# Tank and helical coil sizing (fluids.geometry)¶

class fluids.geometry.TANK(D=None, L=None, horizontal=True, sideA=None, sideB=None, sideA_a=0, sideB_a=0, sideA_f=1.0, sideA_k=0.06, sideB_f=1.0, sideB_k=0.06, sideA_a_ratio=0.25, sideB_a_ratio=0.25, L_over_D=None, V=None)[source]

Bases: object

Class representing tank volumes and levels. All parameters are also attributes.

Notes

For torpsherical tank heads, the following f and k parameters are used in standards. The default is ASME F&D .

f k
2:1 semi-elliptical 0.9 0.17
ASME F&D 1 0.06
ASME 80/6 0.8 0.06
ASME 80/10 F&D 0.8 0.1
DIN 28011 1 0.1
DIN 28013 0.8 0.154

Examples

Total volume of a tank:

>>> TANK(D=1.2, L=4, horizontal=False).V_total
4.523893421169302


Volume of a tank at a given height:

>>> TANK(D=1.2, L=4, horizontal=False).V_from_h(.5)
0.5654866776461628


Height of liquid for a given volume:

>>> TANK(D=1.2, L=4, horizontal=False).h_from_V(.5)
0.44209706414415384


Surface area of a tank with a conical head:

>>> T1 = TANK(V=10, L_over_D=0.7, sideB='conical', sideB_a=0.5)
>>> T1.A, T1.A_sideA, T1.A_sideB, T1.A_lateral
(24.94775907657148, 5.118555935958284, 5.497246519930003, 14.331956620683192)


Solving for tank volumes, first horizontal, then vertical:

>>> TANK(D=10., horizontal=True, sideA='conical', sideB='conical', V=500).L
4.699531057009147
>>> TANK(L=4.69953105701, horizontal=True, sideA='conical', sideB='conical', V=500).D
9.999999999999407
>>> TANK(L_over_D=0.469953105701, horizontal=True, sideA='conical', sideB='conical', V=500).L
4.69953105700979

>>> TANK(D=10., horizontal=False, sideA='conical', sideB='conical', V=500).L
4.699531057009147
>>> TANK(L=4.69953105701, horizontal=False, sideA='conical', sideB='conical', V=500).D
9.999999999999407
>>> TANK(L_over_D=0.469953105701, horizontal=False, sideA='conical', sideB='conical', V=500).L
4.699531057009791

Attributes: table : bool Whether or not a table of heights-volumes has been generated h_max : float Height of the tank, [m] V_total : float Total volume of the tank as calculated [m^3] heights : ndarray Array of heights between 0 and h_max, [m] volumes : ndarray Array of volumes calculated from the heights, [m^3] A : float Total surface area of the tank, [m^2] A_sideA : float Surface area of sideA, [m^2] A_sideB : float Surface area of sideB, [m^2] A_lateral : float Surface area of the lateral side, [m^2] c_forward : ndarray Coefficients for the Chebyshev approximations in calculating V from h, [-] c_backward : ndarray Coefficients for the Chebyshev approximations in calculating h from V, [-]

Methods

 V_from_h(h[, method]) Method to calculate the volume of liquid in a fully defined tank given a specified height h. h_from_V(V[, method]) Method to calculate the height of liquid in a fully defined tank given a specified volume of liquid in it V. set_chebyshev_approximators([deg_forward, …]) Method to derive and set coefficients for chebyshev polynomial function approximation of the height-volume and volume-height relationship. set_misc() Set more parameters, after the tank is better defined than in the __init__ function. set_table([n, dx]) Method to set an interpolation table of liquids levels versus volumes in the tank, for a fully defined tank. solve_tank_for_V() Method which is called to solve for tank geometry when a certain volume is specified.
V_from_h(h, method='full')[source]

Method to calculate the volume of liquid in a fully defined tank given a specified height h. h must be under the maximum height. If the method is ‘chebyshev’, and the coefficients have not yet been calculated, they are created by calling set_chebyshev_approximators.

Parameters: h : float Height specified, [m] method : str One of ‘full’ (calculated rigorously) or ‘chebyshev’ V : float Volume of liquid in the tank up to the specified height, [m^3]
chebyshev = False
h_from_V(V, method='spline')[source]

Method to calculate the height of liquid in a fully defined tank given a specified volume of liquid in it V. V must be under the maximum volume. If the method is ‘spline’, and the interpolation table is not yet defined, creates it by calling the method set_table. If the method is ‘chebyshev’, and the coefficients have not yet been calculated, they are created by calling set_chebyshev_approximators.

Parameters: V : float Volume of liquid in the tank up to the desired height, [m^3] method : str One of ‘spline’, ‘chebyshev’, or ‘brenth’ h : float Height of liquid at which the volume is as desired, [m]
set_chebyshev_approximators(deg_forward=50, deg_backwards=200)[source]

Method to derive and set coefficients for chebyshev polynomial function approximation of the height-volume and volume-height relationship.

A single set of chebyshev coefficients is used for the entire height- volume and volume-height relationships respectively.

The forward relationship, V_from_h, requires far fewer coefficients in its fit than the reverse to obtain the same relative accuracy.

Optionally, deg_forward or deg_backwards can be set to None to try to automatically fit the series to machine precision.

Parameters: deg_forward : int, optional The degree of the chebyshev polynomial to be created for the V_from_h curve, [-] deg_backwards : int, optional The degree of the chebyshev polynomial to be created for the h_from_V curve, [-]
set_misc()[source]

Set more parameters, after the tank is better defined than in the __init__ function.

Notes

Two of D, L, and L_over_D must be known when this function runs. The other one is set from the other two first thing in this function. a_ratio parameters are used to calculate a values for the heads here, if applicable. Radius is calculated here. Maximum tank height is calculated here. V_total is calculated here.

set_table(n=100, dx=None)[source]

Method to set an interpolation table of liquids levels versus volumes in the tank, for a fully defined tank. Normally run by the h_from_V method, this may be run prior to its use with a custom specification. Either the number of points on the table, or the vertical distance between steps may be specified.

Parameters: n : float, optional Number of points in the interpolation table, [-] dx : float, optional Vertical distance between steps in the interpolation table, [m]
solve_tank_for_V()[source]

Method which is called to solve for tank geometry when a certain volume is specified. Will be called by the __init__ method if V is set.

Notes

Raises an error if L and either of sideA_a or sideB_a are specified; these can only be set once D is known. Raises an error if more than one of D, L, or L_over_D are specified. Raises an error if the head ratios are not provided.

Calculates initial guesses assuming no heads are present, and then uses fsolve to determine the correct dimentions for the tank.

Tested, but bugs and limitations are expected here.

table = False
class fluids.geometry.HelicalCoil(Dt, Do=None, pitch=None, H=None, N=None, H_total=None, Do_total=None, Di=None)[source]

Bases: object

Class representing a helical coiled tube, as are found in many heated tanks and some small nuclear reactors. All parameters are also attributes.

One set of the following parameters is required; inner tube diameter is optional.

• Tube outer diameter, coil outer diameter, pitch, number of coil turns
• Tube outer diameter, coil outer diameter, pitch, height
• Tube outer diameter, coil outer diameter, number of coil turns, height
Parameters: Dt : float Outer diameter of the tube wound to make up the helical spiral, [m] Do : float, optional Diameter of the spiral as measured from the center of the coil on one side to the center of the coil on the other side, [m] Do_total : float, optional Diameter of the spiral as measured from one edge of the tube to the other edge; equal to Do + Dt; either Do or Do_total may be specified and the other will be calculated [m] pitch : float, optional Height change from one coil to the next as measured from the middles of the tube, [m] H : float, optional Height of the spiral, as measured from the middle of the bottom of the tube to the middle of the top of the tube, [m] H_total : float, optional Height of the spiral as measured from one edge of the tube to the other edge; equal to H_total + Dt; either may be specified and the other will be calculated [m] N : float, optional Number of coil turns; may be specified along with pitch instead of specifying H or H_total, [-] Di : float, optional Inner diameter of the tube; if specified, inside and annulus properties will be calculated, [m]

Notes

Do must be larger than Dt.

References

 [1] El-Genk, Mohamed S., and Timothy M. Schriener. “A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes.” Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693.

Examples

>>> C1 = HelicalCoil(Do=30, H=20, pitch=5, Dt=2)
>>> C1.N, C1.tube_length, C1.surface_area
(4.0, 377.5212621504738, 2372.0360474917497)


Same coil, with the inputs one would physically measure from the coil, and a specified inlet diameter:

>>> C1 = HelicalCoil(Do_total=32, H_total=22, pitch=5, Dt=2, Di=1.8)
>>> C1.N, C1.tube_length, C1.surface_area
(4.0, 377.5212621504738, 2372.0360474917497)
>>> C1.inner_surface_area, C1.inlet_area, C1.inner_volume, C1.total_volume, C1.annulus_volume
(2134.832442742575, 2.5446900494077327, 960.6745992341587, 1186.0180237458749, 225.3434245117162)

Attributes: tube_circumference : float Circumference of the tube as measured though its center, not inner or outer edges; $$C = \pi D_o$$, [m] tube_length : float Length of tube used to make the helical coil; $$L = \sqrt{(\pi D_o\cdot N)^2 + H^2}$$, [m] surface_area : float Surface area of the outer surface of the helical coil; $$A_t = \pi D_t L$$, [m^2] inner_surface_area : float Surface area of the inner surface of the helical coil; calculated if Di is supplied; $$A_{inside} = \pi D_i L$$, [m^2] inlet_area : float Area of the inlet to the helical coil; calculated if Di is supplied; $$A_{inlet} = \frac{\pi}{4} D_i^2$$, [m^2] inner_volume : float Volume of the tube as would be filled by a fluid, useful for weight calculations; calculated if Di is supplied; $$V_{inside} = A_i L$$, [m^3] annulus_area : float Area of the annulus (wall of the pipe); calculated if Di is supplied; $$A_a = \frac{\pi}{4} (D_t^2 - D_i^2)$$, [m^2] annulus_volume : float Volume of the annulus (wall of the pipe); calculated if Di is supplied, useful for weight calculations; $$V_a = A_a L$$, [m^3] total_volume : float Total volume occupied by the pipe and the fluid inside it; $$V = D_t L$$, [m^3] helix_angle : float Angle between the pitch and coil diameter; used in some calculations; $$\alpha = \arctan \left(\frac{p_t}{\pi D_o}\right)$$, [radians] curvature : float Coil curvature, useful in some calculations; $$\delta = \frac{D_t}{D_o[1 + 4\pi^2 \tan^2(\alpha)]}$$, [-]
class fluids.geometry.PlateExchanger(amplitude, wavelength, chevron_angle=45, width=None, length=None, thickness=None, d_port=None, plates=None)[source]

Bases: object

Class representing a plate heat exchanger with sinusoidal ridges. All parameters are also attributes.

Parameters: amplitude : float Half the height of the wave of the ridges, [m] wavelength : float Distance between the bottoms of two of the ridges (sometimes called pitch), [m] chevron_angle : float or tuple(2), optional Angle of the plate corrugations with respect to the vertical axis (the direction of flow if the plates were straight), between 0 and 90. Many plate exchangers use two alternating patterns; use a tuple of the two angles for that situation [degrees] width : float, optional Width of the plates in the heat exchanger, between the gaskets, [m] length : float, optional Length of the heat exchanger as measured from one port to the other, excluding the diameter of the ports themselves (little useful heat transfer happens there), [m] thickness : float, optional Thickness of the metal making up the plates, [m] d_port : float, optional The diameter of the ports in the plates, [m] plates : int, optional The number of plates in the heat exchanger, including the two not used for heat transfer at the beginning and end [-]

Notes

Only wavelength and amplitude are required as inputs to this function.

References

 [1] Amalfi, Raffaele L., Farzad Vakili-Farahani, and John R. Thome. “Flow Boiling and Frictional Pressure Gradients in Plate Heat Exchangers. Part 1: Review and Experimental Database.” International Journal of Refrigeration 61 (January 2016): 166-84. doi:10.1016/j.ijrefrig.2015.07.010.

Examples

>>> PlateExchanger(amplitude=5E-4, wavelength=3.7E-3, length=1.2, width=.3,
... d_port=.05, plates=51)
<Plate heat exchanger, amplitude=0.0005 m, wavelength=0.0037 m, chevron_angles=45/45 degrees, area enhancement factor=1.16119, width=0.3 m, length=1.2 m, port diameter=0.05 m, heat transfer area=20.4833 m^2, 51 plates>

Attributes: chevron_angles : tuple(2) The two specified angles (repeated value if only one specified), [degrees] chevron_angle : float The averaged angle of the chevrons, [degrees] inclination_angle : float 90 - chevron_angle, used in many publications instead of chevron_angle, [degrees] plate_corrugation_aspect_ratio : float The aspect ratio of the corrugations $$\gamma = \frac{4a}{\lambda}$$, [-] plate_enlargement_factor : float The extra surface area multiplier as compared to a flat plate caused the corrugations, [-] D_eq : float Equivalent diameter of the channels, $$D_{eq} = 4a$$ [m] D_hydraulic : float Hydraulic diameter of the channels, $$D_{hyd} = \frac{4a}{\phi}$$ [m] length_port : float Port center to port center along the direction of flow, [m] A_plate_surface : float The surface area of one plate in the heat exchanger, including the extra due to corrugations (excluding the bit between the ports), $$A_p = L\cdot W\cdot \phi$$ [m^2] A_heat_transfer : float The total surface area available for heat transfer in the exchanger, the multiple of A_plate_surface by the number of plates after removing the two on the edges, [m^2] A_channel_flow : float The area for the fluid to flow in one channel, $$W\cdot b$$ [m^2] channels : int The number of plates minus one, [-] channels_per_fluid : int Half the number of total channels, [-] plate_exchanger_identifier : str Method to create an identifying string in format ‘L’ + wavelength + ‘A’ + amplitude + ‘B’ + chevron angle-chevron angle.

Methods

 plate_enlargement_factor_analytical(…) Calculates the enhancement factor of the sinusoidal waves of the plate heat exchanger.
static plate_enlargement_factor_analytical(amplitude, wavelength)[source]

Calculates the enhancement factor of the sinusoidal waves of the plate heat exchanger. This is the multiplier for the flat plate area to obtain the actual area available for heat transfer. Obtained from the following integral:

\begin{align}\begin{aligned}\phi = \frac{\text{Effective area}}{\text{Projected area}} = \frac{\int_0^\lambda\sqrt{1 + \left(\frac{\gamma\pi}{2}\right)^2 \cos^2\left(\frac{2\pi}{\lambda}x\right)}dx}{\lambda}\\\gamma = \frac{4a}{\lambda}\end{aligned}\end{align}

The solution to the integral is:

$\phi = \frac{2E\left(\frac{-4a^2\pi^2}{\lambda^2}\right)}{\pi}$

where E is the complete elliptic integral of the second kind, calculated with SciPy.

Parameters: amplitude : float Half the height of the wave of the ridges, [m] wavelength : float Distance between the bottoms of two of the ridges (sometimes called pitch), [m] plate_enlargement_factor : float The extra surface area multiplier as compared to a flat plate caused the corrugations, [-]

Notes

This is the exact analytical integral, obtained via Mathematica, Maple, and quite a bit of trial and error. It is confirmed via numerical integration. The expression normally given is an approximation as follows:

\begin{align}\begin{aligned}\phi = \frac{1}{6}\left(1+\sqrt{1+A^2} + 4\sqrt{1+A^2/2}\right)\\A = \frac{2\pi a}{\lambda}\end{aligned}\end{align}

Most plate heat exchangers approximate a sinusoidal geometry only.

Examples

>>> PlateExchanger.plate_enlargement_factor_analytical(amplitude=5E-4, wavelength=3.7E-3)
1.1611862034509677

plate_exchanger_identifier

Method to create an identifying string in format ‘L’ + wavelength + ‘A’ + amplitude + ‘B’ + chevron angle-chevron angle. Wavelength and amplitude are specified in units of mm and rounded to two decimal places.

class fluids.geometry.RectangularFinExchanger(fin_height, fin_thickness, fin_spacing, length=None, width=None, layers=None, plate_thickness=None, flow='crossflow')[source]

Bases: object

Class representing a plate-fin heat exchanger with straight rectangular fins. All parameters are also attributes.

Parameters: fin_height : float The total distance between the two metal plates sandwiching the fins and holding them together (abbreviated h), [m] fin_thickness : float The thickness of the material the fins were formed from (abbreviated t), [m] fin_spacing : float The unit cell spacing from one fin to the next; the space between the sides of two fins plus one thickness (abbreviated s), [m] length : float, optional The total length of the flow passage of the plate-fin exchanger (abbreviated L), [m] width : float, optional The total width of the space the fins are in; this is also $$N_{fins}\times s$$ (abbreviated W), [m] layers : int, optional The number of layers in the plate-fin exchanger; note these HX almost always single-pass only, [-] plate_thickness : float, optional The thickness of the metal separator between layers, [m] flow : str, optional One of ‘counterflow’, ‘crossflow’, or ‘parallelflow’

Notes

The only required parameters are the fin geometry itself; fin_height, fin_thickness, and fin_spacing.

References

 [1] Yang, Yujie, and Yanzhong Li. “General Prediction of the Thermal Hydraulic Performance for Plate-Fin Heat Exchanger with Offset Strip Fins.” International Journal of Heat and Mass Transfer 78 (November 1, 2014): 860-70. doi:10.1016/j.ijheatmasstransfer.2014.07.060.
 [2] Sheik Ismail, L., R. Velraj, and C. Ranganayakulu. “Studies on Pumping Power in Terms of Pressure Drop and Heat Transfer Characteristics of Compact Plate-Fin Heat Exchangers-A Review.” Renewable and Sustainable Energy Reviews 14, no. 1 (January 2010): 478-85. doi:10.1016/j.rser.2009.06.033.

Examples

>>> PFE = RectangularFinExchanger(0.03, 0.001, 0.012)
>>> PFE.Dh
0.01595

Attributes: channel_height : float The height of the channel the fluid flows in $$\text{channel height } = \text{fin height} - \text{fin thickness}$$, [m] channel_width : float The width of the channel the fluid flows in $$\text{channel width } = \text{fin spacing} - \text{fin thickness}$$, [m] fin_count : int The number of fins per unit length of the layer, $$\text{fin count} = \frac{1}{\text{fin spacing}}$$, [1/m] blockage_ratio : float The fraction of the layer which is blocked to flow by the fins, $$\text{blockage ratio} = \frac{s\cdot h - s\cdot t - t(h-t)}{s\cdot h}$$, [m] A_channel : float Flow area of a single channel in a single layer, $$\text{channel area} = (s-t)(h-t)$$, [m] P_channel : float Wetted perimeter of a single channel in a single layer, $$\text{channel perimeter} = 2(s-t) + 2(h-t)$$, [m] Dh : float Hydraulic diameter of a single channel in a single layer, $$D_{hydraulic} = \frac{4 A_{channel}}{P_{channel}}$$, [m] layer_thickness : float The thickness of a single layer - the sum of a fin height and a plate thickness, [m] layer_fin_count : int The number of fins in a layer; rounded to the nearest whole fin, [-] A_HX_layer : float The surface area including fins for heat transfer in one layer of the HX, [m^2] A_HX : float The total surface area of the heat exchanger with all layers combined, [m^2] height : float The height of all the layers of the heat exchanger combined, plus one extra plate thickness, [m] volume : float The product of the height, width, and length of the HX, [m^3] A_specific_HX : float The specific surface area of the heat exchanger - square meters per meter cubed, [m^3]

Methods

 set_overall_geometry
set_overall_geometry()[source]
class fluids.geometry.RectangularOffsetStripFinExchanger(fin_length, fin_height, fin_thickness, fin_spacing, length=None, width=None, layers=None, plate_thickness=None, flow='crossflow')[source]

Methods

 set_overall_geometry
class fluids.geometry.HyperbolicCoolingTower(H_inlet, D_outlet, H_outlet, D_inlet=None, D_base=None, D_throat=None, H_throat=None, H_support=None, D_support=None, n_support=None, inlet_rounding=None)[source]

Bases: object

Class representing the geometry of a hyperbolic cooling tower, as used in many industries especially the poewr industry. All parameters are also attributes.

H_inlet, D_outlet, and H_outlet are always required. Additionally, one set of the following parameters is required; H_support, D_support, n_support, and inlet_rounding are all optional as well.

• Inlet diameter
• Inlet diameter and throat diameter
• Inlet diameter and throat height
• Inlet diameter, throat diameter, and throat height
• Base diameter, throat diameter, and throat height

If the inlet diameter is provided but the throat diameter and/or the throat height are missing, two heuristics are used to estimate them (to avoid these heuristics simply specify the values):

• Assume the throat elevation is 2/3 the elevation of the tower.
• Assume the throat diameter is 63% the diameter of the inlet.
Parameters: H_inlet : float Height of the inlet zone of the cooling tower (also called rain zone), [m] D_outlet : float The inside diameter of the cooling tower outlet (top of the tower; the elevation the concrete section ends), [m] H_outlet : float The height of the cooling tower outlet (top of the tower;the elevation the concrete section ends), [m] D_inlet : float, optional The inside diameter of the cooling tower inlet at the elevation the concrete section begins, [m] D_base : float, optional The diameter of the cooling tower at the very base of the tower (the bottom of the inlet zone, at the elevation of the ground), [m] D_throat : float, optional The diameter of the cooling tower at its minimum section, called its throat; where the two hyperbolas meet, [m] h_throat : float, optional The elevation of the cooling tower’s throat (its minimum section; where the two hyperbolas meet), [m] inlet_rounding : float, optional Radius of an optional rounded protrusion from the lip of the cooling tower shell base, which curves upwards from the lip (used to reduce the dead zone area rather than having a flat lip), [m] H_support : float, optional The height of each support column, [m] D_support : float, optional The diameter of each support column, [m] n_support : int, optional The number of support columns of the cooling tower, [m]

Notes

Note there are two hyperbolas in a hyperbolic cooling tower - one under the throat and one above it; they are not necessarily the same.

A hyperbolic cooling tower is not the absolute optimal design, but is is close. The optimality is determined by the amount of material required to build it while maintaining its rigidity. For thermal design purposes, a hyperbolic model covers any minor variation quite well.

References

 [1] Chen, W. F., and E. M. Lui, eds. Handbook of Structural Engineering, Second Edition. Boca Raton, Fla: CRC Press, 2005.
 [2] Ansary, A. M. El, A. A. El Damatty, and A. O. Nassef. Optimum Shape and Design of Cooling Towers, 2011.

Examples

>>> ct = HyperbolicCoolingTower(D_outlet=89.0, H_outlet=200, D_inlet=136.18, H_inlet=14.5)
>>> ct
<Hyperbolic cooling tower, inlet diameter=136.18 m, outlet diameter=89 m, inlet height=14.5 m, outlet height=200 m, throat diameter=85.7934 m, throat height=133.333 m, base diameter=146.427 m>
>>> ct.diameter(5)
142.84514486126062

Attributes: b_lower : float The b parameter in the hyperbolic equation for the lower section of the cooling tower, [m] b_upper : float The b parameter in the hyperbolic equation for the upper section of the cooling tower, [m]

Methods

 diameter(H) Calculates cooling tower diameter at a specified height, using the formulas for either hyperbola, depending on the height specified.
 plot
diameter(H)[source]

Calculates cooling tower diameter at a specified height, using the formulas for either hyperbola, depending on the height specified.

$D = D_{throat}\frac{\sqrt{H^2 + b^2}}{b}$

The value of H and b used in the above equation is as follows:

• H_throat - H and b_lower if under the throat
• H - H_throat and b_upper, if above the throat
Parameters: H : float Height at which to calculate the cooling tower diameter, [m] D : float Diameter of the cooling tower at the specified height, [m]
plot(pts=100)[source]
fluids.geometry.SA_partial_sphere(D, h)[source]

Calculates surface area of a partial sphere according to [1]. If h is half of D, the shape is half a sphere. No bottom is considered in this function. Valid inputs are positive values of D and h, with h always smaller or equal to D.

\begin{align}\begin{aligned}a = \sqrt{h(2r - h)}\\A = \pi(a^2 + h^2)\end{aligned}\end{align}
Parameters: D : float Diameter of the sphere, [m] h : float Height, as measured from the cap to where the sphere is cut off [m] SA : float Surface area [m^2]

References

 [1] (1, 2) Weisstein, Eric W. “Spherical Cap.” Text. Accessed December 22, 2015. http://mathworld.wolfram.com/SphericalCap.html.

Examples

>>> SA_partial_sphere(1., 0.7)
2.199114857512855

fluids.geometry.V_partial_sphere(D, h)[source]

Calculates volume of a partial sphere according to [1]. If h is half of D, the shape is half a sphere. No bottom is considered in this function. Valid inputs are positive values of D and h, with h always smaller or equal to D.

\begin{align}\begin{aligned}a = \sqrt{h(2r - h)}\\V = 1/6 \pi h(3a^2 + h^2)\end{aligned}\end{align}
Parameters: D : float Diameter of the sphere, [m] h : float Height, as measured up to where the sphere is cut off, [m] V : float Volume [m^3]

References

 [1] (1, 2) Weisstein, Eric W. “Spherical Cap.” Text. Accessed December 22, 2015. http://mathworld.wolfram.com/SphericalCap.html.

Examples

>>> V_partial_sphere(1., 0.7)
0.4105014400690663

fluids.geometry.V_horiz_conical(D, L, a, h, headonly=False)[source]

Calculates volume of a tank with conical ends, according to [1].

\begin{align}\begin{aligned}\begin{split}V_f = A_fL + \frac{2aR^2}{3}K, \;\;0 \le h < R\\\end{split}\\\begin{split}V_f = A_fL + \frac{2aR^2}{3}\pi/2,\;\; h = R\\\end{split}\\V_f = A_fL + \frac{2aR^2}{3}(\pi-K), \;\; R< h \le 2R\\K = \cos^{-1} M + M^3\cosh^{-1} \frac{1}{M} - 2M\sqrt{1 - M^2}\\M = \left|\frac{R-h}{R}\right|\\Af = R^2\cos^{-1}\frac{R-h}{R} - (R-h)\sqrt{2Rh - h^2}\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] L : float Length of the main cylindrical section, [m] a : float Distance the cone head extends on one side, [m] h : float Height, as measured up to where the fluid ends, [m] headonly : bool, optional Function returns only the volume of a single head side if True V : float Volume [m^3]

References

 [1] (1, 2, 3) Jones, D. “Calculating Tank Volume.” Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF

Examples

Matching example from [1], with inputs in inches and volume in gallons.

>>> V_horiz_conical(D=108., L=156., a=42., h=36)/231
2041.1923581273443

fluids.geometry.V_horiz_ellipsoidal(D, L, a, h, headonly=False)[source]

Calculates volume of a tank with ellipsoidal ends, according to [1].

\begin{align}\begin{aligned}V_f = A_fL + \pi a h^2\left(1 - \frac{h}{3R}\right)\\Af = R^2\cos^{-1}\frac{R-h}{R} - (R-h)\sqrt{2Rh - h^2}\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] L : float Length of the main cylindrical section, [m] a : float Distance the ellipsoidal head extends on one side, [m] h : float Height, as measured up to where the fluid ends, [m] headonly : bool, optional Function returns only the volume of a single head side if True V : float Volume [m^3]

References

 [1] (1, 2, 3) Jones, D. “Calculating Tank Volume.” Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF

Examples

Matching example from [1], with inputs in inches and volume in gallons.

>>> V_horiz_ellipsoidal(D=108, L=156, a=42, h=36)/231.
2380.9565415578145

fluids.geometry.V_horiz_guppy(D, L, a, h, headonly=False)[source]

Calculates volume of a tank with guppy heads, according to [1].

\begin{align}\begin{aligned}V_f = A_fL + \frac{2aR^2}{3}\cos^{-1}\left(1 - \frac{h}{R}\right) +\frac{2a}{9R}\sqrt{2Rh - h^2}(2h-3R)(h+R)\\Af = R^2\cos^{-1}\frac{R-h}{R} - (R-h)\sqrt{2Rh - h^2}\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] L : float Length of the main cylindrical section, [m] a : float Distance the guppy head extends on one side, [m] h : float Height, as measured up to where the fluid ends, [m] headonly : bool, optional Function returns only the volume of a single head side if True V : float Volume [m^3]

References

 [1] (1, 2, 3) Jones, D. “Calculating Tank Volume.” Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF

Examples

Matching example from [1], with inputs in inches and volume in gallons.

>>> V_horiz_guppy(D=108., L=156., a=42., h=36)/231.
1931.7208029476762

fluids.geometry.V_horiz_spherical(D, L, a, h, headonly=False)[source]

Calculates volume of a tank with spherical heads, according to [1].

\begin{align}\begin{aligned}V_f = A_fL + \frac{\pi a}{6}(3R^2 + a^2),\;\; h = R, |a|\le R\\V_f = A_fL + \frac{\pi a}{3}(3R^2 + a^2),\;\; h = D, |a|\le R\\V_f = A_fL + \pi a h^2\left(1 - \frac{h}{3R}\right),\;\; h = 0, \text{ or } |a| = 0, R, -R\\V_f = A_fL + \frac{a}{|a|}\left\{\frac{2r^3}{3}\left[\cos^{-1} \frac{R^2 - rw}{R(w-r)} + \cos^{-1}\frac{R^2 + rw}{R(w+r)} - \frac{z}{r}\left(2 + \left(\frac{R}{r}\right)^2\right) \cos^{-1}\frac{w}{R}\right] - 2\left(wr^2 - \frac{w^3}{3}\right) \tan^{-1}\frac{y}{z} + \frac{4wyz}{3}\right\} ,\;\; h \ne R, D; a \ne 0, R, -R, |a| \ge 0.01D\\V_f = A_fL + \frac{a}{|a|}\left[2\int_w^R(r^2 - x^2)\tan^{-1} \sqrt{\frac{R^2-x^2}{r^2-R^2}}dx - A_f z\right] ,\;\; h \ne R, D; a \ne 0, R, -R, |a| < 0.01D\\Af = R^2\cos^{-1}\frac{R-h}{R} - (R-h)\sqrt{2Rh - h^2}\\r = \frac{a^2 + R^2}{2|a|}\\w = R - h\\y = \sqrt{2Rh-h^2}\\z = \sqrt{r^2 - R^2}\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] L : float Length of the main cylindrical section, [m] a : float Distance the spherical head extends on one side, [m] h : float Height, as measured up to where the fluid ends, [m] headonly : bool, optional Function returns only the volume of a single head side if True V : float Volume [m^3]

References

 [1] (1, 2, 3) Jones, D. “Calculating Tank Volume.” Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF

Examples

Matching example from [1], with inputs in inches and volume in gallons.

>>> V_horiz_spherical(D=108., L=156., a=42., h=36)/231.
2303.9615116986183

fluids.geometry.V_horiz_torispherical(D, L, f, k, h, headonly=False)[source]

Calculates volume of a tank with torispherical heads, according to [1].

\begin{align}\begin{aligned}\begin{split}V_f = A_fL + 2V_1, \;\; 0 \le h \le h_1\\ V_f = A_fL + 2(V_{1,max} + V_2 + V_3), \;\; h_1 < h < h_2\\ V_f = A_fL + 2[2V_{1,max} - V_1(h=D-h) + V_{2,max} + V_{3,max}] , \;\; h_2 \le h \le D\end{split}\\V_1 = \int_0^{\sqrt{2kDh - h^2}} \left[n^2\sin^{-1}\frac{\sqrt {n^2-w^2}}{n} - w\sqrt{n^2-w^2}\right]dx\\V_2 = \int_0^{kD\cos\alpha}\left[n^2\left(\cos^{-1}\frac{w}{n} - \cos^{-1}\frac{g}{n}\right) - w\sqrt{n^2 - w^2} + g\sqrt{n^2 - g^2}\right]dx\\V_3 = \int_w^g(r^2 - x^2)\tan^{-1}\frac{\sqrt{g^2 - x^2}}{z}dx - \frac{z}{2}\left(g^2\cos^{-1}\frac{w}{g} - w\sqrt{2g(h-h_1) - (h-h_1)^2}\right)\\V_{1,max} = v_1(h=h_1)\\v_{2,max} = v_2(h=h_2)\\v_{3,max} = \frac{\pi a_1}{6}(3g^2 + a_1^2)\\a_1 = fD(1-\cos\alpha)\\\alpha = \sin^{-1}\frac{1-2k}{2(f-k)}\\n = R - kD + \sqrt{k^2D^2-x^2}\\g = r\sin\alpha\\r = fD\\h_2 = D - h_1\\w = R - h\\z = \sqrt{r^2- g^2}\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] L : float Length of the main cylindrical section, [m] f : float Dish-radius parameter; fD = dish radius [1/m] k : float knuckle-radius parameter ; kD = knuckle radius [1/m] h : float Height, as measured up to where the fluid ends, [m] headonly : bool, optional Function returns only the volume of a single head side if True V : float Volume [m^3]

References

 [1] (1, 2, 3) Jones, D. “Calculating Tank Volume.” Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF

Examples

Matching example from [1], with inputs in inches and volume in gallons.

>>> V_horiz_torispherical(D=108., L=156., f=1., k=0.06, h=36)/231.
2028.626670842139

fluids.geometry.V_vertical_conical(D, a, h)[source]

Calculates volume of a vertical tank with a convex conical bottom, according to [1]. No provision for the top of the tank is made here.

\begin{align}\begin{aligned}V_f = \frac{\pi}{4}\left(\frac{Dh}{a}\right)^2\left(\frac{h}{3}\right),\; h < a\\V_f = \frac{\pi D^2}{4}\left(h - \frac{2a}{3}\right),\; h\ge a\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] a : float Distance the cone head extends under the main cylinder, [m] h : float Height, as measured up to where the fluid ends, [m] V : float Volume [m^3]

References

 [1] (1, 2, 3) Jones, D. “Calculating Tank Volume.” Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF

Examples

Matching example from [1], with inputs in inches and volume in gallons.

>>> V_vertical_conical(132., 33., 24)/231.
250.67461381371024

fluids.geometry.V_vertical_ellipsoidal(D, a, h)[source]

Calculates volume of a vertical tank with a convex ellipsoidal bottom, according to [1]. No provision for the top of the tank is made here.

\begin{align}\begin{aligned}V_f = \frac{\pi}{4}\left(\frac{Dh}{a}\right)^2 \left(a - \frac{h}{3}\right),\; h < a\\V_f = \frac{\pi D^2}{4}\left(h - \frac{a}{3}\right),\; h \ge a\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] a : float Distance the ellipsoid head extends under the main cylinder, [m] h : float Height, as measured up to where the fluid ends, [m] V : float Volume [m^3]

References

 [1] (1, 2, 3) Jones, D. “Calculating Tank Volume.” Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF

Examples

Matching example from [1], with inputs in inches and volume in gallons.

>>> V_vertical_ellipsoidal(132., 33., 24)/231.
783.3581681678445

fluids.geometry.V_vertical_spherical(D, a, h)[source]

Calculates volume of a vertical tank with a convex spherical bottom, according to [1]. No provision for the top of the tank is made here.

\begin{align}\begin{aligned}V_f = \frac{\pi h^2}{4}\left(2a + \frac{D^2}{2a} - \frac{4h}{3}\right),\; h < a\\V_f = \frac{\pi}{4}\left(\frac{2a^3}{3} - \frac{aD^2}{2} + hD^2\right),\; h\ge a\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] a : float Distance the spherical head extends under the main cylinder, [m] h : float Height, as measured up to where the fluid ends, [m] V : float Volume [m^3]

References

 [1] (1, 2, 3) Jones, D. “Calculating Tank Volume.” Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF

Examples

Matching example from [1], with inputs in inches and volume in gallons.

>>> V_vertical_spherical(132., 33., 24)/231.
583.6018352850442

fluids.geometry.V_vertical_torispherical(D, f, k, h)[source]

Calculates volume of a vertical tank with a convex torispherical bottom, according to [1]. No provision for the top of the tank is made here.

\begin{align}\begin{aligned}V_f = \frac{\pi h^2}{4}\left(2a_1 + \frac{D_1^2}{2a_1} - \frac{4h}{3}\right),\; 0 \le h \le a_1\\V_f = \frac{\pi}{4}\left(\frac{2a_1^3}{3} + \frac{a_1D_1^2}{2}\right) +\pi u\left[\left(\frac{D}{2}-kD\right)^2 +s\right] + \frac{\pi tu^2}{2} - \frac{\pi u^3}{3} + \pi D(1-2k)\left[ \frac{2u-t}{4}\sqrt{s+tu-u^2} + \frac{t\sqrt{s}}{4} + \frac{k^2D^2}{2}\left(\cos^{-1}\frac{t-2u}{2kD}-\alpha\right)\right] ,\; a_1 < h \le a_1 + a_2\\V_f = \frac{\pi}{4}\left(\frac{2a_1^3}{3} + \frac{a_1D_1^2}{2}\right) +\frac{\pi t}{2}\left[\left(\frac{D}{2}-kD\right)^2 +s\right] +\frac{\pi t^3}{12} + \pi D(1-2k)\left[\frac{t\sqrt{s}}{4} + \frac{k^2D^2}{2}\sin^{-1}(\cos\alpha)\right] + \frac{\pi D^2}{4}[h-(a_1+a_2)] ,\; a_1 + a_2 < h\\\alpha = \sin^{-1}\frac{1-2k}{2(f-k)}\\a_1 = fD(1-\cos\alpha)\\a_2 = kD\cos\alpha\\D_1 = 2fD\sin\alpha\\s = (kD\sin\alpha)^2\\t = 2a_2\\u = h - fD(1-\cos\alpha)\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] f : float Dish-radius parameter; fD = dish radius [1/m] k : float knuckle-radius parameter ; kD = knuckle radius [1/m] h : float Height, as measured up to where the fluid ends, [m] V : float Volume [m^3]

References

 [1] (1, 2, 3) Jones, D. “Calculating Tank Volume.” Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF

Examples

Matching example from [1], with inputs in inches and volume in gallons.

>>> V_vertical_torispherical(D=132., f=1.0, k=0.06, h=24)/231.
904.0688283793511

fluids.geometry.V_vertical_conical_concave(D, a, h)[source]

Calculates volume of a vertical tank with a concave conical bottom, according to [1]. No provision for the top of the tank is made here.

\begin{align}\begin{aligned}V = \frac{\pi D^2}{12} \left(3h + a - \frac{(a+h)^3}{a^2}\right) ,\;\; 0 \le h < |a|\\V = \frac{\pi D^2}{12} (3h + a ),\;\; h \ge |a|\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] a : float Negative distance the cone head extends inside the main cylinder, [m] h : float Height, as measured up to where the fluid ends, [m] V : float Volume [m^3]

References

 [1] (1, 2, 3) Jones, D. “Compute Fluid Volumes in Vertical Tanks.” Chemical Processing. December 18, 2003. http://www.chemicalprocessing.com/articles/2003/193/

Examples

Matching example from [1], with inputs in inches and volume in gallons.

>>> V_vertical_conical_concave(D=113., a=-33, h=15)/231
251.15825565795188

fluids.geometry.V_vertical_ellipsoidal_concave(D, a, h)[source]

Calculates volume of a vertical tank with a concave ellipsoidal bottom, according to [1]. No provision for the top of the tank is made here.

\begin{align}\begin{aligned}V = \frac{\pi D^2}{12} \left(3h + 2a - \frac{(a+h)^2(2a-h)}{a^2}\right) ,\;\; 0 \le h < |a|\\V = \frac{\pi D^2}{12} (3h + 2a ),\;\; h \ge |a|\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] a : float Negative distance the eppilsoid head extends inside the main cylinder, [m] h : float Height, as measured up to where the fluid ends, [m] V : float Volume [m^3]

References

 [1] (1, 2, 3) Jones, D. “Compute Fluid Volumes in Vertical Tanks.” Chemical Processing. December 18, 2003. http://www.chemicalprocessing.com/articles/2003/193/

Examples

Matching example from [1], with inputs in inches and volume in gallons.

>>> V_vertical_ellipsoidal_concave(D=113., a=-33, h=15)/231
44.84968851034856

fluids.geometry.V_vertical_spherical_concave(D, a, h)[source]

Calculates volume of a vertical tank with a concave spherical bottom, according to [1]. No provision for the top of the tank is made here.

\begin{align}\begin{aligned}V = \frac{\pi}{12}\left[3D^2h + \frac{a}{2}(3D^2 + 4a^2) + (a+h)^3 \left(4 - \frac{3D^2 + 12a^2}{2a(a+h)}\right)\right],\;\; 0 \le h < |a|\\V = \frac{\pi}{12}\left[3D^2h + \frac{a}{2}(3D^2 + 4a^2) \right] ,\;\; h \ge |a|\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] a : float Negative distance the spherical head extends inside the main cylinder, [m] h : float Height, as measured up to where the fluid ends, [m] V : float Volume [m^3]

References

 [1] (1, 2, 3) Jones, D. “Compute Fluid Volumes in Vertical Tanks.” Chemical Processing. December 18, 2003. http://www.chemicalprocessing.com/articles/2003/193/

Examples

Matching example from [1], with inputs in inches and volume in gallons.

>>> V_vertical_spherical_concave(D=113., a=-33, h=15)/231
112.81405437348528

fluids.geometry.V_vertical_torispherical_concave(D, f, k, h)[source]

Calculates volume of a vertical tank with a concave torispherical bottom, according to [1]. No provision for the top of the tank is made here.

\begin{align}\begin{aligned}V = \frac{\pi D^2 h}{4} - v_1(h=a_1+a_2) + v_1(h=a_1 + a_2 -h),\; 0 \le h < a_2\\V = \frac{\pi D^2 h}{4} - v_1(h=a_1+a_2) + v_2(h=a_1 + a_2 -h),\; a_2 \le h < a_1 + a_2\\V = \frac{\pi D^2 h}{4} - v_1(h=a_1+a_2) + 0,\; h \ge a_1 + a_2\\v_1 = \frac{\pi}{4}\left(\frac{2a_1^3}{3} + \frac{a_1D_1^2}{2}\right) +\pi u\left[\left(\frac{D}{2}-kD\right)^2 +s\right] + \frac{\pi tu^2}{2} - \frac{\pi u^3}{3} + \pi D(1-2k)\left[ \frac{2u-t}{4}\sqrt{s+tu-u^2} + \frac{t\sqrt{s}}{4} + \frac{k^2D^2}{2}\left(\cos^{-1}\frac{t-2u}{2kD}-\alpha\right)\right]\\v_2 = \frac{\pi h^2}{4}\left(2a_1 + \frac{D_1^2}{2a_1} - \frac{4h}{3}\right)\\\alpha = \sin^{-1}\frac{1-2k}{2(f-k)}\\a_1 = fD(1-\cos\alpha)\\a_2 = kD\cos\alpha\\D_1 = 2fD\sin\alpha\\s = (kD\sin\alpha)^2\\t = 2a_2\\u = h - fD(1-\cos\alpha)\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] f : float Dish-radius parameter; fD = dish radius [1/m] k : float knuckle-radius parameter ; kD = knuckle radius [1/m] h : float Height, as measured up to where the fluid ends, [m] V : float Volume [m^3]

References

 [1] (1, 2, 3) Jones, D. “Compute Fluid Volumes in Vertical Tanks.” Chemical Processing. December 18, 2003. http://www.chemicalprocessing.com/articles/2003/193/

Examples

Matching example from [1], with inputs in inches and volume in gallons.

>>> V_vertical_torispherical_concave(D=113., f=0.71, k=0.081, h=15)/231
103.88569287163769

fluids.geometry.a_torispherical(D, f, k)[source]

Calculates depth of a torispherical head according to [1].

\begin{align}\begin{aligned}a = a_1 + a_2\\\alpha = \sin^{-1}\frac{1-2k}{2(f-k)}\\a_1 = fD(1-\cos\alpha)\\a_2 = kD\cos\alpha\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] f : float Dish-radius parameter; fD = dish radius [1/m] k : float knuckle-radius parameter ; kD = knuckle radius [1/m] a : float Depth of head [m]

References

 [1] (1, 2, 3) Jones, D. “Calculating Tank Volume.” Text. Accessed December 22, 2015. http://www.webcalc.com.br/blog/Tank_Volume.PDF

Examples

Example from [1].

>>> a_torispherical(D=96., f=0.9, k=0.2)
25.684268924767125

fluids.geometry.SA_ellipsoidal_head(D, a)[source]

Calculates the surface area of an ellipsoidal head according to [1]. Formula below is for the full shape, the result of which is halved. The formula also does not support D being larger than a; this is ensured by simply swapping the variables if necessary, as geometrically the result is the same. In the equations

\begin{align}\begin{aligned}SA = 2\pi a^2 + \frac{\pi c^2}{e_1}\ln\left(\frac{1+e_1}{1-e_1}\right)\\e_1 = \sqrt{1 - \frac{c^2}{a^2}}\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] a : float Distance the ellipsoidal head extends, [m] SA : float Surface area [m^2]

References

 [1] (1, 2) Weisstein, Eric W. “Spheroid.” Text. Accessed March 14, 2016. http://mathworld.wolfram.com/Spheroid.html.

Examples

Spherical case

>>> SA_ellipsoidal_head(2, 1)
6.283185307179586

fluids.geometry.SA_conical_head(D, a)[source]

Calculates the surface area of a conical head according to [1].

$SA = \frac{\pi D}{2} \sqrt{a^2 + \left(\frac{D}{2}\right)^2}$
Parameters: D : float Diameter of the main cylindrical section, [m] a : float Distance the conical head extends, [m] SA : float Surface area [m^2]

References

 [1] (1, 2) Weisstein, Eric W. “Cone.” Text. Accessed March 14, 2016. http://mathworld.wolfram.com/Cone.html.

Examples

>>> SA_conical_head(2, 1)
4.442882938158366

fluids.geometry.SA_guppy_head(D, a)[source]

Calculates the surface area of a guppy head according to [1]. Some work was involved in combining formulas for the ellipse of the head, and the conic section on the sides.

$SA = \frac{\pi D}{4}\sqrt{D^2 + a^2} + \frac{\pi D}{2}a$
Parameters: D : float Diameter of the main cylindrical section, [m] a : float Distance the conical head extends, [m] SA : float Surface area [m^2]

References

 [1] (1, 2) Weisstein, Eric W. “Cone.” Text. Accessed March 14, 2016. http://mathworld.wolfram.com/Cone.html.

Examples

>>> SA_guppy_head(2, 1)
6.654000019110157

fluids.geometry.SA_torispheroidal(D, fd, fk)[source]

Calculates surface area of a torispherical head according to [1]. Somewhat involved. Equations are adapted to be used for a full head.

\begin{align}\begin{aligned}SA = S_1 + S_2\\S_1 = 2\pi D^2 f_d \alpha\\S_2 = 2\pi D^2 f_k\left(\alpha - \alpha_1 + (0.5 - f_k)\left(\sin^{-1} \left(\frac{\alpha-\alpha_2}{f_k}\right) - \sin^{-1}\left(\frac{ \alpha_1-\alpha_2}{f_k}\right)\right)\right)\\\alpha_1 = f_d\left(1 - \sqrt{1 - \left(\frac{0.5 - f_k}{f_d-f_k} \right)^2}\right)\\\alpha_2 = f_d - \sqrt{f_d^2 - 2f_d f_k + f_k - 0.25}\\\alpha = \frac{a}{D_i}\end{aligned}\end{align}
Parameters: D : float Diameter of the main cylindrical section, [m] fd : float Dish-radius parameter = f; fD = dish radius [1/m] fk : float knuckle-radius parameter = k; kD = knuckle radius [1/m] SA : float Surface area [m^2]

References

 [1] (1, 2, 3) Honeywell. “Calculate Surface Areas and Cross-sectional Areas in Vessels with Dished Heads”. https://www.honeywellprocess.com/library/marketing/whitepapers/WP-VesselsWithDishedHeads-UniSimDesign.pdf Whitepaper. 2014.

Examples

Example from [1].

>>> SA_torispheroidal(D=2.54, fd=1.039370079, fk=0.062362205)
6.00394283477063

fluids.geometry.V_from_h(h, D, L, horizontal=True, sideA=None, sideB=None, sideA_a=0, sideB_a=0, sideA_f=None, sideA_k=None, sideB_f=None, sideB_k=None)[source]

Calculates partially full volume of a vertical or horizontal tank with different head types according to [1].

References

 [1] (1, 2) Jones, D. “Compute Fluid Volumes in Vertical Tanks.” Chemical Processing. December 18, 2003. http://www.chemicalprocessing.com/articles/2003/193/

Examples

>>> V_from_h(h=7, D=1.5, L=5., horizontal=False, sideA='conical',
... sideB='conical', sideA_a=2., sideB_a=1.)
10.013826583317465

fluids.geometry.SA_tank(D, L, sideA=None, sideB=None, sideA_a=0, sideB_a=0, sideA_f=None, sideA_k=None, sideB_f=None, sideB_k=None, full_output=False)[source]

Calculates the surface are of a cylindrical tank with optional heads. In the degenerate case of being provided with only D and L, provides the surface area of a cylinder.

Examples

Cylinder, Spheroid, Long Cones, and spheres. All checked.

>>> SA_tank(D=2, L=2)
18.84955592153876
>>> SA_tank(D=1., L=0, sideA='ellipsoidal', sideA_a=2, sideB='ellipsoidal',
... sideB_a=2)
28.480278854014387
>>> SA_tank(D=1., L=5, sideA='conical', sideA_a=2, sideB='conical',
... sideB_a=2)
22.18452243965656
>>> SA_tank(D=1., L=5, sideA='spherical', sideA_a=0.5, sideB='spherical',
... sideB_a=0.5)
18.84955592153876

fluids.geometry.sphericity(A, V)[source]

Returns the sphericity of a particle of surface area A and volume V. Sphericity is the ratio of the surface area of a sphere with the same volume as the particle (equivalent diameter) to the actual surface area of the particle.

$\Psi = \frac{\text{A of sphere with } V_p } {{A}_p} = \frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p}$
Parameters: A : float Surface area of particle, [m^2] V : float Volume of particle, [m^3] Psi : float Sphericity [-]

Notes

All non-spherical particles have spericities less than 1 but greater than 0. Many common geometrical shapes have their results calculated exactly in [2].

References

 [1] Rhodes, Martin J., ed. Introduction to Particle Technology. 2E. Chichester, England ; Hoboken, NJ: Wiley, 2008.
 [2] (1, 2) “Sphericity.” Wikipedia, March 8, 2017. https://en.wikipedia.org/w/index.php?title=Sphericity&oldid=769183043

Examples

>>> sphericity(10., 2.)
0.767663317071005


For a cube of side length a=3, the surface area is 6*a^2=54 and volume a^3=27. Its sphericity is then:

>>> sphericity(A=54, V=27)
0.8059959770082346

fluids.geometry.aspect_ratio(Dmin, Dmax)[source]

Returns the aspect ratio of a shape with minimum and maximum dimension, Dmin and Dmax.

$A_R = \frac{D_{min}}{D_{max}}$
Parameters: Dmin : float Minimum dimension, [m] Dmax : float Maximum dimension, [m] a_r : float Aspect ratio [-]

Examples

>>> aspect_ratio(.2, 2)
0.1

fluids.geometry.circularity(A, P)[source]

Returns the circularity of a shape with area A and perimeter P.

$f_{circ} = \frac {4 \pi A} {P^2}$

Defined to be 1 for a circle. Used to characterize particles. Any non-circular shape must have a circularity less than one.

Parameters: A : float Area of the shape, [m^2] P : float Perimeter of the shape, [m] f_circ : float Circularity of the shape [-]

Examples

Square, side length = 2 (all squares are the same):

>>> circularity(A=(2*2), P=4*2)
0.7853981633974483


Rectangle, one side length = 1, second side length = 100

>>> D1 = 1
>>> D2 = 100
>>> A = D1*D2
>>> P = 2*D1 + 2*D2
>>> circularity(A, P)
0.030796908671598795

fluids.geometry.A_cylinder(D, L)[source]

Returns the surface area of a cylinder.

$A = \pi D L + 2\cdot \frac{\pi D^2}{4}$
Parameters: D : float Diameter of the cylinder, [m] L : float Length of the cylinder, [m] A : float Surface area [m^2]

Examples

>>> A_cylinder(0.01, .1)
0.0032986722862692833

fluids.geometry.V_cylinder(D, L)[source]

Returns the volume of a cylinder.

$V = \frac{\pi D^2}{4}L$
Parameters: D : float Diameter of the cylinder, [m] L : float Length of the cylinder, [m] V : float Volume [m^3]

Examples

>>> V_cylinder(0.01, .1)
7.853981633974484e-06

fluids.geometry.A_hollow_cylinder(Di, Do, L)[source]

Returns the surface area of a hollow cylinder.

$A = \pi D_o L + \pi D_i L + 2\cdot \frac{\pi D_o^2}{4} - 2\cdot \frac{\pi D_i^2}{4}$
Parameters: Di : float Diameter of the hollow in the cylinder, [m] Do : float Diameter of the exterior of the cylinder, [m] L : float Length of the cylinder, [m] A : float Surface area [m^2]

Examples

>>> A_hollow_cylinder(0.005, 0.01, 0.1)
0.004830198704894308

fluids.geometry.V_hollow_cylinder(Di, Do, L)[source]

Returns the volume of a hollow cylinder.

$V = \frac{\pi D_o^2}{4}L - L\frac{\pi D_i^2}{4}$
Parameters: Di : float Diameter of the hollow in the cylinder, [m] Do : float Diameter of the exterior of the cylinder, [m] L : float Length of the cylinder, [m] V : float Volume [m^3]

Examples

>>> V_hollow_cylinder(0.005, 0.01, 0.1)
5.890486225480862e-06

fluids.geometry.A_multiple_hole_cylinder(Do, L, holes)[source]

Returns the surface area of a cylinder with multiple holes. Calculation will naively return a negative value or other impossible result if the number of cylinders added is physically impossible. Holes may be of different shapes, but must be perpendicular to the axis of the cylinder.

$A = \pi D_o L + 2\cdot \frac{\pi D_o^2}{4} + \sum_{i}^n \left( \pi D_i L - 2\cdot \frac{\pi D_i^2}{4}\right)$
Parameters: Do : float Diameter of the exterior of the cylinder, [m] L : float Length of the cylinder, [m] holes : list List of tuples containing (diameter, count) pairs of descriptions for each of the holes sizes. A : float Surface area [m^2]

Examples

>>> A_multiple_hole_cylinder(0.01, 0.1, [(0.005, 1)])
0.004830198704894308

fluids.geometry.V_multiple_hole_cylinder(Do, L, holes)[source]

Returns the solid volume of a cylinder with multiple cylindrical holes. Calculation will naively return a negative value or other impossible result if the number of cylinders added is physically impossible.

$V = \frac{\pi D_o^2}{4}L - L\frac{\pi D_i^2}{4}$
Parameters: Do : float Diameter of the exterior of the cylinder, [m] L : float Length of the cylinder, [m] holes : list List of tuples containing (diameter, count) pairs of descriptions for each of the holes sizes. V : float Volume [m^3]

Examples

>>> V_multiple_hole_cylinder(0.01, 0.1, [(0.005, 1)])
5.890486225480862e-06