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 = 2.8006E+04
Graetz Number = 2145.9305
Nu Developing = 197.5216
Nu Fully = 188.7375
Coeff h = 4843.2298 W/m2K
Transfer Q = 30430.9101 W
T_out = 34.88 C
Entry Len L_h = 0.2500 m
Entry Len L_t = 0.2500 m
--- LENGTH SWEEP ---
x[m] Gz Nu_local h_local[W/m2K]
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0.0020 2.146E+06 1294.5852 31743.2297
0.0853 5.034E+04 268.7081 6588.7232
0.1685 2.547E+04 238.3735 5844.9182
0.2517 1.705E+04 226.2122 5546.7232
0.3350 1.281E+04 219.4196 5380.1695
0.4183 1.026E+04 215.0046 5271.9124
0.5015 8.558E+03 211.8703 5195.0588
0.5847 7.340E+03 209.5124 5137.2430
0.6680 6.425E+03 207.6642 5091.9250
0.7512 5.713E+03 206.1703 5055.2969
0.8345 5.143E+03 204.9339 5024.9798
0.9178 4.677E+03 203.8909 4999.4052
1.0010 4.288E+03 202.9973 4977.4941
1.0842 3.958E+03 202.2218 4958.4776
1.1675 3.676E+03 201.5413 4941.7921
1.2508 3.431E+03 200.9386 4927.0142
1.3340 3.217E+03 200.4005 4913.8194
1.4173 3.028E+03 199.9166 4901.9542
1.5005 2.860E+03 199.4787 4891.2178
1.5838 2.710E+03 199.0803 4881.4486
1.6670 2.575E+03 198.7160 4872.5152
1.7502 2.452E+03 198.3813 4864.3095
1.8335 2.341E+03 198.0727 4856.7416
1.9167 2.239E+03 197.7870 4849.7362
2.0000 2.146E+03 197.5216 4843.2298
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 $h_x$:
$$h_x = \frac{\text{Nu}_x k}{D} = \frac{7.71 \times 0.0275}{0.025} = 8.48\text{ W/m}^2\text{K}$$