Convection to jacketed vessels (ht.conv_jacket)

ht.conv_jacket.Lehrer(m, Dtank, Djacket, H, Dinlet, rho, Cp, k, mu, muw=None, isobaric_expansion=None, dT=None, inlettype='tangential', inletlocation='auto')

Calculates average heat transfer coefficient for a jacket around a vessel according to [1] as described in [2].

$$Nu_{S,L} = \left[\frac{0.03Re_S^{0.75}Pr}{1 + \frac{1.74(Pr-1)} {Re_S^{0.125}}}\right]\left(\frac{\mu}{\mu_w}\right)^{0.14}$$

$$d_g = \left(\frac{8}{3}\right)^{0.5}\delta$$

$$v_h = (v_Sv_{inlet})^{0.5} + v_A$$

$$v_{inlet} = \frac{Q}{\frac{\pi}{4}d_{inlet}^2}$$

$$v_s = \frac{Q}{\frac{\pi}{4}(D_{jacket}^2 - D_{tank}^2)}$$

For Radial inlets:

$$v_A = 0.5(2g H \beta\delta \Delta T)^{0.5}$$

For Tangential inlets:

$$v_A = 0$$

Parameters

mfloat

Mass flow rate of fluid, [kg/s]

Dtankfloat

Outer diameter of tank or vessel surrounded by jacket, [m]

Djacketfloat

Inner diameter of jacket surrounding a vessel or tank, [m]

Hfloat

Height of the vessel or tank, [m]

Dinletfloat

Inner diameter of inlet into the jacket, [m]

rhofloat

Density of the fluid at Tm [kg/m^3]

Cpfloat

Heat capacity of fluid at Tm [J/kg/K]

kfloat

Thermal conductivity of fluid at Tm [W/m/K]

mufloat

Viscosity of fluid at Tm [Pa*s]

muwfloat, optional

Viscosity of fluid at Tw [Pa*s]

isobaric_expansionfloat, optional

Constant pressure expansivity of a fluid, [m^3/mol/K]

dTfloat, optional

Temperature difference of fluid in jacket, [K]

inlettypestr, optional

Either ‘tangential’ or ‘radial’

inletlocationstr, optional

Either ‘top’ or ‘bottom’ or ‘auto’

Returns

hfloat

Average heat transfer coefficient inside the jacket [W/m^2/K]

Notes

If the fluid is heated and enters from the bottom, natural convection assists the heat transfer and the Grashof term is added; if it were to enter from the top, it would be subtracted. The situation is reversed if entry is from the top.

References

1

Lehrer, Isaac H. “Jacket-Side Nusselt Number.” Industrial & Engineering Chemistry Process Design and Development 9, no. 4 (October 1, 1970): 553-58. doi:10.1021/i260036a010.

2(1,2,3)

Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010.

Examples

Example as in [2], matches completely.

>>> Lehrer(m=2.5, Dtank=0.6, Djacket=0.65, H=0.6, Dinlet=0.025, dT=20.,
... rho=995.7, Cp=4178.1, k=0.615, mu=798E-6, muw=355E-6)
2922.128124761829

Examples similar to in [2] but covering the other case:

>>> Lehrer(m=2.5, Dtank=0.6, Djacket=0.65, H=0.6, Dinlet=0.025, dT=20.,
... rho=995.7, Cp=4178.1, k=0.615, mu=798E-6, muw=355E-6,
... inlettype='radial', isobaric_expansion=0.000303)
3269.4389632666557

ht.conv_jacket.Stein_Schmidt(m, Dtank, Djacket, H, Dinlet, rho, Cp, k, mu, muw=None, rhow=None, inlettype='tangential', inletlocation='auto', roughness=0.0)

Calculates average heat transfer coefficient for a jacket around a vessel according to [1] as described in [2].

$$l_{ch} = \left[\left(\frac{\pi}{2}\right)^2 D_{tank}^2+H^2\right]^{0.5}$$

$$d_{ch} = 2\delta$$

$$Re_j = \frac{v_{ch}d_{ch}\rho}{\mu}$$

$$Gr_J = \frac{g\rho(\rho-\rho_w)d_{ch}^3}{\mu^2}$$

$$Re_{J,eq} = \left[Re_J^2\pm \left(\frac{|Gr_J|\frac{H}{d_{ch}}}{50} \right)\right]^{0.5}$$

$$Nu_J = (Nu_A^3 + Nu_B^3 + Nu_C^3 + Nu_D^3)^{1/3}\left(\frac{\mu} {\mu_w}\right)^{0.14}$$

$$Nu_J = \frac{h d_{ch}}{k}$$

$$Nu_A = 3.66$$

$$Nu_B = 1.62 Pr^{1/3}Re_{J,eq}^{1/3}\left(\frac{d_{ch}}{l_{ch}} \right)^{1/3}$$

$$Nu_C = 0.664Pr^{1/3}(Re_{J,eq}\frac{d_{ch}}{l_{ch}})^{0.5}$$

$$\text{if } Re_{J,eq} < 2300: Nu_D = 0$$

$$Nu_D = 0.0115Pr^{1/3}Re_{J,eq}^{0.9}\left(1 - \left(\frac{2300} {Re_{J,eq}}\right)^{2.5}\right)\left(1 + \left(\frac{d_{ch}}{l_{ch}} \right)^{2/3}\right)$$

For Radial inlets:

$$v_{ch} = v_{Mit}\left(\frac{\ln\frac{b_{Mit}}{b_{Ein}}}{1 - \frac{b_{Ein}}{b_{Mit}}}\right)$$

$$b_{Ein} = \frac{\pi}{8}\frac{D_{inlet}^2}{\delta}$$

$$b_{Mit} = \frac{\pi}{2}D_{tank}\sqrt{1 + \frac{\pi^2}{4}\frac {D_{tank}^2}{H^2}}$$

$$v_{Mit} = \frac{Q}{2\delta b_{Mit}}$$

For Tangential inlets:

$$v_{ch} = (v_x^2 + v_z^2)^{0.5}$$

$$v_x = v_{inlet}\left(\frac{\ln[1 + \frac{f_d D_{tank}H}{D_{inlet}^2} \frac{v_x(0)}{v_{inlet}}]}{\frac{f_d D_{tank}H}{D_{inlet}^2}}\right)$$

$$v_x(0) = K_3 + (K_3^2 + K_4)^{0.5}$$

$$K_3 = \frac{v_{inlet}}{4} -\frac{D_{inlet}^2v_{inlet}}{4f_d D_{tank}H}$$

$$K_4 = \frac{D_{inlet}^2v_{inlet}^2}{2f_d D_{tank} H}$$

$$v_z = \frac{Q}{\pi D_{tank}\delta}$$

$$v_{inlet} = \frac{Q}{\frac{\pi}{4}D_{inlet}^2}$$

Parameters

mfloat

Mass flow rate of fluid, [kg/m^3]

Dtankfloat

Outer diameter of tank or vessel surrounded by jacket, [m]

Djacketfloat

Inner diameter of jacket surrounding a vessel or tank, [m]

Hfloat

Height of the vessel or tank, [m]

Dinletfloat

Inner diameter of inlet into the jacket, [m]

rhofloat

Density of the fluid at Tm [kg/m^3]

Cpfloat

Heat capacity of fluid at Tm [J/kg/K]

kfloat

Thermal conductivity of fluid at Tm [W/m/K]

mufloat

Viscosity of fluid at Tm [Pa*s]

muwfloat, optional

Viscosity of fluid at Tw [Pa*s]

rhowfloat, optional

Density of the fluid at Tw [kg/m^3]

inlettypestr, optional

Either ‘tangential’ or ‘radial’

inletlocationstr, optional

Either ‘top’ or ‘bottom’ or ‘auto’

roughnessfloat, optional

Roughness of the tank walls [m]

Returns

hfloat

Average transfer coefficient inside the jacket [W/m^2/K]

Notes

[1] is in German and has not been reviewed. Multiple other formulations are considered in [1].

If the fluid is heated and enters from the bottom, natural convection assists the heat transfer and the Grashof term is added; if it were to enter from the top, it would be subtracted. The situation is reversed if entry is from the top.

References

1(1,2,3)

Stein, Prof Dr-Ing Werner Alexander, and Dipl-Ing (FH) Wolfgang Schmidt. “Wärmeübergang auf der Wärmeträgerseite eines Rührbehälters mit einem einfachen Mantel.” Forschung im Ingenieurwesen 59, no. 5 (May 1993): 73-90. doi:10.1007/BF02561203.

2(1,2)

Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010.

Examples

Example as in [2], matches in all but friction factor:

>>> Stein_Schmidt(m=2.5, Dtank=0.6, Djacket=0.65, H=0.6, Dinlet=0.025,
... rho=995.7, Cp=4178.1, k=0.615, mu=798E-6, muw=355E-6, rhow=971.8)
5695.2041698088615