1.2 HEAT TRANSFER IN FOOD PROCESSING .1 M ODES OF H EAT T RANSFER.1 MODESOF HEAT TRANSFER
1.2.3 C ONVECTIVE H EAT T RANSFER
Convection uses the movement of fl uids to transfer heat. For example, when cold air fl ows over the warm surface of food in a refrigerator, the cold air close to the surface is heated as it comes in contact with the surface. As the air fl ows, the air heated at the surface mixes and exchanges heat with the free stream air. The movement, which causes heat transfer, may occur in a natural or forced form.
Natural convection creates the fl uid movement by the difference between fl uid densities due to the temperature difference. Forced convection uses external means such as agitators, pumps, and fans to produce fl uid movement. Whenever a fl uid moves past a solid surface, it may be observed that the fl uid velocity varies from zero adjacent to the solid surface to a free fl uid velocity at some distance away from the surface.
Convective heat transfer is the major mode of heat transfer between the surface of a solid material and the surrounding fl uid. For analyzing heat transfer by convec-tion, a boundary layer is normally assumed near the surface of the solid material as shown in Figure 1.1b. Heat is transferred by conduction through this layer. The layer contains almost all of the resistance to heat transfer because of the relatively low thermal conductivity of the layer and rapid heat transfer from the outer edge of the boundary layer into the bulk of the fl uid. Using the boundary layer concept, the rate of convective heat transfer may be written as
s s
(T T ) (T T )
q kA A
k
∞ ∞
− −
= =
d d (1.14)
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where
k and d are the thermal conductivity and thickness of the boundary layer, respectively
A is the boundary surface area
Ts and T∞ are the surface temperature and the temperature of bulk fl uids, respectively
However, as the thickness of the boundary layer, d, can neither be predicted nor measured easily, the thermal resistance of the boundary layer cannot be determined.
d/k is thus replaced with the term 1/hc, in which hc is a convective heat transfer coef-fi cient. Equation 1.14 can then be rewritten as
c ( s )
q h A T= −T∞ (1.15)
The convective heat transfer coeffi cient is a function of the fl uid velocity, the differ-ence in temperature between the fl uid and the solid surface, the orientation of the surface, the roughness of the surface, and the properties of the fl uid. The values of convective heat transfer are usually determined experimentally and they can also be calculated mathematically from empirical formulas. The typical values of convec-tive heat transfer are given in Table 1.3 (Singh and Heldman, 2001).
In many heating/cooling applications, conductive and convective heat transfer may occur simultaneously. An example is heat loss through the wall of a cold storage room that circulates cold air at a temperature lower than the temperature of the envi-ronment surrounding the outside of the room. In this case, heat must be transferred fi rst from the surrounding environment to the outer surface of the room by natural or free convection, by conduction through the wall material from the outer surface to the inside surface of the wall, and fi nally by forced convection from the inside surface of the wall to the inside cold air. The heat transfer is thus realized through three layers in series. The total thermal resistance in the pathway of heat transfer is
t i w o
R =R +R +R (1.16)
TABLE 1.3
Some Approximate Values of Convective Heat Transfer Coeffi cients
Fluid
Convective Heat Transfer Coeffi cient (W/m2 °C)
Air natural convection 5–25
Air forced convection 10–200
Water natural convection 20–100
Water forced convection 50–10,000
Boiling water 3,000–100,000
Condensing water vapor 5,000–100,000
Source: From Singh, R.P. and Heldman, D.R., in Introduction to Food Engineering, Academic Press, San Diego, 2001. With permission.
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where
i i i
R 1
= h A (1.17)
w a
R x kA
= ∆ (1.18)
o o o
R 1
=h A (1.19)
In the above equations,
hi and ho are the inside and outside convective heat transfer coeffi cients, respectively
Ai, Ao, and Aa are inside, outside, and average surface areas, respectively Dx is the thickness of the wall
k is the thermal conductivity of wall material
Ri, Rw, and Ro are the thermal resistances to the inside convection, the wall con-duction, and the outside convection, respectively
Example 1.4
A 0.5 cm thick steel pipe with 5 cm inside diameter is used to convey steam from a boiler to a process equipment for a distance of 50 m. The inside steam temperature is 115°C and the outside ambient temperature is 30°C. The inside and outside convective heat transfer coeffi cients are 100 W/m2 °C and 10 W/m°C, respectively.
The thermal conductivity of steel is 42 W/m°C. Calculate the total heat loss to the surroundings under a steady-state condition.
Solution 1.4
Using Equations 1.17 through 1.19, the thermal resistances of the inside convective layer, the steel pipe wall, and the outside convective layer are
= = = × −
⎛ ⎞
× π × ×⎜⎝ ⎟⎠ ×
o 3 i
i i 2 o
1 1
1.27 10 C/W
100 [W/m C] 5[cm] 1[m] 50 [m]
100 [cm]
R h A
( ) ( )
−= = = ×
π π × ×
o i 5 o
w o
ln ln 0.03 0.025
1.38 10 C/W
2 2 50 [m] 42[W/m C]
R r r Lk
= = = × −
⎛ ⎞
× π × ×⎜⎝ ⎟⎠ ×
o 2
2 o o
o o
1 1
1.06 10 C/W
10 [W/m C] 6[cm] 1[m] 50 [m]
100 [cm]
R h A
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Therefore, the overall thermal resistance is
− − − −
= + + = × 3+ × 5+ × 2= × 2 o
t i w o 1.27 10 1.38 10 1.06 10 1.19 10 C/W
R R R R
From Equation 1.10, we can obtain the overall heat transfer rate at
−
∆ −
= − = − = −
×
o o
2 o
115 C 30 C
7143 W 1.19 10 C/W
q T R
From the above equation, we know that the outside convection is the dominant contribution to the overall thermal resistance. The wall conduction makes a negli-gible contribution to the overall thermal resistance. In order to decrease the overall heat transfer rate, insulation should be considered to increase the thermal resis-tance of the wall conduction. However, in order to increase the overall heat transfer rate, the outside convection should be enhanced to reduce its thermal resistance.
For convenience in calculation, the concept of overall heat transfer coeffi cient, Ui or Uo (based on inside area or outside area), is frequently used. The overall heat transfer coeffi cient based on inside area is defi ned as (Singh and Heldman, 2001)
o i
i w o
i i i i o o
1 1 ln 1
2 r
r
R R R
U A h A Lk h A
⎛ ⎞
⎝ ⎠
= + + = + +
π (1.20)
or o
i i
i
i i o o
1 1 ln
r r r r
U h k h r
⎛ ⎞
⎝ ⎠
= + + (1.21)
The overall heat transfer rate is thus
i i
q=U A T∆ (1.22)
The selection of the area over which to calculate the overall heat transfer is arbitrary.
The overall heat transfer coeffi cient based on outside area is defi ned as
o i
i w o
o o i i o o
1 1 ln 1
2 r
r
R R R
U A h A Lk h A
⎛ ⎞
⎝ ⎠
= + + = + +
π (1.23)
or
o
o i
o
o i i o
1 ln 1
r
r r
r
U h r k h
⎛ ⎞
⎝ ⎠
= + + (1.24)
The overall heat transfer rate is thus
o o
q=U A∆T (1.25)
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Example 1.5
What is the overall heat transfer coeffi cient based on inside area and outside area for Example 1.4?
Solution 1.5
From Example 1.4, we have
− °
= × 3
i 1.27 10 C/W R
− °
= × 5
w 1.38 10 C/W R
− °
= × 2
o 1.06 10 C/W R
Therefore, the overall heat transfer coeffi cient based on inside area is
− − − °
= +i w+ o= × 3+ × 5+ × 2=
i i
1 R R R 1.27 10 1.38 10 1.06 10 0.0119 C/W
U A
= 2°
i 10.70 W/m C U
Similarly, the overall heat transfer coeffi cient based on outside area is
= 2°
o 8.92 W/m C U