Reference no: EM132590359

Assignment - Engineering Mathematics Questions

Show detailed steps of your work in each problem. Write your solutions on blank paper, and clearly label each problem number.

Q1. A cup of coffee is 82^oC when it is poured in a room with ambient temperature 21^oC. Fifteen minutes later, its temperature is 60^oC.

(a) Find U(t), the temperature of the thermometer at time t, assuming the coffee loses heat according to Newton's Law of Cooling.

(b) Determine when the coffee will reach 30^oC.

Q2. Heat flows radially, by conduction, through a sphere of radius a. Derive a differential equation that models the equilibrium temperature of the sphere (U) as a function of the radius (r). DO NOT SOLVE.

Q3. Consider a cylindrical heat fin with cross-sectional radius a, and heat flows linearly through the cylinder. Derive a differential equation model for the equilibrium temperature U(x).

Q4. Consider a cylindrical shell with inner radius a and outer radius b. The inner surface gains heat according to Newton's Law of Cooling, where the surrounding air has temperature u_s and h is the Newton cooling coefficient. The temperature on the outer surface is maintained at u_a. Write the two boundary conditions in terms of the temperature U (i.e. do not write them in terms of J). DO NOT SOLVE.

Q5. Consider a cylindrical shell with inner radius a and outer radius b. We have seen the equilibrium temperature U at radius r can be modeled by the equation d/dr(r dU/dr) = 0. The inner surface (at r = a) gains heat with flux q Watts per unit area, and the outer surface (at r = b) is held at a constant temperature u_a. Therefore, the boundary conditions are: J(a) = q and U(b) = u_a. Let k represent the conductivity of the material.

(a) Find the general solution to the differential equation d/dr (r dU/dr) = 0.

(b) Apply the boundary conditions to find the two constants of integration.

(c) Use your solution to find the equilibrium temperature of the inner surface.

(d) What does the equilibrium temperature of the inner surface approach as the conductivity approaches infinity?

(e) What does the equilibrium temperature of the inner surface approach as the conductivity approaches zero?

Q6. Heat flows inside a rectangular slab from left to right. Inside the slab, an electric current generates heat at a constant rate of Q₀ W/m³. Assuming the slab is NOT at thermal equilibrium, derive a partial differential equation model for the non-equilibrium temperature U(x, t).

Q7. The heat equation is used to model oscillating soil temperatures: ∂U/∂t = α ∂²U/∂x², with boundary conditions U(0, t) = u₁cos(ωt), lim_x→∞U(x, t) = 0. To solve this problem, we modify the first boundary condition to U(0, t) = u₁e^iωt. We have seen the general solution to the heat equation is the real part of the complex-valued function U^{^}(x, t) = (c₁e^bx + c₂e^-bx)e^iωt, where b = √(ω/2α)(1+i) . Apply the modified boundary conditions: U^{^}(0, t) = u₁e^iωt, lim_x→∞U^{^}(x, t) = 0 to the general solution to find c₁ and c₂. You do not need to solve the original model - you only need to find expressions for c₁ and c₂.