# What measurement accuracy/precision is required to detect a 10% relative change in the measured quantity

I'm trying to design a measurement system, the primary objective of which is to reliably detect a 10% change in the the volume of fluid flowing through a pipe.

I'm struggling to come up with a required accuracy/precision for the system based on something on something other than "This is simply as accurate as I could make it".

Note I am only really interested in relative change, i'm not sure how this effects the system requirements in terms of required precision/accuracy. For example, there two primary sources of error:

• Measurement error (accuracy)
• Estimated internal diameter of the pipe (precision)

I'm assuming I can completely negate the error introduced from the estimated internal diameter of the pipe and place the requirements solely on the "measurement error"

Is there a statistical technique to determine the accuracy required to reliably detect X change in some variable?

• Is your question about converting the common language "reliably" into actual engineering tolerances bracket? That will heavily depend on the application. There are accuracies/reliabilities appropriate for different purposes, and there are techniques of measuring the actual measurement accuracy to determine if it fits within the bracket. Also usually once you've determined what 'reliably' should mean in your case, you just shop for ready-made parts that provide given reliability, measured and guaranteed by the manufacturer. – SF. Nov 28 '18 at 14:36

# Direct Measurement of Volumetric Flow

Let's handle the (easy) statement of accuracy first. Validating the accuracy of the measurement of volume flow requires that you calibrate your chosen measuring device with a second approach. That second approach must be at least as precise if not better than when you put the device in to practice. A best method will use a standards test, for example as defined by NIST or other agencies, as the calibration. Alternatively, when you intend to buy a device to measure volume flow and just use it directly without in-house calibration, you should insist on getting a calibration test report.

Now, let's handle the issue of sensitivity. You want to measure a change in volumetric flow $$\Delta \dot{V}$$ within a standard of $$\Delta \dot{V} / \dot{V} \geq 0.1$$. This is akin to asking to what you need in order to measure a signal $$S$$ over a noise level $$N$$. A guideline to reliable measurement in this case is that $$S/N \geq 3$$ to state that you have a signal above the noise. For a hypothetic system with no inherent flow noise (the volumetric flow is perfectly stable), the "noise" is the precision of the device itself. At a minimum, you must therefore assure that $$\Delta \dot{V} / \dot{V} \leq 0.1/3 = 0.033$$. You will want a device with a relative measurement precision of at least 3.3%, preferably (much) better. When you intend to buy a device, you should confirm that its measurement precision fits your needs.

# Indirect Measurement of Volumetric Flow

Suppose you are measuring mass flow of an incompressible fluid and you want to correlate this with volume flow in a pipe. To first order, this is an application of linear propagation of uncertainties. The starting equation is

$$\dot{V} = \frac{\dot{m}}{\rho}$$

You know that you can measure $$\dot{m}$$ with a device that has a relative measurement precision $$\delta \dot{m}$$. You know that your fluid has a density that is reported with a precision of $$\delta \rho$$. To first order, the uncertainty propagation equation is as follows:

$$\left(\frac{\Delta\dot{V}}{\dot{V}}\right)^2 = \left(\frac{\delta\dot{m}}{\dot{m}}\right)^2 + \left(\frac{\delta\rho}{\rho}\right)^2$$

To reliably measure a 10% change in volumetric flow, you should have a total uncertainty in your measurement system $$\frac{\Delta\dot{V}}{\dot{V}}$$ that is significantly below 10%. This is a statement of the precision of your system.

As a second example, when you are measuring velocity to obtain volumetric flow, the equation for velocity and volumetric flow

$$\dot{V} = v \pi d^2 / 4$$

gives this

$$\left(\frac{\Delta\dot{V}}{\dot{V}}\right)^2 = \left(\frac{\delta v}{v}\right)^2 + 4\left(\frac{\delta d}{d}\right)^2$$

The precision of your measurement depends on the relative precision of the velocity flow device and the relative uncertainty of the tube diameter.

As with direct measurements, the accuracy of your system depends on how well it is calibrated. Calibration might in this case be done by using a separate device that measures volumetric flow and making a plot of volumetric flow versus mass flow. A perfect linear correlation is expected to validate the accuracy of your correlation equation.