How to Escape Stress With a Bubble Bath (The Right Way)

At the end of the semester, every student is trying to find a way to unwind. My favorite way to distress is to take a hot bath at the end of the day.  However, if I set my tap to run water at a comfortable temperature, by the time my bath is full the water it is too cold!  This is caused by the tub and the water coming to thermal equilibrium.  This means that when these two objects that have different internal energy form a system through physical contact, the temperature becomes spatially uniform over time.  I want to be able to approximate just how much my bathwater will cool by the time the tub and the water reach thermal equilibrium.  Additionally, I want to find out at what temperature I need to run my bath for my bathwater to be perfect once the tub and water reach equilibrium. 

Assumptions to be made:

Comfortable bathwater temperature = T_W = 104˚F = 40˚C                                                                                      

The average bath uses between 30-50 gallons of water.  I take full baths, so I will assume that the baths that I take use 40 gallons of water.
Volume of water in bath = V_W= 40 gallons = 0.15 m³

The density of water = ρ_w= 1000 kg/m³

The specific heat capacity of water between 15˚C -60˚C = Cp_w = 4.18 kJ/kgK = 4180 J/kgK

Initial temperature of the bathtub = T_T = 25˚C

I am not taking into account how much the water will be cooled by the air around it as it is pouring into the tub and as it sits in the tub.

My tub is made of acrylic.  After some searching, I found that the average volume of acrylic used to make a tub is V_T = 0.07 m³.

The density of acrylic = ρ_T = 1400 kg/m³

The specific heat capacity of acrylic at 25˚C= Cp_T = 1470 J/kgK

Part 1: What will my bathwater be if I run my water at a comfortable temperature?

We know that T_W > T_B.  Therefore,  T_W > T_F > T_B. 

We also know that the overall change in temp of the water will be equal to the temperature of the tap water minus the final temperature of the bathtub and water (ΔT_W = T_W – T_F).   

Additionally, the change in temperature of the bathtub will be equal to the final temperature of the bathtub and water minus the initial temperature of the bathtub ((ΔT_B = T_F – T_B).

The heat transferred in a system is represented by  Q = mCpΔT_, where m represents mass, Cp represents specific heat capacity and ΔT represents the change in temperature. This can be converted to Q = ρVCpΔT as m=ρV.   Therefore, the heat lost by the tap water is equal to Q_W= ρ_WV_WCp_W(T_WI-T_F) and the heat gained by the bathtub is equal to Q_T= ρ_TV_TCp_T(T_TF-T_TI).

Finally, thermal energy is conserved, so Q_T=Q_W and therefore:

ρ_TV_TCp_T(T_F-T_TI) = ρ_WV_WCp_W(T_WI-T_F)

If we plug all of our known variables into this equation, we have:

1400 kg/m³*0.07 m³*1470 J/kgK (T_F- 25˚C) = 1000 kg/m³_*0.15 m3*4180 J/kgK (40˚C-T_F)

With this, I found that the final temperature is 37˚C.

Part 2: What temperature do I need the tap water to be for me to have a comfortable bath?

Using the same equation as before, I can find the temperature that the tap water should be running at for me to have a comfortable bath once the tub and water reach equilibrium.

ρ_TV_TCp_T(T_F-T_TI) = ρ_WV_WCp_W(T_WI-T_F)

If we plug all of our known variables into this equation, we have:

1400 kg/m³*0.07 m³*1470 J/kgK (40˚C - 25˚C) = 1000 kg/m³_*0.15 m3*4180 J/kgK (T_WI-40˚C)

With this, I found that the temperature that I need to run my bathwater at is 43.4 ˚C! It seems like running my bath only a few degrees hotter will make all the difference.

References:

Room temperature: http://lamar.colostate.edu/~hillger/temps.htm

Bathtub dimensions: http://www.rempros.com/dimensions/bathtub_sizes.html, http://www.ehow.com/info_12030066_average-volume-bathtub.html#ixzz2mircgDxy and http://www.steps.ie/cmspages/getfile.aspx?guid=aa19bda2-c366-47eb-94e9-26a37305c950&forceattachment=1

Density of Acrylic: http://www.avlandesign.com/density_construction.htm

Density and specific heat capacity of water: http://www.engineeringtoolbox.com/water-thermal-properties-d_162.html

Specific heat capacity of acrylic: http://www.builditsolar.com/References/Glazing/physicalpropertiesAcrylic.pdf

Image Source:

http://www.easepain.net/erc/bathing_and_mosturizing.htm