Heat transfer coefficient

# Heat transfer coefficient

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Encyclopedia
The heat transfer coefficient, in thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

and in mechanical
Mechanical engineering
Mechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the production and usage of heat and mechanical power for the...

and chemical engineering
Chemical engineering
Chemical engineering is the branch of engineering that deals with physical science , and life sciences with mathematics and economics, to the process of converting raw materials or chemicals into more useful or valuable forms...

, is used in calculating the heat transfer
Heat transfer
Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...

, typically by convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

or phase change between a fluid and a solid: $h = \frac\left\{q\right\} \left\{A \cdot \Delta T \right\}$ whereNEWLINE
NEWLINE
q = heat flow in input or lost heat flow , J/s = W
NEWLINE
h = heat transfer coefficient, W/(m2K)
NEWLINE
A = heat transfer surface area, m2$\Delta T$ = difference in temperature between the solid surface and surrounding fluid area, K
NEWLINE From the above equation, the heat transfer coefficient is the proportional coefficient between the heat flux
Heat flux
Heat flux or thermal flux is the rate of heat energy transfer through a given surface. The SI derived unit of heat rate is joule per second, or watt. Heat flux is the heat rate per unit area. In SI units, heat flux is measured in W/m2]. Heat rate is a scalar quantity, while heat flux is a vectorial...

that is aheat flow per unit area, q/lund, and the thermodynamic driving force for the flow of heat (i.e., the temperature difference, ΔT). The heat transfer coefficient has SI
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

units in watts per meter squared-kelvin: W/(m2K). There are numerous methods for calculating the heat transfer coefficient in different heat transfer modes, different fluids, flow regimes, and under different thermohydraulic
Thermal hydraulics
Thermal hydraulics is the study of hydraulic flow in thermal systems. A common example is steam generation in power plants and the associated energy transfer to mechanical motion and the change of states of the water while undergoing this process.The common adjectives are "thermohydraulic",...

conditions. Often it can be estimated by dividing the thermal conductivity
Thermal conductivity
In physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....

of the convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

fluid by a length scale. The heat transfer coefficient is often calculated from the Nusselt number
Nusselt number
In heat transfer at a boundary within a fluid, the Nusselt number is the ratio of convective to conductive heat transfer across the boundary. Named after Wilhelm Nusselt, it is a dimensionless number...

(a dimensionless number). There are also online calculators available specifically for heat transfer fluid applications.

## Derivation of Convective heat transfer coefficient

An understanding of convection boundary layers is necessary to understanding convective heat transfer between a surface and a fluid flowing past it. A thermal boundary layer develops if the fluid free stream temperature and the surface temperatures differ. A temperature profile exists due to the energy exchange resulting from this temperature difference. The heat transfer rate can then be written as, $\left\{\left\{q\right\}_\left\{y\right\}\right\}=hA\left\left( \left\{\left\{T\right\}_\left\{s\right\}\right\}-\left\{\left\{T\right\}_\left\{\infty \right\}\right\} \right\right)$ And because heat transfer at the surface is by conduction, $\left\{\left\{q\right\}_\left\{y\right\}\right\}=-kA\frac\left\{\partial \right\}\left\{\partial y\right\}\left\{\left\{\left. \left\left( T-\left\{\left\{T\right\}_\left\{s\right\}\right\} \right\right) \right|\right\}_\left\{y=0\right\}\right\}$ These two terms are equal; thus $-kA\frac\left\{\partial \right\}\left\{\partial y\right\}\left\{\left\{\left. \left\left( T-\left\{\left\{T\right\}_\left\{s\right\}\right\} \right\right) \right|\right\}_\left\{y=0\right\}\right\}=hA\left\left( \left\{\left\{T\right\}_\left\{s\right\}\right\}-\left\{\left\{T\right\}_\left\{\infty \right\}\right\} \right\right)$ Rearranging, $\frac\left\{h\right\}\left\{k\right\}=\frac The$heat transfer coefficient, in thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

and in mechanical
Mechanical engineering
Mechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the production and usage of heat and mechanical power for the...

and chemical engineering
Chemical engineering
Chemical engineering is the branch of engineering that deals with physical science , and life sciences with mathematics and economics, to the process of converting raw materials or chemicals into more useful or valuable forms...

, is used in calculating the heat transfer
Heat transfer
Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...

, typically by convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

or phase change between a fluid and a solid: $h = \frac\left\{q\right\} \left\{A \cdot \Delta T \right\}$ whereNEWLINE
NEWLINE
q = heat flow in input or lost heat flow , J/s = W
NEWLINE
h = heat transfer coefficient, W/(m2K)
NEWLINE
A = heat transfer surface area, m2$\Delta T$ = difference in temperature between the solid surface and surrounding fluid area, K
NEWLINE From the above equation, the heat transfer coefficient is the proportional coefficient between the heat flux
Heat flux
Heat flux or thermal flux is the rate of heat energy transfer through a given surface. The SI derived unit of heat rate is joule per second, or watt. Heat flux is the heat rate per unit area. In SI units, heat flux is measured in W/m2]. Heat rate is a scalar quantity, while heat flux is a vectorial...

that is aheat flow per unit area, q/lund, and the thermodynamic driving force for the flow of heat (i.e., the temperature difference, ΔT). The heat transfer coefficient has SI
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

units in watts per meter squared-kelvin: W/(m2K). There are numerous methods for calculating the heat transfer coefficient in different heat transfer modes, different fluids, flow regimes, and under different thermohydraulic
Thermal hydraulics
Thermal hydraulics is the study of hydraulic flow in thermal systems. A common example is steam generation in power plants and the associated energy transfer to mechanical motion and the change of states of the water while undergoing this process.The common adjectives are "thermohydraulic",...

conditions. Often it can be estimated by dividing the thermal conductivity
Thermal conductivity
In physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....

of the convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

fluid by a length scale. The heat transfer coefficient is often calculated from the Nusselt number
Nusselt number
In heat transfer at a boundary within a fluid, the Nusselt number is the ratio of convective to conductive heat transfer across the boundary. Named after Wilhelm Nusselt, it is a dimensionless number...

(a dimensionless number). There are also online calculators available specifically for heat transfer fluid applications.

## Derivation of Convective heat transfer coefficient

An understanding of convection boundary layers is necessary to understanding convective heat transfer between a surface and a fluid flowing past it. A thermal boundary layer develops if the fluid free stream temperature and the surface temperatures differ. A temperature profile exists due to the energy exchange resulting from this temperature difference. The heat transfer rate can then be written as, $\left\{\left\{q\right\}_\left\{y\right\}\right\}=hA\left\left( \left\{\left\{T\right\}_\left\{s\right\}\right\}-\left\{\left\{T\right\}_\left\{\infty \right\}\right\} \right\right)$ And because heat transfer at the surface is by conduction, $\left\{\left\{q\right\}_\left\{y\right\}\right\}=-kA\frac\left\{\partial \right\}\left\{\partial y\right\}\left\{\left\{\left. \left\left( T-\left\{\left\{T\right\}_\left\{s\right\}\right\} \right\right) \right|\right\}_\left\{y=0\right\}\right\}$ These two terms are equal; thus $-kA\frac\left\{\partial \right\}\left\{\partial y\right\}\left\{\left\{\left. \left\left( T-\left\{\left\{T\right\}_\left\{s\right\}\right\} \right\right) \right|\right\}_\left\{y=0\right\}\right\}=hA\left\left( \left\{\left\{T\right\}_\left\{s\right\}\right\}-\left\{\left\{T\right\}_\left\{\infty \right\}\right\} \right\right)$ Rearranging, $\frac\left\{h\right\}\left\{k\right\}=\frac The$heat transfer coefficient, in thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

and in mechanical
Mechanical engineering
Mechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the production and usage of heat and mechanical power for the...

and chemical engineering
Chemical engineering
Chemical engineering is the branch of engineering that deals with physical science , and life sciences with mathematics and economics, to the process of converting raw materials or chemicals into more useful or valuable forms...

, is used in calculating the heat transfer
Heat transfer
Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...

, typically by convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

or phase change between a fluid and a solid: $h = \frac\left\{q\right\} \left\{A \cdot \Delta T \right\}$ whereNEWLINE
NEWLINE
q = heat flow in input or lost heat flow , J/s = W
NEWLINE
h = heat transfer coefficient, W/(m2K)
NEWLINE
A = heat transfer surface area, m2$\Delta T$ = difference in temperature between the solid surface and surrounding fluid area, K
NEWLINE From the above equation, the heat transfer coefficient is the proportional coefficient between the heat flux
Heat flux
Heat flux or thermal flux is the rate of heat energy transfer through a given surface. The SI derived unit of heat rate is joule per second, or watt. Heat flux is the heat rate per unit area. In SI units, heat flux is measured in W/m2]. Heat rate is a scalar quantity, while heat flux is a vectorial...

that is aheat flow per unit area, q/lund, and the thermodynamic driving force for the flow of heat (i.e., the temperature difference, ΔT). The heat transfer coefficient has SI
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

units in watts per meter squared-kelvin: W/(m2K). There are numerous methods for calculating the heat transfer coefficient in different heat transfer modes, different fluids, flow regimes, and under different thermohydraulic
Thermal hydraulics
Thermal hydraulics is the study of hydraulic flow in thermal systems. A common example is steam generation in power plants and the associated energy transfer to mechanical motion and the change of states of the water while undergoing this process.The common adjectives are "thermohydraulic",...

conditions. Often it can be estimated by dividing the thermal conductivity
Thermal conductivity
In physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....

of the convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

fluid by a length scale. The heat transfer coefficient is often calculated from the Nusselt number
Nusselt number
In heat transfer at a boundary within a fluid, the Nusselt number is the ratio of convective to conductive heat transfer across the boundary. Named after Wilhelm Nusselt, it is a dimensionless number...

(a dimensionless number). There are also online calculators available specifically for heat transfer fluid applications.

## Derivation of Convective heat transfer coefficient

An understanding of convection boundary layers is necessary to understanding convective heat transfer between a surface and a fluid flowing past it. A thermal boundary layer develops if the fluid free stream temperature and the surface temperatures differ. A temperature profile exists due to the energy exchange resulting from this temperature difference. The heat transfer rate can then be written as, $\left\{\left\{q\right\}_\left\{y\right\}\right\}=hA\left\left( \left\{\left\{T\right\}_\left\{s\right\}\right\}-\left\{\left\{T\right\}_\left\{\infty \right\}\right\} \right\right)$ And because heat transfer at the surface is by conduction, $\left\{\left\{q\right\}_\left\{y\right\}\right\}=-kA\frac\left\{\partial \right\}\left\{\partial y\right\}\left\{\left\{\left. \left\left( T-\left\{\left\{T\right\}_\left\{s\right\}\right\} \right\right) \right|\right\}_\left\{y=0\right\}\right\}$ These two terms are equal; thus $-kA\frac\left\{\partial \right\}\left\{\partial y\right\}\left\{\left\{\left. \left\left( T-\left\{\left\{T\right\}_\left\{s\right\}\right\} \right\right) \right|\right\}_\left\{y=0\right\}\right\}=hA\left\left( \left\{\left\{T\right\}_\left\{s\right\}\right\}-\left\{\left\{T\right\}_\left\{\infty \right\}\right\} \right\right)$ Rearranging, $\frac\left\{h\right\}\left\{k\right\}=\frac\left\{\left\{\left\{\left. \frac\left\{\partial \left\left( \left\{\left\{T\right\}_\left\{s\right\}\right\}-T \right\right)\right\}\left\{\partial y\right\} \right|\right\}_\left\{y=0\right\}\right\}\right\}\left\{\left\{\left\left( \left\{\left\{T\right\}_\left\{s\right\}\right\}-\left\{\left\{T\right\}_\left\{\infty \right\}\right\} \right\right)\right\}\right\}$ Making it dimensionless by multiplying by representative length L, $\frac\left\{hL\right\}\left\{k\right\}=\frac\left\{\left\{\left\{\left. \frac\left\{\partial \left\left( \left\{\left\{T\right\}_\left\{s\right\}\right\}-T \right\right)\right\}\left\{\partial y\right\} \right|\right\}_\left\{y=0\right\}\right\}\right\}\left\{\frac\left\{\left\left( \left\{\left\{T\right\}_\left\{s\right\}\right\}-\left\{\left\{T\right\}_\left\{\infty \right\}\right\} \right\right)\right\}\left\{L\right\}\right\}$ The right hand side is now the ratio of the temperature gradient at the surface to the reference temperature gradient. While the left hand side is similar to the Biot modulus.This becomes the ratio of conductive thermal resistance to the convective thermal resistance of the fluid, otherwise known as the Nusselt number, Nu. $\mathrm\left\{Nu\right\} = \frac\left\{h L\right\}\left\{k\right\}$

## 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. As such, 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
Film Temperature
In heat transfer and fluid dynamics, the film temperature is an approximation to the temperature of a fluid inside a convection boundary layer...

$T_f$, which is the average of the surface $T_s$ and the surrounding bulk temperature, $\left\{\left\{T\right\}_\left\{\infty \right\}\right\}$. $\left\{\left\{T\right\}_\left\{f\right\}\right\}=\frac\left\{\left\{\left\{T\right\}_\left\{s\right\}\right\}+\left\{\left\{T\right\}_\left\{\infty \right\}\right\}\right\}\left\{2\right\}$

#### External flow, Vertical plane

Churchill and Chu correlation for natural convection adjacent to vertical planes. NuL applies to all fluids for both laminar and turbulent flows. L is the characteristic length with respect to the direction of gravity. $\left\{\mathrm\left\{h\right\}\right\} \ = \frac\left\{k\right\} \left\{L\right\}\left[\left\{0.825 + \frac\left\{0.387 \mathrm\left\{Ra\right\}_L^\left\{1/6\right\}\right\}\left\{\left\left[1 + \left(0.492/\mathrm\left\{Pr\right\}\right)^\left\{9/16\right\} \, \right\right]^\left\{8/27\right\} \,\right\}\right\}\right]^2$ For laminar flows in the range of $Ra_L<10^9$, the following equation can be further improved. $\left\{\mathrm\left\{h\right\}\right\} \ = \frac\left\{k\right\} \left\{L\right\} \left[0.68 + \frac\left\{0.67 \mathrm\left\{Ra\right\}_L^\left\{1/4\right\}\right\}\left\{\left\left[1 + \left(0.492/\mathrm\left\{Pr\right\}\right)^\left\{9/16\right\} \, \right\right]^\left\{4/9\right\}\right\}\right] \, \quad \mathrm\left\{Ra\right\}_L \le 10^9$

#### 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 D. The correlations for vertical plane walls can be used when $\frac\left\{D\right\}\left\{L\right\}\ge \frac\left\{35\right\}\left\{Gr_\left\{L\right\}^\left\{\frac\left\{1\right\}\left\{4\right\}\right\}\right\}$

#### External flow, Horizontal plates

W.H. McAdams suggested the following correlations. 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, $\left\{\mathrm\left\{h\right\}\right\} \ = \frac\left\{k 0.54 \mathrm\left\{Ra\right\}_L^\left\{1/4\right\}\right\} \left\{L\right\} \, \quad 10^5 \le \mathrm\left\{Ra\right\}_L \le 2*10^7$ $\left\{\mathrm\left\{h\right\}\right\} \ = \frac\left\{k 0.14 \mathrm\left\{Ra\right\}_L^\left\{1/3\right\}\right\} \left\{L\right\} \, \quad 2*10^7 \le \mathrm\left\{Ra\right\}_L \le 3*10^\left\{10\right\}$ For a hot surface facing down or a cold surface facing up, $\left\{\mathrm\left\{h\right\}\right\} \ = \frac\left\{k 0.27 \mathrm\left\{Ra\right\}_L^\left\{1/3\right\}\right\} \left\{L\right\} \, \quad 3*10^5 \le \mathrm\left\{Ra\right\}_L \le 10^\left\{10\right\}$ The length is the ratio of the plate surface area to perimeter. If the plane surface is inclined at an angle θ, the equations for vertical plane by Churchill and Chu may be used for θ up to $60^o$. When boundary layer flow is laminar, the gravitational constant g is replaced with g cosθ for calculating the Ra in the equation for laminar flow

#### 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 $1 \le \mathrm\left\{Ra\right\}_D \le 10^5$ $\left\{\mathrm\left\{Nu\right\}\right\}_D \ = 2+ 0.43 \mathrm\left\{Re\right\}_D^\left\{1/4\right\}$

#### Internal flow, Laminar flow

Sieder and Tate has the following correlation for laminar flow in tubes where D is the internal diameter, μ_b is the fluid viscosity at the bulk mean temperature, μ_w is the viscosity at the tube wall surface temperature. $N\left\{\left\{u\right\}_\left\{D\right\}\right\}=\frac\left\{1.86\right\}\left\{\left\{\left\{\left\left( RePr \right\right)\right\}^\left\{\left\{\right\}^\left\{2\right\}\!\!\diagup\!\!\left\{\right\}_\left\{3\right\}\;\right\}\right\}\right\}\left\{\left\{\left\left( \frac\left\{D\right\}\left\{L\right\} \right\right)\right\}^\left\{\left\{\right\}^\left\{1\right\}\!\!\diagup\!\!\left\{\right\}_\left\{3\right\}\;\right\}\right\}\left\{\left\{\left\left( \frac\left\{\left\{\left\{\mu \right\}_\left\{b\right\}\right\}\right\}\left\{\left\{\left\{\mu \right\}_\left\{w\right\}\right\}\right\} \right\right)\right\}^\left\{0.14\right\}\right\}$

#### Internal flow, Turbulent flow

The Dittus–Boelter correlation (1930) 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 (in the turbulent pipe flow range), when the fluid's Prandtl number is between 0.7 and 120, for a location far from the pipe entrance (more than 10 pipe diameters; more than 50 diameters according to many authors) 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 as: $h=\left\{\left\{k_w\right\}\over\left\{D_H\right\}\right\}Nu$ where$k_w$ - thermal conductivity
Thermal conductivity
In physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....

of the bulk fluid$D_H$ - $D_i$ - Hydraulic diameterNEWLINE
NEWLINE
Nu - Nusselt number
Nusselt number
In heat transfer at a boundary within a fluid, the Nusselt number is the ratio of convective to conductive heat transfer across the boundary. Named after Wilhelm Nusselt, it is a dimensionless number...

$Nu = \left\{0.023\right\} \cdot Re^\left\{0.8\right\} \cdot Pr^\left\{n\right\}$   (Dittus-Boelter correlation)
NEWLINE
Pr - Prandtl number
NEWLINE
Re - Reynolds number
NEWLINE
n = 0.4 for heating (wall hotter than the bulk fluid) and 0.33 for cooling (wall cooler than the bulk fluid).
NEWLINE The fluid properties necessary for the application of this equation are evaluated at the bulk temperature
Bulk temperature
In fluid dynamics, the bulk temperature, or the average fluid bulk temperature, is a convenient reference point for evaluating properties related to convective heat transfer, particularly in applications related to flow in pipes and ducts....

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 x is the distance from the edge and L is the height of the boundary layer, a mean Nusselt number can be calculated using the Colburn analogy. $\mathrm\left\{Nu\right\}_L = 0.036 \mathrm\left\{Re\right\}_L^\left\{4/5\right\} \mathrm\left\{Pr\right\}^\left\{1/3\right\}$

## Thom correlation

There exist simple fluid-specific correlations for heat transfer coefficient in boiling. The Thom correlation is for flow boiling of water (subcooled or saturated at pressures up to about 20 MPa) 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: $\Delta T_\left\{sat\right\} = 22.5 \cdot \left\{q\right\}^\left\{0.5\right\} \exp \left(-P/8.7\right)$ where:$\Delta T_\left\{sat\right\}$ is the wall temperature elevation above the saturation temperature, KNEWLINE
NEWLINE
q is the heat flux, MW/m2
NEWLINE
P is the pressure of water, MPa
NEWLINE Note that 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. Selecting to base the heat flux on the pipe inner diameter, and assuming that the pipe wall thickness is small in comparison with the pipe inner diameter, then the heat transfer coefficient for the pipe wall can be calculated as if the wall were not curved: $h_\left\{wall\right\} = \left\{k \over x\right\}$ where k is the effective thermal conductivity of the wall material and x is the wall thickness. If the above assumption does not hold, then the wall heat transfer coefficient can be calculated using the following expression: $h_\left\{wall\right\} = \left\{2k \over \left\{d_i\ln\left(d_o/d_i\right)\right\}\right\}$ where di and do are the inner and outer diameters of the pipe, respectively. The thermal conductivity of the tube material usually depends on temperature; the mean thermal conductivity is often used.

## Combining heat transfer coefficients

For two or more heat transfer processes acting in parallel, heat transfer coefficients simply add: $h = h_1 + h_2 + \dots$ For two or more heat transfer processes connected in series, heat transfer coefficients add inversely: $\left\{1\over h\right\} = \left\{1\over h_1\right\} + \left\{1\over h_2\right\} + \dots$ For example, consider a pipe with a fluid flowing inside. The rate of heat transfer between the bulk of the fluid inside the pipe and the pipe external surface is: $q=\left\left( \left\{1\over\left\{\left\{1 \over h\right\}+\left\{t \over k\right\}\right\}\right\} \right\right) \cdot A \cdot \Delta T$ whereNEWLINE
NEWLINE
q = heat transfer rate (W)
NEWLINE
h = heat transfer coefficient (W/(m2·K))
NEWLINE
t = wall thickness (m)
NEWLINE
k = wall thermal conductivity (W/m·K)
NEWLINE
A = area (m2)$\Delta T$ = difference in temperature.
NEWLINE

## Overall heat transfer coefficient

The overall heat transfer coefficient $U$ is a measure of the overall ability of a series of conductive and convective barriers to transfer heat. It is commonly applied to the calculation of heat transfer in heat exchangers, but can be applied equally well to other problems. For the case of a heat exchanger, $U$ can be used to determine the total heat transfer between the two streams in the heat exchanger by the following relationship: $q = UA \Delta T_\left\{LM\right\}$ where$q$ = heat transfer rate (W)$U$ = overall heat transfer coefficient (W/(m²·K))$A$ = heat transfer surface area (m2)$\Delta T_\left\{LM\right\}$ = log mean temperature difference
Log mean temperature difference
The log mean temperature difference is used to determine the temperature driving force for heat transfer in flow systems, most notably in heat exchangers. The LMTD is a logarithmic average of the temperature difference between the hot and cold streams at each end of the exchanger. The larger the...

(K) The overall heat transfer coefficient takes into account the individual heat transfer coefficients of each stream and the resistance of the pipe material. It can be calculated as the reciprocal of the sum of a series of thermal resistances (but more complex relationships exist, for example when heat transfer takes place by different routes in parallel): $\frac \left\{1\right\} \left\{UA\right\} = \sum \frac\left\{1\right\} \left\{hA\right\} + \sum R$ whereNEWLINE
NEWLINE
R = Resistance(s) to heat flow in pipe wall (K/W)
NEWLINE
Other parameters are as above.
NEWLINE The heat transfer coefficient is the heat transferred per unit area per kelvin. Thus area is included in the equation as it represents the area over which the transfer of heat takes place. The areas for each flow will be different as they represent the contact area for each fluid side. The thermal resistance due to the pipe wall is calculated by the following relationship: $R = \frac\left\{x\right\}\left\{k \cdot A\right\}$ whereNEWLINE
NEWLINE
x = the wall thickness (m)
NEWLINE
k = the thermal conductivity of the material (W/(m·K))
NEWLINE
A = the total area of the heat exchanger (m2)
NEWLINE This represents the heat transfer by conduction in the pipe. The thermal conductivity
Thermal conductivity
In physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....

is a characteristic of the particular material. Values of thermal conductivities for various materials are listed in the list of thermal conductivities. As mentioned earlier in the article the convection heat transfer coefficient for each stream depends on the type of fluid, flow properties and temperature properties. Some typical heat transfer coefficients include: NEWLINE
NEWLINE
• Air - h = 10 to 100 W/(m2K)
• NEWLINE
• Water - h = 500 to 10,000 W/(m2K)
NEWLINE

## Thermal resistance due to fouling deposits

Surface coatings can build on heat transfer surfaces during heat exchanger operation due to fouling
Fouling
Fouling refers to the accumulation of unwanted material on solid surfaces, most often in an aquatic environment. The fouling material can consist of either living organisms or a non-living substance...

. These add extra thermal resistance to the wall and may noticeably decrease the overall heat transfer coefficient and thus performance. (Fouling can also cause other problems.) The additional thermal resistance due to fouling can be found by comparing the overall heat transfer coefficient determined from laboratory readings with calculations based on theoretical correlations. They can also be evaluated from the development of the overall heat transfer coefficient with time (assuming the heat exchanger operates under otherwise identical conditions). This is commonly applied in practice, e.g. The following relationship is often used: $\frac\left\{1\right\}\left\{U_\left\{exp\right\}\right\}$ = $\frac\left\{1\right\}\left\{U_\left\{pre\right\}\right\}+R_f$ where$U_\left\{exp\right\}$ = overall heat transfer coefficient based on experimental data for the heat exchanger in the "fouled" state, $\frac\left\{W\right\}\left\{m^2K\right\}$$U_\left\{pre\right\}$ = overall heat transfer coefficient based on calculated or measured ("clean heat exchanger") data, $\frac\left\{W\right\}\left\{m^2K\right\}$$R_f$ = thermal resistance due to fouling, $\frac\left\{m^2K\right\}\left\{W\right\}$

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• Convective heat transfer
Convective heat transfer
Convective heat transfer, often referred to as convection, is the transfer of heat from one place to another by the movement of fluids. The presence of bulk motion of the fluid enhances the heat transfer between the solid surface and the fluid. Convection is usually the dominant form of heat...

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• Convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

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• Churchill-Bernstein Equation
Churchill-Bernstein Equation
In convective heat transfer, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number for a cylinder in cross flow at various velocities. The need for the equation arises from the inability to solve the Navier–Stokes equations in the turbulent flow regime, even for...

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• Heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

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• Heat pump
Heat pump
A heat pump is a machine or device that effectively "moves" thermal energy from one location called the "source," which is at a lower temperature, to another location called the "sink" or "heat sink", which is at a higher temperature. An air conditioner is a particular type of heat pump, but the...

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• Heisler Chart
Heisler Chart
Heisler Charts are a set of two charts per included geometry introduced in 1947 by M. P. Heisler which were supplemented by a third chart per geometry in 1961 by H. Gröber...

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• Thermal conductivity
Thermal conductivity
In physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....

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• Thermal-hydraulics
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• Fourier number
Fourier number
In physics and engineering, the Fourier number or Fourier modulus, named after Joseph Fourier, is a dimensionless number that characterizes heat conduction. Conceptually, it is the ratio of the heat conduction rate to the rate of thermal energy storage. Together with the Biot number, it...

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• Nusselt number
Nusselt number
In heat transfer at a boundary within a fluid, the Nusselt number is the ratio of convective to conductive heat transfer across the boundary. Named after Wilhelm Nusselt, it is a dimensionless number...

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