Flow rate and velocity are related, but quite different, physical quantities. To make the distinction clear, consider the flow rate of a river. The greater the velocity of the water, the greater the flow rate of the river. But flow rate also depends on the size and shape of the river. A rapid mountain stream carries far less water than the Amazon River in Brazil, for example. Figure illustrates the volume flow rate. The volume flow rate is [latex]Q=\frac{dV}{dt}=Av,[/latex] where [latex]A[/latex] is the cross-sectional area of the pipe and v is the magnitude of the velocity.

The volume of fluid passing by a given location through an area during a period of time is called flow rate Q, or more precisely, volume flow rate. In symbols, this is written as

where [latex]\rho[/latex] is the density, A is the cross-sectional area, and v is the magnitude of the velocity. The mass flow rate is an important quantity in fluid dynamics and can be used to solve many problems. Consider Figure. The pipe in the figure starts at the inlet with a cross sectional area of [latex]{A}_{1}[/latex] and constricts to an outlet with a smaller cross sectional area of [latex]{A}_{2}[/latex]. The mass of fluid entering the pipe has to be equal to the mass of fluid leaving the pipe. For this reason the velocity at the outlet [latex]({v}_{2})[/latex] is greater than the velocity of the inlet [latex]({v}_{1})[/latex]. Using the fact that the mass of fluid entering the pipe must be equal to the mass of fluid exiting the pipe, we can find a relationship between the velocity and the cross-sectional area by taking the rate of change of the mass in and the mass out:

Since liquids are essentially incompressible, the equation of continuity is valid for all liquids. However, gases are compressible, so the equation must be applied with caution to gases if they are subjected to compression or expansion.

Another method for representing fluid motion is a streamline. A streamline represents the path of a small volume of fluid as it flows. The velocity is always tangential to the streamline. The diagrams in Figure use streamlines to illustrate two examples of fluids moving through a pipe. The first fluid exhibits a laminar flow (sometimes described as a steady flow), represented by smooth, parallel streamlines. Note that in the example shown in part (a), the velocity of the fluid is greatest in the center and decreases near the walls of the pipe due to the viscosity of the fluid and friction between the pipe walls and the fluid. This is a special case of laminar flow, where the friction between the pipe and the fluid is high, known as no slip boundary conditions. The second diagram represents turbulent flow, in which streamlines are irregular and change over time. In turbulent flow, the paths of the fluid flow are irregular as different parts of the fluid mix together or form small circular regions that resemble whirlpools. This can occur when the speed of the fluid reaches a certain critical speed.

This blog covers TPR valves in detail including what TPR valves are, what they do, how to inspect them and step by step instructions on how to replace them.

We can use the relationship between flow rate and speed to find both speeds. We use the subscript 1 for the hose and 2 for the nozzle.

where A is the cross-sectional area and [latex]v[/latex] is the average speed. The relationship tells us that flow rate is directly proportional to both the average speed of the fluid and the cross-sectional area of a river, pipe, or other conduit. The larger the conduit, the greater its cross-sectional area. Figure illustrates how this relationship is obtained. The shaded cylinder has a volume [latex]V=Ad[/latex], which flows past the point P in a time t. Dividing both sides of this relationship by t gives

We note that [latex]Q=V\text{/}t[/latex] and the average speed is [latex]v\text{ }=d/t[/latex]. Thus the equation becomes [latex]Q=Av[/latex].

Thermostatic mixingvalve

Figure shows an incompressible fluid flowing along a pipe of decreasing radius. Because the fluid is incompressible, the same amount of fluid must flow past any point in the tube in a given time to ensure continuity of flow. The flow is continuous because they are no sources or sinks that add or remove mass, so the mass flowing into the pipe must be equal the mass flowing out of the pipe. In this case, because the cross-sectional area of the pipe decreases, the velocity must necessarily increase. This logic can be extended to say that the flow rate must be the same at all points along the pipe. In particular, for arbitrary points 1 and 2,

The first part of this chapter dealt with fluid statics, the study of fluids at rest. The rest of this chapter deals with fluid dynamics, the study of fluids in motion. Even the most basic forms of fluid motion can be quite complex. For this reason, we limit our investigation to ideal fluids in many of the examples. An ideal fluid is a fluid with negligible viscosity. Viscosity is a measure of the internal friction in a fluid; we examine it in more detail in Viscosity and Turbulence. In a few examples, we examine an incompressible fluid—one for which an extremely large force is required to change the volume—since the density in an incompressible fluid is constant throughout.

PressureRelief Valve

Many figures in the text show streamlines. Explain why fluid velocity is greatest where streamlines are closest together. (Hint: Consider the relationship between fluid velocity and the cross-sectional area through which the fluid flows.)

A nozzle with a diameter of 0.500 cm is attached to a garden hose with a radius of 0.900 cm. The flow rate through hose and nozzle is 0.500 L/s. Calculate the speed of the water (a) in the hose and (b) in the nozzle.

Velocity vectors are often used to illustrate fluid motion in applications like meteorology. For example, wind—the fluid motion of air in the atmosphere—can be represented by vectors indicating the speed and direction of the wind at any given point on a map. Figure shows velocity vectors describing the winds during Hurricane Arthur in 2014.

The solution to the last part of the example shows that speed is inversely proportional to the square of the radius of the tube, making for large effects when radius varies. We can blow out a candle at quite a distance, for example, by pursing our lips, whereas blowing on a candle with our mouth wide open is quite ineffective.

Tpvalve

We gush over technological advances to our shiny Sub-Zeros and glitzy Viking ranges, but when was the last time you heard someone rave about the modern features found in their new water heater. Water heaters are often taken for granted and don’t possess the “Bling Factor” of other home appliances but their attributes are attractive in their own way.

Water heaters can start to shake just like your clothes steam and moisture produced is greater than normal. The steam increases the pressure inside your water heater causing it to become unstable. Water heaters can start to shake just like your clothes washer does when it’s overloaded or unbalanced. This is where the TPR Valve comes into play. When something goes wrong, the valve automatically opens to release excess pressure through a discharge line.

Water emerges straight down from a faucet with a 1.80-cm diameter at a speed of 0.500 m/s. (Because of the construction of the faucet, there is no variation in speed across the stream.) (a) What is the flow rate in [latex]{\text{cm}}^{3}\text{/s}[/latex]? (b) What is the diameter of the stream 0.200 m below the faucet? Neglect any effects due to surface tension.

The rate of flow of a fluid can also be described by the mass flow rate or mass rate of flow. This is the rate at which a mass of the fluid moves past a point. Refer once again to Figure, but this time consider the mass in the shaded volume. The mass can be determined from the density and the volume:

Figure is also known as the continuity equation in general form. If the density of the fluid remains constant through the constriction—that is, the fluid is incompressible—then the density cancels from the continuity equation,

a. 12.6 m/s; b. [latex]0.0800\,{\text{m}}^{3}\text{/s}[/latex]; c. No, the flow rate and the velocity are independent of the density of the fluid.

The TPR valve is a special safety valve and it’s responsible for making sure your hot temperature-pressure relief valve water tank stays within its designed temperature and pressure limits.

What is the average flow rate in [latex]{\text{cm}}^{3}\text{/s}[/latex] of gasoline to the engine of a car traveling at 100 km/h if it averages 10.0 km/L?

[latex]Q={A}_{1}{\overset{\text{–}}{v}}_{1}={A}_{2}{\overset{\text{–}}{v}}_{2},[/latex] or [latex]\pi \frac{{d}_{1}^{2}}{4}{\overset{\text{–}}{v}}_{1}=\pi \frac{{d}_{2}^{2}}{4}{\overset{\text{–}}{v}}_{2}\Rightarrow {\overset{\text{–}}{v}}_{2}={\overset{\text{–}}{v}}_{1}({d}_{1}^{2}\text{/}{d}_{2}^{2})={\overset{\text{–}}{v}}_{1}{({d}_{1}\text{/}{d}_{2})}^{2}[/latex]

Water heaterrelief valvedripping

What is the fluid speed in a fire hose with a 9.00-cm diameter carrying 80.0 L of water per second? (b) What is the flow rate in cubic meters per second? (c) Would your answers be different if salt water replaced the fresh water in the fire hose?

It’s located on top or on the side near the top of your water heater. The valve has a lever that can be lifted up or down and a discharge pipe that runs from the valve straight down to the bottom of your water heater.

Consider two different pipes connected to a single pipe of a smaller diameter, with fluid flowing from the two pipes into the smaller pipe. Since the fluid is forced through a smaller cross-sectional area, it must move faster as the flow lines become closer together. Likewise, if a pipe with a large radius feeds into a pipe with a small radius, the stream lines will become closer together and the fluid will move faster.

Water heater pressurerelief valvefailure symptoms

Prove that the speed of an incompressible fluid through a constriction, such as in a Venturi tube, increases by a factor equal to the square of the factor by which the diameter decreases. (The converse applies for flow out of a constriction into a larger-diameter region.)

This is called the equation of continuity and is valid for any incompressible fluid (with constant density). The consequences of the equation of continuity can be observed when water flows from a hose into a narrow spray nozzle: It emerges with a large speed—that is the purpose of the nozzle. Conversely, when a river empties into one end of a reservoir, the water slows considerably, perhaps picking up speed again when it leaves the other end of the reservoir. In other words, speed increases when cross-sectional area decreases, and speed decreases when cross-sectional area increases.

Water is moving at a velocity of 2.00 m/s through a hose with an internal diameter of 1.60 cm. (a) What is the flow rate in liters per second? (b) The fluid velocity in this hose’s nozzle is 15.0 m/s. What is the nozzle’s inside diameter?

They have a wide range of fancy new built-in features, but star billing goes to the safety feature that protects our homes and families 24/7; the Temperature Pressure Relief valve or TPR Valve.

Hot water tank pressurerelief valve

This little device is vital to the overall safety of your water heater, your home and your family and it needs to be checked, inspected, maintained and/or replaced to prevent overheating and excessive pressurization.

The heart of a resting adult pumps blood at a rate of 5.00 L/min. (a) Convert this to [latex]{\text{cm}}^{3}\text{/s}[/latex]. (b) What is this rate in [latex]{\text{m}}^{3}\text{/s}[/latex]?

A speed of 1.96 m/s is about right for water emerging from a hose with no nozzle. The nozzle produces a considerably faster stream merely by constricting the flow to a narrower tube.

If your water heater doesn’t have a TPR valve or if the TPR valve ceases to function properly, your water heater can literally blow up and put the safety of your family and home at risk.

The SI unit for flow rate is [latex]{\text{m}}^{3}\text{/s,}[/latex] but several other units for Q are in common use, such as liters per minute (L/min). Note that a liter (L) is 1/1000 of a cubic meter or 1000 cubic centimeters [latex]({10}^{-3}{\,\text{m}}^{3}\text{or}\,{10}^{3}\,{\text{cm}}^{3}).[/latex]

Proudly Serving the South Bay with exceptional plumbing services since 1963, including Culver City, El Segundo, Los Angeles, Hollywood, Beverly Hills, Redondo Beach, Mar Vista, Westchester, Santa Monica, Pacific Palisades, Manhattan Beach, Hermosa Beach, and Woodland Hills.

(a) Estimate the time it would take to fill a private swimming pool with a capacity of 80,000 L using a garden hose delivering 60 L/min. (b) How long would it take if you could divert a moderate size river, flowing at [latex]{5000\,\text{m}}^{3}\text{/s}[/latex] into the pool?

The Huka Falls on the Waikato River is one of New Zealand’s most visited natural tourist attractions. On average, the river has a flow rate of about 300,000 L/s. At the gorge, the river narrows to 20-m wide and averages 20-m deep. (a) What is the average speed of the river in the gorge? (b) What is the average speed of the water in the river downstream of the falls when it widens to 60 m and its depth increases to an average of 40 m?