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Imagine two bodies, say a body A with an infinite heat capacity(reservoir) and the another body B with some finite heat capacity($C = \alpha \ T_{B}(t) $).

They come into direct contact with each other for a limited time, as B moves along the surface of A with a constant velocity v.

Given the thermal conductivity $\kappa$ of the moving object B, the temperatures of the reservoir $T_{A}$ and the moving object $T_{B}(t)$.

How to calculate the amount of heat transferred in the process as a function of time?

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  • $\begingroup$ What does $C \propto \alpha \ T_{B}(t)$ mean precisely? What is $\alpha$? $\endgroup$ – Gert Jul 24 '20 at 17:27
  • $\begingroup$ actually $\alpha $ is just a constant that depends on the type of the material of B $\endgroup$ – EverydayFoolish Jul 24 '20 at 20:01
  • $\begingroup$ In thermodynamics we normally write: $\Delta Q=mc_p\Delta T$: the heat absorbed (or shed) by a change of temperature of an object of mass $m$ and specific heat capacity $c_p$. $\endgroup$ – Gert Jul 24 '20 at 20:06
  • $\begingroup$ I made another edit. $\endgroup$ – Gert Jul 24 '20 at 22:14
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How to calculate the amount of heat transferred in the process as a function of time?

The calculation will first of all depend on the Biot Number:

$$\text{Bi}=\frac{hL}{\kappa}$$

where $h$ is the convective heat transfer coefficient, $\kappa$ the thermal conductivity and $L$ a characteristic length of the object (a length or diameter, typically)

  1. Low Biot Number:

The interior of the object may be considered of uniform temperature distribution (no or low temperature gradient inside the object $B$)

Here lumped thermal analysis can be used by means of Newton's Law of Cooling. It states:

$$\dot{Q}=hA[T(t)-T_A]=hA\Delta T(t)\tag{1}$$ where:

  • $\dot{Q}$ is the rate of heat transfer out of the body ($B$)
  • $A$ is the surface area shared by $A$ and $B$
  • $T(t)$ the temperature of body $B$ and $T_A$ the (constant) temperature of $A$

Note that due to cooling (or heating) of $B$, $\Delta T(t)\neq \text{constant}$. To calculate the heat transfer in a given amount of time, the evolution of $T(t)$ has to be (and can be) calculated.

$\text{NLC}$ is mathematically simple to use.

Eq. $(1)$ is a separable differential equation that solves to:

$$\Delta T(t)=(T_0-T_A)e^{-kt}\text{ where }k=\frac{hA}{mc_p}$$

($T_0$ is the initial temperature of $B$)

So with $(1)$ we get:

$$\dot{Q}=\frac{\text{d}Q}{\text{d}t}=hA(T_0-T_A)e^{-kt}$$

The heat transferred in a time-interval $\Delta t$ is:

$$Q=\int_0^{\Delta t}hA(T_0-T_A)e^{-kt}\text{d}t$$

Or:

$$Q=-mc_p(T_0-T_A) e^{-k\Delta t}$$

  1. High Biot Number:

Here significant temperature gradients in the body $B$ must be assumed and heat conduction will play a significant part in the heat transfer process.

The 'go to' equation here is Fourier's Heat Equation:

$$\frac{\partial T}{\partial t}=\alpha \nabla^2T\text{ with }\alpha=\frac{\kappa}{c_p\rho}$$

which is a partial differential equation (PDE) and will require boundary conditions and an initial condition. From the obtained spatial distribution of temperature can then be calculated the heat transferred in a given time-interval.

In many cases solving Fourier's equation is mathematically very demanding, requiring higher calculus. Analytical solutions are really only possible for simple geometries (rod, plate, sphere for instance)

An example for an internally heated sphere can be found here.

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