An engineer is developing an electric water heater to provide a continuous (\"on
ID: 2237351 • Letter: A
Question
An engineer is developing an electric water heater to provide a continuous ("on demand") supply of hot water. One trial design is shown in the figure. (Figure 1) Water is flowing at the rate , the inlet thermometer registers , the voltmeter reads , and the ammeter reads current . Then the power (i.e., the heat generated per unit time by the heating element) is . Assume that the heat capacity of water is and that the heat capacity of the heater apparatus is . When a steady state is finally reached, what is the temperature reading of the outlet thermometer? What is the power at which the heater operates?Explanation / Answer
make no claim for the accuracy of this answer as I don't have a picture to understand how the heating element works in the system. This answer is based entirely on dimensional arguments. Heat capacity in SI units is in Joules/(kg*Kelvin degree) This means that if you apply some energy (in Joules) E, for a heat capactity C and mass m, you get a change in temperature T. E=m*C*T. Our heating element supplies V*I Watts (Joules per second) so V*I*t (time it is applied) gives the energy E supplied. VIt = mCT. Now the change in temperature is the difference between our initial temperature T1 and our steady state temperature Tf. T = (Tf-T1). This means V*I*t = m*C*(Tf-T1). Now the water is flowing in at some rate F which I'll assume is in units of a mass of water over a certain time. This would be m/t so in our expression: V*I= (m/t)*C*(Tf-T1) so we can rewrite (m/t) as F. giving us: V*I = F*C*(Tf-T1) If that assumption is incorrect then make any appropriate modifications to substitute sme variation of F with (m/t). Solving this expression for our outlet temperature we have: ANSWER: [Tf = [(V*I)/(F*C)]+T1] Now you may ask, what about C_h? Well the heater apparatus initially turns on, some of the energy in the heater (due to the current I) goes into heating the element up which means the water won't get as hot since the system is still heating both parts. When the element itself reaches its optimal operating temperature, then the energy generated goes to the water. More specifically, all energy generated goes into heating the heater apparatus and since it is then hotter than the water, that heat continues on into the water. Assuming there are no energy losses in the heater (all energy is converted into heat) then the heat capacity of the water is all that matters.