Solving Heat Exchanger Problem: Finding # of Tubes in Bundle

[cwj8820]

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

We are given:
$\dot{m}$_steam = 4.38kg/s
T_steam = 378K
the steam is saturated 90%
T_c of cooling water = 293.15K
T_h of cooling water = 296.5K

For the tubes in the bundle:
Inner diameter = 2.16cm
Outer diameter = 2.67cm
Re_max = 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:
h_f = 438 kJ/kg
h_fg = 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:
T_c = 293.15K
T_h = 296.5K
ΔT = (296.5-293.15) = 3.35K
C_p = 4.18 kJ/(kg*K)
8850 kJ/s = m * C_p * ΔT
8850 kJ/s = m * 4.18 kJ/(kg*K) * 3.35K
$\dot{m}$_water = 632kg/s

After I found $\dot{m}$_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.
Re_max = 10,000
L = Inner diameter = 2.16cm = 0.0216m
μ_water = 1.002 * 10^-3 kg/m*s
ρ_water = 998kg/m³
v_water = (Re * μ)/(ρ * L)
v_water = (10,000 * 1.002*10^-3) / ( 998kg/m³ * 0.0216m)
v_water = 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:
$\dot{m}$_water = 632kg/s
v_water = 0.46m/s
ρ_water = 998kg/m³
A_required = $\dot{m}$_water/[ρ_water * v_water]
A_required = 632kg/s / [998kg/m * 0.46m/s]
A_required ≈ 1.38m² = 13,800cm

A_{one tube} = (pi/4)*(2.16cm)² = 3.66cm²

So the required number of tubes N_t would be
N_t = (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!

 

[Valkarie]

I think the problem is that you didn't take into account the total area of the bundle of tubes. The total area of a tube-in-shell condenser is actually the sum of the inner and outer diameters of the tubes. So you need to calculate the total area of the bundle instead of just the area of one tube. The total area of the bundle can be calculated as:Atotal = Nt * (pi/4) * (Douter2 - Dinner2)Where Nt is the number of tubes, Douter is the outer diameter of the tubes, and Dinner is the inner diameter of the tubes.Then you can calculate the velocity of the cooling water in the bundle using this equation:Vwater = \dot{m}water / [Atotal * ρwater]This should give you a much more reasonable velocity that you can use to calculate the number of tubes necessary to keep the Reynolds number below 10,000.