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Heat flow -- Dynamics of a 1D rod

Developed by L. Engelhardt

In introductory physics, students learn two equations that involve heat: Q=mcΔT, describes the amount of heat transferred in calorimetry; and QΔt=ktAΔTΔx describes the rate of steady state heat flow through a window. In this Exercise Set we combine these two equations to explore the dynamics of heat flow and temperature change in one dimension. Specifically, we will explore the temperature of a frying pan handle as a function of both position and time, and we will see how this temperature profile depends on the material properties of the handle.

Subject Area Thermal & Statistical Physics
Levels First Year and Beyond the First Year
Available Implementations IPython/Jupyter Notebook and Python
Learning Objectives

Students will be able to:

  • Mathematically derive an equation for the small slice of a 1D rod at the end of the rod (Exercise 1)
  • Create 1D arrays to store the discretized values of both position and time (Exercise 2)
  • Look up the relevant material properties for heat transfer through a rod (Exercise 3)
  • Compute the heat transfer constant, r (Exercise 4)
  • Set initial values in a 2D array – T(x,t) – to represent a 1D rod at time t=0(Exercise 5)
  • Use the hyperbolic tangent function to model the increasing temperature of a frying pan, and use this to set boundary conditions for T(x=0,t) (Exercise 6)
  • Convert temperatures between Fahrenheit and Celsius (Exercise 7)
  • Write code to compute T(x,t) for all x, t (Exercises 8 and 9)
  • Plot T(x) for various values of t (Exercise 10)
  • Test for convergence in T(x,t) (Exercise 11)
  • Create animations of T(x,t) (Exercise 12)
  • Create contour plots of T(x,t) (Exercise 13)
  • Interpret simulated data of T(x,t) from plots, animations, and contour plots (Exercise 14)
  • Carry out the analysis described above for multiple materials (Exercise 15)
Time to Complete 120 min
Exercise 1-3

In the exercises below, you will simulate heat transfer along a L=15 cm long frying pan handle for a frying pan that is heated on a stove for 10 minutes.

Exercise 1: Mathematical Derivation

Derive an equation for the temperature of slice number N at the end of a one-dimensional rod, TNj. Your derivation, and the resulting equation, will be very similar to the derivation and result for the other slices of the rod (for iN1). The only difference is that there is no heat flow in/out of the right side of this slice.

Exercise 2: x and t

Write lines of code to generate two one-dimensional arrays: one for the discrete values of position, xi, and one for the discrete values of time, tj.

Exercise 3: Material Properties

You will simulate the handle of a frying pan, using handles made of (1) stainless steel and (2) Bakelite. Look up the relevant material properties for both stainless steel and Bakelite. What are the numerical values for each of these properties for each material?

Exercise 4-6

Exercise 4: Dimensionless constant, r

Compute the constant “r” for stainless steel. What value do you obtain? If it is greater than 0.5, you will need to adjust your discretization of the position and/or the time to decrease the values of r.

Exercise 5: 2D array and initial temperature

Generate a 2D array that will be used to store temperature as a function of both position and time, Tij.

The temperature of the frying pan handle will begin at a room temperature of Ti0=72 F. Store this initial (j=0) value in the 2D array for all slices of the rod (for all i).

Exercise 6: Temperature of the pan

You will need to generate an array of temperatures for the frying pan (at the left edge of the handle) as a function of time, T0(t). The frying pan will start out at room temperature at time t=0. A typical frying pan will then heat up quicky for the first minute or two after a stove is turned on, and then it will reach a constant temperature of around 350 to 400 F. This behavior can be reproduced using the equation

(24)T0(t)=Troom+ΔTstove×tanh(tτ)

with Troom=72 F, ΔTstove=300 F, and τ=60 seconds. “tanh” is the “hyperbolic tangent” function, which is one of the standard built-in functions for any computer math library.

Use this equation to compute an array of T0(t) values, and then plot T0 versus t to verify that the temperature of you frying pan agrees with the following plot:

Once you have verified that T0(t) is correct, use the 1D array of T0(t) values to set the temperature of the left (i=0) edge of the handle in the 2D array, T0j, for all values of time (for all j).

Exercise 7-9

Exercise 7: Conversions

You will need to convert the 2D array, Tij from units of Fahrenheit to Celsius before beginning the simulation. Then convert back from Celsius to Fahrenheit after the simulation. What will you need to do in order to perform both of these conversions?

Exercise 8: Computing for many i and j

Complete the line of code to compute Tij for all positions, 1iN1, and all times, j>0.

Exercise 9: End of the handle

Complete the line of code to compute TNj at the end of the handle for all times, j>0. (You derived this equation in Exercise 1.)

Exercise 10-12

Exercise 10: Plotting

Plot the temperature of the rod versus position, T(x), for a few different values of time, t. Make sure that your results seem reasonable. Discuss if/why the results seem reasonable.

Note: It is likely that your program won’t work right away! If your code generates errors, or if the results just don’t seem physically reasonable, you will need to debug your code.

Exercise 11: Convergence

Decrease the values of Δx and Δt, and re-run the simulation to check for convergence. If the temperature results change when you change the discretization, that means that Δx and/or Δt are too big, and you need to make them smaller. (But make sure that r<0.5.)

Exercise 12: Animating

Create an animation showing temperature versus position at many different moments in time. Again, make sure that the results seem reasonable, and check for convergence. Discuss.

Exercise 13-15

Exercise 13: Contour plots

Create a contour plot that shows temperature as a function of both position and time in a single static plot. Again, make sure that the results seem reasonable, and check for convergence. Discuss.

Exercise 14: Analysis

At a temperature of 150 F, an object will quickly cause a burn if touched. (You can also get burned by cooler objects, but not immediately – only if you continue to touch the object for an extended period of time.) Based on the results of your simulations, discuss where you would – and would not – be able to safely touch the handle of the pan at different moments in time.

Exercise 15: Repeat for other material

Repeat the exercises above for a frying pan handle that is made of Bakelite rather than stainless steel. Discuss your results.

Dynamics of a 1D rod.py

# # Author: Larry Engelhardt (July, 2016)

# Material properties for a few different materials are listed below

#

# 1. Thermal conductivity: `k_t`

# 2. Specific heat capacity: `c`

# 3. Density: `rho`

#

# Stainless Steel

k_t = 16 # W / (m C)

c = 500 # J / (kg C)

rho = 8000 # kg / m^3

# Bakelite

k_t = 0.2 # W / (m C)

c = 920 # J / (kg C)

rho = 1300 # kg / m^3

# Oak

k_t = 0.17 # W / (m C)

c = 2000 # J / (kg C)

rho = 700 # kg / m^3

# Fiberglass

k_t = 0.04 # W / (m C)

c = 700 # J / (kg C)

rho = 2000 # kg / m^3

# BEGINNING THE PROGRAM:

from pylab import *

# Discretizing space

L = 0.15 # Length of handle in meters

Nx = 76 # Number of positions in space

x = linspace(0, L, Nx) # Array of positions

dx = L / (Nx - 1) # Values of delta x (meters)

# Discretizing time

tMax = 600 # Maximum time in seconds

Nt = 1801 # Number of time steps

t = linspace(0, tMax, Nt) # Array of time values

dt = tMax / (Nt - 1) # Size of time step (seconds)

T0 = 72 # Initial temperature (Farenheit)

dT_stove = 300 # Temperature increase of the stove (F)

# Stainless Steel's material properties

k_t = 16 # W / (m C)

c = 500 # J / (kg C)

rho = 8000 # kg / m^3

r = k_t * dt / (c * rho * dx**2) ## THE PARAMETER! (dimensionless) ##

print('r:', r) # Check to make sure that r < 0.5

# Preparing 2D array of temperatures

T = zeros((Nt, Nx)) # 2D array for temperature. NOTE THE ORDER: [time, space]

T[0, :] = T0 # Initial values (time t = 0)

tau = 60 # Used in tanh function, units of seconds

T[:, 0] = T0 + dT_stove * tanh(t/tau) # Edge of rod (at x=0) in deg. F

T = (T - 32) / 1.8 # Convert all temperatures to Celsius

# Verifying that \(T(t)\) for the pan (\(x=0\)) looks correct:

Tpan = T0 + dT_stove * tanh(t/tau)

plot(t, Tpan, lw=2)

grid(True)

xlabel('Time (sec)')

ylabel('Pan Temperature (F)')

grid(True); show()

# Computing the temperature of the handle for all `x` and `t`:

for j in range(1, Nt): # Loop thru TIME

for i in range(1, Nx-1): # Loop thru SPACE

T[j,i] = r*T[j-1, i+1] + (1 - 2*r)*T[j-1,i] + r*T[j-1, i-1]

T[j, -1] = (1-r)*T[j-1,i] + r*T[j-1,i-1] # End of rod (x=L)

T = 1.8*T + 32 # Convert back to F from C

x = x * 100 # Convert from meters to cm

# Plotting \(T(x)\) for a few moments in time

jValues = [30, 180, 540, -1] # A few values of the time index (-1 for last element)

for j in jValues:

plot(x, T[j,:], linewidth=2, label='Time (seconds): '+str(t[j]))

legend()

title('Stainless Steel')

xlim([0, 15])

xlabel('Position (cm)')

ylabel('Temperature (\(^{\circ}\)F)')

grid(True);

savefig('StainlessSteel_T_vs_x.png')

show()

# Creating a contour plot of \(T(x, t)\)

colorvals = linspace(70, 160, 91) # colors for contour plot (min, max, num)

contourf(x, t, T, colorvals) # (x-axis, y-axis, z(x,y))

title('Stainless Steel')

xlabel('x (cm)')

ylabel('Time (sec)')

colorbar()

grid(True)

savefig('StainlessSteel_contour.png')

show()

# Repeating using Bakelite

# Discretizing space

L = 0.15 # Length of handle in meters

Nx = 76 # Number of positions in space

x = linspace(0, L, Nx) # Array of positions

dx = L / (Nx - 1) # Values of delta x (meters)

# Discretizing time

tMax = 600 # Maximum time in seconds

Nt = 1801 # Number of time steps

t = linspace(0, tMax, Nt) # Array of time values

dt = tMax / (Nt - 1) # Size of time step (seconds)

T0 = 72 # Initial temperature (Farenheit)

dT_stove = 300 # Temperature increase of the stove (F)

# Bakelite's material properties

k_t = 0.2 # W / (m C)

c = 920 # J / (kg C)

rho = 1300 # kg / m^3

r = k_t * dt / (c * rho * dx**2) ## THE PARAMETER! (dimensionless) ##

print('r:', r) # Check to make sure that r < 0.5

# Preparing 2D array of temperatures

T = zeros((Nt, Nx)) # 2D array for temperature. NOTE THE ORDER: [time, space]

T[0, :] = T0 # Initial values (time t = 0)

tau = 60 # Used in tanh function, units of seconds

T[:, 0] = T0 + dT_stove * tanh(t/tau) # Edge of rod (at x=0) in deg. F

T = (T - 32) / 1.8 # Convert all temperatures to Celsius

for j in range(1, Nt): # Loop thru TIME

for i in range(1, Nx-1): # Loop thru SPACE

T[j,i] = r*T[j-1, i+1] + (1 - 2*r)*T[j-1,i] + r*T[j-1, i-1]

T[j, -1] = (1-r)*T[j-1,i] + r*T[j-1,i-1] # End of rod (x=L)

T = 1.8*T + 32 # Convert back to F from C

x = x * 100 # Convert from meters to cm

jValues = [30, 180, 540, -1] # A few values of the time index (-1 for last element)

for j in jValues:

plot(x, T[j,:], linewidth=2, label='Time (seconds): '+str(t[j]))

legend()

title('Bakelite')

xlim([0, 15])

xlabel('Position (cm)')

ylabel('Temperature (\(^{\circ}\)F)')

grid(True);

savefig('StainlessSteel_T_vs_x.png')

show()

colorvals = linspace(70, 160, 91) # colors for contour plot (min, max, num)

contourf(x, t, T, colorvals) # (x-axis, y-axis, z(x,y))

title('Bakelite')

xlabel('x (cm)')

ylabel('Time (sec)')

colorbar()

grid(True)

savefig('Bakelite_contour.png')

show()

 

Translations

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Software Requirements

SoftwareRequirements


Android iOS Windows MacOS
with best with Chrome Chrome Chrome Chrome
support full-screen? Yes. Chrome/Opera No. Firefox/ Samsung Internet Not yet Yes Yes
cannot work on some mobile browser that don't understand JavaScript such as.....
cannot work on Internet Explorer 9 and below

 

Credits

Fremont Teng; Loo Kang Wee

 

Version

  1. https://www.compadre.org/PICUP/exercises/exercise.cfm?I=138&A=heatflow_1d 
  2. http://weelookang.blogspot.com/2018/06/heat-flow-dynamics-of-1d-rod.html 

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