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 = 9.5432E+03
Graetz Number = 338.7851
Nu Developing = 32.4701
Nu Fully = 28.9182
Coeff h = 17.0792 W/m2K
Transfer Q = 167.0615 W
T_out = 48.93 C
Entry Len L_h = 0.5000 m
Entry Len L_t = 0.5000 m
--- LENGTH SWEEP ---
x[m] Gz Nu_local h_local[W/m2K]
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0.0010 3.388E+05 476.0657 250.4106
0.0426 7.948E+03 61.2542 32.2197
0.0843 4.021E+03 48.9884 25.7679
0.1259 2.691E+03 44.0710 23.1814
0.1675 2.023E+03 41.3245 21.7367
0.2091 1.620E+03 39.5393 20.7977
0.2507 1.351E+03 38.2719 20.1310
0.2924 1.159E+03 37.3185 19.6295
0.3340 1.014E+03 36.5712 19.2364
0.3756 9.019E+02 35.9672 18.9187
0.4173 8.119E+02 35.4672 18.6558
0.4589 7.383E+02 35.0455 18.4339
0.5005 6.769E+02 34.6841 18.2439
0.5421 6.249E+02 34.3706 18.0789
0.5837 5.804E+02 34.0954 17.9342
0.6254 5.417E+02 33.8517 17.8060
0.6670 5.079E+02 33.6341 17.6915
0.7086 4.781E+02 33.4385 17.5886
0.7502 4.516E+02 33.2614 17.4955
0.7919 4.278E+02 33.1003 17.4108
0.8335 4.065E+02 32.9530 17.3333
0.8751 3.871E+02 32.8177 17.2621
0.9168 3.696E+02 32.6929 17.1965
0.9584 3.535E+02 32.5774 17.1357
1.0000 3.388E+02 32.4701 17.0792
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 $h_x$:
$$h_x = \frac{\text{Nu}_x k}{D} = \frac{7.71 \times 0.0275}{0.025} = 8.48\text{ W/m}^2\text{K}$$