# 3.8: Untitled Page 34

- Page ID
- 18167

## Chapter 3

In order to relate this volume to the surface area of the control volume, we need to make use of the projected area theorem (Stein and Barcellos, 1992, Sec. 17.1).

*Figure 3‐9*. Volume of fluid crossing the surface *dA* in a time *t *.

*Figure 3‐10*. Volume leaving at the control surface during a time *t*

This theorem allows us to express the cross sectional area at an *exit* according to

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*dA*

**λ ****n** *dA , *

projected area theorem

*cs*

(3‐20)

and from this we see that the differential volume takes the form *V*

v *t ***λ ****n** *dA *

(3‐21)

Since the fluid velocity vector is given by,

**v ** v **λ **

(3‐22)

we see that the volume of fluid that crosses the surface area element *per unit time* is given by

*V*

volumetric flow rate across

**v ****n** *dA , *

(3‐23)

*t*

an *arbitrary* area *dA*

This expression is identical to that given earlier by Eq. 3‐15; however, in this case we have used the projected area theorem to demonstrate that this is a generally valid result. From Eq. 3‐23 we determine that the rate at which mass crosses the differential surface area is

*V*

*m*

mass flow rate across

**v ****n** *dA , *

(3‐24)

*t*

*t*

an *arbitrary* area *dA *

This representation for *m*

*/ * *t * is identical *in form* to that given for the *special case* illustrated in Figure 3‐7 where the velocity vector and the unit normal vectors were parallel. The result given by Eq. 3‐24 is *entirely general* and it indicates that

**v ****n ** represents the *mass flux* (mass per unit time per unit area) at any exit.

We are now ready to return to Eq. 3‐8 and express the rate at which mass *leaves* the control volume according to

rate at which mass

*flows out of the*

**v n** *dA*

(3‐25)

control volume

*A exit*

It should be clear that over the *exits* we have the condition **v ****n ** 0 since **n** is always taken to be the outwardly directed unit normal. At the *entrances*, the velocity vector and the normal vector are related by **v ****n ** 0 , and this requires that the rate at which mass enters the control volume be expressed as

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