- Heat Transfer - Section 1
- Heat Transfer - Section 2
- Heat Transfer - Section 3
- Heat Transfer - Section 4
- Heat Transfer - Section 5
- Heat Transfer - Section 6
- Heat Transfer - Section 7
- Heat Transfer - Section 8
- Heat Transfer - Section 9
- Heat Transfer - Section 10
- Heat Transfer - Section 11


Heat Transfer - Engineering
Q1: A long iron rod initially at a temperature of 20°C has one end dipped in boiling water (100°C) at time, t = 0. The curved surface of the rod is insulated so that heat conduction is one dimensional in the axial direction. The temperature at a distance 100 mm from the dipped end becomes 40°C at time, t = 200 s. The same temperature is achieved at a distance of 200 mm from the dipped end at timeA t = 283 s
B t = 356 s
C t = 400 s
D t = 800 s
ANS:D - t = 800 s To solve this problem, we can use the one-dimensional heat conduction equation in cylindrical coordinates: ∂x2∂2T=α1∂t∂T where T is the temperature, x is the distance along the rod, and α is the thermal diffusivity of the material. Given that the rod is initially at 20°C and one end is dipped in boiling water at 100°C, we can define the boundary condition at =0x=0 as (0,)=100T(0,t)=100 and at =x=L (the length of the rod) as (,)=20T(L,t)=20. We can solve this partial differential equation subject to these boundary conditions to find the temperature distribution along the rod as a function of time. Once we have the temperature distribution, we can determine the time at which the temperature at a distance of 200 mm from the dipped end reaches 40°C. Given that the temperature at a distance of 100 mm from the dipped end becomes 40°C at =200t=200 s, we can use this information to estimate the thermal diffusivity (α) using the formula: =24⋅α=4⋅td2 where d is the diameter of the rod (assuming the rod is cylindrical). Let's calculate α: =(0.1)24⋅200=1.25×10−4 m2/sα=4⋅200(0.1)2=1.25×10−4m2/s Now, using this value of α, we can solve the heat conduction equation to find the temperature distribution along the rod and determine the time at which the temperature at a distance of 200 mm from the dipped end reaches 40°C. Let's solve the equation to find the time. To solve the problem, we first need to find the thermal diffusivity (α) of iron using the provided information: =24⋅α=4⋅td2 Given that the distance from the dipped end to where the temperature is 40°C is 100 mm (0.1 m), and the time taken for this temperature to be reached is 200 s, we can calculate: =(0.1)24⋅200=1.25×10−4 m2/sα=4⋅200(0.1)2=1.25×10−4m2/s Now, using the one-dimensional heat conduction equation: ∂2∂2=1∂∂∂x2∂2T=α1∂t∂T We can solve this equation subject to the boundary conditions: At =0x=0: (0)=100T(0,t)=100 At =x=L: (,)=20T(L,t)=20 We'll use the method of separation of variables and apply appropriate boundary conditions to find the temperature distribution (,)T(x,t) along the rod. After obtaining the temperature distribution, we'll find the time t at which the temperature at a distance of 200 mm from the dipped end reaches 40°C. Let's solve the heat conduction equation to find the time. Given the thermal diffusivity =1.25×10−4 m2/sα=1.25×10−4m2/s, and the boundary conditions, we'll solve the one-dimensional heat conduction equation using separation of variables. Let's denote (,)=⋅()T(x,t)=X(x)⋅T(t). Then the heat conduction equation becomes: 1dx2d2X=α1T1dtdT Since the left side of the equation depends only on x and the right side depends only on t, they must be equal to a constant, denoted by −2−λ2, to satisfy the equality for all x and t. This leads to two ordinary differential equations:
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