Basic Thermodynamics ~ J. Pohl © www.THERMOspokenhere.com (D116) -  (D117)

Wave-Range Cooker ©

The "Wave-Range Cooker," newly available appliance for cooking, incorporates microwave technology from above with standard range or oventop heating below.

Fig-1 shows the device in operation. Initially (a) the pot contained four liters of soup is at a gentle boil. The chef intends to thickened the soup, slowly, by boiling by use of the microwave (above) operating at 800 Watts while the range (below) operates at 1000 Watts. The final condition, 30% of its water boiled away, Fig-2 (b).

Estimate the Least Time of heating is required to thicken the soup?

♦  We select the 4 liters of water as a closed system, Fig-1 shows the soup initially and finally.

Some call this type event a "batch process." The event "commences" and at a later time attains its second conditions. Initially the soup is boiling. It continues to boil until 30% is changed to vapor. Although good soup is not water, we approximate the soup to have the properties of pure water. As water heats to vapor it disperses into the kitchen.

The kitchen is at atmospheric pressure, hence the heat of the "thickening" process occurs at the constant pressure of the surrounding atmosphere. Upon boiling the water (initially) and water plus vapor (finally) experience near zero changes of kinetic or potential energies. However the boiled water will experience a change of volume. Work happens by virtue of the expansion of part of the water - as it becomes vapor at one atmosphere.

A basic energy equation, written in rate-form is:

(1)e01

The "work" and "heat" of Eqn-1 are written preceded by "summation sign" to remind us to address terms carefully. All substances are at least "simple compressible." "Other work" for the soup does not apply. Compression work (rate-form) is written as below-right.

(2)

Eqn (2) above but with summations.

Above, the heat of the soup is that arriving from the range and that passing from the boiling water into the surrounding kitchen space/air. At our level of study, the heat from the water to the kitchen cannot be calculated. Consequently we must assume it is "small or zero" and strike it out (/). Yet by neglecting this heat (which is not zero) we realize the answer obtained will be the "minimum" time for the event. Our energy equation is:

(3)

This is a first-order ordinary differential equation. The next step toward solution is to separate variables.

(4)

4

By definitions of calculus, the compression work term reduces to be the boundary pressure times differential volume. Also the boundary pressure of the soup equals the ambient pressure of the air.

(5)

5

Some algebra along with the definition of enthalpy (h = u + pv) yield a final differential equation form.

(6)

6

Next apply the linear integration operator to the equation. The limits of the integral "left-of-equality" are the initial and final state of enthalpy of the water. Right-of-equality the limits of integration are the corresponding times, commencement of heating and the "unknown," time of completion of the event.

(7)

7

To proceed we must cast the initial and final enthalpies in terms their masses and specific enthalpies. 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. Some algebra is requires:

(8)

8

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. (9)9

The enthalpy of phase change (some call this "latent heat of caporization at one atmosphere") is:

hg - hf = (2675 - 419) J/g (10)10

Entering these numbers we obtain the "least" time required.

(11)11

Thermodynamic calculations do not provide correct answers. Only 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.

Chef Thickens the Soup

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?