How are heat transfer rate and heat capacity constant of a material related? How are heat transfer rate and heat capacity of a given material is related?
When I was a kid I imagined it like "Capacity is like a container or bucket of some sort, if it gets filled easily it can give away heat easily. Metaphorically overflowing". Or is it related at all?
 A: Heat capacity is defined as$$
C ~\equiv~ \frac{\Delta Q}{\Delta T}
\,;$$see "Heat Transfer", Holman.
It is a property that refers to the amount of energy required to increase the temperature of an object.  This is slightly different than specific heat$$
C_p ~{\equiv}~ \frac{1}{m} \frac{\Delta Q}{\Delta T}
\,;$$
see "Specific heat", HyperPhysics.
Heat transfer rate is much more complex because this term is not quite defined.  If you mean the rate at which thermal energy flows, you need to solve the heat transfer equation.  Take a look at this for a summary of equations for heat transfer methods.  Energy can be transfer via conduction, radiation, convection, or even advection.
Perhaps you are thinking of thermal diffusivity, $\alpha$.  Thermal diffusivity is specifically the rate at which thermal conduction can change the temperature of an object:  $\alpha = \frac {k}{\rho \times c_p}$.  It has units of meters-squared/sec ($ \frac {m^2}{sec}$).
I like using thermal diffusivity to estimate the rate at which an thermal system approaches steady state.  Heat transfer has a tendency to look like:$$
T\left(t\right)~=~ T_{\text{steady}} \, \left(1 - \exp{\left(\frac{-t} {\tau}\right)}\right)
\,.$$
Given a 1-dimensional distance that heat is traveling, it comes to equilibrium in a few "$\tau$" or time constants where
$$\tau = \frac{s}{\sqrt\alpha}\,,$$
where $s$ is a characteristic distance in your system.
A: Heat capacity is the ability of a material to store heat, higher the heat capacity higher the amount to heat stored by the material. Heat transfer usually varies inversely with heat capacity, i.e heat transfer will decrease with increase in heat capacity and vice versa. Thermal diffusivity is the ratio of thermal conductivity to the heat capacity, it says how fast (or slow) heat is transferred inside a material. Usually thermal diffusivity is directly proportional to the heat transfer rate, since heat capacity appears on the denominator, heat transfer is inversely proportional to the heat capacity. 
A: Heat, a measure of thermal energy, can be transferred from one point to another. Heat flows from the point of higher temperature to one of lower temperature. The heat content, $Q,$ of an object depends upon its specific heat, $c,$ and its mass, $m.$ Heat transfer is the measurement of the thermal energy transferred when an object having a defined specific heat and mass undergoes a defined temperature change.$$
\left[\text{heat transfer}\right]
~=~ \left[\text{mass}\right] \times \left[\text{specific heat}\right] \times \left[\text{temperature change}\right]
\,,$$or$$
Q ~=~ mc \, \Delta T
\,,$$where:


*

*$Q$ is heat content in Joules;

*$m$ is mass;

*$c$ is specific heat in $\frac{\mathrm{J}}{\mathrm{g} \cdot \mathrm{K}};$

*$T$ is temperature, and

*$\Delta T$ is change in temperature,
as described here.
Heat transfer takes place in 3 main ways: conduction, convection and radiation. The process of conduction in thermal dynamics is dependent on the factors such as heat transfer coefficient of the material, area, thickness through which the heat is transferred and the change in temperature.
A: I don't think that the heat capacity and thermal conductivity of a material that closely related. Think about systems rather than materials for a moment. If you take a lump of material and stick it in a thermos, then the heat capacity stays the same, but the thermal conductivity will plummet. By varying how good the thermos is, you can get a wide range of thermal conductivities.
Heat capacity is about storing thermal energy, and thermal conductivity is about moving thermal energy. As the thermos example shows, those two things needn't be closely related.
You can come up with an approximate kinetic formula for thermal conductivity of a material (see, for example, Ziman's Electrons and Phonons: The theory of Transport Phenomena in Solids):
$$\kappa \propto C \bar{v} \Lambda$$
Here, $\kappa$ is the thermal conductivity, $C$ is the specific heat, $\bar{v}$ is the average velocity of the particles carrying the heat (electrons, phonons, molecules, etc.), and $\Lambda$ is the average distance those particles go before scattering off something (i.e. being deflected in another direction).
So, all things being equal, larger specific heat means a larger thermal conductivity. However, $\bar{v}$ and $\Lambda$ can vary a lot from material to material, so it's hard to say anything predictive. I haven't seen a scatter plot of $\kappa$ vs $C$ for different materials, but at least for solids, I'm guessing that it will basically be a big blob without that much structure.
A: An analogy that could work is an electric circuit that has a battery, a capacitor and a resistor. The higher the capacity (which depends on surface area and other factors) the more charge it can store, and, the more time it requires for it fully charge.
The resistor affects (restricts) the flow rate of electric charge (The opposite of coefficient of heat transfer). the higher the value of the resistor, the less the value of the electric current (i.e. it increases the time required to charge the capacitor).
as you can see here, there is no relationship between the capacity of the capacitor and the value of the resistor. you can change the resistor and the capacity of the capacitor stays the same.
returning to heat transfer; a material "A" can have a high coefficient of heat transfer and a high heat capacity. A different material "B" can have low coefficient and high capacity. in this case, both "A" and "B" require the same amount of heat to get them to a specific temperature. but "A" will reach that temperature quicker due to the high heat transfer coefficient.
A: Heat transfer rate is $dQ/dt$ where Q is heat and t is time.
Specific heat has to do with "how much heat do I need to raise the temperature of this object" (dependent on mass too). Water has a high specific heat since it requires a lot to raise the temperature of a given mass of water. Iron has a low specific heat since it requires little to raise the temperature of a given mass of iron (usually 1 gram is the de facto). Let's call this $c$.
We know that $Q=mc\Delta t$ If we take the derivative on both sides with respect to time, we can see that $dQ/dt$ is a function of m,c,and how fast temperature is changing.
