# Time required to cool an object using a fan

#### samiers92

Hello,

A flat object with a known surface area is at a temperature of 285F. The ambient temperature in the room is 80F. The object needs to be cooled down to 100F within a time limit of 5 minutes. To accelerate the heat transfer process, there is a fan in the setup.

The objective is to calculate the air flow rate required in the fan (in CFM) to achieve the above mentioned cooling process.

#### benit13

Whenever you have a heat transfer problem, consider the following:

1. Whether the system is going to be in thermal equilibrium or not
2. What kind of heat transfer processes there (i.e. radiation, conduction and convection)
3. Whether the system is isolated or not

In your case, I would recommend a system that is not in thermal equilibrium, is not isolated by the surroundings and then consider convection heat transfer from a substance to the surroundings.

Then, you can consider a control volume around your system of interest (draw a diagram!), draw the heat transfer processes occurring in the your volume and determine a conservation of energy equation describing your volume.

In your case, you probably have something like a lump of mass, $$\displaystyle M$$, with initial temperature, $$\displaystyle T_0$$, and heat capacity, $$\displaystyle c_p$$, surrounded by air with a constant ambient temperature, $$\displaystyle T_a$$. Then:

$$\displaystyle Q_{system} = Q_{gen} + Q_{in} - Q_{out}$$

Where
$$\displaystyle Q_{gen} = 0$$ (no internal heating),
$$\displaystyle Q_{in} = 0$$ (no heat gain by any sources), and
$$\displaystyle Q_{out} = h_cA(T_a - T)$$ (heat loss due to convection)

where $$\displaystyle h_c$$ is the convective heat transfer coefficient and $$\displaystyle A$$ is the surface area.

The heat transfer to the control volume can be sensible (not latent), so

$$\displaystyle Q_{system} = Mc_p \frac{\partial T}{\partial t}$$

Substituting into the formula above gives

$$\displaystyle Mc_p \frac{\partial T}{\partial t} = - h_cA(T_a - T)$$

This gives a first order separable ODE that can be solved for temperature as a function of time with the boundary condition $$\displaystyle T = T_0$$ when $$\displaystyle t = 0$$.

Note that the convective heat transfer coefficient is given as a function of air velocity, so you can solve the equation above to get the heat transfer coefficient with the boundary condition $$\displaystyle T = T_{target}$$ when t = $$\displaystyle t_{target}$$. You can then relate it to the air velocity required and finally the fan volume flow rate.