The thermal conductivity of water is much higher than the one of e.g. air (0.609 and 0.026 Wm-1K-1, respectively [1]). This means that heat will leave human body faster in cold water than in air of the same temperature, so you definitely don’t want your bath water to cool down. If you prefer to avoid cutting down on your bath time, you have two options: adding warm water or some bubbles.
Adding more water requires you to first drain a portion of the old water and then refill with a new hot one, which is a waste of water, energy and money. Although individual preferences differ, the recommended temperature for the bath water is 1 – 2°C above your average body temperature[2], so let’s imagine you have a bath with the initial water temperature of 40° and you let it drop for 2°C, in a 200 liters bathtub. Even if you set the temperature of the tap water to the maximum (50°C[3]), mixing it with the cold water means that you will need to refill 33l of new hot water if you want to achieve the original temperature of the bath!
(T*( m1c1 + m2c2 ) = m1c1T1 + m2c2T2 🡪 313K[40°C] * 200L = m1*311K[38°C] + (200L-m1)*323K[50°C] 🡪 m1 = 166.67L 🡪 m2 = 33.33L
Another possibility mentioned above is to add some bubbles - foam. You read it well – the bubble bath will not only bring your experience closer to Hollywood, but will also keep your water warm for longer. The soap bubbles will trap air, which is a great insulator. Therefore, a thick and stable layer of these bubbles will thermally insulate the top surface of the water. This will slow down the heat escaping through surface via radiation, convection and evaporation[4]. An experiment[4] on a smaller scale showed this works. Two identical bowls were filled with hot water, one was left like that and the other one was turned into a small bubble bath. Despite both started at 44°, after one hour the water in the bowl with bubbles was at 38°, while the one without was at 34°! So now you have a good reason to pamper yourself with some foam in bath!
However, if your bath loses heat suspiciously fast, adding more soap won’t help since the tub itself may actually be the problem, letting the heat transfer too quickly through its walls. It all depends on the heat retention qualities of the material. Some (e.g. fiberglass) have moderate heat retention, while others (e.g. stone resin) has excellent heat retention[5]. If you are a real bath lover, you should definitely check that out. Naturally, the quality will reflect on the price, too. You may also want to check insulation around the tub. If the cavities between the tub and walls or floor are not insulated already, consider filling the gaps with foam or some other insulation. This, however, might be a painful process if your bathtub is fully installed already!
Since filling up the bathtub takes time, you can also turn up the temperature of the water higher than you actually desire. This will not only give room for the water to cool down to a comfortable temperature by the time the bath is ready, but will aslo preheat the interior surface of the tub. Just don’t overdo it with the temperature or with the time spent bathing (once the skin gets wrinkled it’s time to go!).
(Thank you @Bianka Trenčanová - Sustainable Energy Master Student, for this great article!)
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Sources:
[0]wikipedia
[1] wikipedia
[2] my-home-zen-spa.com; victoriaplum
[3] childsafetyeurope
[4] thenakedscientists
[5] badeloftusa
