Answer:
the absolute pressure in the smaller pipe = 19.63 psi
Explanation:
Let A be the diameter of the first pipe = 3 inches
Let B be the diameter of the second pipe. = 1.5 inches
To feet (ft) ; we have
Diameter of the pipe A [tex]D_1 = (\dfrac{3}{12})ft = 0.25 \ ft[/tex]
Diameter of pipe B [tex]D_1 = (\dfrac{1.5}{12})ft = 0.125 \ ft[/tex]
Temperature T = 120° F = (120+ 460)°R
= 580 ° R
The pressure gage to atmospheric pressure ; we have:
[tex]P_{Absolute }=P _{Atm} + P_{guage}[/tex]
where;
atmospheric pressure = 1.47 psi
pressure gage = 20 psi
[tex]P_{Absolute }=(1.47+20)psi[/tex]
[tex]P_{Absolute }=34.7 \ psi[/tex]
To lb/ft²; we have:
[tex]P_{Absolute }=(34.7 *144 ) lb/ft^2[/tex]
[tex]P_{Absolute }=[/tex] 4.998.6 fb/ft²
The density of carbon dioxide can be calculated by using the relation
[tex]\rho = \dfrac{P}{RT}[/tex]
[tex]\rho = \dfrac{4996.8}{(1130 \ lb /slug ^0 R)*(580{^0} R)}[/tex]
[tex]\rho = 7.64*10^{-3}\ slug /ft^3[/tex]
Formula for calculating cross sectional area is
[tex]A = \dfrac{\pi}{4}D[/tex]
For diameter of pipe [tex]D_1 = 0.025[/tex]
A₁ = [tex]\dfrac{\pi}{4}*0.25^2[/tex]
A₁ = 0.04909 ft²
For diameter of pipe [tex]D_2 - 0.0125[/tex]
A₂ [tex]=\dfrac{\pi}{4}*0.125^2[/tex]
A₂ = 0.012227 ft²
Using the continuity equation to determine the velocities V₁ and V₂ respectively.
For V₁
Q = A₁V₁
V₁ = Q₁/ A₁
V₁ = 1.5/0.04909
V₁ = 30.557 ft/s
For V₂
Q = A₂V₂
V₂= Q₂/ A₂
V₂ = 1.5/0.04909
V₂ = 30.557 ft/s
Finally; using Bernoulli's Equation to the flow of the carbon dioxide from the larger pipe to the smaller pipe ; we have:
[tex]p_1 + \dfrac{\rho V_1^2}{2}+\gamma Z_1= p_2 + \dfrac{\rho V_2^2}{2}+\gamma Z_2[/tex]
Since the pipe is horizontal then;
[tex]\gamma Z_1= \gamma Z_2[/tex]
So;
[tex]p_1 + \dfrac{\rho V_1^2}{2}= p_2 + \dfrac{\rho V_2^2}{2}[/tex]
[tex]p_2 =p_1 +\dfrac{1}{2} \rho(V_1^2-V_2^2)[/tex]
[tex]p_2 =4996.8+\dfrac{1}{2} *7.624*10^{-3}(30.557^2-122.23^2)[/tex]
[tex]p_2 =4943.41 \ lb/ft^2[/tex]
To psi;
[tex]p_2 =\dfrac{4943.41 }{144}psi[/tex]
[tex]p_2 =34.33 \ psi[/tex] gage
The absolute pressure in the smaller pipe can be calculated as:
[tex]p_2 _{absolute} = 34.33 - 14.7[/tex]
[tex]p_2 _{absolute} = 19.63 \ \ absolute[/tex]
Hence, the absolute pressure in the smaller pipe = 19.63 psi
