### Hydraulic Diameter

The hydraulic diameterDH, is a commonly used term when handling flow in non-circular tubes and channels. Using this term, one can calculate many things in the same way as for a round tube. It is defined as

DH = 4A/P

where

A is the cross-sectional area of the flow,

P is the wetted perimeter of the cross-section.

More intuitively, the hydraulic diameter can be understood as a function of the hydraulic radius RH, which is defined as the cross-sectional area of the channel divided by the wetted perimeter. Here, the wetted perimeter includes all surfaces acted upon by shear stress from the fluid.

RH = A/P

DH = 4RH

Note that for the case of a circular pipe,

DH = 4πR2 /2πR = 2R

The need for the hydraulic diameter arises due to the use of a single dimension in case of dimensionless quantity such as Reynolds number, which prefers a single variable for flow analysis rather than the set of variables as listed in the table. The Manning formula contains a quantity called the hydraulic radius. Despite what the name may suggest, the hydraulic diameter is not twice the hydraulic radius, but four times larger.

### Reynold’s Number

The Reynolds number (Re) is an important dimensionless quantity in fluid mechanics used to help predict flow patterns in different fluid flow situations. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers turbulence results from differences in the fluid’s speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow (eddy currents). These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of cavitation.

The Reynolds number is the ratio of inertial forces to viscous forces within a fluid which is subjected to relative internal movement due to different fluid velocities. A region where these forces change behaviour is known as a boundary layer, such as the bounding surface in the interior of a pipe. A similar effect is created by the introduction of a stream of high-velocity fluid into a low-velocity fluid, such as the hot gases emitted from a flame in the air. This relative movement generates fluid friction, which is a factor in developing turbulent flow. Counteracting this effect is the viscosity of the fluid, which tends to inhibit turbulence. The Reynolds number quantifies the relative importance of these two types of forces for given flow conditions and is a guide to when the turbulent flow will occur in a particular situation.

With respect to laminar and turbulent flow regimes:

• laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion;
• turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities

The Reynolds number is defined as

Re = ρuD/ μ = uD/ ν

where:

• ρ is the density of the fluid (SI units: kg/m3)
• u is the flow speed (m/s)
• D is the diameter of the tube (m)
• μ is the dynamic viscosity of the fluid (Pa·s or N·s/m2 or kg/(m·s))
• ν is the kinematic viscosity of the fluid (m2/s).

### Colburn J Factor

Chilton–Colburn J-factor analogy is a successful and widely used analogy between heat, momentum, and mass transfer. The basic mechanisms and mathematics of heat, mass, and momentum transport are essentially the same. Among many analogies (like Reynolds analogy, Prandtl–Taylor analogy) developed to directly relate heat transfer coefficients, mass transfer coefficients, and friction factors Chilton and Colburn J-factor analogy proved to be the most accurate.

It is written as follows,

j= (Rea)-0.49 * (ᶿ/90)0.27 * (Pf/Pl)-0.14 * (Lf/Pl)0.29 * (Dl/Pl)-0.23 * (Ll/Pl)0.68 * (Pt/Pl)-0.28 * (tf/Pl)-.05

This equation permits the prediction of an unknown transfer coefficient when one of the other coefficients is known. The analogy is valid for fully developed turbulent flow in conduits with Re > 10000, 0.7 < Pr < 160, and tubes where L/d > 60 (the same constraints as the Sieder–Tate correlation). The wider range of data can be correlated by Friend–Metzner analogy.

### Nusselts Number

In fluid dynamics, the Nusselt number (Nu) is the ratio of convective to conductive heat transfer at a boundary in a fluid. Convection includes both advection (fluid motion) and diffusion (conduction). The conductive component is measured under the same conditions as the convective but for a hypothetically motionless fluid. It is a dimensionless number, closely related to the fluid’s Rayleigh number.

A Nusselt number of value one represents heat transfer by pure conduction. A value between one and 10 is characteristic of slug flow or laminar flow. A larger Nusselt number corresponds to more active convection, with turbulent flow typically in the 100–1000 range. The Nusselt number is named after Wilhelm Nusselt, who made significant contributions to the science of convective heat transfer.

The Nusselt number is the ratio of convective to conductive heat transfer across a boundary. The convection and conduction heat flows are parallel to each other and to the surface normal of the boundary surface and are all perpendicular to the mean fluid flow in the simple case.

NuL = (Convective heat transfer / Conductive heat transfer)=hL/k

where h is the convective heat transfer coefficient of the flow, L is the characteristic length, k is the thermal conductivity of the fluid.

• Selection of the characteristic length should be in the direction of growth (or thickness) of the boundary layer; some examples of characteristic length are the outer diameter of a cylinder in (external) crossflow(perpendicular to the cylinder axis), the length of a vertical plate undergoing natural convection, or the diameter of a sphere. For complex shapes, the length may be defined as the volume of the fluid body divided by the surface area.
• The thermal conductivity of the fluid is typically (but not always) evaluated at the film temperature, which for engineering purposes may be calculated as the mean average of the bulk fluid temperature and wall surface temperature.

### Efficiency of Fin

The efficiency of a fin is defined as the ratio of the actual heat transfer from the fin to that the heat that would be dissipated if the whole surface of the fin is maintained at base temperature.

ηfin=Actual heat transferred by the fin (Qfin)Maximum heat that would be transferred if the whole surface of the fin is maintained at the base temperature (Qmax)

According to the definition, the efficiency of the fin of infinite length is given as below

ηfin=h.P.K.Ac(ts-ta)h.As(ts-ta)

ηfin=h.P.K.Ach.As

### Effectiveness of fin (εfin):

It is defined as the ratio of the actual heat transfer that takes place from the fin to the heat that would be dissipated from the same surface area without fin.

By above definition ε for infinite length fin is given by

εfin=h.P.K.Ac(ts-ta)h.Ac(ts-ta)

εfin=P.Kh.Ac

#### Factors affecting fin effectiveness

• P.Kh.Ac should be greater than unity if the rate of heat transfer from the primary surface is to be improved.
• If the ratio of P and Ac is increased, the effectiveness of fin is improved.
• Use of fin will be more effective with materials of large thermal conductivities.

The radiator dissipates heat by transferring the heat to the air. Hence This is a long process involving conduction and convection.

At first, the heat generated by the drive train (motor and controller) is transferred through the coolant that flows through the cooling jacket provided in the motor and controller respectively. The Cp (specific heat capacity of water is higher than air and hence for a given amount of heat to be carried, the temperature rise is water will be lesser than air. Hence a coolant with a higher specific heat capacity is used.

This heat is then transferred to the radiator. In the radiator the liquid flows through the tubes and the heat is transferred to the fins through the radiator tube walls. This takes place through the process of conduction.

The fins interact with the flowing air and the heat is carried away by the flowing air by the process of convection. Hence it is important that the flow of air available to the radiator should be maximum and should be laminar.

The temperature of the ambient air is also important to determine the performance of the radiator. For a given amount of air, the heat sustained or carried away will be maximum if the temperature difference between the air and the radiator fins is maximum, Hence while designing the worst-case condition should be chosen.

Hence the heat is transferred from the water to the air, hence we neglect the losses caused by the aluminium. We can assume that

QDT = Qw = Qair

Hence we know the heat generated by the drive train system and we know the amount of heat to be dissipated by the radiator through the water and air interaction.

We know the constraints from the system, i.e. the temperature range in which the system should work, hence we know the water inlet and outlet temperature of the water. The ambient air temperature is known to us and hence we can calculate the air exit temperature if we know the mass flow rates of the air and water.

The mass flow rate of air can be calculated by the vehicle velocity and the actual area of radiator exposed to air.

Hence using the formulae mentioned before we can proceed with the calculations.

Further, we need to calculate the effective heat transfer coefficient for air and water to calculate the amount of heat transferred from the water to the fins and from the fins to the air.

The following formulae will help you to calculate the heat rejection from the radiator using the Effectiveness-NTU method for a single pass cross flow heat exchanger. A random example of an automobile radiator has been chosen to carry out the calculations with an example.

An excel or MatLab based calculator should be used to compare different iterations of the radiator designs.

Care should be taken that the fin density and tube sizes should be confirmed with the manufacturer before beginning the design.

### Air Side Calculations

 AIR properties at 30.00 ˚C Viscosity (µa) 0.00001870  Ns/sq.m Density (ρa) 1.17  kg/cu. M Prandtl Number (Pra) 0.712 Specific Heat (Ca) 1,005.00  J/kgK Velocity Of Air (Va) 5m/s Mass flow rate 0.9046 kg/s Air Inlet Temperature (°C) 30
• Reynolds no. of Air, Rea = ρa*Da*Vau a
• Hydraulic Dia of Air, Da = 4*Af/Pf  ;

Colburn Factor,

• j= (Rea)-0.49 * (ᶿ/90)0.27 * (Pf/Pl)-0.14 * (Lf/Pl)0.29 * (Dl/Pl)-0.23 * (Ll/Pl)0.68 * (Pt/Pl)-0.28 * (tf/Pl)-.05
• Convection Heat Transfer Coefficient, ha = (j*ρa*Va*Cpa*1000)/Pra0.66
• m factor = ((ha*Pfc)/(Afc*KcAl))0.5
 Hydraulic Diameter (Dha) 0.002192 m Reynold’s Number (Rea) 683.3936 Colburn Factor (ja) 0.040529 Heat Transfer Coefficient (ha) 297.1401 m (fin factor) 163.9849

### Waterside calculations

 Water properties at 95.00 ˚C Viscosity (µw) 0.00029700 N-s/sq.m Density (ρw) 962.00 kg/cu.m Prandtl Number (Prw) 1.96 Thermal Conductivity (kW) 0.6768  W/m-K Specific Heat (Cw) 4,213.00  J/kg-K Flow rate 6 lpm @ 4000 RPM Mass flow rate 0.4329 kg/s Velocity Of Water through each tube( Vw) 0.2151  m/s

• Hydraulic Dia of Water, Dw = 4*At/Pt /1000
• Reynolds no. of Water, Rew = ρw*Dw*Vww
• Friction factor, f = (0.79*ln (Rew)-1.64)-2
• Nusselts no, Nuw = ((f/8)*(Rew-1000)*Prw)/(1+{12.7*(f/8)0.5(Prw0.66-1)})
• Convection through Tubes, hw = Nuw* Kcw/Dw

 Hydraulic Diameter (Dhw) 0.002557 m Reynold’s Number (Rew) 1782.052 Friction Factor (fw) 0.054755 Nusselt’s Number (Nuw) 6.60862 Heat Transfer Coefficient (hw) 1749.079 W/sq.mK

### Efficiency Calculations

• Fin Efficiency, ŋf = (tan h (m*lf/2)/ (m*lf2))*100 = 87.76 %
• Total Efficiency, ŋt = (1 – (Asf/ Ar)*(1 – ŋf ))*100 = 89.4 %

### Heat Transfer Calculation

• Overall Heat Transfer Coefficient, U={hf+1/hw+tw/KcAl+Aps/(At*ha*ηt)}-1= 726.5287W/m2K
• Heat Capacity Rate of Water, Cw (Cmax) = mw*Cpw = 405.2906 W/K
• Heat Capacity Rate of Air, Ca (Cmin) = ma*Ca = 154.1612675 W/K
• NTU = U*A/(tCmin*1000000) = 0.7195
• For Cross-Flow, Unmixed Heat Exchanger,
• Capacity Ratio, Cr = Cmin/Cmax = 0.3804
• Effectiveness, ε = 1 – e^(Cr(NTU)0.22 (e^-Cr(NTU)0.78 -1)) = 0.4638

Heat rejection per Radiator = 4.648 kW

• Heat Rejection, Q = 9.296 kW
• Water Inlet Temperature,  Twi  = Q/(ɛ* Cmin ) + Tai = 95.00°C
• Water Outlet Temperature, Two = Twi – Qmw*Cpw = 83.53°C
• Air Outlet Temperature, Tao = Tai + Qma*Cpa  = 60.15°C

### Limitations of theoretical calculations

• The calculations don’t account for a pressure drop of fluids on passage through the radiator.
• Steady-state operation of the engine is assumed i.e. water has a large heat capacity and it takes time to respond to quick changes in heat load, this is evident from the logged data shown below.
• The properties of fluids change with temperature, which is not accounted for in the calculations.

### Reference

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