Qualitative Major Findings and Framework Refinement

47 then equation (3.41) becomes

𝑃₂ = 𝑃₁−^{𝑢𝑔𝑆𝑚𝑎𝑠𝑠}

𝐴 𝑥 −^{𝑃 𝜏𝑤}

𝐴 𝑥 − 𝛼_𝑔𝜌_𝑔𝑔_𝑥𝑥 +^{𝑆𝑚𝑜𝑚}

𝐴 𝑥 (3.42)

48

Equation (3.36) will provide the pressure at discrete points along the flash tube and this can be used to predict the flow velocity profile of the continuous phase and using the basic thermodynamic equation;

𝑃₁𝑉 ₁ 𝑇1 =^{𝑃2𝑉 2}

𝑇2 (3.46)

where , 𝑉 and 𝑇 are pressure, volumetric flow rate, and temperature and for a pipe section with uniform cross sectional area:

𝑢_𝑔2= ^{𝑃1𝑢𝑔1𝑇2}

𝑃2𝑇1 (3.47)

Where, 𝑉 = 𝐴𝑢_𝑔 (A = pipe cross sectional area).

It should be emphasized that absolute values of both pressure and temperature must always be used in these equations. Most data for these values, such as that for minimum conveying air velocity are generally determined experimentally or from operating experience. It is for the purposes of this work, important to take the presence of the particles into account because in accelerating the material at zero velocity at the feed point to some value along the flow line requires momentum exchange between the particles and the continuous phase.

In dilute phase conveying, with particles in suspension in the air, the mechanism of conveying is one of drag force. The velocity of the particles, therefore, will be lower than that of the conveying air. It is a difficult and complex process to measure particle velocity, and apart from research purposes, particle velocity is rarely measured. Once again it is generally only the velocity of the air that is ever referred to in pneumatic conveying.

In a horizontal pipeline the velocity of the particles will typically be about 80% of that of the air. This is usually expressed in terms of a slip ratio, defined in terms of the

49

velocity of the particles divided by the velocity of the air transporting the particles, and in this case it would be 0.8. The value depends upon the particle size, shape and density, and so the value can vary over an extremely wide range. In vertically upward flow in a pipeline a typical value of the slip ratio will be about 0.7.

These values relate to steady flow conditions in pipelines remote from the point at which the material is fed into the pipeline, bends in the pipeline and other possible flow disturbances and shall be used as a ball pack check on the result of analytical methods. At the point at which the material is fed into the pipeline, the material will essentially have zero velocity. The material will then be accelerated by the conveying air to its slip velocity value. This process will require a significant pipeline length and this is referred to as the acceleration length. The actual distance will depend once again on particle size, shape and density.

There is a pressure drop associated with acceleration of the particles in the air stream and it has to be taken into account by some means. It is not only at the material feed point that there is an acceleration pressure drop. It is likely to occur at all bends in the pipeline. In traversing a bend the particles will generally make impact with the

bend wall and so be retarded. The slip velocity at exit from a bend will be lower than that at inlet and so the particles will have to be re-accelerated back to their steady-state value. This additional element of the pressure drop is usually incorporated in the overall loss associated with a bend.

The momentum coupling source term (per unit volume) due to the reverse effect of particles can be expressed as suggested by Hamed M. H (2005):

𝑆_𝑚𝑜𝑚 = −𝑁_𝑝¹

2𝐶_𝐷^𝜋𝑑𝑝

2

4 𝜌_𝑝𝑥 𝑢_𝑔− 𝑢_𝑠 𝑢_𝑔 − 𝑢_𝑠 (3.48)

50 3.3.5 Energy Equation

In writing the energy equations for a multi phase flow, it is necessary to construct an energy equation for each of the phases or components. First total energy density (per unit mass) 𝑒_𝑁^∗ is defined for each component such that

𝑒_𝑁^∗ = 𝑒_𝑁 +¹

2𝑢_𝑁𝑖𝑢_𝑁𝑖 + 𝑔𝑥 (3.49)

Then the appropriate statement of the first law of thermodynamics for each phase becomes:

Rate of heat addition to N from outside control volume, 𝑄_𝑁 + Rate of work done to N by the exterior surroundings, 𝑊𝐴_𝑁 + Heat transfer to N within the control volume, 𝑄𝐼_𝑁

+ Rate of work done to N by the other component in the control volume, 𝑊𝐼_𝑁

= Rate of increase of total kinetic energy of N in control volume

+ Net flux of internal energy of N out of the control volume (3.50)

The second term on the RHS of equation (3.50) contains two contributions: (i) minus the rate of work done by the stress acting on the component of N on the surface of the control volume and (ii) the rate of external shaft work, 𝑊_𝑁, done on the component N.

In evaluating the first of these, the same modifications to the control volume as we did for the momentum equation are made; specifically small deformations is made to the control volume so that its boundaries lie wholly within the continuous phase.

Then using continuous phase stress tensor,𝜍_𝑔𝑖𝑗, as defined earlier the expression for 𝑊𝐴_𝑁 becomes:

𝑊𝐴_𝑔 = 𝑊_𝑔+ ^𝜕

𝜕𝑥_𝑗 𝑢_𝑔𝑖𝜍_𝑔𝑖𝑗 (3.51)

51 And

𝑊𝐴_𝑠 = 𝑊_𝑁 (3.52) Also the last two terms of equation (3.3.60) can be written as

^𝜕

𝜕𝑡 𝜌_𝑔𝛼_𝑔𝑒_𝑁𝑔^∗ + ^𝜕

𝜕𝑥_𝑖 𝜌_𝑔𝛼_𝑔𝑒_𝑔^∗𝑢_𝑔𝑖 (3.53) Then the energy equation can be written as:

𝜕

𝜕𝑡 𝜌_𝑔𝛼_𝑔𝑒_𝑔^∗ + ^𝜕

𝜕𝑥_𝑖 𝜌_𝑔𝛼_𝑔𝑒_𝑔^∗𝑢_𝑔𝑖 = 𝑄_𝑔− 𝑊_𝑔 + 𝑄𝐼_𝑔+ 𝑊𝐼_𝑔+ 𝛿_𝑔 ^𝜕

𝜕𝑥_𝑗 𝑢_𝑔𝑖𝜍_𝑔𝑖𝑗 (3.54) Note that the two terms involving internal exchange of energy between the phases may be combined into an energy interaction term given by

𝑆_{𝑒𝑛𝑒𝑟𝑔𝑦 ,𝑇} = 𝑄𝐼_𝑔+ 𝑊𝐼_𝑔 (3.55) It then follows that

𝑄𝐼_𝑔

𝑁

= 0

and

𝑊𝐼_𝑔

𝑁

= 0

and

𝑆_{𝑒𝑛𝑒𝑟𝑔𝑦 ,𝑇}

𝑁

= 0

Moreover, the work done terms, 𝑊𝐼_𝑁, may clearly be related to the interaction forces, 𝑆_𝑚𝑜𝑚. In a two phase flow with one dispersed phase:

𝑄𝐼_𝑔 = −𝑄𝐼_𝑠 , 𝑊𝐼_𝑔 = −𝑊𝐼_𝑠 = −𝑢_𝑠𝑖𝐹_𝑠𝑖 , 𝑆_{𝑒𝑛𝑒𝑟𝑔𝑦 ,𝑔} = −𝑆_{𝑒𝑛𝑒𝑟𝑔𝑦 ,𝑠} (3.56) When the left hand side of equation (3.54) are expanded and use is made of continuity equations and momentum equation , it results in the thermodynamic form of the energy equation. Using expression (3.54) and the relation

𝑒_𝑔 = 𝑐_𝑉𝑔𝑇_𝑔+ 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (3.57)

52

Between the internal energy, 𝑒_𝑔 , the specific heat capacity at constant volume, 𝑐_𝑉𝑔, and the temperature, 𝑇_𝑔, of the continuous phase, the energy equation can be written as

𝜌_𝑔𝛼_𝑔𝐶_𝑣𝑔 ^𝜕𝑇𝑔

𝜕𝑡 + 𝑢_𝑔^𝜕𝑇𝑔

𝜕𝑥𝑖 = 𝛿_𝑁𝜍_𝑔𝑖𝑗 ^{𝜕𝑢𝑔𝑖}

𝜕𝑥𝑗 + 𝑄_𝑔+ 𝑊_𝑔 + 𝑄𝐼_𝑔+ 𝑆_𝑚𝑜𝑚 𝑢_𝑠− 𝑢_𝑔 − 𝑒_𝑔^∗− 𝑢_𝑔² 𝑆_{𝑚𝑎𝑠𝑠} (3.58)

In equation (3.58) it has been assumed that the specific heat,𝑐_𝑣𝑁, is constant and uniform. Finally the one –dimensional duct flow equation for energy balance is:

𝜕

𝜕𝑡 𝜌_𝑔𝛼_𝑔𝑒_𝑔^∗ +¹

𝐴

𝜕

𝜕𝑥 𝐴𝜌_𝑔𝛼_𝑔𝑒_𝑔^∗𝑢_𝑔 = 𝑄_𝑔 + 𝑊_𝑔+ 𝑄𝐼_𝑔 + 𝑊𝐼_𝑔 + 𝛿_𝑔 ^𝜕

𝜕𝑥 𝑝𝑢_𝑔 (3.59) In simplifying the last term on the RHS of equation (3.59) notice that for continuous phase, 𝛿_𝑁 = 1 while for the dispersed phase, 𝛿_𝑁 = 0. And that for the flow situation under consideration, there no shaft work done on the gas component therefore

𝜕

𝜕𝑡 𝜌_𝑔𝛼_𝑔𝑒_𝑔^∗ +1 𝐴

𝜕

𝜕𝑥 𝐴𝜌_𝑔𝛼_𝑔𝑒_𝑔^∗𝑢_𝑔 = 𝑄_𝑔+ 𝑄𝐼_𝑔+ 𝑊𝐼_𝑔+ 𝜕

𝜕𝑥 𝑝𝑢_𝑔

Since there is no energy accumulation on the control volume, this further simplifies to

1 𝐴

𝜕

𝜕𝑥 𝐴𝜌_𝑔𝛼_𝑔𝑒_𝑔^∗𝑢_𝑔 = 𝑄_𝑔+ 𝑄𝐼_𝑔 + 𝑊𝐼_𝑔 + ^𝜕

𝜕𝑥 𝑝𝑢_𝑔 (3.60) Now

𝜕

𝜕𝑥 𝐴𝜌_𝑔𝛼_𝑔𝑒_𝑔^∗𝑢_𝑔

=𝜕𝐴

𝜕𝑥 𝜌_𝑔𝛼_𝑔𝑒_𝑔^∗𝑢_𝑔 +𝜕𝜌_𝑔

𝜕𝑥 𝐴𝛼_𝑔𝑒_𝑔^∗𝑢_𝑔 +𝜕𝛼_𝑔

𝜕𝑥 𝐴𝜌_𝑔𝑒_𝑔^∗𝑢_𝑔 +𝜕𝑒_𝑔^∗

𝜕𝑥 𝐴𝜌_𝑔𝛼_𝑔𝑢_𝑔 +𝜕𝑢_𝑔

𝜕𝑥 𝐴𝜌_𝑔𝛼_𝑔𝑒_𝑔^∗ but

𝜕𝐴

𝜕𝑥 = 0 ^𝜕𝜌𝑔

𝜕𝑥 = 0 ^𝜕𝛼𝑔

𝜕𝑥 = 0

So LHS of equation (3.60)

53

1 𝐴

𝜕

𝜕𝑥 𝐴𝜌_𝑔𝛼_𝑔𝑒_𝑔^∗𝑢_𝑔 =^𝜕𝑒𝑔

∗

𝜕𝑥 𝜌_𝑔𝛼_𝑔𝑢_𝑔 +^𝜕𝑢𝑔

𝜕𝑥 𝜌_𝑔𝛼_𝑔𝑒_𝑔^∗ equation (3.60) becomes:

𝜌_𝑔𝛼_𝑔 𝑢_𝑔^𝜕𝑒𝑔

∗

𝜕𝑥 + 𝑒_𝑔^{∗ 𝜕𝑢𝑔}

𝜕𝑥 = 𝑄_𝑔 + 𝑄𝐼_𝑔+ 𝑊𝐼_𝑔+ ^𝜕

𝜕𝑥 𝑝𝑢_𝑔 (3.61)

𝜕

𝜕𝑥 𝑝𝑢_𝑔 = 𝑢_𝑔𝜕𝑝

𝜕𝑥+ 𝑝𝜕𝑢_𝑔

𝜕𝑥 let

𝑑𝑢_𝑔

𝑑𝑥 = 𝑆_{𝑚𝑎𝑠𝑠} 𝜌_𝑔𝛼_𝑔 = 𝑎 𝑑𝑝_𝑔

𝑑𝑥 = −𝛼_𝑔𝜌_𝑔𝑔_𝑥 −𝑃 𝜏_𝑤

𝐴 +𝑆_𝑚𝑜𝑚

𝐴 − 𝑢_𝑔𝑆_{𝑚𝑎𝑠𝑠} = 𝑏

Since the values of ^𝑑𝑢𝑔

𝑑𝑥 and ^𝑑𝑝𝑔

𝑑𝑥 are coefficients their values can be replaced with a and b for convenience

𝜕

𝜕𝑥 𝑝𝑢_𝑔 = 𝑢_𝑔𝑏 + 𝑝𝑎 Equation (3.61) can be rewritten as

𝜌_𝑔𝛼_𝑔 𝑢_𝑔𝜕𝑒_𝑔^∗

𝜕𝑥 + 𝑒_𝑔^∗𝑆_{𝑚𝑎𝑠𝑠}

𝜌_𝑔𝛼_𝑔 = 𝑄_𝑔+ 𝑄𝐼_𝑔+ 𝑊𝐼_𝑔 + 𝑢_𝑔𝑏 + 𝑝𝑎 rearranging

𝜌_𝑔𝛼_𝑔𝑢_𝑔𝜕𝑒_𝑔^∗

𝜕𝑥 + 𝑒_𝑔^∗𝑆_{𝑚𝑎𝑠𝑠} = 𝑄_𝑔 + 𝑄𝐼_𝑔 + 𝑊𝐼_𝑔+ 𝑢_𝑔𝑏 + 𝑝𝑎 and

𝜕𝑒_𝑔^∗

𝜕𝑥 + 𝑒_𝑔^{∗ 𝑆𝑚𝑎𝑠𝑠}

𝜌𝑔𝛼𝑔𝑢𝑔 = ¹

𝜌𝑔𝛼𝑔𝑢𝑔 𝑄_𝑔+ 𝑄𝐼_𝑔+ 𝑊𝐼_𝑔 + 𝑢_𝑔𝑏 + 𝑝𝑎 (3.62) Again, since the entire LHS is a constant, it is denoted with m for convenience

1

𝜌_𝑔𝛼_𝑔𝑢_𝑔 𝑄_𝑔+ 𝑆_{𝑒𝑛𝑒𝑟𝑔𝑦} + 𝑢_𝑔𝑏 + 𝑝𝑎 = 𝑚

54 Equation (3.62) becomes

𝜕𝑒_𝑔^∗

𝜕𝑥 + 𝑒_𝑔^∗ 𝑆_{𝑚𝑎𝑠𝑠}

𝜌_𝑔𝛼_𝑔𝑢_𝑔 = 𝑚

𝜕𝑒_𝑔^∗

𝜕𝑥 = 𝑚 − 𝑒_𝑔^∗ 𝑆_{𝑚𝑎𝑠𝑠} 𝜌_𝑔𝛼_𝑔𝑢_𝑔

𝜕𝑒_𝑔^∗

𝜕𝑥 = −𝑚𝜌_𝑔𝛼_𝑔𝑢_𝑔

𝑆_{𝑚𝑎𝑠𝑠} +𝑒_𝑔^∗ − 𝑆_{𝑚𝑎𝑠𝑠} 𝜌_𝑔𝛼_𝑔𝑢_𝑔

𝜕𝑒𝑔∗ 𝜕𝑥 𝑒_𝑔^∗−𝑚 𝜌 𝑔 𝛼𝑔 𝑢 𝑔

𝑆𝑚𝑎𝑠𝑠

= − ^{𝑆𝑚𝑎𝑠𝑠}

𝜌𝑔𝛼𝑔𝑢𝑔 (3.63)

Equation (3.63) is variable separable and integrating gives 𝑙𝑛 𝑒_𝑔^∗−𝑚𝜌_𝑔𝛼_𝑔𝑢_𝑔

𝑆_{𝑚𝑎𝑠𝑠} = − 𝑆_{𝑚𝑎𝑠𝑠}

𝜌_𝑔𝛼_𝑔𝑢_𝑔𝑥 + 𝐶 𝑒_𝑔^∗ = ^{𝑚 𝜌𝑔𝛼𝑔𝑢𝑔}

𝑆𝑚𝑎𝑠𝑠 + 𝐶𝑒⁻

𝑆𝑚𝑎𝑠𝑠 𝜌 𝑔 𝛼𝑔 𝑢 𝑔𝑥

(3.64) Using the following initial conditions

𝑥 = 0 ; 𝑒_𝑔^∗ = 𝑒_𝑔1^∗ Then

𝐶 = 𝑒_𝑔1^∗ −𝑚𝜌_𝑔𝛼_𝑔𝑢_𝑔 𝑆_{𝑚𝑎𝑠𝑠} 𝑒_𝑔^∗ = 𝑚𝜌_𝑔𝛼_𝑔𝑢_𝑔

𝑆_{𝑚𝑎𝑠𝑠} + 𝑒_𝑔1^∗ −𝑚𝜌_𝑔𝛼_𝑔𝑢_𝑔 𝑆_{𝑚𝑎𝑠𝑠} 𝑒⁻

𝑆_{𝑚𝑎𝑠𝑠} 𝜌_𝑔𝛼_𝑔𝑢_𝑔𝑥

𝑒_𝑔^∗ = 𝑐_𝑉𝑔𝑇_𝑔+𝑢_𝑔² 2 + 𝑔𝑥 𝑇_𝑔2= 𝑚𝜌_𝑔𝛼_𝑔𝑢_𝑔

𝑆_{𝑚𝑎𝑠𝑠} + 𝑒_𝑔1^∗ −𝑚𝜌_𝑔𝛼_𝑔𝑢_𝑔 𝑆_{𝑚𝑎𝑠𝑠} 𝑒⁻

𝑆_{𝑚𝑎𝑠𝑠} 𝜌_𝑔𝛼_𝑔𝑢_𝑔𝑥

−𝑢_𝑔²

2 − 𝑔𝑥 𝑐_𝑉𝑔

𝑇_𝑔2 = ^{𝑚𝜌𝑔𝛼𝑔𝑢𝑔}

𝑆_{𝑚𝑎𝑠𝑠} + 𝑐_𝑉𝑔𝑇_𝑔1+^𝑢𝑔1

2

2 + 𝑔𝑥 −^{𝑚 𝜌𝑔𝛼𝑔𝑢𝑔}

𝑆_{𝑚𝑎𝑠𝑠} 𝑒⁻

𝑆𝑚𝑎𝑠𝑠 𝜌 𝑔𝛼 𝑔𝑢 𝑔𝑥

−^𝑢𝑔2

2 − 𝑔𝑥 𝑐_𝑉𝑔 (3.65)

55 3.3.6 Constitutive Relationships (Energy)

The energy coupling source term for the total energy equation involves convective heat transfer and the work due to particle drag as suggested by Hamed M. H (2005) is expressed as:

𝑆_{𝑒𝑛𝑒𝑟𝑔𝑦} = −𝑁_𝑝𝑕𝐴𝜒𝜋𝑑_𝑝² 𝑇_𝑔− 𝑇_𝑠 + 𝑆_𝑚𝑜𝑚𝑢_𝑠 (3.66) The dispersed phase is introduced into the dispersing phase at a point along the flow path; the feed point which is always upstream of the flash tube. At this point the dispersed phase temperature is much smaller than the dispersing phase temperature.

Heat transfer between the phases tends to reduce the difference in temperature.

Therefore it is necessary to characterize the rate of equilibration of the particle and fluid temperatures by defining a temperature relaxation time, 𝑡_𝑇. This temperature relaxation time can be obtained by equating the rate of heat transfer from the continuous phase to the particle with the rate of increase of heat stored in the particle.

The heat transfer to the particle can occur as a result of conduction, convection or radiation and there are practical flows in which each of these mechanisms are important but for the situation at hand the radiation component shall be neglected.

If the relative motion between the particles and the fluid is sufficiently small, the only contributing mechanism is conduction and it is limited by the thermal conductivity, 𝑘_𝑔 of the gas since the thermal conductivity of the particle is usually much higher.

Then the rate of heat transfer to the particle of radius, R will be given approximately by

2𝜋𝑅𝑘_𝑔(𝑇_𝑔 − 𝑇_𝑠) (3.67)

where 𝑇_𝑔 𝑎𝑛𝑑 𝑇_𝑠are respectively temperatures of the gas phase and the particle.

Since the situation being modelled involves conveyance and drying, the relative motion that is slip velocity can only be low to the extent that it guarantees conveyance

56

and in this situation conveyance takes precedence over heat transfer which drives drying.

In determining the convective heat transfer coefficient the empirical approach was used. The usual drawback of using the empirical approach is that it requires a large number of experiments to obtain the required data. This challenge is overcome by the use of dimensionless numbers. To formulate this approach, first the required dimensionless numbers are identified: Reynolds number, Re, Nusselt number, Nu, and Prantl number, Pr

To add the component of heat transfer by convection caused by relative motion is done by defining the Nusselt number, Nu, as twice the ratio of the rate of heat transfer with convection to that without convection. Then the rate of heat transfer becomes Nu times the above result for conduction.

The convective heat transfer coefficient, h, was calculated from Nusselt number, Nu, which is expressed as a function of Reynold number, Re_p and Prantl number, Pr, which are defined as:

𝑅𝑒 = 2 𝑢_𝑠− 𝑢_𝑔 𝑅/𝑣_𝑔 (3.68) 𝑃𝑟 =^{𝜌𝑔 𝑣𝑔𝐶𝑝𝑔}

𝑘𝑔 (3.69) Various empirical correlations that can be used to calculate the heat transfer coefficient has been proposed and are listed below.

 Frantz correlation (Radford R. D., 1997)

The correlation was used by Radford to calculate the heat transfer coefficient in pneumatic conveying dryer.

𝑁𝑢 = 0.015𝑅𝑒_𝑝^1.6𝑃𝑟^0.667 (3.70)

 De Brandt correlation (Fyhr C. and Rasmuson A., 1997 )

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The correlation was developed for pneumatic drier,

𝑁𝑢 = 0.16𝑅𝑒_𝑝^1.6𝑃𝑟^0.667 (3.71)

 De Brand correlation (Debrand S., 1974 ) The correlation was developed for a pneumatic dryer,

𝑁𝑢 = 0.035𝑅𝑒_𝑝^1.15𝑃𝑟^0.333 (3.72)

 Bayeans et al. Correlation (Baeyens et al, 1995) The correlation was developed for large scale pneumatic conveyor

𝑁𝑢 = 0.15𝑅𝑒_𝑝 (3.73)

 Modified Ranz-Marshall correlation (Levy and Borde, 1999) The correlation was developed for simple droplet/wet particle and it takes into account the resistance of the liquid vapour around the particle to the heat transfer by Spalding number, B.

𝑁𝑢 =^{2+0.6𝑅𝑒𝑝}

0.5𝑃𝑟^0.333

1+𝐵 ^0.7 (3.74) 𝐵 =^{𝐶𝑝𝑤𝑣 𝑇𝑔−𝑇𝑠}

𝐻𝑓𝑔 (3.75)

 Modified Weber correlation (Kemp et al, 1994) An additional term proportional to Rep0.8

was added to Ranz-Marshall correlation to account for turbulent flow.

𝑁𝑢 = 2 + 0.5𝑅𝑒_𝑝^0.5+ 0.06𝑅𝑒_𝑝^0.8 𝑃𝑟^0.333 (3.76)

 Ranz and Marshall correlation

𝑁𝑢 = 2 + 0.6𝑅𝑒^1/2𝑃𝑟^1/3 (3.77)

The correlation above reduces to pure conduction result, Nu = 2, when the second term on the right hand is small. Assuming that the particle temperature has a roughly uniform value of 𝑇_𝑠, it follows that

𝑄𝐼_𝑠 = 2𝜋𝑅𝑘_𝑔𝑁𝑢 𝑇_𝑔− 𝑇_𝑠 𝑛_𝑠 = 𝜌_𝑠𝛼_𝑠𝑐_𝑠^𝐷𝑇𝑠

𝐷𝑡 (3.78)

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where the material derivative D/Dt, follows the particle. This provides the equation that must be solved for 𝑇_𝑠, namely

𝐷𝑇𝑠 𝐷𝑡 =^𝑁𝑢

2 𝑇_𝑔−𝑇_𝑠

𝑡𝑇 (3.79)

where,

𝑡_𝑇 = 𝑐_𝑠𝜌_𝑠𝑅²/3𝑘_𝑔 (3.80)

 Singh and Heldman correlation (Singh and Heldman, 2001)

For a flow past a single sphere, when the single sphere may be heated or cooled, the following equation will apply:

𝑁𝑢 = 2 + 0.60𝑅𝑒¹²𝑃𝑟¹³ for 1 < 𝑅𝑒 < 70000 and 0.6 < 𝑃𝑟 < 400 (3.81) where the characteristic dimension, 𝑑_𝑝, is the outside diameter of the sphere.

The correlation suggested by Singh and Heldman (2001) shall be used for this work.

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