Flow Across Banks of Tubes
Evaluate convection coefficients and pressure drop across inline or staggered tube banks using the Zukauskas correlation.
Aligned vs Staggered Bundles
In crossflow over tube banks, the configuration (inline or staggered) alters the maximum velocity $V_{max}$ inside the bundle channels. Max velocity dictates the Reynolds number ($\text{Re}_{max}$) and the heat transfer rates.
Parameters Setup
Results & Curves
📈 Convection coefficient h vs Velocity
Engine Output
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FLOW ACROSS BANKS OF TUBES
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--- INPUTS ---
Configuration = STAGGERED
Number of Rows N = 10
Tube Diameter D = 0.0250 m
Transverse Pitch S_T = 0.0500 m
Longitudinal Pitch S_L = 0.0500 m
Approach Velocity V = 5.000 m/s
T_inf = 25.00 C
T_surface = 100.00 C
--- GEOMETRY ---
S_T / D = 2.000
S_L / D = 2.000
Maximum Velocity V_max = 10.000 m/s
--- RESULTS ---
Reynolds Re_max = 1.5905E+04
C1 (Zukauskas) = 0.3500
m (Zukauskas) = 0.6000
C2 (row correction) = 0.9700
Nusselt Number Nu = 99.59
Convection Coeff h = 104.7696 W/m2K
Pressure Drop dP = 48.47 Pa
dP per row = 4.85 Pa
--- VELOCITY SWEEP ---
V[m/s] Vmax[m/s] Re_max Nu h[W/m2K] dP[Pa]
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0.50 1.00 1.591E+03 25.02 26.317 0.70
1.31 2.62 4.175E+03 44.64 46.958 4.14
2.12 4.25 6.760E+03 59.60 62.700 10.04
2.94 5.88 9.344E+03 72.38 76.145 18.21
3.75 7.50 1.193E+04 83.80 88.160 28.55
4.56 9.12 1.451E+04 94.27 99.169 40.95
5.38 10.75 1.710E+04 104.01 109.416 55.36
6.19 12.38 1.968E+04 113.17 119.059 71.73
7.00 14.00 2.227E+04 121.87 128.207 90.01
7.81 15.62 2.485E+04 130.17 136.939 110.17
8.62 17.25 2.744E+04 138.13 145.314 132.17
9.44 18.88 3.002E+04 145.80 153.380 155.98
10.25 20.50 3.261E+04 153.20 161.171 181.58
11.06 22.12 3.519E+04 160.38 168.720 208.94
11.88 23.75 3.778E+04 167.35 176.049 238.04
12.69 25.38 4.036E+04 174.13 183.180 268.87
13.50 27.00 4.294E+04 180.73 190.131 301.40
14.31 28.62 4.553E+04 187.18 196.917 335.62
15.12 30.25 4.811E+04 193.49 203.550 371.51
15.94 31.88 5.070E+04 199.66 210.042 409.06
16.75 33.50 5.328E+04 205.71 216.402 448.25
17.56 35.12 5.587E+04 211.64 222.641 489.07
18.38 36.75 5.845E+04 217.46 228.765 531.51
19.19 38.38 6.104E+04 223.18 234.782 575.55
20.00 40.00 6.362E+04 228.80 240.697 621.19
--- CORRELATIONS ---
Zukauskas: Nu = C1*C2*Re_max^m * Pr^0.36 * (Pr/Pr_s)^0.25
Ref: Incropera Ch.7 Eq.7.58, Kothandaraman Ch.8 Sec.8.6
Calculation Methodology
Mathematical Model & Theory
Heat transfer across tube bundles is modeled using the Zukauskas correlation:
$$\text{Nu}_D = C_1 C_2 \text{Re}_{max}^m \text{Pr}^{0.36} \left(\frac{\text{Pr}}{\text{Pr}_s}\right)^{0.25}$$
- $C_1$ and $m$ are constants determined from the Reynolds number range ($\text{Re}_{max} = \rho V_{max} D / \mu$).
- $C_2$ is a row correction factor for banks with fewer than 20 rows ($N_L < 20$).
For aligned arrangement, maximum velocity is:
$$V_{max} = V \frac{S_T}{S_T - D}$$
For staggered arrangement, if the diagonal flow area is smaller than the transverse area, then:
$$V_{max} = V \frac{S_T}{2(S_D - D)}$$
Aerodynamic pressure drop is calculated using friction factors:
$$\Delta P = N \chi f \frac{\rho V_{max}^2}{2}$$
Academic References:
- Incropera, F. P., & DeWitt, D. P. (2011). Fundamentals of Heat and Mass Transfer. 7th Edition, John Wiley & Sons.
- Zukauskas, A. (1972). Heat Transfer from Tubes in Crossflow. Advances in Heat Transfer.
Worked Engineering Example
Air at $25^\circ\text{C}$ flows at $5\text{ m/s}$ across an inline tube bank of 10 rows. Tubes of diameter $25\text{ mm}$ are arranged with pitches $S_T = S_L = 50\text{ mm}$. Surface temp is $100^\circ\text{C}$ ($T_f = 62.5^\circ\text{C}$). Find the convection coefficient $h$.
Step-by-step Solution:
1. Evaluate air properties at film temperature:
- $\rho = 1.177\text{ kg/m}^3, \mu = 1.85 \times 10^{-5}\text{ Pa·s}, k = 0.0263\text{ W/m·K}, Pr = 0.71$.
2. Calculate maximum velocity $V_{max}$:
$$V_{max} = V \frac{S_T}{S_T - D} = 5.0 \times \frac{0.05}{0.05 - 0.025} = 10.0\text{ m/s}$$ 3. Calculate maximum Reynolds number:
$$\text{Re}_{max} = \frac{\rho V_{max} D}{\mu} = \frac{1.177 \times 10.0 \times 0.025}{1.85 \times 10^{-5}} = 15,905$$ 4. From Zukauskas constants ($10^3 < \text{Re}_{max} < 2 \times 10^5$, inline):
- $C_1 = 0.27, m = 0.63$. For 10 rows, $C_2 = 0.97$.
$$\text{Nu}_D = 0.27 \times 0.97 \times (15,905)^{0.63} \times (0.71)^{0.36} \times 1.0 = 93.5$$ 5. Calculate convective coefficient $h$:
$$h = \frac{\text{Nu}_D k}{D} = \frac{93.5 \times 0.0263}{0.025} = 98.4\text{ W/m}^2\text{K}$$