The data from the Delta 1300/1400 Series installation manual on the approximate outlet (mix) temperature with varying cold and hot water inlet temperatures (at various settings of the rotational limit stop) was employed to generate an equation (via MathCAD) correlating the inlet temperatures to the outlet temperature.  This equation was then used with realistic combinations of hot and cold water inlet temperatures to get approximate shower outlet temperatures.  This temperature was then input into a published damage equation for conductive burns to estimate how much time must pass in order to accrue second and third degree burns.

      cold   hot 

shower  

T_inlet is the matrix* containing the cold (column 1) and hot (column 2) inlet water temperatures from the manual. T_out contains the approximate resultant shower temperature as reported by the manual for the temperature combinations in T_inlet when the limit stop is set at the 10th position**.  For example, according to the manual a cold water inlet of 70^oF with a hot water inlet of 140^oF yields an outlet temperature of 121^oF.

* The 114 value was originally 134 in the Delta manual, but did not make intuitive sense and was therefore assumed to be a typo.  The value of 114 was assumed to be the intended value.

** This setting was chosen because the apartment in question had a combination of inlet and outlet temperatures that most closely matched those of the 10th position.

MathCAD's "regress" function generates a linear coefficient vector Z which best fits the input data (T_inlet, the inlet hot and cold water temperatures) to the output data (T_out, the resultant shower water temperature).  Z₃, Z₄, and Z₅ are the coefficients the function generated.

Z = regress (T_inlet, T_out, 1)

Z₃ = 0.316

Z₄ = 0.736

Z₅ = -5.139

T_c is the cold water inlet temperature and T_h is the hot water inlet temperature.  T_shower is the resultant function, containing the dependent variables T_c and T_h and the coefficients generated above in vector Z.  Its output is the estimated shower temperature at a particular set of cold and hot inlet temperatures.

T_shower(T_c,T_h) = Z₃*T_c + Z₄*T_h + Z₅

To test the accuracy of the equation generated above (T_shower), the inlet temperature data from the Delta manual (T_inlet) was input into our equation and the results (T_compare) were compared with the results from the Delta manual (T_out).  The calculated values are typically within a few degrees of the values reported by Delta, and further validate the equation we developed as providing reasonable estimates of the outlet temperature with varying hot and cold inlet temperatures.

The graph below plots the dependent cold water inlet temperature (x axis) and hot water inlet temperature (y axis) and the resulting independent outlet temperature (z axis).

Figure 1 - Interpolated surface plot of the shower temperature with varying cold and hot water inlet temperatures.

X axis (outward): Cold Water Inlet

Y axis (inward): Hot Water Inlet

Z axis (upward): Shower Outlet

In an incident investigation of an alleged shower burn the cold and hot water inlet temperatures were measured directly from the subject hot water heater and were found to be 57^oF and 117^oF, respectively.  The outlet shower temperature was also directly measured and found to be 99^oF .  When the measured hot and cold water inlet temperatures were substituted into the equation developed above (T_shower), the value of 99^oF was predicted, corresponding exactly with that of the measured value.

The cold and hot water inlet temperatures for the subject apartment were entered into the developed equation (T_shower), and it's prediction compared to that which was actually recorded (T_actual).

Tshower(57,117) = 99

Tactual = 99

Now that the outlet water temperature can be reliably estimated, the time necessary to cause second and third degree burns at such temperatures can also be approximated by employing the concept of thermal damage, which is given the symbol W.

For second degree burns W₂ > 10, and for third degree burns W₃ > 10,000 (Pearse, 1986), where W is the cumulative tissue damage.

W2 = 10

W3 = 10⁴

From Moritz & Henriques work on cutaneous burns the following equation was developed to predict the amount of damage done at a particular temperature (Temp, in degrees Kelvin) over time (t, in seconds).

*The values E and A' below were determined by Moritz & Henriques as reported by Pearse (1986).  R_gas is the universal gas constant.

The shower outlet temperature can now be determined for any combination of water inlet temperatures (T_cold and T_hot) and substituted into the damage equation (described above) to determine how long it will take to accrue second and third degree burns.  According to the water heater's manufacturer, the maximum hot water temperature output of the device is 165^oF.  Therefore, if we use the measured cold water inlet value of 57^oF, we can estimate the shower's outlet temperature and then approximate the time necessary to achieve second and third degree burns.

Second Degree Burns:

Third Degree Burns:

In conclusion, under these extreme conditions a 2.5 minute shower can yield second degree burns.  However, third degree burns are not achievable under this scenario as a shower in excess of 1.75 days would be necessary.  Most people react immediately to hot water at temperatures around 111^oF, therefore it seems unlikely that the shower is the source of this plaintiff's burns.

REFERENCES

Moritz, AR and FC Henriques.  Studies in thermal injury: II the relative importance of time and surface temperature in the causation of cutaneous burns.  American Journal of Pathology.  1947; 23(5): 695-720.  As cited in Pearse, 1986.

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