4. Project of a tank with heat exchange by vertical tubular baffles

Nu ¼ 0:09Rea

tubular baffle dimensions. The model is presented in Eq. (48).

continuous operation, as illustrated in Eq. (50).

number, as presented in Eq. (47).

50 Heat Exchangers– Design, Experiment and Simulation

<sup>0</sup>:<sup>65</sup>Pr<sup>0</sup>:<sup>33</sup> <sup>μ</sup>

Nu ¼ 0:208Rea

Nu ¼ 0:494Rea

expression similar to Eq. (48). The obtained function is presented in Eq. (49).

Nu ¼ 0:542Rea

Nu ¼ 17:88Rea

Nu ¼ 25:03Rea

however, this time using a turbine radial impeller. The model is shown in Eq. (51).

μw

Havas et al. [37] carried out experiments in a tank with 0.4 and 0.8 m of diameter for water and fuel oils heating with five-tube bank and radial impeller equipped with six flat blades. The operation was conducted on discontinuous mode. Based on the experimental data, the researchers concluded that the effect caused by the impellers' diameter and by the amount of tube bank is negligible in relation to the turbulence generated by the mechanical impeller. The obtained model incorporates the effects, which are directly disregarded on the Reynolds

<sup>0</sup>:<sup>4</sup> Da

<sup>0</sup>:<sup>65</sup>Pr<sup>0</sup>:<sup>33</sup> <sup>μ</sup>

<sup>0</sup>:<sup>67</sup>Pr<sup>0</sup>:<sup>33</sup> <sup>μ</sup>

<sup>0</sup>:<sup>65</sup>Pr<sup>0</sup>:<sup>33</sup> <sup>μ</sup>

<sup>0</sup>:<sup>27</sup>Pr<sup>0</sup>:<sup>29</sup> <sup>μ</sup>

<sup>0</sup>:<sup>38</sup>Pr<sup>0</sup>:<sup>11</sup> <sup>μ</sup>

Karcz and Strek [38] defined a model to determine the external convection coefficients using an axial mechanical impeller with three inclined blades in nonstandard conditions for the vertical

Lukes [39] expanded the studies carried by Karcz and Strek [38], using an axial impeller with three inclined blades and a standard axial impeller with four blades inclined 45°, obtaining an

Rosa et al. [40] carried experiments in a tank with 0.4 m diameter for heating sucrose solutions at 20% and 32% (concentration?) using hot water, through four-tube bank using an axial impeller with four blades inclined 45°; however, this is the first model presented operating in

Rosa et al. [41] carried out experiments under the same conditions that Rosa et al. [35],

μw <sup>0</sup>:<sup>4</sup>

μw <sup>0</sup>:<sup>14</sup>

μw <sup>0</sup>:<sup>40</sup>

μw <sup>0</sup>:<sup>37</sup>

μw <sup>0</sup>:<sup>20</sup>

Dt

<sup>0</sup>:<sup>33</sup> 2

nb <sup>0</sup>:<sup>2</sup>

(46)

(47)

(48)

(49)

(50)

(51)

Specify the heat exchange system model with vertical tube baffles for a tank with useful volume of 3 m<sup>3</sup> on continuous operation, as shown in Figure 1. The vertical tube baffles are made of brass composed by vertical pipelines (one baffle/three pipelines). The unit must heat an aqueous solution with 20% in mass of sucrose and a 2.0 m3 /h flow (V\_ sol) from 20 to 42°C, in steady-state operation. The liquid available for heating is water at 90°C with a 10 m3 /h flow. The impeller, which has four blades, inclined 45°, works at 150 rpm. Data: (a) solutions' specific mass (ρsol) of 1074.2 kg/m<sup>3</sup> , (b) solutions' specific heat (Cpsol) of 3650 J/kg K, (c) water specific mass (ρwater) of 1000 kg/m3 , (d) water specific heat (Cpwater) of 4180 J/kg K, (e) water viscosity (μwater) of 0.001 kg/(m s), (f) solutions' viscosity (μsol)of 0.0017 kg/(m s), (g) solutions' thermal conductivity (ksol) of 0.43 W/(m °C), and (h) combined fouling factor (Rd) of 0.001 (h ft<sup>2</sup> °F)/Btu.

The described system has as project equation for heat transfer area, Eq. (52) (nonisothermal continuous operation) (see Eq. (1)).

Figure 1. Tank with axial impeller and four blades inclined by 45° and system of heat exchange by vertical tubular baffles. Continuous operation [42].

$$A = \mathbb{Q}/\text{ULMTD} \tag{52}$$

The solution of Eq. (52) includes the determination of the overall coefficient of heat transfer. Due to the high thermal conductivity of brass, the conduction will be disregarded in this process; hence, the Ud coefficient will only depend on the convection coefficients (Eq. (53)).

$$1/\mathcal{U}\_d = 1/h\_{\rmio} + 1/h\_o \tag{53}$$

The hi coefficient will be calculated from the expression proposed by Bondy and Lippa [6] (Eq. (12) of item 2)—Eq. (54)—and corrected regarding the pipeline's external surface (Eq. (55)).

$$h\_i = 1429 \left( 1 + 0.0146.\overline{T} \right) .u^{0.8} / D\_i^{0.2} \tag{54}$$

$$h\_{\rm bi} = h\_i \frac{D\_i}{D\_e} \tag{55}$$

The average speed u in the interior of the vertical tubular baffles' pipeline is calculated from the continuity equation (V\_ <sup>¼</sup> <sup>u</sup>:A); however, the pipeline's diameter is not known yet. This procedure involves the determination of the vessel's internal geometry, which will be calculated from the standards recommended by Rushton et al. [32]. With 3 m<sup>3</sup> volume and applying the geometric relations, the following is obtained:

$$V = \left(\pi D\_t^2 / 4\right) \text{H} = 1.56 \,\text{m} \tag{56}$$

$$D\_t = H = 1.56\,\text{m} \tag{57}$$

$$D\_t/D\_a = \mathfrak{Z} \to D\_a = 0.52\,\text{m} \tag{58}$$

$$E/D\_a = 1 \to E = 0.52\,\text{m} \tag{59}$$

$$W/D\_a = 1/5 \to W = 0.104 \,\text{m} \tag{60}$$

$$X = W/0.707 \to X = 0.147 \,\text{m} \tag{61}$$

$$\text{J/}D\_t = 0.1 \to \text{J} = 0.156\,\text{m} = 6.14\,\text{in.}\,\text{\#6}\,\text{in} \tag{62}$$

Since the baffle is composed by three pipelines, their external (De) and internal (Di) diameters can be specified:

$$D\_{\mathfrak{e}}\mathfrak{a}\mathfrak{G}\text{in.}/3\mathfrak{a}\mathfrak{2}\text{in.}\tag{63}$$

Considering a commercial pipeline with DN of 11/2 in., Sch 40S and Di of 1.61 in., and thickness (ep) of 0.145 in. therefore

Design of Heat Transfer Surfaces in Agitated Vessels http://dx.doi.org/10.5772/66729 53

$$D\_t = D\_i + 2\,\text{ep} = 0.04826\,\text{m} \tag{64}$$

$$D\_i = 0.040894 \,\text{m} \tag{65}$$

The average temperature T is calculated from the arithmetic average between the hot fluid's inlet and outlet (T<sup>1</sup> and T2), respectively. However, on non-isothermal conditions, the temperature T<sup>2</sup> also varies over time. Considering that the vessel is perfectly insulated (isolated?), in such a way that all the heat given by the hot fluid (water) will be transferred to the cold fluid (solution), the temperature T<sup>2</sup> and the heat flow Q can be obtained by an energy balance, as shown in Eq. (66).

$$
\dot{w}\_h c p\_h \frac{dT}{d\Theta} = \dot{w}\_c c p\_c \frac{d t\_b}{d\Theta} \tag{66}
$$

Integrating in θ = 0 with T = T<sup>1</sup> and tb = tb1, the following is obtained:

A ¼ Q=U:LMTD (52)

1=Ud ¼ 1=hio þ 1=ho (53)

<sup>t</sup> <sup>=</sup><sup>4</sup> <sup>H</sup> <sup>¼</sup> <sup>1</sup>:56m (56)

Dt ¼ H ¼ 1:56m (57)

Dt=Da ¼ 3 ! Da ¼ 0:52m (58)

E=Da ¼ 1 ! E ¼ 0:52m (59)

W=Da ¼ 1=5 ! W ¼ 0:104m (60)

X ¼ W=0:707 ! X ¼ 0:147m (61)

De≅6in:=3≅2in: (63)

J=Dt ¼ 0:1 ! J ¼ 0:156m ¼ 6:14in:≅6in (62)

<sup>0</sup>:<sup>2</sup> (54)

(55)

=Di

The solution of Eq. (52) includes the determination of the overall coefficient of heat transfer. Due to the high thermal conductivity of brass, the conduction will be disregarded in this process; hence, the Ud coefficient will only depend on the convection coefficients

The hi coefficient will be calculated from the expression proposed by Bondy and Lippa [6] (Eq. (12) of item 2)—Eq. (54)—and corrected regarding the pipeline's external surface

hi <sup>¼</sup> 1429 1 <sup>þ</sup> <sup>0</sup>:0146:<sup>T</sup> :u<sup>0</sup>:<sup>8</sup>

hio ¼ hi

<sup>V</sup> <sup>¼</sup> <sup>π</sup>D<sup>2</sup>

the geometric relations, the following is obtained:

The average speed u in the interior of the vertical tubular baffles' pipeline is calculated from the continuity equation (V\_ <sup>¼</sup> <sup>u</sup>:A); however, the pipeline's diameter is not known yet. This procedure involves the determination of the vessel's internal geometry, which will be calculated from the standards recommended by Rushton et al. [32]. With 3 m<sup>3</sup> volume and applying

Since the baffle is composed by three pipelines, their external (De) and internal (Di) diameters

Considering a commercial pipeline with DN of 11/2 in., Sch 40S and Di of 1.61 in., and thickness

Di De

(Eq. (53)).

52 Heat Exchangers– Design, Experiment and Simulation

(Eq. (55)).

can be specified:

(ep) of 0.145 in. therefore

$$T2 = \frac{\dot{w}\_c c p\_c}{\dot{w}\_{h \text{ cph}}} (t\_{b2} - t\_{b1}) + T\_1 = 85.9^{\circ} \text{C} \tag{67}$$

$$Q = \dot{w}\_h c p\_h \frac{dT}{d\theta} = 48180 \, W \tag{68}$$

Hence, the average temperature Tis given by Eq. (69) and the average speed u by Eq. (70).

$$
\overline{T} = 87.95^{\circ} \text{C} \tag{69}
$$

$$
\mu = \frac{4.\dot{V}}{\pi D\_i^2} = 2.11 \,\mathrm{m/s} \tag{70}
$$

Replacing Eqs. (65), (69), and (70) in Eq. (54), the hi coefficient (Eq. (71)) is obtained, and replacing the value of hi and of the Eqs. (64) and (65) in Eq. (55), the hio coefficient is given (Eq. (72)).

$$h\_i = 11238.20 \,\text{W/m}^2 \,\text{°C} \tag{71}$$

$$h\_{\rm ib} = 9522.90 \,\mathrm{W/m^2 \, ^\circ C} \tag{72}$$

The ho coefficient for the current project example will be calculated with the expression given by Rosa et al. [40] (Eq. (73)).

$$Nu = 17.88 Re\_a^{0.27} Pr^{0.29} \left(\frac{\mu}{\mu\_w}\right)^{0.37} \tag{73}$$

Eqs. (74) and (75) show the calculation for the Reynolds (Rea) and Prandtl numbers. The relation μ/μwwill be assumed as 1, considering that fluid temperature in the tank tb will be equal to the wall temperature tw.

$$Re\_4 = \frac{D\_a^2 N \rho\_{\rm sol}}{\mu\_{\rm sol}} = 424654.5\tag{74}$$

$$\text{Pr} = \frac{\text{C}\_{\text{p}}\mu\_{\text{sol}}}{K\_{\text{sol}}} = 14.52 \tag{75}$$

Replacing Eqs. (74) and (75) in Eq. (73), the following is given:

$$Nu = 1247.14\tag{76}$$

$$h\_0 = \frac{Nu K\_{\text{sol}}}{D\_t} = 343.76 \,\text{W/m}^2 \,\text{°C} \tag{77}$$

$$\text{CL}\_{\text{C}} = \frac{h\_{\text{i}0}h\_{\text{0}}}{h\_{\text{i}0} + h\_{\text{0}}} = 331.78 \,\text{W/m}^2 \,\text{°C} \tag{78}$$

$$\mathbf{U}\_D = \mathbf{313.47W/m^2} \,\mathrm{°C} \tag{79}$$

The LMTD will be calculated considering the agitation system operating on countercurrent. Eqs. (80) and (81) present the LMTD calculation.

$$\text{LMTD} = \frac{\Delta t\_q - \Delta t\_f}{\ln \left( \Delta t\_q / \Delta t\_f \right)} = \frac{\Delta t\_f - \Delta t\_q}{\ln \left( \Delta t\_f / \Delta t\_q \right)} \tag{80}$$

$$\text{LMTD} = 56.4^{\circ}\text{C} \tag{81}$$

Finally, replacing Eqs. (68), (79), and (81) in Eq. (52), the necessary heat exchange area for this project is found (Eq. (82)).

$$A = 2.72 \,\text{m}^2\tag{82}$$

The total pipeline length (L), the number of pipelines (Nt), and the number of tubes per baffle (Nc) are given by Eqs. (83)–(85), respectively.

$$L = \frac{A}{\pi D\_\epsilon} = 17.9 \,\mathrm{m} \tag{83}$$

$$N\_t = \frac{L}{H} = 11.5 \,\text{tubes} \overline{\text{}12 \,\text{tubes}}\tag{84}$$

$$N\_b = \frac{N\_t}{4\,\text{baffles}}\,\text{3\frac{tubes}{baffles}}\tag{85}$$

Hence, the vessel described by the given example must have four vertical tubular baffles, and each one must have three tubes. If the tank's heating were to be carried out with agitation promoted by a radial impeller, the ho coefficient should be calculated by the equation proposed by Dunlap and Rushton [36], shown in item 3.4 (Eq. (51)), obtaining a heat transfer area of just 0.91 m<sup>2</sup> .
