Line

class grheat.line_source.Line(yp, zp, tp=0, diffusivity=1.4557999999999998e-07, capacity=4184000.0, boundary='infinite')[source]

Bases: object

Provides Green’s function heat transfer solutions for a line source.

The Line class encapsulates the behavior of a line source situated in a semi-infinite medium with a surface defined at z=0. The line source extends horizontally along all x-values passing through coordinates (yp, zp). At time tp, the line source delivers a heat impulse of 1 Joule per meter along its length.

Boundary conditions at z=0 can be:

‘infinite’: No boundary (infinite medium).
‘adiabatic’: No heat flow across the boundary.
‘zero’: Boundary is fixed at T=0.

yp

y-coordinate of the line source. [meters]

Type:: scalar

zp

z-coordinate of the line source. [meters]

Type:: scalar

diffusivity

Thermal diffusivity of the medium. [m^2/s]

Type:: scalar

capacity

Volumetric heat capacity of the medium. [J/degree/m^3]

Type:: scalar

boundary

Boundary condition at z=0. [‘infinite’, ‘adiabatic’, ‘zero’]

Type:: str

Methods Summary

`continuous`(y, z, t)	Calculate temperature rise due to a 1W/m continuous x-line source.
`instantaneous`(y, z, t)	Calculate the temperature rise due to a 1 J/m instant x-line source.
`pulsed`(y, z, t, t_pulse)	Calculate temperature rise due to a 1 J/m pulsed x-line source.

Methods Documentation

continuous(y, z, t)[source]

Calculate temperature rise due to a 1W/m continuous x-line source.

The x-line source turns on at t=0 and passes through the coordinates (yp, zp).

Reference: Carslaw and Jaeger (1959), page 261, Equation 10.4(5).

Parameters:

y (scalar) – The y-coordinate for the desired temperature location. [meters]
z (scalar) – The z-coordinate for the desired temperature location. [meters]
t (scalar or array) – Time(s) of desired temperature. [seconds]

Returns:

scalar or array – Temperature increase in degrees Celsius [°C].

Example

The following example demonstrates how to use this method to calculate and plot the temperature rise over time due to a continuous line source 1mm deep that turned on at t=0:

import grheat
import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(0, 500, 100) / 1000   # seconds
y, z = 0, 0                           # meters
yp, zp = 0, 0.001                     # meters

line = grheat.Line(yp, zp)
T = line.continuous(y, z, t)

plt.plot(t * 1000, T, color='blue')
plt.xlabel("Time (ms)")
plt.ylabel("Surface Temperature Increase (°C)")
plt.title("Continuous Line Source at 1mm depth")
plt.show()

instantaneous(y, z, t)[source]

Calculate the temperature rise due to a 1 J/m instant x-line source.

This method computes the temperature rise at a specified location and time due to an instant x-line source. The line source is parallel to the x-axis and passes through the coordinates (yp, zp).

Parameters:

y (scalar or array) – The y-coordinate(s) for the temperature location. [meters]
z (scalar or array) – The z-coordinate(s) for the temperature location. [meters]
t (scalar or array) – The time(s) of the temperature. [seconds]

Returns:

scalar – Temperature increase in degrees Celsius [°C].

Example

The following example demonstrates how to use this method to calculate and plot the temperature rise over time due to an instant line source:

import grheat
import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(0, 500, 100) / 1000   # seconds
y, z = 0, 0                           # meters
yp, zp = 0, 0.001                     # meters

line = grheat.Line(yp, zp)
T = line.instantaneous(y, z, t)

plt.plot(t * 1000, T, color='blue')
plt.xlabel("Time (ms)")
plt.ylabel("Surface Temperature Increase (°C)")
plt.title("Instantaneous Line Source at 1mm depth")
plt.show()

pulsed(y, z, t, t_pulse)[source]

Calculate temperature rise due to a 1 J/m pulsed x-line source.

This method computes the temperature rise at a specified location and time due to a pulsed x-line source. 1 J/m of heat is deposited along the x-line passing through the coordinates (yp, zp) from t=0 to t=t_pulse.

Parameters:

y (scalar or array) – The y-coordinate for the desired temperature location. [meters]
z (scalar or array) – The z-coordinate for the desired temperature location. [meters]
t (scalar or array) – Time(s) of desired temperature. [seconds]
t_pulse (scalar) – Duration of the pulse. [seconds]

Returns:

scalar or array – Temperature increase in degrees Celsius [°C].

Example

The following example demonstrates how to use this method to calculate and plot the temperature rise over time due to a pulsed line source 1mm deep:

import grheat
import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(0, 500, 100) / 1000   # seconds
y, z = 0, 0                           # meters
yp, zp = 0, 0.001                     # meters
t_pulse = 0.3

line = grheat.Line(yp, zp)
T = line.pulsed(y, z, t, t_pulse)

plt.plot(t * 1000, T, color='blue')
plt.xlabel("Time (ms)")
plt.ylabel("Surface Temperature Increase (°C)")
plt.title("Pulsed Line Source at 1mm depth")
plt.show()