What is the diameter of the top section, d?

In the figure, water flows in the pipe as shown. What is the diameter of the top section, d?
d₁=5cm
v=4m/s
P=100kPa
P₂=31.5kPa
h=1.5m
d=?

vA=v₂A₂
vd₁²=v₂d²
v₂=vd₁²/d²
P+½ρv²=P₂+½ρv₂²+rgh
P+½ρv²=P₂+½ρ(vd₁²/d²)²+ρgh
P-P₂-ρgh=½ρv²d₁⁴/d⁴-½ρv²=½ρv²(d₁⁴/d⁴-1)
(P-P₂-ρgh)/(½ρv²)=d₁⁴/d⁴-1
(P-P₂-ρgh)/(½ρv²)+1=d₁⁴/d⁴
∜{(P-P₂-ρgh)/(½ρv²)+1} = d₁/d

d/d₁=∜{1/[(P-P₂-ρgh)/(½ρv²)+1]}
d  = d₁ ∜{1/[(P-P₂-ρgh)/(½ρv²)+1]}

Calculation

5cm*(1/( (100kPa-31.5kPa-1000kg/m^3*9.81m/s^2*1.5m)/(1000kg/m^3*(4m/s)^2)+1  ))^(1/4)=3.4598676 cm≈3.5cm

or
d₁⁴/d⁴-1= (P-P₂-ρgh)/(½ρv²)=
=(100kPa-31.5kPa-1000kg/m^3*9.81m/s^2*1.5m)/( 1000kg/m^3*(4m/s)^2)=
=3.3615625
d₁/d=(3.3615625+1)^(1/4)= 1.44514200657
d=d₁/∙1.44514200657= 5cm /1.44514200657=3.4598676 cm≈3.5cm

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