Head loss also known as pressure loss is used as a measure to compute total energy per unit weight above a particular point of reference. Usually, pressure loss is the summation of three constituent; elevation head (relative potential energy in terms of an altitude), velocity head (kinetic energy generated by the motion of uid) and, pressure head (corresponding gauge pressure of a column of water at the base of the piezometer where piezometer is a well with a small diameter used to measure the hydraulic pressure of underground water). The pressure loss occurs in a pipe is the loss of ow energy due to the friction or turbulence.
Moreover, the pressure loss is divided into two components namely major loss and minor loss [Crowe et al., 2010]. Both major and minor head losses depend on the properties of the uid and the material of the pipe. The total head loss is the combination of pipe head loss and component head loss. The viscous or the shear stress that act on the fully developed owing uid is known as the pipe head loss. Component head loss is occurred when the uid ows through pipe such as valves, bends, and tees. Sometimes the major head loss is known as the pipe head loss and minor head loss is known as component head loss. Major losses are easily calculated by using Darcy-Weisbach equation as expressed in the Section 4.1 [Crowe et al., 2010].
For example when the pipes are long such as in water distribution pipes in a city the major pressure loss might be large and moreover, the minor pressure loss will also be dominant because of the large number of bends and valves in the network. Therefore, in reality the pressure drop is inevitable as the pipes are not perfectly smooth with no friction and no uid exists without turbulence.
Pressure loss of a uid ow is directly proportional to the length of the pipe, the square of the velocity of the uid ow and a proportionality constant known as a friction factor while diameter is inversely proportional.
4.1 Pressure Drop Accross the Heat Exchanger
Fluids are required to be pumped through the heat exchangers in most of the industrial applications. It is signicant to determine the uid pumping power required as a part of designing the system and in cost operation analysis. Pumping power of the uid in a device is directly proportional to the uid pressure drop, which is associated
with friction of the uid and other pressure drop inuences along the path of uid ow. Pressure drop of the uid has a direct association with heat exchanger heat transfer, process, dimension, mechanical features and other relevant aspects including economic considerations. Objective is to outline the methods for pressure drop analysis in heat exchangers and related ow devices. Determination of pressure drop across a heat exchanger is essential for many applications for at least two reasons: The heat transfer rate can be inuenced signicantly by the saturation temperature change for a condensing/evaporating uid if there is a large pressure drop associated with the ow.
This is because saturation temperature changes with changes in saturation pressure and in turn aects the temperature potential for heat transfer [Saari, 2010].
In this section Darcy-Weisbach equation [Crowe et al., 2010], which is one of the most useful equations in uid mechanics is presented. It is used to calculate the head loss which occurs in a uid in a straight ow pipe. Darcy-Weisbach equation can be used if the ow is completely developed and steady. It is used for either laminar ow or turbulent ow and for either circular pipes or non circular pipes such as a rectangular pipe. Purpose of using the Darcy-Weisbach equation is to calculate a value for the friction factor. As stated in the equation the friction factor, the pipe-length-to-diameter ratio, and the mean velocity squared inuence the head loss.
hf =f.L D.V2
2g (4.1)
hf is the head loss due to friction which will be in meters in SI units. Therefore, it is converted into a form with the term ∆P which will give the pressure drop in Pascals.
∆P =ρ·g·hf (4.2)
By combinning Equation (4.1) and (4.2) an equation to compute the pressure drop across the heat exchanger can be dervied as follows.
∆P =f · L
D ·ρ·V2
2 (4.3)
Where;
∆P : Pressure loss due to friction across the heat exchanger f : Darcy friction factor
ρ : Density of the uid V : Velocity
D : Diameter of the pipe L : Length of the pipe
The simplication to the pressure drop across the heat exchanger can be derived by the combination of Equation 4.3 and Equation 4.7 as follows:
∆P = 0.1582·ρ3/4·L·D−5/4·µ1/4·V7/4 (4.4)
4.1.1 Darcy Friction Factor
Friction factor is the ratio between the shear stress acting at the wall and the kinetic pressure as shown in the Equation (7.11).
f ≡ 4·τ0
ρ·V2/2 (4.5)
Friction factor is also known as Darcy friction factor, Darcy-Weisbach friction factor and the resistance coecient is a dimensionless quantity. Moreover, friction factor for laminar ow depends only on the Reynolds number as shown in below [Crowe et al., 2010].
f = 64
Re (4.6)
There are many correlations such as: Colebrook-White equation, Haaland equation, Swamee¤-Jain equation, Serghides's solution which were determined by experimental data to obtain formulas for friction factor and one such important correlation which is called Blasius correlation is of interest as follows.
f = 0.3164
Re1/4 (4.7)
Where;
Re= ρV D
µ (4.8)
The Blasius correlation [Nunn, 1989] is valid for turbulent ows with up to Reynolds number 105.
4.2 Pressure Drop at Dividers (Valves)
Figure 4.1: Cross-section of a control valve
The dividers of the system is considered to have a control valve for the division of the uid ow into given proportions. Control valve is a component that controls or regulates a ow of uid by opening, closing or partially obstructing. There are many types of valves such as hydraulic valve, pneumatic valve, manual valve, solenoid valve and motor valve. Control valves are important in applications such as in controlling water for irrigation, industrial processes, taps, oil reneries, gas, petroleum processes and etc [Liptak, 2005].
Daniel Bernoulli, introduced a relationship to express the relation between the pressure drop occurs at a valve and the velocity by using the principle of conservation of energy.
The pressure drop across the valve is directly proportional to the square of the velocity while the specic gravity of the uid is inversely proportional. If the velocity of the owing uid across the valve is higher the pressure drop is higher and greater the density the lower the velocity and hence pressure drop. The valve sizing coecient (Cv) computed by experiments is dependent on the size and the type of the valve. The pressure drop across the valve is as follows [McAllister, 2013] [Liptak, 2005]:
Q=Cv
G :Specic gravity of uids (Water is normally1.0000) Q : Capacity in Gallons per minute
Cv: Valve sizing constant determined experimentally
∆P: Pressure dierence in psi 1psi= 6894.75729P a
4.3 Pressure Drop at Mixers (T-junction)
The mixers of the heat exchanger network are considered to have T-junctions and to compute the pressure drop at the mixers formulas modeled for the T-junctions are used.
When computing pressure loss at a T-junction head loss coecient for T-junction plays an important role.
The pressure loss which occurs at T-junction depends on few factors, such as velocity of incoming and outgoing uid at the junction, pipe diameters and the angle of the
branches at the junction. Some classical formulas for the pressure drop at a T-junction have been derived by Benedict [1980].
There are few assumptions made at the junction to compute pressure loss. A T-junction can be considered as a combination of two pipe components such as two elbows or an elbow and a sudden contraction [Vasava, 2007]. This section will state the classical formulas and the relevant formulas that were derived based on the above assumptions [Vasava, 2007]. Possible pipe arrangement for a mixer is illustrated in Figure 4.2.
Figure 4.2: Pipe arrangement of a mixer
Loss coecient for combining uid ows
The formula in this section was derived by considering the combination of the uid ows. Fluid ow may combine, combining from two or more pipes at the inlet to one pipe at outlet. The formula for the loss coecient of the uid within the branch1 and 0is given by, whereK depends on the kinetic energy of the combined ow in branch0,V denes the velocities of the respective branches,λ3 is dened by Figure 4.3, to be equal to 0.61at a T-junction, and α′ and β′ are dened by the same graph as described by equation (4.12) [Benedict, 1980],
where;
α′ = 141α−0.00594α2
β′ = 141α−0.00594β2 (4.12)
Data from T-junction states that there is no variation of the loss coecient for the uids with Reynolds number RD >1000.
Figure 4.3: Plot of value of λ3 across the pipe angle
Pressure drop for combining uid ows
As shown in the Figure 4.2, branch 1 and branch 2 will combine together to form branch0. Let the velocities and volumetric ow rates beV1, V2, V0 anda·V1, a·V2, a·V0 respectively. The loss coecient along the branch1 to0 is given by:
K1,0 =λ3 along branch 1to 0is given by:
P1−P0 = 2·g·K1,0
(V12) (4.14)
Similarly, pressure drop along branch 2 to0 is given by:
P2−P0 = 2·g·K2,0
(V22) (4.15)
4.4 Pressure Drop according to Bernoulli Equation
Bernoulli equation is an important principle in uid dynamics and is derived from the principle of conservation law of energy. Further, it states that, in a steady state uid ow, the combination of energy appearing in all forms remains constant along the streamline. Bernoulli equation includes the energy in forms of potential energy, kinetic energy and internal energy. In addition, it includes some limitations of the uid such
as: steady state, incompressible uid with negligible friction losses [Sleigh and Noakes, 2009].
Figure 4.4: Bernoulli Eect
Z1+ P1
ρ·g + V12
2·g =Z2+ P2
ρ·g + V22
2·g (4.16)
Where;
Z1,2: Elevation above the reference level and in this work all the pipes are considered to be at the same level for the simplication and simplied form of the Bernoulli equation to compute the pressure drop can be stated as follows.
P1−P2 = ρ
2·(V22−V12) (4.17)