Chapter 6: Convection Heat Transfer

Vocabulary practice flashcards covering fundamental definitions, boundary layer scales, governing equations, similarity solutions, and dimensionless numbers for Chapter 6 Convection Heat Transfer.

Convection Heat Transfer

The mode of heat transfer that comprises the combined effects of bulk fluid motion (advection, where $U \neq 0$) and microscopic molecular motion (diffusion or conduction, where $U = 0$).

Advection

The mechanism of heat transport accomplished by the macroscopic, bulk movement of a flowing fluid ($U \neq 0$).

Diffusion (Conduction in Fluids)

The mechanism of heat transport that occurs via microscopic molecular motion in the absence of bulk velocity ($U = 0$).

Newton's Law of Cooling

The empirical rate equation for convection defined by $h = \frac{\dot{q}''}{T_s - T_\infty} = \frac{\dot{Q}}{A_s(T_s - T_\infty)}$, where $\dot{q}''$ is the heat flux at the fluid-solid interface.

Convection Heat Transfer Coefficient ($h$)

A non-material property parameter defining convection intensity, which depends on fluid properties (kinematic viscosity $\nu$, thermal diffusivity $\alpha$), flow strength ($Re$), flow direction, and interface geometry.

Prandtl Number ($Pr$)

The dimensionless ratio of momentum diffusivity to thermal diffusivity, defined as $Pr = \frac{\nu}{\alpha}$.

No-Slip Condition

The physical condition at an impermeable solid-fluid interface where the fluid velocity relative to the stationary surface is zero ($u = 0$), causing advection to vanish and heat transfer at the boundary to occur solely by conduction.

Fluid-Solid Interface Heat Flux Equality

The boundary requirement that convective heat flux equals conductive heat flux at the surface: $\dot{q}_{\text{conv}} = \dot{q}_{\text{cond}}$, leading to $h(T_s - T_\infty) = -k_f \left(\frac{\partial T}{\partial y}\right)_{y=0}$.

Flow Strengthening Convection Effect

The physical process where increasing flow velocity diminishes boundary layer thickness ($\Delta x$ or $\delta$), thereby steepening the surface temperature gradient ($\nabla T$) and increasing the heat flux ($\dot{q}$).

Local Convection Heat Transfer Coefficient ($h_x$)

The position-dependent heat transfer coefficient given by $h_x = \frac{-k_f \left(\frac{\partial T}{\partial y}\right)_{y=0}}{T_s - T_\infty}$, which varies with spatial coordinates $x$ and $z$ as well as time $t$.

Surface-Averaged Convection Coefficient ($\bar{h}$)

The mean heat transfer coefficient over an entire surface area $A_s$, defined as $\bar{h} = \frac{1}{A_s} \int_{A_s} h \, dA_s$, or $\bar{h} = \frac{1}{L} \int_0^L h(x) \, dx$ for one-dimensional variations.

Thin Boundary Layer Assumption

The scale requirement for boundary layer flow stating that hydrodynamic and thermal boundary layer thicknesses are much smaller than the characteristic plate length: $\delta \ll L$ and $\delta_T \ll L$.

2D Incompressible Continuity Equation

The differential conservation of mass equation for two-dimensional steady boundary layer flow: $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$.

$x$-Momentum Boundary Layer Equation

The differential momentum balance in the streamwise direction: $u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = -\frac{1}{\rho}\frac{\partial P}{\partial x} + \nu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right)$.

$y$-Momentum Boundary Layer Equation

The differential momentum balance normal to the surface: $u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = -\frac{1}{\rho}\frac{\partial P}{\partial y} + \nu \left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2}\right) + \rho g_y$.

2D Boundary Layer Energy Equation

The differential thermal energy equation: $u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \alpha \left(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right) + \Phi$, where $\Phi$ represents viscous dissipation.

Leading-Edge Boundary Conditions

The upstream flow state specified at $x = 0$: $u(0, y) = U_\infty$, $v(0, y) = 0$, $T(0, y) = T_\infty$, and $P(0, y) = P_\infty$.

Solid Wall Boundary Conditions ($y = 0$)

The physical conditions specified along an impermeable, stationary, isothermal solid boundary: $u(x, 0) = 0$, $v(x, 0) = 0$, and $T(x, 0) = T_s$.

Free-Stream Boundary Conditions ($y \rightarrow \infty$)

The conditions in the undisturbed fluid far from the plate: $u(x, \infty) = U_\infty$, $v(x, \infty) = 0$, $T(x, \infty) = T_\infty$, and $P(x, \infty) = P_\infty$.

Scale Analysis

A method of estimating orders of magnitude within a physical domain by approximating derivatives as ratios of finite scale differences (e.g., $\frac{\partial T}{\partial y} \sim \frac{\Delta T}{\Delta y}$) to identify dominant forces and scale relationships.

Transverse Velocity Scale ($V$)

The order of magnitude of the normal velocity component in a thin boundary layer, established from the continuity equation as $V \sim \frac{U_\infty \delta}{L}$.

Transverse Pressure Variation in Boundary Layers

The scaling outcome showing that transverse pressure changes scale to negligible values ($\frac{\partial P}{\partial y} \sim 0$), establishing that static pressure throughout the boundary layer is governed solely by the free-stream pressure: $P \approx P_\infty(x)$.

Simplified Boundary Layer Momentum Equation (Prandtl, 1905)

The reduced boundary layer equation for laminar flow over a flat plate with zero pressure gradient: $u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2}$.

Hydrodynamic Boundary Layer Thickness ($\delta$)

The normal distance from the solid surface to the point where the streamwise velocity reaches $99\%$ of the free-stream velocity: $u \approx 0.99 U_\infty$.

Momentum Boundary Layer Scale Relation

The scale analysis result for laminar boundary layer growth along a flat plate: $\delta(x) \sim x (Re_x)^{-1/2}$, or $\delta \sim L (Re_L)^{-1/2}$.

Leading-Edge Boundary Layer Growth Rate

The derivative of boundary layer thickness with respect to streamwise position at the leading edge ($x=0$), which tends to infinity: $\left.\frac{d\delta}{dx}\right|_{x=0} = \infty$.

Irrotational (Inviscid) Flow Region

The region located outside the momentum boundary layer ($y > \delta$) where viscous shear forces are negligible and the flow is effectively inviscid.

Wall Shear Stress ($\tau_s$)

The tangential frictional shear force per unit area exerted on a solid boundary, defined as $\tau_s = \left.\mu \frac{\partial u}{\partial y}\right|_{y=0} \sim \mu \frac{U_\infty}{\delta(x)}$.

Skin Friction Coefficient ($C_f(x)$) Scale

The dimensionless local friction parameter defined as $C_f(x) = \frac{\tau_s}{\rho U_\infty^2 / 2}$, which scales proportionally to $(Re_x)^{-1/2}$.

Thermal Boundary Layer Thickness ($\delta_T$)

The normal distance from the surface where the temperature difference relative to the wall reaches $99\%$ of the total temperature difference: $(T - T_s) \approx 0.99(T_\infty - T_s)$.

Isothermal Flow Region

The region situated outside the thermal boundary layer ($y > \delta_T$) where temperature is uniform and equal to the free-stream value: $T = T_\infty$.

Thermal Boundary Layer Scaling for $Pr < 1$

The scale analysis relationship for fluids with $Pr < 1$ where the thermal boundary layer is thicker than the momentum boundary layer: $\delta_T(x) \sim x (Re_x Pr)^{-1/2} \sim \delta(x) Pr^{-1/2}$.

Thermal Boundary Layer Scaling for $Pr > 1$

The scale analysis relationship for fluids with $Pr > 1$ where the thermal boundary layer is thinner than the momentum boundary layer: $\delta_T(x) \sim x (Re_x)^{-1/2} Pr^{-1/3} \sim \delta(x) Pr^{-1/3}$.

Liquid Metals Prandtl Number Range

A characteristic Prandtl number range of $0.004$ to $0.030$, indicating exceptionally rapid thermal diffusion relative to momentum diffusion.

Gases Prandtl Number Range

A characteristic Prandtl number range of $0.7$ to $1.0$, which causes thermal and momentum boundary layers to develop at roughly equal rates ($\delta_T \approx \delta$).

Water Prandtl Number Range

A characteristic Prandtl number range of $1.7$ to $13.7$, meaning momentum diffusivity moderately exceeds thermal diffusivity.

Light Organic Fluids Prandtl Number Range

A characteristic Prandtl number range of $5$ to $50$, reflecting moderately high viscosity relative to thermal conduction.

Oils Prandtl Number Range

A characteristic Prandtl number range of $50$ to $100{,}000$, producing thermal boundary layers that are substantially thinner than momentum boundary layers.

Glycerin Prandtl Number Range

A characteristic Prandtl number range of $2000$ to $100{,}000$, representing highly viscous fluids with slow thermal diffusion.

Nusselt Number ($Nu_x$)

The dimensionless parameter defined as $Nu_x = \frac{h_x x}{k_f} \sim \frac{x}{\delta_T(x)}$, representing the ratio of convective to conductive heat transfer across a characteristic length.

Nusselt Number Scaling for $Pr < 1$

The scale relationship for laminar convective heat transfer in fluids with low Prandtl number: $Nu_x \sim (Re_x Pr)^{1/2}$.

Nusselt Number Scaling for $Pr > 1$

The scale relationship for laminar convective heat transfer in fluids with high Prandtl number: $Nu_x \sim Re_x^{1/2} Pr^{1/3}$.

Eddy Diffusivities

Turbulent transport coefficients that quantify the apparent, greatly increased diffusion of momentum and energy caused by macroscopic fluid eddies, resulting in enhanced mixing and transport compared to laminar molecular diffusion.

Turbulent Wall Velocity Gradient

The steep velocity gradient at the wall ($(\partial \bar{u}/\partial y)_{y=0}$) characteristic of turbulent boundary layers, which is significantly larger than in laminar flow, leading to higher skin friction and heat transfer.

Blasius Similarity Variable ($\eta$)

The dimensionless coordinate defined as $\eta = y \sqrt{\frac{U_\infty}{\nu x}}$, which collapses the two-dimensional laminar boundary layer equations into a single ordinary differential equation.

Blasius Equation

The ordinary differential equation for laminar boundary layer flow: $2 f''' + f f'' = 0$, where $f(\eta)$ is related to the stream function via $\psi = \sqrt{\nu x U_\infty} f(\eta)$.

Blasius Skin Friction Coefficient Formula

The exact similarity solution for local skin friction on an isothermal laminar flat plate: $C_{f,x} = 0.664 Re_x^{-1/2}$, with boundary layer thickness $\delta(x) = 4.91 x Re_x^{-1/2}$.

Pohlhausen Solution for Local Nusselt Number

The exact similarity solution for laminar heat transfer on an isothermal flat plate with $Pr > 0.6$: $Nu_x = 0.332 Re_x^{1/2} Pr^{1/3}$, with $\delta_T(x) = \frac{4.91 x}{Pr^{1/3} Re_x^{1/2}}$.

Stanton Number ($St_x$)

A dimensionless heat transfer parameter expressing convection relative to thermal capacity of the flow: $St_x = \frac{Nu_x}{Re_x Pr} = \frac{h_x}{\rho c_p U_\infty}$.

Chilton-Colburn Analogy ($j_H$-Factor)

The momentum-heat transfer analogy defined by $j_H = St_x Pr^{2/3} = \frac{C_{f,x}}{2} = 0.332 Re_x^{-1/2}$, which directly relates the skin friction coefficient to convective heat transfer.