| Basic Thermodynamics ~ J. Pohl © | www.THERMOspokenhere.com (169-D116) |
A pot on a range contains four liters of soup and is at a gentle boil. The chef intends that the soup be thickened slowly by boiling away 30% of its water. The rating of the stove element is 1800 watts.
What is the least time required to thicken the soup?
♦ The water (of the soup) that is boiled to vapor which then disperses into the kitchen is the system. The kitchen is at atmospheric pressure, hence the heat of the "thickening" process occurs at the constant pressure of the surrounding atmosphere. Since we do not know temperatures in the kitchen, we cannot calculate the precise time required. However, with the assumption that ALL of the power of the stove heating element transfers (as heat) to the soup, the LEAST time to thicken the soup can be determined.
Water initially liquid in a pot is boiled and becomes water vapor floating in the kitchen atmosphere. Upon boiling that water experiences near zero changes of its kinetic or potential energies. However the boiled water will experience a change of volume. The heated water must have extra energy, in order to "push-away" the surrounding atmospheric air. The energy equation required is:
| (1)1 |
Since the process occurs at a constant pressure of one atmosphere, we have within the integral that: pdV = d(pV). Apply that fact, then separate the variables of the equation.
| (2)2 |
Next, integrate. Deal with the integral of heat rate by use of the Mean Value Theorem.
| (3)3 |
To proceed we must cast each enthalpy, H, in terms of its mass and its specific enthalpy mh. The initial state is saturated liquid at one atmosphere. In the final state, some water remains as saturated liquid and some has "boiled-off" as saturated vapor. The subscripts " f " and " g " are used for the saturated liquid and vapor states, respectively.
| (4)4 |
Some algebra will provide H2 - H1:
| (5)5 |
Thirty percent of the original mass of saturated liquid is heated to become vapor. Thus the mass of vapor generated is:
| mg,2 = 0.30 [4000 cm3(1 g/cm3] = 1200 grams. | (6)6 |
The enthalpy of phase change (some call this "latent heat of boiling") is:
| hg - hf = (2675 - 419) J/g | (7)7 |
Entering these numbers we obtain the "least" time required.
| (8)8 |
Thermodynamic calculations do not provide correct answers. Approximate answers are obtained. In this case this time is the least because we realize some of the heat of the cooking range will pass to the surroundings.
A pot on a range contains four liters of soup and is at a gentle boil. The chef intends that the soup be thickened slowly by boiling away 30% of its water. The rating of the stove element is 1800 watts.
What is the least time required to thicken the soup?