown in Figure 1, two cross-sections are taken on a variable diameter pipe, Sections 1 and 2. Section 1 is the cross-section before the contraction of the stream, and Section 2 is the cross-section after the contraction of the stream. According to the Bernoulli Equation of the incompressible ideal fluid and continuity equation of the incompressible fluid constant flow:  
\[\frac{{p_{1}}^{\prime}}{{\rho_{1}}}+\frac{{u_{1}}^{2}}{2}=\frac{{p_{2}}^{ \prime}}{{\rho_{2}}}+\frac{{u_{2}}^{\prime 2}}{2} \tag{1}\]  
\[A_{1}u_{1}=A_{2}u_{2} \tag{2}\]  
where: \({p_{1}}^{\prime}\) and \({p_{2}}^{\prime}\) are the average pressures of the fluid at Sections 1 and 2; \(\rho_{1}\) and \(\rho_{2}\) are the density of the fluid at Sections 1 and 2; \(u_{1}\) and \({u_{2}}^{\prime}\) are the flow velocities at Sections 1 and 2; \(A_{1}\) and \(A_{2}\) are the areas of Sections 1 and 2. For the same coolant, \(\rho_{1}=\rho_{2}\). Substitute Equation (2) into Equation (1), and the flow velocity at Section 2 can be derived as follows:  
\[{u_{2}}^{\prime}=\frac{1}{\sqrt{1-(A_{2}/A_{1})^{2}}}\sqrt{\frac{2}{\rho}({p_{1 }}^{\prime}-{p_{2}}^{\prime})} \tag{3}\]  
The flow velocity at Section 2 \({u_{2}}^{\prime}\) can be rewritten as follows:  
\[{u_{2}}^{\prime}=\frac{1}{\sqrt{1-\mu^{2}\beta^{4}}}\sqrt{\frac{2}{\rho}({p_{ 1}}^{\prime}-{p_{2}}^{\prime})} \tag{4}\]  
where: \(\mu\) is the contraction coefficient of the stream; \(\beta\) is the diameter ratio of the throttle device, and \(\beta=\sqrt{A_{2}/A_{1}}\). Because \({p_{1}}^{\prime}\) and \({p_{2}}^{\prime}\) are the average pressures of the fluid at Sections 1 and 2, the pressure drop \(p_{1}-p_{2}\) is read at the pipe wall according to a certain way of taking pressure in the actual measurement. Therefore, a pressure correction coefficient \(\psi\) needs to be introduced, and \({p_{1}}^{\prime}-{p_{2}}^{\prime}=\psi(p_{1}-p_{2})\). When the taking pressure method is different, the pressure correction coefficient \(\psi\) is different from coefficients of other pressure methods. In engineering practice, there is flow velocity loss in the fluid flow, which is different from the assumption of the isentropic constant flow. Therefore, a coefficient \(\zeta\) is introduced to modify the flow velocity \({u_{2}}^{\prime}\). The volume flow of the incompressible fluid after modification can be expressed as follows:  
\[G=u_{2}\mu A_{0}=\frac{\mu\zeta\sqrt{\psi}}{\sqrt{1-\mu^{2}\beta^{4}}}A_{0} \sqrt{\frac{2}{\rho}(p_{1}-p_{2})} \tag{5}\]  
where: \(p_{1}-p_{2}\) is the actual pressure drop measured. The mass flow of the incompressible fluid [21] after modification can be expressed as follows:  
\[G=u_{2}\mu A_{0}=\frac{\mu\zeta\sqrt{\psi}}{\sqrt{1-\mu^{2}\beta^{4}}}A_{0} \sqrt{2\rho(p_{1}-p_{2})} \tag{6}\]  
where: \(G\) is the flow of coolant in the pipe.