Heat Transfer · Chapter 6
0%
Heat Transfer · Chapter Six

Fundamentals of Convection

Convection is heat transfer through a fluid in motion. The flow field and the temperature field are inseparable here — to predict the heat rate we must understand the boundary layers that form along a surface.

This chapter explains the physical mechanism of convection, classifies flows, and develops the velocity and thermal boundary layers. It introduces the Reynolds, Prandtl, and Nusselt numbers, derives the governing convection equations, solves flow over a flat plate, and closes with the analogies that link friction to heat transfer.

Learning objectives

  • Explain how wall conduction and fluid advection combine to produce convection.
  • Classify flows by driving mechanism, geometry, regime, and boundary condition.
  • Interpret velocity and thermal boundary layers through \(Re\), \(Pr\), \(Nu\), and \(St\).
  • Select and validate a convection correlation using geometry, characteristic length, property temperature, and range.
  • Calculate heat transfer and friction quantities while auditing units, signs, and physical limits.
6–1

Physical Mechanism of Convection

Convection combines conduction at the wall with energy carried away by bulk fluid motion. Faster flow sweeps heated fluid away more quickly, steepening the temperature gradient at the surface and raising the heat rate.

Convection = Conduction + Advection

At the wall the fluid is stationary (no-slip), so heat first crosses by pure conduction. Fluid motion then transports that energy downstream. The presence of motion is what distinguishes convection from conduction through a still fluid.

Nusselt Number

The Nusselt number is the dimensionless convection coefficient: the ratio of convective heat transfer across a fluid layer to the conduction that would occur across the same layer if it were motionless. \(Nu = 1\) means pure conduction; larger values mean more effective convection.

\[ Nu = \frac{h L_c}{k} \]

Physical meaning

Newton's law, \(\dot Q=hA_s(T_s-T_\infty)\), defines an average coefficient rather than a fluid property. The value of \(h\) embeds geometry, velocity, regime, boundary condition, and temperature-dependent fluid behavior. The wall heat flux is also \(q''=-k(\partial T/\partial y)_s\), linking \(h\) directly to the wall gradient.

Convection rate and wall-gradient definition
\[\dot Q=hA_s(T_s-T_\infty),\qquad h(T_s-T_\infty)=-k\left.\frac{\partial T}{\partial y}\right|_s\]
Use a declared sign convention; \(h\) itself is nonnegative.
EXAMPLE 1Calibrating an Air-Jet Convection Coefficient
Problem. An electrically heated equipment panel exposes \(0.180\ \text{m}^2\) to an air jet. At steady state, \(420\ \text{W}\) passes to air at \(25^\circ\text{C}\) while the nearly isothermal surface is \(60^\circ\text{C}\). Neglect radiation and back losses. Determine the area-average \(h\).

Strategy

Measured steady heat rate, surface area, and bulk-to-wall temperature difference close Newton's law directly; no predictive correlation is needed.

Givens

  • \(\dot Q=420\ \text{W}\), \(A_s=0.180\ \text{m}^2\).
  • \(T_s=60^\circ\text{C}\), \(T_\infty=25^\circ\text{C}\); radiation and parasitic losses neglected.

Physical/model interpretation

The control surface is the exposed panel. Positive heat is defined from panel to air, and the inferred \(h\) averages spatially varying jet behavior.

Step-by-step solution
1. Balance. Steady electrical input equals convection: \(\dot Q=hA_s(T_s-T_\infty)\).
2. Temperature difference. \(\Delta T=60-25=35\ \text{K}\).
3. Solve symbolically. \(h=\dot Q/(A_s\Delta T)\).
4. Calculate. \(h=420/[0.180(35)]=66.7\ \text{W/m}^2\text{K}\).

Unit/sign/limit checks

Units: \(\text{W}/(\text{m}^2\text{K})\) are the required units of \(h\).

Signs: \(T_s>T_\infty\), so positive heat leaves the panel.

Limits: Holding heat rate fixed, doubling area or \(\Delta T\) would halve inferred \(h\); the value is plausible for forced air.

Final answer: \(\boxed{\bar h=66.7\ \text{W/m}^2\text{K}}\).

Engineering insight

If radiation or backside loss exists, assigning all 420 W to convection overestimates \(h\). Calibration work therefore needs a complete uncertainty and loss budget.

6–2

Classification of Fluid Flows

Convection problems are sorted along several axes, each affecting which correlations apply.

Forced vs. Natural

Forced convection is driven by an external device (fan, pump, wind). Natural (free) convection arises from buoyancy when heated fluid becomes lighter and rises.

Problem-solving strategy

  1. Recognize forced versus buoyancy-driven flow and external versus internal geometry.
  2. Choose the correlation's characteristic length and distinguish local from average output.
  3. Evaluate external-flow properties at film temperature \(T_f=(T_s+T_\infty)/2\), unless the correlation specifies otherwise.
  4. Compute governing dimensionless groups before selecting a correlation; never select by equation shape alone.
  5. Confirm range, boundary condition, surface roughness, and property-ratio restrictions, then audit heat direction and magnitude.

External vs. Internal; Other Distinctions

Flow may be external (over a plate or cylinder) or internal (through a pipe or duct). Flows are further classified as viscous/inviscid, compressible/incompressible, laminar/turbulent, and steady/unsteady.

6–3

Velocity Boundary Layer

As fluid flows over a surface, the no-slip condition holds the layer touching the wall at rest. Velocity rises from zero at the wall to the free-stream value over a thin region called the velocity boundary layer.

Wall shear stress and friction coefficient
\[ \tau_s = \mu \frac{\partial u}{\partial y}\bigg|_{y=0}, \qquad \tau_s = C_f\,\frac{\rho\,u_\infty^2}{2} \]
boundary layer edge u∞ (free stream) no-slip wall (u = 0)
Figure 6–1The velocity boundary layer: speed rises from zero at the wall to the free-stream value \(u_\infty\) over a thin growing layer.
6–4

Thermal Boundary Layer

Just as velocity develops a boundary layer, so does temperature. The thermal boundary layer is the region where fluid temperature transitions from the wall value to the free-stream value.

Prandtl Number

The Prandtl number is the ratio of momentum diffusivity to thermal diffusivity. It governs the relative thickness of the velocity and thermal boundary layers: gases have \(Pr \approx 1\) (similar thicknesses), oils have \(Pr \gg 1\) (thin thermal layer), liquid metals have \(Pr \ll 1\) (thick thermal layer).

\[ Pr = \frac{\nu}{\alpha} = \frac{\mu c_p}{k} \]

For laminar flat-plate flow, a useful scaling is \(\delta_t/\delta\sim Pr^{-1/3}\). It is a trend, not a universal profile formula. Variable properties are especially important for oils and gases with large wall-to-free-stream temperature differences.

6–5

Laminar and Turbulent Flows

At low speeds the flow is smooth and orderly (laminar); above a critical speed it becomes chaotic and mixed (turbulent). The Reynolds number predicts the transition.

Reynolds Number

The Reynolds number is the ratio of inertial forces to viscous forces. Below a critical value viscous forces keep the flow laminar; above it inertial forces dominate and the flow becomes turbulent. For flow over a flat plate the transition occurs near \(Re \approx 5\times10^{5}\).

\[ Re = \frac{\rho\,u_\infty L_c}{\mu} = \frac{u_\infty L_c}{\nu} \]

Key Idea Turbulent flow mixes fluid vigorously, so it transfers heat far better than laminar flow — at the cost of higher friction. Many engineering surfaces are deliberately roughened or tripped to trigger turbulence where heat transfer matters.

A critical Reynolds number is not a universal material constant. Free-stream turbulence, pressure gradients, roughness, vibration, and leading-edge shape shift transition. Correlations must state whether they assume laminar-from-leading-edge, fully turbulent, or mixed flow.

6–6

Heat and Momentum Transfer in Turbulent Flow

Turbulent eddies transport momentum and energy far more effectively than molecular motion. This is modeled by adding eddy (turbulent) diffusivities to the molecular ones.

Effective (total) shear and heat flux
\[ \tau_{\text{total}} = (\mu + \mu_t)\frac{\partial u}{\partial y}, \qquad \dot{q}_{\text{total}} = -(k + k_t)\frac{\partial T}{\partial y} \]
\(\mu_t\) and \(k_t\) are eddy viscosity and eddy conductivity — flow properties, not fluid properties, and they vanish at the wall.
6–7

Differential Convection Equations

Applying conservation of mass, momentum, and energy to a differential fluid element yields the governing equations of convection. For steady, two-dimensional, incompressible flow with constant properties:

Continuity (mass conservation)
\[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \]
Momentum (x-direction, boundary-layer form)
\[ \rho\left(u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}\right) = \mu\frac{\partial^2 u}{\partial y^2} - \frac{\partial P}{\partial x} \]
Energy
\[ u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} = \alpha\frac{\partial^2 T}{\partial y^2} \]
6–8

Solutions of Convection Equations for a Flat Plate

Blasius transformed the laminar boundary-layer equations for a flat plate into a single ordinary differential equation using a similarity variable, yielding the classic results for boundary-layer thickness and friction.

Laminar boundary-layer thickness and local friction
\[ \delta = \frac{4.91\,x}{\sqrt{Re_x}}, \qquad C_{f,x} = \frac{0.664}{\sqrt{Re_x}} \]

The displayed Nusselt relation is local. For an isothermal plate that remains laminar from its leading edge, \(\overline{Nu}_L=0.664Re_L^{1/2}Pr^{1/3}\), typically used for \(Re_L<5\times10^5\) and approximately \(Pr\gtrsim0.6\). Use the flow-direction length in \(Re_L\) and in \(h=\overline{Nu}_Lk/L\).

EXAMPLE 2Cooling Load on an Instrument Cover
Problem. Air at \(20^\circ\text{C}\) flows at \(8.0\ \text{m/s}\) over one side of a \(0.60\)-m-long, \(0.40\)-m-wide isothermal cover at \(70^\circ\text{C}\). At the \(45^\circ\text{C}\) film temperature, use \(\nu=1.80\times10^{-5}\ \text{m}^2/\text{s}\), \(k=0.0280\ \text{W/mK}\), and \(Pr=0.70\). Find average \(h\) and heat rate.

Strategy

The cues are external forced flow over an isothermal flat plate. Determine regime at the trailing edge, then use the matching average correlation.

Givens

  • \(U_\infty=8.0\ \text{m/s}\), \(L=0.60\ \text{m}\), width \(=0.40\ \text{m}\).
  • Surface and free-stream temperatures; supplied properties at \(T_f=45^\circ\text{C}\).

Physical/model interpretation

Steady incompressible flow, smooth plate, zero pressure gradient, laminar leading edge, negligible radiation, and one exposed face are assumed.

Step-by-step solution
1. Regime. \(Re_L=U_\infty L/\nu=8(0.60)/(1.80\times10^{-5})=2.67\times10^5<5\times10^5\).
2. Correlation. \(\overline{Nu}_L=0.664Re_L^{1/2}Pr^{1/3}=0.664(2.67\times10^5)^{1/2}(0.70)^{1/3}=304\).
3. Coefficient. \(\bar h=\overline{Nu}_Lk/L=304(0.0280)/0.60=14.2\ \text{W/m}^2\text{K}\).
4. Heat rate. \(A=0.60(0.40)=0.240\ \text{m}^2\); \(\dot Q=\bar hA(70-20)=14.2(0.240)(50)=170\ \text{W}\).

Unit/sign/limit checks

Units: \(Nu\) and \(Re\) are dimensionless; \(k/L\) supplies \(\text{W/m}^2\text{K}\).

Signs: Heat leaves the hotter cover, so the outward rate is positive.

Limits: The correlation range includes both \(Re_L\) and \(Pr\); \(h\) is reasonable for moderate-speed air.

Final answer: \(\boxed{\bar h\approx14.2\ \text{W/m}^2\text{K},\quad \dot Q\approx170\ \text{W}}\).

Engineering insight

Because laminar \(\bar h\propto U_\infty^{1/2}\), doubling fan speed does not double cooling and may increase acoustic and pumping costs substantially.

Local Nusselt number, laminar flat plate
\[ Nu_x = 0.332\,Re_x^{1/2}\,Pr^{1/3} \qquad (Pr > 0.6) \]
6–9

Nondimensionalized Convection Equations and Similarity

Recasting the governing equations in dimensionless variables reveals that the solutions depend only on a small set of dimensionless groups — the basis for similarity and for generalizing experimental data.

Similarity

When two flows share the same dimensionless groups (Reynolds, Prandtl) and geometry, their dimensionless velocity and temperature fields are identical. This lets results from a small model transfer directly to a full-scale system.

Functional dependence revealed by nondimensionalization
\[ Nu = f(Re, Pr), \qquad C_f = f(Re) \]

Similarity requires matched geometry and boundary conditions as well as dimensionless groups. A model with the correct Reynolds number but a different Prandtl number does not reproduce the thermal field.

6–10

Functional Forms of Friction and Convection Coefficients

Because of similarity, correlations for the average Nusselt number and friction coefficient can be written as power-law functions of the governing dimensionless groups, with constants fit to data.

Typical correlation forms
\[ Nu = C\,Re^{m}\,Pr^{n}, \qquad \overline{C_f} = C\,Re^{m} \]
The constants \(C, m, n\) depend on geometry and flow regime and are determined experimentally or from boundary-layer solutions.
6–11

Analogies between Momentum and Heat Transfer

The momentum and energy equations have the same form when \(Pr = 1\). This similarity lets us infer heat transfer from more easily measured friction data — a powerful practical shortcut.

Reynolds Analogy

For flow with \(Pr = 1\) and negligible pressure gradient, the friction coefficient and the heat transfer are directly linked: knowing one gives the other. It connects the wall friction to the Stanton number.

\[ \frac{C_f}{2} = St = \frac{Nu}{Re\,Pr} \qquad (Pr = 1) \]

Modified (Chilton–Colburn) analogy
\[ \frac{C_f}{2} = St\,Pr^{2/3} \qquad (0.6 < Pr < 60) \]
The \(Pr^{2/3}\) correction extends the Reynolds analogy to fluids whose Prandtl number differs from one.
Course Note

The Chilton–Colburn analogy is genuinely useful in the field: if you can measure or estimate drag (friction) on a surface, you can estimate its heat transfer coefficient without a separate thermal test. Keep the \(0.6 < Pr < 60\) validity range in mind — it covers air, water, and most oils, but not liquid metals.

EXAMPLE 3Estimating Heat Transfer from a Friction Test
Problem. A turbulent water-channel test over a \(0.50\)-m surface reports an average skin-friction coefficient \(\overline C_f=0.0060\) at \(Re_L=1.50\times10^5\). At the bulk-film reference temperature, \(Pr=5.0\) and \(k=0.62\ \text{W/mK}\). Using the Chilton–Colburn analogy, estimate \(\overline{Nu}_L\) and \(\bar h\).

Strategy

Friction data and a moderate Prandtl number invite the momentum–heat analogy. First recover Stanton number, then Nusselt number.

Givens

  • \(\overline C_f=0.0060\), \(Re_L=1.50\times10^5\), \(Pr=5.0\).
  • \(k=0.62\ \text{W/mK}\), \(L=0.50\ \text{m}\); \(0.6

Physical/model interpretation

The analogy treats turbulent momentum and heat transport as similar. It assumes a compatible friction definition, attached flow, and no dominant pressure-gradient, roughness, or buoyancy effects.

Step-by-step solution
1. Formulate. \(\overline C_f/2=St\,Pr^{2/3}\), so \(St=(0.0060/2)/5^{2/3}\).
2. Stanton number. Since \(5^{2/3}=2.924\), \(St=1.026\times10^{-3}\).
3. Nusselt number. \(St=Nu/(Re\,Pr)\), hence \(\overline{Nu}_L=(1.026\times10^{-3})(1.50\times10^5)(5.0)=769\).
4. Coefficient. \(\bar h=\overline{Nu}_Lk/L=769(0.62)/0.50=954\ \text{W/m}^2\text{K}\).

Unit/sign/limit checks

Units: \(St,Re,Pr,Nu\) are dimensionless; \(k/L\) gives \(\text{W/m}^2\text{K}\).

Signs: \(C_f\), \(St\), \(Nu\), and \(h\) are positive; heat direction still depends on \(T_s-T_b\).

Limits: The supplied \(Pr=5\) lies within the analogy range, and the large water-side \(h\) is physically plausible.

Final answer: \(\boxed{\overline{Nu}_L\approx769,\quad \bar h\approx9.54\times10^2\ \text{W/m}^2\text{K}}\).

Engineering insight

This is a screening estimate, not a certified design value. Surface roughness, developing flow, conjugate conduction, and the test's friction convention must be reconciled before specifying hardware.

Chapter recap

  • Convection begins with molecular conduction at the wall and is shaped by fluid transport in the boundary layer.
  • \(Re\) selects regime, \(Pr\) compares momentum and thermal diffusion, and \(Nu\) converts boundary-layer behavior into \(h\).
  • A valid correlation must match mechanism, geometry, length, local/average definition, boundary condition, properties, and range.

Before moving on

  • Can you classify a flow and state where its properties should be evaluated?
  • Can you audit correlation validity, heat direction, dimensions, and limiting behavior?
← Chapter 5 · Numerical Methods All chapters →