Question

(Laminar flow) A fluid flows through two horizontal pipes of equal length which are connected together to form a pipe of length 2l. The flow is laminar and fully developed. The pressure drop for the first pipe is 1.44 times greater than it is for the second pipe. If the diameter of the first pipe is D, determine the diameter of the second pipe.

Answer 1

The diameter of the second pipe is
`D_2 = 1.095 D`

Step-by-step explanation:

From the question we are told that

The length of the connected pipe is
`d = 2L`

The pressure drop for the first pipe is
`\Delta p __(1)} = 1.44* \Delta p__(2)}`

The diameter of the pipe is
`D`

The rate at which the fluid flows for laminar flow is mathematically represented as

`\r m = (\pi D^4 \Delta p)/(128 \mu L)`

Where L is the length of the pipe

`\mu` is the dynamic viscosity

`\Delta p` is the difference in pressure

`\r m` is the flow rate of the fluid

From the equation of continuity

`\r m_ 1 = \r m_2`

Where
`\r m_1` is the flow rate in pipe one

`\r m_2` is the flow rate in pipe two

So

`(\pi D^4 \Delta p_1)/(128 \mu L) = (\pi D^4_2 \Delta p)/(128 \mu L)`

Where
`D_2` is the diameter of the second pipe

=>
`(\pi D^4 (1.44 \Delta p_2))/(128 \mu L) = (\pi D^4_2 \Delta p)/(128 \mu L)`

=>
`1.44 D^4 = D_2 ^4`

`D_2 =\sqrt[4]{ (D ^4 )/(1.44 ) }`

`D_2 = 1.095 D`