MECH3610 · Advanced Thermofluids
Internal Flow & Pipe Convection
Flow inside ducts and pipes differs from external flow because the fluid is bounded and its bulk temperature changes along the length. This chapter covers the hydraulic diameter, entry-length effects, the constant-flux and constant-wall-temperature cases (with the internal log-mean temperature difference), the laminar Nusselt constants and the turbulent Dittus-Boelter correlation. These pipe-flow calculations appear in the Week-3/4 tutorials and on the 30% open-book mid-term.
What this chapter covers
- 01Hydraulic diameter D_h = 4A_c/P for non-circular ducts
- 02Developing vs fully-developed flow and the mean (bulk) temperature
- 03Energy balance Q = m_dot*c_p*(T_e - T_i) for the whole tube
- 04Constant heat flux: mean temperature rises linearly, T_m(x) = T_i + (P*q''_s/(m_dot*c_p))*x
- 05Constant surface temperature: internal LMTD deltaT_lm and Q = h*A_s*deltaT_lm
- 06Fully-developed laminar Nu_D = 4.36 (constant flux), Nu_D = 3.66 (constant wall temperature)
- 07Turbulent Dittus-Boelter Nu_D = 0.023 Re_D^{4/5} Pr^n with n = 0.4 heating, n = 0.3 cooling
- 08Validity limits of Dittus-Boelter: 0.6 <= Pr <= 160, Re_D >~ 10000, L/D >~ 10
Turbulent pipe flow: Dittus-Boelter Nusselt number and h
- +1Reynolds number Re_D = rho*u_m*D/mu = 992*1.0*0.025/6.53e-4 = 24.8/6.53e-4 = 3.80e4. This exceeds 10000, so the flow is turbulent and Dittus-Boelter applies.
- +1The wall is hotter than the fluid, so the water is being heated and the exponent is n = 0.4 (it would be 0.3 for cooling).
- +1Nu_D = 0.023 Re_D^{0.8} Pr^{0.4} = 0.023 * (3.80e4)^{0.8} * (4.32)^{0.4} = 0.023 * 4610 * 1.796 = 190.
- +1Convection coefficient h = Nu_D*k/D = 190*0.631/0.025 = 4.81e3 W/m2-K. (Check the validity box: Pr = 4.32 is in [0.6, 160], Re_D > 10000, and a long tube gives L/D > 10.)
Key terms
- Hydraulic diameter (D_h)
- D_h = 4A_c/P, an equivalent diameter that lets non-circular ducts use the same correlations as round pipes; for a circular tube it reduces to the actual diameter D.
- Mean (bulk) temperature (T_m)
- The flow-weighted average fluid temperature at a cross-section; internal-flow properties are evaluated here (unlike external flow, which uses the film temperature). It rises along a heated tube.
- Fully-developed flow
- The state, reached beyond the entry length, where the velocity (and, thermally, the scaled temperature) profile no longer changes with distance; here the laminar Nusselt numbers take their constant values.
- Laminar Nusselt constants
- For fully-developed laminar pipe flow Nu_D = 4.36 under constant surface heat flux and Nu_D = 3.66 under constant surface temperature; they are pure numbers, independent of Re and Pr.
- Dittus-Boelter correlation
- Nu_D = 0.023 Re_D^{4/5} Pr^n for fully-developed turbulent pipe flow, with n = 0.4 for heating and 0.3 for cooling; valid for 0.6 <= Pr <= 160, Re_D >~ 10000 and L/D >~ 10.
- Internal LMTD
- For a tube at constant wall temperature the driving difference decays exponentially, so the mean is the log-mean deltaT_lm = (deltaT_e - deltaT_i)/ln(deltaT_e/deltaT_i) with deltaT = T_s - T_m; then Q = h*A_s*deltaT_lm.
Internal Flow & Pipe Convection FAQ
When do I use Nu = 4.36 or 3.66 instead of a correlation?
Use the constant Nusselt numbers when the flow is laminar and fully developed: 4.36 for a constant-heat-flux wall and 3.66 for a constant-temperature wall. They do not depend on Re or Pr. Only switch to Dittus-Boelter once the flow is turbulent (Re_D above about 10000).
Is the exponent n in Dittus-Boelter 0.4 or 0.3?
n = 0.4 when the fluid is being heated (the wall is hotter than the fluid) and n = 0.3 when it is being cooled. A quick check: decide the direction of heat flow first, then pick the exponent. Choosing the wrong one is a frequent slip.
Why does internal flow use the mean temperature, not the film temperature?
Because the fluid is bounded, its bulk temperature changes along the tube, so properties are evaluated at the local mean (bulk) temperature rather than a surface-based film temperature. The mean temperature also drives the energy balance Q = m_dot*c_p*(T_e - T_i).
How is pipe convection examined in MECH3610?
As Week-3/4 tutorial and mid-term problems: compute Re_D, choose laminar constant or turbulent Dittus-Boelter, find h, and often close an energy balance or an internal-LMTD calculation. Because the exam is open-book, the marks are for the regime and correlation choice. Confirm the coverage on Moodle.
Exam move
Split internal flow into two decisions: laminar or turbulent (from Re_D vs ~10000-4000), and constant-flux or constant-wall-temperature (which sets the laminar constant 4.36 vs 3.66 and whether you use a linear mean-temperature rise or the internal LMTD). Practise Dittus-Boelter with the correct heating/cooling exponent and always evaluate properties at the mean temperature. Keep the validity box (Pr range, Re threshold, L/D) beside the correlation and tick it. Rehearse closing the energy balance Q = m_dot*c_p*(T_e - T_i) to link h to outlet temperature. Confirm the exam format on Moodle.
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