I have flow in a pipe with a constant surface temperature boundary condition. The velocity profile is fully developed and the temperature profile is also fully developed. In my textbook, it says that the Nusselt number stays at a constant 3.66 value when these conditions exist. For fully-developed laminar flow, the Nusselt numbers are constant-valued. Convection with uniform surface heat flux. From Incropera & DeWitt 4, Nu D = 4.36 Convection with uniform surface temperature. For the case of constant surface temperature 4, Nu D = 3.66 See also. Simple derivation of the Nusselt number from Newton's law of cooling (Accessed 23 Sept 2009) Sherwood number (mass transfer Nusselt number).
In fluid dynamics, the Nusselt number (Nu) is the ratio of convective to conductiveheat transfer at a boundary in a fluid. Convection includes both advection (fluid motion) and diffusion (conduction). The conductive component is measured under the same conditions as the convective but for a hypothetically motionless fluid. It is a dimensionless number, closely related to the fluid's Rayleigh number.[1]
A Nusselt number of value one represents heat transfer by pure conduction.[2] A value between one and 10 is characteristic of slug flow or laminar flow.[3] A larger Nusselt number corresponds to more active convection, with turbulent flow typically in the 100–1000 range.[3] The Nusselt number is named after Wilhelm Nusselt, who made significant contributions to the science of convective heat transfer.[4]
A similar non-dimensional parameter is Biot number, which concerns thermal conductivity for a solid body rather than a fluid. The mass transfer analogue of the Nusselt number is the Sherwood number.
- 4Empirical Correlations
- 4.1Free convection
- 4.3Forced convection in turbulent pipe flow
- 4.4Forced convection in fully developed laminar pipe flow
Definition[edit]
The Nusselt number is the ratio of convective to conductive heat transfer across a boundary. The convection and conduction heat flows are parallel to each other and to the surface normal of the boundary surface, and are all perpendicular to the mean fluid flow in the simple case.
where h is the convectiveheat transfer coefficient of the flow, L is the characteristic length, k is the thermal conductivity of the fluid.
- Selection of the characteristic length should be in the direction of growth (or thickness) of the boundary layer; some examples of characteristic length are: the outer diameter of a cylinder in (external) cross flow (perpendicular to the cylinder axis), the length of a vertical plate undergoing natural convection, or the diameter of a sphere. For complex shapes, the length may be defined as the volume of the fluid body divided by the surface area.
- The thermal conductivity of the fluid is typically (but not always) evaluated at the film temperature, which for engineering purposes may be calculated as the mean-average of the bulk fluid temperature and wall surface temperature.
In contrast to the definition given above, known as average Nusselt number, local Nusselt number is defined by taking the length to be the distance from the surface boundary[5] to the local point of interest.
The mean, or average, number is obtained by integrating the expression over the range of interest, such as:[6]
Context[edit]
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.
Thermal Boundary Layer
The heat transfer rate can then be written as,
And because heat transfer at the surface is by conduction,
These two terms are equal; thus
Rearranging,
Making it dimensionless by multiplying by representative length L,
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.
Derivation[edit]
The Nusselt number may be obtained by a non-dimensional analysis of Fourier's law since it is equal to the dimensionless temperature gradient at the surface:
- , where q is the heat transfer rate, k is the constant thermal conductivity and T the fluidtemperature.
Indeed, if: , and
we arrive at
then we define
so the equation becomes
By integrating over the surface of the body:
,
where
Empirical Correlations[edit]
Typically, for free convection, the average Nusselt number is expressed as a function of the Rayleigh number and the Prandtl number, written as:
Otherwise, for forced convection, the Nusselt number is generally a function of the Reynolds number and the Prandtl number, or
Empirical correlations for a wide variety of geometries are available that express the Nusselt number in the aforementioned forms.
Free convection[edit]
Free convection at a vertical wall[edit]
Cited[7] as coming from Churchill and Chu:
Free convection from horizontal plates[edit]
If the characteristic length is defined
where is the surface area of the plate and is its perimeter.
Then for the top surface of a hot object in a colder environment or bottom surface of a cold object in a hotter environment[7]
And for the bottom surface of a hot object in a colder environment or top surface of a cold object in a hotter environment[7]
Forced convection on flat plate
Flat plate in laminar flow[edit]
The local Nusselt number for laminar flow over a flat plate, at a distance downstream from the edge of the plate, is given by[8]
The average Nusselt number for laminar flow over a flat plate, from the edge of the plate to a downstream distance , is given by[8]
Forced convection in turbulent pipe flow[edit]
Gnielinski correlation[edit]
Gnielinski's correlation for turbulent flow in tubes:[8][10]
where f is the Darcy friction factor that can either be obtained from the Moody chart or for smooth tubes from correlation developed by Petukhov:[8]
The Gnielinski Correlation is valid for:[8]
Dittus-Boelter equation[edit]
The Dittus-Boelter equation (for turbulent flow) is an explicit function for calculating the Nusselt number. It is easy to solve but is less accurate when there is a large temperature difference across the fluid. It is tailored to smooth tubes, so use for rough tubes (most commercial applications) is cautioned. The Dittus-Boelter equation is:
where:
- is the inside diameter of the circular duct
- is the Prandtl number
- for the fluid being heated, and for the fluid being cooled.[7]
The Dittus-Boelter equation is valid for[11]
Example The Dittus-Boelter equation is a good approximation where temperature differences between bulk fluid and heat transfer surface are minimal, avoiding equation complexity and iterative solving. Taking water with a bulk fluid average temperature of 20 °C, viscosity 10.07×10−4 Pa·s and a heat transfer surface temperature of 40 °C (viscosity 6.96×10−4, a viscosity correction factor for can be obtained as 1.45. This increases to 3.57 with a heat transfer surface temperature of 100 °C (viscosity 2.82×10−4 Pa·s), making a significant difference to the Nusselt number and the heat transfer coefficient.
Sieder-Tate correlation[edit]
The Sieder-Tate correlation for turbulent flow is an implicit function, as it analyzes the system as a nonlinear boundary value problem. The Sieder-Tate result can be more accurate as it takes into account the change in viscosity ( and ) due to temperature change between the bulk fluid average temperature and the heat transfer surface temperature, respectively. The Sieder-Tate correlation is normally solved by an iterative process, as the viscosity factor will change as the Nusselt number changes.[12]
- [7]
where:
- is the fluid viscosity at the bulk fluid temperature
- is the fluid viscosity at the heat-transfer boundary surface temperature
The Sieder-Tate correlation is valid for[7]
Forced convection in fully developed laminar pipe flow[edit]
For fully developed internal laminar flow, the Nusselt numbers tend towards a constant value for long pipes.
For internal Flow:
where:
- Dh = Hydraulic diameter
- kf = thermal conductivity of the fluid
- h = convectiveheat transfer coefficient
Convection with uniform temperature for circular tubes[edit]
From Incropera & DeWitt,[13]
Convection with uniform heat flux for circular tubes[edit]
For the case of constant surface heat flux,[13]
OEIS sequence A282581 gives this value as .
See also[edit]
- Sherwood number (mass transfer Nusselt number)
External links[edit]
- Simple derivation of the Nusselt number from Newton's law of cooling (Accessed 23 September 2009)
References[edit]
- ^Çengel, Yunus A. (2002). Heat and Mass Transfer (Second ed.). McGraw-Hill. p. 466.
- ^Çengel, Yunus A. (2002). Heat and Mass Transfer (Second ed.). McGraw-Hill. p. 336.
- ^ ab'The Nusselt Number'. Whiting School of Engineering. Retrieved 3 April 2019.
- ^Çengel, Yunus A. (2002). Heat and Mass Transfer (Second ed.). McGraw-Hill. p. 336.
- ^Yunus A. Çengel (2003). Heat Transfer: a Practical Approach (2nd ed.). McGraw-Hill.
- ^E. Sanvicente; et al. (2012). 'Transitional natural convection flow and heat transfer in an open channel'. International Journal of Thermal Sciences. 63: 87–104. doi:10.1016/j.ijthermalsci.2012.07.004.
- ^ abcdefIncropera, Frank P.; DeWitt, David P. (2000). Fundamentals of Heat and Mass Transfer (4th ed.). New York: Wiley. p. 493. ISBN978-0-471-30460-9.
- ^ abcdeIncropera, Frank P.; DeWitt, David P. (2007). Fundamentals of Heat and Mass Transfer (6th ed.). Hoboken: Wiley. pp. 490, 515. ISBN978-0-471-45728-2.
- ^Incropera, Frank P. Fundamentals of heat and mass transfer. John Wiley & Sons, 2011.
- ^Gnielinski, Volker (1975). 'Neue Gleichungen für den Wärme- und den Stoffübergang in turbulent durchströmten Rohren und Kanälen'. Forsch. Ing.-Wes. 41 (1): 8–16.
- ^Incropera, Frank P.; DeWitt, David P. (2007). Fundamentals of Heat and Mass Transfer (6th ed.). New York: Wiley. p. 514. ISBN978-0-471-45728-2.
- ^'Temperature Profile in Steam Generator Tube Metal'(PDF). Retrieved 23 September 2009.
- ^ abIncropera, Frank P.; DeWitt, David P. (2002). Fundamentals of Heat and Mass Transfer (5th ed.). Hoboken: Wiley. pp. 486, 487. ISBN978-0-471-38650-6.
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Nusselt_number&oldid=893231262'
In fluid dynamics, the Nusselt number (Nu) is the ratio of convective to conductiveheat transfer at a boundary in a fluid. Convection includes both advection (fluid motion) and diffusion (conduction). The conductive component is measured under the same conditions as the convective but for a hypothetically motionless fluid. It is a dimensionless number, closely related to the fluid's Rayleigh number.[1]
A Nusselt number of value one represents heat transfer by pure conduction.[2] A value between one and 10 is characteristic of slug flow or laminar flow.[3] A larger Nusselt number corresponds to more active convection, with turbulent flow typically in the 100–1000 range.[3] The Nusselt number is named after Wilhelm Nusselt, who made significant contributions to the science of convective heat transfer.[4]
A similar non-dimensional parameter is Biot number, which concerns thermal conductivity for a solid body rather than a fluid. The mass transfer analogue of the Nusselt number is the Sherwood number.
Definition
The Nusselt number is the ratio of convective to conductive heat transfer across a boundary. The convection and conduction heat flows are parallel to each other and to the surface normal of the boundary surface, and are all perpendicular to the mean fluid flow in the simple case.
where h is the convectiveheat transfer coefficient of the flow, L is the characteristic length, k is the thermal conductivity of the fluid.
- Selection of the characteristic length should be in the direction of growth (or thickness) of the boundary layer; some examples of characteristic length are: the outer diameter of a cylinder in (external) cross flow (perpendicular to the cylinder axis), the length of a vertical plate undergoing natural convection, or the diameter of a sphere. For complex shapes, the length may be defined as the volume of the fluid body divided by the surface area.
- The thermal conductivity of the fluid is typically (but not always) evaluated at the film temperature, which for engineering purposes may be calculated as the mean-average of the bulk fluid temperature and wall surface temperature.
In contrast to the definition given above, known as average Nusselt number, local Nusselt number is defined by taking the length to be the distance from the surface boundary[5] to the local point of interest.
The mean, or average, number is obtained by integrating the expression over the range of interest, such as:[6]
Context
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.
Thermal Boundary Layer
The heat transfer rate can then be written as,
And because heat transfer at the surface is by conduction,
These two terms are equal; thus
Rearranging,
Making it dimensionless by multiplying by representative length L,
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.
Derivation
The Nusselt number may be obtained by a non-dimensional analysis of Fourier's law since it is equal to the dimensionless temperature gradient at the surface:
- , where q is the heat transfer rate, k is the constant thermal conductivity and T the fluidtemperature.
Indeed, if: , and
we arrive at
then we define
so the equation becomes
By integrating over the surface of the body:
,
where
Empirical Correlations
Typically, for free convection, the average Nusselt number is expressed as a function of the Rayleigh number and the Prandtl number, written as:
Otherwise, for forced convection, the Nusselt number is generally a function of the Reynolds number and the Prandtl number, or
Empirical correlations for a wide variety of geometries are available that express the Nusselt number in the aforementioned forms.
Free convection
Free convection at a vertical wall
Cited[7] as coming from Churchill and Chu:
Free convection from horizontal plates
If the characteristic length is defined
where is the surface area of the plate and is its perimeter.
Then for the top surface of a hot object in a colder environment or bottom surface of a cold object in a hotter environment[7]
And for the bottom surface of a hot object in a colder environment or top surface of a cold object in a hotter environment[7]
Forced convection on flat plate
Flat plate in laminar flow
The local Nusselt number for laminar flow over a flat plate, at a distance downstream from the edge of the plate, is given by[8]
- 0.6)'>
The average Nusselt number for laminar flow over a flat plate, from the edge of the plate to a downstream distance , is given by[8]
- 0.6)}'>
Forced convection in turbulent pipe flow
Gnielinski correlation
Gnielinski's correlation for turbulent flow in tubes:[8][10]
where f is the Darcy friction factor that can either be obtained from the Moody chart or for smooth tubes from correlation developed by Petukhov:[8]
The Gnielinski Correlation is valid for:[8]
Dittus-Boelter equation
The Dittus-Boelter equation (for turbulent flow) is an explicit function for calculating the Nusselt number. It is easy to solve but is less accurate when there is a large temperature difference across the fluid. It is tailored to smooth tubes, so use for rough tubes (most commercial applications) is cautioned. The Dittus-Boelter equation is:
where:
- is the inside diameter of the circular duct
- is the Prandtl number
- for the fluid being heated, and for the fluid being cooled.[7]
The Dittus-Boelter equation is valid for[11]
Example The Dittus-Boelter equation is a good approximation where temperature differences between bulk fluid and heat transfer surface are minimal, avoiding equation complexity and iterative solving. Taking water with a bulk fluid average temperature of 20 °C, viscosity 10.07×10−4 Pa·s and a heat transfer surface temperature of 40 °C (viscosity 6.96×10−4, a viscosity correction factor for can be obtained as 1.45. This increases to 3.57 with a heat transfer surface temperature of 100 °C (viscosity 2.82×10−4 Pa·s), making a significant difference to the Nusselt number and the heat transfer coefficient.
Sieder-Tate correlation
The Sieder-Tate correlation for turbulent flow is an implicit function, as it analyzes the system as a nonlinear boundary value problem. The Sieder-Tate result can be more accurate as it takes into account the change in viscosity ( and ) due to temperature change between the bulk fluid average temperature and the heat transfer surface temperature, respectively. The Sieder-Tate correlation is normally solved by an iterative process, as the viscosity factor will change as the Nusselt number changes.[12]
- [7]
where:
- is the fluid viscosity at the bulk fluid temperature
- is the fluid viscosity at the heat-transfer boundary surface temperature
The Sieder-Tate correlation is valid for[7]
Forced convection in fully developed laminar pipe flow
For fully developed internal laminar flow, the Nusselt numbers tend towards a constant value for long pipes.
For internal Flow:
where:
- Dh = Hydraulic diameter
- kf = thermal conductivity of the fluid
- h = convectiveheat transfer coefficient
Convection with uniform temperature for circular tubes
From Incropera & DeWitt,[13]
Convection with uniform heat flux for circular tubes
For the case of constant surface heat flux,[13]
OEIS sequence A282581 gives this value as .
See also
- Sherwood number (mass transfer Nusselt number)
External links
- Simple derivation of the Nusselt number from Newton's law of cooling (Accessed 23 September 2009)
References
- ^Çengel, Yunus A. (2002). Heat and Mass Transfer (Second ed.). McGraw-Hill. p. 466.
- ^Çengel, Yunus A. (2002). Heat and Mass Transfer (Second ed.). McGraw-Hill. p. 336.
- ^ ab'The Nusselt Number'. Whiting School of Engineering. Retrieved 3 April 2019.
- ^Çengel, Yunus A. (2002). Heat and Mass Transfer (Second ed.). McGraw-Hill. p. 336.
- ^Yunus A. Çengel (2003). Heat Transfer: a Practical Approach (2nd ed.). McGraw-Hill.
- ^E. Sanvicente; et al. (2012). 'Transitional natural convection flow and heat transfer in an open channel'. International Journal of Thermal Sciences. 63: 87–104. doi:10.1016/j.ijthermalsci.2012.07.004.
- ^ abcdefIncropera, Frank P.; DeWitt, David P. (2000). Fundamentals of Heat and Mass Transfer (4th ed.). New York: Wiley. p. 493. ISBN 978-0-471-30460-9.
- ^ abcdeIncropera, Frank P.; DeWitt, David P. (2007). Fundamentals of Heat and Mass Transfer (6th ed.). Hoboken: Wiley. pp. 490, 515. ISBN 978-0-471-45728-2.
- ^Incropera, Frank P. Fundamentals of heat and mass transfer. John Wiley & Sons, 2011.
- ^Gnielinski, Volker (1975). 'Neue Gleichungen für den Wärme- und den Stoffübergang in turbulent durchströmten Rohren und Kanälen'. Forsch. Ing.-Wes. 41 (1): 8–16.
- ^Incropera, Frank P.; DeWitt, David P. (2007). Fundamentals of Heat and Mass Transfer (6th ed.). New York: Wiley. p. 514. ISBN 978-0-471-45728-2.
- ^'Temperature Profile in Steam Generator Tube Metal'(PDF). Retrieved 23 September 2009.
- ^ abIncropera, Frank P.; DeWitt, David P. (2002). Fundamentals of Heat and Mass Transfer (5th ed.). Hoboken: Wiley. pp. 486, 487. ISBN 978-0-471-38650-6.
The Biot number (Bi) is a dimensionless quantity used in heat transfer calculations. It is named after the eighteenth century French physicist Jean-Baptiste Biot (1774–1862), and gives a simple index of the ratio of the heat transfer resistances inside of a body and at the surface of a body. This ratio determines whether or not the temperatures inside a body will vary significantly in space, while the body heats or cools over time, from a thermal gradient applied to its surface.
In general, problems involving small Biot numbers (much smaller than 1) are thermally simple, due to uniform temperature fields inside the body. Biot numbers much larger than 1 indicate more difficult problems due to non-uniformity of temperature fields within the object. It should not be confused with Nusselt number, which employs the thermal conductivity of the fluid and hence is a comparative measure of conduction and convection, both in the fluid.
The Biot number has a variety of applications, including transient heat transfer and use in extended surface heat transfer calculations.
Characteristic lengthIn physics, a characteristic length is an important dimension that defines the scale of a physical system. Often, such a length is used as an input to a formula in order to predict some characteristics of the system.
Examples:
Reynolds Number
Biot number
Nusselt numberIn computational mechanics, a characteristic length is defined to force localization of a stress softening constitutive equation. The length is associated with an integration point. For 2D analysis, it is calculated by taking square root of the area. For 3D analysis, it is calculated by taking cubic root of the volume associated to the integration point.
Churchill–Bernstein equationIn 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 a Newtonian fluid. When the concentration and temperature profiles are independent of one another, the mass-heat transfer analogy can be employed. In the mass-heat transfer analogy, heat transfer dimensionless quantities are replaced with analogous mass transfer dimensionless quantities.
This equation is named after Stuart W. Churchill and M. Bernstein, who introduced it in 1977. This equation is also called the Churchill–Bernstein correlation.
Convective mixingIn fluid dynamics, convective mixing is the vertical transport of a fluid and its properties. In many important ocean and atmospheric phenomena, convection is driven by density differences in the fluid, e.g. the sinking of cold, dense water in polar regions of the world's oceans; and the rising of warm, less-dense air during the formation of cumulonimbus clouds and hurricanes.
Dimensionless quantityIn dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned, also known as a bare, pure, or scalar quantity or a quantity of dimension one, with a corresponding unit of measurement in the SI of one (or 1) unit that is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities to which dimensions are regularly assigned are length, time, and speed, which are measured in dimensional units, such as metre, second and metre per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.
Film temperatureIn heat transfer and fluid dynamics, the film temperature () is an approximation to the temperature of a fluid inside a convection boundary layer. It is calculated as the arithmetic mean of the temperature at the surface of the solid boundary wall () and the free-stream temperature ():
The film temperature is often used as the temperature at which fluid properties are calculated when using Prandtl number, Nusselt number, Reynolds number or Grashof number to calculate a heat transfer coefficient, because it is a reasonable first approximation to the temperature within the convection boundary layer.
Somewhat confusing terminology may be encountered in relation to boilers and heat exchangers, where the same term is used to refer to the limit (hot) temperature of a fluid in contact with a hot surface.
Heat transfer coefficientThe heat transfer coefficient or film coefficient, or film effectiveness, in thermodynamics and in mechanics is the proportionality constant between the heat flux and the thermodynamic driving force for the flow of heat (i.e., the temperature difference, ΔT):
The overall heat transfer rate for combined modes is usually expressed in terms of an overall conductance or heat transfer coefficient, U. In that case, the heat transfer rate is:
where:
- : surface area where the heat transfer takes place, m2
- : temperature of the surrounding fluid, K
- : temperature of the solid surface, K.
The general definition of the heat transfer coefficient is:
where:
- q: heat flux, W/m2; i.e., thermal power per unit area, q = d/dA
- h: heat transfer coefficient, W/(m2•K)
- ΔT: difference in temperature between the solid surface and surrounding fluid area, K
It is used in calculating the heat transfer, typically by convection or phase transition between a fluid and a solid. The heat transfer coefficient has SI units in watts per squared meter kelvin: W/(m2K).
The heat transfer coefficient is the reciprocal of thermal insulance. This is used for building materials (R-value) 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 (a dimensionless 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 (e.g. ).
Heat transfer enhancementHeat transfer enhancement is the process of increasing the effectiveness of heat exchangers. This can be achieved when the heat transfer power of a given device is increased or when the pressure losses generated by the device are reduced. A variety of techniques can be applied to this effect, including generating strong secondary flows or increasing boundary layer turbulence.
Micro heat exchangerMicro heat exchangers, Micro-scale heat exchangers, or microstructured heat exchangers are heat exchangers in which (at least one) fluid flows in lateral confinements with typical dimensions below 1 mm. The most typical such confinement are microchannels, which are channels with a hydraulic diameter below 1 mm. Microchannel heat exchangers can be made from metal, ceramic,Microchannel heat exchangers can be used for many applications including:
high-performance aircraft gas turbine engines
heat pumps
air conditioning
Natural convectionNatural convection is a type of flow, of motion of a liquid such as water or a gas such as air, in which the fluid motion is not generated by any external source (like a pump, fan, suction device, etc.) but by some parts of the fluid being heavier than other parts. The driving force for natural convection is gravity. For example if there is a layer of cold dense air on top of hotter less dense air, gravity pulls more strongly on the denser layer on top, so it falls while the hotter less dense air rises to take its place. This creates circulating flow: convection. As it relies of gravity, there is no convection in free-fall (inertial) environments, such as that of the orbiting International Space Station. Natural convection can occur when there are hot and cold regions of either air or water, because both water and air become less dense as they are heated. But, for example, in the world's oceans it also occurs due to salt water being heavier than fresh water, so a layer of salt water on top of a layer of fresher water will also cause convection.
Natural convection has attracted a great deal of attention from researchers because of its presence both in nature and engineering applications. In nature, convection cells formed from air raising above sunlight-warmed land or water are a major feature of all weather systems. Convection is also seen in the rising plume of hot air from fire, plate tectonics, oceanic currents (thermohaline circulation) and sea-wind formation (where upward convection is also modified by Coriolis forces). In engineering applications, convection is commonly visualized in the formation of microstructures during the cooling of molten metals, and fluid flows around shrouded heat-dissipation fins, and solar ponds. A very common industrial application of natural convection is free air cooling without the aid of fans: this can happen on small scales (computer chips) to large scale process equipment.
Péclet numberThe Péclet number (Pe) is a class of dimensionless numbers relevant in the study of transport phenomena in a continuum. It is named after the French physicist Jean Claude Eugène Péclet. It is defined to be the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an appropriate gradient. In the context of species or mass transfer, the Péclet number is the product of the Reynolds number and the Schmidt number. In the context of the thermal fluids, the thermal Peclet number is equivalent to the product of the Reynolds number and the Prandtl number.
The Péclet number is defined as:
For mass transfer, it is defined as:
For heat transfer, the Péclet number is defined as:
where L is the characteristic length, u the local flow velocity, D the mass diffusion coefficient, and α the thermal diffusivity,
where k is the thermal conductivity, ρ the density, and cp the heat capacity.
In engineering applications the Péclet number is often very large. In such situations, the dependency of the flow upon downstream locations is diminished, and variables in the flow tend to become 'one-way' properties. Thus, when modelling certain situations with high Péclet numbers, simpler computational models can be adopted.
A flow will often have different Péclet numbers for heat and mass. This can lead to the phenomenon of double diffusive convection.
In the context of particulate motion the Péclet number has also been called Brenner number, with symbol Br, in honour of Howard Brenner.
Rayleigh numberIn fluid mechanics, the Rayleigh number (Ra) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free or natural convection. It characterises the fluid's flow regime: a value in a certain lower range denotes laminar flow; a value in a higher range, turbulent flow. Below a certain critical value, there is no fluid motion and heat transfer is by conduction rather than convection.
The Rayleigh number is defined as the product of the Grashof number, which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number, which describes the relationship between momentum diffusivity and thermal diffusivity. Hence it may also be viewed as the ratio of buoyancy and viscosity forces multiplied by the ratio of momentum and thermal diffusivities. It is closely related to the Nusselt number.For most engineering purposes, the Rayleigh number is large, somewhere around 106 to 108. It is named after Lord Rayleigh, who described the property's relationship with fluid behaviour.
Sherwood numberThe Sherwood number (Sh) (also called the mass transfer Nusselt number) is a dimensionless number used in mass-transfer operation. It represents the ratio of the convective mass transfer to the rate of diffusive mass transport, and is named in honor of Thomas Kilgore Sherwood.
It is defined as follows
where
Using dimensional analysis, it can also be further defined as a function of the Reynolds and Schmidt numbers:
For example, for a single sphere it can be expressed as:
where is the Sherwood number due only to natural convection and not forced convection.
A more specific correlation is the Froessling equation:
This form is applicable to molecular diffusion from a single spherical particle. It is particularly valuable in situations where the Reynolds number and Schmidt number are readily available. Since Re and Sc are both dimensionless numbers, the Sherwood number is also dimensionless.
These correlations are the mass transfer analogies to heat transfer correlations of the Nusselt number in terms of the Reynolds number and Prandtl number. For a correlation for a given geometry (e.g. spheres, plates, cylinders, etc.), a heat transfer correlation (often more readily available from literature and experimental work, and easier to determine) for the Nusselt number (Nu) in terms of the Reynolds number (Re) and the Prandtl number (Pr) can be used as a mass transfer correlation by replacing the Prandtl number with the analogous dimensionless number for mass transfer, the Schmidt number, and replacing the Nusselt number with the analogous dimensionless number for mass transfer, the Sherwood number.
As an example, a heat transfer correlation for spheres is given by the Ranz-Marshall Correlation:
This correlation can be made into a mass transfer correlation using the above procedure, which yields:
This is a very concrete way of demonstrating the analogies between different forms of transport phenomena.
Skin friction dragSkin friction drag is a component of profile drag, which is resistant force exerted on an object moving in a fluid. Skin friction drag is caused by the viscosity of fluids and is developed from laminar drag to turbulent drag as a fluid moves on the surface of an object. Skin friction drag is generally expressed in term of the Reynolds number, which is the ratio between inertial force and viscous force.
Stanton numberThe Stanton number, St, is a dimensionless number that measures the ratio of heat transferred into a fluid to the thermal capacity of fluid. The Stanton number is named after Thomas Stanton (engineer) (1865–1931). It is used to characterize heat transfer in forced convection flows.
Thermal entrance lengthThe thermal entrance length Leh is used to outline the boundary between the fully developed heat flow and the non-fully developed heat flow in a heated/cooled pipe flow of a fluid.
Transport phenomenaIn engineering, physics and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mechanics and thermodynamics, it places a heavy emphasis on the commonalities between the topics covered. Mass, momentum, and heat transport all share a very similar mathematical framework, and the parallels between them are exploited in the study of transport phenomena to draw deep mathematical connections that often provide very useful tools in the analysis of one field that are directly derived from the others.
The fundamental analysis in all three subfields of mass, heat, and momentum transfer are often grounded in the simple principle that the sum total of the quantities being studied must be conserved by the system and its environment. Thus, the different phenomena that lead to transport are each considered individually with the knowledge that the sum of their contributions must equal zero. This principle is useful for calculating many relevant quantities. For example, in fluid mechanics, a common use of transport analysis is to determine the velocity profile of a fluid flowing through a rigid volume.
Transport phenomena are ubiquitous throughout the engineering disciplines. Some of the most common examples of transport analysis in engineering are seen in the fields of process, chemical, biological, and mechanical engineering, but the subject is a fundamental component of the curriculum in all disciplines involved in any way with fluid mechanics, heat transfer, and mass transfer. It is now considered to be a part of the engineering discipline as much as thermodynamics, mechanics, and electromagnetism.
Transport phenomena encompass all agents of physical change in the universe. Moreover, they are considered to be fundamental building blocks which developed the universe, and which is responsible for the success of all life on earth. However, the scope here is limited to the relationship of transport phenomena to artificial engineered systems.
Wilhelm NusseltErnst Kraft Wilhelm Nußelt (Nusselt in English; born November 25, 1882 in Nuremberg – died September 1, 1957 in München) was a German engineer. Nusselt studied mechanical engineering at the Munich Technical University (Technische Universität München), where he got his doctorate in 1907. He taught in Dresden from 1913 to 1917.
During this teaching tenure he developed the dimensional analysis of heat transfer, without any knowledge of the Buckingham π theorem or any other developments of Lord Rayleigh.
In so doing he opened the door for further heat transfer analysis. After teaching and working in Switzerland and Germany between 1917 and 1925, he was appointed to the Chair of Theoretical Mechanics in München. There he made important developments in the field of heat exchangers. He held that position until 1952, being succeeded in the job by another important figure in the field of heat transfer, Ernst Schmidt.
The Nusselt number used in Fluid Mechanics and Heat Transfer is named in his honour.
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