The next question concerns the Heat Equation Here, a > 0 is a positive real number called the thermal diffusivity and w(2,1) is the tem perature at position z and time 1. The thermal difusivity is different for different materials and mediums. In this problem, the spatial dimension is measured in milimeters (mm), the emporal dimension t' is measured in seconds (s), the temperature is measure in degrees elsius (°C) and the thermal diffusivity has dimensions of m³/s. a. A bar of length 1000 milimeters is assumed to be perfectly insulated in the lateral direction, allowing us to model it as one dimensional. Then, both ends of the bar are instantaneously submerged in an ice buth, which keeps them at the constant temperature of C. The rod is made of a carbon composite, which has a thermal diffusivity of approximately 200mm³/s. i Write down the boundary conditions for this problem. solution to this particaler moblem in the form =(2, 1) – Σ =(2, 1) – Σ (4. sim (5.(z)) + B_ cos (g_{z}}} <^~|)_ That is you do NOT need to apply separation of variables to find the solution in the above form but you do need to state the functions g.(2) and A.(1), and any known values of the constants A. or B. i. What will happen to the temperature in the bar after a very long time? Justify your
The next question concerns the Heat Equation Here, a > 0 is a positive real number called the thermal diffusivity and w(2,1) is the tem perature at position z and time 1. The thermal difusivity is different for different materials and mediums. In this problem, the spatial dimension is measured in milimeters (mm), the emporal dimension t' is measured in seconds (s), the temperature is measure in degrees elsius (°C) and the thermal diffusivity has dimensions of m³/s. a. A bar of length 1000 milimeters is assumed to be perfectly insulated in the lateral direction, allowing us to model it as one dimensional. Then, both ends of the bar are instantaneously submerged in an ice buth, which keeps them at the constant temperature of C. The rod is made of a carbon composite, which has a thermal diffusivity of approximately 200mm³/s. i Write down the boundary conditions for this problem. solution to this particaler moblem in the form =(2, 1) – Σ =(2, 1) – Σ (4. sim (5.(z)) + B_ cos (g_{z}}} <^~|)_ That is you do NOT need to apply separation of variables to find the solution in the above form but you do need to state the functions g.(2) and A.(1), and any known values of the constants A. or B. i. What will happen to the temperature in the bar after a very long time? Justify your
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![The next question concerns the Heat Equation
Here, a > 0 is a positive real number called the thermal diffusivity and u(z,t) is the tem
perature at position z and time t. The thermal difusivity is different for different materials
and mediums. In this problem, the spatial dimension is measured in milimeters (mm), the
temporal dimension t is measured in seconds (s), the temperature u is measure in degrees
celsius (°C) and the thermal diffusivity has dimensions of s
a. A bar of length 1000 milimeters is assumed to be perfectly insulated in the lateral
direction, allowing us to model it as one dimensional. Then, both ends of the bar are
instantaneously submerged in an ice bath, which keeps them at the constant temperature
of °C. The rod is made of a carbon composite, which has a thermal diffusivity of
approximately 200mm³/s.
i Write down the boundary conditions for this problem.
solution to this particula problem in the form
(2, 1) - Σ(2, 1) - Σ[A₂ sitt (g₂(2)) + B₂ cos (9₂(1)}]<^«(?),
That is, you do NOT need to apply separation of variables to find the solution in
the above form but you do need to state the functions 9,(2) and A,(†), and any
known values of the constants A, or B.
ii. What will happen to the temperature in the bar after a very long time? Justify your
answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F997ee88e-1f1b-44c0-8082-48c5bdbd6861%2F326cb971-ac5d-4a56-80d3-427681fe6c9f%2Falj4sdg_processed.png&w=3840&q=75)
Transcribed Image Text:The next question concerns the Heat Equation
Here, a > 0 is a positive real number called the thermal diffusivity and u(z,t) is the tem
perature at position z and time t. The thermal difusivity is different for different materials
and mediums. In this problem, the spatial dimension is measured in milimeters (mm), the
temporal dimension t is measured in seconds (s), the temperature u is measure in degrees
celsius (°C) and the thermal diffusivity has dimensions of s
a. A bar of length 1000 milimeters is assumed to be perfectly insulated in the lateral
direction, allowing us to model it as one dimensional. Then, both ends of the bar are
instantaneously submerged in an ice bath, which keeps them at the constant temperature
of °C. The rod is made of a carbon composite, which has a thermal diffusivity of
approximately 200mm³/s.
i Write down the boundary conditions for this problem.
solution to this particula problem in the form
(2, 1) - Σ(2, 1) - Σ[A₂ sitt (g₂(2)) + B₂ cos (9₂(1)}]<^«(?),
That is, you do NOT need to apply separation of variables to find the solution in
the above form but you do need to state the functions 9,(2) and A,(†), and any
known values of the constants A, or B.
ii. What will happen to the temperature in the bar after a very long time? Justify your
answer.
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