Expressions for surface heat transfer coefficients

The flow is generated by external forces and the criteria for laminar flow is

Pr > 0.6 Re < 2300

(6.8)

The characteristic length for calculation of the Reynolds number is the hydraulic diameter of the duct which for non-circular geometries can be calculated as four times the section area A divided by the length of the perimeter P of the interior duct section.

d

_h

 4 A

P

(6.9)

It follows that the hydraulic diameter for a rectangular duct with sides a and b will be

d

_h

 4 a  b

2  a  b 

_(6.10)

and if a>>b the hydraulic diameter will become 2 b. This condition is typical for ventilated air gaps in the exterior part of insulated constructions.

The following expression gives the average Nusselt number along the surface.

Nu  3.65  0.067(Re

^.

Pr

^.

d

_h

/ L)

(1 0.045(Re

^.

Pr

^.

d

_h

/ L)

^{2 / 3}

)

(6.11) L is the length of the duct in the direction of the flow, m.

6.3.2 Forced turbulent flow of air in a duct

Figure 6.2

0.7 < Pr < 100 l/d > 60

Re > 10000

The following expression is valid for a duct with smooth surfaces with constant surface temperature. The characteristic length for calculation of the Reynolds number is the hydraulic diameter of the duct which for non-circular geometries can be calculated as four times the section area divided by the length of the perimeter of the interior duct section.

Nu = 0.023

^.

Re

^0.8

Pr

ⁿ

= h

c

·d

h

/   

^(6.12)

n=0.4 if the surface is warmer than the air n=0.3 if the surface is colder than the air

The expression can for air also be used approximately in the interval 2300 < Re < 10000 if n is set equal to 0.4.

6.3.3 Forced flow along flat surfaces

Figure 6.3

l u d

l u

The convective surface heat transfer at the exterior surfaces of outer walls and the roof is usually governed by the wind generated air flow along the surfaces. The characteristic length here is the length of the surface in the direction of the air flow at the surface i.e.

the length or width of a roof or a wall. A complication here is that the air velocity at different locations around the house will differ substantially from the free wind speed observed at a distance from the building. Calculations of this types are however seldom carried out to get exact values for the heat transfer coefficient at the exterior surfaces of a building but rather to study the influence of different parameters.

6.3.4 Forced laminar flow along a flat surface

The conditions for laminar flow are as follows

0.6 < Pr < 2000 and Re < 10

⁵ (6.13) And the expression for the Nusselt number becomes:

Nu = 0.664 Re

^1/2

Pr

^1/3

=( h

c

·l)/   

^(6.14)

l, which is the length of the surface in the direction of the flow, m, is also the characteristic length to be used in the calculation of the Reynolds number.



6.3.5 Forced convection with turbulent flow along a flat surface The criteria for turbulent flow along a flat surface are as follows:

0.6 < Pr

5

^.

10

⁵

< Re < 10

⁷

Nu  0.037  Re

^0.8

Pr

1 2.443

^.

Re

^0.1

(Pr

^{2 / 3}

 1)  h

_c^.

l



_(6.15)

Expression (6.15) is mostly used to estimate the dependency of the convective surface heat transfer coefficient on the convective surface heat transfer coefficients on the exterior surfaces of buildings. Using those equations one has to bear in mind that the air velocity around buildings usually is different from the meteorological wind and there are many elements around the building that can disturb the air flow.

If a building is 12 m wide Reynolds number will exceed 5^.10⁵ at 0.5 m/s and 10⁷ at 10 m/s. The average wind velocity in Stockholm is about 3 m/s.

6.3.6 Natural convection on room surfaces

T T

+ -

Figure 6.4

The following equations apply for the natural convection surface heat transfer coefficients on room surfaces i.e. floors, walls and ceilings. The characteristic length used in the calculation of the Grashof and Nusselt numbers is usually the height of the wall or the ratio Area/Perimeter of the floor/ceiling, which is being considered. All cases can be expressed with the same basic equation (6.20) where A and B have different values for different situations.

Nu  A(Gr Pr)

^B

 h

_c

 l

   

^(6.16)

For laminar flow B=1/4 and for turbulent flow B=1/3.

6.3.6.1 Vertical walls

For vertical walls we can use the same expression independent on whether the wall is colder or warmer than the room air. The mode of flow is determined from the Grashof number and the characteristic length to be used in the calculation of the Grashof number is the wall height.

Laminar flow (Gr^.Pr<10⁹) A=0.59 B=1/4 (6.17) Turbulent flow (Gr^.Pr>10⁹) A=0.13 B=1/3 (6.18) 6.3.6.2 Horizontal surfaces

For horizontal surfaces the heat transfer will depend not only on the temperature difference but also on the thermal stability at the surface. For a ceiling colder than the room air the density of the air at the surface will be higher than below which will generate turbulent air movements at relatively low Grashof numbers. Similar instability will appear at warm floors where the density of the air in the vicinity of the surface is lower than for the room air. The characteristic length will be the floor or ceiling area divided by the perimeter of the floor or the ceiling. As an example consider a room

3×6×2.4 m. The floor area is 18 m² and the perimeter is 18 m which will give the characteristic length 1 m.

For a relatively warm floor or a cold ceiling:

Laminar flow (Gr^.Pr< 2^.10⁷) A=0.54 B=1/4 (6.19) Turbulent flow (Gr^.Pr>2^.10⁷) A=0.15 B=1/3 (6.20) For a relatively cold floor or a warm ceiling we expect conditions to be stable up to high Grashof numbers.

Laminar flow (Gr^.Pr< 3^.10¹⁰) A=0.27 B=0.25 (6.21) 6.3.7 Natural air convection within an enclosure

It is not always easy to make a distinction between a room and an enclosure. By an enclosure we mean a space where the distance between the surfaces is so small that the convection flow generated at one surface affects the other. The most common examples are thin non-ventilated air layers in building constructions and the air gaps between the panes in a multi-glazed window.

6.3.7.1 Horisontal gap with upwards heat flow

The thickness of the gap d, m, is the characteristic length to be used in the calculation of the Grashof number

Figure 6.5 The following expressions are valid for air

4 /

195

1

.

0 Gr

Nu  

3.7^.10⁴<Gr<3.7^.10⁵ (6.22)

3 /

068

1

.

0 Gr

Nu  

3.7^.10⁵<Gr<3.7^.10⁷ (6.23)

And a more general expression for fluids is

Nu0.069



GrPr



^1/3Pr^0.074 1.5^.10⁵<Gr^.Pr<10⁹ (6.24)

The heat transfer coefficient hcis in this case given from surface to surface. When Nu = 1 the heat transfer coefficient is given as

q T

T+

-d

h

c

= /d   

^(6.25)

which means that the air in the gap is standing still and the heat transfer in the air is taking place by conduction only.

6.3.7.2 Vertical gap with limited height H and horizontal heat flow

The thickness of the gap, d, m , is the characteristic length to be used in the calculation of the Grashof number

2·10

⁴

< Gr < 2·10

⁵

H>3d

Nu = 0.18·Gr

^1/4

(H/d)

^–1/9 (6.26)

2 .

10

⁵

< Gr < 2·10

⁷

H>3d

Nu = 0.065·Gr

^1/3

(H/d)

^–1/9 (6.27)

Expressions (6.25) and (6.26) are used for instance for the calculation of the convective heat transfer between the panes of a multi-glazed window.

The heat transfer coefficient is given as

h

_c

 Nu  

d

(6.28)

and the average density of heat flow rate between the surfaces can be calculated as

q  h

_c

  T

_

 T

_



_(6.29

₎

At high Grashof numbers the air in the cavity will start rotating due to the density differences and we can assume that we have downward flow of air on the cold side and upward flow of air on the warm side. This means that there will be a temperature and heat flow gradient along the surfaces.

In document Building Physics (Pldal 50-55)