Thermal Entry Length Calculator
Analyze hydrodynamic and thermal boundary layer development in circular pipes.
Entrance Region Convection
Near the inlet, the boundary layer is thin, and the convection coefficient ($h_x$) is extremely high. As the boundary layers merge, the flow becomes fully developed, and Nusselt number approaches a constant asymptotic value ($3.66$ for laminar flow under uniform wall temperature).
Parameters Setup
Results & Curves
📈 Local Nusselt number along tube length (x)
Engine Output
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INTERNAL FLOW ENTRY LENGTH HEAT TRANSFER
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Reynolds Re = 3.3060E+01
Graetz Number = 628.1400
Nu Developing = 14.3265
Nu Fully = 3.6600
Coeff h = 105.5634 W/m2K
Transfer Q = 244.1649 W
T_out = 44.95 C
Entry Len L_h = 0.0314 m
Entry Len L_t = 15.7035 m
--- LENGTH SWEEP ---
x[m] Gz Nu_local h_local[W/m2K]
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0.0005 6.281E+05 146.1959 1077.2327
0.0213 1.474E+04 42.9682 316.6075
0.0421 7.456E+03 34.2789 252.5812
0.0629 4.990E+03 29.9474 220.6652
0.0838 3.750E+03 27.1705 200.2040
0.1046 3.004E+03 25.1720 185.4778
0.1254 2.505E+03 23.6334 174.1412
0.1462 2.148E+03 22.3956 165.0200
0.1670 1.881E+03 21.3680 157.4482
0.1878 1.672E+03 20.4948 151.0145
0.2086 1.505E+03 19.7394 145.4483
0.2294 1.369E+03 19.0764 140.5630
0.2502 1.255E+03 18.4876 136.2246
0.2711 1.159E+03 17.9596 132.3341
0.2919 1.076E+03 17.4822 128.8164
0.3127 1.004E+03 17.0475 125.6133
0.3335 9.417E+02 16.6493 122.6788
0.3543 8.864E+02 16.2825 119.9760
0.3751 8.372E+02 15.9430 117.4747
0.3959 7.932E+02 15.6276 115.1504
0.4168 7.536E+02 15.3333 112.9824
0.4376 7.178E+02 15.0580 110.9534
0.4584 6.852E+02 14.7995 109.0486
0.4792 6.554E+02 14.5561 107.2555
0.5000 6.281E+02 14.3265 105.5634
Calculation Methodology
Mathematical Model & Theory
The hydrodynamic entry length ($L_h$) and thermal entry length ($L_t$) for laminar flow ($\text{Re} \le 2300$) are calculated as: $$L_h \approx 0.05 \text{Re} D, \quad L_t = L_h \text{Pr}$$ For turbulent flows, both boundary layers develop much quicker due to eddy mixing: $$L_h \approx L_t \approx 10 D$$ In the laminar developing region, the local Nusselt number under constant wall temperature (UWT) is given by the Hausen correlation: $$\text{Nu}_x = 3.66 + \frac{0.0668 \text{Gz}_x}{1 + 0.04 \text{Gz}_x^{2/3}}$$ where $\text{Gz}_x = (D/x)\text{Re}\text{Pr}$ is the local Graetz number.
Academic References:
- Incropera, F. P., & DeWitt, D. P. (2011). Fundamentals of Heat and Mass Transfer. 7th Edition, John Wiley & Sons.
- Hausen, H. (1943). Darstellung des Wärmeüberganges in Rohren durch verallgemeinerte Potenzbeziehungen. VDI Z.
Worked Engineering Example
Air at $25^\circ\text{C}$ enters a $25\text{ mm}$ diameter tube at $0.5\text{ m/s}$. The wall is kept at $80^\circ\text{C}$. Find the local heat transfer coefficient at $x = 0.1\text{ m}$ ($T_f = 52.5^\circ\text{C}$).
Step-by-step Solution:
1. Evaluate air properties at film temperature:
- $\nu = 1.83 \times 10^{-5}\text{ m}^2\text{/s}, k = 0.0275\text{ W/m·K}, Pr = 0.70$.
2. Calculate Reynolds number:
$$\text{Re} = \frac{V_0 D}{\nu} = \frac{0.5 \times 0.025}{1.83 \times 10^{-5}} = 683 \quad \text{(Laminar)}$$ 3. Compute Graetz number at $x = 0.1\text{ m}$:
$$\text{Gz}_x = \frac{D}{x} \text{Re} \text{Pr} = \frac{0.025}{0.1} \times 683 \times 0.7 = 119.5$$ 4. Compute Nusselt number using Hausen correlation:
$$\text{Nu}_x = 3.66 + \frac{0.0668 \times 119.5}{1 + 0.04 \times (119.5)^{2/3}} = 3.66 + \frac{7.98}{1 + 0.97} = 7.71$$ 5. Convective coefficient $h_x$:
$$h_x = \frac{\text{Nu}_x k}{D} = \frac{7.71 \times 0.0275}{0.025} = 8.48\text{ W/m}^2\text{K}$$