What is the cause and solution for the error in pipe bending by pipe bending machine?

The pipe bending machine has errors when bending the pipe, so how to eliminate the working errors of the pipe bending machine? From the error derivation formula caused by the differential pressure transmitter, it can be seen that the primary reason is the existence of the height difference when calculating the differential pressure, that is, H≠0. In order to eliminate the existence of H, it is required that the two pressure points of the elbow sensor are in the same horizontal plane. That is to say, in principle, the horizontal layout plan may be adopted. However, when the horizontal plan cannot be adopted, the length of the differential pressure pipeline should be shortened and the online real-time compensation plan should be adopted. In the discussion of the formula of the curvature radius ratio caused by thermal expansion, it can be seen that: when L1=L2, the violation length of R′ and R is the smallest, that is, it is required to keep the straight pipe length on both sides of the elbow as flat as possible. In addition, in order to reduce the concentration of stress at the elbow flowmeter due to the elongation of the pipeline, the elongation diversion method can be adopted, that is, expansion joints are separately set on both sides of the elbow flowmeter beyond the length required for the elbow flowmeter installation, or the π-type installation method is considered. The fluid flowing through the elbow undergoes centripetal acceleration due to the centripetal force exerted by the pipe wall, forming an overall spatial rotation movement. (Flow pattern description) The intensity of the centripetal acceleration of the activity is characterized by the centripetal acceleration of the fluid, and can be determined by measuring the differential pressure signal of the fluid inside and outside the elbow. Under the condition of knowing the elbow diameter ratio (degree of tortuosity) and the density of the fluid medium, the flow rate can be calculated by the formula when the differential pressure of the inner and outer diameters is measured.

The elbow flowmeter uses a 90° standard elbow sensor as a flow sensor, and measures the flow rate by measuring the differential pressure signal at 45° points inside and outside the sensor. The fluid flow rate is generally determined by the following formula

V=α[（R/D）×ΔP/ρ1）][1/2]

Where V is the average flow rate; α is the flow rate coefficient; R/D is the elbow diameter ratio; ρ1 is the fluid density in the pipe; ΔP is the differential pressure signal that causes the centripetal movement of the fluid.

The differential pressure signal ΔP is:

ΔP=ΔP1-ΔP2

ΔP1=P1-P10, ΔP2=P2-P20,

In the formula, ΔP1 is the pressure increment of the fluid at point A on the outer bend side of the 45° point of the elbow sensor; ΔP2 is the pressure increment of the fluid at point B on the inner bend side of the 45° point of the elbow sensor; P1 is the pressure at point A when the fluid flows at a flow rate V; P10 is the static pressure at point A when the fluid is stationary; P2 is the pressure at point B when the fluid flows at a flow rate V; P20 is the static pressure at point B when the fluid is stationary.

When the fluid is stationary, the static pressure difference (P10-P20) between points A and B is determined by the height difference H between the two points and the density of the fluid. That is: ΔP=P1-P2-ρ1gH

In practical use, elbow flowmeters are often used in hot water pipes and other environments where there is a temperature difference compared to the design conditions. Because of environmental changes, there will be errors in measurement.

The following is an analysis of the causes of errors in pipe bending machines.

(1) Errors caused by differential pressure transmitters

In the calculation formula of the elbow flowmeter, P is the differential pressure signal of the points on the inner and outer walls of the elbow. At present, various differential pressure transmitters are generally used to measure the differential pressure signal ΔP. The typical measurement system is shown in Figure 2. Pressure P1 is taken from point A, and pressure P2 is taken from point B. They are connected to the A' and B' ends of the differential pressure transmitter G through the pressure pipes L1 and L2 respectively. The height difference between A and A', and between B and B' is H and H'. The differential pressure signal ΔP' directly measured by the differential pressure transmitter is:

ΔP'=P'1-P'2=（P'1+H1ρ0g）-（P2+H2ρ0g）

ΔP'=-ΔP+（ρ0-ρ1）gH

In the formula, ρ0 is the density of water in the pressure pipe, which is a function of temperature at normal pressure. Under normal temperature conditions, the water temperature in the main pipe is the same as the water temperature in the pressure pipe, that is, they have the same density, that is,

ρ0=ρ1

When the water temperature in the main pipe is higher than the ambient temperature, due to the existence of heat conduction, the water density in the main pipe and the water in the pressure pipe is different.

Experiments have shown that the water in the pressure pipe uses heat radiation as the primary conduction method, and the temperature of the water in the pressure pipe decays rapidly with the extension of the length of the pressure pipe. When the length of the conduit reaches a certain length L0, the water temperature in the conduit is the same as the ambient temperature, that is, the density does not change after L0. Within the range of L0, ρ0 is a function of temperature and also a function of length.

The above formula shows that the differential pressure signal directly measured by the differential pressure transmitter contains not only the differential pressure signal ΔP that causes the centripetal acceleration of hot water, but also the differential pressure error signal item caused by the height difference H of the two pressure points and the difference in the density of hot water in the main pipe and the density of water in the pressure pipe.

(2) Curvature error caused by thermal expansion

Because the elbow itself is a flexible element, the thermal expansion of the heating pipe will inevitably concentrate on the elbow, so the elbow will bend and deform, which will cause the curvature radius of the elbow to be different from the designed curvature radius.

To further understand the error that occurs, we make the following assumptions based on the actual pipe network system (as shown in Figure 3) for calculation: (1) Ignore the resistance of the elbow, that is, the internal force of the elbow resists the bending deformation and the elongation of the L1 and L2 sections is reduced to 0; (2) Assume that the elbow is still an arc after deformation and is still tangent to the pipelines on both sides; (3) The L1 and L2 sections of the pipeline are treated as rods, and all deformations are small deformation ranges. Assume that the deformations of L1 and L2 are ΔL1 and ΔL2, respectively. According to the assumed conditions, L1 and L2 can be regarded as cantilever beams with A and B as fixed points, and the deflections caused by temperature changes are YC and YD; ΔL1=ηL1Δt; ΔL2=η2L2Δt;

where η is the linear expansion coefficient of the pipeline material

According to the continuity of the pipeline, it is approximated as follows:

YC=ΔL2; YD=ΔL1

Based on the geometric relationship and the knowledge of material mechanics, the rotation angles of points C and D are:

θC=PDL12/2EI; θD=PCL22/2EI

The deflections of points C and D are:

YC=PDL13/3E1; YD=PCL23/3EI