air flows through a main duct at a mass flow rate of 4kg/s the main duct enters a junction that splits into two branch ducts one with a cross section of 20cm x 30cm and one with a cross section og 40cm x 60cm if the mass flow rate in the large branch is 2.8kg/s what is the flow rate in the small branch? if the density of air is p=1.16kg/m what is the velocity in each branch
To find the flow rate in the smaller branch, we can use the principle of conservation of mass. The sum of the flow rates in the branches must equal the flow rate in the main duct.
Let's call the flow rate in the smaller branch Qsmall and the flow rate in the larger branch Qlarge. Given that the mass flow rate in the main duct is 4 kg/s and the mass flow rate in the larger branch is 2.8 kg/s, we can set up the following equation:
Qsmall + Qlarge = 4 kg/s (Equation 1)
Qlarge = 2.8 kg/s (Equation 2)
From Equation 2, we can solve for Qlarge:
Qlarge = 2.8 kg/s
Now, we can substitute this value into Equation 1 to find Qsmall:
Qsmall + 2.8 kg/s = 4 kg/s
Subtracting 2.8 kg/s from both sides:
Qsmall = 4 kg/s - 2.8 kg/s
Qsmall = 1.2 kg/s
Therefore, the flow rate in the small branch is 1.2 kg/s.
To find the velocity in each branch, we can use the equation for the mass flow rate:
Q = ρ * A * V
Where:
Q is the mass flow rate
ρ is the density of air
A is the cross-sectional area of the duct
V is the velocity of air
For the small branch, we have:
Qsmall = ρ * Asmall * Vsmall
Rearranging the equation, we can solve for Vsmall:
Vsmall = Qsmall / (ρ * Asmall)
Given that the density of air (ρ) is 1.16 kg/m^3, the cross-sectional area of the small branch (Asmall) is (0.2 m * 0.3 m) = 0.06 m^2, and the flow rate in the small branch is 1.2 kg/s, we can substitute these values into the equation to find the velocity in the small branch:
Vsmall = 1.2 kg/s / (1.16 kg/m^3 * 0.06 m^2)
Vsmall = 17.24 m/s
Therefore, the velocity in the small branch is 17.24 m/s.
Similarly, we can find the velocity in the large branch using the same equation:
Vlarge = Qlarge / (ρ * Alarge)
Given that the density of air (ρ) is 1.16 kg/m^3, the cross-sectional area of the large branch (Alarge) is (0.4 m * 0.6 m) = 0.24 m^2, and the flow rate in the large branch is 2.8 kg/s, we can substitute these values into the equation to find the velocity in the large branch:
Vlarge = 2.8 kg/s / (1.16 kg/m^3 * 0.24 m^2)
Vlarge = 10.64 m/s
Therefore, the velocity in the large branch is 10.64 m/s.