Convection coefficients will be studied under two sections, firstly, natural convection in which movements occur due to density differences on heating or cooling; and secondly, forced convection, in which an external source of energy is applied to create movement. In many practical cases, both mechanisms occur together.
Heat transfer by natural convection occurs when a fluid is in contact with a surface hotter or colder than itself. As the fluid is heated or cooled it changes its density. This difference in density causes movement in the fluid that has been heated or cooled and causes the heat transfer to continue.
There are many examples of natural convection in the food industry. Convection is significant when hot surfaces, such as retorts which may be vertical or horizontal cylinders, are exposed with or without insulation to colder ambient air. It occurs when food is placed inside a chiller or freezer store in which circulation is not assisted by fans. Convection is important when material is placed in ovens without fans and afterwards when the cooked material is removed to cool in air.
It has been found that natural convection rates depend upon the physical constants of the fluid, density r, viscosity m, thermal conductivity k, specific heat at constant pressure cp and coefficient of thermal expansion b (beta) which for gases = l/T by Charles' Law. Other factors that also affect convection-heat transfer are, some linear dimension of the system, diameter D or length L, a temperature difference term, DT, and the gravitational acceleration g since it is density differences acted upon by gravity that create circulation. Heat transfer rates are expressed in terms of a convection heat transfer coefficient hc, which is part of the general surface coefficient hs, in eqn. (5.5).
Experimentally, if has been shown that convection heat transfer can be described in terms of these factors grouped in dimensionless numbers which are known by the names of eminent workers in this field:
and in some cases a length ratio (L/D).
we assume that these ratios can be related by a simple power function
we can then write the most general equation for natural convection:
Experimental work has evaluated K, k, m, n, under various conditions. For a discussion, see for example McAdams ( 1954) . Once K, k, m, n, are known for a particular case, together with the appropriate physical characteristics of the fluid, the Nusselt number can be calculated. From the Nusselt number we can find hc and so determine the rate of convection-heat transfer by applying eqn. (5.5). In natural convection equations, the values of the physical constants of the fluid are taken at the mean temperature between the surface and the bulk fluid. The Nusselt and Biot numbers look similar: they differ in that for Nusselt, k and h both refer to the fluid, for Biot k is in the solid and h is in the fluid.
These are related to a characteristic dimension of the body (food material for example) being considered, and typically this is a length for rectangular bodies and a diameter for spherical/cylindrical ones.
(1) Natural convection about vertical cylinders and planes, such as vertical retorts and oven walls
For air these equations can be approximated respectively by:
Equations (5.17) and (5.18) are dimensional equations and are in standard units (DT in °C and L (or D) in metres and hc in J m-2 s-1 °C-1). The characteristic dimension to be used in the calculation of (Nu) and (Gr) in these equations is the height of the plane or cylinder.
(2) Natural convection about horizontal cylinders such as a steam pipe or sausages lying on a rack
equations can be employed in the case of air, which is so often encountered
in contact with hotter or colder foods giving again:
(3) Natural convection from horizontal planes, such as slabs of cake cooling
The corresponding cylinder equations may be used, employing the length of the plane instead of the diameter of the cylinder whenever D occurs in (Nu) and (Gr). In the case of horizontal planes, cooled when facing upwards, or heated when facing downwards, which appear to be working against natural convection circulation, it has been found that half of the value of hc in eqns. (5.19) - (5.21) corresponds reasonably well with the experimental results.
Note carefully that the simplified equations are dimensional. Temperatures must be in °C and lengths in m and then hc will be in J m-2 s-1 °C-1. Values for s, k and m are measured at the film temperature, which is midway between the surface temperature and the temperature of the bulk liquid.
First it is necessary
to establish the value of (Pr.Gr).
(Pr.Gr) = (cpm
= [(1.2)3 x (1.12)2 x 9.81 x 32 x 1.0 x 103 )/(308 x 1.9 x 10-5 x 0.025)
= 5 x 109
Therefore eqn. (5.18) is applicable.
Total area of vessel
wall = pDL
x 0.9 x 1.2 = 3.4 m2
Therefore heat loss rate = hc A(T1 - T2)
= 4.3 x 3.4 x 32
When a fluid is forced past a solid body and heat is transferred between the fluid and the body, this is called forced convection heat transfer. Examples in the food industry are in the forced-convection ovens for baking bread, in blast and fluidized freezing, in ice-cream hardening rooms, in agitated retorts, in meat chillers. In all of these, foodstuffs of various geometrical shapes are heated or cooled by a surrounding fluid, which is moved relative to them by external means.
The fluid is constantly being replaced, and the rates of heat transfer are, therefore, higher than for natural convection. Also, as might be expected, the higher the velocity of the fluid the higher the rate of heat transfer. In the case of low velocities, where rates of natural convection heat transfer are comparable to those of forced convection heat transfer, the Grashof number is still significant. But in general the influence of natural circulation, depending as it does on coefficients of thermal expansion and on the gravitational acceleration, is replaced by dependence on circulation velocities and the Reynold’s number.
As with natural convection, the results are substantially based on experiment and are grouped to deal with various commonly met situations such as fluids flowing in pipes, outside pipes, etc.
(1) Heating and cooling inside tubes, generally fluid foods being pumped through pipes
In cases of moderate temperature differences and where tubes are reasonably long, for laminar flow it is found that:
and where turbulence is developed for (Re) > 2100 and (Pr) > 0.5
For more viscous liquids, such as oils and syrups, the surface heat transfer will be affected, depending upon whether the fluid is heating or being cooled. Under these cases, the viscosity effect can be allowed, for (Re) > 10,000, by using the equation:
In both cases, the fluid properties are those of the bulk fluid except for ms, which is the viscosity of the fluid at the temperature of the tube surface.
Since (Pr) varies little for gases, either between gases or with temperature, it can be taken as 0.75 and eqn. (5.23) simplifies for gases to:
In this equation the viscosity ratio is assumed to have no effect and all quantities are evaluated at the bulk gas temperature. For other factors constant, this becomes hc = k' v0.8, as in equation (5.28)
(2) Heating or cooling over plane surfaces
Many instances of foods approximate to plane surfaces, such as cartons of meat or ice cream or slabs of cheese. For a plane surface, the problem of characterizing the flow arises, as it is no longer obvious what length to choose for the Reynolds number. It has been found, however, that experimental data correlate quite well if the length of the plate measured in the direction of the flow is taken for D in the Reynolds number and the recommended equation is:
For the flow of air over flat surfaces simplified equations are:
(3) Heating and cooling outside tubes
examples in food processing are water chillers, chilling sausages, processing
The powers n and m vary with the Reynolds number. Values for D in (Re) are again a difficulty and the diameter of the tube, over which the flow occurs, is used. It should be noted that in this case the same values of (Re) cannot be used to denote streamline or turbulent conditions as for fluids flowing inside pipes.
For gases and for liquids at high or moderate Reynolds numbers:
whereas for liquids at low Reynolds numbers, 1 < (Re) < 200:
As in eqn. (5.23), (Pr) for gases is nearly constant so that simplified equations can be written. Fluid properties in these forced convection equations are evaluated at the mean film temperature, which is the arithmetic mean temperature between the temperature of the tube walls and the temperature of the bulk fluid.
Mean film temperature = (74 + 24)/2 = 49°C.
Properties of water
at 49°C are:
(Re) = (Dvr/µ)
(Pr) = (cp m/k)
(Nu) = (hcD/k)
(Re) = (Lvr/m)
Pr = (cpm/k)
(hcL/k) = 0.036(2.34 x 104)0.815.30.33
Assuming the vessel
is equivalent to a flat plate then from eqn. (5.27)
= 5.7 + (3.9 x 61)/60
= 9.7 J m-2 s-1°C-1
So with A = 3.4m2, DT = 32°C,
q = 9.7 x 3.4 x 32
= 1055 J s-1
Heat-Transfer Theory > OVERALL HEAT TRANSFER COEFFICIENTS
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