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')[source]¶
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)[source]¶
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