Design and Optimization of Thermal Systems Episode 1 Part 8 doc

For instance, conduction in a given region is governed by the integral equation R T Further Simplification of Governing Equations After the governing equations are assembled, along with t

a set of simultaneous ordinary differential equations arise For instance, the

tem-peratures T1, T2, T3,zn of n components of a given system are governed by a

system of equations of the form

where the F’s are functions of the temperatures and thus couple the equations

These equations can be solved numerically to yield the temperatures of the ous components as functions of time T (see Example 2.6).

vari-Partial differential equations are obtained for distributed models Thus, tion (3.4) is the applicable energy equation for three-dimensional, steady conduc- tion in a material with constant properties Similarly, one-dimensional transient conduction in a wall, which is large in the other two dimensions, is governed by the equation

Equa-R T

C T

x k

T x

t

t t

t t

T yp

¤

¦¥

³ µ´ 

t

t t

¤

¦¥

2 2 2 2

³³

where Cp is the specific heat of the fluid at constant pressure and u and v are the velocity components in the x and y directions, respectively Partial differential

equations that govern most practical thermal systems are amenable to a solution

by analytical methods in very few cases and numerical methods are generally necessary Finite-difference and finite-element methods are the most commonly employed techniques for partial differential equations Ordinary differential equations can often be solved analytically, particularly if the equation is linear The integral formulation is based on an integral statement of the conservation laws and may be applied to a small finite region, from which the finite-element and finite volume methods are derived, or to the entire domain For instance, conduction in a given region is governed by the integral equation

R T

Further Simplification of Governing Equations

After the governing equations are assembled, along with the relevant boundary conditions, employing the various approximations and idealizations outlined here, further simplification can sometimes be obtained by a consideration of the various terms in the equations to determine if any of them are negligible This

is generally based on a nondimensionalization of the governing equations and evaluation of the governing parameters, as given earlier in Equation (3.6) For

instance, cooling of an infinite heated rod moving continuously at speed U along the axial direction x [Figure 1.10(d) and Figure 3.6(a)] is governed by the dimen-

Q Q Pe

X

where 2 is the Laplacian operator in cylindrical coordinates,

nondimensional-ized by the rod diameter D, and the Peclet number Pe is given by Pe  UD/A.

The dimensionless temperature Q is defined as Q  T/Tref, where the reference

temperature T may be the temperature at x  0 Also, dimensionless time T` is

defined here as T`  AT/D2, where T is physical time If Pe is very small, Pe  1, the second term from the left, which represents convective transport due to rod movement, may be neglected, reducing the given circumstance to a simple con- duction problem Similarly, at low Reynolds number Re in a flow, Re  1, where

Re  UL/N, N being the kinematic viscosity, the inertia or convection terms can

be neglected in the momentum equation This is the creeping flow tion, which is used in film lubrication modeling Several such approximations are well known and frequently employed in modeling, as discussed in standard textbooks.

approxima-Summary

The preceding discussion gives a step-by-step approach that may be applied to the system or its parts in order to develop the appropriate mathematical model Generally, modeling is first applied to the various components and then these sub- models are assembled to obtain an overall model for the system In many cases, rigorous proofs and appropriate estimates cannot be easily obtained to determine

if a particular approximation is applicable In such cases, approximations and simplifications are made without adequate justification in order to derive relatively

(d)(c)

IsothermsStreamlines

V2

ρ1V1A1 = ρ2V2A2

V1

FIGURE 3.13 Differential formulations (a) Flow in an enclosed region due to inflow and

outflow of the fluid; and (b) temperature distribution due to conduction in a solid body Also shown are a few integral formulations: (c) flow in a pipe and (d) flow through a turbine

simple mathematical models The results from analysis and numerical solution

of these models can then be used to verify if the approximations made are valid

In addition, the approximations may be gradually relaxed to obtain models that are more accurate Thus, one may go from simple to increasingly complicated models, if needed Clearly, modeling requires a lot of experience, practice, under- standing, and creativity The following simple examples illustrate the use of the approach given here to develop suitable mathematical models.

Example 3.1

Consider typical thermodynamic systems such as

A power plant, shown in Figure 2.17, with the thermodynamic cycle in Figure 2.15(a)

A vapor compression cooling system, shown in Figure 1.8, with the namic cycle in Figure 2.21

thermody-An internal combustion engine, with the thermodynamic cycle in Figure 2.15(b).Discuss the development of simple mathematical models for these in order to cal-culate the energy transport rates and the overall performance

Solution

In all of these commonly used systems, as well as in many others like them, the major focus is on the heat input or removal rate and on the work done Many of the details, such as the temperature and velocity distributions in the various com-ponents, are not critical Similarly, though the transients are important in control-ling the system as well as at start-up and shutdown, the system performance under steady operation is of particular interest for system analysis and design

Keeping the preceding considerations in mind, the two main assumptions that can be made for each component are:

Steady-state conditions

Lumped flow and temperature

This implies that time dependence is neglected and uniform conditions are assumed

to exist within each system component Energy loss to or gain from the ment may be neglected for idealized conditions, which will yield the best possible performance and can thus be used for calculating the efficiency

environ-Then, considering the vapor compression system of Figure 2.21, we obtain for a mass flow rate of m

Heat rejected at the condenser  m(h2 3)

Heat removed at the evaporator  m(h1 4)

Work done on the compressor  m(h2 1)

yielding the coefficient of performance (COP), given in Equation (2.19), as

(h1 4)/ (h2 1)

Similarly, for the power plant given by the cycle in Figure 2.15(a), the heat input

in the boiler or condenser is m(hout ) and work done by the turbine or the pump

is m(hin out ), where in and out refer to conditions at the inlet and outlet of the

component Thus, boiler heat input is positive, condenser heat input is negative (heat rejected), work done by the turbine is positive, and work done by the pump

is negative (work done on the pump) The internal combustion engine and other thermodynamic systems may be similarly analyzed to yield the net heat input and work done, allowing subsequent design and optimization of the system The design considerations are discussed in Chapter 5

Example 3.2

For common heat exchangers, such as the parallel and counterflow heat exchangers shown in Figure 1.5, discuss the development of a simple mathematical model to analyze the system

Solution

In heat exchangers, the main physical aspect of interest is the overall heat transfer between the two fluids The velocity and temperature distributions at various cross-sections of the heat exchanger are generally of little interest Similarly, transient aspects, although important in some cases, are usually not critical Thus, energy transfer under steady flow, as a function of the operating conditions and the heat exchanger design, is generally needed With this in mind as the major consider-ation, we can assume the following:

The flow is lumped across the cross-sections of the channels or tubes

The temperature is also uniform across these cross-sections

Steady-state conditions exist

With these assumptions, the temperature in, say, the inner tube or channel of the heat exchangers in Figures 1.5(a) and (b) varies only with distance in the axial direction The overall energy balance is



m c Cp,c (T c,out c,in) Q where Q is the rate of heat input to the colder fluid, indicated by the subscript c, over

the entire length If energy loss to the ambient is neglected, we have for the hotter

fluid, which is indicated by subscript h,

Example 3.3

In the design of a hot water storage system, it is given that a steady flow of hot water

at 75nC and a mass flow rate m of 113.1 kg/h enters a long circular pipe of eter 2 cm, with convective heat loss at the outer surface of the pipe to the ambient medium at 15nC with a heat transfer coefficient h of 100 W/m2K The density R,

diam-specific heat at constant pressure C p , and thermal conductivity k of water are given

as 103 kg/m3, 4200 J/kgK, and 0.6 W/mK, respectively Develop a simple ematical model for this process and calculate the water temperature after the flow has traversed 10 m of pipe

math-Solution

The problem can be simplified considerably by assuming steady-state conditions and lumped velocity and temperature conditions across any cross-section of the pipe This approximation applies for turbulent flow in a pipe of relatively small

diameter In addition, interest lies in the average temperature at any given x, where

x  0 is the inlet and x is the distance along the pipe, as shown in Figure 3.14 The average velocity U in the flow is

D

R P( 2/ )4 3600 0.1 m/s

where D is the pipe diameter The Reynolds number Re  UD/N  (0.1)(0.02)/(5.5 r 10 )

 3636 Turbulent flow arises in the pipe at this high value of the Reynolds ber The Peclet number Pe  UD/A  (0.1)(0.02)/(1.5 r 10 ) 1.3 r 104 Therefore, convection dominates and axial diffusion effects may be neglected; see Equation (3.20)

num-With the above approximations, the governing equation for the temperature T(x)

is obtained from energy balance over a region of length $x, as shown in Figure 3.14 The reduction in thermal energy transported in the pipe equals the convective loss

to the ambient This gives the decrease in temperature $T over an axial distance

h, T a

Therefore, with $x l 0, we obtain the differential equation

RC UA dT

where A is the cross-sectional area (PD2/4) and P is the perimeter (PD) This gives

the simple mathematical model for this problem The inlet temperature is given as 75nC and the ambient temperature Ta 15nC This equation may be solved analyti-cally to give

tempera-The preceding three examples present relatively simple models of some monly encountered thermal systems These included thermodynamic systems like heating/cooling systems and flows through channels as in heat exchangers Steady-state conditions could be assumed in these cases, along with lumping to further simplify the models The resulting models yielded algebraic equations and first-order ordinary differential equations, which could be easily solved ana- lytically to yield the desired results However, many practical thermal systems are more involved than these and spatial and temporal variations have to be con- sidered Then the resulting equations are partial differential equations, which generally require numerical methods for the solution In a few cases, these equa- tions can be simplified or idealized to obtain ordinary differential equations, which may again be solved analytically The following two examples illustrate such problems that would generally need numerical methods for the solution and that may be idealized to obtain analytical results in some cases for validation of the numerical scheme.

com-Example 3.4

A large cylindrical gas furnace, 3 m in diameter and 5 m in height, is being lated for design and optimization Its outer wall is made of refractory material, covered on the outside with insulation, as shown in Figure 3.15 The wall is 20 cm

simu-thick and the insulation is 10 cm simu-thick The variations of the thermal conductivity k,

specific heat at constant pressure C p, and density R of the wall material with perature are represented by best fits to experimental data on properties as

tem-k 2.2 (1 1.5 r r

Cp 900 (1 10 r $T)

R  2500 (1 6 r 10 r $T)

where $T is the temperature difference from the reference temperature of 300 K

and all the values are in S.I units The temperature difference across the wall

is not expected to exceed 200 K The properties of the insulation may be taken

as constant Develop a mathematical model for the time-dependent temperature distribution in the wall and in the insulation Solve the governing equations for the temperature distribution in the idealized steady-state circumstance, with the

thermal conductivity of the insulation given as 1.0 W/mK, temperature (T w)1 at the

inner surface of the wall as 500 K, and temperature (T i)2 at the outer surface of the insulation as 300 K

FIGURE 3.15 The cylindrical furnace, with the wall and insulation, considered in

Example 3.4

WallInsulation

The ratio of the wall thickness to the furnace diameter is 0.2/3.0, which gives 0.067 Similarly, the ratio of the insulation thickness to the furnace diameter is 0.1/3, or 0.033 Since both of these ratios are much less than 1.0, the curvature effects can be neglected, i.e., the wall and insulation may be treated as flat surfaces

The ratio of the furnace height to the wall thickness is 5.0/0.2, or 25, and that

to the insulation thickness is 50 In addition, the circumference is much larger than these thicknesses If there is good circulation of gases in the furnace, the thermal conditions on the inner surface of the wall can be assumed uniform Then, the wall, as well as the insulation, may be modeled as one-dimensional, with transient diffusion occurring across the thickness and uniform conditions in the other two directions

The material properties are given as constant for the insulation However, these vary with temperature for the wall material Considering a maximum temperature difference of 200 K across the wall, the ratios $k/ko,$C p /(C p)o and $R/(R)o may be calculated as 0.3, 0.02, and 0.012, respectively, where $k, $Cp, and $R are the differ-

ences in these quantities due to the temperature difference The reference values k o,

(C p)o, and Ro are used instead of the average values because the actual temperature

levels are not known From these calculations, it is evident that the variations of C p

and R with temperature may be neglected over the temperature range of interest

However, the variation of k is important and must be included.

The governing equations for the wall and the insulation are thus obtained as, respectively,

RT

RT

T x

p w w

w

p i i

t

t 

tt

tt

t 

tt

x i i

2 2

where the corresponding temperatures and material properties are used, denoted

by subscripts w and i for the wall and the insulation, respectively, and x is the

coor-dinate distance normal to the surface, i.e., in the radial direction for the furnace; see Figure 3.16 Heat transfer conditions at the inner and outer surfaces of the wall-insulation assembly give the required boundary conditions for these equations In addition, at the interface between the wall and the insulation

T x

w i i

ttand

Therefore, the governing equations for the wall and the insulation may be solved, with the appropriate boundary conditions, to yield the time-dependent temperature distributions in these two parts of the thermal system Because of the variation

of k w with temperature, the two partial differential equations, which are coupled through the boundary conditions, are nonlinear Therefore, numerical modeling will generally be needed to solve these equations

The simpler steady-state problem, with temperatures specified at the inner and outer surfaces of the wall-insulation combination, is an idealized circumstance and may be solved analytically The equations that apply in the wall and the insulation for this case are

w

i i

where all the temperatures are taken as differences from the reference value of

300 K to simplify the analysis and the C’s are constants to be determined from

the boundary conditions shown in Figure 3.16 At the interface, the heat flux and the temperature in the two regions match, as given previously The temperature distribution in the insulation is linear, with 0 K at the outer surface, and that in the wall is nonlinear, with 200 K at the inner surface The temperature distributions are obtained as

collector system The tank loses energy to the ambient air at temperature T a with

a convective heat transfer coefficient h at the outer surface of the tank wall The

temperature range in the system may be taken as 20nC to 90nC Develop a ematical model for the storage tank to determine the temperature distribution in the water Also use nondimensionalization to obtain the governing parameters Then solve the steady-state problem

math-Solution

The temperature range being relatively small, the variation in material properties may be taken as negligible because parameters such as $R/Ravg,$k/kavg, etc., where

R is the density and k is the thermal conductivity, are much less than 1.0 Because of

the thinness of the stainless steel wall and its high thermal conductivity compared

to water, the ratio being 23.59, the energy storage and temperature drop in the wall may be neglected compared to those in water This is justified from the ratio of the wall thickness, 5 mm, to the tank diameter, 40 cm

A substantial simplification of the problem is obtained by assuming that the temperature distribution across any horizontal cross-section in the tank is uniform This is based on axisymmetry, which reduces the original three-dimensional prob-lem to two dimensions and the effect of buoyancy forces that tend to make the tem-perature distribution horizontally uniform Because hot water is discharged at the top, the water in the tank is stably stratified, with lighter fluid lying above denser fluid This curbs recirculating flow in the tank and promotes horizontal tempera-

ture uniformity Therefore, the temperature T in the water is taken as a function only of the vertical location z, i.e., T(z) The vertical velocity in the tank is also

taken as uniform across each cross-section, by employing the average value This is obviously an approximation because the velocity at the walls is zero due to the no-slip condition Therefore, the problem is substantially simplified because the flow

field is taken as a uniform vertical downward velocity, which can easily be obtained from the flow rate Without this simplification, the coupled convective flow has to

be determined, making the problem far more involved

The governing energy equation for thermal transport in the water tank may be written with the above simplifications as

RT

where R is the density of the fluid, C p is its specific heat at constant pressure, T is

the physical time, w is the average vertical velocity in the tank, k is the fluid mal conductivity, A is the cross-sectional area, and P is the perimeter of the tank;

ther-see Figure 3.17 The problem is treated as transient because the time-dependent behavior can be important in such energy storage systems The initial and boundary conditions may be taken as

At 0: ( )

TT

Storagetank

Flow of

hot water

P P

A A

T(z) z

h, T a

h, T a

FIGURE 3.17 The hot-water storage system considered in Example 3.5, along with the

simplified model obtained