The next question concerns the Heat Equation Ut = AUxx. Here, a > 0 is a positive real number called the thermal diffusivity and u(x, t) is the tem- perature at position x and time t. The thermal difusivity is different for different materials and mediums. In this problem, the spatial dimension x 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 mm²/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 0°C. The rod is made of a carbon composite, which has a thermal diffusivity of approximately 200mm²/s. i Write down the boundary conditions for this problem. ii. We solved a very similar problem during the Math1151 classes. Using what you have done in tutorials/lectures, and the information given above, write down the solution to this particular problem in the form u(x, t) = [un(x, t) = Σ[An sin (gn (x)) + B₁ cos (9n(x))] e¹n(t)¸ NEZ NEZ 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 gn(x) and hn(t), and any known values of the constants An or Bn ii. What will happen to the temperature in the bar after a very long time? Justify your answer.
The next question concerns the Heat Equation Ut = AUxx. Here, a > 0 is a positive real number called the thermal diffusivity and u(x, t) is the tem- perature at position x and time t. The thermal difusivity is different for different materials and mediums. In this problem, the spatial dimension x 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 mm²/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 0°C. The rod is made of a carbon composite, which has a thermal diffusivity of approximately 200mm²/s. i Write down the boundary conditions for this problem. ii. We solved a very similar problem during the Math1151 classes. Using what you have done in tutorials/lectures, and the information given above, write down the solution to this particular problem in the form u(x, t) = [un(x, t) = Σ[An sin (gn (x)) + B₁ cos (9n(x))] e¹n(t)¸ NEZ NEZ 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 gn(x) and hn(t), and any known values of the constants An or Bn ii. What will happen to the temperature in the bar after a very long time? Justify your answer.
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
Ut = AUxx.
Here, a > 0 is a positive real number called the thermal diffusivity and u(x, t) is the tem-
perature at position x and time t. The thermal difusivity is different for different materials
and mediums. In this problem, the spatial dimension x 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 mm²/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 0°C. The rod is made of a carbon composite, which has a thermal diffusivity of
approximately 200mm²/s.
i Write down the boundary conditions for this problem.
ii. We solved a very similar problem during the Math1151 classes. Using what you
have done in tutorials/lectures, and the information given above, write down the
solution to this particular problem in the form
u(x, t) = Σ un(x, t) = Σ[An sin (9n(x)) + Bn cos (In(x))] e¹n(t).
NEZ
NEZ
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 gn(x) and hn(t), and any
known values of the constants An or Bn
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%2F38e693cf-3870-4eab-9920-bd211e6a8ee6%2F1d412836-76f1-4642-adaf-4666fce5dad4%2Fjwzz9n_processed.png&w=3840&q=75)
Transcribed Image Text:The next question concerns the Heat Equation
Ut = AUxx.
Here, a > 0 is a positive real number called the thermal diffusivity and u(x, t) is the tem-
perature at position x and time t. The thermal difusivity is different for different materials
and mediums. In this problem, the spatial dimension x 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 mm²/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 0°C. The rod is made of a carbon composite, which has a thermal diffusivity of
approximately 200mm²/s.
i Write down the boundary conditions for this problem.
ii. We solved a very similar problem during the Math1151 classes. Using what you
have done in tutorials/lectures, and the information given above, write down the
solution to this particular problem in the form
u(x, t) = Σ un(x, t) = Σ[An sin (9n(x)) + Bn cos (In(x))] e¹n(t).
NEZ
NEZ
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 gn(x) and hn(t), and any
known values of the constants An or Bn
ii. What will happen to the temperature in the bar after a very long time? Justify your
answer.
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