Heat exchanger problem

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Okay, I'm having trouble with a heat transfer HW problem. We are designing a tube-in-shell steam condenser.

We are given:
[itex]\dot{m}[/itex]steam = 4.38kg/s
Tsteam = 378K
the steam is saturated 90%
Tc of cooling water = 293.15K
Th of cooling water = 296.5K

For the tubes in the bundle:
Inner diameter = 2.16cm
Outer diameter = 2.67cm
Remax = 10,000

The problem I'm having is in calculating the necessary number of tubes in the bundle to keep the Re low enough

I started by calculating the heat duty for the condensing steam:
hf = 438 kJ/kg
hfg = 2245 kJ/kg
h1 = 438 + (0.90 * 2245) = 2458 kJ/kg
h2 = 438 kJ/kg
Δh = 2458 - 438 = 2020 kJ/kg
Q = 4.38 * 2020 kJ/kg ≈ 8850 kJ/s = 8850 kW

Next, I calculated the necessary mass flow of the cooling water:
Tc = 293.15K
Th = 296.5K
ΔT = (296.5-293.15) = 3.35K
Cp = 4.18 kJ/(kg*K)
8850 kJ/s = m * Cp * ΔT
8850 kJ/s = m * 4.18 kJ/(kg*K) * 3.35K
[itex]\dot{m}[/itex]water = 632kg/s

After I found [itex]\dot{m}[/itex]water I tried to calculate the maximum velocity in one tube based on the maximum reynolds number. This is where I think I went wrong. This is what I did.
Remax = 10,000
L = Inner diameter = 2.16cm = 0.0216m
μwater = 1.002 * 10-3 kg/m*s
ρwater = 998kg/m3
vwater = (Re * μ)/(ρ * L)
vwater = (10,000 * 1.002*10-3) / ( 998kg/m3 * 0.0216m)
vwater = 0.46m/s

I thought that the velocity seemed pretty low, but I went even further to calculate the necessary number of tubes in the bundle:
[itex]\dot{m}[/itex]water = 632kg/s
vwater = 0.46m/s
ρwater = 998kg/m3
Arequired = [itex]\dot{m}[/itex]water/[ρwater * vwater]
Arequired = 632kg/s / [998kg/m * 0.46m/s]
Arequired ≈ 1.38m2 = 13,800cm

Aone tube = (pi/4)*(2.16cm)2 = 3.66cm2

So the required number of tubes Nt would be
Nt = (13,800cm2)/(3.662) ≈ 3770

This number that I'm getting is way beyond the number of tubes were are allowed to use, which supposed to be in the hundreds. I feel like my mistake is somewhere in my calculation of the velocity, but I've double and triple checked and can't find my mistake. Help would be greatly appreciated!!!
 

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