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 =

Volume of water in bath =

**V**_{W}= 40 gallons = 0.15 m^{3}
The
density of water =

**ρ**_{w= }1000 kg/m^{3}_{}
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 m^{3}.
The
density of acrylic =

**ρ**_{T}**= 1400 kg/m**^{3}
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_{W}V_{W}Cp_{W}(T_{WI}-T_{F}) and the heat gained by the bathtub is equal to Q_{T}= ρ_{T}V_{T}Cp_{T}(T_{TF}-T_{TI}).
Finally,
thermal energy is conserved, so Q

_{T}=Q_{W}and therefore:
ρ

_{T}V_{T}Cp_{T}(T_{F}-T_{TI}) = ρ_{W}V_{W}Cp_{W}(T_{WI}-T_{F})
If
we plug all of our known variables into this equation, we have:

1400
kg/m

^{3}*0.07 m^{3}*1470 J/kgK (T_{F}- 25˚C) = 1000 kg/m^{3}_{*}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

_{T}V_{T}Cp_{T}(T_{F}-T_{TI}) = ρ_{W}V_{W}Cp_{W}(T_{WI}-T_{F})
If
we plug all of our known variables into this equation, we have:

1400
kg/m

^{3}*0.07 m^{3}*1470 J/kgK (40˚C - 25˚C) = 1000 kg/m^{3}_{*}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

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