18.6.2. GATE 2012 Example

The pipe is there and there is a joint which is called a T-joint like this. P is inflow that is coming. Q is going out from this. R is going from this out. The pipes is having branching of P, Q, R. The diameters are given. The velocities V and V are given. V to be estimated which is very simplified problem.

In this problem, we have a T-joint pipe system where fluid flows into one inlet (P) and branches out to two outlets (Q and R). The challenge is to find out the velocity (V) in the branch pipe R, given the diameters and velocities of the other branches. This setup is common in fluid mechanics as it introduces branching flow, and will utilize the principles of conservation of mass.

Think of a garden hose splitting into two sprinkler heads. If you adjust the flow in one hose, you may change how water flows to the sprinklers. This is similar to how we analyze the flow in the T-joint: understanding how water flows into one pipe and splits into two helps us predict how fast it will come out of a smaller pipe.

Flow classification:

• One dimensional
• Steady
• Laminar
• Fixed control volume
• Incompressible flow

Here we categorize the flow as one-dimensional (flow is uniform across the cross-section, with no variation in the direction perpendicular to flow), steady (conditions do not change with time), laminar (the flow is smooth with layers that slide past each other), fixed control volume (the analysis looks at a specific volume in space, not changing with time), and incompressible (the fluid density remains constant despite pressure changes). These classifications help us apply the right equations to analyze the flow.

Imagine syrup pouring from a bottle. The syrup flows smoothly (laminar), consistently at a steady rate (steady), and it takes the shape of its container but doesn't change its thickness (incompressible). You would analyze it the same regardless of how much syrup is in the bottle.

Data Given:

• D = 4m (diameter at P)
• V = 6 m/s (velocity at P)
• D = 4m (diameter at Q)
• V = 5 m/s (velocity at Q)
• D = 2m (diameter at R)
• V = ? m/s (velocity at R)

The data provided gives us the diameters of the pipes at points P, Q, and R, and the velocities of fluid flow in the branches P and Q, which are necessary for calculating the flow rate. The question marks indicate we need to find the unknown velocity at R. This data allows us to establish the relationship between the inflow and outflows using the principles of mass conservation.

Consider filling three containers with different diameters using one pipe. If you know how fast the water enters the first two containers, you can calculate how fast it will fill the third, smaller one, as long as the speed and volume of water remain consistent.

The mass inflow what is coming, rate of mass inflow is what is coming in, it should be equal to rate of mass inflow going out from this control volume. Being a steady problem, what we have.

In fluid mechanics, one fundamental principle is conservation of mass, which states that the total mass input into a system must equal the total mass output. For this T-joint, the sum of the mass inflow should equal the sum of the mass outflow. For steady conditions, the mass flow rates in and out can be directly computed from their respective velocities and cross-sectional areas of the branches.

Think of a bathtub faucet (inlet) and a drain (outlet). If you fill the bathtub at a steady rate but lose some water through the drain, knowing the rates at which water flows in and out will help you predict whether the bathtub will overflow or not.

So, numerically that is what is coming. For the steady flow this becomes zero. So, you have in and out. As you know, this in will be negative and both out will be positives, and substituting these Q values for all the cases, with V unknown. So finally, substituting to this equations R will give V equal to 4 m/s.

To find the velocity at R, we apply the equation of continuity which states the total inflow must equal the total outflow. We will calculate the mass flow rates using the relationship Q = A × V, where A is the area and V is the velocity. By setting the inflow equal to the sum of the outflows, we can rearrange the equation to solve for the unknown velocity at R. After substituting in the known values for branches P and Q, we can solve for velocity V at branch R.

Imagine a water balloon that keeps filling from a water source; if you know how fast water is going in and how much is being released from it, you can calculate how full the balloon will get at any moment. This is what we do with our pipe analysis to find the flow at R.

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