Chapter 4. Model Development
4.2 Proposed Design Model
The relationship between the overall heat transfer coefficient Uov and the individual heat transfer resistances (Figure 4.2) is derived from heat bal-
Figure 4.1: Sketch of a Vertical Wiped Film Evaporator. The heat added to the system generates evaporation at the surface of the falling liquid and the rotating blades generate turbulence at the surface.
ances around the heating medium, the wall, and liquid. q = ho(To− TW o) q = δwall kwall (TW o− TW L) q = hp(TW L− TL)
where q is the heat flux per unit area at each interface.
Figure 4.2: Heat transfer resistances in a wiped film evaporator.
The previous equations state that the amount of heat transferred from the medium to the wall must be equal to the amount passing through the wall
and the amount transferred to the liquid. Equating all the heat terms and solving for q, the following expression for the overall heat transfer coefficient results: 1 Uov = 1 ho + δwall kwall + 1 hp (4.1) where Uov is the overall heat transfer coefficient (W/m2-K), ho is the heat transfer coefficient for the heating medium (W/m2-K), kwall is the thermal resistance of the wall (W/m-K), δwall is the thickness of the wall (m), and hp is the heat transfer coefficient for the liquid film (W/m2-K).
As mentioned in Chapter 3, the process side HTC, hp, can be calculated from experimental data. Equation 3.13 is used to calculate Uov, then Equation 3.16 (steam) or 3.17 (other fluid) is used to calculate the hot fluid side HTC, ho. The wall resistance is readily calculated using the thermal conductivity of the wall as well as it thickness. Finally, Equation 3.15 is used to calculate the process side HTC, hp.
The present research was focused on modeling hp (the heat transfer coefficient inside the WFE) and kW F E
L (mass transfer coefficient inside the WFE). Preliminary studies had indicated that hp in WFEs is a function of the number of blades, the speed of rotation, and the physical properties of the system (i.e., viscosity, thermal conductivity, etc.)
Considering the WFE as a stage-wise unit (i.e., dividing the length into small “stages”, see Figure 4.1) and assuming plug flow (i.e., no backmixing), the performance of a WFE can be predicted using the equations below.
Applying mass balance, energy balance, and equilibrium considerations to the stage, the amount of generated vapor (∆V , kg/s) can be calculated.
Mass balance: Ln+ Vn = F + Vn−1 (4.2) Lnxn+ Vnyn = F xF + Vn−1yn−1 (4.3) Equilibrium: Kn = yn xn (4.4) Energy balance: LnhL,n+ VnhV,n = F hL,F + Vn−1hV,n−1+ q Vn−1 = LnhL,n+ VnhV,n− F hL,F − q hV,n−1 (4.5) where q = UovA∆Tlm
The possibility of correcting correlations for falling film evaporator and applying them to wiped film evaporators was analyzed, and was found that it can be used. Additional experimental data were needed in order to verify this approach.
The initial approach was to use an enhancement factor β, defined as the ratio of the WFE heat or mass transfer coefficient to the FFE heat or mass transfer coefficient. Since little information has been reported on WFE mass
transfer, the enhancement factor was initially evaluated based on reported WFE and FFE heat transfer information.
The heat transfer enhancement factor, βh, is defined as follows: βh = hW F E p hF F E p (4.6) where hW F Ep is the heat transfer coefficient for the WFE, and hF F Ep is for the FFE.
Since wiped film evaporation is generally used in liquid phase controlled systems (e.g. viscous mixtures), our models are based on the prediction of liquid phase coefficients for heat and mass transfer. From the published models for the prediction of the heat transfer coefficient for WFE, two were selected: Bott and Romero [11], Bott and Sheikh [14]. These correlations are of the form: N u = f Rea1 f Re a2 NP r a3Na4 b (D/L) a5 Na6 b (4.7) where the parameters a1 to a6 were correlated using heat transfer coefficient data. Nb is the number of blades, D is the diameter, L is the length, N is the rotational speed, and the dimensionless numbers are:
N u = hpD
k is the Nusselt number Ref =
4F
πDµ is the film Reynolds number ReN =
D2N ρ
µ is the rotational Reynolds number P r = Cpµ
The expression for each particular WFE heat transfer coefficient model is as follows.
Bott and Romero [11]:
N u = 0.018Re0.46f Re0.6N P rL0.87(D/L)0.48Nb0.24 (4.8)
Bott and Sheikh [14]:
N u = 0.65Re0.25f Re0.43N P r0.3L Nb0.33 (4.9)
Two FFE heat transfer coefficient models for different N u values were used: Ahmed and Kaparthi [3], and Numrich [73]. The expression for each model is as follows.
Ahmed and Kaparthi [3]
N u = 6.92 × 10−3Re0.345f P r0.4L (4.10)
Numrich [73]
N u = 0.003Re0.44f P r0.4L (4.11)
In these models, the Nusselt number is defined as: N u = hδL k (4.12) where δL= µ2 ρ2g 1/3
= the characteristic length.
Figures 4.3, 4.4, and 4.5 show the variation of the heat transfer en- hancement factor (βh) with the film Reynolds number, rotational Reynolds
number, and Prandtl number using the four possible combinations of models for the heat transfer coefficient (two for WFEs and two for FFEs).
Figure 4.3 shows that as the film Reynolds number (Re) increases, the heat transfer enhancement factor decreases, having a high value at low Re. This means that the performance of the equipment will be expected not to change significantly after a critical Re is achieved. For this particular case, the value is around 2000.
Figure 4.4 shows that as the rotational Reynolds number (ReN) in- creases, the heat transfer enhancement factor increases. This is due to the in- crease in the speed of rotation, which also increases the HTC for the wiped film evaporator. This is consistent with what other authors have found [14, 42, 64]. There is a region of the rotational speed where the evaporator is operated typ- ically, highlighted by the square box.
Figure 4.5 presents a sharp increase in the enhancement factor as a function of the Prandtl number (P r). This is because as the Prandtl number increases, usually the viscosity increases, and the HTC in a falling film evap- orator will decrease, while in a wiped film evaporator, the HTC will increase.