# Lecture 30 Review of Fluid Flow and Heat Transfer

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1 Objectives In this lecture you will learn the following We shall summarise the principles used in fluid mechanics and heat transfer. It is assumed that the student has already been exposed to courses in heat transfer and fluid flow /18

2 Fluid Flow in Circular ducts Flow and heat transfer inside circular passages are the most common applications of Fluid Mechanics and Heat transfer. Flow and heat transfer in between rod bundles are the most common application in Nuclear Engineering. The Objective of this lecture is to review relevant portions of fluid flow and heat transfer so that we can solve some simple problems in nuclear reactors. Before reading the chapter, the students are advised to refresh their concepts in fluid mechanics and heat transfer studied prior to this course. When velocity of fluid increases beyond a critical value, several whirls called vortices are formed. This is called Turbulent Flow. In this case the velocity and temperature continuously fluctuate with time. The transition to turbulent is governed by a non-dimensional quantity called the Reynolds number. V Velocity (m/s) ρ Density (kg/m 3 ) d Diameter (m) µ Dynamic Viscosity (Ns/m 2 ) Its value in circular ducts is typically Now we shall review analysis of fully developed laminar flows in circular ducts. The approach is to integrate the Navier-Stokes equations. Prior to the analysis let us understand the concept of fully developed flows /18

3 Fully Developed Flow In the above figure the fluid velocity profile is shown at different sections in a pipe. It has a flat profile at the entrance and develops into a fully developed profile after some distance, called entrance length. Fully developed flow implies that the velocity profile does not change along length. Detailed analysis indicates that the non-dimensional entrance length (L h /D) is ~ 0.06 Re. This is small in turbulent flow (L h /D ~ 6-10). The subscript h is used to denote hydrodynamic entrance length. Since velocity profile is same, it implies that wall shear stress is same. Laminar Flow in Pipes For axi-symmetric, incompressible, steady flow V = V(r,z). If the flow is also fully developed, we can write the following: Continuity Equation V r and V z represent the velocities in r and z directiions respectively. Since flow is fully developed, = 0 We can calculate the following: rv r is independent of r. Since V r at r = R is 0, V r is 0 everywhere. Hence there is only V z, and V z is only a function of r.

4 r - Momentum Equation Since V r is zero everywhere, we can write. This implies that p is only a function of z /18

5 z - Momentum Equation Since V r, are zero, The left hand side (LHS) of the above equation is only a function of z and the right hand side (RHS) is only a function of r. This demands that both LHS and RHS be each equal to a constant. Transposing r, we can write Integrating once with r Using the boundary condition that flow is symmetric as r 0 Hence /18

6 Transposing r, we can write Integrating with r Using the boundary condition V z = 0 at r = R Hence /18

7 Velocity distribution is parabolic V z = V z (max) at r = 0 The velocity distribution is sketched below /18

8 Average Velocity The average velocity can be obtained as follows. From above we see that average velocity is equal to one half of the maximum velocity /18

9 Shear Stress The shear stress in the z direction can be obtained as follows. (Since V r = 0 every where) Thus shear stress varies linearly with radius. Force Balance Referring to the figure shown below, Note the answer is same, but the sign is opposite as we have already assigned its direction /18

10 Fanning Friction Factor We can introduce Non-dimensional shear stress in the form of Fanning Friction factor. It is defined as Further and /18

11 Pressure Drop From force balance /18

12 Darcy Friction Factor Many books and papers use alternate form of friction factor called Darcy friction factor. In this definition, the pressure gradient is computed using the relation By comparison with the expression given earlier, we get /18

13 Fluid Flow in Complex Passages Often in engineering applications such as nuclear reactors the passage is not circular. We could use circular tube correlation by introducing the concept of hydraulic diameter, In turbulent flow, the method works reasonably well. However, specialized correlations exist for several shapes in literature and they may be used for better prediction /18

14 Fully Developed Heat Transfer The figure above shows the development of temperature profile in a pipe subjected to constant heat flux. It should be noted here that, unlike in velocity development, where the velocities do not change is the fully developed region, the temperature will continue to change. However the shape will be similar. The entrance length (L t /D) is ~ 0.05 Re Pr. The subscript 't' denotes thermal entry length. Pr = Prandtl number and is computed using where C p is the specific heat at constant pressure, μ is the dynamic viscosity and K is the thermal conductivity of the fluid. The entrance length is large for oils (Pr>>1). For turbulent flow L t /D ~ /18

15 Convective Heat Transfer in Pipes In convective heat transfer, the fluid carries away heat from a wall subjected to constant heat flux by virtue of its motion. In such systems, it is conventional to define a term called heat transfer coefficient, h. It can be defined as the rate of heat carried out per unit surface area and unit temperature difference. The purpose of defining it that way is to estimate the wall temperature, if the heat flux and the fluid temperature are known. As shown above, the temperature of the fluid continuously varies all along the radius. In such cases, we need a reference temperature to be defined for the sake of defining the heat transfer coefficient. The most common definition used is the thermodynamic bulk coolant temperature. This temperature is also called the mixing cup temperature or mixed mean temperature /18

16 Thermodynamic Mean Temperature Mathematically, the total energy carried by a flowing fluid in a pipe can be written as In the above expression C is the specific heat at constant pressure. For the sake of argument, let us assume that the specific heat, C, is constant. Now we want to hypothetically define a temperature T B that will be homogeneous at a cross section such that the energy carried is same as above. In such a case the above integral will become Thus T B reduces to For varying specific heat (if C is a strong function of temperature) this would be modified as /18

17 Energy Balance For the constant heat flux case, the energy balance over a length Δx of pipe gives To make the understanding simple, we shall assume a constant c p. The equation can now be written as where HOT stands for higher order terms of Taylor series and unit length on the tube. represents heat per For steady flow and constant c p, when we shrink the length to 0, we get The above stated equation is the simplified energy equation that we shall use subsequently /18

18 Heat Transfer Coefficient Now we can define heat transfer coefficient as In heat transfer analysis, a non-dimensional heat transfer coefficient, called Nusselt number, denoted by Nu is defined Where, h - Heat Transfer Coeff. (W/m 2 -K ) d Diameter (m) k -Thermal Conductivity (W/m-K) /18

19 Heat Transfer in Internal Passages In internal passages, boundary layers develop and merge. This leads to fully developed regions. In the developed region, both friction factor and Nussselt number are constant. In laminar flow, the values of Nu = 4.36 for constant wall heat flux case and is 3.66 for constant wall temperature case. Skipping details of analysis, we shall only state some of useful correlations. If we employ modified Reynolds analogy, we can show that In the above equation, Pr is the Prandtl number, The above equation is modified to give the Dittus-Boelter Equation and is the most common correlation used in turbulent flows n = 0.4 for heating (T W >T B ), n = 0.3 for cooling (T W <T B ) The properties are calculated at mean bulk coolant temperature. The validity of the above has been checked for 0.7 < Pr < 160 Re D > 10,000 L/D > 10 It is about +/- 15% For high temperature difference between the wall and the bulk fluid, Sieder-Tate equation is mostly used The properties are calculated at mean bulk coolant temperature, except for μ w that is taken at wall Temperature. The validity of the above has been checked for 0.7 < Pr < 16,700

20 Re D > 10,000 L/D > 10 It is about +/- 15% For more accurate equations, one can refer text books and reference books in heat transfer /18

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