[教學] Background knowledge on heat transfer (30% completed)

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The Dimensionless numbers below are the ones most commonly used in heat transfer:

A.Reynolds Number
B.Nusselt Number
C.Prandtl Number
D.Grashof Number
E.Rayleigh Number

Some background information,

1. Density
Mass of fluid contained in a unit volume. Its units are Kg/m^3 or slugs/ft^3.
Typical values:
Water = 1000 kg/m^3,
Mercury = 13546 kg/m^3,
Air = 1.23 kg/m^3,
Paraffin Oil = 800 kg/m^3. (at pressure =1.013e+5 Pascals and Temperature = 288.15 K.)

2. Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms (and for fluids only), viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity. Put simply, the less viscous the fluid is, the greater its ease of movement (fluidity).

3. Thermal Conductivity
Thermal conductivity is a measure of the ability of a material to conduct heat. It is defined using the Fourier's law of condution which, relates the rate of heat transfer by conduction to the temperature gradient:



where k is the thermal conductivity. Using the Fourier's law we can define the thermal conductivity as the rate of heat transfer through a unit thickness of a material per unit area and per unit temperature difference. A good conductor of heat has a high value of thermal conductivity. The thermal conductivity is expressed in the units of (energy rate/(length.Temperature). In metric system, its unit is W/m.K.

Thermal conductivity of most material vary with temperature. For example:

T (K) Copper Aluminum
100      482          302
200      413          237
300      401          237
400      393          240
600      379          231
800      366          218

For both cases the thermal conductivity decreases with temperature. Thermal conductivity of most liquids decrease with increasing temperature. Water is, however, an exception to this rule. According to the kinetic theory of gases, the thermal conductivity of gases is proportional to the square root of the absolute temperature and inversely proportional to the square root of the molar mass. It is obvious that the thermal conductivity of a gas increases with the increasing temperature.

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4. Specific Heat
Specific heat is the amount of heat that is required to raise the temperature of a unit mass of a substance by one degree. In a constant pressure process



The units for the specific heat are kJ/kg.K (or C). Typical values of Cp for various materials (at 300 K) are shown below:

Material : Cp (kJ/kg.K)
Aluminum (pure) : 903
Copper (pure) : 385
Gold : 129
Silicon : 712
Water : 4180
Air : 1005

5. Coefficient of Thermal Expansion

6. Thermal disffusivity

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Section A : Reynolds Number

When the Reynolds number is large, the inertia forces are in command. Viscous forces dominate the boundary layer when the Reynolds number is small. Now, how does this relate to transition from laminar to turbulent flow?

Any real flow of fluid contains small disturbances that will grow given enough opportunities. as long as the viscous forces dominate these disturbances are under control. As the inertia forces get bigger, the viscosity can no longer maintain order and these tiny disturbances grow into trouble makers and we transition to turbulent flow.

Another important quantity of the boundary layer that is influenced by the Reynolds number is its thickness. As the Reynolds number increases, the viscous layer gets squeezed into a smaller distance from the surface.

The value of Reynolds number beyond which the flow is no longer considered laminar is called the critical Reynolds number. For flow over a flat plate, the critical Reynolds number is observed to vary between 1e+5 to 3e+6 depending on the turbulence level in the free stream and the roughness of the surface. We normally use 5e+5 as the critical Reynolds number for flow over flat plates.

Calculation of the Reynolds number is easy as long as you:

Identify the characteristic length
Pick the right velocity
Use a consistent set of units

For flow over a flat plate, the characteristic length is the length of the plate and the characteristic velocity is the free stream velocity. For pipes the characteristic length is the pipe diameter and the characteristic velocity is the average velocity through the pipe obtained by dividing the volumetric flow rate by the cross-sectional area (we are assuming that the pipe is full, of course). For pipes with a non-circular cross-section, the characteristic length is the Hydraulic Diameter defined as 4A/P, where A is the cross-sectional area of the duct and P is the wetted perimeter. You can easily verify that for a circular pipe the hydraulic diameter equals the pipe diameter. For non-cicular pipes the average velocity is used as the characteristic velocity. The situation gets messy when you are dealing with a problem that has many velocity and length scales such as the flow inside a computer cabinet. You must decide, based on your design objectives, which length and velocity length scales make sense for calculation of the Reynolds number.

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Section B : Nusselt Number

Nusselt number is the dimensionless heat transfer coefficient and appears when you are dealing with convection. It, therefore, provides a measure of the convection heat transfer at the surface. It is defined as hL/k where, h is the heat transfer coefficient, L is a characteristic length and k is the thermal conductivity. But, what does this grouping mean from a physical standpoint? Let's find out.

I am afraid that we have to look at the boundary layer in order to explain the concept of Nusselt number. We will, of course, cover the basics of the boundary layer in a separate tutorial but for now it suffices to say that when a fluid flows over a solid surface, the first layer of the fluid stick to the boundary (we even have a name for this thing called, no slip condition). This causes the flow to retard in the vicinity of the wall. As we move away from the wall the effect of this no slip thing gets smaller and smaller up to a point where it is no longer felt by the fluid. To get to this point, though, we have had to go through a layer of fluid who still knows about the wall. This layer is called the boundary layer. This was the effect of the wall on the velocity (or momentum). A similar argument applies when, for example, a cold fluid flows over a hot surface. The first layer of the fluid (which is now stuck to the surface) gets its heat from the surface through pure conduction. It then gives its newly acquired energy to all of the other fluid molecules that it comes in contact with as they pass by it (this is convection). As we move further and further away from the wall, the effect of the hot wall is felt less and less (it, of course, depends on the thermal conductivity of the fluid). Eventually, there comes a point where the fluid does not have a clue about the hot wall. The layer of fluid between the wall and this point is called the thermal boundary layer. It is where all of the action is taking place (as far as heat transfer between the solid and fluid is concerned). Before continuing with the Nusselt number, let us define another dimensionless property:

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Section C: Prandtl Number

Heat transfer gurus have invented another dimensionless number called the Prandtl number which is a grouping of the properties of the fluid but it has a significance to our discussion. Prandtl number is defined as:



It is the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. It can be related to the thickness of the thermal and velocity boundary layers. It is actually the ratio of velocity boundary layer to thermal boundary layer. When Pr=1, the boundary layers coincide. Typical values of the Prandtl number are:

Material : Pr
Liquid metals : 0.004-0.03
Gases : 0.7-1.0
Water : 1.7-13.7
Oils : 50-100,000

When Pr is small, it means that heat diffuses very quickly compared to the velocity (momentum). This means the thickness of the thermal boundary layer is much bigger than the velocity boundary layer for liquid metals.

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Section D : Grashof Number

You see this number and you should think of natural or free convection. The Grashof number is the ratio of buoyancy forces to the viscous forces.

For pipes

For vertical flat plates

For bluff bodies

where the L and D subscripts indicates the length scale basis for the Grashof Number.
g = acceleration due to Earth's gravity
β = volumetric thermal expansion coefficient (equal to approximately 1/T, for ideal fluids, where T is absolute temperature)
Ts = surface temperature
T∞ = bulk temperature
L = length
D = diameter
ν = kinematic viscosity

In natural convection the Grashof number plays the same role the is played by the Reynolds number in forced convection. The buoyant forces are fighting with viscous forces and at some point they overcome the viscous forces and the flow is no longer nice and laminar. For a vertical plate, the flow transitions to turbulent around a Grashof number of 10^9.

Section E : Rayleigh Number

The Rayleigh number is the product of Grashof and Prandtl numbers. It turns out that in natural convection the Nusselt number scales with Rayleigh rather than just Grashof. Most correlations in natural convection are of the form:



x = Characteristic length (in this case, the distance from the leading edge)
Rax = Rayleigh number at position x
Grx = Grashof number at position x
Pr = Prandtl number
g = acceleration due to gravity
Ts = Surface temperature (temperature of the wall)
T∞ = Quiescent temperature (fluid temperature far from the surface of the object)
ν = Kinematic viscosity
α = Thermal diffusivity
β = Thermal expansion coefficient

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This is an edited article from http://www.coolingzone.com/library.php?read=481 with some minor changes.

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yaocc0456

太深澳小學雞睇唔明

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回復 8# yaocc0456

If you were to do a decent DIY liquid system, you will need to calculate all of the above to define the duty and mechanical design of your system. It is indeed very tricky, but it is essential.

Some typical example : http://www.cambridge.org/us/engi ... s/EXAMPLE_6.2-3.pdf

chan1996

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