Basic Thermodynamics ~ J. Pohl © www.THERMOspokenhere.com (111-C240)

Work to Move a Wall

A tank contains a shallow body of water. Its bottom, left wall, and ends are fixed. The right wall can be pushed to the left and will remain vertical should that happen. Seals prevent all leakage of water. Figure 1. shows the wall in its initial and final positions.

Calculate the work required to push the movable wall a short distance to the left. Note that the depth of the tank (into the page) is L.

Discussion:  The event might proceed in a number of ways. As one example, the wall might be promptly shoved a short distance with this action being repeated. There are many ways the event might proceed, the work of the worker cannot be calculated. So, with that being the case, is there anything we can calculate and of what value might that calculation be?

We can calculate the "least work" of the event by use of a "frictionless" assumption which is to assume the system matter begins (State (1)) in a equilibrium and proceeds to an equilibrium State (2), with all intermediate states also (by assumption) "in equilibrium." Some term such events as "ideal," "quasi-static," or "frictionless."

The "number" provided by an "ideal solution" will serve as a "bound" for the work. In this case, the work calculated will be much smaller than any actual value. Having an ideal answer, we can say "the work of every actual event (of this manner) is "greater than" our answer.

Solutions:  In theory work can be calculated by either of two methods. One method, commonly called "Work/Energy" employs the energy equation. This method is possible when all energy changes of the event can be calculated. Whereupon work is the only unknown of the energy equation (Eqn 1 - below). Since the "other" terms of the energy equation are scalars, this evaluation is easier.

The second method, called "Force/Displacement," is a direct evaluation of work wherein the force and differential displacement are written as the vectors they are, scalar multiplied then integrated (Eqn 2 - below). This method can be tedious.

(1) Work is calculated "indirectly" as the sum of system
energy changes of the event.		 (2) Work is calculated "directly" by integration
of the force through its displacement.

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Work/Energy Solution:  Since the "wall and water" move during the event, our inclination is to select them as the system. But as the wall moves, the water surface moves upward moving the air above it. So perhaps a system as "water, wall and atmosphere" is needed. Work occurs "at the system boundary." Where is the boundary of the atmosphere? So we are obliged to create a special, imaginative system.

Figure 2 identifies our system as the masses within the volume bounded by the "dashed-line." This volume (of depth "L") is fixed in space. It does not move and we assume mass neither enters nor leaves it. Looking at (1) and (2), one realizes that with leftward movement of the wall, the water level rises to diminish the space above it and space behind, to the right of the wall, is created. So as the wall moves, air on the left (within the dashed-line system) passes over the top of the wall to reside in the vacated space right of the wall. To let that happen, our system selection includes a small space above the wall, a passageway of sorts, very small in height ,δ.

The system boundary is the "dashed-line."  F is the force
of the worker. R is the reaction force of the surroundings.
The distance δ, is a little space above for air to move left to right,
over the wall.

The figure shows the center of mass of the air, of the water and of the wall (initially and finally). Also shown is the least force, F, applied by the worker (to effect the event) and a second, symbolic force: the reaction of the surroundings (notated as R) which is required to maintain equilibrium of the system. This is our system for the work/energy method.

Notice: Gravity forces acting on the wall, water, and air are not shown in the system sketch. That those forces are not shown means (though it is usually not stated) that Earth is part of the system. With Earth as part of the system, system masses have potential energy. The gravity forces "Earth on mass" and "mass on Earth" are being equal, opposite and inside the system sum to zero.

Analysis:  The energy types of the wall, water and air are kinetic and potential. By our system selection, there is but one work effect, the force applied by the worker as it acts through its displacement.

(3) The sum of energy changes of three substances
equals the work of their event.

Since the process proceeds very slowly, the kinetic energies, initially, intermediately and finally are negligibly small, or zero - Eqn 4.

(4) The sum of potential energy changes of three
substances equals the work of their event.

Potential energy changes are written as mass times local gravity times elevation change.

(5) Only "water" and "air" experience energy change.

Figure 3 is used to specify the volumes and initial and final elevations of centers of mass. We enter these facts carefully. We avoid needless algebra.

(6a) The volumes needed have the dimension, L, into the page.

(6b) Keep the algebra clean and simple.

The terms have been reduced in the equation. The equation is rearranged to have work left-of-equality.

(6c) Check the units!

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Force/Displacement Solution:  With this method the system is the wall. The initial and final positions are shown (Fig 1). As the wall is moved to the left, the depth of water increases hence the force of the worker must increase. Figure 3 shows how the water depth changes with wall location. From that figure we deduce the relation H(x) (the water depth at the arbitrary position x*) to equal 2L²/x*. A check of this relation is made (below right).

   (7) Water depth equation is correct!

To write the relation "force-needed" as a function of position, we draw a free body diagram of the wall at an arbitrary position, x* Figure 4. The diagram shows the "pressure-force distributions" and other forces acting. The relation of forces for movement are prescribed by its momentum, mV, the property of Newton's 2'nd Law.

(8) C240_eqn_07

The derivative of momentum of the wall has a right slash, "/", which means it is taken as zero, by assumption. Right-of-equality, the first two terms represent forces (acting on the wall left and right respectively. These forces result from distributions of contact of the fluids air and water. The next force in the equation is that exerted by the worker. Since these forces change as the wall moves, they are functions of x. Eqn 8 is written for the position x*.

For equilibrium of the wall, three more forces are relevant. There is the gravity force, the normal force (or support by the ground) and the buoyant force. The directions of these forces are in the vertical and their sum is zero. Our momentum equation reduces to three terms which we write in "magnitude-direction" form with dependence indicated as x*.

(9)C240_eqn_07b

To express the magnitude, F(x*), of the distributed pressure of fluid over its contact area requires a the set up of integrals. Eqn 10 outlines the changes needed.

(10)C240_eqn_07aa

Our task becomes the evaluation of the pressure forces of fluid contact on the left and right surfaces of the wall as shown in Figure 5; the first and second evaluations noted implicitly in Eqn. 7. By brief inspection it is apparent that the force contributions of the top and on both sides of the wall are equal and opposite. A new, simpler free-body diagram will be useful, Figure 7.

The figure shows the pressures at the bottom and the top of the acting fluids. An equation of pressure with vertical value of "z" is:

(11)C240_eqn_07aaa

The next step is to insert the expression (11) into Eqn 8. Using it once with water as the fluid (left side of the wall) and again for air as the fluid (right side).

(12a)C240_eqn_11a

Below the integrations have been performed but limits have not been applied.

(12b)C240_eqn_11b

Do the "apply limit" step carefully.

(12c)C240_eqn_11c

Integrations are completed and the equation is "tidied-up". The next equation has only the force required of the worker as unknown. The equation applies at wall location x* but since this is an arbitrary location, the equation applies at all values of x.

(12d)12d

Terms of the above equation are forces. The work of a movement of the forces is determined by scalar multiplication of the equation by the displacement, dx(-I) (the wall displacement is in the direction, negative I.

(13a)11e

The integration is perfunctory. With a step or two we arrive below.

(13b)13

So, we arrive at an answer.

(14)13

Thankfully, this the same answer as obtained by the "Work/Energy" method.