# Heat transfer coefficient

The **heat transfer coefficient**, in thermodynamics and in mechanical and chemical engineering, is used in calculating the heat transfer, typically by convection or phase change between a fluid and a solid:

- <math>\Delta Q=h \cdot A \cdot \Delta T \cdot \Delta t</math>

where

*ΔQ*= heat input or heat lost, J*h*= overall heat transfer coefficient, W/(m^{2}K)*A*= heat transfer surface area, m^{2}- <math>\Delta T</math> = difference in temperature between the solid surface and surrounding fluid area, K
- <math>\Delta t</math> = time period, s

From the above equation, the heat transfer coefficient is the proportionality coefficient between the heat flux, *Q*/(*AΔt*), and the thermodynamic driving force for the flow of heat (i.e., the temperature difference, *ΔT*).

The heat transfer coefficient has SI units in watts per meter squared-kelvin (W/(m^{2}K)).

Heat transfer coefficient can be thought of as an inverse of thermal resistance.

There are numerous correlations for calculation of 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 (a dimensionless number).

## Dittus–Boelter correlation (forced convection)

A common and particularly simple correlation useful for many applications is the Dittus–Boelter heat transfer correlation for fluids in turbulent flow. 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 liquid flowing in a straight circular pipe with a Reynolds number between 10 000 and 120 000 (in the turbulent pipe flow range), when the liquid'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^{[1]}) or other flow disturbances, and when the pipe surface is hydraulically smooth, the heat transfer coefficient can be expressed as:

- <math>h={{k_w}\over{D_H}}Nu</math>

where

- <math>k_w</math> = thermal conductivity of water
- <math>D_H</math> = Hydraulic diameter
*Nu*= Nusselt number- = <math>{0.023} \cdot Re^{0.8} \cdot Pr^{n}</math> => Dittus-Boelter correlation
*Pr*= Prandtl number = <math>{C_p \cdot \mu}\over{k_w}</math>*Re*= Reynolds number = <math>{\dot m \cdot D_H}\over{\mu \cdot A }</math>- <math>\dot m</math> = mass flow rate
*μ*= fluid viscosity*C*_{p}= heat capacity at constant pressure*A*= cross-sectional area of flow*n*= 0.4 for heating (wall hotter than the bulk fluid) and 0.3 for cooling (wall cooler than the bulk fluid)^{[2]}.

The fluid properties necessary for the application of this equation are evaluated at the bulk temperature thus avoiding iterations.

## 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:^{[3]}

<math>\Delta T_{sat} = 22.5 \cdot {q}^{0.5} \exp (-P/8.7)</math>

where:

- <math>\Delta T_{sat}</math> is the wall temperature elevation above the saturation temperature, K
- q is the heat flux, MW/m
^{2} - P is the pressure of water, MPa

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:

- <math>h_{wall} = {k \over t}</math>

where k is the effective thermal conductivity of the wall material and t is the wall thickness.

If the above assumption does not hold, then the wall heat transfer coefficient can be calculated using the following expression:

- <math>h_{wall} = {2k \over {d_i\ln(d_o/d_i)}}</math>

where d_{i} and d_{o} 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:

- <math>h = h_1 + h_2 + \dots</math>

For two or more heat transfer processes connected in series, heat transfer coefficients add inversely. This means that the overall heat transfer coefficient is a harmonic mean of the partial heat transfer coefficients:

- <math>{1\over h} = {1\over h_1} + {1\over h_2} + \dots</math>

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:

- <math>Q=\left( {1\over{{1 \over h}+{t \over k}}} \right) \cdot A \cdot \Delta T</math>

where

*Q*= heat transfer rate (W)*h*= heat transfer coefficient (W/m^{2}.K)*t*= wall thickness (m)*k*= wall thermal conductivity (W/m.K)*A*= area (m^{2})- <math>\Delta T</math> = difference in temperature.

## Overall Heat Transfer Coefficient

The **overall heat transfer coefficient**, **U**, is the reciprocal of the overall thermal resistance across a heat exchanger. It is used to determine the total heat transfer between the two streams in the heat exchanger by the following relationship:

- <math>Q = U \cdot A \cdot \Delta T</math>

where

*Q*= heat transfer rate (W)*U*= overall heat transfer coefficient (W/m^{2}.K)*A*= area (m^{2})*ΔT*= log mean temperature difference (K)

The overall heat transfer coefficient takes into account the individual heat transfer coefficients of each stream and the resistance of the pipe material.

- <math>\frac {1} {UA} = \Sigma \frac{1} {hA} + \Sigma R </math>

where

- R = Resistance(s) to heat flow in pipe wall (K/W)
- Other parameters are as above.
^{[4]}

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=x/k.A

where

- x = The wall thickness (m)
- k = the thermal conductivity of the material (W/mk)
- A = The total area of the heat exchanger (m
^{2}

This represents the heat transfer by conduction in the pipe.

The *thermal conductivity* is a characteristic of the particular material.

Some typical themal conductivity values include:

- - Polypropylene - k = 0.12 W/mk
- - Stainless steel - k = 21 W/mk
^{[5]}

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:

- - Air - h = 10 to 100 W/m
^{2}K - - Water - h = 500 to 10 000 W/m
^{2}K

## Resistance due to Fouling

Surface coatings known as foul can build up in heat exchangers, which add extra thermal resistance to the wall thus decreasing the overall heat transfer coefficent. Fouling also increases pumping costs. The resistance due to fouling is found by comparing calculations of the overall heat transfer coefficient from laboratory readings with calculations based on predicted theoretical correlations. The following relationship is used:

1/Uexp = 1/Upre+Rf

where,

- Uexp = Overall Heat transfer coefficient based on experimental data (W/m
^{2}K) - Upre = Overall Heat transfer coefficient based on predicted data (W/m
^{2}K) - Rf = The resistance due to fouling (m
^{2}W.K)

## References

- ↑ S.S. Kutateladze and V.M. Borishanskii, "A Concise Encyclopedia of Heat Transfer", Pergamon Press, 1966.
- ↑ F.Kreith (editor), "The CRC Handbook of Thermal Engineering", CRC Press, 2000.
- ↑ W.Rohsenow, J.Hartnet, Y.Cho, "Handbook of Heat Transfer", 3rd edition, McGraw-Hill, 1998.
- ↑ Coulson and Richardson, "Chemical Engineering", Volume 1,Elsevier, 2000
- ↑ The Engineering ToolBox, www.EngineeringToolBox.com, 2005