Heat Transfer · Chapter 7
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Heat Transfer · Chapter Seven

External Forced Convection

External forced convection couples a driven free stream to boundary layers that develop over, and sometimes separate from, a surface.

The central engineering skill is not memorizing a Nusselt correlation; it is matching geometry, characteristic length, boundary condition, property temperature, and Reynolds-number range before using one.

Learning objectives

  • Distinguish skin-friction and pressure drag and connect both to boundary-layer behavior.
  • Select characteristic length, area, and local or average coefficients for plates and bluff bodies.
  • Evaluate fluid properties at a defensible film temperature and verify correlation limits.
  • Compute heat transfer from plates, cylinders, spheres, and tube banks.
  • Audit units, heat-flow direction, limiting behavior, and design sensitivity.
7–1

Drag and Heat Transfer in External Flow

An imposed stream creates velocity and thermal boundary layers. Wall shear produces skin-friction drag; separation creates a wake and pressure drag. Streamlined bodies are usually friction-dominated, while bluff bodies are usually form-dominated. The same near-wall gradients set shear stress and convection.

Recognition and model selection

First identify an external flow: the fluid has an approach velocity and is not confined by a duct. Then identify geometry, the correlation's area convention, whether a local or surface-average result is needed, and whether the surface temperature or heat flux is prescribed.

Drag and convection definitions
\[F_D=C_DA_{\rm ref}\frac{\rho V_\infty^2}{2},\qquad \overline{Nu}_L=\frac{\bar hL}{k},\qquad \dot Q=\bar hA_s(T_s-T_\infty)\]
Use the reference area specified with \(C_D\); use exposed heat-transfer area in Newton's law. Properties are commonly evaluated at \(T_f=(T_s+T_\infty)/2\) unless the selected correlation states otherwise.

Physical meaning

\(Nu\) compares convection with conduction through a fluid layer of thickness \(L\). Larger velocity thins the boundary layer, increases the wall gradient, and generally raises both drag and heat transfer.

Problem-solving strategy

  1. Recognize the geometry and define \(V_\infty\), \(L_c\), exposed area, and requested average or local quantity.
  2. Evaluate properties at the correlation's reference temperature, usually the film temperature.
  3. Compute \(Re\) and \(Pr\), then choose only a correlation whose geometry, boundary condition, and validity range match.
  4. Find \(Nu\), then \(h=Nu\,k/L_c\), and apply the energy or drag relation.
  5. Audit dimensions, heat-flow sign, regime, and whether radiation or natural convection is also important.
7–2

Parallel Flow over Flat Plates

For a smooth plate with zero pressure gradient, \(Re_x=V_\infty x/\nu\) grows downstream. A laminar boundary layer commonly transitions near \(Re_x\approx5\times10^5\), but roughness and free-stream turbulence can shift transition. Local \(h_x\) falls as the boundary layer thickens; the average \(\bar h_L\) over \(0\le x\le L\) is therefore larger than the local value at \(L\).

Isothermal smooth plate, average correlations
\[\overline{Nu}_L=0.664Re_L^{1/2}Pr^{1/3}\quad(Re_L<5\times10^5),\]\[\overline{Nu}_L=(0.037Re_L^{0.8}-871)Pr^{1/3}\quad(5\times10^5\lesssim Re_L\lesssim10^7)\]
These are average, not local, values and are commonly used for \(Pr\gtrsim0.6\). The second expression includes the laminar leading edge.
EXAMPLE 1Air cooling of an instrument panel
Problem. Air at 20 °C flows at 4.0 m/s over one face of a 0.60 m by 0.40 m isothermal calibration panel at 70 °C, along the 0.60 m length. Given film-temperature properties \(k=0.0277\) W/m·K, \(\nu=1.75\times10^{-5}\) m²/s, and \(Pr=0.70\), determine \(\bar h\) and convective heat removal.

Strategy

A driven stream parallel to a smooth plate calls for an external flat-plate correlation; calculate \(Re_L\) before choosing laminar or mixed flow.

Givens

  • \(V=4.0\) m/s, \(L=0.60\) m, width \(b=0.40\) m, \(T_s-T_\infty=50\) K.
  • Properties supplied at \(T_f=45^\circ\)C; find \(\bar h,\dot Q\).

Physical/model interpretation

Steady, one-sided, isothermal plate; negligible edge effects and radiation. Positive \(\dot Q\) is heat leaving the panel.

Numbered derivation steps
1. Classify. \(Re_L=VL/\nu=(4.0)(0.60)/(1.75\times10^{-5})=1.37\times10^5\), below the stated transition criterion.
2. Correlate. \(\overline{Nu}_L=0.664(1.37\times10^5)^{1/2}(0.70)^{1/3}=218\).
3. Recover the coefficient. \(\bar h=\overline{Nu}_Lk/L=(218)(0.0277)/0.60=10.1\ {\rm W/m^2K}\).
4. Apply the area. \(A_s=Lb=0.240\ {\rm m^2}\), so \(\dot Q=\bar hA_s\Delta T=(10.1)(0.240)(50)=121\) W.

Unit/sign/limit checks

Units: \(Nu\,k/L\) gives W/m²K and \(hA\Delta T\) gives W.

Signs: \(T_s>T_\infty\), so 121 W leaves the panel as defined.

Limit: \(Re_L\) is comfortably laminar; as \(V\to0\), this forced-flow model trends toward zero and natural convection must replace it.

Final answer: \(\boxed{\bar h=10.1\ {\rm W/m^2K},\quad \dot Q=121\ {\rm W\ outward}}\)

Engineering insight

The modest air-side coefficient makes exposed area and velocity influential. A real thermal budget must add radiation rather than hiding it inside \(h\).

7–3

Flow across Cylinders and Spheres

Curvature, separation, and a wake prevent a single boundary-layer solution from describing a bluff body. Diameter is the characteristic length, and correlations return a surface-average \(Nu_D\). Never substitute a flat-plate expression merely because the upstream flow is uniform.

Churchill–Bernstein cylinder correlation
\[Nu_D=0.3+\frac{0.62Re_D^{1/2}Pr^{1/3}}{[1+(0.4/Pr)^{2/3}]^{1/4}}\left[1+\left(\frac{Re_D}{282000}\right)^{5/8}\right]^{4/5}\]
Use \(Re_D=V_\infty D/\nu\), film-temperature properties, and check \(Re_DPr>0.2\). For spheres, use a sphere-specific relation such as Whitaker, including its viscosity correction and range.
EXAMPLE 2Crosswind cooling of a heated sensor mast
Problem. A 2.0 m long, 50 mm diameter cylindrical sensor mast has a uniform surface temperature of 100 °C in 20 °C air crossing it at 12 m/s. At the 60 °C film temperature, \(k=0.0283\) W/m·K, \(\nu=1.85\times10^{-5}\) m²/s, \(Pr=0.70\). Estimate side-surface convection loss.

Strategy

Cross-flow around a cylinder is separation dominated, so use diameter and Churchill–Bernstein, then multiply the average coefficient by cylindrical side area.

Givens

  • \(D=0.050\) m, \(L=2.0\) m, \(V=12\) m/s, \(\Delta T=80\) K.
  • Film properties as supplied; ends neglected.

Physical/model interpretation

Steady two-dimensional cross-flow, uniform surface temperature, negligible end effects and radiation. Heat leaving the mast is positive.

Numbered derivation steps
1. Establish regime. \(Re_D=VD/\nu=(12)(0.050)/(1.85\times10^{-5})=3.24\times10^4\), and \(Re_DPr=2.27\times10^4>0.2\).
2. Evaluate correlation. Substitution in Churchill–Bernstein gives \(Nu_D=104.9\).
3. Convert to \(h\). \(\bar h=Nu_Dk/D=(104.9)(0.0283)/0.050=59.4\ {\rm W/m^2K}\).
4. Compute heat loss. \(A_s=\pi DL=0.3142\ {\rm m^2}\), so \(\dot Q=(59.4)(0.3142)(80)=1.49\times10^3\) W.

Unit/sign/limit checks

Units: \(k/D\) supplies W/m²K and the final product is W.

Signs: The mast is hotter, so the computed 1.49 kW is outward.

Limit: \(Nu_D\gg1\) is consistent with strong forced convection; a zero-speed cylinder would approach a conduction/natural-convection limit, not this result.

Final answer: \(\boxed{\dot Q_{\rm conv}=1.49\ {\rm kW\ outward}}\)

Engineering insight

The cylinder correlation captures wake effects but not gusts, mounting interference, or radiation; those require separate allowances in equipment qualification.

7–4

Flow across Tube Banks

Tube-bank flow accelerates between neighboring tubes. Consequently, heat-transfer and pressure-drop correlations use the maximum gap velocity, not the approach velocity. Arrangement, pitch ratios, row count, and property correction are part of the model; constants cannot be transferred between tables or Reynolds ranges.

Aligned-bank velocity and supplied correlation form
\[V_{\max}=\frac{S_T}{S_T-D}V,\qquad Nu_D=CRe_{D,\max}^{m}Pr^{0.36}\left(\frac{Pr}{Pr_s}\right)^{1/4}\]
For staggered banks the narrowest passage may be diagonal. Apply the published finite-row correction when required.
EXAMPLE 3Preliminary coefficient for an air-heater bank
Problem. Air approaches an aligned bank of 20 mm tubes at 3.0 m/s; transverse pitch is 40 mm. For the applicable range, a supplied bank correlation has \(C=0.27\), \(m=0.63\). Use \(\nu=1.90\times10^{-5}\) m²/s, \(k=0.0290\) W/m·K, \(Pr=0.70\), and \(Pr_s=0.69\) to estimate the uncorrected average \(h\).

Strategy

The repeated cylinders identify tube-bank flow. Accelerate the approach flow through the transverse gap, then use the supplied arrangement-specific constants.

Givens

  • \(D=0.020\) m, \(S_T=0.040\) m, \(V=3.0\) m/s.
  • \(C=0.27,m=0.63,\nu,k,Pr,Pr_s\) supplied; row correction requested to be excluded.

Physical/model interpretation

Steady aligned bank with uniform approach flow. The result is a correlation estimate, not a complete exchanger rating or pressure-drop design.

Numbered derivation steps
1. Find gap velocity. \(V_{\max}=S_TV/(S_T-D)=(0.040)(3.0)/(0.020)=6.0\) m/s.
2. Form Reynolds number. \(Re_{D,\max}=V_{\max}D/\nu=(6.0)(0.020)/(1.90\times10^{-5})=6.32\times10^3\).
3. Apply the supplied fit. \(Nu_D=0.27(6.32\times10^3)^{0.63}(0.70)^{0.36}(0.70/0.69)^{0.25}=59.1\).
4. Recover the coefficient. \(\bar h=Nu_Dk/D=(59.1)(0.0290)/0.020=85.7\ {\rm W/m^2K}\).

Unit/sign/limit checks

Units: \(Nu_D\) is dimensionless and \(k/D\) gives W/m²K.

Signs: \(h\) is positive; actual heat direction follows \(T_s-T_\infty\), which was not specified.

Limit: \(V_{\max}=2V\) follows from a gap half the pitch; using approach speed would underpredict \(Re\) and \(h\).

Final answer: \(\boxed{\bar h_{\rm uncorrected}=85.7\ {\rm W/m^2K}}\)

Engineering insight

A production design still needs the actual row correction, pressure drop, fouling allowance, fan operating point, and uncertainty of the selected correlation.

Chapter recap

  • External-flow selection begins with geometry, characteristic length, boundary condition, and local versus average output.
  • Evaluate properties at the specified reference temperature and confirm \(Re\), \(Pr\), and geometry limits.
  • Flat plates, isolated bluff bodies, and tube banks require different models.
  • Before moving on, be able to compute \(Re\), select and audit a correlation, recover \(h\), and interpret heat direction and uncertainty.
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