# Modeling a wine cooler heat loss to ambient

I'm trying to model the steady state heat loss to ambient, in W, for a wine cooler similar to the following: For the modeling, I will need the following variables/constants:

• $T_a$ [K]: Ambient temperature (typically 300 K)
• $T_c$ [K]: Temperature inside the wine cooler enclosure (typically 280-290 K)
• $H, W, D$ [m]: height, width and depth of the cooler (facing from the front of the cooler, about the same perspective as the picture above)
• $L_w$ [m]: thickness of the thermal isolation material on the walls
• $k_w$ [W/(m K)]: thermal conductivity of the isolation material on the walls
• $Q_w$ [W]: heat conducted through the walls
• $L_g$ [m]: thickness of the cooler's front glass window
• $k_g$ [W/(m K)]: thermal conductivity of the front glass window
• $Q_g$ [W]: heat conducted through the front glass window
• $Q_c$ [W]: total heat conducted through the walls and front glass window
• $\varepsilon$: emissivity of the wine cooler walls
• $\sigma = 5.67 \times 10^{-8}$ [W/(m$^2$ K$^4$)]: Stefan-Boltzmann constant
• $Q_r$ [W]: heat loss by radiation
• $Q$ [W]: total heat loss to ambient

First define $H' = H - 2L_w$, $W' = W - 2L_w$ and $D' = D - 2L_w$, which are the dimensions of the thermal isolation walls, disregarding corners (to simplify the analysis).

Applying Fourier's law of heat conduction to the walls of the cooler, we get:

$$Q_w = k_w \frac{2 H' D' + 2 W' D' + H' W'}{L_w} (T_a - T_c)$$

Again, applying Fourier's law but now to the front glass window, we get

$$Q_g = k_g \frac{H' W'}{L_g} (T_a - T_c)$$

The total heat loss by conduction is thus

$$Q_c = Q_w + Q_g$$

However, since there is a front glass window on the cooler, we should also consider the effect of heat transfer by radiation. This is where it gets hard for me. This depends on the area of emission, which I understand to be the five internal walls of the cooler. Assuming this, we get:

$$Q_r = \varepsilon \sigma (2 H' D' + 2 W' D' + H' W') (T_a^4 - T_c^4)$$

$$Q = Q_c + Q_r = Q_w + Q_g + Q_r$$

The problem is that I have a wine cooler with a TEC1-12706 thermoelectric module connected to a constant 12 V voltage source. From the datasheet (linked above), assuming a $\Delta T = 10$ K (a strict lower bound to the actual $\Delta T$), the maximum heat pumping capability, reading from the graph, would be somewhat below 50 W (I estimate 48-49 W). The actual $\Delta T$ should be a little higher due to the heatsink being at a higher than ambient temperature, and hence the heat pumping capability should be smaller than the estimate above.

However, plugging in $T_a = 300.15$ K and $T_c = 290.15$ K (values that can actually be attained in practice by this cooler), and assuming $\varepsilon = 0.9$ since the inside walls are painted black, I get $Q_r = 47.1$ W. Hence all of the TEC's heat pumping capability should be used to pump the heat loss due to radiation, with nothing left to pump the heat loss due to conduction. This is clearly not the case as the wine cooler actually works at the claimed parameters, so I've got an error somewhere in my calculations. I am assuming it has to do with the fact that the front glass window is not perfectly transparent, so it should reflect some of the incoming radiation. However, I don't know how to account for this -- just multiply $Q_r$ by a constant representing the percentage of reflected radiation?

So here are my questions:

• Is the modeling of heat conduction correct?
• How should I fix my model of heat loss by radiation?
• Have you considered the heat gained from radiation by the environment? Heat loss due to radiation would result in the cooler cooling itself, with the thermoelectric element attempting to keep it warm (and failing)... – Floris Sep 21 '14 at 16:16
• Sorry, I worded this incorrectly. Everywhere in my post that I've used the term "heat loss" I actually meant to say "heat gain". – swineone Sep 21 '14 at 16:20
• the area to use for radiation is the window - not the walls. It is across the window that heat transfer by radiation could occur. But windows do not transmit all wavelengths equally so it is a bit more complicated (think greenhouse effect). Also, there is reflection off the window because of the refractive index mismatch. So cut the area to just $H\cdot W$ and reduce $\epsilon$ to 0.5 and you will be a bit closer to the truth... That's still spitballing it. – Floris Sep 21 '14 at 16:26
• Thanks, that looks much closer to a plausible value. Make that into an answer and I'll upvote and accept it. – swineone Sep 21 '14 at 18:39

## 1 Answer

The main problem with your approach is that you are using the wrong area for the radiation "window". The area over which the exchange of radiative energy can take place is just the window in the door - the walls "see each other" and that part of the radiation has no net effect. So you want to use just $H\cdot W$ for the area.

Secondly the window is partially reflective which again lowers the rate of heat exchange. Calculating this precisely is quite hard, but setting the emission to 0.5 seems like a reasonable way to approximate this.