A11100 NBS

PUBLICATIONS

NBSIR 81-2303

NBS/ESDS - A Computer Model

for the Simulation of Hot Water

Systems

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Consumer Product Technology

Washington, DC 20234

September 1 981

—QC~

10U

. U56

)ared for

>. Department of Commerce

81-2303

1981

c. 2

NBSIR 81-2303

U-< NBS/ESDS - A COMPUTER MODEL

FOR THE SIMULATION OF HOT WATER

SYSTEMS

NATIONAL BUREAU

09 STANDARDS

library

OEC 7 1981

Quoo

. LLSZ>

no, 3 '

C. A. Wan

c, aU.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Consumer Product Technology

Washington, DC 20234

September 1 981

Prepared for U.S. Department of Commerce

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
VJk&n'JH KHOir t <t

"l

'

F 3& V 030
NBSIR 81-2303

NBS/ESDS - A COMPUTER MODEL

FOR THE SIMULATION OF HOT WATER

SYSTEMS

C. A. Wan

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Consumer Product Technology

Washington, DC 20234

September 1981

Prepared for U.S. Department of Commerce

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Director

Table of Contents

Page

Abstract ...... ........ i

I. Introduction ................... 1

II. Bulk Behavior of the Water Heaters .............. 2

III. Modeling of the Water Heater and its Distribution Pipes ... 5

A. Conduction Model 6

B. Verification of the Conduction Model . 8

C. Pipe Flow Model 9 D. Film Coefficients . 13

E. Draw Model ............... 15

1. Option 1 15

2. Option 2 16

3. Option 3 ..... ........ 16

F. Buoyancy Stacking 18

IV. Parametric Studies 19

A. Description of the Water Heater and the Distribution Pipe. 19

B. Modeling of the Water Heater ...... 20

1. Basic Water Heater (Water Heater A) . 20

2. Insulated Water Heater (Water Heater B) 22

C. Modeling of the Distribution Pipe ...... 22

1. Distribution Pipe (Pipe A) 22

Film Coefficients 22

a. Vertical Sections 22

b. Horizontal Sections ...... 23

c. Combined Vertical/Horizontal Sections ...... 23

d. Water Side Film Coefficients 23

2. Insulated Distribution Pipe (Pipe B) 24

D. Modeling of the System 24

1. Water Heater-Pipe Connection 24

2. Heat Traps 24

3. Draw Schedule 24

4. Quasi-Steady State System Modeling 25

E. Summary of Parametric Studies 25

Case A. Basic System (Water Heater A, Pipe A) 25

Case B. Insulated Water Heater, Bare Pipe (WH B, Pipe A . 25

Case C. Insulated Water Heater, Insulated Pipe

(WH B, Pipe B) 25

Case A-l. Basic System, Schedule B 25

Case A-2. Basic System With a Heat Trap 25

Case A-3. Basic System, Schedule B, Reduced Thermostat

Settings 25
Page

V. Results and Discussions ............. 25

A. Basic System (Case A) . . 25

1. Schedule B (Case A-l) . .......... 28

2. Effects of a Heat Trap (Case A-2) ........... 28

B. Insulated Water Heater (Case B) ....... . 28

c. Pipe Insulation (Case C) 29

D. Thermostat Setback (Case A-3) 29

E. Water Delivery Temperature 29

VI. Conclusions ...... ......... 30

References . . ..... ........ 32

List of Tables

Table 1 Selected Heat Transfer Parameters for Modeling .... 33

Table 2 Draw Schedule A 34

Table 3 Schedule B (Two-peak Schedule) ....... 35

Table 4 Energy Consumptions (in MJ) . 36

Table 5 Distribution of Energy Consumption .......... 37

List of Figures

Figure 1 Bulk Standby of an Electric Water Heater ...... 38

Figure 2 Nodal Scheme for the Conduction Model . 39

Figure 3 Verification of the Conduction Model of ESDS .... 40

Figure 4 Two-layer Pipe Model 42

Figure 5 Nodal Scheme for the Pipe Flow Model 43

Figure 6 Draw Model, Option 1 . . 44

Figure 7 Draw Model, Option 2 45

Figure 8 Draw Model, Option 3 ........... 47

Figure 9 Water Node Temperature for Small Draw ........ 48

Figure 10 Heat Leak Through a Stand Pipe 49

Figure 11 Nodal Diagram for the Basic Water Heater (Water Heater

A), Dimensions in mm. (1) water (2) cement (3) steel,

and (4) insulation, X heating source, V thermostat . 50

Figure 12 Nodal Diagram for the Pipe and its Connection to the

Water Tank 51

Figure 13 A 24-hour Operation of Case A in which Draw Schedule

A is used 53

Figure 14 The Rates of Heat Loss of Water Heater A 55

Figure 15 Faucet Temperatures of Draws , Case A 56

Figure 16 The Rates of Heat Loss of Water Heater B ...... 57

Figure 17 Faucet Temperatures of Dishwashing Draws, Cases B and

C. Elapsed Time for Piston Flow of Hot Water to Travel

to the Faucet: 0.00238 hour 58
ABSTRACT

A computer model, entitled NBS/ESDS, has been developed using a finite

difference method to simulate the generation, storage and distribution of a hot

water in a system commonly found in many residences, and some commercial

facilities.

The conduction heat equation which governs the heat transfer in the water

heater during all modes of operation, and the heat transfer in the distribution

pipe network during its cooldown periods, is solved. A solution is also

obtained for the pipe flow equation which governs the heat transfer in the

distribution pipe network during the draw periods. Thermal stratification of

water is modeled, and slug flow is used to simulate the draw of heated water

from the water heater.

A parametric study is also conducted on a given hot water system for a

given draw schedule during a 24-hour period. The effects studied include

insulating the hot water heater, insulating the distribution pipe, and

incorporating a heat trap. Effects of concentrating the distributed draws to

two short periods of time of the day on the hot water delivery and the heat loss

with the same energy consumption are also studied.

i

I. Introduction

Domestic water heating is the second largest energy use in the home—ranked

only after space heating. It accounts for about 15# of the total residential

energy consumption and 3# of the total U.S. energy use (l) . In addition, waterheating accounts for about 8# of commercial energy consumption. In total, water

heating accounts for more than 4# of the U.S. energy consumption. The energy

cost for water heating, however, is often over-shadowed by the costs of space

heating, air conditioning and limiting—it is hidden in the utility bills

covering the more significant energy usage.

Of the annual energy dollars spent for water heating in the domestic,

storage-type water heater, a significant portion is lost during standby just to

maintain the hot water ready for use. The heat lost during the standby phase

constitutes more than 15# of the total energy for the electric water heater and

more than 25# for the gas water heater (2).

Another significant loss is due to the distribution pipes. Their heat loss occurs during both the draw and the non-draw cycles in a typically daily hot

water use cycle. During a draw cycle, hot water enters and flows along the

distribution pipes from the hot water heater. As a segment of hot water travels

inside the pipe it loses heat both to the pipe walls and through the pipe walls

to the ambient. Thus, when this segment of water reaches its draw points, its

temperature is much lower than when it leaves the water heater. Immediately

after a draw cycle terminates, the average temperature of water in the pipe is

between the water temperature in the water heater and the water temperature at

the draw point—-usually much higher than the ambient. A good portion of the

thermal energy of the water in the pipe is lost to the ambient if this non-draw

cycle lasts more than a few minutes. A few quantitative questions need to be

addressed on the pipe heat transfer:

1. How much energy is lost during a draw cycle?

2. How much energy is lost during a non-draw (or cooldown) cycle?

3* How effective is the pipe insulation in reducing heat loss during the

draw and non-draw cycles?

4- How may the hot water delivery be affected by the different draw

schedules?

5* How does the configuration of the distribution pipe (including the

heat trap) may affect the heat loss?

The loss from one system in the house may be considered as a gain for

another system. For example, the heat loss from the hot water system may be

considered as a heat gain to the living quarters during a heating season. During a cooling season, on the other hand, the same heat loss is not only a

1
loss of the hot water system hut also an added load to the cooling system of the

living quarters. To address the question of what constitutes heat loss the

entire house (including all the subsystems) should be studied as one integrated

system. Only by such means could the interaction among all the subsystems may

accurately determined. While the present model of the hot water system does not

consider the cooling and heating system of the house as a part of the computing

iteration loop, it does compute accurately the amount and distribution of heat

loss to the house. This heat loss could then be used as an input to a model of

the cooling and heating system of the house.

II. Bulk Behavior of the Water Heaters

A basic understanding of water heater standby and recovery may be obtained

by considering the bulk behavior of the water heater as given in this section.

This approach has many limitations. A more detailed study of the water heater

system which considers the dynamics of draw, temperature dependent conductance

and variable heat capacitance of the water heater composite wall along with a

parametric study of the tank structure and distribution system are given in the

following sections.

Suppose only the bulk behavior of the water heaters is to be studied, then

the water in the storage tank may be considered as at one uniform temperature,

T. If the energy in the water stored above room temperature is Q and the heat

transfer rate through the walls, is q, then during cooldown

§=-<! = UA(T-T

a

) (1)

where U is the overall heat transfer coefficient, A its associated area, © time

and Ta the ambient temperature. The product UA is sometimes called overall

conductance and may be expressed as

UA = (UA) t + (UA)t

+ (UA) o(2)

where the subscripts t, b and o denote top, bottom and lateral surfaces,

respectively. A typical overall lateral conductance (UA) may be expressed as;

(UA) o

= ^ + R

2 + R

3

"+-R

4

-+"'R

5

(3)

where the R's are the individual thermal resistances of the composite layers of

the water heater. Namely, R, is the resistance between the water and tank

walls, Bp ‘the resistance of the tank walls, the insulation, PL exterior

shell ana PL the resistance between the exterior shell and ambient air. The

internal energy, Q, of the water relative to the ambient temperature is

expressed as

Q = P o V (T-T ) (4) d

2
where p is the water density, c the water specific heat and v the volume of the

storage tank. Equation (l) is rewritten using equation (4) as

(5)

(6)

is the cooldown rate, and is defined as a fraction of the instantaneous

stSred energy lost per hour. This cooldown rate is different from the standby

loss. The standby loss is a measure of average heat loss of the water heater

and includes the efficiency of the water heater’s heat source to replenish the

heat lost. This n takes into account only energy lost during the cooldown

phase. If the resistances in equation 2 are constants, n is a constant, and

equation (6) can then be integrated as

where Q is the internal energy content of water at the beginning of the

cooldown period, and 0 is the time constant equal to l/n . c s The heat transfer between the water and the tank walls and between the

exterior shell and the ambient are functions of temperature. The former is

mainly natural convection and the latter is a combination of natural convection

and radiation. Therefore, experimental cooldown curves of water heaters do not

follow an exponential curve precisely.

A more general form of Equation (6) is

where qg

is a constant strength heat source (or sink). Equation (6), then, is a

special case of equation (7) in which the heat source has zero strength.

Equation (7) has the following solution,

where A is an arbitrary constant. Suppose Equation (7) is subject to the

initial conditions of

-0

Q = Q

0

e

(7)

( 8 )

Q = Q

1

at Q = 0,

3
then , thus

%

A = An

1 n

s

- n

s 0

. q e

-n 0

Q = Q-je + (1-e

s ), or

Q _ Q

1 ' n s® 1 q

(

o ''o n Q Qs %

With typical values of:

-n 0

(l-e

s

)

( 9 )

00 )

Q

1

/Q0 = 0.90

n =0 .Ol/hour

s

= 0.4l/hour

y

o

Then the Q/Qo may he tabulated with respect to 0 (Figure 1).

© (hours) 0.0 0.1 0.2 0.2497 0.5 0.75 1.0

Q/Q0 0.9 0.94 0.98 1.0000 1.10 1.1996 1.299

Suppose the heater is turned off at 0 = 0.2497 hour, then the governing equation

(10) reverts to

-n 0'

Q/Q0

= e

s (11)

where 9’ is time in hours counting from 9 = 0.2497 hour. With ng = O.Ol/hr the

cooldown curve may he tabulated thus

9' (hours) 0 2.0 4.0 6.0

Q/Q

o 1.0 0.980 0.961 0.942

8.0 10.536

0.923 0.900

Using equation

0

2

(9), the duration of recovery, is

( 12 )

The two ways to shorten the duration of recovery are:

l) by reducing n and/or

s

4
2) by increasing qg

.

The duration between the thermostat cutout and the subsequent thermostat

cutin, the duration of cooldown 0^, may be derived from the cooldown equation,

6, =-iln&. (13) 1

<7s Q

1

One way to lengthen 0, is to reduce n by adding insulation.

_L o There are many reasons why the above analysis does not describe the water

heater systems adequately, among them:

1. This analysis does not lend itself to the study of the

dynamic behavior of draw.

2. The overall conductance is not a constant, but rather a

complex function of temperatures.

3. The heat capacity of the composite walls of the water

heater is not taken into consideration.

For these reasons and also in order to study the effects of the distribution

pipe, a more elaborate model and computer code is developed.

This computer simulation model, called NBS/ESDS (Energy Storage and

Distribution System) developed at the National Bureau of Standards (NBS) , can be

used for studying the operation and designs of storage-type hot water systems.

A separate users’ manual (3) has been prepared and includes a description of the

subroutines, a listing of the computer program as implemented on the NBS UNIVAC

1108 and the requirements for the actual preparations of input data set.

III. Modeling of the Water Heater and Its 'Distribution Pipes

The water heater currently considered is assumed to be an electric water

heater having cylindrical, axi symmetrical storage tank. Each layer of its

composite wall is assumed to form a rigjrt cylinder. Cold supply water is

assumed to enter the tank from the bottom and hot water withdrawn from the top.

The distribution pipe consists of one single run of pipe with no branches and

has constant diameter and thickness.

The basic elements of the modeling for the water heater and the associated

distribution pipes consists of a conduction model, a pipe flow model, and a

series of auxiliary models.

The conduction model is used to calculate the unsteady conduction heat

transfer for an axi symmetric solid body. It is used to simulate the heat

transfer of a water heater in which the water is temporarily considered a solid.

5
'The same conduction model is also used to calculate the heat transfer of a

distribution pipe during the non-draw periods. The water inside the pipe is

also temporarily considered as a solid.

The pipe flow model is used to calculate the heat transfer from the

distribution pipe during the periods of the draw. The governing equations are

two simultaneous (for single wall pipe) partial differential equations involving

only first order terms in terms of time and longitudinal distances. The water

temperature is considered to be a function of time and the longitudinal

distances and not of the radial distances. The water is considered to be

flowing at a velocity uniform across the entire pipe inner cross-section.

Therefore, the distribution pipe is modeled alternately with a conduction model

and a pipe flow model depending on the mode of pipe operation. During the non- draw periods the former is used; during the draw periods the latter is used.

Since the non-draw periods and the draw periods are interspersed in a 24 hour

cycle, the solution of the conduction model forms the initial conditions of the

subsequent pipe flow model; and the solution of the pipe flow model forms the

initial conditions of the subsequent conduction model.

The other important features incorporated in the model are:

1. Buoyancy stacking. It is used to simulate the process of thermal

stratification, in which water elements at higher temperatures float to a higher

elevation and water elements at lower temperatures sink to a lower elevation in

the storage tank of the water heater. In vertical sections of the distribution

pipes, the same type of stratification occurs.

2. Heating elements. The heat source terms are included in the heat

equation of the conduction model to simulate the heating elements.

3. Draw schedule. Depending on the flowrate of a draw cycle and the

size of the time step of the water heater conduction model, different algorithms

are used to simulate a draw of hot water.

A. Conduct ion -Model

There are two basic approaches for solving the transient heat conduction

equation governing the heat transfer in a water heater which is cylindrical and

axi symmetr i cal

.

6
1. The first approach is to apply a finite difference technique to the

heat transfer equation (14). Essentially, the scheme involves approximating the

various partial derivatives with their discrete counterpart:

2 2 1 3T_ _ 3 T

+ 1_ +

3 T

oc 30 3r 2 r 3r dT1

where may be approximated by

~

’

(14)

3 2 T

9ry may be approximated by

Ti+1 ,j - 2Ti ,j + Ti-1 ,j , Ar z T . .n+1 T . .n Ti ,1 Ti ,j

A0

3T may be approximated by

90

is the thermal diffusivity of the medium.

and

This approach has the advantage of being quite straightforward for a

homogeneous body of simple geometry. The disadvantage is its awkwardness to

handle a body of mixed materials with complex geometry.

2. The second approach is to divide the physical model into a number of

small elements (nodes). A conservation of energy equation is written for each

of the nodes, relating the temperature change in time in a central node with

temperatures in its neighboring nodes. Denoting Tq as the temperature of the

central node and T, through T. as those of its neighboring nodes, Figure 2, then

the conservation or energy equation for node 0 may be written as

C

1

(TrT

0 )^2

(T2-T0 )^

3

(T

3

-T0)+C

4

(T4-T0)=(T0'-T0

)C

5

/A9 (15)

where the coefficients, C's are functions of thermal properties and geometrical

parameters and (Tq'-Tq) is the temperature change of the node 0 in a time step

of aQ. This approach is versatile for handling a model of mixed materials and

complex geometry since any change from node to node, of properties, volumes, areas or heat transfer modes can easily be incorporated into the conservation of

energy equation. Therefore, this approach is more suitable for the modeling of

water heaters and was chosen for this study.

For time-advancing, either an explicit or an implicit method may be used.

A detailed discussion of the problems of convergence and stability and other

merits of each method is beyond the scope of this report and has been presented

elsewhere (4). The implicit method is used because it provides unconditional

stability regardless of the size of the nodes or the time step. The two

different methods are summarized briefly in the following paragraphs.

If an explicit method is used then all the unprimed temperatures in

equation (15) are known because the starting point of the time-advancing is the

initial condition, which is given. The primed temperature , evaluated at one time step in the future, is the only unknown. Therefore, the temperature of a

node one time step in the future may be calculated using equation (15) if the

7
present temperature in the five nodes are known- Writing a similar equation for

each of the nodes, the temperature one time step in the future for all the nodes

may be found if the present temperature and the boundary conditions are known.

On the other hand, if an implicit method is used, then all the temperatures on the left-hand side and T ' on the rigjht-hand side of equation (15), are evaluated at a future time, while T

Q on the right-hand side is evaluated at the

present time. Unlike in the explicit method T~’ cannot be evaluated directly

using this one equation, since there are 5 unknowns in this one equation.

Writing one conservaton of energp equation for each of the nodes results in a

system of algebraic equations whose coefficient matrix consists of five diagonal

non-zero terms. With given boundary conditions, there are an equal number of

unknowns and equations. The temperature of all the nodes may be calculated by

solving this system of equations.

Solving this system, however, may be laborious and time-consuming. An

alterating direction method (5) may be used resulting in solutions for two

sequential tri-diagonal systems of equations. Equation (15) is then written as two equations:

c

i

(

T

i*~v)+c2

( t2*~to* )+wv+(yvT

o

) =

C

5

(T0*-T0

)/(l/2)ae (16)

and

C

l

(Tl*_V)+C

2

(T2*‘V)+C

3

(Tj'"T

o'

)+C4

( V-t

o'

C

5

(T0'-T0

*)/(l/2 )a9 (17)

In the above equations the unsuperscripted terms are temperatures evaluated

at the present, the starred terms at one-half time-step in the future, and the

primed terms at one full time step in the future. Equation (16), which for this

case is implicit in the r-direction only and written out fully for all the

nodes, may be used to find the temperature distribution at one half time step in

the future. It involves solving a system of equations whose coefficient matrix

consists of only three diagonal non-zero elements (hence tridiagonal matrix)

.

Equation (17), which is implicit in the z-direction only, may be solved using

the same method, since the starred temperatures have already been calculated

from equation ( 16 )

.

B. Verification' of the Conduction Model

In order to verify the basic numerical techniques of the NBS/ESDS program,

a test problem was solved analytically as well as using ESDS. The test problem

is to solve the transient temperature distribution of a cylindrical rod 2.0

units in diameter and 2.0 units in height, Figure 5- At the initial time its

temperature is T^. From that time on, the rod is so submerged in a fluid that

the film coefficients on the end surfaces, h , are 1 .0; and the film ’ z’

8
coefficients for the lateral surfaces, h , are 0.5 • Mathematically the problem

is

3T _ 8 2T+

1 3T + 8 2 T (18)

90 r 9r dZ?

with initial and boundry conditions of

I. c. T = T. 0<r<l, 0<z<2 , 0=0

B. C. h = 2.0 0<r<X, z = 2, 9>0

z — —

h

z = 2.0 0<r<l, z = 0, 9>_0 = 0.5 r = 1, Oizil, 9>0

The analytical solution of equation (18) is given in the literature, e.g.

(6) and is plotted in Figure 3* The temperature distribution for the same problem was calculated by running KBS/ESDS and is also plotted in Figure 3* The

agreement indicates the correctness of the numerical scheme.

C . Pipe -Flow-Model

As the hot water flows in the distribution pipe, it losses heat through the

walls of the pipe to the environment. The temperature of the water is a

function of time, 0; and distance, z, i.e., T=T(z,0). Problems of this nature

have been studied by many authors (for example, 7, 8, 9, 10). In general, under

the following assumptions:

1. The velocity and temperature of the fluid do not vary in the radial

direction (piston or plug flow).

2. The temperature of the fluid is a function of time and of the

distance from the pipe entrance.

3* Both the diameter and the wall thickness of the pipe are constant

throughout the system.

4. Each pipe layer is made of material of constant properties.

5. Heat transfer in the radial direction is much greater than that in

the longitudinal direction.

9
The governing equations for the temperatures of the hot water and the two

layers of the pipe shown in Figure 4 are as follows

*5 2c

f

p f H

= mc

f lr - 2”

1

U

i (-u+T) (19)

‘ r

l

2) p u

cu It

= ZTrr.U. (T-U)+2TiroU

0 (y-u) (20)

2 2 3 7T(r - r ) p c = 2irr U (u-y)+27rr U (T ,-y) (21) s o y y 30 o o

J * ss amb J v '

In the above equation set, T, u, y are the temperatures of water, the inner

and outer layer of the pipe, respectively- The r is the various radii of the

pipe; c the specific heat; p the density; m the mass flow rate; U the unit

overall conductance; and T , the ambient temperature. Note that in equation

(19) the term on the left n§nd side represents the rate of change of internal

thermal energy locally. The first term on the right hand side represents the

rate of change of internal thermal energy due to flow. The second term on the

right hand side represents the heat transfer to the wall from the water. The

internal thermal energy reduction (or increase) of an element of the hot water

is due to the flow and heat transfer with the wall. Equations (20) and (21)

indicate that the rate of change of internal thermal energy locally is due to

heat transfer in the radial direction with the neighboring medium.

To calculate the temperature distribution of pipe flow, an explicit method

is used. The explicit method involves a staggered nodal system, Figure 5-

For equation (19) at nodal point 2,

T

' T 9T v 2-2 (22)

30 A0

where T„ is the temperature at 2 at the current time, 9; Tp' the temperature at

2 at a future time, ©+&9. And

T - T

3_T % __2 21 (23)

3x Ax

where Ax is the distance betewen nodal points 1 and 2. And

%

T

1 + T

2

(T - u) = (- - 2 -uA

) (24)

The staggered nodal system aligis the (T-u) and 3T/ 3x.

10
For the inner layer of the pipe described by equation 20,

9U 2, (25)

UA - UA

30 A0

T n + T 9

(T-u) ^ uA

)

and y

- u = (yD

- uA

) (26)

And similarly for the outer layer of the pipe described by equation 21,

3y ^ (27)

~

30 A0

u - y = (uA

- yD)

and

amb

(28)

(29)

As a result of the above approximation or substitution, equation (19)

be rewritten as

^

2

' ~ ^2 (^

a_ 2^) +

1"i (a-^b) + bu^

where , _ m A 0

a - n~

TTT z AXp H

, b =

21)

i A0

W7

Equation (20) may be rewritten as

u

a"

= U A

(l-c-d)f^- c T

2 + 1 c Ti + dyjwhere

c

=

U

i n. A0

i(r

0

^-r

i ")Axp

u

c

u

’ d

=

ir(r

0 ^-r^)Axpu c

u

U A A0

0 0

(30)

(31)

with A

i = 2 tt r

i

Ax,

and A = 2 it r ax.

o o

Equation (21) may be rewritten as

yD

= 7

D

(1 -P-9> + PU A

+ P T

amb < 32 >

may

11
where P

U A A0

o o

Tr(r 2 -r 2 )Axp c

’ ^

S o y y

U A A0

s s

tt (r 2 -r 2 )Axp c

s o y y

The overall unit conductance, U. , is a function of the water side film

coefficient, h.. Likewise, U is a function of the air side film coefficient,

h . 1 s

s Using equation (30), and for given Tp and boundary condition T,, Tp' is

calculated. Similarly, T ' and the rest of the temperature distribution of the

fluid nodes are calculated.

Using equation (31), and for given u

A

, y, Tp and T^, u

A

' is calculated.

Using equation (32), and for given y^, u

A

, and T^, y^' is calculated.

Dusinberre (8) discussed the criterion for selecting the time-step, ^>0.

The criterion is that all the coefficients of temperatures on the right hand

side of equations (30), (3l), and (32) be positive. In case of equations (31)

and (32) the criterion specifies that

,U.A.+U A . A0 (11 00)

it (r z -r

.

1

) Axp c u u

< 1

or A0 < r

2 -r 2 Mo i

) Axp c u u

(U 11. A. + U A ) 00

(33)

In case of thin copper pipes as commonly used, *6 has to be extremely small

to satisfy the criterion. To circumvent the expensive computation involving

extremely small time-step, the equations (20) and (21) are reconsidered. For

pipe flow in thin-walled pipes with high conductances the left hand side terms

are small in comparison with the right hand side terms so if the left hand side

terms are neglected, equations (20) and (21) may then be rewritten as,

r

i

Uj.(T-u) + r

0

U

0

( y

-u ) = 0

r U (u-y) + r U (T . -y) = 0. o o

v J s amb J '

The temperatures of the wall layers may then be calculated for the nodes shown

in Figure 4, using the following substitution

(34)

(35)

T,+T.

(t-u) = "12

2

U

A’ (24)

(y-u) = W (36)

and

^amb

-^" (37)

12
The temperatures may he calculated,

T

2

’ = T

2 (l-a-l/2 h) + T-^a-l/2 h) + huA

T '+T ' r.U. 11 (-——~) + r U yn 2 ooD

U

A

=

r U +r.U.

o o 11

( 30 )

(38)

r U U

'+ r U T , o o A s s amb

yD r U +r 0U

o o S s (39)

The procedure is

1) calculate T' first using the transient heat transfer equation

30;

2) calculate u^, using the fluid temperatures as calculated above,

the current y^ and using equation 38;

3) calculate y’^ using the temperature of inner wall as calculated

above in equation 39

D. Film- Coefficients

The film coefficients, h, for both the water heater model and the pipe

model (conduction and flow) consist of two components: convective (h ) and

radiant (h ), or

0

h = h + h . c r The convective film coefficient as modeled in the MBS/ESDS may be one of

the following:

1 )

2 )

given constant,

a calculated value which is a function of temperatures as,

a) in case of turbulent natural convection,

h =

c where

,

C(T -T ) s a

/ 1/3

(41)

13
T

g = the surface temperature

T

a

= the ambient temperature

h) in case of laminar natural convection,

h = C(T -T ) 1//4

. (42)

c ' s a'

The radiant film coefficient, expressing the temperatures is in terms

of absolute temperatures, calculated as:

h

r F

l_2

0

4 4

(T -T ) s a

(T - T ) s a

(43)

where

,

F,_P = the gray body shape and emissivity factor,

a = the Stefan-Boltzmann constant.

Two assumptions are made in determining F-^,.

l) In case of the pipe and the lateral surfaces of the water heater, F^^

is calculted as that of two concentric cylinders of infinite length,

F 1-2

(44)

where

,

e ^ = the emissivity of the concerned cylinder surface

e 2

= the emissivity of the walls of the room where the cylinderis

located.

r = the characteristic radius of the room a

r

Q = the radius of the cylinder.

2) In case of the top and the bottom surfaces of the water heater,

Ff_2 is calculated as that of two infinite parallel plates,

fcl +

t2

The procedure of iteration is:

14
a) calculate the film coefficients with the current surface

temperatures and the ambient temperature with equations 41, 42 and

43;

h) calculate the temperature distribution according to the methods

described in the conduction model or the pipe flow model with

equations 30, 38 and 39;

c) calculate the surface temperatures from the temperature

distribution of the nodes at the appropriate boundry.

E. Draw Model

The draw model is dependent on the node size of the water region and the

time step. Different algorithms are used depending on the draw volume of one single time step.

l) Option 1. Draw in one time step greater than that contained in each

horizontal section of the tank (one row of horizontal nodes). A r-z cross- section of a tank full of water before draw is shown in Figure 6(a). The axis

of the cylindrical tank coincides with the left-hand edge of the cross-section.

Note that the spacings of the grid line are not uniform in order to make all the

nodes equal volume! . In Figure 6(b) the bottom portion of the stored water is

displaced by cold supply water (cross-hatched), while the volume of draw is

shown on top of the cross-section.

Figure 6(c) shows the temperature distribution of one typical column

of nodes after one A0 of draw. The bottom most node (node l) is cross-hatched.

The water there is displaced by the supply water and the new temperature is the

temperanre of the supply water. This process is indicated by T.+ T . The node

2 has a new temperature which is a combination of the old temperaturl of the

node 1, T^, and the supply temperature T

g

. This process is indicated by T

2

«- (T,/T ). xThis algorithm is called a mixed shift and the actual calculation is

V x T + (V - Vj x T,

T (46)

where V is the volume of each of the nodes. V is the volume of supply water

that has displaced the hot water in the node 2. The node 3 has a new

temperature T~ (T^/T^), etc. The delivered water from this column has a

temperature which is a combination of Tg (whole node) and Tg (partial node)

.

Figure 6(d) shows the temperature distribution at the end of the second A0

of a continuous draw. As in the first A0 of the draw the bottom portion of the

column is displaced by the supply water (cross-hatched), while the cross-hatched

area of Figure 6(c) is now pushed upwards and is now diagonal hatched. This is

because the old supply water which has been resident in the water tank for one

A0 and has been subject to heat transfer in the tank. The procedure to

calculate the new temperature distribution is same as in the first A0.

15
2) Option 2. Draw in one time step greater than the volume of one node

hut less than a section of the water region. Figure 7(a) shows the water region

prior to a draw. Figure 7(h) shows the temperature distribution at the end of

the first A0 of a draw. The cross-hatched area indicates the nodes or partial

node where the temperatures have been replaced by that of the supply water. The

delivered water, which is considered as outside the water region is shown at top

of Figure 7(b). In the first two columns of nodes (counting from the axis of

the water region) , the temperature of both of the first nodes are replaced by

the supply water temperature, T, •*- T . The temperatures of both of the second

nodes are replaced by the old temperature of the first nodes, T~ T, , This

algorithm is called a simple shift as mentioned previously. In the third

column, the temperature of the first node is also replaced by the temperature of

the supply water. The second node of the same column, however, is merely

partially displaced by the supply water. It's temperature, then, is replaced by

the combination of the old temperature of the first node and the supply water,

Tp + (T-i/T ). The new temperatures of all the nodes in the third column

except the first node are calculated using the mixed shift algorithm.

Figure 7(c) shows the temperature disribution at the end of the second A0

of a continuous draw. This time the first nodes of the fourth and the fifth

columns are filled with the supply water (simple shift). The first node of the

first column is also filled with the supply water, the second node of the first

column is partially filled with the supply water (mixed shift) . In the

meantime, the supply water residing in the water region for one time step has

been subject to local heat transfer.

Figure 7(d) shows the temperature distribution at the end of the third A©

of the continuous draw. The same procedure is used to calculate the current

temperature distribution. The supply water residing in the water region for two

A0 is shown dotted. The supply water residing in the water region for one AO is

shown diagonal hatched indicating continuous heat transfer during the three A0

draws.

3) Option 3* Draw of one time step less than the volume of one node.

Figure 8(a) shows the temperature distribution of the water region at the end of

a time step during a draw. This option has the same initial temperature

distribution prior to draw, Figure 7(a). The first node of the first column now

has a new temperature which is a combination of the old T^ and T , T-j^(T,/T ) (mixed shift). At the end of the second A0 of a continuous drawf Figure 8b,

sthe

first node of the second column now has a new temperature as a result of a mixed

shift. In the meantime, the water in the first node of the first column has

undergone local heat transfer.

If the draw is continuing for more than five A0 , under the current grid

system, the first nodes of all the columns will have been subject to a mixed

shift. At the end of the sixth A0 , the second node of the first column will

have undergone a mixed shift for the calculation of its new temperature. The

number of A0 in a draw schedule using Option 3 may not exceed the total number

of water nodes (20) in the case of the grid system shown in Figure 8.

16
Two approaches may be used to accommodate longer durations of a draw with

small flow rate: i) using a finer mesh or grid system in the water region; and

ii) reducing the time step.

Erroneous calculations will result under the current modeling if the first

node of the first column is used again at the sixth A0 • The cause for error

may be illustrated as follows:

a. Assume the temperature of the first node of the first column

prior to draw as T, , Figure 9; its volume as 1.0 the draw in one A0 as a fraction of tne nodal volume as (x).

b. At the end of one A0 of a draw the volume of hot water is (lx), Figure 9(a); the volume of supply water is (x)

;

the new

temperature of the node then:

T,

^ = (l-x)T, + xT . 1 1 s

c. At the end of a second A0 of a continuous draw, Figure 9c, the

volume of hot water is (l-2x); the volume of supply water residing in

the node for one time step, (x); the volume of fresh supply water,

(x); the new temperature of the node then:

T^2 ) = (l-2x)T

1 + xT

g

' + xT

g

.

This calculation would be correct if the old water remained at T^

during the two time steps, but it changed as explained below.

There are two reasons why nodal temperatures during a draw

should not be calculated as above, i) it requires additional

computer memory (for T, , etc.); and, ii) it requires heat transfer

calculation for a fractional node. Heat transfer calculation for a

fractonal node is not done by definition under the current context.

Under the current modeling, when the algorithm mixed shift is

performed,

T-l = (1-x)^ + xT

g

The old water temperature is lost, because the quantity on the

right-hand-side of the equation has been stored in the location "T^".

Therefore, when the mixed shift is performed on the node again

T-, = (l-x)T + xT -L S = (l-x)[(l-x)T

1

+xT

g

] + xT

g

,

17
The result is different from the correct calculation as in (c) above.

F. Buoyancy- Stacking

A sorting algorithm is used to generate a thermal stratification both in

the water region of the tank or in vertical sections of distribution pipes

during standby. The sorting algorithm merely reorders a series of numbers to a

decreasing order. For example, in processing the temperatures of the nodes in a

column , the sorting algorithm will reorder the numbers in a decreasing order, so that the highest temperature will be associated with node at the top of the

column, the lowest at the bottom.

To model the heat loss due to a stand pipe for the purpose of simulating a

heat trap or the lack of it, the conservation of energy has to be considered in

addition to the buoyancy stacking. The first vertical segnent of pipe,

sometimes referred to as a stand pipe, connecting the distribution pipe network

with the water heater causes heat leaks due to boyancy effects. These heat

leaks may be reduced by the installation of heat traps.

Shown in Figure 10(a) is a temperature distribution of a column of the

water in the tank and a contiguous stand pipe. The sorting algorithm reorders

the temperatures in such a way so that TP^’ > TP-^' > T.' —> T^'. In order to

satisfy the conservation of energy the following^adjusunent is made. Let e be

the energy of the stand pipe before the buoyancy stacking; e the energy of-^the

column of water before the buoyancy stacking; e ' and e ’ th$ corresponding

energy after the buoyancy stacking. The total Energy of the system is e^., or

6 , = e + e (47)

t p w

k

where e = E pc

P i-1

VP. TP.

l l (48)

and kw

e = E pc W

i-1

V. T.

l l (49)

In the above equations P and c are the density and specific heat of water,

respectively; YP. , TP. the nodal volume and temperature of the stand pipe,

respectively; V.

1 and $. the nodal volume and temperature of the water column,

respectively; k

1 and k1 are the number of nodes in the stand pipe and the water

column, respectively. w

The energy of the stand pipe and the water

stacking, respectively, are:

e P

pc vp. tp:

i 1

ew

pc V.

1

t:

i

column after the buoyancy

(50)

(51)

18
The energy of the water column after the thermal stacking should he

e.-e

t p

? ( 52 )

The difference, because the water nodes and the pipe nodes have different

volumes , is

ti w

( 53 )

In order to reduce the above difference to zero—satisfying the

conservation of energy principle—an adjustment to the water column temperature

is made. The adjustment, as calculated below, is made only to a number of nodes

at the bottom of the water column. This number equals to the number of nodes in

the stand pipe,

AT

w w

.z; pc vp. 1=1 X

( 54 )

Then rp II _ m I i 11 ~ 1

rp M rp I i x2 2

T, " = T,

' kp kp

A T

A T

+ A T

The T^”, Tp

buoyancy-sx

", . .., T^ ", •••» T, ’ are then the correct,

acked, tem^eratuPe"

L distr?bution of the water column.

IV. Parametric Studies

A . Description- of the 'Water Heater and the Distribution Pipe

A commercially available "52 gallon" electric water heater was modeled for

these studies. The actual capacity is 191 L (49*8 gal*)* The exterior shell

measures 1.45 m (57*2 in) in height and 0.58 m (22.9 in) in diameter. The shell

is made of 21 gage sheet carbon steel and consists of a right cylinder with a

top and a bottom cover. The cylindrical water tank with convex top and bottom

is made of 12 gage sheet carbon steel having an interior anti-corrosion cement

lining 1.8 cm thick. A fitting for connecting a relief value is located at the

center of the top. Three legs support the water tank on the bottom of the

exterior shell. There are two heating elements, two thermostats, water outlet

and inlet, junction box and wiring located on the side walls of the water tank.

In this particular water heater there is also a built-in heat trap connected to

19
the hot water outlet. The annular gap varying from 2.5 cm (1.0 in) to 6.0 cm

(2.4 in) is filled with fiberglass insulation.

A typical residential hot water distribution piping system consists of one main line and a few branch lines. In the current study, the distribution piping

system is assumed to consist of one single pipe, 6.71 m (22 ft.) long, 1.68 cm

(0.6619 in.) OD, 1.60 cm (0.631 in.) ID. This pipe has two vertical sections

and one horizontal section.

B. Modeling of- the -Water Heater

1. Basic Water Heater (Water Heater A)

The exterior shell and the water tank of the water heater are considered as

right cylinders. The actual outer diameters of the water tanks are used, the

heights are derived from the measured volumes of the tanks. The steel walls and

the insulation are taken to have uniform thicknesses. Their thermal properties

are considered constant. Different thermal conductivities for insulation are

used, depending on the temperature range any particular layer will experience to

account for the thermal conductivity variation as a function of temperature.

The thermal conductivity of an insulation layer is evaluated at the average of

its temperature range during the cooldown period of the water heater. Water

heater fittings and leakages are not independently considered but are lumped

with the mean thermal conductivity of the insulation which is summarized in

Table 1. Internal water current has been considered by using a water thermal

conductivity 100 times higher* than the actual value to account for internal

mixing. Radiative and convective heat transfer parameters are given in Table 1.

An initial condition of 71.1 C (160°F) is used for all the nodes while a

constant ambient temperature of 21.1 C (70°F) together with the surface heat

transfer coefficients are used to describe the boundary conditions.

In modeling the electric water heater, 207 nodes are used, Figure 11. It

was established in earlier work (2) that 104 nodes were sufficient for accurate

modeling. The 207 nodes were chosen to provide sufficiently small nodal volumes

of the water region to accommodate small flowrate during the draw. In the water

region the nodes have equal volume. There are 10 grid lines in the radial

direction and 24 grid lines in the axial direction [(lO-l)x( 24-1) =207] • There

are 6 radial grid lines in the water region, two in the cement lining, two in

the tank walls, two in the insulation and two in the exterior shell; among these

14, four are common grid lines [ 14-4=10] which constitute boundries between

layers. From the bottom to the top of the water heater there are two axial grid

lines in the exterior shell bottom, two in the insulation, two in the tank

bottom, two in the lower cemenet lining, , 16 in the water, two in the upper

cement lining, two in tank top, two in the upper insulation, and two in the

upper exterior shell top; among the 32, 8 are common [32-8= 24].

A time step of .01 hours is sufficient for accurate modeling of the water

heater (2).

•k Footnote: This value which can easily be adjusted was chosen arbitrarily

in the absence of experimental data on the internal mixing.

20
For the exterior shell of this water heater, the product of Grashof and

Prandtl number, X, exceeds lCr; thus, the film coefficient for the vertical

surfaces with turbulent flow is expressed as in equation 41:

h = C(T-T)1^

o o o a ,

the film coefficient for the horizontal top surface with laminer flow is

detemined from equation 42 as:

h

t = C

t<-

T,-T

U"

* 1/4

and

the film coefficient for the horizontal bottom surface is also for laminer flow

and determined from equation 42 as:

h

b

c ( VT

a) l/4

The coefficients of the above correlations as used in the NBS/ESDS are summarized in Table 1.

Two 4*5 kW heating elements—same number and ratings as in the actual water

heater—are modeled. Each of them is considered as an evenly distributed heat

source in a row of four contiguous nodes, the second through the fifth nodes

from the center of the water heater, Figure 11. the upper heating element is

located at the 15th row of nodes from the bottom of the water heater which is

86.1 cm (54-0 in) from the bottom of the water tank; the lower heating element

at the eighth row of nodes is 34-5 cm (13*5 in) from the bottom of the water

tank.

Two thermostats are modeled. Both thermostats are located on the outer

surface of the water tank. The upper thermostat is located at the eighteenth

node, 93*5 cm (36.8 in), from the bottom of the water heater and the lower

thermostat at the eighth node, 34*3 cm (13*5 in), from the bottom of the water

heater. The upper thermostat controls the upper heating element and the lower

thermostat the lower heating element. The thermostats and the heating elements

are arranged in such a way that only one heating element operates at one time.

The upper limit for both of the thermostats is 71.1°C (160°F) and he lower limit

68.3°C (155°F). If the water temperature is so low after a heavy draw or during

a startup that both thermostats sense temperature below the lower limit of the

thermostats, the upper thermostat takes precedence. The thermostats sense only

the temperature of the outer tank wall. Since there is a cement lining with low

thermal conductivity between the water and the tank wall, the thermostat does

not respond directly to the water temperature. For example, the lower

thermostat is located at the tank wall node of the eighth row of nodes.

21
Counting from the center of the water heater, the second through the sixth are

the water nodes, the seventh the cement node and the eighth the tank wall node.

In a recovery mode at a certain time the temperature of the sixth node (water)

reaches 71. mC (160°F), the thermostat node, however, does not reach this

temperature until a few times steps later. In fact the water node may reach

73.9°C (165°F) before the thermostat acts to cutout the heater. The reverse of

this occurs before the cutin point of the thermostat.

2. Insulated Water Heater (Water Heater B)

The same 207 node model is used to study an electric water with an extra

2.5 cm (l in) of insulation applied to the exterior shell (top and side walls).

The only difference from water heater A is that the steel casing regions on the

top and on the side walls of the model are replaced with insulated regions. The

presence of the thin layer of the casing has very little effect on the heat

transfer properties of the water heater model. The same film coefficients and

mean thermal conductivity for water heater A insulation are used, while the

extra insulation has a thermal conductivity compatible with its type.

C. Modeling' of ~ the' Distribution Pipe

1. Distribution Pipe (Pipe A)

The distribution pipe, as shown in Figure 12 is 671 cm (264 in) long. It

consists of two vertical sections of 91*4 cm, (36 in) and one 488 cm (192 in)

horizontal section. The radii given in section IVA are used. The copper walls

are taken to have uniform thickness. Their thermal properties are considered

constant. Internal water mixing during standby in the horizontal section of the

pipe is considered to be minimal. The standard text book value of the thermal

conductivity is used for the water. In the vertical section of the pipe, the

process of the buoyancy stacking is assumed to place.

Film Coefficients. Film coefficients for the pipe are calculated from

"corFelations recommended by McAdams (ll) for different value of X--the product

of Grashof and Prandtl numbers.

a) Vertical Section

Preliminary studies indicate that the maximum surface

temperature for bare pipe is about 70C (158°F) and^the

corresponding ^ ( =p 2 gBcp /uk ) is approximately 10 . The

corresponding X = 7«04 x 10 for a 61 cm (24 in) vertical pipe

section. Therefore, for laminar natural convection (from

equation 42)

h = 1.42(4* )°“ 25

(54) V 1/ p where h is in watt/m K, L in m, a T the temperature difference

betweenv

pipe surface and ambient in K. For insulated vertical

22
pipes, the maximum surfaee temperature is approximately 46.1 C

(115°F), V = 1.2 x 10°, and X = 4.3 x 10° for 61 cm (2 ft.)

sections. Therefore, equation (54) is also applicable for

insulated vertical pipe sections.

b) Horizontal Sections

Preliminary studies indicate that the maximum surface

temperature for bare p^pe is 66. 7C (152°F) . The

corresponding ip is 10 , X = 5 x 104

. therefore, for laminar

natural convection from equation (42)

.

h

h

= 1.52(^)

0-25

(55)

where D is the diameter of the pipe. For the insulated

horizontal pipe, equation (55) may also be used.

c) Combination Vertical/Horizontal Sections

Where a node consists of a horizontal and vertical

segment, a prorated film coefficient may be used.

d) Water Side Film Coefficient

McAdams (ll) recommends for calculation of the water side

film coefficient for pipe flow the following formula:

h

w

. 1057 [1.52 + -018 T

b

]

where h is in terms of watt/m K; T^, the bulk fluid

temperature in C; V, the fluid velocity, in m/sec; D, the

inside diameter of pipe, in m. Preliminary studies gave the

following typical values.

Tb

= 48.9°C (120°F)

V = 0.17 m/sec (0-55 ft/sec), and

D = 0.016 m ( 0.63 in)

Th^ typical value of h , then is 1288 Watt/M^K (227 Btu/hr

ft U

F) which was used in the modeling.

As discussed previously, during a non-draw cycle the distribution pipe is

considered as a conduction model. The water inside the pipe is considered as

solid. Only a model of buoyancy stacking is used to simulate the fact that in

vertical sections, the warmer portion of the water floats to the top and the

colder portion sinks to the bottom. During a draw cycle, the pipe flow model is

23
used to model simulate the distribution pipe, i.e., a fluid flows inside the

pipe. In the latter case a fluid side film coefficient is used.

2. Insulated Distribution Pipe (Pipe B)

The nodal diagram of the insulated pipe model is same as that of the basic pipe

model except that the radii are different. The radius of the water region, r.,

(Figure 5) remains the same. The radius of the copper pipe, r , now is 1.68 cm

(0.661 in). The radius of the insulation (polychloroprene) r

° is now 2.79 cm

(l.l in). The properties used for polychloroprene, given in fable 1, are taken

from (12)

.

D. Modeling of- the System

1. Water Heater-Pipe Connection

The water heater pipe connection is shown in Figure 12(b). The water

connection is made at the second node of the top row of the water region. The

pipe connection is made at the second node of the top of the tank wall region.

Mathematically, the contiguous water temperature becomes the boundary condition

for calculating pipe water temperature in both of the pipe flow model and the

pipe conduction model. Similarly, the temperature of the contiguous tank wall

region becomes the boundary condition in calculating pipe wall temperature in

the pipe conduction model. The heat transfer in the pipe conduction model at

the connection, however, is negligible.

2 . Heat Traps

The lack of a heat trap is modeled by the modified buoyancy stacking

algorithm. The heat leaking due to the lack of a heat trap occurs only during a

non-draw cycle. Hot water in the second column of the water region of the water

heater floats, through the connection between the pipe and the water tank, to

the second (counting from the connection) node of the pipe model, and then cools

off.

3* Draw Schedule

The energy consumption for supplying hot water is highly dependent on the

daily draw schedule for the same total draw. A basic draw schedule (A) is taken

from one particular sample of a field survey (13). Since there is neither

identification of the purpose nor the flowrate of the draws, the flowrates of

the chosen sample (Townhouse #5, day #88 of Ref. 13) are determined based on the

time of the day of a draw and a suggested usage/flowrate relationship (14).

Draw schedule A is given in Table 2.

In Table 2, the starting time of a draw is given in column 1, the flow rate

in column 2, the duration of a draw in column 3, the volume of the draw in

column 4 and the probable purpose in column 5. The durations of the draws are in units of hundredths of hours to conform with the time step of the water

heater model.

24
An alternate draw schedule (B) is used to demonstrate the capability of the

program to handle substantially different draw schedules. This alternate draw schedule is given in Table 3* It allows the same number of draws and same amount of total, draw in a 24 hour period as Schedule (A)

.

The prominent feature

of this draw schedule is its two concentrated peaks (thus 2-peak schedule), one in the morning between 6 am and 9 am, the second one in the evening, between 6

pm and 10 pm.

4. Quasi-Steady State System Modeling

For the current parametric studies as described below quasi-steady state

operation of the system for a typical 24-hour period was used. The assumption

was made that the system is operated the same way day after day. In order to

achieve this quasi-steady state, the system is modeled to operate more than a

24-hour period. The results then are checked for repeatability. If the results

obtained at a time are nearly the same as that obtained 24 hours earlier, a

quasi-steady state solution is achieved.

E. Summary - of - Parametric Studies

Case A. Basic System (Water Heater A, Pipe A). The basic system consists

of the basic water heater without a heat trap, the basic pipe and draw

schedule A.

Case B. Insulated Water Heater, Bare Pipe (WH B, Pipe A) The water heater

is without a heat trap. Draw schedule A is used.

Case C. Insulated Water Heater, Insulated Pipe (WH B, Pipe B)

.

Case A-l. Basic System, Schedule (B)

.

Same as Case A, draw schedule B is

used.

Case A-2. Basic System with a Heat Trap. Same as Case A, the water heater

having a heat trap.

Case A-3. Basic System, Schedule B, Reduced Thermostat Settings. Same as Case A-l except the thermostat settings are reduced to maintain the same average faucet temperature for dishwashing draws.

V. Results and Discussions'

A. Basic System (Case A)

The basic system consists of a water heater (WH A) with the thermostats set

at 68.3°C (155°F) cutin and 71.1°C (160°P) cutout, and a distribution pipe (Pipe

A). The system is operated on draw schedule A which is plotted in Figure 13(a).

The flowrates are plotted versus time of the day. The numerals atop the spikes

are durations of the individual draws, in hundredths of hours. The three draws

25
starting at 20.00 hour have been identified as dishwashing draws. More details

about the dishwashing draws are discussed later.

The thermal energy content of the water in the water heater referenced to

21.1°C (70°F) is plotted in Figure 13(b). It decreases exponentially during

cooldown periods, and increases in a complex function—a combination of

exponential terms during the recovery periods. During the periods of draws, the

thermal energy content of water decreases in an approximately linear fashion.

If during a period of draw the heating element is concurrently on, then of

course, the thermal energy content of water decreases less rapidly (or may even

increase)

.

The power input to the water heater is plotted in Figure 13(c). Since only

one heating element may be operating at one time and each of the heating

elements is rated at 4*5 kW, the power input is 4*5 kW anytime the power is on. The area under the power input curve is the total energy consumption of the

water heater.

The average faucet temperature of the individual draws are plotted in

Figure 13(d). The average temperature of a draw is a complex function of many

factors: l) the flowrate and the duration of the draw, 2J the flowrate and the

duration of the previous draw, 3) the time elapsed between the previous draw and

the current draw, 4) the heat transfer properties of the distribution pipe, 5)

the temperature of the water before it enters the pipe, and 6) the ambient

temperature. The factors (2) and (3) above give the initial condition for the

pipe flow equations. The factors (5) and (6) give the boundary conditions for

the pipe flow equations.

The three identical draws at .16 L/sec (2.5 gpm) starting at the 20.00 hour

mark and lasting for .04 hours have been identified as probable dishwasher

draws. The first draw achieves an average faucet temperature of 56. 7C (134 F)

,

the second 65°C (149°F)» the third 67.2°C (153°F)- The series of three

identical draws at 12.00, 13-00 and 14-00 hour form another interesting case. The first one achieves an average faucet temperature of 43-3 C (110 F) , the

second 30.6°C (87.0°F), the third 26.1 C (79-0°F). The rising or falling of the

average faucet tonperature for three consecutive draws depends on the

temperatures of the pipe at the beginning of each of the draws (or the initial

conditions) as previously discussed. This property becomes clearer when the

variations of the faucet temperature are discussed.

During non-draw periods the thermal energy content of water in the

distribution pipe decreases in an exponential fashion similar to water heater

cooldown. Upon the start of a draw the thermal energy content of the pipe

increases rapidly. Similarly, the rate of heat loss from the pipe decreases in

an exponential fashion during non draw periods since it is directly related to

the thermal energy content of the water in the pipe. This is demonstrated in

Figure 13(e) at the 8.40 hour mark where after achieving a peak due to three

heavy draws, the rate of heat loss starts to decrease exponentially until the

onset of the next draw. At that time, it increases rapidly to a high value.

26
The rates of heat loss from the water heater are plotted against time of

the day in Figure 14 . The rate of heat loss through the top surface remains

almost constant throughout the day. This is bgcause ijhe water in the upper

portion of the tank remains approximately 71.1 C (160°F) due to the

stratification which is modeled. The cold supply water is delivered to the

bottom of the tank. If the stored water heat content were substantially

depleted, the water temperature in the upper portion of the tank would drop

significantly below 71-1°C (160°F). The fact that the hot water content is not

substantially depleted may be demonstrated by Figure 13(b) where the water

energy at the lowest point is still 68fo of the fully recovered value. The rates

of heat loss of the lateral surfaces and of the bottom, on the other hand, are

sensitive to the draws as shown in Figure 14 . Note the synchronous variation of

the rates of heat loss of the lateral surfaces and the bottom, and the water

energy curves.

The faucet temperature variations of the selected draws are plotted in

Figure 15. A typical faucet temperature variation is that of the 7.00 hour

draw. Starting at 28. 7C (83.7°F) , it dips slightly and momentarily before

increases rapidly in the first .01 hour. Its rate of increase gradually reduces

until at large time the faucet temperature reaches a steady-state value before

the draw terminates. For shorter and smaller draws, the faucet temperature for

the 18.00 hour draw is typical. Starting at 25.9°C (78.7°F), it reduces to

25.6°C (78°F) .009 hour later. It then increases to 27. 30 (81.2°F) at .02 hour

after the start of the draw. The reduction of the faucet temperature right

after the onset of a draw is a characteristic of the distribution pipe with a

vertical section at the faucet end of the pipe. Because of the vertical

section, a portion of its water temperature is warmer than the rest of the pipe

due to the buoyancy effect. When the faucet is opened, the warmer water flows

out first. After the segment of warmer water is exhausted the colder water

flows out of the faucet, before the hot water arrives from the water heater.

The falling of the average faucet temperatues of the three draws at 12:00,

13:00 and 14:00 hour and the rising of the average faucet temperatures of the

three dishwashing draws can now be clarified. The former three draws have

decreasing initial temperatures while the latter three draws have increasing

initial temperatures.

Table 4 summarizes for all cases studied, the electrical energy consumption

(Electricity) , the thermal energy in the delivered water relative to the supply

water (Hot Water), the total heat loss (Loss) for a 24-hour period, and the net

difference in the energy balance over the 24^hour period (Convergence). The

electrical energy consumption is that consumed by the electrical heating

elements and the hot water delivery is the energy delivered at the faucet.

The total heat loss consists of the heat loss through the jacket of the

water heater (top, bottom, and lateral surfaces), and the walls of the

distribution pipe. The convergence is the change of the total energy of the

system in a 24-hour period. The same energy consumptions are presented in Table

27
5 as the percentages of the electrical energy consumptions. The total heat

loss, as a percentage of the electrical energy consumption, indicates the

inefficiency of the operation of a given system.

1. Schedule B (Case A-l)

The "basic (Case A) model is used to study of the effect of changes in

draw schedule. The same total volume draw of 438.5 L (115*8 gallon), is

delivered by the system, the only difference being the time schedule of the

draws. The differences in energy supplied, losses, and in hot water

between the two draw schedules are shown in Table 4.

The differences in the efficiency with a concentrated draw schedule

may be slightly less pronounced in a multi-branching pipe than in the

single pipe network. This is because in a single-pipe network, the entire

length of pipe contributes to changing the hot water delivery and reducing

the pipe heat loss when a concentrated draw schedule is used. In a multibranching pipe network, however, only the main line of the network

contributes to the change when a concentrated draw schedule is used.

2. Effects of a Heat Trap (Case A-2)

A distribution pipe is usually connected to a water heater with a

vertical section. The water in the vertical section cools off much faster

than the water in the water heater, because 1) the pipe section is usually

not insulated that 2) the pipe has a surface-to-volume ratio which is much

greater than that of the water heater. The effect of the heat trap is

studied when it is modeled on the Case A system.

If there is no heat trap installed at the pipe to water heater

connection, the colder water in the vertical pipe section sinks to the

water heater. The same amount of hot water rises from the water heater to

the pipe section to take the place of the colder water. The hot water

again loses heat and be replaced by hot water from the water heater. This

thermal syphon process is taking place continuously during the non-draw

periods. An ideal heat trap stops this process.

The effectiveness of a heat trap is demonstrated in Table 4 when

comparing the results of Case A-2 with those of Case A. The heat loss is

reduced by 3*06 MJ (2900 Btu) . In terms of the distribution of the energy

consumptions (Table 5), the heat loss is reduced from 25.8$ of the

electricity consumption to 23*7$.

B . insulated Water - Heater - ( Case B)

The rates of heat loss from the insulated water heater are plotted in

Figure 16. As might be expected the rates of heat loss of the lateral and the

bottom are synchronous with the thermal energy of the water in the heaters in

28
Case A. The values of these rates of heat loss for Case B are much lower than

the corresponding values for Case A except for the rate of heat loss of the

hottom which is not affected (no insulation was added to the bottom of the

tank) . The insulating of the water heater causes the heat loss to reduce from

29.01 MJ (27500 Btu) to 26.27 MJ (24900 Btu), and the daily electricity

consumption to reduce from 51*28 kWh to 30.64 kWh. In terms of electricity

consumption, the heat loss decreases from 25-8$ to 23.8$.

The reduction in heat loss is different from that of a simpler calculation

discussed in section II, because the simpler calculation does not account for:

the heat loss of the distribution pipe, the heat capacity of the insulating

material, the temperature dependence of the overall conductance and the

temperature stratification in the tank.

C. Pipe- Insulation- (Case C)

Tables 4 and 5 show that when the pipe is insulated in addition to the

water heater the heat loss is reduced substantially. For this example, losses

were reduced from 23.8$ to 20$ of the electrical energy used in a 24 hour

period.

D. Thermostat ~ Setback ( Case ' A-5)

The effects of a 5*6°C (10°F) thermostat setback are also shown in Tables 4

and 5 . For this example, the energy required to meet the requirements of draw

schedule A are reduced from 31.56 to 28.76 kWh and the losses are reduced

proportionally.

E. Water Delivery Temperature- (Faucet Temperature)

To illustrate the capability of the model to predict the changes in water

delivery temperature with time, the water delivery temperatures for three

successive identical 22.7 L (6 gal.) draws were computed, Figure 17, with and

without pipe insulation (Cases C and B, respectively). The first draw starts at

18.00 hour mark, and lasts until 18.04 hour mark; the second draw starts at

18.25 and lasts until 18.29 hour mark; the third draw starts at 18.35 and lasts

until 18. 37 hour mark. Thus, the three draws are separated by two cooldown

periods. For this example, it is noted that the insulated pipe maintains higher

delivery temperature than that of the bare pipe during the entire period of the

first draw. In the second draw, however, the bare pipe gives higher delivery

temperature than the insulated pipe initially. Later on, the insulated pipe

again delivers hotter water.

The reason: subsequent to the termination of the first draw, in case of

the bare pipe, the water is losing heat via the pipe wall to the ambient, in

case of the insulated pipe, however, the water is losing heat, via the pipe, to

the insulation which, having a thermal diffusivity 1000 times greater than that

of copper, is still relatively cold. In case of the bare pipe, the convective

heat transfer between the pipe and the ambient is the controlling factor

29
determining the rate of heat loss, in case of insulated pipe, the conduction

heat transfer between the pipe and the insulation is the controlling factor

determining the rate of heat loss. The former heat transfer is smaller than the

latter heat transfer during the first cooldown period. The net result: the

bare pipe has hotter water than the insulated pipe at the start of the second

draw. A similar situation occurs for the third draw. If a cooldown period

lasts for a long time giving enough time for the pipe insulation to heat up (so

insulation is no more a heat sink), then the overall thermal resistance between

the water and the ambient, now the determining factor, is higher in the

insulated pipe than that in the bare pipe. The net result: the insulated pipe

has hotter water than a bare pipe, e.g. , the first draw.

VI. Conclusions

A computer model has been developed to study the dynamic performance of the

hot water system of a residence and some commercial facilities.

The model may be used to study the effects of changing the following

parameters on the performance of the system:

A. Pertaining' to 'the water 'heater :

1) the size of tank,

2) the diameter-to-heigjht ratio of the tank,

3) the location and size of the heating elements,

4) the location of the thermostats and their settings,

5) the property and thickness of the tank wall and, particularly

the insulation.

B. Pertaining to~ distribution- pipe :

1) the length and the size of the pipe,

2) the location of horizontal and vertical sections of the pipe,

3) the material of the pipe,

4) the insulation of the pipe.
c. Pertaining to the System:

1) the draw schedule,

2) the presence of a heat trap,

3) the location of the connection of the distribution pipe

to the water heater,

4) the temperature of the supply water,

5) the ambient temperature.

A parametric study has been performed to illustrate quantitatively the

effects of

1) insulating the water heater,

2) insulating the distribution pipe,

3 ) changing the draw schedule,

4) installing a heat trap, and

5) thermostats setback.

It is recommended that an experimental study be performed to verify or refine the analytical dynamic model. A verified analytical model could form the

basis of simpler models for use as design tools in predicting thermal

performance of water heating systems.

31
REFERENCES

1) "Patterns of Energy Consumption in the U.S.", Office of Science and

Technology, Washington, D.C., Jan. 1972.

2) Wan, C. A., "Computer Simulation of Water Heater Standby Loss, Heat

Transfer in Energy Conservation", Heat Transfer in Energy Conservation,

Goldstein, R.J., Ed., ASME Winter Annual Meeting, Atlanta, GA, Nov. 1977.

3) Wan, C.A., Users' Manual for the ESDS Computer Model, NBSIR under

preparation.

4) Richtmeyer, R.D., and Morton, K.W., Difference Methods for Initial-Value

Problems , 2nd - ed

.

, Interscience Publishers, New York. 1967.

5) Douglas, J. Jr., and Peaseman, D. W., "Numerical Solution of TwoDimensional Heat Flow Problems," AICHE Journal, Yol. 1, 1955

6) Boelter, L.M.K. , Cherry, V.H., Johnson, H.A., and Martinelli, R.C., Heat

Transfer Notes, University of California Press, Berkeley and Los Angeles,

13487 —

7) Rizika, J.W., "Thermal Lags in Flowing Systems Containing Heat Capacitors",

Trans. ASME, Vol. 76, 1954.

8) Dussinberre, G.M., "Calculation of Transient Temperatures in Pipes and Heat

Exchangers by Numerical Methods", Transaction ASME Vol. 76, 1954

9) Sherik, H. , Jr., Fortran Methods in Heat 'Flow, (Ronald Press Co., New York,

N.Y. , 1963).

10) Chi, J. , "DEPAF - A computer Model for Design and Performance Analysis of

Furnaces", ASME/AICHE National Heat Transfer Conf . , Salt Lake City, Aug.

1977.

11) McAdams, W. H. , Heat Transmission, 3rd ed. , McGraw-Hill Book Co., New York,

1954.

12) Brandrup, J. , and Limmergut, E.H., Ed., Polymer Handbook, Interscience

Publishers, N.Y. 1972.

13) Grot, R.A., "Field Performance of Gas and Electric Water Heaters",

Proceedings of the Conf. on Major Home Appl. Tech for Energy Conservation,

Purdue University, Feb. 29 - March 1, 1978.

14) Guinn, R.S., Jr., The Effect of Increased Capital Expenditure as a Method

of Reducing Electricity Demand for Hot Water Generation in New Home", M.S.

Thesis, Univ. of Tenn., Aug. 1972.

32
Coefficients for

Equations 41 and 42

C

o Ct Cb

Thermal

Conductivity (

, watt k'"ST

Insulation

Extra

Insulation Copper

Water Heater A 5.5 2,6 4.5 .048 — —

Water Heater B 5.5 2.6 4.5 .048 .046 —

Pipe A See text — — — — 418

Pipe B See text — — — .23 418

Characteristic Dimension of Room = 4.5 m

Emmissivity of Water Heater Jacket = 0.6

Emissivity of Walls of Room = 0.9

Emissivity of Pipe Surface =0.6

Table 1 Selected Heat Transfer Parameters for Modeling

33
Time of Day

(Hour)

Flowrate

L/sec (GPM)

Duration

(Hour)

Volume

L (Gallon)

Purpose

0.00 0.032 (0.5) 0.01 1.14 (0.3)

1.00 0.095 (1.50) 0.03 10.2(2.7)

2.00 0.032 (0.50) 0.01 1.14(0.3)

3.00 0.032 (0.50) 0.08 9.09(2.40)

4.00 0.032 (0.50) 0.02 2.27 (0.60)

5.00 0.032(0.50) 0.02 2.27(0.60)

6.00 0. 032 (0. 50) 0.02 2.27(0.60)

7.00 0.088 (1.40) 0.23 73.1(19.32) Bath

8.00 0.17 (2.70) 0.05 30.7(8.10) Clothes

Washing

8.33 0.17 (2.70) 0.05 30.7(8.10) Clothes

Washing

8.42 0.17 (2.70) 0.05 30.7 (8.10) Clothes

Washing

11.00 0.095 (1.50) 0.03 10.2 (2.70)

12.00 0.032 (0.50) 0.01 1.14 (0.30)

13.00 0.032(0.50) 0.01 1.14(0.30)

14.00 0.032 (0.50) 0.01 1.14 (0.30)

15.00 0.095 (1.50) 0.08 27.30 (7.20)

17.00 0.032 (0.50) 0, 01 1.14 (0.30)

18.00 0.032 (0.50) 0.02 2.27 (0.60)

20.00 0.16 (2.50) 0.04 22.7 (6.00) Dish

Washing

20.25 0.16 (2.50) 0.04 22.7(6. 00) Dish

Washing

20 . 33 0.16 (2.50) 0.04 22.7(6.00) Dish

Washing

21.00 0.088 (1.40) 0.11 35.0(9.24) Shower

22.00 0.088 (1.40) 0.11 35.0 (9.24) Shower

23.00 0.16 (2.50) 0.11 62.5(16.50) Bath

Table 2 Draw Schedule A

34
Time of Day

(Hour)

Flowrate

L/Sec (GPM)

Duration

(Hour)

Draw Volume

L (Gallon)

Purpose

6.80 0.032 (0.50) 0.01 1.14(0.3)

6.90 0.032 (0.5) 0.01 1.14 (0.3)

7.00 0.088(1.40) 0.23 73.1(19.32) Bath

7 .50 0.17 (2.70) 0.05 30.7(8.10) Clothes

Washing

7.83 0. 17 (2 . 70) 0.05 30.7(8.10) Clothes

Washing

7 .92 0.17 (2.70) 0.05 30.7 (8.10) Clothes

Washing

8.00 0.095(1.50) 0.03 10.2(2.70)

8.10 0.032(0.5) 0.01 1.14(0.30)

18.50 0.095(1.50) 0.08 27.3 (7.20)

18.60 0.032 (0.50) 0.01 1.14 (0.30)

18.70 0.032 (0.50) 0.02 2.27(0.60)

19 . 00 0.088 (1.40) 0.11 35.0(9.24) Shower

19.50 0.088(1.40) 0.11 35.0(9.24) Shower

19.70 0.095(1.50) 0.03 10.2(2.70)

19.80 0.032 (0.50) 0.08 9.09 (2.40)

20.00 0.16 (2.50) 0.11 62.5(16.50) Bath

20.50 0.16 (2.50) 0.04 22.7 (6.00) Dishwashing

20.75 0.16 (2.50) 0.04 22.7 (6.00) Dishwashing

20.83 0.16 (2.50) 0.04 22.7 (6.00) Dishwashing

21.00 0,032 (0.50) 0.01 1.14 (0.30)

21.10 0.032 (0.50) 0.01 1.14 (0.30)

21.20 0.032(0.50) 0.02 2.27(0.60)

21.30 0.032(0.50) 0.02 2.27(0.60)

21.40 0.032(0.50) 0.02 2.27(0.60)

Table 3 Schedule B (Two-Peak Schedule)

35
CASES

CO t—l

uo O'

r-. 00 OO » 9 o o o\ rH o o 00 CM 1—1

'vO CO 40

CO ON CM 5 1 o © o 9 o CO vD o

rH 00 CM \

1

CM vO r^ 04

vO <r 40 » © © CO CNI o o o 00 04

rH a.gd

4-1 cuG o u c u

D© •> CM 1—1 U0 40 G O <2 ©^

LO I

1 d

G

a e • • • 4-1 CD G

CO LO o JT 1—©O o 00 04 C <3 3 4-1 rH G "G G

04G co “ r—1 ©G

<3 3 0 4-1 T3 CO G o aj 4-1 i—1 ©d «© co 3 0 0) o 3 C/4 G a o CO 00 04 04 *H d o in o O' ©G * P© G 0 « o « O 04 ©G CO O'! 04 o CO P© T3 4-1 rH 00 04 •rl o cq rH ©> (O, 4-1 CU 3d

04 GO)

G 5

Q© CJ i1 i

1 d

•H *rl 3 3 4-1 [*4 G CO (0 3 0 G d G 4-1 O a ©Q -d ©G G rd

•H

O

G

CO •> ©'CO ©d ©d CO CN i—1 LO

ba WH WH

h

4-1 v£3 o o ©G -H

® * o © 4-1 4-1 £

CM 00 04 o » 3 3 rl •rl rH 00 04 K U) 4) !5 13 r-l rH 3 4-1 4-1 G G <3 1 <3 <3 O i

1 i—1 •H 3 3 04 G G

CO CO CO CO CO CO G d G G G G

©O -H -rl CJ CJ CJ

>> a) 1 1 1 1 1 1 U a

•rl

d

O

g QJ I—1

CM CO

•H 4-1 00 1 j 1 U G u A B C A <3 <3

4-1 12 04 a co > 04 0) 04 04

GG

Q) 4-1 co

G CO CO CO CO co co rH O O o G G G G G G w 53 hJ cj CJ CJ CJ U CJ CJ

Table 4

Energy

Consumptions

(in

MJ)
CASES

B-? 6^ &-? 5^ O MO o « • • 0 o o o o r- CN 1—1

s-? B-? BM! S'? o O CO CM • • • « o v£> CO O o

pH

CN

&-? S'? O m 00

co • • « •

<! o CTs o\ o o 1—1 .H so

•H

CUBO

co

tioa

>>

00

!-i <U 6s? &-? S-2 O CM \—1 • • a • O V£> CO O o

rH

CN

*4Ho

IdO

•H

4J3

rQ

•H

J-4 4-1 CO •HO

S'? S'? &-? S'? o CM o 00 • • • • o ON o O o

1—1

CM

S'? S'? S'? B-? O CM 00 o • • • • O in o o r*- (Ml

;n <U

4-1

a

•H fd CJ cu a)

•rH 4-1 oo H cfl u

4-1 rs cu u CO >

a) 4-1 CO id i—i O o o w dd l-J a

37
Figure

1

Bulk

standby

of

an

electric

water

heater.
39

Figure

2

Nodal

scheme

for

the

conduction

model.
C\J

II N

Oii M

40

Figure

3

Verification

of

the

conduction

model

of

ESDS.
T

(

r,z,fl)

-Too

Ti-Too

Figure 3 Verification of the conduction model of ESDS.

41
I

42
QJ H3Oeso0) Cl

•H

Cl0> -Cuo

4-1

<UB

<u JCu

cn i—i

to TSo

'Z

u~>

<UM3

00

•H

Pli

43
ta an

5“

“ T" _r_

6 1 8 1 1 .

t© an

IH* 1H"

— «/> y> 45 ?\p£j££|^

i

«s $m

MM"s? r*t TM . •"

t M M I

«® an ^ f>3 H I*- h* HSNI «-

Figure

6

Draw

model.

Option
45

Figure

7

Draw

model.

Option
CM

46

Figure

7

Draw

model,

Option
old

T.I

old T

47

Figure 8

Draw

model,

Option
48
TP2 tp 2

TP TP

T 4 Ti

t 3 T3

t 2 t2

Ti T,'

(a) (b)

Figure 10 Heat leak through a stand pipe,

49
0.8 ©

25 i

3

r

18 © > k V

X X x X

1110 ill ©=

<

X X X x V

18 ©

3 V

25 m

0.8 (?)

« 245 > 18 3 53 0.8

Figure 11 Nodal diagram for the basic water heater (water heater A)

Dimensions in mm. (1) water, (2) cement, (3) steel,

(4) insulation. X heating source, V thermostat.

50
ra

51

Figure

12

Nodal

diagram

for

the

pipe

and

its

connection

to

the

water

tank.
// // // // /X-

/

/

/

//

// / 77 ~7 y

/ • ,V. • 9

•

a • , # •

• • V V. V

T

•I;

• •

; ;

-

j

.*• /

• • * • 7

• •

g ° k

>

a

»• /

I 7

• # S

* «

:z\.f » yvr. 22 /

// // // '/7

~

(b)

Figure 12 Nodal diagram for the pipe and its connection

to the water tank.
nia oooi

i L

10 o

(MX)

indNi

U3M0d

o

CO CM

co cm —

cmas —i

co

BBS O'

CM

CO

CO —

in

LO :

IT) CO

CM

J-l 1 .1

CM

CM

CM

CO

>-<Q

I—

ID CO CM

LD

CM

CO

(riAl! A9H3N3 U3JLVM (s/1) 31VUM013

53

Figure

13

A

24-hour

operation

of

Case

A,

in

which

Draw

Schedule

A

is

used
q/nia

Jo

Q <0 &

£3 O O

m o m ®

(Mi 3dld JO

SS01 IV3H 30 31VH

(Do) dlAI31

130nvd 3AV

54

Figure

13

A

24-hour

operation

of

Case

A

in

which

Draw

Schedule

A

is

used.
RATE

OF

HEAT

LOSS

(W)

800

600

400

200

0

55

(BTU/h)
Figure 15 Faucet temperatures of draws. Case A.

56
TIME OF DAY (h)

Figure 16 The rates of heat loss of water heater B.

57

(BTU/h)
Figure 17 Faucet temperatures of dishwashing draws. Cases B and C. Elapsed time for piston flow of hot water to travel to the

faucet = 0.00238 hour.

58
NBS-114A (rev. #-78)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 81-2303

%. Recipient's Ac-cmiort

I11M

4. TITLE AND SUBTITLE

NBS/ESDS - A Computer Model for the

Simulation of Hot Water Systems

5. Publication Date October 1981

$. Performing Organization Code

7. AUTHOR(S) 8. Performing Organ. Report No. —

— J

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, state, ZIP) 13. Type of Report & Period Covered

15. SUPPLEMENTARY NOTES

| | Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or teas factual summary of most significant information. If document includes a significant bibliography or

literature survey, mention it here.) A computer model, entitled NBS/ESDS, has been developed using a finite difference

method to simulate the generation, storage and distribution of a hot water system

commonly found in many residences, commercial facilities or industrial facilities.

The conduction heat equation which governs the heat transfer in the water heater

during all modes of operation, and the heat transfer in the distribution pipe

network during its cooldown periods, is solved. A solution is also obtained for

the pipe flow equation which governs the heat transfer in the distribution pipe

network during the draw periods. Thermal stratification of water is modeled. A

plug flow model is used to simulate the draw of heated water from the water heater,

A parametric study is also conducted on a given hot water system for a given draw

schedule during a 24-hour period. The effects studied include insulating the hot

water heater, insulating the distribution pipe, and incorporating a heat trap.

Effects of concentrating the distributed draws to two short periods of time of

the day on the hot water delivery and the heat loss with the same energy

consumption are also studied.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons)

Computer model, Distribution pipe, Dynamic response. Electricity, Energy, Explicit

method. Faucet temperature. Finite difference, Heat loss. Implicit method,

Insulation, Water heater.

18. AVAILABILITY [J\ Unlimited

I | For Official Distribution. Do Not Release to NTIS

I | Order From Sup. of Doc., U.S. Government Printing Office, Washington, DC

20402, SO Stock No. SN003-003-

pC| Order From National Technical Information Service (NTIS), Springfield, VA. 22161

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF

PRINTED PAGES

63

22. Price

$ 8.00

U S COMM* DC


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