Fusion Method Forward Selection Model

Determination of the rate of heat transfer due to convection is complicated because of the presence of fluid motion. However there is a useful procedure called the empirical approach which shall be adopted in this work in the determination of the rate of convective heat transfer. The only drawback of this approach is that it requires large experimental data input. However that could be avoided by the use of appropriate dimensionless numbers as suggested by Singh and Heldman (2001) and adopted in

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this work. The relevant dimensionless numbers are Reynold Number, Nusselt Number and Prantl number.

The Reynolds number provides an indication of the inertial and viscous forces present in a fluid. Reynolds Number is calculated as follows:

𝑁_𝑅𝑒 = ^{𝜌𝑢 𝐷}

𝜇 = ^4𝑚

𝜇𝜋𝐷 (3.108)

where 𝜌 = fluid density; 𝑢 = fluid velocity; 𝐷 = pipe diameter; 𝜇 =fluid viscosity and 𝑚 = fluid mass flowrate.

The second required dimensionless number is the Nusselt number which is the dimensionless form of convective heat transfer coefficient, 𝑕. Nusselt number may be considered as the enhancement in the rate of heat transfer caused by convection over the conduction mode. The Nusselt number is calculated as follows:

𝑁_𝑁𝑢 ≡^𝑕𝑑𝑐

𝑘 (3.109)

where 𝑕 = convective heat transfer coefficient, 𝑑_𝑐= inside diameter of the pipe and 𝑘

= thermal conductivity

The third required dimensionless number for this analysis is the Prantl number which describes the thickness of the hydrodynamic boundary layer compared with the thermal boundary layer. It is essentially the ratio between molecular diffusivity of momentum to the molecular diffusivity of heat. Prantl number is calculated with the following expression:

𝑁_𝑃𝑟 = ^𝜇𝐶𝑝

𝑘 (3.110)

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The basis of this analysis using the dimensionless numbers is the relationship established between the dimensionless numbers and given as:

𝑁_𝑁𝑢 = 𝐶𝑁_𝑅𝑒^𝑚𝑁_𝑃𝑟^𝑛 (3.111)

where C, m and n are constants.

By substituting experimentally obtained constants into the equation above, we obtain empirical correlation specific to a given condition. Previous works have determined the experimental correlations for a variety of operating conditions such as flow in a pipe, flow over a pipe or over a sphere. Different relations are obtained depending on whether the flow is laminar or turbulent.

In all, this work will adopt the steps outlined in Singh and Heldman (2001) in the determination of convective heat transfer coefficient using empirical correlations as follows;

 Identify flow geometry

 Identify fluid and determine its properties

 Calculate Reynolds number

 Select an appropriate empirical correlation

 Calculate Nusselt number

 Calculate convective heat transfer coefficient Step 1: Identify flow geometry

The flow situation for this work is that heated air is pumped through a pipe and particles are introduced into the air stream. The point of focus is the heat transfer between the particles introduced into the air stream and the body of fluid flowing through the pipe. Essentially we have flow over a spherical body (the particles of TMe 419)

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Step 2: Identify fluid and determine its properties

Air is a mixture of several constituent gases. The composition of air varies slightly depending on the geographical location and altitude. For scientific purposes, the commonly acceptable composition is referred to as standard air. The composition of standard air as used in this work and as suggested in http://www.grc.nasa.gov/WWW/K-12/airplane/airprop.html is given in table 3.1.

Table 3.1: Composition of Standard Air

Constituents Percentage by Volume

Nitrogen 78.084000

Oxygen 20.947600

Argon 0.934000

Carbon dioxide 0.031400

Neon 0.001818

Helium 0.000524

Other gases 0.000658

100.000000

In addition to the physical composition of air, the thermodynamic properties have to be determined for drying air fed into the flash tube. It has also been emphasised that the properties of air varies over time and so some form of average has to be used for the purposes of this analysis. The air properties are taken as suggested by http://www.jazminesmeralda.ifunnyblog.com/averageweatherfornigeria for the yearly average conditions for Nigeria:

Dry bulb temperature = 25.8°C = 298.95K

Wet bulb temperature = 24.351°C = 297.501K

Dew point: = 23.854°C = 297.004K

Humidity = 50%

Air moisture concentration (c0_air) = 0.0135*rho_air (appendix 4-3) Specific moisture capacity (C_m_air) =

The air is driven through the heat exchanger where it is heated up. Heating or cooling of air is accomplished without addition or removal of moisture (Singh and Heldman,

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2001). Thus the humidity ratio remains constant. Consequently the air properties as it exits the heat exchanger and enters the flash tube will be as follows:

Dry bulb temperature = 160°C = 433.15K

Wet bulb temperature = 134.041 °C = 407.191K (appendix 4-4)

Dew point: = 133.433°C = 406.583K

Humidity: = 50%

Air moisture concentration = 0.0135*rho_air (appendix 4-3)

The phenomenon of adiabatic saturation of air is applicable to convective drying of food materials. The adiabatic saturation process can be visualised by considering a well-insulated chamber with an inlet and an outlet. The chamber prevents the gain or loss of heat to the surrounding (adiabatic conditions). Air enters the chamber and blows over water inside the chamber and exits through the outlet. In the process, part of the sensible heat of the entering air is transformed into latent heat.

For the condition described above, the process of evaporating water into the air results in saturation by converting part of the sensible heat of the air into latent heat in the process referred to as adiabatic saturation.

When heated air is forced through a bed of moist granular food, the drying process can be described on the psychrometric chart as an adiabatic saturation process. The heat of evaporation required to dry the product is supplied only by the drying air; no transfer of heat occurs due to conduction or radiation from the surroundings. As air passes through the granular mass, a major part of the sensible heat is converted to latent heat, as water is held in the air as vapour.

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(Source: http://www.f150forum.com/f70/ecoboost-condensate-drain-hole-post-your- results-here-223824/index26/)

Fig. 3.8: Psychrometric Chart

The drying process happens at constant enthalpy and the two points on the red line shows the state of the drying air at the beginning and at the end of the convective drying. The use of psychrometric chart though simple shall not be adopted for this work, but the relevant correlations shall be used to suit the modelling process.

Step 3: Calculate Reynolds number

In pneumatic conveying drying, the air is driven by a blower through a heat exchanger, so that it is heated up and has improved capacity to retain moisture, before it is then introduced into the flash tube. To understand the nature and properties of the

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air that is entering the flash tube through the heat exchanger the air data have to be provided.

It should be noted that air inlet velocity shall be varied to determine its effect on the drying rate of cassava mash but the inlet velocity must be such that it is above the minimum carrying velocity determined experimentally. It is also important to note that nature of the flow, laminar or turbulent, for our situation is for a material that is being conveyed as it is dried. At entry the velocity of the solid particle is zero and the nature of the flow is determined based on the velocity of the gas phase only. But an instant later the flowing stream imparts momentum on the particle to cause it to move by transferring momentum to it; at that point the nature of the flow is determined by the difference in the velocity between the gas phase and the solid dispersed phase.

The program shall be designed to monitor the nature of flow and apply the appropriate correlations for determining the variable.

𝑅𝑒 =^𝑉𝐷𝐻

𝜈 = ^{𝜌𝑎𝑉𝑑𝑝}

𝜇𝑎 = 𝑅𝑒𝑝 = 𝜌_𝑔𝑑_𝑝 𝑢_𝑔 − 𝑢_𝑠 /𝜇_𝑔 (3.112) The drag coefficient is calculated by the relation suggested by Han T. et al (2000)

𝐶_𝐷 = ²⁴

𝑅𝑒𝑝 Rep ≤ 1 (3.113) 𝐶_𝐷 = ²⁴

𝑅𝑒_𝑝^0.646 1< Rep≤ 400 (3.114) 𝐶_𝐷 = 0.5 400 < Rep< 3 x 10⁵ (3.115)

𝐶_𝐷 = 0.5 (3.116)

Diffusivity of water vapour in air, 𝐷_{𝑤𝑣 ,𝑎} = 0.26 × 10⁻⁴ Step 4: Select the appropriate correlation

𝑆𝑐 = ^𝜈

𝐷𝑤𝑣 ,𝑎 (3.117)

` For 𝑅𝑒 < 5 × 10⁵- (laminar flow)

93 Step5: Calculate Nusselt Number:

For flow past a single sphere, when th single sphere may be heated or cooled, the Nusselt number is evaluated as follows:

𝑁_𝑁𝑢 = 2 + 0.60𝑅𝑒^0.5𝑃𝑟^1/3 For 1<Re<70000 ; 0.6<Pr<400

𝑆𝑕 = ^𝑕𝑚𝐿

𝐷_{𝑤𝑣 ,𝑎} = 𝐿 = 𝑑_𝑝 (3.118)

𝑕_𝑚 = ^{0.664𝑅𝑒1 2𝑆𝑐1 3𝐷𝑤𝑣 ,𝑎}

𝐿 (3.119)

𝜒 = sphericity =^{𝐴𝑠𝑜𝜌𝑠𝑎𝑑𝑝}

6 (3.120)

During constant-rate drying period, the product surface temperature remains at the wet bulb temperature of the heated air. The magnitude of water vapour transfer 𝑚 , during constant-rate drying is described by the following mass transfer expression (Singh and Heldman, 2001):

𝑚 =^{𝑕𝑚𝜒𝜋 𝑑𝑝}

2𝑀𝑤𝑃

0.622𝑅𝑇𝑔 𝑊_𝑠− 𝑊_𝑔 (3.121) 𝑅_𝑔 = gas constant = 8.314 m³ KPa/(kg mol)

𝑀_𝑤 = molecular weight of water vapour =18 kg/kg mol 𝑃 = atmospheric pressure (kPa) = 101.325

𝑊₁= humidity ratio at product surface (kg water/kg dry air) = humidity ratio for saturated air at 𝑇_𝑠.

The maximum amount of water vapour in the air is achieved when 𝑝_𝑤 = 𝑝_𝑤𝑠 the saturation pressure of water vapour. At the actual temperature the following expression can be used:

𝑊₁ = 0.62198𝑝_𝑤𝑠/ 𝑝_𝑎 − 𝑝_𝑤𝑠 (3.122) where

𝑊₁ = specific humidity at saturation (kgwater/kg_air) 𝑝_𝑤𝑠 = saturation pressure of water vapour 𝑝_𝑎= atmospheric pressure of moist air

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𝑊₂ = humidity ratio for air (kg water/kg dry air) = humidity ratio at 𝑇_𝑔 and at air relative humidity

Alternatively, 𝑚 can be determined by the method of partial pressures 𝑒 = Euler number = 2.718281828459

𝑝_𝑣𝑜= saturated H2O vapour pressure at 𝑇_𝑠(𝐾) = e (77.3450+0.0057T-7235/T)

/T^{8.2 =} 3.13030 kPa

𝑝_𝑤 = 𝑝_𝑤𝑏 − ^{𝑝𝐵−𝑝𝑤𝑏 𝑇𝑑𝑏−𝑇𝑤𝑏}

1555 .56−0.722𝑇_𝑠 (3.123)

𝑝_𝑤𝑏=𝑝_𝑣𝑜 water vapour saturation pressure at 𝑇_𝑠 (kPa) 𝑝_𝐵= barometric pressure (kPa) =

𝑇_𝑑𝑏= dry bulb temperature (°C) 𝑇_𝑤𝑏= wet bulb temperature (°C)

𝑀_𝑤= molecular weight of water vapour =18 kg/kg mol ℜ = universal gas constant = 8.314 m³ kPa/(kg mol K) 𝑝_𝑣𝑔 = 𝑝_𝑤= partial pressure of water at 𝑇_𝑑𝑏(kPa) 𝑚 _𝑠 = 𝑕_𝑚𝜒𝜋𝑑_𝑝^{2 𝑀𝑤𝑝𝑣𝑜}

ℜ𝑇𝑠 −^{𝑀𝑤𝑝𝑣𝑔}

ℜ𝑇𝑔 (kg/s)

In document Fusing Small-footprint Waveform LiDAR and Hyperspectral Data for Canopy-level Species Classification and Herbaceous Biomass Modeling in Savanna Ecosystems (Page 48-57)