Heat transfer coefficient


In thermodynamics, the heat transfer coefficient or film coefficient, or film effectiveness, is the proportionality constant between the heat flux and the thermodynamic driving force for the flow of heat. It is used to calculate heat transfer between components of a system; such as by convection between a fluid and a solid. The heat transfer coefficient has SI units in watts per square meter per kelvin.
The total heat transfer rate for combined modes and system components is usually expressed in terms of an [|overall heat transfer coefficient], thermal transmittance or U-value. The heat transfer coefficient is the reciprocal of thermal insulance. This is used for building materials and for clothing insulation.
There are numerous methods for calculating the heat transfer coefficient in different heat transfer modes, different fluids, flow regimes, and under different thermohydraulic conditions. Often it can be estimated by dividing the thermal conductivity of the convection fluid by a length scale. The heat transfer coefficient is often calculated from the Nusselt number. There are also online calculators available specifically for Heat-transfer fluid applications. Experimental assessment of the heat transfer coefficient poses some challenges especially when small fluxes are to be measured.

Definition

The general definition of the heat transfer coefficient is:
where:
The heat transfer coefficient replaces the thermal conductivity within a generalization of Fourier's law postulated to also describe convection flows. Upon reaching a steady state of flow, the heat transfer rate is:
where :
In much practical application, a heat transfer coefficient has a relatively constant value over its specified temperature range of usefulness.

Composition

A simple method for determining an overall heat transfer coefficient that is useful to find the heat transfer through a sequence of simple elements such as walls in buildings or across heat exchangers is shown below. This method most readily accounts for conduction and convection. Effects of radiation can be similarly estimated, but introduce non-linear temperature dependence. The method is as follows:
Where:
As the areas for each surface approach being equal the equation can be written as the transfer coefficient per unit area as shown below:
or
Often the value for is referred to as the difference of two radii where the inner and outer radii are used to define the thickness of a pipe carrying a fluid, however, this figure may also be considered as a wall thickness in a flat plate transfer mechanism or other common flat surfaces such as a wall in a building when the area difference between each edge of the transmission surface approaches zero.
In the walls of buildings the above formula can be used to derive the formula commonly used to calculate the heat through building components. Architects and engineers call the resulting values either the U-Value or the R-Value of a construction assembly like a wall. Each type of value are related as the inverse of each other such that R-Value = 1/U-Value and both are more fully understood through the concept of an overall heat transfer coefficient described in lower section of this document.

Convective heat transfer correlations

Although convective heat transfer can be derived analytically through dimensional analysis, exact analysis of the boundary layer, approximate integral analysis of the boundary layer and analogies between energy and momentum transfer, these analytic approaches may not offer practical solutions to all problems when there are no mathematical models applicable. Therefore, many correlations were developed by various authors to estimate the convective heat transfer coefficient in various cases including natural convection, forced convection for internal flow and forced convection for external flow. These empirical correlations are presented for their particular geometry and flow conditions. As the fluid properties are temperature dependent, they are evaluated at the film temperature, which is the average of the surface and the surrounding bulk temperature,.

External flow, vertical plane

Recommendations by Churchill and Chu provide the following correlation for natural convection adjacent to a vertical plane, both for laminar and turbulent flow. k is the thermal conductivity of the fluid, L is the characteristic length with respect to the direction of gravity, RaL is the Rayleigh number with respect to this length and Pr is the Prandtl number.
For laminar flows, the following correlation is slightly more accurate. It is observed that a transition from a laminar to a turbulent boundary occurs when RaL exceeds around 109.

External flow, vertical cylinders

For cylinders with their axes vertical, the expressions for plane surfaces can be used provided the curvature effect is not too significant. This represents the limit where boundary layer thickness is small relative to cylinder diameter. For fluids with Pr ≤ 0.72, the correlations for vertical plane walls can be used when

where is the Grashof number.
And in fluids of Pr ≤ 6 when
Under these circumstances, the error is limited to up to 5.5%.

External flow, horizontal plates

W. H. McAdams suggested the following correlations for horizontal plates. The induced buoyancy will be different depending upon whether the hot surface is facing up or down.
For a hot surface facing up, or a cold surface facing down, for laminar flow:

and for turbulent flow:
For a hot surface facing down, or a cold surface facing up, for laminar flow:
The characteristic length is the ratio of the plate surface area to perimeter. If the surface is inclined at an angle θ with the vertical then the equations for a vertical plate by Churchill and Chu may be used for θ up to 60°; if the boundary layer flow is laminar, the gravitational constant g is replaced with g cos θ when calculating the Ra term.

External flow, horizontal cylinder

For cylinders of sufficient length and negligible end effects, Churchill and Chu has the following correlation for.

External flow, spheres

For spheres, T. Yuge has the following correlation for Pr≃1 and.

Vertical rectangular enclosure

For heat flow between two opposing vertical plates of rectangular enclosures, Catton recommends the following two correlations for smaller aspect ratios. The correlations are valid for any value of Prandtl number.
For :
where H is the internal height of the enclosure and L is the horizontal distance between the two sides of different temperatures.
For :
For vertical enclosures with larger aspect ratios, the following two correlations can be used. For 10 < H/''L'' < 40:
For :
For all four correlations, fluid properties are evaluated at the average temperature—as opposed to film temperature—, where and are the temperatures of the vertical surfaces and.

Forced convection

See main article Nusselt number and Churchill–Bernstein equation for forced convection over a horizontal cylinder.

Internal flow, laminar flow

Sieder and Tate give the following correlation to account for entrance effects in laminar flow in tubes where is the internal diameter, is the fluid viscosity at the bulk mean temperature, is the viscosity at the tube wall surface temperature.
For fully developed laminar flow, the Nusselt number is constant and equal to 3.66. Mills combines the entrance effects and fully developed flow into one equation

Internal flow, turbulent flow

The Dittus-Bölter correlation is a common and particularly simple correlation useful for many applications. This correlation is applicable when forced convection is the only mode of heat transfer; i.e., there is no boiling, condensation, significant radiation, etc. The accuracy of this correlation is anticipated to be ±15%.
For a fluid flowing in a straight circular pipe with a Reynolds number between 10,000 and 120,000, when the fluid's Prandtl number is between 0.7 and 120, for a location far from the pipe entrance or other flow disturbances, and when the pipe surface is hydraulically smooth, the heat transfer coefficient between the bulk of the fluid and the pipe surface can be expressed explicitly as:
where:
The fluid properties necessary for the application of this equation are evaluated at the bulk temperature thus avoiding iteration.

Forced convection, external flow

In analyzing the heat transfer associated with the flow past the exterior surface of a solid, the situation is complicated by phenomena such as boundary layer separation. Various authors have correlated charts and graphs for different geometries and flow conditions.
For flow parallel to a plane surface, where is the distance from the edge and is the height of the boundary layer, a mean Nusselt number can be calculated using the Colburn analogy.

Thom correlation

There exist simple fluid-specific correlations for heat transfer coefficient in boiling. The Thom correlation is for the flow of boiling water under conditions where the nucleate boiling contribution predominates over forced convection. This correlation is useful for rough estimation of expected temperature difference given the heat flux:
where:
This empirical correlation is specific to the units given.

Heat transfer coefficient of pipe wall

The resistance to the flow of heat by the material of pipe wall can be expressed as a "heat transfer coefficient of the pipe wall". However, one needs to select if the heat flux is based on the pipe inner or the outer diameter.
If the heat flux is based on the inner diameter of the pipe, and if the pipe wall is thin compared to this diameter, the curvature of the wall has a negligible effect on heat transfer. In this case, the pipe wall can be approximated as a flat plane, which simplifies calculations. This assumption allows the heat transfer coefficient for the pipe wall to be calculated as:
where
However, when the wall thickness is significant enough that curvature cannot be ignored, the heat transfer coefficient needs to account for the cylindrical shape. Under this condition, the heat transfer coefficient can be more accurately calculated using :
where
The thermal conductivity of the tube material usually depends on temperature; the mean thermal conductivity is often used.