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viscosity (viscous dissipation)
From thermodynamics
l 4 This equation ignores changing electrical fields, microwave energy, etc.
Chapter 3: The Energy Balances 159
d o = (\$\$dv+(%)vdT
= [ - p + T (8)J dV + CQ dT
Observing that
1 DP 1 DP
utilizing the continuity equation
v . ( p v ) + x dP = 0
and rewriting, illustrating using rectangular coordinates
(3.2.3 -3)
(3.2.3 -4)
(3.2.3-5)
(3.2.3 -6)
(3.2.3-7)
DP
p ( v . v ) + 1 5 i . = 0 (3.2.3-8)
- F E 1 DP = ( v - v )
(3.2.3 -9)
Substituting Equation (3.2.3-9) into Equation (3.2.3-5), the result in Equation
(3.2.3-4), and thence into Equation (3.2.3-2)
I60 Chapter 3: The Energy Balances
Pcvm DT = -(V.q)-T(n)P(V.v)-(r:Vv) aP
(3.2.3- 10)
As a frrst illustration, for constant thermal conductivity one obtains after
substituting Fourier\u2019s law of heat conduction
Where
ye = thermal energy generation
= - [ (V q) - k V\u2019T] - (T : V v)
(3.2.3-11)
(3.2.3 - 12)
The thermal energy generation term indicated contains both the effects of viscous
dissipation and of externally imposed sources such as electromagnetic fields that
represent non-conductive contributions to the heat flux.
As a second illustration, ignoring viscous dissipation and non-conductive
contributions to the heat flux, rewriting Equation (3.2.3-1 1) in component form
for a rectangular coordinate system yields
After inserting Fourier\u2019s law
(3.2.3- 13)
aT
q, = - k x (3.2.3- 14)
one obtains
Chapter 3: The Energy Balances
(3.2.3-15)
where, since we have not assumed the thermal conductivity to be constant in
space, it continues to reside inside the derivatives.
Third, in cylindrical coordinates, for unsteady-state heat transfer with flow in
only in the z-direction, with transfer
only by conduction in the r-direction,
only by convection in the z-direction (neglecting conduction
in the z-direction compared to convection in the zdirection)
for
an incompressible fluid (for which e = 2: v )
with
constant k,
the following equation is obtained
which is the differential equation describing
(3.2.3 - 16)
unsteady-state temperature of an
incompressible fluid flowing in a tube with no thermal energy generation in one
dimensional axial flow at constant pressure, where k is the constant coefficient
of thermal conductivity. For a solid rod fixed in space, as opjmsed to fluid in a
tube, with no temperature gradient in the z-direction so that there still is no
conduction in this direction, vz is zero and the second term drops out of the
equation. Solutions of this equation are discussed in Chapter 7.
162 Chapter 3: The Energy Balances
Chapter 3 Problems
3.1 Initially, a mixing tank has fluid flowing in at a steady rate w l and
temperature T1. The level remains constant and the exit temperature is T2 = T1.
Then at time zero, Q Btdsec flows through tbe tank walls and heats the fluid (Q-
constant). Find T2 as a function of time. Use dH = C, dT and dU = C,dT
where Cp = C,.
3.2 A perfectly mixed tank with dimension as in the sketch of liquid is at an
initial temperature of To= 100°F at time t = 0 an electrical heating coil in the
tank is turned on which applies 500,000 B T U h , and at the same time pumps
are turned on which pump liquid into the tank at a rate of 100 gal/min. and a
temperature of T1 = 80°F and out of the tank at a rate of 100 gaymin.
Determine the temperature of the liquid in the tank as a function of time.
Properties of liquid (assume constant with temperature):
p = 62.0 lbm/ft3
C, = 1.1 BTU/(lbm°F
4 ft diameter
w, T1
4 ft
Chapter 3: The Energy Balances I63
3.3 Two streams of medium oil are to be mixed and heated in a steady process.
For the conditions shown in the sketch, calculate the outlet temperature T. The
specific heat at constant pressure is 0.52 BTUhbmOF and the base temperature
for zero enthalpy is 32OF.
yI 200 Ibm /min w2 . 100 Ibm / min
0=20.000
40'F
3.4 Two fluids are mixed in a tank while a constant rate of heat of 350 kW is
added to the tank. The specific heat of each stream 2000 J/(kg K). Determine the
outlet temperature.
w1 = 200 lbmasdmin. T1 = 25OC
w2 = 100 lbmass/min. T2 = 10°C
w2 T2
kW
3.5 An organic liquid is being evaporated in a still using hot water flowing in a
cooling coil. Water enters the coil at 1300F at the rate of 1,OOO lbdmin., and
exits at 140OF. The organic liquid enters the still at the same temperature as the
exiting vapor. Calculate the steady-state vapor flow rate in l b d s .
164 Chapter 3: The Energy Balances
3.6 Steam at 200 psia, 60WF is flowing in a pipe. Connected to this pipe
through a valve is an evacuated tank. The valve is suddenly opened and the tank
fills with steam until the pressure is 200 psia and the valve is closed. If the
whole process is insulated (no heat transfer) and K and fare negligible, determine
the final internal energy of the steam.
r'L1
A t 200 psia, 600' F
A
Hz 1321.4 Btu 1 Jbm
3.7 Oil (sp. gr. = 0.8) is draining from the tank shown. Determine
a. efflux velocity as a function of height
b. initial volumetric flow rate (gal/min.)
c. mass flow rate (lbds)
How long will it take to empty the upper 5 ft of the tank?
Chapter 3: The Energy Balances 165
3.8 Fluid with sp. gr. = 0.739 is draining from the tank in the sketch. Starting
with the mechanical energy balance deternine the volumetric flow rate and of the
tank.
3.9 The system in the sketch shows an organic liquid (p = 50 lbm/ft3) being
siphoned from a tank. The velocity of the liquid in the siphon is 4 ft/s.
a. Write down the macroscopic total energy balance for this
problem and simplify it for this problem (state your
assumptions).
b. What is the magnitude of the lost work lw.
c. If the lw is assumed linear with the length of the siphon,
what is the pressure in psi at l?
3.10 A siphon as shown in the sketch is used to drain gasoline (density = 50
lbm/ft3) from a tank open to the atmosphere. The liquid is flowing through the
tube at a rate of 5.67 fds.
166 Chapter 3: The Energy Balances
a. What is tbe lost work lw?
b. If the lw term is assumed to be linear with tube length,
what is the pressure inside the bend at point A?
3.11 Cooling water for a heat exchanger is pumped from a lake through an
insulated 3-in. diameter Schedule 40 pipe. The lake temperature is 50°F and the
vertical distance from the lake to the exchanger is 200 ft. The rate of pumping
is 150 gaymin. The electric power input to the pump is 10 hp.
a. Draw the flow system showing the control surfaces and
control volume with reference to a zero energy reference plane.
b. Find the temperature at the heat exchanger inlet where the
pressure is 1 psig.
3.12 It is proposed to generate hydroelectric power from a dammed river as
shown below.
-I- e2o i t
Chapter 3: The Energy Balances 167
a. Draw a control volume for the system and label all control
S\lrfaces.
b. Find the power generated by a 90 percent efficient water
turbine if lw = 90 ft. 1bfAbm.
3.13 River water at 700F flowing at 10 mph is diverted into a 30' x 30' channel
which is connected to a turbine operated 500 ft below the channel entrance. The
water pressure at the channel inlet is 25 psia and the pressure 20 ft below the
turbine is atmospheric. An estimate of the lost work in the channel is 90 ft
1bfAbm.
a. Draw a flow diagram showing all control volumes and
control surfaces with reference to a zero energy plane.
b. Calculate the horsepower from a 90 percent efficient water
turbine.
3.14 Water flows from the bottom of a large tank where the pressure is 90 psia
to a turbine which produces 30 hp. The turbine is located 90 f t below