NonLumpedCylinder¶

class pychemengg.heattransfer.transient.NonLumpedCylinder(radius=None, surfacearea=None, volume=None, density=None, specificheat=None, thermalconductivity=None, thermaldiffusivity=None, heattransfercoefficient=None, T_infinity=None, T_initial=None)[source]¶

Bases: object

Model for nonlumped analysis of cylindrical solid object.

Parameters

radiusint or float: Radius of solid object.
surfaceareaint or float: Surface area of solid object.
volumeint or float: Volume of solid object.
densityint or float: Density of solid object.
specificheatint or float: Specific heat of solid object.
thermalconductivityint or float: Thermal conductivity of solid object.
thermaldiffusivityint or float: Thermal diffusivity of solid object.
heattransfercoefficientint or float: Heat transfer coefficient between solid object and surrounding.
T_infinityint or float: Temperature of surroundings.
T_initialint or float: Temperature of solid object at time = 0.

Examples

First import the module transient

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import transient
>>> cylinder=transient.NonLumpedCylinder(radius=10e-2, surfacearea=1, T_initial=600, volume=np.pi*10e-2**2*1, T_infinity=200, density=7900, thermaldiffusivity=None, specificheat=477, heattransfercoefficient=80, thermalconductivity=14.9)
# This will create an instance of 'NonLumpedCylinder' with a name 'cylinder' 

Attributes

See “Parameters”. All parameters are attributes. Additional attributes are listed below.
massint or float: Mass of solid object computed as (volume * density) of solid object.

__init__(radius=None, surfacearea=None, volume=None, density=None, specificheat=None, thermalconductivity=None, thermaldiffusivity=None, heattransfercoefficient=None, T_infinity=None, T_initial=None)[source]¶

Methods

`__init__`([radius, surfacearea, volume, …])
`calc_Bi`()	Computes Biot number.
`calc_Fo`([time])	Computes Fourier number.
`calc_eigenvalues`([numberof_eigenvalues_desired])	Computes eigen values of characteristic equation for Cylindrical geometry.
`calc_heatrateof_conv_at_time_t`()	Heat rate of convection between object and surroundings at a given time = t.
`calc_maxheattransferpossible`()	Maximum possible heat transfer between solid object and surroundings.
`calc_temperature_of_solid_at_time_t`([…])	Calculates temperature of solid object at a given time = t and radius = r.
`calc_totalheat_transferred_during_interval_t`()	Heat transferred between solid object and surroundings during time interval = 0 to t.

calc_Bi()[source]¶

Computes Biot number.

Parameters

None_requiredNone: Attributes that are already defined are used in calculation.

Returns

Biint or float: Biot number

Notes

Biot number is calculated using the following formula.

\[Bi = \frac {h L_{c}} {k} \]

where:

h = heat transfer coefficient

k = thermal conductivity of solid object

\(L_c\) = characteristic length of solid object = radius

Bi = Biot number

References

[1] Y. A. Cengel and A. J. Ghajar, “Heat And Mass Transfer Fundamentals and Applications”, 6th Edition. New York, McGraw Hill Education, 2020.

Examples

First import the module transient

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import transient
>>> cylinder=transient.NonLumpedCylinder(radius=10e-2, surfacearea=1, T_initial=600, volume=np.pi*10e-2**2*1, T_infinity=200, density=7900, thermaldiffusivity=None, specificheat=477, heattransfercoefficient=80, thermalconductivity=14.9)
# This will create an instance of 'NonLumpedSlab' with a name 'plate' 
# Next call calc_Bi
>>> cylinder.calc_Bi()
0.5369127516778524

calc_Fo(time=None)[source]¶

Computes Fourier number.

Parameters

timeint or float: Time at which temperature or heat transfer is to be evaluated.

Returns

Foint or float: Fourier number

Notes

Fourier number is calculated using the following formula.

\[Fo = \frac {\alpha t} {L_c^2} \]

where:

\(\alpha\) = thermal diffusivity

t = time at which temperature or heat transfer is to be evaluated

\(L_c\) = characteristic length = radius

Fo = Fourier number

References

[1] Y. A. Cengel and A. J. Ghajar, “Heat And Mass Transfer Fundamentals and Applications”, 6th Edition. New York, McGraw Hill Education, 2020.

Examples

First import the module transient

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import transient
>>> cylinder=transient.NonLumpedCylinder(radius=10e-2, surfacearea=1, T_initial=600, volume=np.pi*10e-2**2*1, T_infinity=200, density=7900, thermaldiffusivity=None, specificheat=477, heattransfercoefficient=80, thermalconductivity=14.9)
# This will create an instance of 'NonLumpedSlab' with a name 'plate' 
# Next call calc_Fo assuming temperature is required at 7 min
>>> cylinder.calc_Fo(time=7*60)
0.16606958044741657

calc_eigenvalues(numberof_eigenvalues_desired=10)[source]¶

Computes eigen values of characteristic equation for Cylindrical geometry.

Parameters

numberof_eigenvalues_desiredint or float (default = 10): Number of eigen values desired for the characteristic equation.

Returns

eigenvaluesnp.array of int or float: Eigen values

Notes

Eigen values are calculated as roots of the following equation.

\[\lambda_n \frac{J_1(\lambda_n)}{J_0(\lambda_n)} - Bi = 0 , n = 1 \hspace{2pt} to \hspace{2pt} \infty \]

where:

\(J_0\) = Bessel function of first kind of order 0

\(J_1\) = Bessel function of first kind of order 1

\(\lambda_n\) = nth eigen value

Bi = Biot number

References

[1] Y. A. Cengel and A. J. Ghajar, “Heat And Mass Transfer Fundamentals and Applications”, 6th Edition. New York, McGraw Hill Education, 2020.

Examples

First import the module transient

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import transient
>>> cylinder=transient.NonLumpedCylinder(radius=10e-2, surfacearea=1, T_initial=600, volume=np.pi*10e-2**2*1, T_infinity=200, density=7900, thermaldiffusivity=None, specificheat=477, heattransfercoefficient=80, thermalconductivity=14.9)
# This will create an instance of 'NonLumpedSlab' with a name 'plate' 
# Next call calc_Bi
>>> cylinder.calc_Bi()
0.5369127516778524
# Let first 5 eigen values be required
>>> cylinder.calc_eigenvalues(numberof_eigenvalues_desired=5)
array([ 0.97061535,  3.96852663,  7.0915602 , 10.22605944, 13.36390715])

calc_heatrateof_conv_at_time_t()[source]¶

Heat rate of convection between object and surroundings at a given time = t.

Parameters

None_requiredNone: Attributes that are already defined are used in calculation.

Returns

heat rate of convectionint or float ; Positive: Heat is gained by object, Negative: Heat is lost by object: Heat rate of convection between solid object and surroundings at time = t.

Notes

Heat rate of convection is calculated using the following formula:

\[q_{t} = h A_s (T_{infinity} - T_{t}) \]

where:

t = time at which temperature is to be computed

h = heat transfer coefficient

\(T_{infinity}\) = temperature of surrounding fluid

\(T_{t}\) = temperature of surface of solid object at time = t

\(A_s\) = surface area of solid object

\(q_{t}\) = heat rate of convection at time = t

References

[1] Y. A. Cengel and A. J. Ghajar, “Heat And Mass Transfer Fundamentals and Applications”, 6th Edition. New York, McGraw Hill Education, 2020.

Examples

First import the module transient

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import transient
>>> cylinder=transient.NonLumpedCylinder(radius=10e-2, surfacearea=1, T_initial=600, volume=np.pi*10e-2**2*1, T_infinity=200, density=7900, thermaldiffusivity=None, specificheat=477, heattransfercoefficient=80, thermalconductivity=14.9)
# This will create an instance of 'NonLumpedCylinder' with a name 'cylinder' 
# Next call calc_Bi
>>> cylinder.calc_Bi()
0.5369127516778524
# Next call calc_Fo assuming temperature is required at 7 min
>>> cylinder.calc_Fo(time=7*60)
0.16606958044741657
# Let default (=10) eigen values be required
>>> cylinder.calc_eigenvalues()
array([ 0.97061535,  3.96852663,  7.0915602 , 10.22605944, 13.36390715,
16.50318456, 19.64320399, 22.78365791, 25.92438812, 29.06530494])
>>> calc_heatrateof_conv_at_time_t()
-24040.54791137568

calc_maxheattransferpossible()[source]¶

Maximum possible heat transfer between solid object and surroundings.

Parameters

None_requiredNone: Attributes that are already defined are used in calculation.

Returns

maximum heat transfer possible: int or float; Positive: Heat is gained by object, Negative: Heat is lost by object: Maximum heat transfer posssible between object and surroundings.

Notes

Maximum heat transfer possible between solid object and surroundings is calculated using the following formula. This is based on the assumption that final object temperature will eventually reach surrounding temperature of \(T_{infinity}\)

\[q_{max} = m C_p (T_{infinity} - T_{initial}) \]

where:

m = mass of solid object

\(C_{p}\) = specific heat of solid object

\(T_{infinity}\) = temperature of surrounding, which the solid object will eventually attain

\(T_{initial}\) = temperature of solid object at time = initial

\(q_{max}\) = max heat transfer possible

References

[1] Y. A. Cengel and A. J. Ghajar, “Heat And Mass Transfer Fundamentals and Applications”, 6th Edition. New York, McGraw Hill Education, 2020.

Examples

>>> from pychemengg.heattransfer import transient
>>> cylinder=transient.NonLumpedCylinder(radius=10e-2, surfacearea=1, T_initial=600, volume=np.pi*10e-2**2*1, T_infinity=200, density=7900, thermaldiffusivity=None, specificheat=477, heattransfercoefficient=80, thermalconductivity=14.9)
>>> cylinder.calc_maxheattransferpossible()
-47353854.386089675
# negative value indicates heat is being lost by the solid object

calc_temperature_of_solid_at_time_t(rposition_tofindtemp=None)[source]¶

Calculates temperature of solid object at a given time = t and radius = r.

Parameters

timeint or float: Time instant from begining of process, at which temperature of solid object is to be found.
rposition_tofindtempint or float: Radius from center of cylindrical object where temperature is to be found.

Returns

temperatureint or float: Temperature of solid object at time = t and radius = r.

Notes

Temperature of solid object at time = t and radius = r is calculated using the following formula:

\[T(t) = T_{infinity} + (T_{initial} - T_{infinity}) \displaystyle\sum_{n=1}^\infty \cfrac{2}{\lambda_n} \left( \frac{J_1(\lambda_n)}{J_0^2(\lambda_n) + J_1^2(\lambda_n)} \right) e^{- \lambda_n^2 \tau} J_0(\lambda_n r/r_{outside}) \]

where:

\(T_{infinity}\) = temperature of surrounding fluid

\(T_{initial}\) = intitial temperature of solid object

\(J_0\) = Bessel function of first kind of order 0

\(J_1\) = Bessel function of first kind of order 1

\(\lambda_n\) = \(n^{th}\) eigen value of \(x_n tan(x_n) - Bi = 0\) , n = 1 to \(\infty\)

Bi = Biot number

\(\tau\) = Fourier number

r = radius from center of solid cylinder where temperature is required (r = 0 for center of cylinder)

\(r_{outside}\) = outer radius of the cylinder

Examples

First import the module transient

Units used in this example: SI system

However, any consistent units can be used

>>> from pychemengg.heattransfer import transient
>>> cylinder=transient.NonLumpedCylinder(radius=10e-2, surfacearea=1, T_initial=600, volume=np.pi*10e-2**2*1, T_infinity=200, density=7900, thermaldiffusivity=None, specificheat=477, heattransfercoefficient=80, thermalconductivity=14.9)
# This will create an instance of 'NonLumpedSlab' with a name 'plate' 
# Next call calc_Bi
>>> cylinder.calc_Bi()
0.5369127516778524
# Next call calc_Fo assuming temperature is required at 7 min
>>> cylinder.calc_Fo(time=7*60)
0.16606958044741657
# Let default (=10) eigen values be required
>>> cylinder.calc_eigenvalues()
array([ 0.97061535,  3.96852663,  7.0915602 , 10.22605944, 13.36390715,
16.50318456, 19.64320399, 22.78365791, 25.92438812, 29.06530494])
>>> cylinder.calc_temperature_of_solid_at_time_t(rposition_tofindtemp=0)
578.8399893522001

calc_totalheat_transferred_during_interval_t()[source]¶

Heat transferred between solid object and surroundings during time interval = 0 to t.

Parameters

None_requiredNone: Attributes that are already defined or calculated are used in calculation.

Returns

total heat transferredint or float; Positive: Heat is gained by object, Negative: Heat is lost by object: Total heat transferred between object and surroundings during interval 0 to t