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.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
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