Heat transfer coefficient
The heat transfer coefficient, often denoted as h or \alpha, is a proportionality factor that characterizes the rate of convective heat transfer between a solid surface and an adjacent fluid, as expressed in Newton's law of cooling: q'' = h (T_s - T_f), where q'' is the heat flux, T_s is the surface temperature, and T_f is the fluid temperature.[1][2] This coefficient quantifies the effectiveness of heat exchange in processes involving fluid flow over surfaces, such as in pipes, heat exchangers, and cooling systems, and its value depends on fluid properties (e.g., viscosity, thermal conductivity), flow characteristics (e.g., velocity, turbulence), and surface geometry.[1][2] In engineering applications, the convective heat transfer coefficient typically ranges from 10–100 W/m²·K for gases like air in free convection to 500–10,000 W/m²·K for liquids like water in forced convection, with higher values in turbulent flows due to enhanced mixing and a thinner boundary layer.[3][2] For composite systems involving multiple layers or fluids, the overall heat transfer coefficient U is used, defined by Q = U A \Delta T, where Q is the total heat transfer rate and it accounts for both convective resistances on either side and conductive resistances through walls, calculated as \frac{1}{U} = \frac{1}{h_i} + \sum \frac{\delta}{k} + \frac{1}{h_o} for plane walls.[3][1] Dimensionless groups like the Nusselt number (Nu = \frac{[h](/page/H+) L}{k}), Reynolds number, and Prandtl number are essential for predicting [h](/page/H+) in various flow regimes, enabling correlations for design in thermal systems.[1] The SI unit for both [h](/page/H+) and U is watts per square meter per kelvin (W/m²·K), reflecting the heat transfer rate per unit area per unit temperature difference, with common conversions to imperial units like Btu/(h·ft²·°F).[1][3]Fundamentals
Definition and Newton's Law of Cooling
The heat transfer coefficient, denoted as h, is defined as the proportionality factor between the convective heat flux at a solid surface and the temperature difference driving the heat transfer process. In the context of Newton's law of cooling, the rate of heat transfer q from a surface of area A to a surrounding fluid is given by q = h A (T_s - T_\infty), where T_s is the temperature of the surface and T_\infty is the temperature of the fluid far from the surface. This formulation quantifies the convective heat transfer under conditions where the fluid motion enhances heat removal compared to pure conduction.[4] Isaac Newton first articulated the foundational idea in 1701 in his paper "Scala graduum Caloris," published in the Philosophical Transactions of the Royal Society, where he stated that the rate of cooling of a warm body suspended in cooler air is proportional to the difference in temperature between the body and the surrounding air. This empirical observation laid the groundwork for convective heat transfer analysis, though Newton's original experiments involved natural convection with limited control over flow conditions. Modern interpretations refine the law to address non-linear temperature dependencies that arise in cases of large temperature gradients, where radiation or variable fluid properties may influence the process.[5][6] Physically, h represents the effectiveness of convection in bridging the thermal resistance between the surface and the bulk fluid, encapsulating the combined effects of molecular diffusion near the surface and bulk fluid motion. Its magnitude depends on intrinsic fluid properties, such as thermal conductivity, viscosity, and density, as well as extrinsic flow characteristics like velocity and turbulence intensity, which determine how efficiently heat is carried away from the surface. Higher values of h indicate more vigorous convective transport, often achieved through forced flow or enhanced surface features.[4][7] The derivation of h conceptually stems from equating the conductive heat flux at the fluid-solid interface, as described by Fourier's law of conduction, to the overall convective heat flux. Fourier's law gives the local heat flux as q'' = -k \frac{\partial T}{\partial y} \big|_{y=0}, where k is the fluid's thermal conductivity and y is the direction normal to the surface. Setting this equal to the convective expression q'' = h (T_s - T_\infty) yields h = -\frac{k \frac{\partial T}{\partial y} \big|_{y=0}}{T_s - T_\infty}, illustrating how h normalizes the near-wall temperature gradient to the far-field driving potential. This approach assumes steady-state conditions, constant thermophysical properties, negligible radiative contributions, and a linear temperature profile approximation valid for small Biot numbers.[8][9]Local versus Average Coefficients
The local heat transfer coefficient, h(x), represents the convective heat transfer at a specific point on a surface and is defined by Newton's law of cooling as h(x) = \frac{q''(x)}{T_s(x) - T_\infty}, where q''(x) is the local heat flux, T_s(x) is the local surface temperature, and T_\infty is the free-stream fluid temperature. This coefficient varies spatially along the surface due to the development of the thermal boundary layer in convective flows; as the boundary layer thickens in the downstream direction, the temperature gradient at the wall decreases, resulting in a corresponding reduction in h(x). For instance, in external flows over a flat plate, h(x) is highest near the leading edge where the boundary layer is thinnest and progressively diminishes along the plate length.[10][11] In contrast, the average heat transfer coefficient, \bar{h}, provides a spatially integrated measure suitable for overall performance evaluation and is computed as the surface integral of the local coefficient divided by the total surface area: \bar{h} = \frac{1}{A} \int_A h \, dA. For one-dimensional flows, such as along a plate of length L, this reduces to \bar{h} = \frac{1}{L} \int_0^L h(x) \, dx. Local coefficients are employed in detailed design scenarios, such as predicting temperature hotspots or optimizing non-uniform surface conditions, whereas average coefficients facilitate practical engineering calculations of total heat transfer rates across entire surfaces. In developing flows, like the entrance region of a duct, h(x) starts high due to thin boundary layers and decreases toward a constant fully developed value, making the average \bar{h} typically higher than the local value at the trailing edge.[12] For cases where the bulk fluid temperature varies significantly along the surface, such as in heat exchangers, the average coefficient is often determined using the logarithmic mean temperature difference (LMTD) to account for non-constant driving potentials: the total heat transfer Q = \bar{h} A \Delta T_{lm}, with \Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln \left( \frac{\Delta T_1}{\Delta T_2} \right)}, where \Delta T_1 and \Delta T_2 are the temperature differences at the inlet and outlet. This approach yields an effective \bar{h} that represents the integrated performance without requiring point-wise integration of varying local values.[13]Dimensionless Framework
Nusselt Number and Its Relation to h
The Nusselt number, denoted as \mathrm{Nu}, is defined as the dimensionless form of the heat transfer coefficient h, given by the equation \mathrm{Nu} = \frac{h L}{k}, where L is the characteristic length of the surface (such as the plate length or pipe diameter) and k is the thermal conductivity of the fluid.[14] This formulation arises in convective heat transfer problems to normalize the convective resistance against conductive resistance across the fluid boundary layer.[15] Physically, the Nusselt number represents the ratio of the total heat transfer (convective plus conductive) to the purely conductive heat transfer across a fluid layer of thickness equal to the characteristic length L.[16] A value of \mathrm{Nu} = 1 indicates heat transfer dominated by conduction alone, as in a stagnant fluid, while \mathrm{Nu} > 1 signifies enhancement due to convection, with higher values corresponding to more effective convective mixing within the boundary layer.[14] This interpretation underscores the Nusselt number's role in quantifying the relative importance of convection in boundary layer flows.[16] The Nusselt number emerges from dimensional analysis of the convection boundary layer equations, where the heat transfer coefficient h is expressed in terms of fluid properties, flow velocity, and geometry.[15] By applying the Buckingham π theorem to the energy equation and boundary conditions, the dimensionless temperature gradient at the wall yields \mathrm{Nu} as the key group linking convective heat flux to conduction, directly tied to the thermal boundary layer thickness.[15] This approach connects to Prandtl's boundary layer theory, where convection effects scale with velocity and temperature profiles near the surface. Similar to the heat transfer coefficient, the Nusselt number can be defined locally or as an average over a surface. The local Nusselt number \mathrm{Nu}_x at position x is \mathrm{Nu}_x = \frac{h_x x}{k}, capturing variations along the flow direction due to boundary layer development.[17] The average Nusselt number \overline{\mathrm{Nu}}_L over length L is then obtained by integration: \overline{\mathrm{Nu}}_L = \frac{1}{L} \int_0^L \mathrm{Nu}_x \, dx, which parallels the averaging for h. From the definition, the heat transfer coefficient is explicitly recovered as h = \frac{\mathrm{Nu} \, k}{L}, allowing empirical or theoretical correlations for \mathrm{Nu} (often as functions of Reynolds and Prandtl numbers) to directly provide values of h for engineering calculations.[14] These correlations, derived from experiments or similarity solutions, thus bridge dimensionless analysis to practical convective heat transfer rates.[15] The Nusselt number formulation assumes constant fluid thermal conductivity k and negligible viscous dissipation effects, typically valid when the Eckert number \mathrm{Ec} \ll 1, ensuring that mechanical energy dissipation does not significantly alter the temperature field.[18] These assumptions align with low-speed flows and moderate temperature differences in standard convection problems.[18]Key Dimensionless Groups (Reynolds, Prandtl, Grashof)
In the context of heat transfer, dimensionless groups provide a framework for scaling and correlating convective heat transfer coefficients by capturing the essential physics of fluid flow and thermal transport without dependence on specific units. The Reynolds, Prandtl, and Grashof numbers are pivotal in this framework, particularly for expressing the Nusselt number (Nu), which directly relates to the heat transfer coefficient h through \mathrm{Nu} = \frac{h L}{k}, where L is a characteristic length and k is the fluid thermal conductivity. These groups arise from dimensional analysis and experimental correlations, enabling predictions of h across different geometries, fluids, and flow conditions. The Reynolds number (\mathrm{Re}), introduced by Osborne Reynolds in his 1883 experimental investigation of pipe flow transitions, quantifies the ratio of inertial forces to viscous forces in a fluid flow. It is defined as \mathrm{Re} = \frac{\rho u L}{\mu} = \frac{u L}{\nu}, where \rho is fluid density, u is a characteristic velocity, L is a characteristic length, \mu is dynamic viscosity, and \nu is kinematic viscosity. Physically, low \mathrm{Re} (typically below 2300 for pipe flows) indicates laminar flow dominated by viscous effects, while high \mathrm{Re} signals turbulent flow where inertia promotes mixing and enhanced heat transfer. In forced convection correlations for the Nusselt number, \mathrm{Re} determines the flow regime and is a primary independent variable, as in the Dittus-Boelter equation for turbulent pipe flow: \mathrm{Nu} = 0.023 \mathrm{Re}^{0.8} \mathrm{Pr}^{0.4}, which scales h with flow speed and geometry. This role underscores \mathrm{Re}'s influence on boundary layer development and convective enhancement in external and internal flows.[19][20][21] The Prandtl number (\mathrm{Pr}), named after Ludwig Prandtl for his foundational 1904 boundary layer theory, measures the relative thickness of the momentum (velocity) boundary layer to the thermal boundary layer in convective flows. It is defined as \mathrm{Pr} = \frac{\nu}{\alpha} = \frac{\mu c_p}{k}, where \alpha is thermal diffusivity and c_p is specific heat capacity at constant pressure. For gases, \mathrm{Pr} \approx 0.7; for water, \mathrm{Pr} \approx 7; and for oils, \mathrm{Pr} > 100, reflecting how momentum diffuses faster than heat in low-\mathrm{Pr} fluids, leading to thicker thermal layers and potentially reduced h. In both forced and natural convection, \mathrm{Pr} appears in Nusselt correlations to account for fluid property effects on heat diffusion relative to momentum transport, such as in the aforementioned Dittus-Boelter form or natural convection relations like \mathrm{Nu} = C (\mathrm{Gr} \mathrm{Pr})^n, where it modulates the overall convective resistance. Prandtl's boundary layer concept highlights how \mathrm{Pr} influences the interplay between conduction within the layer and bulk convection, critical for accurate h predictions across fluid types.[22] The Grashof number (\mathrm{Gr}), attributed to Franz Grashof's early contributions to fluid mechanics in the late 19th century, characterizes the ratio of buoyancy-driven forces to viscous forces in natural convection, where flow arises from density differences due to temperature gradients. It is defined as \mathrm{Gr} = \frac{g \beta \Delta T L^3}{\nu^2}, where g is gravitational acceleration, \beta is the thermal expansion coefficient, and \Delta T is the temperature difference driving buoyancy. High \mathrm{Gr} (e.g., >10^9) promotes turbulent natural convection, enhancing h through stronger circulation, while low \mathrm{Gr} yields laminar regimes. In correlations, \mathrm{Gr} replaces \mathrm{Re} for buoyancy-dominated flows, often combined with \mathrm{Pr} into the Rayleigh number (\mathrm{Ra} = \mathrm{Gr} \mathrm{Pr}) for Nusselt expressions, such as \mathrm{Nu} = 0.59 \mathrm{Ra}^{1/4} for laminar vertical plates, directly linking h to geometry, temperature difference, and fluid properties. This group's role is essential for applications like electronics cooling or solar collectors, where forced flow is absent.[23]Natural Convection Correlations
Vertical Flat Plates and Planes
In natural convection heat transfer from vertical flat plates and planes, buoyancy-driven flow arises from density variations caused by temperature differences between the surface and the surrounding fluid, leading to the formation of a boundary layer along the plate. The velocity boundary layer, where fluid velocity adjusts from zero at the wall to the free-stream value, develops concurrently with the thermal boundary layer, where temperature gradients occur. For low Prandtl number fluids (Pr ≪ 1), such as liquid metals, the thermal boundary layer is thicker than the velocity boundary layer due to higher thermal diffusivity relative to momentum diffusivity, resulting in broader temperature profiles and reduced local heat transfer rates compared to high-Pr fluids like oils.[24] Empirical correlations for the average heat transfer coefficient h on vertical flat plates express it through the average Nusselt number \overline{\mathrm{Nu}}_L = \frac{h L}{k}, where L is the plate height and k is the fluid thermal conductivity. The Rayleigh number \mathrm{Ra}_L = \mathrm{Gr}_L \mathrm{Pr} governs the flow regime, incorporating the Grashof number \mathrm{Gr}_L (ratio of buoyancy to viscous forces) and Prandtl number \mathrm{Pr} (ratio of momentum to thermal diffusivity). For laminar flow over $10^4 < \mathrm{Ra}_L < 10^9, the correlation for isothermal surfaces is \overline{\mathrm{Nu}}_L = 0.59 \mathrm{Ra}_L^{1/4}, while turbulent transition occurs around \mathrm{Ra}_L \approx 10^9, yielding \overline{\mathrm{Nu}}_L = 0.10 \mathrm{Ra}_L^{1/3} for higher Rayleigh numbers. These relations follow the general form h = \frac{k}{L} C \mathrm{Ra}_L^n, with constants C and exponent n varying by regime (C = 0.59, n = 1/4 for laminar; C = 0.10, n = 1/3 for turbulent).[25] For uniform heat flux boundary conditions, where the surface heat flux q'' is constant rather than the temperature, the correlations adopt similar forms but require adjustment, often using an effective temperature difference derived iteratively from q'' = h (T_s - T_\infty); dedicated expressions, such as those approximating \overline{\mathrm{Nu}}_L \approx 0.60 \mathrm{Ra}_L^{1/4} for laminar cases, account for the altered temperature profile and yield heat transfer coefficients about 5-15% higher than isothermal predictions. These distinctions arise because uniform flux maintains a monotonically increasing surface temperature along the plate, thickening the boundary layer differently than the fixed-temperature case.[25][26] The correlations stem from boundary layer analyses and experiments with gases like air and liquids such as water and oils, validated across Prandtl numbers from 0.7 to 10,000 and plate heights up to several meters; they provide engineering accuracy of ±10-20% within the specified Rayleigh ranges, with the Churchill-Chu framework encompassing both laminar and turbulent regimes for broad applicability.[25]Horizontal Cylinders and Plates
In natural convection from horizontal surfaces, the orientation significantly influences flow stability and heat transfer rates. For upward-facing hot plates or downward-facing cold plates, the buoyancy forces create an unstable configuration where lighter fluid rises, promoting the development of thermal plumes that enhance mixing and heat transfer compared to stable orientations.[27] For horizontal plates, the average Nusselt number for the upper surface of a hot plate (or lower surface of a cold plate) is correlated as Nu_L = 0.54 Ra_L^{1/4} for 10^4 < Ra_L < 10^7, where Ra_L is the Rayleigh number based on the characteristic length L, typically taken as the plate's side length for square plates or the average of length and width for rectangular ones.[28] This relation, recommended by McAdams, yields the average heat transfer coefficient h via h = (Nu_L k)/L, with k as the fluid thermal conductivity, and applies to fluids like air where Prandtl numbers are around 0.7.[29] For broader applicability, Churchill extended similar forms to account for edge effects and varying plate sizes, emphasizing L as the hydraulic diameter or area-to-perimeter ratio for non-rectangular shapes to capture diameter or length influences on boundary layer development.[30] Horizontal cylinders exhibit analogous behavior, with the plume forming along the top surface. The widely adopted correlation by Churchill and Chu for the average Nusselt number over the cylinder surface, valid for a broad range of Rayleigh numbers (10^{-2} < Ra_D < 10^{12}) and Prandtl numbers (0.5 < Pr < ∞), is \overline{\mathrm{Nu}}_D = \left\{ 0.60 + \frac{0.387 \mathrm{Ra}_D^{1/6}}{\left[1 + \left( \frac{0.559}{\mathrm{Pr}} \right)^{9/16} \right]^{8/27}} \right\}^2, where the subscript D denotes the cylinder diameter as the characteristic length.[30] This expression provides the average h as h = (\overline{\mathrm{Nu}}_D k)/D and outperforms earlier laminar-only models by incorporating turbulent transitions. McAdams' earlier recommendations for cylinders align closely for moderate Ra but are limited to narrower ranges.[31] These correlations are essential in engineering applications such as horizontal pipes conveying fluids in ambient air, where natural convection governs cooling, and extended surface heat exchangers like finned tubes, optimizing designs for passive thermal management without forced flow.[32] The Rayleigh number determines the regime applicability, with lower values indicating laminar plume-dominated flow.[28]Spheres and Vertical Enclosures
In natural convection around spheres, the heat transfer coefficient h is determined using Nusselt number correlations based on the Rayleigh number \mathrm{Ra}_D, where the characteristic length is the sphere diameter D. For laminar flow at low Rayleigh numbers, a widely used correlation is \overline{\mathrm{Nu}}_D = 2 + 0.43 \mathrm{Ra}_D^{1/4}, valid for \mathrm{Ra}_D \lesssim 10^9 and applicable to gases like air. This form approaches the conduction limit of \mathrm{Nu}_D = 2 as \mathrm{Ra}_D \to 0, reflecting pure diffusive heat transfer across the stagnant fluid layer surrounding the sphere. At higher Rayleigh numbers leading to turbulent flow (\mathrm{Ra}_D > 10^9), the correlation extends to forms like \overline{\mathrm{Nu}}_D = 2 + 0.50 \mathrm{Ra}_D^{1/4}, accounting for enhanced mixing and boundary layer instability, though more comprehensive expressions incorporate Prandtl number dependence for broader fluids. The heat transfer coefficient is then obtained as h = \overline{\mathrm{Nu}}_D k / D, where k is the fluid thermal conductivity. For vertical enclosures, such as those formed by parallel plates with heated and cooled vertical walls separated by a small gap, natural convection is characterized by counterflowing boundary layers along the walls, leading to an effective thermal conductivity k_\mathrm{eff} that quantifies the enhanced heat transfer across the enclosure. The characteristic length is the plate spacing L, and for aspect ratios H/L \approx 1 (where H is the enclosure height), a key correlation is k_\mathrm{eff}/k = 0.22 \left[ \Pr \mathrm{Ra}_L / (0.2 + \Pr) \right]^{0.28}, applicable for $10^4 < \mathrm{Ra}_L < 10^7 and moderate Prandtl numbers typical of air or water. This Nusselt number-based form, where \mathrm{Nu}_L = h L / k = (k_\mathrm{eff}/k) \cdot (H/L), assumes H \approx L for square-like cavities, yielding h = (k_\mathrm{eff}/k) \cdot k / H. Confinement in narrow gaps reduces h compared to isolated plates due to boundary layer interference, suppressing circulation and limiting the convective enhancement to near-conductive levels at very small L. These correlations originate from experimental studies on isothermal boundaries and are validated for applications like building insulation panels and electronic component cooling, where buoyancy-driven flows dominate without external forcing.Forced Convection Correlations
External Flows over Bodies
In external flows over bodies, forced convection heat transfer coefficients are determined using the Nusselt number (Nu), which relates the convective heat transfer to conduction across the fluid boundary layer, with the coefficient given by h = \frac{Nu \cdot k}{L}, where k is the fluid thermal conductivity and L is a characteristic length (e.g., plate length or body diameter). The Reynolds number (Re) is based on the free-stream velocity U_\infty and characterizes the flow regime, influencing boundary layer development and thus h. Laminar flows occur at low Re (e.g., Re < 5 \times 10^5 for flat plates), where viscous forces dominate and the boundary layer remains smooth, leading to lower h; transitional regimes (around 5 \times 10^5 < Re < 10^6) exhibit intermittent turbulence, increasing h; and turbulent flows at higher Re promote mixing, significantly enhancing h through chaotic eddies. Prandtl number (Pr) effects are captured in correlations, typically showing h increasing with Pr^{1/3} for Pr > 0.6. For a flat plate in laminar forced convection parallel to the surface, the local Nusselt number at distance x from the leading edge is Nu_x = 0.332 Re_x^{1/2} Pr^{1/3} for constant wall temperature and Pr ≥ 0.6, derived from the similarity solution to the boundary layer energy equation. The average Nusselt number over length L is then Nu_L = 0.664 Re_L^{1/2} Pr^{1/3} for Re_L < 5 \times 10^5, doubling the local value at the trailing edge due to boundary layer growth. These apply to isothermal or uniform heat flux conditions in gases and liquids, providing h values that scale with U_\infty^{1/2}, emphasizing the role of developing thermal boundary layers in external flows. Cross-flow over cylinders, common in heat exchanger design, uses the Churchill-Bernstein correlation for the average Nusselt number:Nu_D = 0.3 + \frac{0.62 Re_D^{1/2} Pr^{1/3}}{\left[1 + (0.4/Pr)^{2/3}\right]^{1/4}} \left[1 + \left(\frac{Re_D}{282000}\right)^{5/8}\right]^{4/5}
for Re_D Pr > 0.2, valid across laminar, transitional, and turbulent regimes up to Re_D ≈ 10^7 and 0.5 < Pr < 10^2, based on diameter D. This empirical fit unifies prior data, showing h rising sharply with Re_D due to boundary layer separation and wake formation, with turbulent effects dominating at Re_D > 10^3. For spheres in uniform cross-flow, the Whitaker correlation gives the average Nusselt number as
Nu_D = 2 + \left[ 0.4 Re_D^{1/2} + 0.06 Re_D^{2/3} \right] [Pr](/page/PR)^{0.4} \left( \frac{\mu_\infty}{\mu_s} \right)^{1/4}
for $3.5 \leq Re_D \leq 7.6 \times 10^4, $0.71 \leq [Pr](/page/PR) \leq 380, and $1 \leq \frac{\mu_\infty}{\mu_s} \leq 3.2, where \mu_\infty and \mu_s are the fluid viscosities at free-stream and surface temperatures, respectively (the ratio is 1 for constant properties).[33] This extends the stagnant fluid limit (Nu = 2) with forced convection enhancement. Here, h = Nu_D k / D reflects symmetric boundary layer growth until separation at the rear, with lower h than cylinders due to reduced drag and wake intensity. In cylinder cross-flows, unsteadiness from vortex shedding (prominent at 40 < Re_D < 10^5) periodically disrupts the wake, enhancing local h on the rear surface by up to 20-30% through improved mixing, though average correlations like Churchill-Bernstein account for this statistically.