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 = 7.9527E+02
Graetz Number = 47.0535
Nu Developing = 5.7261
Nu Fully = 3.6600
Coeff h = 6.0238 W/m2K
Transfer Q = 6.1773 W
T_out = 46.23 C
Entry Len L_h = 0.9941 m
Entry Len L_t = 0.7058 m
--- LENGTH SWEEP ---
x[m] Gz Nu_local h_local[W/m2K]
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0.0003 4.705E+04 62.8156 66.0820
0.0128 1.104E+03 17.6459 18.5635
0.0253 5.585E+02 13.7086 14.4214
0.0378 3.738E+02 11.7787 12.3912
0.0502 2.809E+02 10.5699 11.1195
0.0627 2.250E+02 9.7212 10.2267
0.0752 1.877E+02 9.0839 9.5563
0.0877 1.609E+02 8.5836 9.0299
0.1002 1.409E+02 8.1779 8.6031
0.1127 1.253E+02 7.8409 8.2487
0.1252 1.128E+02 7.5558 7.9486
0.1377 1.025E+02 7.3106 7.6908
0.1501 9.401E+01 7.0973 7.4664
0.1626 8.679E+01 6.9097 7.2690
0.1751 8.061E+01 6.7432 7.0938
0.1876 7.524E+01 6.5942 6.9371
0.2001 7.054E+01 6.4601 6.7961
0.2126 6.640E+01 6.3386 6.6683
0.2251 6.272E+01 6.2280 6.5519
0.2376 5.942E+01 6.1268 6.4454
0.2500 5.645E+01 6.0338 6.3475
0.2625 5.377E+01 5.9480 6.2572
0.2750 5.133E+01 5.8685 6.1737
0.2875 4.910E+01 5.7948 6.0961
0.3000 4.705E+01 5.7261 6.0238
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 $h_x$:
$$h_x = \frac{\text{Nu}_x k}{D} = \frac{7.71 \times 0.0275}{0.025} = 8.48\text{ W/m}^2\text{K}$$