A reboiler is required to supply 15,000 lb/h of vapor to a distillation column that separates cyclohexane as the bottom product. The heating medium will be steam at a design pressure of 18 psia. The temperature and pressure below the bottom tray in the column are 182º F and 16 psia. Physical

Question

A reboiler is required to supply 15,000 lb/h of vapor to a distillation column that separates cyclohexane as the bottom product. The heating medium will be steam at a design pressure of 18 psia. The temperature and pressure below the bottom tray in the column are 182º F and 16 psia. Physical property data for cyclohexane at these conditions are given in the following table:

Property	Liquid	Vapor
ρ (lbm/ft3)	45	0.2
μ (cp)	0.4	0.0086
[latex]C_ {p}[/latex] (Btu/lbm . ºF)	0.45	–
k (Btu/h . ft .ºF)	0.086	–
σ (lbf/ft)	0.00124	–
λ (Btu/lbm)	154	–
Pr	5.063	–

The vapor pressure of cyclohexane is given by the following equation [17], where [latex]P_ {sat }[/latex] ∝ torr and T ∝ K:

[latex]P_{ ext {sat }}=\exp \left[15.7527-\frac{2766.63}{T-50.50} ight][/latex]

The critical pressure of cyclohexane is 590.5 psia.
The feed line to the reboiler will consist of 100 equivalent feet of 6-in. schedule 40 pipe, and the return line will consist of 50 equivalent feet of 10-in. schedule 40 pipe. Design a recirculating vertical thermosyphon reboiler for this service.

Accepted Answer

(a) Make initial specifications.
(i) Fluid placement
Cyclohexane will flow in the tubes with steam in the shell.
(ii) Tubing
One-inch, 14 BWG tubes with a length of 8 ft are specified. Relatively short tubes are used in order to minimize the liquid height in the column sump.
(iii) Shell and head types
A TEMA E-shell is chosen for a vertical thermosyphon reboiler. Since condensing steam is a clean fluid, a fixed-tubesheet configuration can be used. Channel-type heads are selected for ease of tubesheet access. Thus, an AEL configuration is specified. A somewhat less expensive NEN configuration could also be used,
(iv) Tube layout
A triangular layout with a tube pitch of 1.25 in. is specified since mechanical cleaning of the external tube surfaces is not required.
(v) Baffles
Segmental baffles with a 35% cut and a spacing B/[latex]D_{s}[/latex] ≅ 0.4 are specified based on the recommendation for condensing vapors given in Figure 5.3.
(vi) Sealing strips
None are required for a fixed-tubesheet exchanger.
(vii) Construction materials
Since neither stream is corrosive, plain carbon steel is specified for all components.
(b) Energy balance and steam flow rate.
The reboiler duty is obtained from the vapor generation rate and the latent heat of vaporization for cyclohexane:

[latex]q=\dot{m}_{V} \lambda=15,000 imes 154=2.31 imes 10^{6} Btu / h[/latex]

From Table A.8, the latent heat of condensation for steam at 18 psia is 963.7 Btu/lbm. Therefore, the steam flow rate is:

[latex]\dot{m}_{ ext {steam }}=q / \lambda_{ ext {steam }}=2.31 imes 10^{6} / 963.7=2397 lbm / h[/latex]

(c) Mean temperature difference.
From Table A.8, the temperature of saturated steam at 18 psia is 222.4º F. Assuming that cyclohexane vaporizes at a constant temperature of 182º F, i.e., neglecting pressure effects in the reboiler system, we have:

[latex]\Delta T_{m}=222.4-182=40.4^{\circ} F[/latex]

(d) Heat-transfer area and number of tubes.
Based on Table 3.5, an overall heat-transfer coefficient of 250 Btu/h . ft².º F is assumed. The required area is then:

[latex]A=\frac{q}{U_{D} \Delta T_{m}}=\frac{2.31 imes 10^{6}}{250 imes 40.4}=228.7 ft ^{2}[/latex]

TABLE 3.5 Typical Values of Overall Heat-Transfer Coefficients in Tubular Heat Exchangers. U = Btu/h · ft² · °F-cont'd
Shell Side	Tube Side	Design U	Includes Total Dirt
Vaporizers
Anhydrous ammonia	Steam condensing	150–300	0.0015
Chlorine	Steam condensing	150–300	0.0015
Chlorine	Light heat-transfer oil	40–60	0.0015
Propane, butane, etc	Steam condensing	200–300	0.0015
Water	Steam condensing	250–400	0.0015

The corresponding number of tubes is:

[latex]n_{ t }=\frac{A}{\pi D_{o} L}=\frac{228.7}{\pi(1 / 12) imes 8} \cong 109[/latex]

(e) Number of tube passes and actual tube count.
A single tube pass is used for a vertical thermosyphon reboiler. From Table C.6, the closest tube count is 106 tubes in a 15.25-in. shell.
This completes the preliminary design of the reboiler system. Since the piping configuration was specified in the problem statement, sizing of the feed and return lines is not required here. The circulation rate is calculated in the steps that follow; only the final iteration is presented.
(f) Estimated circulation rate.
Assume an exit vapor fraction of 13.2%, i.e.,[latex]x_{e}[/latex] = 0.132. The corresponding circulation rate is:

[latex]\dot{m}_{i}=\frac{\dot{m}_{V}}{x_{e}}=\frac{15,000}{0.132}=113,636 lbm / h[/latex]

(g) Friction factors.
The internal diameters for the tubes, inlet line, and exit line are obtained from Tables B.1 and B.2:

[latex]D_{t}=0.834 ext { in. }=0.0695 ft[/latex]

[latex]D_{ ext {in }}=6.065 in .=0.5054 ft[/latex]

[latex]D_{e x}=10.02 ext { in. }=0.835 ft[/latex]

The corresponding Reynolds numbers are computed next, based on all-liquid flow:

[latex]R e_{ t }=\frac{4 \dot{m}_{i}}{n_{ t } \pi D_{ t } \mu_{L}}=\frac{4 imes 113,636}{106 imes \pi imes 0.0695 imes 0.4 imes 2.419}=20,297[/latex]

[latex]R e_{ ext {in }}=\frac{4 \dot{m}_{i}}{\pi D_{ ext {in }} \mu_{L}}=\frac{4 imes 113,636}{\pi imes 0.5054 imes 0.4 imes 2.419}=295,866[/latex]

[latex]R e_{L O, e x}=\frac{4 \dot{m}_{i}}{\pi D_{e x} \mu_{L}}=\frac{4 imes 113,636}{\pi imes 0.835 imes 0.4 imes 2.419}=179,079[/latex]

Equations (4.8) and (5.2) are used to calculate the friction factors for the pipes and tubes, respectively:

[latex]f_{t}=0.4137 Re_{t}^{-0.2585} =0.4137(20,297)^{-0.2585} =0.0319 [/latex]

[latex]f_{ ext {in }}=0.3673 R e_{ ext {in }}^{-0.2314}=0.3673(295,866)^{-0.2314}=0.0199[/latex]

[latex]f_{e x}=0.3673 R e_{L O, e x}^{-0.2314} =0.3673(179,079)^{-0.2314}=0.0224[/latex]

(h) Sensible heating zone.
(i) Slope of saturation curve
Conditions in the column sump are first checked by calculating the vapor pressure of cyclohexane at the given temperature of 182º F = 356.7 K:

[latex]P_{ ext {sat }}=\exp \left[15.7527-\frac{2766.63}{356.7-50.50} ight]=826.6 ext { torr }[/latex]

[latex]P_{ ext {sat }}=826.6 ext { torr } imes \frac{14.696 psi / atm }{760 ext { torr } / atm }=15.98 psia[/latex]

This value is in close agreement with the stated pressure of 16 psia below the bottom tray in the column. Next, the vapor pressure is calculated at a somewhat higher temperature, 192º F = 362.2 K:

[latex]P_{ ext {sat }}=\exp \left[15.77527-\frac{2766.63}{362.2-50.50} ight]=969.5 ext { torr }[/latex]

[latex]P_{ ext {sat }}=969.5 imes 14.696 / 760=18.75 psia[/latex]

The required slope is obtained as follows:

[latex](\Delta T / \Delta P)_{s a t}=\frac{192-182}{18.75-15.98}=3.61^{\circ} F / psi[/latex]

(ii) Pressure gradient
The pressure gradient in the sensible heating zone is estimated using Equation (10.21), neglecting the friction loss term:

[latex]-(\Delta P / L)=\frac{ ho_{ L }\left(g / g_{c} ight)}{144}=\frac{45 imes 1.0}{144}=0.3125 psi / ft[/latex]

(iii) Temperature gradient
To estimate the temperature gradient in the sensible heating zone, the heat-transfer coefficient for all-liquid flow in the tubes is calculated using the Seider–Tate equation:

[latex]h_{L O}=\left(k_{L} / D_{t} ight) imes 0.023 \operatorname{Re}_{t}^{0.8} \operatorname{Pr}_{L}^{1 / 3}\left(\mu / \mu_{w} ight)^{0.14}[/latex]

[latex]=(0.086 / 0.0695) imes 0.023(20,297)^{0.8}(5.063)^{1 / 3}(1.0)[/latex]

[latex]h_{L O}=136 Btu / h \cdot ft ^{2} \cdot{ }^{\circ} F[/latex]

The overall coefficient is calculated assuming a film coefficient (including fouling) of 1500 Btu/h . ft².º F for steam and a fouling allowance for cyclohexane of 0.001 h . ft².ºF/Btu:

[latex]U_{D}=\left[\left(D_{o} / D_{t} ight)\left(\frac{1}{h_{L O}}+R_{D i} ight)+\frac{D_{o} \ln \left(D_{o} / D_{t} ight)}{2 k_{ ext {tube }}}+\left(1 / h_{o}+R_{D o} ight) ight]^{-1}[/latex]

[latex]=\left[(1.0 / 0.834)\left(\frac{1}{136}+0.001 ight)+\frac{(1.0 / 12) \ln (1.0 / 0.834)}{2 imes 26}+\frac{1}{1500} ight]^{-1}[/latex]

[latex]U_{D} \cong 91 Btu / h \cdot ft ^{2} \cdot{ }^{\circ} F[/latex]

The temperature gradient is calculated using Equation (10.22) with a mean temperature difference of approximately 40º F:

[latex]\Delta T / L=\frac{n_{t} \pi D_{o} U_{D} \Delta T_{m}}{\dot{m}_{i} C_{P, L}}=\frac{106 \pi(1.0 / 12) imes 91 imes 40}{113,636 imes 0.45}=1.975^{\circ} F / ft[/latex]

(iv) Length of sensible heating zone
The fractional length of the sensible heating zone is estimated using Equation (10.20):

[latex]\frac{L_{B C}}{L_{B C}+L_{C D}}=\frac{(\Delta T / \Delta P)_{ ext {sat }}}{(\Delta T / \Delta P)_{ ext {sat }}-\frac{(\Delta T / L)}{(\Delta P / L)}}[/latex]

[latex]\frac{L_{B C}}{8}=\frac{3.61}{3.61+\frac{1.975}{0.3125}}=0.364[/latex]

[latex]L_{B C} \cong 2.9 ft[/latex]

It follows that [latex]L_{ CD} ≅ L_{ AC}[/latex] ≅ 5.1 ft. It is assumed that the liquid level in the column sump is maintained at approximately the elevation of the upper tubesheet in the reboiler.
(i) Average two-phase density.
The two-phase density is calculated at a vapor fraction of [latex]x_{e}[/latex]/3 = 0.044. The Lockhart–Martinelli parameter is calculated using Equation (9.37):

[latex]X_{tt}=\left(\frac{1-x}{x} ight)^{0.9}\left( ho_{V} / ho_{L} ight)^{0.5}\left(\mu_{L} / \mu_{V} ight)^{0.1}[/latex]

[latex]=\left(\frac{1-0.044}{0.044} ight)^{0.9}(0.2 / 45)^{0.5}(0.4 / 0.0086)^{0.1}[/latex]

[latex]X_{ tt }=1.563[/latex]

Since this value is greater than unity, the Chisholm correlation, Equation (9.63), gives the slip ratio as:

[latex]S R=\left( ho_{L} / ho_{ hom } ight)^{0.5}[/latex]

The homogeneous density is given by Equation (9.51):

[latex] ho_{ hom }=\left[x / ho_{V}+(1-x) / ho_{L} ight]^{-1}=[0.044 / 0.2+0.956 / 45]^{-1}[/latex]

[latex] ho_{ hom }=4.1452 lbm / ft ^{3}[/latex]

Substitution into the above equation gives the slip ratio:

[latex]S R=(45 / 4.1452)^{0.5}=3.295[/latex]

Next, the void fraction is computed using Equation (9.59).

[latex]\epsilon_{V}=\frac{x}{x+S R(1-x) ho_{V} / ho_{L}}=\frac{0.044}{0.044+3.295 imes 0.956 imes 0.2 / 45}[/latex]

[latex]\epsilon_{V}=0.7586[/latex]

Finally, the two-phase density is computed from Equation (9.54):

[latex]\bar{ ho}_{t p}=\epsilon_{V} ho_{V}+\left(1-\epsilon_{V} ight) ho_{L}=0.7586 imes 0.2+0.2414 imes 45[/latex]

[latex]\bar{ ho}_{ tp }=11.01 lbm / ft ^{3}[/latex]

(j) Average two-phase multiplier.
The two-phase multiplier is calculated at a vapor fraction of 2[latex]x_{e}[/latex]/3 = 0.088. The Müller–Steinhagen and Heck (MSH) correlation, Equation (9.53), is used here

[latex]\bar{\phi}_{L O}^{2}=Y^{2} x^{2}+\left[1+2 x\left(Y^{2}-1 ight) ight](1-x)^{1 / 3}[/latex]

The Chisholm parameter, Y, is calculated using Equation (9.42) with n = 0.2585 for heat-exchanger tubes:

[latex]Y=\left( ho_{L} / ho_{V} ight)^{0.5}\left(\mu_{V} / \mu_{L} ight)^{n / 2}=(45 / 0.2)^{0.5}(0.0086 / 0.4)^{0.2585 / 2}[/latex]

Y = 9.13

Substituting in the MSH correlation gives:

[latex]\bar{\phi}_{L O}^{2}=(9.13)^{2}(0.088)^{3}+\left\{1+2 imes 0.088\left((9.13)^{2}-1 ight) ight\}(0.912)^{1 / 3}[/latex]

[latex]\bar{\phi}_{L O}^{2}=15.08[/latex]

(k) Two-phase multiplier for exit line.
The above calculation is repeated with x = [latex]x_{e}[/latex] = 0.132. For the exit pipe, however, the Chisholm parameter is calculated with n = 0.2314. Thus

[latex]Y=(45 / 0.2)^{0.5}(0.0086 / 0.4)^{0.2314 / 2}=9.62[/latex]

[latex]\phi_{L O, e x}^{2}=(9.62)^{2}(0.132)^{3}+\left\{1+2 imes 0.132\left((9.62)^{2}-1 ight) ight\}(0.868)^{1 / 3}[/latex]

[latex]\phi_{L O,e x}^{2}=24.22[/latex]

(l) Exit void fraction.
At x =[latex]x_{e}[/latex]= 0.132, the Lockhart–Martinelli parameter is:

[latex]X_{t t}=\left(\frac{1-0.132}{0.132} ight)^{0.9}(0.2 / 45)^{0.5}(0.4 / 0.0086)^{0.1}=0.533[/latex]

Since this value is less than 1.0, the Chisholm correlation gives the slip ratio as:

[latex]S R=\left( ho_{L} / ho_{V} ight)^{0.25}=(45 / 0.2)^{0.25}=3.873[/latex]

From Equation (9.59), the void fraction is:

[latex]\epsilon_{V, e}=\frac{x_{e}}{x_{e}+S R\left(1-x_{e} ight) ho_{V} / ho_{L}}=\frac{0.132}{0.132+3.873 imes 0.868 imes 0.2 / 45}[/latex]

[latex]\epsilon_{V, e}=0.8983[/latex]

(m) Acceleration parameter.
The acceleration parameter, γ , is given by Equation (10.12):

[latex]\gamma=\frac{\left(1-x_{e} ight)^{2}}{1-\epsilon_{V,}}+\frac{ ho_{L} x_{e}^{2}}{ ho_{V} \epsilon_{V,}}-1=\frac{0.868}{0.1017}+\frac{45(0.132)^{2}}{0.2 imes 0.8983}-1[/latex]

[latex]\gamma=10.77[/latex]

(n) Circulation rate.
Equation (10.15) is used to obtain a new estimate of the circulation rate. Due to the complexity of the equation, the individual terms are computed separately, starting with the numerator

numerator = [latex]3.2 imes 10^{10} D_{t}^{5} s_{L}\left(g / g_{C} ight)\left( ho_{L} L_{A C}-\bar{ ho}_{t p} L_{C D} ight)[/latex]

=[latex]3.2 imes 10^{10}(0.0695)^{5}(45 / 62.43)(1.0)(45 imes 5.1-11.01 imes 5.1)[/latex]

numerator = 6,483,575

Each of the four terms in the denominator is computed next:

[latex] ext { term } 1=6.6150 imes 10^{-5}[/latex]

[latex] ext { term } 2=f_{i n} L_{i n}\left(D_{t} / D_{i n} ight)^{5}=0.0199 imes 100(0.0695 / 0.5054)^{5}[/latex]

[latex] ext { term } 2=9.7859 imes 10^{-5}[/latex]

[latex] ext { term } 3=\left(f_{t} / n_{t}^{2} ight)\left(L_{B C}+L_{C D} \bar{\phi}_{L O}^{2} ight)=\frac{0.0319}{(106)^{2}}(2.9+5.1 imes 15.08)[/latex]

[latex] ext { term } 3=2.2658 imes 10^{-4}=22.658 imes 10^{-5}[/latex]

[latex] ext { term } 4=f_{e x} L_{e x} \phi_{L O e x}^{2}\left(D_{t} / D_{e x} ight)^{5}=0.0224 imes 50 imes 24.22(0.0695 / 0.835)^{5}[/latex]

[latex] ext { term } 4=1.0836 imes 10^{-4}=10.836 imes 10^{-5}[/latex]

Substituting the above values into Equation (10.15) gives:

[latex]\dot{m}_{i}^{2}=\frac{6,483,575}{(6.6150+9.7859+22.658+10.836) imes 10^{-5}}=1.2994 imes 10^{10}[/latex]

[latex]\dot{m}_{i}=113,991 lbm / h[/latex]

This value agrees with the assumed flow rate of 113,636 lbm/h to within about 0.3%, which is more than adequate for convergence. The average of the assumed and calculated values is taken as the final value, i.e.,

[latex]\dot{m}_{i}=\frac{113,991+113,636}{2} \cong 113,814 lbm / h[/latex]

(o) Mist flow limit.
The mass flux at the onset of mist flow is given by Equation (10.23a):

[latex]G_{t, m i s t}=1.8 imes 10^{6} X_{t t}=1.8 imes 10^{6} imes 0.533=959,400 lbm / h \cdot ft ^{2}[/latex]

The actual mass flux in the tubes is:

[latex]G_{t}=\frac{m_{i}}{n_{t}(\pi / 4) D_{t}^{2}}=\frac{113,814}{106(\pi / 4)(0.0695)^{2}}=283,029 lbm / h \cdot ft ^{2}[/latex]

The actual mass flux is far below the mist flow limit, as would be expected with a vapor fraction of only about 13%.
This completes the circulation rate calculation. The following steps deal with the zone analysis (stepwise calculations). To simplify matters, a single boiling zone is used. In this case, the pressure drop in the tubes is not recalculated and the circulation rate is not adjusted. Therefore, only heat-transfer calculations are involved in the zone analysis.
(p) Duty in boiling zone.
The cyclohexane temperature in the boiling zone is estimated based on the temperature gradient calculated above:

[latex]T_{c \gamma h x}=182+(\Delta T / L) L_{B C}=182+1.975 imes 2.9 \cong 187.7^{\circ} F[/latex]

Hence, the duty in the sensible heating zone is that required to raise the temperature of the liquid by 5.7º F:

[latex]q_{B C}=\dot{m}_{i} C_{P, L} \Delta T_{B C}=113,814 imes 0.45 imes 5.7=291,933 Btu / h[/latex]

The duty for the boiling zone is the total duty minus the duty for the sensible heating zone. Thus,

[latex]q_{C D}=q-q_{B C}=2.31 imes 10^{6}-291,933 \cong 2.018 imes 10^{6} Btu / h[/latex]

(q) Boiling heat-transfer coefficient.
Since the boiling fluid is a pure component, the Liu–Winterton correlation, Equation (9.80), is used to calculate the heattransfer coefficient. The average vapor weight fraction for the zone is used in the calculations, i.e., x = 0.132/2 = 0.066.

[latex]h_{b}=\left[\left(S_{L W} h_{n b} ight)^{2}+\left(E_{L W} h_{L} ight)^{2} ight]^{1 / 2}[/latex]

(i) The enhancement factor
The convective enhancement factor, [latex]E_{L W}[/latex], is given by Equation (9.82):

[latex]E_{L W}=\left[1+x \operatorname{Pr}_{L}\left( ho_{L}- ho_{V} ight) / ho_{V} ight]^{0.35}[/latex]

[latex]=[1+0.066 imes 5.063(45-0.2) / 0.2]^{0.35}[/latex]

[latex]E_{L W}=4.550[/latex]

(ii) The suppression factor
The nucleate boiling suppression factor, [latex]S_{L W}[/latex], is given by Equation (9.81):

[latex]S_{L W}=\left[1+0.055 E_{L W}^{0.1} R e_{L}^{0.16} ight]^{-1}[/latex]

The Reynolds number is calculated for the liquid phase flowing alone in the tubes:

[latex]R e_{L}=\frac{4(1-x)\left(\dot{m}_{i} / n_{t} ight)}{\pi D_{t} \mu_{L}}=\frac{4(1-0.066)(113,814 / 106)}{\pi imes 0.0695 imes 0.4 imes 2.419}[/latex]

[latex]R e_{L}=18,987[/latex]

Substituting into the above equation for [latex]S_{L W}[/latex] gives:

[latex]S_{L W}=\left[1+0.055(4.550)^{0.1}(18,987)^{0.16} ight]^{-1}=0.7636[/latex]

(iii) Convective heat-transfer coefficient
The Dittus–Boelter correlation, Equation (9.75), is used in conjunction with the Liu–Winterton correlation to calculate [latex]h_{L}[/latex].

[latex]h_{L}=0.023\left(k_{L} / D_{t} ight) R e_{L}^{0.8} P r_{L}^{0.4}[/latex]

[latex]=0.023(0.086 / 0.0695)(18,987)^{0.8}(5.063)^{0.4}[/latex]

[latex]h_{L}=144 Btu / h \cdot ft ^{2} \cdot{ }^{\circ} F[/latex]

(iv) Nucleate boiling heat-transfer coefficient
The Cooper correlation in the form of Equation (9.6a) is used to calculate [latex]h_{n b}[/latex]:

[latex]h_{n b}=21 \hat{q}^{0.67} P_{r}^{0.12}\left(-\log _{10} P_{r} ight)^{-0.55} M^{-0.5}[/latex]

For cyclohexane, the molecular weight is M = 84. The pressure in the boiling zone is estimated as the vapor pressure of cyclohexane at 187.7º F = 359.8 K

[latex]P_{s a t}=\exp \left[15.7527-\frac{2766.63}{359.8-50.50} ight]=904.96 ext { torr }=17.5 psia[/latex]

The reduced pressure is then:

[latex]P_{r}=P / P_{c}=17.5 / 590.5=0.0296[/latex]

The heat flux is estimated using the total duty and total tube length, as follows:

[latex]\hat{q} \cong \frac{2.31 imes 10^{6}}{106 \pi imes 0.0695 imes 8}=12,476 Btu / h \cdot ft ^{2}[/latex]

Substituting into the above equation for[latex]h_{n b}[/latex] gives:

[latex]h_{n b}=21(12,476)^{0.67}(0.0296)^{0.12}\left(-\log _{10} 0.0296 ight)^{-0.55}(84)^{-0.5}[/latex]

[latex]h_{n b}=660 Btu / h \cdot ft ^{2} \cdot{ }^{\circ} F[/latex]

(v) Convective boiling coefficient
Substituting the results from the above steps into the Liu–Winterton correlation gives the following result for [latex]h_{b}[/latex]:

[latex]h_{b}=\left[(0.7636 imes 660)^{2}+(4.550 imes 144)^{2} ight]^{1 / 2}=827 Btu / h \cdot ft ^{2} \cdot{ }^{\circ} F[/latex]

(r) Overall coefficient.
Due to the higher velocity and greater agitation in the boiling zone, a fouling factor of 0.0005 h ft².°F/Btu is deemed appropriate for cyclohexane. A film coefficient, including fouling allowance, of 1500 Btu/h . ft².º F is again assumed for steam. The overall heat-transfer coefficient for the boiling zone is then:

[latex]U_{D}=\left[\frac{D_{o}}{D_{i}}\left(\frac{1}{h_{i}}+R_{D i} ight)+\frac{D_{o} \ln \left(D_{o} / D_{i} ight)}{2 k_{ ext {tube }}}+\left(\frac{1}{h_{0}}+R_{D o} ight) ight]^{-1}[/latex]

[latex]=\left[\frac{1.0}{0.834}\left(\frac{1}{827}+0.0005 ight)+\frac{(1.0 / 12) \ln (1.0 / 0.834)}{2 imes 26}+\frac{1}{1500} ight]^{-1}[/latex]

[latex]U_{D}=332.6 Btu / h \cdot ft ^{2} \cdot{ }^{\circ} F[/latex]

(s) Check heat flux and iterate if necessary.
The mean temperature difference for the boiling zone is taken as:

[latex]\Delta T_{m}=T_{ ext {steam }}-T_{c γ h x}=222.4-187.7=34.7^{\circ} F[/latex]

Since the saturation temperature decreases with decreasing pressure, both the steam and cyclohexane temperatures will vary somewhat over the length of the boiling zone due to the pressure drops experienced by the two streams. These effects are neglected here. Thus, the heat flux is:

[latex]\hat{q}=U_{D} \Delta T_{m}=332.6 imes 34.7=11,541 Btu / h \cdot ft ^{2}[/latex]
This value is within 10% of the initial estimate of 12,476 Btu/h . ft². After a few more iterations, the following converged values are obtained:

[latex]h_{n b}=622 Btu / h \cdot ft ^{2} \cdot{ }^{\circ} F[/latex]

[latex]h_{b}=809 Btu / h \cdot ft ^{2} \cdot{ }^{\circ} F[/latex]

[latex]U_{D}=329 Btu / h \cdot ft ^{2} \cdot{ }^{\circ} F[/latex]

[latex]\hat{q}=11,416 Btu / h \cdot ft ^{2}[/latex]

(t) Tube length.
The tube length required for the boiling zone is calculated as follows:

[latex]L_{ ext {req }}=\frac{q_{C D}}{n_{t} \pi D_{o} U_{D} \Delta T_{m}}=\frac{2.018 imes 10^{6}}{106 \pi(1.0 / 12) imes 329 imes 34.7} \cong 6.4 ft[/latex]

This value is greater than the available length of 5.1 ft, indicating that the reboiler is somewhat under-sized.
(u) Critical heat flux.
For brevity, the critical heat flux is estimated using Palen’s method as given by Equation (9.84a):

[latex]\hat{q}_{c}=16,070\left(D_{t}^{2} / L ight)^{0.35} P_{c}^{0.61} P_{r}^{0.25}\left(1-P_{r} ight)[/latex]

[latex]=16,070\left[(0.0695)^{2} / 8 ight]^{0.35}(590.5)^{0.61}(0.0296)^{0.25}(1-0.0296)[/latex]

[latex]\hat{q}_{c} \cong 23,690 Btu / h \cdot ft ^{2}[/latex]

[latex]\hat{q} / \hat{q}_{c}=11,416 / 23,690 \cong 0.48[/latex]

Thus, the heat flux is safely below the critical value.
(v) Design modification.
Based on the above calculations, the only problem with the initial design is that the unit is under-sized. The under-surfacing is due to the presence of a significant sensible heating zone that was not considered in the preliminary design. Although a more rigorous analysis using more zones might yield a different result, it is assumed here that some modification of the initial design is required. Three possible design changes are the following:
(i) Increase the tube length from 8 to 10 ft. This change will increase the static head and, hence, the degree of subcooling at the reboiler entrance. It will thus tend to increase the length of the sensible heating zone.
(ii) Increase the number of tubes. From the tube-count table, the next largest unit is a 17.25-in. shell containing 147 tubes. This represents an increase of about 39% in heat-transfer area, whereas the initial design is under-surfaced by only about 20%.
(iii) Raise the steam temperature by 5 to 8°F, corresponding to a steam pressure of 20 to 21 psia. This change will reduce the tube length required in both the sensible heating and boiling zones. Of the three options considered here, this one appears to be the simplest and most cost effective.
Each of the above changes will affect the circulation rate; therefore, verification requires essentially complete recalculation for each case. Due to the lengthiness of the calculations, no further analysis is presented here.

Question 10.4: A reboiler is required to supply 15,000 lb/h of vapor to a d...

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