LECTURE 7

Chapter 8: Heat Balance Method

Chapter 10: Flow, Pumps and Piping Design

1

Overview of the heat balance method

2

For exterior surface:

3

For interior surface:

+ =

Correct this in book

Limitations

qconduction,ext,j,q qconduction,in,j,q , unless steady-state heat transfer conditions prevail. This would be unusual for

cooling load calculations.

Both the interior surface and exterior surfaces may radiate

to several surfaces or objects.

4

Heat balance equation

Assuming that the zone air has negligible thermal storage

capacity, a heat balance on the zone air may be

represented conceptually as

5

Transient conduction heat transfer

1. Lumped parameter methods-treating walls and roofs as a small number of discrete resistances and lumped capacitances

2. Numerical methods-finite difference and finite element methods

3. Frequency response methods-analytical solutions requiring periodic boundary conditions

4. Z-transform methods - methods based on Z-transform theory, including response factors and conduction transfer functions.

6

Conduction Transfer Functions Z-transform methods result in one of two formulations, utilizing either

response factors or conduction transfer functions (CTF).

Response factors may be thought of as time series coefficients relating the current heat flux to past and present values of interior and exterior temperatures.

Conduction transfer functions replace much of the required temperature history with heat flux history. In other words, many of the response factors are replaced with coefficients that multiply past values of heat flux.

While the determination of CTF coefficients is relatively complex, their use is relatively straightforward.

CTF coefficients multiply: (i) present values of interior and exterior surface temperatures, (ii) past values of interior and exterior surface temperatures, and (iii) past values of surface heat flux.

7

The heat flux at the jth surface for time q is:

8

Ex. 8-4

8-4. Determine the wall conduction transfer function

coefficients for a wall composed of 4 in. brick [k = 7 (Btu-

in.)/(hr-ft2-F)], 1 in. regular density sheathing (vegetable

fiber board), 3 in. mineral fiber insulation (R-13), and 1

in. gypsum board.

After running the HvacLoadExplorer program in execute

for room mode, we obtain the following CTF coefficients

for this wall

nXn,

Btu/(hr-ft2-F)Yn,

Btu/(hr-ft2-F)Zn,

Btu/(hr-ft2-F) n

0 4.276507 0.000445 0.642344

1 -5.36497 0.011581 -0.98287 0.638772

2 1.141149 0.011845 0.376555 -0.02179

3 -0.02759 0.001134 -0.01101

4 -7.7E-05 0.000017 -5E-06

Ex. 8-12 For the wall described in Problem 8-4, with an outside

surface temperature profile given by Table 8-5 and a

constant inside surface temperature of 70 F, determine

the inside conduction heat flux for each hour.

11

Sol: To start the calculation, we must

assume something about the past values of

the heat flux.

We will assume that prior to the first day of

the calculation, the heat flux was zero.

For the second day of the calculation, we

will use the values from the first day, and so

on until we reach a converged steady

periodic solution.

Hour Day 1 Day 2 Day 3

1 0.312 1.126 1.126

2 0.463 0.954 0.954

3 0.508 0.804 0.804

4 0.494 0.673 0.673

5 0.454 0.561 0.561

6 0.41 0.475 0.475

7 0.387 0.426 0.426

8 0.402 0.426 0.426

9 0.473 0.487 0.487

10 0.612 0.62 0.62

11 0.82 0.825 0.825

12 1.089 1.092 1.092

13 1.399 1.401 1.401

14 1.715 1.716 1.716

15 1.998 1.999 1.999

16 2.223 2.223 2.223

17 2.362 2.362 2.362

18 2.405 2.405 2.405

19 2.352 2.353 2.353

20 2.215 2.215 2.215

21 2.016 2.016 2.016

22 1.786 1.786 1.786

23 1.551 1.551 1.551

24 1.327 1.327 1.327

12

Outside surface heat balance-opaque surfaces

1. Absorbed Solar Heat Gain

2. Exterior Convection

3. Exterior Radiation

13

Absorbed solar heat gain

Gt is calculated with the ASHRAE Clear Sky Model described in Chapter 7.

Since Gt must be calculated for a specific time, yet represent the entire hour, it is usually calculated at the half hour.

14

Exterior convection

15

For high-rise buildings,

The correlation was based on wind-speeds between 0.5 mph (0.2 m/s) and 9 mph (4 m/s).

Loveday and Taki do not make a recommendation for wind-speeds below 0.5 mph (0.2 m/s), but a minimum convection coefficient of 1.3 Btu/(hr-ft2-F) or 7.5 W/(m2-K) might be inferred from their measurements.

16

Exterior radiation

17

where

Simplified equation for exterior radiation

18

For horizontal surfaces, a = 00, therefore, Fs-g = 0; Fs-sky = 1

For vertical surfaces, a = 900, therefore, Fs-g = Fs-sky = 0.5

For surfaces that are not horizontal, the effective sky temperature will be affected by the path length through the atmosphere:

The initial value of tsky is usually taken as 10.8 F below the outdoor temperature according to the Blast model.

19

Exterior surface heat balance formulation

20

Ex 8-9

A wall has an incident solar radiation of 300 Btu/(hr-ft2), an

outside air temperature of 98 F, and an outside wind speed

of 15 mph. The wall has a solar absorptivity of 0.6, a

thermal emissivity of 0.9, negligible thermal mass, an

outside-surface-to-inside-surface U-factor of 0.1 Btu/(hr-ft2-

F), and an inside surface temperature of 72 F. Determine

the conduction heat flux.

21

22

Interior surface heat balance - opaque surfaces

1. Interior convection

2. Interior surface-to-surface radiation

3. Internal heat gains due to people, lights, etc.

23

Interior convection Interior convection heat transfer in rooms occurs under a

wide range of conditions that may result in natural

convection, mixed convection, and forced convection.

The air flow may be laminar or turbulent.

24

Interior surface-to-surface radiation

For each surface in the room, the model represents all of the other surfaces as a single fictitious surface with a representative area, representative emissivity, and representative temperature [mean radiant temperature (MRT)] that will be seen by surface.

The area of the fictitious surface that exchanges radiation with the jth surface in the room is the sum of the other areas of the other surfaces:

25

Emissivity and temperature of fictitious surface

ef,j of the fictitious surface is an area-weighted average of

each individual surface ei, but not including the jth surface

26

The temperature is an area-emissivity-weighted temperature

The radiation between the interior surface and its

corresponding fictitious surface is analyzed based on

fundamental principles, although the area, emissivity,

temperature, and view factor of the fictitious surface are

approximated.

A radiation interchange factor of the fictitious surface is

approximated as:

View factor of fictitious surface

Radiation coefficient is given as:

where tj and tf,j are given in absolute temperature, R or K,

and tj,avg is the average of tj and tf,j.

The net radiation leaving each surface for the other room

surfaces is then given by

28

Radiation coefficient and net radiation

If a check is made once the net radiation leaving each surface

has been calculated, some imbalance will be found, due to the

approximations made in the method.

Rather than leaving a net imbalance in the radiation, it is

preferable to make a correction, adjusting the radiative heat

flux on each surface slightly using a balancing factor:

Balanced radiation leaving each surface is then given by:

29

Balanced radiation

Internal heat gains - radiation After internal heat gains from people, lights, and

equipment are determined for a given hour, radiative

portions of the heat gains are distributed uniformly on the

interior surfaces:

30

Interior surface heat balance formulation For any given hour, past values of interior surface temperature

and conduction heat flux will be known.

A history term that contains all of the historical terms for the interior CTF equation should be defined:

31

Ex. 8-19 The attic space shown in Fig. 8-11 has H = 6ft, W = 28ft,

and L = 42ft, and all interior surfaces have emissivities of

0.9. For a time when the inside surface temperatures are

t1 = 122 F, t2 = 143 F, t3 = 102 F, t4 = 92 F, and t5 = 95 F,

estimate the net thermal radiation incident on each

surface using the MRT/balance method.

32

Sol.

Surface

No.

Surface

Orientation

Ff Tavg (R) hr,f

Btu/(hr-ft2-F)

qrad-surf-in,j, theta

Btu/(hr-ft2)

1 North roof 0.872 576.0 1.14 12.97

2 South roof 0.872 583.1 1.19 46.37

3 West wall 0.897 567.6 1.12 13.26

4 East wall 0.897 562.7 1.10 24.25

5 Attic floor 0.832 571.4 1.06 35.53

33

FLOW, PUMPS, AND

PIPING DESIGN

34

Chapter 10

Fluid flow basics The adiabatic, steady flow of a fluid in a pipe or conduit is governed

by the first law of thermodynamics, which leads to the equation

The sign convention is such that work done on the fluid is negative.

35

Mass flow rate and Continuity Equation

Conservation of mass:

36

Other forms of Bernoullis equation

37

Total pressure

Total pressure is the sum of the static pressure and velocity

pressure

In terms of head,

Energy equation is written in terms of total head as below:

This form of the equation is much simpler to use with gases

because the term z1z2 is negligible, and when no fan is in the system, the lost head equals the loss in total pressure head.

38

Lost head For incompressible flow in pipes and ducts, lost head is expressed as:

39

Reynolds number

Laminar flow occurs when Re < 2300;

Turbulent flow occurs when Re > 4000;

In the interval between 2300 and 4000, laminar and turbulent flows are

possible ('transition' flows), depending on factors such as pipe roughness

and flow uniformity).

For conduits of non-circular cross-sections, hydraulic diameter Dh is:

40

Absolute roughness e

Moodys friction factor is a function of the Reynolds number (Re) and the relative roughness e/D of the conduit in the transition zone

It is a function of only the Reynolds number for laminar flow, and

It is a function of only relative roughness in full turbulence zone.

41

Friction factors for pipe flow

42

Co

rrect

err

or

It s

ho

uld

be 0

.0004