# Cooling Down Fridge Encased in wooden box

I'm currently working on a project which involves a small fridge encased in a wooden box. It's designed for hotels and the boxes vary slightly, but one common trait is that the box has an open vent towards the back which is meant for ventilation. Unfortunately, the hotels like to jam the box up against a wall, which prevents the vents from working properly. We have no control over how the wooden box is manufactured (in fact, the fridge was made to fit inside this box) and the vent hole varies in height from hotel to hotel.

The point is, the hotels demand that we be able to cool down the temperature inside the box, without any added electricity. I have already looked at Thermoelectric Couples and the temperature difference is about 18°C between the ambient interior of the box and the compressor which isn't enough to run a small fan. I have attached some passive heat sinks to the compressor itself, which did reduce the ambient temperature, but increased the temperature inside the fridge. I'm stuck, and I am thinking of using aluminum foil extending from the fridge to outside of the box to help conduct the heat away. Can anyone suggest any other ideas?

# Theoretical Refridgeration

The coefficient of performance metric for the fridge is

$$CoP_R = \dot{q}_c/\dot{w}_R = T_c/(T_h - T_c)$$

Heat $$\dot{q}_c$$ is removed from the fridge with power $$\dot{w}_R$$. For an ideal case, this is defined by the temperature in the fridge $$T_c$$ and the temperature at the external coils $$T_h$$. You want to keep $$T_c$$ as low as possible. You control $$\dot{w}_R$$ through the electrical power to the fridge $$P$$ and the (irreversible) efficiency of the motor $$\epsilon_M$$ as $$\dot{w}_R = \epsilon P$$. The system defines $$\dot{q}_c$$ as the heat flow that enters the fridge from the outside. You must remove at least that to sustain the temperature $$T_c$$. The temperature $$T_h$$ is a design setting for the compressor and fluid that are used in the cooling coils themselves. Finally, $$\dot{q}_c$$ is related to the heat that must be pulled from the coils $$\dot{q}_h$$ by the expression $$\dot{q}_c + \dot{w}_R = \dot{q}_h$$. Heat transfer into the fridge and away from the coils are addressed as theoretical problems in heat transfer.

# Theoretical Heat Transfer

The heat flow into the fridge $$\dot{q}_i$$ has to be removed as $$\dot{q}_c$$ by the heat pump (cooling cycle). How can it be decreased? Consider it from heat transfer as below.

$$\dot{q}_i = U A (T_a - T_c)$$

Here, $$U$$ is the overall heat transfer coefficient for the system, $$A$$ is the essentially external area of the fridge, and $$T_a$$ is the surrounding air temperature. We can expand $$U$$ as a sum of resistors for external convection $$R_a = 1/h_a A$$, wall conduction $$R_w = w/kA$$, and internal convection $$R_c = 1/h_cA$$. In these expressions, $$h_j$$ is the air or internal convection coefficient, $$w$$ is the wall thickness, and $$k$$ is the thermal conduction of the wall.

At the back side, convection around the coil to the surrounding air pulls out the heat that is dumped to the air. This affects the temperature of the coil as below.

$$\dot{q}_h = h_a A_c (T_h - T_a)$$

In this, $$A_c$$ is the area of the coils.

# Practical Insights

Let us assume that you have no control on $$T_a$$.

As two first steps to lower $$T_c$$ with all else the same, you can improve the efficiency of your motor to increase $$\epsilon$$ or you can increase the power to the motor to increase $$P$$.

In the heat transfer realm, you can make changes to decrease the heat transfer into the fridge to lower $$\dot{q}_i$$. Ideally, you would want to DECREASE the convection around and inside the fridge to lower $$h_a$$ and $$h_i$$. This means that a fridge sitting in stagnate air is the best case. Alternatively, you can increase the thickness of the wall to the fridge $$w$$ or decrease the thermal conductivity of the insulation in the walls $$k$$.

The opposing case is to make changes that will increase $$\dot{q}_h$$. Here, the immediate inverse recommendation arises to INCREASE $$h_a$$ as convection around the coils or to increase the area of the coils $$A_c$$.

# Practical Suggestions

One suggestion is to change the insulation in your fridge to a material with a higher $$R$$ value. This will decrease $$U$$ and lower the heat flow into the fridge from the outside.

Another suggestions is to work on increasing the convection flow around the coils. Ask the hotels to have holes drilled in the bottom back of the box. This will improve the natural convective flow of air along the coils at the back of the fridge. The wider the holes or the opening is for air flow, the better. The best option is when the entire bottom of the box is a mesh that is mechanically strong enough to hold the weight of the fridge yet open enough to allow full air flow from below the fridge. This will only work when the box also includes a wide enough hole at the top to vent the hot air.

Absent the option to change the box design, design a channel system attachment to the fridge. The channel should allow natural convection to pull cooler air from the front along the bottom, run it up the coils in the back, and vent it out the back of the box.

An alternative answer is to redesign the fridge to include cooling fans to blow the air. The power could be pulled from a part of the power to the fridge itself. This will be a better cooling system than the passive one above. The downside is, this may generate more noise than a hotel will tolerate.

# Other Thoughts

When the hotel has their specifications on cooling demand, you should set your specifications on ventilation to meet that cooling demand. Otherwise it seems that you put yourself in a box that you allow them to dictate unilaterally that your system must meet their criteria regardless of where they put it. I know of no heat transfer system that functions the same always independently of its surroundings.

• Wonderful answer, and I agree with what you've said. Unfortunately the hotels manufacture their own boxes. We have no control over the specification and how the wooden box will end up looking. And as for drawing some power additional power I will look into it, but the way the fridge functions is optimized for the power at hand. Thank you. – Hossain Noor May 10 '19 at 12:54
• @HossainNoor My main note is, when the hotel sets specifications on the units, you should set specifications on the surroundings to the unit in order to meet those targets. The downsides may be, either you lose their business or you pull out your hair trying to meet ever-changing and potentially un-realistic configurations on the surroundings. It seems the latter is your approach now. – Jeffrey J Weimer May 10 '19 at 13:02
• Thank you for the feedback Jeffery, I hope I can work something out, but you're right I can't meet an ever changing demand with one solution. At least I don't see it with the restraints I have at hand. – Hossain Noor May 10 '19 at 13:08
• @HossainNoor I have expanded my answer with additional theory. I hope this might prove conclusive enough to complete your question. – Jeffrey J Weimer May 11 '19 at 13:43
• Thank you so much for this detailed write up! This cleared up a lot of the confusion I had with my project. – Hossain Noor May 13 '19 at 12:31

All the refrigerators have specific clearance behind the fridge radiators.

Many hotels have louvered doors for the cabinet enclosing the fridge, for the exact reasons you mentioned.

I am not sure wrapping the aluminum foil around the radiator will help. It will impede the convection circulation around the fridge.

• I will ask to see if the hotels can maybe manufacture with louvered doors. Also the aluminum foil will impede natural convection, didn't realize it as I am desperate to fix this small issue. I think maybe there might be a better compressor which will generate less heat. – Hossain Noor May 10 '19 at 12:56