For Heat Transfer , please answer as soon as possible give expalanations if poss

Question

For Heat Transfer , please answer as soon as possible

give expalanations if possible please

The Avengers just found out that Loki has buried an internal source of heat generation given by q ?= q ?_0 (1-r/r_o ) deep within the earth, where the earth may be assumed to be a solid sphere of radius ro. To save the earth from blowing up, Thor drops this sphere in a special fluid at a temperature Tf with a coefficient of convective heat transfer of hc.

1)Starting with the Shell Energy Balance,: Derive the Differential Equation that best describes the temperature profile in this situation, assuming that the system has not achieved steady state.

2)Discretize the equation above using the Implicit method

3)Given that you have decided to discretize this system using 10 domains, please write out the BCs in the discretized form that you would teach Thor to use to calculate the temperature profile for this system.

Here is the heat generation

Explanation / Answer

1)    the Differential Equation that best describes the temperature profile in this situation

=> dq/dT = dqo/dT * ( 1 - r/ro)

=>   q = qo * e(-roT)

=>    T =   -(1/ro) * ln(q/qo)

=>   dT/dq =   -(1/qo*ro) * (1/q)

=> dT/dq + (1/qo*ro) * (1/q) = 0

2)    Here, T(q) =   (1/ro) * ln(qo/q)

=>   T(q)   =   (1/ro) * ln(qo) -   (1/ro) * ln(q)

3)    Here,    the BCs in the discretized form

=>   dT/dq + (1/qo*ro) * (1/q) = 0

=>   T(q)   = [ -(1/qo*ro) * (1/q)]dq

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