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NBS TECHNICAL NOTE 1135

NA^L INST OF STAND & TECH

A111D7 EDIBLE

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

Cooling of Bitumen During Construction

of Built-Up Roofing Systems—

A IVIathematical Model

QC

100

.U5753

NO. 1135

1981

c. 2
NATIONAL BUREAU OF STANDARDS

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)

National Bureau of Standards

Library, E-01 Admin. Bidg.

Cooling of Bitumen During Construction ^^^ 3 o 1981

of Built-Up Roofing Systems-

"^Qaoo"" ^'^'

A Mathematical Model ^ ^?

^ no. iizT

Walter J. Rossiter, Jr.' ^' ^' Robert G. Mathey'

Herbert W. Busching^

William C. Cullen^

' Center for Building Technology

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

^ Department of Civil Engineering

Clemson University

Clemson, SC 29613

' Office of Engineering Standards

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

*''»eAo o? V

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

U. : NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued March 1981
National Bureau of Standards Technical Note 1135 1

Nat. Bur. Stand. (U.S.), Tech. Note 1 135, 76 pages (Mar. 1981) CODEN: NBTNAE

!

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1981

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D C 20402

Price (Add 25 percent for other than U.S. mailing)
TABLE OF CONTENTS

Page

LIST OF FIGURES iv

LIST OF TABLES vi

ABSTRACT vll

1

.

INTRODUCTION 1

1.1 Background 2

1.2 Objective 4

2. COOLING OF BITUMEN 4

2.1 Factors Affecting Cooling 4

2.2 Model for Predicting Cooling Time 5

2.2.1 Assumptions In the Development of the Model 5

2.2.2 Finite-Difference Heat Flow Equations 6

3. APPLICATION OF MODEL FOR PREDICTING BITUMEN COOLING 11

3.1 Values Needed to Solve Model 11

3.2 Results for Specific Roofing Systems 13

• 3.3 Uses for Model 16

4. SUMMARY 17

5. ACKNOWLEDGMENTS 18

6. REFERENCES 18

APPENDIX A. LISTING OF FINITE-DIFFERENCE EQUATIONS Al

A. 1 Finite-Difference Equations for the Four-Component System .... A3

A. 2 Finite-Difference Equations for the Three-Component System ... A4

A. 3 Finite-Difference Equations for the Two-Component System A5

APPENDIX B. LISTING OF COMPUTER PROGRAM Bl

B.l Check on Computer Program Execution B13

APPENDIX C. NOTATION CI

111
LIST OF FIGURES

Page

Figure 1. Modes of Heat Transfer During Construction of Bituminous

Built-Up Roofing Systems. The Effect of Solar Radiation

was not Included in the Mathematical Model. Heat Transfer

due to Convection and Radiation from the Underside of Thin

Decks Without Insulation was Included in the Model 26

Figure 2. Diagram of the Elements Within the Bitumen and Top Elements

of the Roofing Component Below the Bitumen 27

Figure 3. Diagram of the Elements and Nodes Used to Develop the

Finite-Difference Equations for Heat Flow Within the

Bitumen, at the Surface of the Bitumen, and at the Bottom

Boundary of the Bitumen 28

Figure 4. Relationship Between Bitumen Temperature and Cooling Time

for the Two-Component Model, Asphalt Applied on Polyurethane Jjj

Foam Insulation. The Quantity of Applied Bitumen is iH

Approximately 20 lbm/100 ft^ (0.98 kg/m^) 29

Figure 5. Relationship Between Bitumen Temperature and Cooling Time

for the Two-Component Model, Asphalt Applied on Fiber Glass

Insulation. The Quantity of Applied Bitumen is Approximately 20 lbm/100 ft^ (0.98 kg/m^) 30

Figure 6. Relationship Between Bitumen Temperature and Cooling Time

for the Two-Component Model, Asphalt Applied on Insulating

Concrete. The Quantity of Applied Bitumen is Approximately

20 lbm/100 ft^ (0.98 kg/m^) 31

Figure 7 . Relationship Between Bitumen Temperature and Cooling Time

for the Two-Component Model, Asphalt Applied on Plywood.

The Quantity of Applied Bitumen is Approximately

20 lbm/100 ft^ (0.98 kg/m^) 32

Figure 8. Relationship Between Bitumen Temperature and Cooling Time

for the Two-Component Model, Asphalt Applied on Concrete.

The Quantity of Applied Bitumen is Approximately

20 lbm/100 ft^ (0.98 kg/m^) 33

Figure 9. Relationship Between Bitumen Temperature and Cooling Time

for the Two-Component Model, Asphalt Applied on Steel. The

Quantity of Applied Bitumen is Approximately 20 lbm/100 ft

(0.98 kg/m^) 34

iv
LIST OF FIGURES (Continued)

Page

Figure 10. Relationship Between Bitumen Temperature and Cooling

Time for the Two-Component Model, Coal Tar Pitch Applied

on Concrete. The Quantity of Applied Bitumen is

Approximately 25 lbm/100 ft^ (1.2 kg/m^) 35

Figure 11. Relationship Between Bitumen Temperature and Cooling Time

for the Three-Component Model, Asphalt Applied on Felt over Polyurethane Foam Insulation. The Quantity of Applied

Bitumen is Approximately 20 lbm/100 ft^ (0.98 kg/m^) 36

Figure 12. Relationship Between Bitumen Temperature and Cooling Time

for the Three-Component Model, Asphalt Applied on Felt over

Plywood. The Quantity of Applied Bitumen is Approximately

20 lbm/100 ft^ (0.98 kg/m^) 37

Figure 13. Relationship Between Bitumen Temperature and Cooling Time

for the Four-Component Model, Asphalt Applied on Felt over Asphalt on Polyurethane Foam Insulation. The Quantity of

Applied Bitumen is Approximately 20 lbm/100 ft^ (0.98 kg/m^) 38

Figure 14. Relationship Between Bitumen Temperature and Cooling Time

for the Four-Component Model, Asphalt Applied on Felt over Asphalt on Concrete. The Quantity of Applied Bitumen is

Approximately 20 lbm/100 ft^ (0.98 kg/m'^) 39

Figure 15. Relationship Between Bitumen Temperature and Cooling Time

for the Four-Component Model, Coal Tar Pitch Applied on Felt

over Coal Tar Pitch on Fiber Glass Insulation. The Quantity

of Applied Bitumen is Approximately 25 lbm/100 ft

(1.2 kg/m^) 40

Figure 16. Effect of Asphalt Thickness on the Cooling Time for the

Three-Component Model, Asphalt Applied on Felt over Polyurethane Foam Insulation 41

Figure 17. Effect of Substrate on the Asphalt Cooling Time 42

Figure 18. Temperature Profiles of the Three-Component Roofing Systems,

Asphalt Applied to Felt on Plywood and Asphalt Applied to

Felt on Polyurethane Foam Insulation, for Various Times

after Bitumen Application. Asphalt at 500°F (260°C) was Applied to the Felt and Substrate at 120°F (49°C) with Zero

Wind Speed 43

Figure 19. Plot of Dimensionless Temperature Versus Time for the

Two-Component Roofing System, Asphalt Applied on Steel 44
LIST OF FIGURES (Continued)

Page

Figure Al. Diagram of the Finite-Elements Within the Components

of the Roofing System Used for the Calculation of

Bitumen Cooling Time by Finite-Difference Techniques A2

LIST OF TABLES

Table 1. Thermal Properties of the Roofing Component Materials for

Use in the Model 21

Table 2. Values of Roofing Component Temperatures and Environmental

Factors Used in the Calculations 22

Table 3. Thickness of the Roofing Component Materials and

Corresponding Number of Elements Used in the Calculations ... 23

Table 4. Configurations of Roofing Components for Which the Cooling

Time of the Applied Bitumen Was Calculated 24

Table 5. Heat Capacities of Various Substrates and Cooling Times for

Asphalt When Applied on Them 23

Table Bl. Temperature Distribution in the Finite Elements of the

Flat Plate as Determined from the Computer Program

and the Temperature Charts B14

Table CI. Notation and Description of Terms CI

vi
COOLING OF BITUMEN DURING

CONSTRUCTION OF BUILT-UP ROOFING

SYSTEMS - A MATHEMATICAL MODEL

by

Walter J. Rossiter, Jr.

Robert G. Mathey

Herbert W. Busching

William C. Cullen

ABSTRACT

Construction of bituminous built-up roofing systems in the United States

generally involves the application of hot bitumen to the roofing components,

including deck, insulation, and felts, to adhere them to each other and to

form a waterproof membrane. Adequate adhesion of the bitumen to the roofing

component materials may be obtained only when the hot bitumen is applied at

a viscosity sufficient to flow uniformly, to cover the component surfaces or substrate completely and to provide the proper thickness. During construction, rapid cooling of hot bitumen increases its viscosity significantly. If

the viscosity becomes too high, poor adhesion between components, voids within

the bitumen, and an excessive and non-uniform thickness of the bitumen may

result.

This report describes a mathematical model based on finite-difference equations for calculating transient heat flow to estimate the cooling time of

hot roofing bitumen. Estimates of the time required for hot bitumen to cool

from its application temperature to SOCF (149*C) were computed as a function

of material and environmental factors, including: quantity of applied bitumen,

bitumen application or contact temperature, air temperature, wind speed, and

thermal properties of the bitumen and of the roofing components. The model

was used to predict cooling times expected for hot asphalt applied to typical

substrates with thermal properties representative of those of polyurethane

foam and glass fiber insulation boards, insulating concrete, plywood, concrete

and steel decks, and roofing felt on decks or insulations. In addition, the

model was used to predict cooling times for hot coal tar pitch applied to

concrete and to felt adhered to glass fiber insulation. The results of the

calculations demonstrate the widely varying bitumen cooling times which

depend upon the component material to which the bitumen is applied and

the environmental conditions during application. Under certain environmental

conditions, hot bitumen applied to some substrates cools extremely rapidly.

In these cases, sufficient time for proper application may not be available.

Key words: Bitumen application temperatures; bituminous roofing; builtup roofing; cooling of roofing bitumens; mathematical model;

roofing.

vli

1. INTRODUCTION

The most widely accepted waterproofing system for low-sloped roofs in the

United States is bituminous built-up roofing. Construction of bituminous

built-up roofing systems generally involves the application of hot bitumen

to the roofing components, including deck, insulation, and felts, to adhere

them to each other and to form a waterproof membrane* Bitumens commonly

used as waterproofing materials and adhesives in Imilt-up membranes are

asphalt and coal tar pitch.

Roofing bitumens must be applied at elevated teiiq>eratures so that their

viscosities will be sufficiently low to enable uniform flow, complete coverage of component surfaces, and proper thickness. After hot bitumen is applied

to the substrate, it cools rapidly. Little time is available to place

roofing felts or insulation on the hot bitumen while its temperature is

sufficiently high and its viscosity is sufficiently low to provide for adequate adhesion and proper thickness. Laboratory data are available on the

relationship between temperature and viscosity of roofing asphalts [1,2].

However, little data were found in the literature on the cooling time of hot

bitumen during built-up roof construction. A field study has been reported

wherein the cooling time of asphalt during cold weather construction was determined [3].

Since the rate at which roofing bitumens cool is important to the quality

of built-up roofing construction, a mathematical model was developed to

enable quantitative assessment of the effects of various factors on the

cooling time of roofing bitumens. The mathematical model is based on wellknown heat transfer principles, uses finite-difference equations [4-6], and

makes use of representative roofing materials and their properties. It also

incorporates the principal environmental variables associated with roof construction. The mathematical model to predict the cooling time of bitumen

assumes one dimensional heat transfer; that is, the roof is taken to be a flat plate with its components at uniform temperature across the horizontal

planes of the plate. During built-up roofing construction, hot bitumen is

applied over large areas of the roof. Because large areas of the roof are

considered to be at uniform temperature under these conditions, the heat

transfer was assumed to occur in the direction perpendicular to the plane of

the roof and not laterally within its plane.

Cooling time of bitumen can in principle be determined experimentally by

means of field studies. However, measurement of cooling times applicable

to many different types of roofing systems under a variety of environmental

conditions are precluded in practice because of the excessive time and

expense required to conduct investigations involving many variables. Nevertheless, selective field studies to obtain data on bitumen cooling times are needed to assess the validity of the model presented.

* Figures in brackets indicate references listed in Section 6.

1
1.1 BACKGROUND

Rosslter and Mathey [1] measured the viscosities of ASTM Type I and Type III

roofing asphalts from different sources over a range of application temperatures. They recommended that roofing asphalts be applied at temperatures

based on viscosity rather than on empirically determined temperature limits.

From a similar investigation, Ducy recommended that roofing asphalts be

applied based on an equiviscous temperature (EVT) [2]. It was suggested

by Rossiter and Mathey [1] that in order to maintain the recommended viscosity

range during application, consideration should be given to factors affecting

the rate of cooling such as air temperature, wind speed, substrate temperature, and the specific heat of the substrate. A viscosity criterion for the

application of asphalt has not generally been used in roofing construction

nor included in specifications. It is noted, for example, that the National

Roofing Contractors Association (NRCA) [7] recommends a minimum application

temperature of 350°F (HT'C) and a maximum heating temperature of 475*F (246**C)

for asphalt. Likewise, the Tri-Service Roofing Manual [8] states that asphalt

should not be heated above ASCF (232**C) and should not normally be lower

than 350°F (lyT'C) when applied to the roof. Current trends in the roofing

industry indicate an acceptance of the selection of application temperatures

based upon the viscosity of the bitumen.

During application, if bitumen cools too rapidly with an acconq)anying

Increase in its viscosity, the bitumen layers may be too thick, uneven, con- tain voids, and result in inadequate adhesion between roof components. These

effects result in unsatisfactory built-up bituminous membrane performance.

A thick or uneven application of interply bitumen can lead to poor adhesion

between plies [1]. Voids between plies or between the membrane and insulation

are sources of blistering [9]. Excessive thickness of between-ply bitumen

under certain conditions may contribute to membrane slippage [10]. Since the

bitumen has a larger coefficient of thermal expansion than other membrane

materials, excessive thickness of bitumen between plies may result in higher

thermally induced forces in the membrane. Lee, Dupuis, and Johnson [11] showed

that temperature-induced forces measured for 2-ply membranes increased as the

quantity of interply asphalt was increased. Temperature-induced forces due

to a temperature change from 70 to -20**F (21 to -29*'C) at an unspecified

rate, measured for "thin, normal, and thick" interply thicknesses of asphalt,

were about 54, 55, and 80 Ibf/in (9.4, 9.6, and 14.0 kN/m), respectively [11].

In a field investigation of cooling times of asphalt applied during cold

weather, Dupuis, Lee, and Johnson [3] measured temperatures of hot asphalt in

kettles and the contact tenq>eratures (temperatures at the time the asphalts

came in contact with the substrates). They reported that during asphalt

application, peak contact ten5)eratures* were usually 50 - 75°F (28 - 42*C)

lower than the temperature in the asphalt kettle. Asphalt cooling times

* Peak contact temperature was defined as the highest temperature recorded

during asphalt application [3].
were recorded at various locations within Insulated roofing systems under

construction In four cities In Wisconsin with air temperatures ranging

from about 20 to 40*F (-7 to 4**C) and wind speeds from about 3 to 14 mph

(1.4 to 6.3 m/s). Under those weather conditions, they found that the

asphalt cooled rapidly. For example, In the application of asphalt to

Insulation, the asphalt temperatures dropped from the contact teoq)erature

to less than 300*F (149*C) In about 2 to 7 seconds.

The rapid cooling of asphalt observed by Dupuls, Lee, and Johnson [3] raised

a question about adhesion of roofing components. They reported that rapid

temperature decay of hot asphalt makes good adhesion extremely difficult to

attain. They also stated that hot asphalt applied at subfreezlng temperatures to light gauge steel decks congeals so rapidly that mechanical fastening Is probably the only effective technique for securely anchoring the

Insulation. With regard to membrane application, they Indicated that raising

peak asphalt contact temperatures during cold weather may Increase the time

the asphalt remains sufficiently hot to adhere a ply of felt adequately.

In presenting the rate of heat loss (time dependency) of roofing asphalts,

Dupuls, Lee, and Johnson [3] expressed the temperature of the asphalt with

time as:

T = Tg + (T^ - Tg) e

-ICT

where T = the asphalt temperature at time t Tg = the temperature of the surroundings

Tq = the Initial temperature of the asphalt at application

ic = an empirical constant

It Is noted that this model was empirical and did not Include terms for all

the factors which affect the cooling time of asphalt. The cooling data

obtained by Dupuls, Lee, and Johnson have not been compared with data

generated by the model presented In this paper.

The roofing Industry In the United States has not employed mathematical

models to predict the cooling time of hot bitumen during construction of

built-up roofing systems. A mathematical model would benefit the roofing

industry since it would provide guidelines for the proper application of

hot bitumen without excessive cooling on various components under a range

of environmental conditions. Mathematical models were developed by Dickson

and Corlew [12,13] for predicting temperatures of hot asphaltlc concrete

pavement layers placed in cold weather. These layers must be compacted

while the viscosity of the asphalt is low enough to enable coiiq>actlon of a pavement of proper density. Dickson and Corlew [12,13] used established

finite-difference equations as a basis for evaluating teiiq)erature profiles

in the hot asphaltlc concrete pavement layers. Several references [14-18]

identify the rationale for preheating the base to permit cold weather highway and airfield paving. The techniques used in estimating teiiq>eratures

of paving mixtures are adaptable, with some modification, for use in estimating the cooling times of bitumen during the application of built-up

roofing systems.
1.2 OBJECTIVE

The objective of this study was to develop a mathematical model for

predicting the cooling time or time interval for hot bitumen to cool from

its application temperature to a specified lower temperature during construction of built-up roofing systems. This paper presents examples of temperature decay charts generated from the model to enable the prediction of

bitumen cooling times for various environmental conditions and for some roofing systems typical of those encountered in built-up roofing construction. The computer program based on the model is given to make it available

to those individuals who wish to calculate bitumen cooling times for built-up

roofing systems other than those presented here. Busching [19] has presented

some early results from this study in a paper concerning the effects of

moisture and temperature on roofing membranes.

Data developed through the use of the model may be useful in applying

viscosity criteria and specifications effectively. In addition, information

on the time of cooling for a wide range of variables may be used to help

assure proper application of hot bitumen and other roofing conponents to

improve construction practice. Data may be obtained from a computerized

model at much lower cost than would be encountered if the same data were

obtained through field experiment. However, before full reliance is placed

on the results of the model calculations, data should be obtained as recommended in section 1 through selective field studies to assess the validity

of the model.

2. COOLING OF BITUMEN

Hot bitumen is normally applied to roofs by mopping or by mechanical

application. The method for applying hot bitumen was not a consideration

in the development of the model to predict cooling times of bitumen. The

model assumes that the temperature of the hot bitumen at the time of contact

with the surface to which it is applied is known. The hot bitumen begins

to cool immediately upon contact with the surface. The cooling time is

dependent on factors associated with environmental conditions and materials

properties.

2.1 FACTORS AFFECTING COOLING

Many factors affect the cooling of hot bitumen during roof construction.

These factors include application temperature and thermal properties of the

bitumen, quantity of bitumen applied per unit area (spread quantity), ambient

air temperature, wind speed on the roof, and temperature, thickness and

thermal properties of the component materials below the hot bitumen.

Thermal properties of the bitumen, substrate, and other component materials

which affect the cooling are thermal conductivity, specific heat, and density.

The application temperature of the hot bitumen is defined as the temperature

at which the bitumen makes contact with the surface to which it is applied.

The factors which affect cooling were incorporated in the mathematical model
for predicting the cooling time of hot bitumen occurring in the construction

of built-up roofing systems.

Most individuals experienced with roofing construction would attest that

many of the factors listed previously affect the cooling of hot bitumen.

However, the effect of these factors on bitumen cooling is only understood

qualitatively. For example, a decrease in either the ambient air or substrate temperature will result in a decrease in the cooling time. The

time of cooling will also decrease with an increase in wind speed. In

addition, an increase in either the application temperature or quantity of

hot bitumen will increase the time available to apply component roofing

materials before the viscosity of the bitumen becomes too high to enable

good adhesion. A mathematical model to predict cooling time of bitumen

would quantify the effect of these factors.

The effect of the thermal properties (thermal conductivity, specific heat,

and density) of the substrate and component materials on the cooling of

hot bitumen may not be as evident to those experienced with roofing con- struction as for the environmental factors. Calculations using the mathematical model also allow quantitative determination of the effect of these

thermal properties on bitumen cooling. Definitions of these thermal

properties and related parameters are as follows:

o Thermal conductivity (k) - Time rate of heat flow through a homogeneous

material under steady-state conditions through unit area, per unit

temperature gradient. Btu*in/h»ft '"F (W/m*K)

o Specific heat (C ) - The quantity of heat required for a one degree

temperature change in a unit mass of material. Btu/lbm***F (J/kg*K)

o Density (p ) - The mass of a material per unit volume. Ibm/ft (kg/m )

o Heat capacity (pC ) - The quantity of heat required for a one degree

temperative change in a unit volume of material. Btu/ft '^F (J/m ^K)

o Thermal diffusivlty (a) - The ratio of a material's capacity to conduct

thermal energy to its capacity to store thermal energy, ft /h (m /s)

Thermal diffusivlty is defined by the relationship:

a = k

PCp

2.2 MODEL FOR PREDICTING COOLING TIME

2.2.1 Assumptions in the Development of the Model

Assumptions made in the development of the mathematical model using

finite-difference equations for predicting cooling time of hot bitumen are

as follows:
o heat transfer from the hot bitumen Is one-dlmenslonal perpendicular to

to the plane of the roof;

o the values of the properties of the bitumen and other roofing system

components remain constant with time as the bitumen cools;

o the thickness of the bitumen does not change as It cools;

o absorbed solar radiation during bitumen application Is negligible and

neglected;

o heat Is lost from the bottom of the substrate by radiation and

convection unless the substrate contains more than 100 elements;

o the elements for each of the roofing system components have the same thickness except for the half elements at the top and bottom surfaces

of the system;

o the temperature of the bitumen was taken as the average of the temperature

of all elements In the bitumen component;

o the temperature of the air and sky are taken to be the same; and

o the thermal resistance at the Interfaces of the roofing system components

Is taken to be zero.

2.2.2 Finite-Difference Heat Flow Equations

2.2.2.1 Description of the System

During the construction of built-up roofing systems, hot bitumen Is applied

to cooler roofing materials and components such as decks, Insulations and

felts. Figure 1 shows schematically the modes of heat transfer that are considered In the analysis. Heat Is lost from the hot bitumen to the cooler

surroundings by convection and radiation at the top surface, and by conduction at the bottom surface which Interfaces with the roofing component to

which the bitumen Is applied. In addition, as heat Is lost from the surfaces

of the bitumen layer, heat Is transfered by conduction from the Interior of

the bitumen layer towards the cooler surfaces.

As shown In figure 1, heat flow from the hot bitumen Is directed up from

Its top surface and down Into the substrate. The heat flow from the upper

surface of the bitumen Is dependent on environmental conditions at the

surface and on the temperature of the bitumen. Heat flow from the lower

surface of the hot bitumen Is dependent upon the temperature and thermal

properties of the substrate as well as the temperature of the bitumen.

Radiant energy from the sun may also be Incident on the roof and partially

absorbed by the surface of the bitumen during built-up roofing construction.

The quantity of radiant energy Is negligible during bitumen application
because of the relatively short time (usually a few seconds) during which

the hot bitumen is e3q>osed to solar radiation prior to being covered with

insulation board or ply of felt.

The mathematical model to predict the cooling time of hot bitumen uses well-known finite-difference equations to calculate heat transfer. The

treatment presented is based in general on that described by Dusinberre [5,6].

The calculations incorporated one-dimensional balanced heat flow between

discrete nodes at specific locations within the hot bitumen. One-dimensional

heat flow equations were developed that included terms which account for heat

transfer by conduction, convection, and radiation depending upon the location

of the discrete nodes selected. Figure 2 presents a diagram of the hot bitumen

layer divided into elements for development of the finite-difference equations.

The diagram in figure 2 shows the location of the nodes, the distance between

nodes, the thickness of the elements, and the notation used to describe the

temperature of the elements. Finite-difference equations were developed for

three types of elements: the elements within the bitumen and the two boundary

elements of the top and bottom surfaces. The development of these

finite-difference equations is given in the sections which follow.

2.2.2.2 Heat Flow Within the Bitumen

Heat transfer by conduction within a material can be described by the partial

differential equation for one-dimensional heat flow: —3T 3^T

= a—

(1)

at 3y^

where a is the thermal diffusivity of the material.

As applied to the cooling of hot bitumen, this equation describes the

relationship between bitumen temperature, T^, time, t, thermal diffusivity

of the bitumen, o. , and a position coordinate, y. The finite-difference

form of equation (1) for the cooling of bitumen can be expressed as:

''^bi "

''^bi « o <T^^(i-l) ~ Tbj) + (Tb(i+1) - Tfej) (2)

At Ay2

where T^j/^_j^\, T^j^, and Tvq+^x are temperatures in adjacent bitumen elements

having a thickness of Ay (figure 2). The terms T^^^ and 1-^^ are the temperatures in bitumen element 1 at times t + At, and t, respectively. Substituting

*lj =^b^^pb'^b ^^^o equation (2) and rearranging* the equation gives:

H<Tb(i-l) - Tbi) ^ ^b(Tb(i+l) - Tbj) _ CpbPb(^y)(Tbi - Tbi)

Ay Ay At ^^^

where subscript b refers to bitumen.
Equation (3) can also be Interpreted as the one-dimensional heat flow balance

between three adjacent nodes whereby heat flows from the two outer nodes Into

the center node, as shown in figure 3a. The sum of the heat flow into node 1

from node i+1 and node i-1 must equal the energy stored in node i during the

time. At, of heat flow. The heat flow per unit area into node i from either

outer node is the temperature difference between nodes (AT. ) divided by the

thermal resistance between nodes (K^ = Ay/k^). The energy stored in node i

during the time of heat flow (At) equals the heat capacity (Cp|jp|j) times

the volume of the element of node i (Ay*l) times the increase in temperature

of node i (T^^^ - T^^),

Equation (3) may be rearranged to yield equation (4) and then equation (5):

(Tb(i-l) - T^i) + (T^d+l) - Tbi) = CpbPb(^y)^ (T£i - T^O <^)

k^At

(Tb(i-l) - T^i) + (Tb(i+i) - T^i) = Mb(Tbi - Tt,i) (5)

2 The term Mv = C jjp^(Ay) /^uAt contains thermal, geometric and time

parameters. In order to provide a stable solution using the finite-difference

method for conduction, values of the modulus, M^, must be chosen to be equal

or greater than 2 [5,6]. The requirement, M^^ > 2, assures that in the heat

flow calculation, the second law of thermodynamics is satisfied whereby heat

flow is from a warm to a cold node. Because of this restriction on Mu,

values of Ay and At can not be selected independently. Solving for T.^

in equation (5) yields the equation to calculate the future temperature of

the bitumen interior element after heat flow during the time. At:

Tbi = Tfci +Y ^^b(i-l) - 2Tbi + T^(i+1)) (6)

b

2.2.2.3 Heat Flow at the Surface Boundary of the Bitumen

The finite-difference equation describing heat flow per unit area at the

surface boundary of the bitumen is obtained by performing a heat balance on the surface node 1 (figure 3b). It is noted that the thickness of the

first element is only one half that of the other bitumen elements. Consequently the volume of this element is equal to (Ay/2).l. For a heat balance

at node 1 (figure 3b), the heat flow from node a to node 1 and the heat flow

from node 2 to node 1 must equal the thermal energy stored at node 1. The

heat flow from node a to node 1 is expressed as the product of the bitumen

surface coefficient of heat transfer (hu) and the temperature difference between

the air and the surface element (T^^ - Tbi)* Heat flow per unit area from

node 2 to node 1 is the temperature difference of the nodes (T^2 ~ ^bl^

divided by the thermal resistance (R^j = Ay/k^^). The expression for stored

thermal energy at node 1 in Figure 3b considers that the volume of the element containing node 1 is only half that of the element containing node i

(figure 3a). The equation for heat balance at the surface boundary is

expressed as follows:
Ay

Equation (7) may be rearranged:

-^ (8) (Ta - Tbi) + (Tb2 T^i) - ? ^^^^

Nb(Ta - Tbl) + (Tb2 " ^bl) = ^

(T^l " Tfei) (9)

The term, N^ = hb^y/kb, defines a Biot or Nusselt number. The term, H^^ has

been previously defined and must be chosen such that Mb _^ (^N^^ + 2) in order

for equation (9) to proAd.de a stable solution and to satisfy the thermodynamic

requirements concerning the direction of heat flow [5,6]. Rearrangement of

equation (9) results in the expression to determine the future temperature

(Tui) of node 1 after heat has flowed during the time. At:

bl =Tbi[l.O -^(Nb + 1.0) + ^ (Tb2 + NbT^) (10)

b

For this study, the surface coefficient of heat transfer, h, is taken as:

h = h^ + h^

where h^, = convection coefficient Btu/h»ft • °F (W/m 'K)

hj. = radiation coefficient = ae(Tg + Tgj^y)(Tg + Tg^^ )

where a = Stefan-Boltzman constant, Btu/h»f t^» "R (W/m^»K^)

e = emittance of the surface, dimensionless

Tg = temperature of the surface, **R (K)

^sky ~ ^^y temperature, °R (K); for purposes of this model, the sky

temperature was taken to be the same as the ambient air

temperature.

In the application of the model, the convection coefficient, h , was maintained constant for each set of given conditions and changed only to

account for the effect of different wind speeds on the bitumen cooling. The

radiation coefficient, h^., was changed at each time increment to account for

heat radiation which is nonlinear ly dependent upon surface tenqierature. It

is noted that the value of the convection coefficient, h , may be difficult

to approximate. The surface coefficient may vary according to the rate of

air movement across the surface, the surface length, and roughness.

2.2.2.4 Heat Flow at the Bottom (Interface) Boundary of the Bitumen

The finite-difference equation for heat flow at the bottom boundary element

of the bitumen is determined by considering the heat balance at this element.

The bottom boundary element interfaces with the top element of the roofing
component to which the hot bitumen is applied, and the thermal resistance

across this interface is considered to be zero. Figure 3c indicates the

heat flow at node n. The expressions for heat flow from node n-1 to node n and for stored energy at node n may be determined similarly to the expressions given in figure 3a. The heat flow per unit area from roofing component

node xl to node n is the temperature difference (T . - T ) divided by the

thermal resistance between the nodes (Ru„) as shown in figure 3c. The

resistance Rv is the sum of the resistance of the bitumen (Ly/lV.,) and

that of the component element (Ay/2k ) and may be expressed as:

From Figure 3c, the heat balance equation at node n is expressed as:

^b^^b(n-l) " y) +

Ay

Rearrangement of equation (12) gives:

2 / ^^x \ (T 1 - Tu ^ - SbPb^^y^^T^n - ^bn)

^K C„vp, (Ay)2(T' - T, „)

(Tb(n-l) - Tbn) + r—V ^^xl " T^n) = ^^^

^ ,. ^^ ^

\ + K V^ (13)

(Tb(n-l) - Tbn) + , ^\ (Txl " Tbn) = Mb(Tin " Tbn) (14)

^b "^ '^x The term M, has been previously defined. Solving for Tv , the future

temperature at node n after heat has flowed for time. At, results in the

finite-difference equation for heat flow at the bottom boundary element of

bitumen:

2.2.2.5 Heat Flow within the Roofing Components Other than Bitumen

The mathematical model, as developed here, is applicable to roof systems

which may contain two, three, or four components. In the four component

system hot bitumen is applied to a felt which is bonded with a bitumen

(referred to as bitumen-cement for purpose of classification of the model)

to a substrate. The three-component system contains hot bitumen, felt, and

substrate; and the two-component system is applicable to hot bitumen applied

to a substrate. The finite-difference equations used to describe heat flow

in the typical elements of the felt, bitumen-cement, and substrate components

may be derived from heat balance equations similar to those presented in

sections 2.2.2.2 through 2.2.2.4. For the convenience of the reader, the

10
typical finite-difference equations used to calculate cooling time of hot

bitumen in the four-, three-, and two-component systems are given in appendix A.

2.2.2.6 Solution to the Model

To calculate the cooling time of hot bitumen during roofing construction,

each roofing component was divided into a number of elements each with

the same thickness, Ay, with the exception of the top and bottom surface

elements which had thickness of Ay/2. The number of elements for the individual components was based on the thickness of the component. Each element

was assigned an appropriate finite-difference equation for calculating the

change in temperature with time within that element. Each element was also

assigned an initial temperature. The temperature of each element as a function of an increment of time. At, was calculated sequentially from the

top surface element of the bitumen to the lowest element of the substrate.

This process started with the initial bitumen temperature and continued

iteratively until the bitumen temperature reached a limiting value. The

temperature of the bitumen was taken as the average of the temperature

of each element in the bitumen component for each increment of time. At.

In order to satisfy the condition that the limiting value of the modulus,

M, be greater than or equal to 2, values of Ay and At cannot be chosen

independently and must be chosen relatively small to provide a stable solution for the heat flow calculation (section 2.2.2.2).

To facilitate the calculations, the model was computerized. The listing of

the computer program is given in appendix B. The program uses the finitedifference equations for two-, three-, and four-component roofing systems

given in appendix A. It is noted that the program contains a control statement that directs the calculation for roofing systems depending on the

nximber of components.

3. APPLICATION OF MODEL FOR PREDICTING BITUMEN COOLING

3.1 VALUES NEEDED TO SOLVE MODEL

Because the mathematical model was developed to predict cooling time of hot

bitumen applied to specific roofing components and under specific environmental conditions, the program prepared for the solution of the model requires

values for each of the factors discussed in section 2.1 which affect cooling.

In this regard, values for the following factors are read into the program

for each set of calculations: initial temperature, number of elements, thermal conductivity, and thermal diffusivity of the roofing components; air

temperature; and convection coefficient for the top surface of the bitumen

and the bottom surface of the substrate. Emissivities of the bitumen and

substrate are constants included in the program and the values of emissivity

must be changed as appropriate for different bitumens and substrates. The

initial temperature of any component is taken to be uniform throughout

its thickness. Other values for operation of the program are the minimum

temperature to which the bittimen cools and the maximum time allowed for

bitumen to cool during program execution. Values are also needed for the

11
number of components, the thickness of the elements, Ay, and the increment

of time, At.

Although not directly included as part of the program input needed for the

cooling calculations, values of the following parameters are necessary since

values of input are dependent on these parameters. The thermal diffusivity

of a material is dependent on its density, thermal conductivity, and specific

heat. The convection coefficient is related to the wind speed. The number

of finite-elements of each component is determined by dividing the component

thickness by the thickness of the element. Ay, with the exception that at

a surface a half element is used in the calculation. It is noted that the

computer program limits the number of substrate elements to 100.

In order to illustrate the applicability of the mathematical model, specific

values of material, environmental, and physical properties for selected model

roofing systems were needed for use in the cooling calculations. Values

for materials properties and convection coefficients as a function of wind

speed were obtained from literature sources. Environmental factors

influencing cooling (air temperature, roof component temperature, and wind

speed) were chosen to represent a range of conditions encountered in builtup roofing applications. Roofing component thicknesses were selected on

the basis of current roofing practice.

The thermal properties of the roofing component materials considered in this

study are given in table 1. Selected values for roofing component temperatures and environmental factors are given in table 2. Table 3 gives thick- nesses of roofing component materials and the number of finite-elements for

each material used in the calculations. The element thickness. Ay, for com- puter program input was selected as 0,00 3 in or 0.00025 ft (0.076 mm). Values

of the time increment, At, ranged from 0.000000045 to 0.000001 h (0.000162

to 0.00 36 s). These values of time were selected to satisfy the conditions

for heat flow calculation by finite-difference techniques [5,6] and to

minimize computation time.

Using input data in tables 1, 2 and 3 and selected values of Ay and At

within the ranges given above, the average bitumen temperature was computed

as a function of time for various model roofing systems (configurations)

given in table 4. The model roofing systems contain two, three, or four

components and included bitumens, felts, insulations, and deck materials.

The solution of the mathematical model for a given set of conditions was

a tabular listing of bitumen temperatures corresponding to half second time

intervals. The temperatures of the applied bitumen during the cooling

period were computed over a range beginning with the selected application

temperature (table 2) to the selected lower limit of 300"? (149°C).

Tabular data from the model were used to prepare the bitumen temperature

versus cooling time plots given in figures 4-17.

12
3.2 RESULTS FOR SPECIFIC ROOFING SYSTEMS

This section presents the results (figures 4-17) of the calculations for

various two-, three-, and four-component model roofing systems (table 4) conducted to illustrate the applicability of the mathematical model. The results

in figures 4-15 give the temperature-time relationships for the cooling of

hot bitumen applied at an application quantity of about 20 lbm/100 ft

(0.98 kg/m^) for asphalt and about 25 lbm/100 ft^ (1.2 kg/m^) for coal tar

pitch, at temperatures of 500, 450, 400, and 350°F (260, 232, 204, and 177*C)

to a lower temperature of 300**F (149*0 under specific environmental conditions. These environmental conditions were air and initial component temperatures of 120, 70, and 20*'F (49, 21, and -7*0 and wind speeds of 0, 10,

20, and 30 mph (0, 4.5, 9.0, and 13.5 m/s). The asphalt and coal tar pitch

application quantities correspond to a thickness of 0.04 in (1.0 mm).

Figure 16 gives results for the three-component roofing system of asphalt

applied to felt on polyurethane foam insulation and shows the effect of the

quantity of applied asphalt (asphalt thickness) on cooling time for application tenperatures of 500 and 400*'F (260 and 204*C) and a wind speed of zero. The air and initial component temperatures in this figure were also

120, 70, and 20 "F (49, 21, and -7''C). Figure 17 compares the cooling

time of asphalt applied at 500*F (260''C) to various substrates. In this

figure the air and initial substrate temperatures were 70*F (21 *C) and

the wind speed was zero.

A comparison of the results given in figures 4-15 indicates the influence

of the various factors that affect the cooling rate of hot bitumen. The

time for the bitumen to cool from 500 to 300''F (260 to 149°C) ranged from

less than 4 to more than 50 seconds depending upon the wind speed, air temperature, initial component material temperatures, and the type of roofing

system. As expected, the cooling time decreased for a particular roofing

system with a decrease in application temperature, an increase in wind speed,

and a decrease in the air and initial conqjonent material tenperatures.

The type of substrate to which the hot asphalt is applied may have a significant effect on the cooling time. Figures 4-9 illustrate this effect. In

comparing these two-component systems, the cooling time was much longer for

the cases of asphalt applied to low thermal conductivity and low heat capacity polyurethane foam and fiber glass insulations than the cooling time for

asphalt applied to the more conductive and higher heat capacity concrete

and steel. The cooling times for asphalt applied to insulating concrete

and plywood, figures 6 and 7, were about the same, and fell between those

for the low and high thermal conductivity and heat capacity materials. It

is noted from table 1 that insulating concrete and plywood have comparable

thermal properties.

Because steel has a higher thermal conductivity and heat capacity than

concrete, it might be expected that asphalt applied on steel would cool

faster than on concrete. From figures 8 and 9 it can be seen that the calculated asphalt cooling times were shorter for a concrete substrate than

for a steel substrate. This can be explained in part by considering the

13
thickness of the substrate. The thin ste^l substrate rapidly heats upon

application of the asphalt until its temperature is nearly uniform through

its thickness. At this point, the steel is less of a heat sink for the

asphalt than the concrete. In the case of asphalt applied to the thick

concrete substrate, the concrete has a considerably larger volume, heats

more slowly, and continuously absorbs heat from the hotter asphalt.

A comparison is made from figures 8 and 10 of the cooling times of asphalt

and coal tar pitch applied to concrete. The slightly longer cooling time

for coal tar pitch is attributed to its lower thermal diffusivity and higher

heat capacity. It is noted that identical values of thermal conductivity

and specific heat were assigned for these two bitumens. The lower

diffusivity of the coal tar pitch is accounted for by its higher density.

Figures 11 and 12 illustrate the use of the mathematical model for

three-component systems, whereby hot bitumen is applied on a felt which is

attached to a substrate. In these examples, the bitumen was asphalt, the

felt was type 15 asphalt organic, and the substrates were polyurethane foam

insulation (figure 11) and plywood (figure 12). The plots in figures 11

and 12 indicate that the felt and substrate combination influence the cooling of the applied bitumen. For these three-component systems, the asphalt

cooled faster when applied to the felt on the substrate with the higher

heat capacity and thermal conductivity (plywood). In these two cases, the

thermal properties of the felt were identical.

The substrate in these cases influences the cooling of the asphalt because

the felt is relatively thin. Heat flowing from the asphalt into the thin

felt flows into the substrate before the asphalt cools to the lower temperature limit of 300°F (149°C). Since heat flows from the hot asphalt first

into the felt and then into the substrate, the substrate below the felt has

little effect on the cooling time in the initial seconds after application

of the asphalt. In comparing figures 11 and 12, it may be seen that the

temperature-time cooling curves for the asphalt applied to felt on polyurethane insulation or to felt on plywood are essentially identical during the

first two to three seconds after application. Then the time of cooling

increases for the case of asphalt applied to felt on polyurethane foam.

The effect of the substrate below the felt on the cooling time of the

applied asphalt may be seen by examining the temperature distribution of

the elements within the roofing system as a function of time. Figure 18

presents temperature profiles through the thickness of the three-component

roofing systems, asphalt applied to felt on either polyurethane foam insulation or plywood substrate. In figure 18, asphalt at 500°F (260°C) was applied to the felt and substrate at 120°F (49°C) with a zero wind speed.

The temperature profiles are obtained by plotting the temperature of the

element versus the number of the element. The thickness of the substrate

was limited to 100 elements. The temperature profiles given in figure 18 are for times of 2, 4, 6, 8, 10, and 20 seconds after application of the

asphalt (figures 18a through 18f, respectively).

14
It can be seen from figure 18 that the temperature profiles for the two

three-component systems are very similar at times corresponding to a few

seconds after application of the asphalt. For example, 2 seconds after

application (figure 18a), the distribution of temperature through the asphalt

and felt components are almost identical for the two systems. The temperatures within the polyurethane substrate are seen to be only slightly higher

than those of the plywood substrate for a few elements below the feltsubstrate interface. Four seconds after application (figure 18b), the temperature distributions within the asphalt components remain similar for

the two three-component systems. As the time after application increases,

the temperature profiles of the two three-component systems exhibit greater

disparity. The temperatures of the asphalt applied to felt on plywood are lower than the temperatures of asphalt applied to felt on polyurethane. In

addition, the temperatures within the polyurethane foam insulation substrate

rise higher than the temperatures within the plywood substrate which has

greater heat capacity and thermal conductivity. The result is a slower

cooling of asphalt applied to felt on polyurethane foam in comparison to

asphalt applied to felt on plywood.

A comparison may be made between figures 4 and 11, or figures 7 and 12 to

show the effect of the felt on the time of bitumen cooling. Figure 4 gives

the calculated results for asphalt applied directly to polyurethane foam,

while in figure 11, the asphalt is applied to felt on polyurethane foam.

The asphalt applied to felt on foam cools more rapidly than that applied

directly on foam, because of the effect of the relatively high heat capacity and thermal conductivity of the felt compared to the heat capacity

and thermal conductivity of the foam. The cooling time for asphalt applied

to plywood, figure 7, was essentially the same as that for asphalt applied

to felt over plywood, figure 12, The thermal properties of the felt and

plywood were considered to be comparable (table 1).

Examples of asphalt cooling time for two four- component roofing systems are presented in figures 13 and 14. The cooling time for hot asphalt applied

to felt on asphalt on concrete was essentially the same as for hot asphalt

applied to felt on asphalt on polyurethane foam, even though the thermal

properties of the concrete and polyurethane foam differ significantly. The

cooling time to reach XO°F (149*'C) for these four-component systems was influenced only by the felt and asphalt and not the substrate.

Figure 15 allows a comparison to be made for a four-component system

containing coal tar pitch instead of asphalt. The cooling times for coal

tar pitch in figure 15 were slightly longer than for the asphalt cooling

times given in figures 13 and 14. Assuming that the substrate has little

influence on the cooling time for a four-component system, this observation

is consistent with the previously noted comparison between figures 8 and

10 for the case of these two bitumens applied to the same substrate.

The effect of the quantity of applied asphalt on the cooling time is

illustrated in figure 16. In figures 4 through 15, the bitumens were applied

at a constant thickness of about 0.04 in (1.0 mm). This corresponds to an

application quantity of about 20 lbm/100 ft (0.98 kg/m ) for asphalt and

15
25 lbm/100 ft^ (1.2 kg/m^) for coal tar pitch. As noted in figure 16,

Increasing the amount of applied bitumen significantly increased the cooling

time to reach a given temperature. For example, in figure 16 the cooling

time to reach 300°F (lAg'C) for asphalt at 500°F (260°C) applied to com- ponents at 20**F (-7**C) ranged from about 8 to 45 seconds for application

quantities of about 15 and 30 lbm/100 ft^ (0.73 and 1.5 kg/m^) , respectively.

These quantities correspond to asphalt thicknesses of about 0.03 and 0.06 in

(0.7 and 1.5 mm).

Figure 17 shows the effect of the substrate on asphalt cooling time for a specific set of environmental conditions, application temperature, and

asphalt thickness. The air and substrate temperatures are 70°F (21°C)

with a zero wind speed and with an asphalt application temperature of 500''F

(260°C). The asphalt thickness corresponds to about 20 lbm/100 ft^

(0.98 kg/m ). In general, the curves in this figure are grouped according

to the thermal properties of the substrate, and in particular, the heat

capacities of the substrates. Table 5 presents the heat capacities of the

substrates and the times for the asphalt to cool to 300°F (149°C) for the

specific set of environmental conditions in figure 17. It can be seen from

table 5 that the asphalt cooling times generally decrease as the heat capacities of the substrates increase. The thickness of the steel substrate is

considerably less then that of the other substrates (table 3). This accounts

for the cooling time being comparable to that for concrete, as shown in

table 5. It is noted that the cooling time of asphalt applied to felt is

influenced by the polyurethane foam insulation substrate. Cooling time of

asphalt applied to felt alone was not calculated, since this is not

applicable to roofing construction.

3.3 USES FOR MODEL

The model for predicting the cooling time of hot bitumen has practical

application to the construction of bituminous built-up roofing systems. The

model can provide quantitative information about bitumen cooling time with

regard to the time available to apply insulations or felts properly. Bitumen cooling is a complex process which depends upon a combination of environmental factors and thermal properties of the materials in the roofing

system.

The model has shown in section 3.2 that for certain roofing systems under

specific conditions bitumen will cool extremely rapidly. In cases where

cooling may be more rapid than expected, there may be insufficient time to

place insulation or to broom felts to achieve adequate adhesion between

roofing components. The model permits a quantitative determination of

bitumen cooling time as opposed to empirical or intuitive approaches. Use

of the model indicates that under some conditions, the application temperature may have to be raised above normally accepted limits to provide sufficient time to apply component materials, or hot bituminous roofing should

not be applied under some conditions.

Although the use of the model was demonstrated in this paper for specific

roofing systems and environmental conditions, it may be applied to other

16
roofing systems and environmental conditions as well. This paper presented

the results of the cooling calculations as plots of bitumen temperature

versus time for specific bitumen application temperatures at various wind

speeds and at specific air and substrate temperatures. Other types of

plots may be used to express the results of the cooling calculations. One

such plot relates the dimensionless temperature parameter, (Tv, - T^^)/

(T - T ), to time. The term T^^ is the bitumen temperature at a given time,

T is the application temperature of the bitumen, and T^^ is the temperature

of the air and the initial temperature of components other than bitumen.

Figure 19 is an example of a plot of dimensionless temperature versus time

for the application of hot asphalt to steel, and was generated from the

results of the cooling calculations from which figure 9 was prepared.

Whereas figure 9 contains forty-eight curves, figure 19 contains only four

curves. These four curves may be used to approximate the bitumen temperature (Tl) at any given time for any bitumen application temperature (T ) and air and initial roof component temperature (T^^,) for each of the four

wind speeds.

4. SUMMARY

This paper presented a mathematical model using finite-difference equations

to predict the cooling times of hot bitumen applied during the construction

of built-up roofing. The application or contact temperature of the hot

bitumen was taken as the temperature at which the bitumen made contact with

the surface to which it was applied. The quality of built-up roofing is

dependent upon the application of bitumen which is hot enough (i.e., has

the proper viscosity) to provide a uniform continuous layer of bitumen and

adequate adhesion between roofing components. A model for the prediction

of cooling times was needed, since it would not be practicable to measure

these rates for the numerous roofing systems to which hot bitumen may be

applied under many different environmental conditions. Up to now, mathematical modeling of the cooling of bitumen has not been employed in the

roofing industry.

The effect of various factors on the cooling times of hot bitumen was

investigated. These factors included materials properties and environmental

conditions. The materials properties were the initial temperature, thermal

conductivity, specific heat, density, and thickness of the roofing components!

Environmental conditions affecting cooling were air temperature and wind

speed. Values for each of these factors were incorporated in the model to

determine quantitatively their effect on the cooling. The model was com- puterized for application to roofing systems having two, three, or four

components.

Examples of the use of the model to predict the cooling times of hot bitumen

applied to various roofing components were given. These components included

felts, insulations such as fiber glass and polyurethane foam boards, and

typical decks such as concrete, steel, and plywood. Times for bitumens to

cool to 300°F (149*'C) were computed for bitumen application temperatures

ranging from 500 to 350°F (260 to 177°C). For the selected examples, cooling

times ranged from less than 4 to more than 50 seconds depending upon the

17
type of model roofing system in question, environmental conditions, and

component material temperatures. Cooling times of the hot bitumen were

graphically presented using computer-generated plots of bitumen temperature

versus time.

The model indicated that for some systems under certain environmental

conditions, hot bitumen cools extremely rapidly. In these cases, sufficient

time may not be available to place insulation or felts properly. To provide

adequate time for proper application, the temperature of the bitumen may need

to be increased above normally accepted limits, or hot bituminous roofing

should not be applied under some conditions.

5. ACKNOWLEDGMENTS

The authors wish to acknowledge the assistance provided by discussion with

Mr. Douglas M. Burch and Mr. Bradley A. Peavy, National Bureau of Standards,

Building Thermal and Service Systems Division; Dr. Geoffrey J. C. Frohnsdorff,

National Bureau of Standards, Structures and Materials Division; Dr. Philip

F. Dickson, Colorado School of Mines, Department of Chemical and Petroleum

Refining Engineering; and. Dr. Edwin Mertz, formerly of the National Roofing

Contractors Association.

Special thanks are given to Dr. James J. Filliben, National Bureau of

Standards, Statistical Engineering Division, for his contributions to this

report. The authors appreciate his valuable assistance in the development

of the computer programs to predict the cooling times of hot bitumen and to

plot the temperature versus cooling time curves.

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18
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Association of Asphalt Paving Technologists, Vol. 39, pp. 377-403 (1970).

14. Corlew, J. S., and Dickson, P. F., Methods for Calculating Temperature

Profiles of Hot-Mix Asphalt Concrete as Related to the Construction of

Asphalt Pavements, Proceedings of the Association of Asphalt Paving

Technologists, Vol. 41, pp. 101-140 (1972).

15. Shah, N. D., and Dickson, P. F., Design Considerations for a Direct-Fired Propane Heater to Preheat the Base for Cold-Weather Paving,

Highway Research Record, No. 549, pp. 55-62, (1975).

16. Corlew, J. S., and Dickson, P. F., Cold-Weather Paving of Thin Lifts

of Hot-Mixed Asphalt on Preheated Asphalt Base, Highway Research Record,

No. 316, pp. 1-6 (1972).

17. Frenzel, Bruce G., Dickson, Philip F., and Corlew, Jack S., Computer

Analysis for Modification of Base Environmental Conditions to Permit

Cold Weather Paving, Proceedings of the Association of Asphalt Paving

Technologists, Vol. 40, pp. 487-508 (1971).

19
18. Foster, Charles R., A Study of Cessation Requirement for Constructing

Hot-Mix Asphalt Pavements, Highway Research Record, No. 316, pp. 70-75

(1970).

19. Busching, Herbert, Effects of Moisture and Temperature on Roofing

Membranes in Thermally Efficient Roofing Systems, Proceedings of the

5th Conference on Roofing Technology, NBS/NRCA, National Roofing

Contractors Association, Oak Park, Illinois, pp. 52-59 (April 1979).

20. Mack, Charles, in Bituminous Materials: Asphalts, Tars, and Pitches,

Hoiberg, Arnold J., Editor, Interscience Publishers, New York, Volume 1 Chapter 2, pp. 25-121 (1964).

21. Nottage, H. B., Thermal Properties of Building Materials Used in Heat

Flow Calculations, Transactions of the American Society of Heating and

Ventilating Engineers, Vol. 5 3, pp. 215-24 3 (1947).

22. ASHRAE Handbook 1977 Fundamentals, American Society of Heating,

Refrigerating and Air-Conditioning Engineers, New York, Chapter 37,

Table 3, pp. 37 . 3 and 37.4 (1977).

23. ASHRAE Handbook 1977 Fundamentals, American Society of Heating,

Refrigerating and Air-Conditioning Engineers, New York, Chapter 22,

Table 3A, pp. 22.13-22.17 (1977).

24. ASHRAE Handbook 1977 Fundamentals, American Society of Heating,

Refrigerating and Air-Conditioning Engineers, New York, Chapter 22,

Figure 1, p. 22.2 (1977).

20
Table 1. Thermal Properties of the Roofing Component Materials for Use In the Model.

Thermal Properties

Roofing

Component Material

Density

P Thermal

Conductivity

k

Specific

Heat ThermalDlffuslvity^'^

a

Heat Capaclty(2) Data

Source Reference

ibm/ft^

(kg/m^)

Btu'ft/h'ft^*"

(W/m«K)

F Btu/lbm«'F

(kJ/kg'K)

ft^/h

(mm^/s)

Btu/ft3«°F

(kJ/m^'K)

Bitumen (Hot)^-^^

Asphalt 56 (896) 0.067

(0.12) 0.53

(2.22) 0.0023

(0.059) (1989)

[20]

Coal Tar Pitch^^^

o Felt

72

(115 3) 0.067

(0.12) 0.53(2.22) 0.0018(0.046)

38 (2560)

[20]

Type 15 - Asphalt

Organic 63

(1008) 0.047(0.08)

0.27

(1.13) 0.0028(0.072)

17 (1139)

[21]

Type 15 - Coal Tar

Pitch Organic

69 (1105) 0.047

(0.08) 0.35(1.46) 0.0019(0.049)

24 (1613)

[21]

Bitumen - Cement

Asphalt 64

(1025) 0.089(0.15) 0.42(1.76) 0.00 33 (0.085)

27 (1804)

[20]

Coal Tar Pltch^*^

Substrate

80

(1281) 0.089

(0.15) 0.42(1.76) 0.0026(0.067)

34 (2255)

[20]

Steel 489

(7829)

26.2

(45.3) 0.12

(0.50) 0.45(11.61)

59 (3915)

[221

Concrete 140 (2241) 0.75

1.30 0.22

(0.92) 0.024(0.62)

31 (2062)

[23]

Insulating

Concrete

40

(640) 0.096(0.17) 0.21(0.88) 0.011

(0.62)

8 (56 3) [23]

Plywood 34 (544) 0.067

(0.12) 0.29(1.21) 0.0068

(0.18)

10 (658)

[23]

Glass Fiber

Insulation Board

6.5 (104) 0.021

(0.0 36) 0.23(0.96) 0.014

(0.36)

1.5 (100)

[23]

Polyurethane Foam

Insulation Board

1.5

(24) 0.013(0.022) 0.38(1.59) 0.023

(0.59) 0.57

(38)

[23]

(1)

(2)

(3)

(4)

Calculated from the relationship, a k/pC .

Calculated from the values of density and specific heat.

Density, thermal conductivity, and specific heat values have been adjusted for the elevated bitumen application temperatures. The adjustment was made for a temperature

of 450°F (232°C).

According to Mack [20], the thermal conductivity and specific heat of coal tar pitches

are of the same order of magnitude as those of asphalts. Thus, for purposes of this paper, values of thermal conductivity and specific heat of coal tar pitch were chosen

to be Identical to those of asphalt.

21
Table 2. Values of Roofing Component Temperatures and Environmental

Factors Used in the Calculations

Factor

Bitumen Application

Temperature

Units

<»F

Numerical Values

500

(260)

450

(232)

400

(204)

350

(177)

Felt, Bitumen-Cement

and Substrate Temperatures

°F

(*C)

120

(49)

70

(21)

20

(-7)

Air Temperature op

(*C)

120 70 20

(49) (21) (-7)

Wind Speed

and

Corresponding

Convection

Coefficient^^^

mph

(m/s)

Btu/h«ft2*°F

(W/m^'K)

(0)

10

(4.5)

1.0 3.6

(5.7) (20.4)

20 30

(9.0) (13.5)

6.1 8.5

(34.6) (48.2)

(1) Wind speeds are those blowing across the surface of the cooling bitumen.

The values of the convection coefficient corresponding to the selected

wind speeds were estimated from the curve for smooth surfaced materials

in figure 1, chapter 22 of ASHRAE Handbook 1977 Fundamentals [24]. The

radiation contribution to the overall surface coefficient has been

subtracted to estimate the convection coefficient.

22
Table 3. Thickness of the Roofing Component Materials and Corresponding Number

of Elements Used In the Calculations*

Roofing Component Material Component Thickness

In mm

Number of Elements (1)

o Bitumen

Asphalt(2)

Coal Tar Pitch(2)

0.0285 0.72

0.0405 1.0 3

0.0495 1.26

0.0585 1.49

0.0405 1.03

10

14

1720

14

o Felt

Type 15 Asphalt Organic 0.033 0.84

Type 15 Coal Tar Pitch

Organic 0.033 0.84

o Bitumen-Cement

Asphalt 0.042 1.1

Coal Tar Pitch 0.042 1.1

o Substrate^

Steel(2) 0.0285 0.72

Concrete 6.0 152.4

Insulating Concrete 4.0 101.6

Plywood 0.75 19.1

Glass Fiber Insulation

Board 1.0 25.4

Polyurethane Foam

Insulation Board 1.0 25.4

11

11

1'+

14

10

100

100

100

100

100

(1)

(2)

The number of finite elements is in general determined by dividing the component

thickness by the thickness of a single element, Ay. In this report, Ay was chosen equal to be 0,00 3 in (0.076 mm).

The component thickness considers the half-thickness element at the surface.

( 3)

^ The computer program limits the number of substrate elements to 100.

23
Table A. Configurations of Roofing Conq>onents for VRilch the

Cooling Time of the Applied Bitumen was Calculated.

Number of

Components

Roofing Configuration

Applied Bitumen

asphalt

asphalt

asphalt

asphalt

asphalt

asphalt

coal tar pitch

Other Roofing Components (1)

polyurethane foam Insulation board

fiber glass insulation board

Insulating concrete

plywood

concrete

steel

concrete

Figure Number^ ^^

4

5

6

7

8

9

10

asphalt

asphalt

felt on polyurethane foam Insulation

board

felt on plywood

11, 16

12

asphalt

asphalt

coal tar pitch

felt on asphalt on polyurethane foam

insulation board

felt on asphalt on concrete

felt on coal tar pitch on fiber glass

insulation board

13

14

15

^ ' For roofing configurations which contain three and four components, the coiqponents are listed in the sequence in which they are located in the system, the first

listed being uppermost.

(2) The figure numbers refer to the figures of bitumen temperature versus cooling time

given in section 3.2.

24
Table 5. Heat Capacities of Various Substrates and Cooling

Times for Asphalt When Applied on Them^^-'

Substrate Heat Capacity

PCp

Btu/ft^.°F

(kJ/m^"K)

Polyurethane Foam

Insulation Board

0.57

(38)

Glass Fiber 1.5

Insulation Board (100)

Insulating Concrete 8

(563)

Plywood 10

(658)

FeltO) 17

(1139)

Felt on Polyurethane

Foam Insulation

_(c)

Concrete 31

(2062)

Steel<^) 59

(3915)

Asphalt Cooling

Tlme^^^

59

49

16

17

_(b)

28

4.5

6.5

(a)

(b)

(c)

(d)

Time to cool to 300*F (149'C) . Asphalt applied at 500*'F (260°C) with

air and Initial substrate temperatures of 70*'F (21°C) and zero wind speed.

The asphalt application quantity corresponds to about 20 lbm/100 ft

(0.98 kg/m^).

Asphalt Is not applied to felt alone, and felt Is not considered to be a substrate.

Heat capacities of the felt and polyurethane foam are listed separately.

The thickness of the steel Is considerably less than that of the other

substrates (table 3)

.

25
zg<< zoho

ILI >zoo

x^X/v z

LUDhmhoI ' :h • -Z

-. liJ (D * . z IL . :,0

,. Q. • c . 5 " ' 1 * " O D l ' (0c . .. z . JC . ' IL o • .• 9Q

•a: z • o .

onu

00 -H C 4J •H CO

<4-l S O 0) X

c« 1 (U

r-\ JJ •H3 C

CQ t-l m T3 O T> c a

•H rH S O 3 fi OOo o 3 -H

4J CO

to -H O CO u OS 00 u C CO •H iH u o 3 c/3 M o

0)

14-1 4J

COu

CO 4-1 H W o <u

J-l T3

4-1

O cO (U •H J2

4-1 4-1 CO

-rA

G

CO PC T3

(U

C 3

CO iH

CJ c c O M

COa

00 0)

0)OS W

0)

!-i 3

00

•H

fa

CJ>oO -H

4-)

O CO +J r-\3

0) CO

S-i

u

14-1

O

CD

X

C! 4J

CO "H H

CO +J ^ CO o

0) <u 3S QC • -H

r-l J2

(U

H

O <4-l S O

26
CONVECTION

LOSS

Z

UJD

h

h

O

I

I8|

o

z

g

h (0

50Q -i Z

.0

o

RADIATION

LOSS

Jii

bs

b(i-i )

bi

Tb(i+1)

"^b(n-i )

bn

NODES

Ay/2

X1

X2

Ay

INTERFACE

Ay

Key: T^ = temperature of the air

"^bl ~ temperature of the first bitumen element

T|3j = temperature of an interior bitumen element

T|3p = temperature of the last bitumen element

Txi= temperature of the first element of the roofing component

below the bitumen

Ay = the thickness of an element and the distance between nodes

Figure 2. Diagram of the Elements Within the Bitumen and Top Elements

of the Roofing Component Below the Bitumen.

27
nodes

node temperature

T ±

Ay

T

JL

^bl

T

i-l

•

i •

• i+1

Qi-1.1 heat flow from node 1-1 to node 1

^^b(l-l) - ''^bi^ „ H^'^b(i-l) - "^bi^ *

stored heat at node i - Sb^b^^^^^^bl - '^bl^ At

Qt+l J = heat flow from node i+1 to node i

, ^^b(i+l) "

"^bl^ „ H^'^b(i+1) " ^bi^

Rh Ay

3a. Heat flow within an element within the bltinnen

^bl

^b2

'b(n-l) .

^bn

^xl

Ay/2

2*

Q- 1 « heat flow from air node a to surface node 1

' \^\ - Tbl)

Stored heat at node 1 = Sb^ b^^^^^^^'^bl - '^bl^

Ql.2

At

heat flow from node 2 to surface node 1

(^b2 - Tbl> . H(^b2 - Tbl>

Ay

3b. Heat flow at the surface boundary element of the bitumen

interfacen-1

>

n 11

1 n+1

>

Qjj_, = heat flow from node n-1 to node n

„ ^^b(n-l) "

'^bn^ _ H^'^b(n-l) " ^bn^

Rb ' Ay

Stored heat at node n = Sb'^b^Ay)^^^ _ T^^) At

Qxl.n heat flow from node xl to node n

(Txl - Tbn) . 2 ^bx

kK^v \ (T^i - T.„)

^b-^x Ay V kb + k. xl 'bn''

3c. Heat flow at the bottom boundary element of the bitumen

Figure 3 Diagram of the Elements and Nodes Used to Develop the FiniteDifference Equations for Heat Flow Within the Bitumen, at the

Surface of the Bitumen, and at the Bottom Boundary of the Bitumen.

28
nODCL ~ ASPHALT ON PUF INSULATIOM >OARD SM

45*

4ee

3S9

see

see

45e

4ee -

ase

3ee

see

45e

4ee

35e

aee

w UIND SPEED

SOLID • e nPH

DOTTED • le ftPH DASHED • ae nPH DASH/DOT • 3e nPH vN^x •••. \\^

\\\y\"--. ^^^^

1 1 I'V'f" v\ T^^T"^ .•1^1 1 ^1-'" i-TTt^-

29 39 se

COOLING TinE. SECONDS AIR AND PUF IMSULATION TEHPERfiTURES ARE lEO F

COOLING TIPIE, SECONDS

AIR AND PUF INSULATION TEHPERATURES ARE 78 F

COOLING TiriE, SECONDS

AIR AND PUF INSULATION TEHPERATURES ARE E0 F

Figure 4. Relationship Between Bitumen Temperature and Cooling Time for the

Two-Component Model, Asphalt Applied on Polyurethane Foam

Insulation. The Quantity of Applied Bitumen is Approximately

20 lbm/100 ft^ (0.98 kg/m^).

29
nODEL ASPHALT ON FIBER CLASS INSULATION BOARD 5M

45«

UIND SPEED

SOLID • e BPH DOTTED • le nPH

DASHED • ee npH DASH/DOT • 30 nPH

aee

see

4se

4ee

35*

3ee

see

450

4ee

350

3«e

COOLING TIHE, SECONDS

AIR AND FIBER CLASS INSULATION TENPERATURES ARE ISO F

COOLING TINE, SECONDS

AIR AND FIBER GLASS INSULATION TEHPERATURES ARE 7e F

COOLING TINE, SECONDS AIR AND FIBER CLASS INSULATION TEHPERATURES ARE 20 F

Figure 5. Relationship Between Bitumen Temperature and Cooling Time for the

Two-Component Model, Asphalt Applied on Fiber Glass Insulation.

The Quantity of Applied Bitumen Is Approximately 20 lbm/100 ft^

(0.98 kg/m2).

30
nODEL — ASPHALT ON INSULATING CONCRETE 5M

45«

3W -1

3M

5M

4S«

UIND SPEED

SOLID NPH DOTTED 10 nPH

DASHED • 20 NPH DASH/DOT • 30 NPH

le 20 30 50 *.„»«. COOLING TIHE. SECONDS

AIR AND INSULATING CONCRETE TEHPERATURES ARE 120 F

3S*

300

500

UIND SPEED

SOLID nPH DOTTED 10 NPH DASHED 20 nPH DASH/DOT • 30 nPH

I I I I 10 20 30 40 50 COOLING TINE. SECONDS

AIR AND INSULATING CONCRETE TEMPERATURES ARE 70 F

UIND SPEED

SOLID

DOTTED

DASHED DASH/DOT

tlPH 10 NPH

20 NPH

30 nPH

20 30 —*—I—I—I—r- 40 50 COOLING TINE, SECONDS AIR AND INSULATING CONCRETE TEMPERATURES ARE 20 F

Figure 6. Relationship Between Bitumen Temperature and Cooling Time for the

Two-Component Model, Asphalt Applied on Insulating Concrete. The

Quantity of Applied Bitumen is Approximately 20 lbm/100 ft

(0.98 kg/m^).

31
nODEL fiSPHALT ON PLVUOOD

see

45e

4ee

3se -'

3ee

UIND SPEED

SOLID • e nPH DOTTED • le flPH DASHED • ae nPH DASH/DOT • 30 nPH

3e 40 COOLING TINE. SECONDS AIR AND PLVUOOD TEnPERATURES ARE 120 F

se

see

45e

4»e

ase

aee

see

4se

4ee

35e

3ee

UIND SPEED SOLID • e nPH DOTTED le nPH DASHED se nPH DASH/DOT • 3e nPH

se 3e 4e COOLING TINE. SECONDS

AIR AND PLVUOOD TEMPERATURES ARE 78 F

UIND SPEED

SOLID e nPH

DOTTED le nPH

DASHED • ae nPH DASH/DOT • 3e HPH

se 3e 50 COOLING TirtE, SECONDS

AIR AND PLVUOOD TEnPERATURES ARE 30 F

Figure 7. Relationship Between Bitumen Temperature and Cooling Time for the

Two-Component Model, Asphalt Applied on Plywood. The Quantity of

Applied Bitumen is Approximately 20 lbm/100 ft^ (0.98 kg/m ).

32
NODEL ~ ftSPMftLT ON CONCRETE 5M

UIND SPEED

45« -

4ee

359

see

see

45e

4ee

3se

see

SOLID e NPH DOTTED • le nPH DASHED • se nPH

DflSH/DOT • 3e nPH

le 26

I I

'

I I I I

' 39 4« se COOLING TIME, SECONDS

AIR AND CONCRETE TEriPERATURES ARE 129 F

UIND SPEED

SOLID • 9 NPH DOTTED • le nPH

DASHED • 29 NPH DASH/DOT • 39 NPH

le se 39 49 59

COOLING TinE, SECONDS

AIR AND CONCRETE TENPERATURES ARE 79 F

see

UIND SPEED

SOLID • e nPH

DOTTED • 19 flPH DASHED • 29 nPH DASH/DOT • 39 NPH

399

19

I I I

'

I I I I

'

I I I I . . 39 39 49 59 COOLING TIHE. SECONDS

AIR AND CONCRETE TEHPERATURES ARE 29 F

Figure 8. Relationship Between Bitumen Temperature and Cooling Time for the

Two-Component Model, Asphalt Applied on Concrete. The Quantity

of Applied Bitumen is Approximately 20 lbm/100 ft^ (0.98 kg/m^).

33
nODEL ~ ASPHALT CM STEEL 5M

45«

4M

3S»

3M

UIND SPEED

SOLID • e nPH

DOTTED • le nPH

DASHED • 20 HPH DASH/DOT • 30 nPH

20 30 40 50 COOLING TinE, SECONDS AIR AND STEEL TENPERATURES ARE 120 F

•

-

UIND SPEED

450

SOLID • nPH

DOTTED • 10 nPH DASHED • 20 nPH DASH/DOT • 30 nPH

4«e

35*

3M Ik , 1 1 1 10 20 30 40 50 COOLING TinE, SECONDS

AIR AND STEEL TEHPERATURES ARE 70 F

SM

UIND SPEED

SOLID • NPH DOTTED • 10 nPH DASHED • 20 nPH

DASH/ DOT • 30 nPH

I I I I 10 20 30 40 50 COOLING TINE, SECONDS AIR AND STEEL TENPERATURES ARE 20 F

Figure 9. Relationship Between Bitumen Temperature and Cooling Time for the

Two-Component Model, Asphalt Applied on Steel. The Quantity of

Applied Bitumen is Approximately 20 lbm/100 ft^ (0.98 kg/voT).

34
nODEL COAL TAR PITCH ON CONCRETE 5M

35*

3M

5M

UIND SPEED SOLID • e NPH DOTTED • 10 nPH DASHED • ee nPH DASH/DOT • 30 NPH

30 30 COOLING TiriE. SECONDS

AIR AND CONCRETE TEHPERATURES ARE IS9 F

UIND SPEED

35*

3M

10 20 30 40 COOLINQ TIME, SECONDS AIR AND CONCRETE TEHPERATURES ARE 70 F

10 ae COOLING TIME, SECONDS

AIR AND CONCRETE TEDPERATURES ARE 20 F

SOLID • FIPH DOTTED • 10 MPH DASHED • SO NPH DASH/DOT • 30 nPH

50

50

^

-

UIND SPEED 4M

SOLID nPH

DOTTED • 10 nPH DASHED • SO nPH DASH/DOT • 30 PIPH

4M

\

35* 1

3M 1 , < '1 1 1

'

1 1 1 ,

1

, , , ,

1 , , , , 1

, , , , 50

Figure 10. Relationship Between Bitumen Temperature and Cooling Time for the

Two-Component Model, Coal Tar Pitch Applied on Concrete. The

Quantity of Applied Bitumen is Approximately 25 lbm/100 ft^

(1.2 kg/m^).

35
nODEL ASPHALT ON FELT CM PUF INSULATION BOARD

see

45e

4ee

3se

aee

see

45e

499

3se

3ee

see

45e

COOLING TIHE, SECONDS

AIR, FELT AND PUF INSULATION TEMPER. 'TURES ARE 120 F

COOLING TinE, SECONDS

AIR. FELT AND PUF INSULATION TEMPERATURES ARE 76 F

35e

- UIMD SPEED

-i SOLID nPH DOTTED • 10 flPH DASHED • 20 PIPH DASH^DOT 30 PIPH 11

^m

\m"-'\

:iSX.

le 20 se COOLING TINE, SECONDS

FELT AND PUF INSULATION TENPERRTURES ARE 29 F

Figure 11. Relationship Between Bitumen Temperature and Cooling Time for the

Three-Component Model, Asphalt Applied on Felt over Polyurethane

Foam Insulation. The Quantity of Applied Bitumen is Approximately

20 lbm/100 ft^ (0.98 kg/m^).

36
nODEL — ASntALT ON FELT ON PLVUOOD

5««

COOLING TINE, SECONDS

AIR, FELT AND PLVUOOD TENPERATURES ARE IB9 F

see

4se

4ee

359

see

see

45e

4ee

ase

aee

COOLING tihe. seconds

AIR, felt and PLVUOOD TENPERATURES ARE 70 F

UIND SPEED

SOLID • nPH DOTTED • 10 nPH DASHED • 20 HPH DASH/DOT 30 HPH

Z9

I I '

I I t I '

I I I 30 40 50 COOLING TINE. SECONDS

AIR. FELT AND PLVUOOD TEMPERATURES ARE 20 F

Figure 12. Relationship Between Bitumen Temperature and Cooling Time for the

Three-Component Model, Asphalt Applied on Felt over Plywood. The

Quantity of Applied Bitumen is Approximately 20 lbm/100 ft

(0.98 kg/m^).

37
ASPHftLT ON FELT ON ASPHAL" ON PUF INSULATION BOARD

UIND SPEED

SOLID e flPH DOTTED 19 nPH DASHED • ae FIPH DASH/DOT 38 nPH

4d9

359 -'

399

3e 50 COOLING TinE, SECONDS

AIR, FELT, ASPHALT AND PUF INSULATION TEPIPERATURES ARE 120 F

SM

UIND SPEED SOLID • • DOTTED • 10 DASHED • 20 DASH/DOT • 30

80 30

I I I

'

I I I 40 50 COOLING TINE, SECONDS AW. FELT, ASPHALT AND PUF INSULATION TEHPERATURES ARE 70 F

500

450

400

350

300

UIND SPEED

SOLID • e NPH DOTTED 10 nPH DASHED • 20 nPH DASH/DOT • 30 nPH

20 30

'

I I I I 40 50 COOLING TINE, SECONDS

AIR, FELT, ASPHALT AND PUF INSULATION TENPERATURES ARE 20 F

Figure 13. Relationship Between Bitumen Temperature and Cooling Time for the

Four-Component Model, Asphalt Applied on Felt over Asphalt on Polyurethane Foam Insulation. The Quantity of Applied Bitumen is

Approximately 20 lbm/100 ft^ (0.98 kg/m^).

38
nODEL — ASPHALT ON FELT ON ASPHALT ON CONCRETE

see

4se

4ee

35e

aee

see

45e

'4ee

ase

see

UIND SPEED

SOLID DOTTED

DASHED DASH/DOT

e nPH

le nPH

ee NPH

3e nPH

30 40 se COOLING TINE. SECONDS

AIR, FELT, ASPHALT AND CONCRETE TENPERATURES ARE 120 F

UIND SPEED

SOLID • e nPH DOTTED • 19 NPH DASHED • 20 HPH DASH/DOT 30 nPH

20 30 S0 COOLING TINE. SECONDS

AIR, FELT, ASPHALT AND CONCRETE TEHPERATURES ARE 70 F

I

UIND SPEED

450 SOLID • nPH

DOTTED • 10 NPH DASHED • 20 NPH DASH/DOT 30 HPH

4ee l\

35e i\

aee Ik , 1, , , 1, , , le ae 30 COOLING TinE, SECONDS

AIR, FELT. ASPHALT AND CONCRETE TEflPERATURES ARE 2* F

Figure 14. Relationship Between Bitumen Temperature and Cooling Time for the

Four-Component Model, Asphalt Applied on Felt over Asphalt on Concrete. The Quantity of Applied Bitumen is Approximately

20 lbm/100 ft^ (0.98 kg/m^).

39
nODEL — COAL TAR PITCH ON FELT ON COAL TAB ON F6 INSULATION

I UIND SPEED

45# \

SOLID • e nPH DOTTED • le nPH DASHED • 2e RPH DASH/DOT 30 NPH

^w \iA'\;\

350 ~\\W\^>;0\N"\ V ^'^•"^•^

V ^^oN^'C^^' .^\

3M «^ '^nXv-'-NL-. >s>

, 1 le ee 3e 40 50 .,„ COOLING TINE, SECONDS

AIR, FELT, COAL TAR AND FS INSULATION TENPERATURES ARE ISO F

SM

UIND SPEED

SOLID

DOTTED

DASHED DASH/DOT

e NPH

10 NPH

eo NPH

30 NPH

3M '

, ,

I ee 30

r I I

'

I I I 40 SO COOLING TINE, SECONDS

AIR, FELT, COAL TAR AND FC INSULATION TENPERATURES ARE 70 F

500

450

400

350

300

UIND SPEED

SOLID • e f*H DOTTED • 10 nPH DASHED • 20 nPH DASH/DOT ' 30 nPH

20

' I I—r I

'

30 40 50 COOLING TINE, SECONDS

AIR, FELT. COAL TAR AND FG INSULATION TENPERATURES ARE 20 F

Figure 15. Relationship Between Bitumen Temperature and Cooling Time for the

Four-Component Model, Coal Tar Pitch Applied on Felt over Coal

Tar Pitch on Fiber Glass Insulation. The Quantity of Applied

Bitumen is Approximately 25 lbm/100 ft^ (1.2 kg/m^).

40
nODEL ~ ASPHALT ON FELT ON PUF INSULATION BOARD

9VV

ASPHALT APPLICATION

456

SOLID • 3a LB/SO DOTTED • 2S LB/SO DASHED • 28 LB/SO DASH/DOT • 15 LB/SO \\v\

4M -\ \ -Nv

•.V\ \ '> "'^^—

35» AvNx ''^-., ••••-....

,

1 ,7""':'t-7-, , , le 2e se COOLING TINE, SECONDS

AIR, FELT AND PUF INSULATION TEHPERATURES ARE 128 F

seo

45*

400 -

350

300

500

ASPHALT APPLICATION

SOLID • 38 LB/SO DOTTED 25 LB/SO DASHED • 28 LB/SO DASH/DOT • IS LB/SO

COOLING TINE, SECONDS

AIB, FELT AND PUF INSULATION TEMPERATURES ARE 78 F

COOLING TINE. SECONDS

AIR, FELT AND PUF INSULATION TEHPERATURES ARE 80 F

Figure 16. Effect of Asphalt Thickness on the Cooling Time for the Three-Component

Model, Asphalt Applied on Felt over Polyurethane Foam Insulation.

41
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:

8

TIME, s

12 16

T|3 = bitumen temperature at a given time

Ta = bitumen application temperature

Tg^c = temperature of air and substrate

Figure 19. Plot of Dimensionless Temperature Versus Time for the Two-Component

Roofing System, Asphalt Applied on Steel.

44
APPENDIX A. LISTING OF FINITE-DIFFERENCE EQUATIONS

The mathematical model for predicting the cooling times of hot bitumen

during built-up roofing construction, as developed here, is applicable to

roof systems which may contain two, three, or four components. The model

contains three types of finite-difference equations for each component of

the roofing system; that is, the model uses 6, 9 or 12 types of equations

for calculating cooling times for systems containing two, three, or four

components, respectively. For the convenience of the reader, this appendix

presents the types of finite-difference equations used for the two-, three-, or four-component roofing systems. These equations may be developed by the

treatment given in section 2.2. The subscripts b, f, c, and s in the equations which follow refer to bitumen, felt, bitumen-cement, and substrate,

respectively. It is noted that the types of finite-difference equations

for similar elements in different components (e.g., the equations within

the bitumen-cement and within the substrate) are identical in form. The

difference between the equations is the values of the constants for the

thermal conductivity, specific heat, and density properties of the conaponents

in question.

Figure Al presents a diagram of the elements and mathematical notation used

to denote the temperature of the elements for the four-component roofing

system. This figure was prepared for development of the finite-difference

equations for the four-component system. Similar diagrams may be prepared

for the three- and two-component systems.

Al
Convection

loss

nodes

AY/2

ROOFING COMPONENT

HOT BITUMEN

FELT

BITUMEN-CEMENT

SUBSTRATE

KEY: Tbi = temperature of any bitumen element

Tbn = temperature of the last bitumen element

Tfi = temperature of any felt element

Tfn = temperature of the last felt element

Tci = temperature of any bitumen-cement element

Ten = temperature of the last bitumen-cement element

Tsi = temperature of any substrate element

Tsn = temperature of the last substrate element

Figure Al. Diagram of the Finite-Elements Within the Components of the

Roofing System Used for the Calculation of Bitumen Cooling

Time by Finite-Difference Techniques.

A2
A.l FINITE-DIFFERENCE EQUATIONS FOR THE FOUR-COMPONENT SYSTEM

* At the bitumen surface, T^-^

bl = Ttifl-O - 2_(Nb + 1.0) + 2_(Tb2 + N^T^)] (Al)

L

M^, M. J

Within the bitumen, T^^

T = T

en en

''^bi = ^bi *'i_^*^b(i-l) " 2Tbi + Tb(i+i)) (A2)

* In the bottom elemental layer of bitumen, Tv

T^n = Tb„ + J_rTb(n-l) " ^ + ^^^ Kn +

fJHf—Vfl] (A3)

^bL \ ^b + ^fl Ub + ^f/ J

* In the top elemental layer of the felt, Tj-,

* Within the felt, T^^

Tf i = Tfi + l_(Tf (i_i) - 2Tfi + Tf (i+i)) (A5)

Mf

* In the bottom elemental layer of felt, T^

* In the top elemental layer of bitumen-cement, T ,

^cLV'^c + V \ l^c + V J

** Within the bitumen-cement, T . ' ci

Kl = Tei + i_(Tc(i-i) - 2T^i + T^(i+1)) (A8)

^c

* In the bottom elemental layer •' of bitumen-cement, T„^ ' en 'kb^-^^-i''^h'(^y-]

""

A3
In the top elemental layer of the substrate, T ,

^81 ' ^Sl + 1 [(t^y--(^^^)^=^^H (AlO)

Within the subtrate, T^^

^Si - Tgi + 1 (Tsd-i) - 2Tgi + T3( .^.1)) (All)

In the bottom elemental layer ' of the substrate, T ' sn T » T sn sn Flo - /|_yN3+ 1)1 + /Z-\asin-l) + Va) <A12)

A. 2 FINITE-DIFFERENCE EQUATIONS FOR THE THREE-COMPONENT SYSTEM

The nine types of finite-difference equations for the three-component system

are:

o At the bitumen surface, Tvi

^bl - Tblfl-O - (l-y^b + 1-0)] + / 2_\(Tb2 + Va^ (A13)

o Within the bitumen, T, . K± - Tbi + 1 (Tb(i-i) - 2Tbi + Tb(i+i)) (A14)

o In the bottom elemental layer of bitumen, T,

+ i-hcn-l) - fl + -i^Vbn -^ f ^^^-Vfll (A15)

MfeL ' ' V k^ + kf/ Vkb + kfj J

^bn " Tb„

o In the top elemental layer of felt, T^,

Tfl - Tfi + 1. \( ^^b Vbn - fl + -^^)Tf1 + Tf2] (A16)

^fLW + kf/ V k^ + ^^f/ J

o Within the felt, T^.

Tfi - Tfi + l.(Tf(i_i) - 2Tfi + Tf(i+i)) (A17)

Mf

A4
o In the bottom elemental layer of felt, T^rn

o In the top elemental layer of the substrate, T -.

+ L.\( ^^f Vfn - ^+ ^^f Tsl = T31 Vsl + T32I (A19)

o Within the substrate, Tg^

^si ="^sl +l.(Ts(i_i) - 2Tg^ + Tg(^^.^p (A20)

Ms

o In the bottom elemental layer of the substrate, T

T = T sn sn fl-O - (l.yNs+ 1)1 + /|-yTs(n-l) + Va> (A21)

s/ J \ s- A. 3 FINITE-DIFFERENCE EQUATIONS FOR THE TWO-COMPONENT SYSTEM

The six types of finite-difference equations for the two component system

are:

o At the bitumen surface, T

bl

^bl = Tblfl-O - (-^Y^b + 1-0>1 + / |-y^b2 + Va) (A22)

o Within the bitumen, T^^^^

''^bi ~ ^bi •" i_('^b(i-l) 2T^^ + T^(i+i)) (^3)

^b

o In the bottom elemental layer of bitumen, Tbn

W = Tbn + l.K(n-l) -

f

1 + ^\ Vbn + ( -J^^sl] (A24)

\l \ ^b + M V^b + ^s/ J

A5
o In the top elemental layer of the substrate, 1 ,

«sL\H + ks/ V ^b + V J

o Within the substrate, T^^

Tsi = Tg^ + 1 (Tg(^_^) - 2Tg^ + Ts(jL+i)) (A26)

M

S

o In the bottom elemental layer ' of the substrate, T ' sn

^sn = Tsnp-0 - (l-VNg + 1)1 + /^yT^^^.^) + N^V (A27)

A6
APPENDIX B. LISTING OF COMPUTER PROGRAM

The computer program for the mathematical model for predicting the cooling

time of hot bitumen applied to various substrates under different environmental conditions was written specifically for the preparation of the

majority of the figures (4 to 16) presented in section 3.2. The Fortran

program is given in this appendix. Only one set of input data was necessary

to produce the 48 distinct curves given in the majority of the figures (4

to 16).

The initial set of input data described a model roofing system in which a given thickness of bitumen at 500°F (260°C) was applied to other components

at 120°F (49°C) with zero wind speed. After completing the calculation for

predicting the cooling time of bitumen under these initial conditions, the

program repeated the cooling calculations using new input data which was generated from the initial input data. The bitumen application temperatures

selected in the study were 500, 450, 400, and 350°F (260, 232, 204, and

177''C) and the air and initial component temperatures were 120, 70, and 20*'F

(49, 21, and -7°C). The wind speeds used in the calculations were 0, 10, 20,

and 30 mph (0, 4.5, 9.0, and 13.5 m/s). The cooling calculation using new

input data repeated in a stepwise fashion until the output to plot the 48

distinct curves in each of the figures 4 to 16 was obtained.

Although the progran was written to produce data from which the sets of

plots in figures 4 to 16 were generated, minor changes may readily be made

to allow the prediction of bitumen cooling times under environmental and

temperature conditions different than those presented here. Also, minor

changes in the program may be made to alter in an incremental fashion the

thickness of the hot bitumen or other roofing components. For example, the curves shown in figure 17 were plotted after the program was altered to

allow an iteration of the quantity of applied bitumen of about 15, 20, 25,

and 30 lbm/100 ft^ (0.73, 0.98, 1.2, and 1.5 kg/m^) in the absence of wind.

The Fortran program contains two subroutines which were useful to the NBS

research staff in the development of the program and the interpretation of

the preliminary output. Because these subroutines were useful, reference

to them was kept within the program reproduced here, but the subroutines are not given. The first of these subroutines is called "TRACEH. " Its purpose

was to facilitate "debugging" the program and to verify whether the program

was operating sequentially as expected. The second subroutine, called "PLOTT,

allowed the generation of a single temperature versus time cooling curve for

a distinct set of environmental conditions and component temperatures.

The Fortran program prepared to predict the cooling times of hot bitumen

on various substrates under different environmental conditions follows.

Bl
• • • • 3

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»-r fV «\J N C\J 3 >- U1 _i_ie • • »-a: Cbibi « bl Pk-k Z»-

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<»/) O B _J kJ O « H wm W-— * • • i/)B ai- • • MO ii bibi NbJbi< M blll33enuJ«M M M XK -ilU O Z M I M _ia.bj o • • • • »-£ »-- NKKZ K ^K XZ Zm 3m<>'<> ft. K«> M»- Z K " t o z tn 1/1 -at" -i II ZZ h- • • • • Z" MM M M M K M M M M M MK Z p< mb: Z z z

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U.M K^ MM Mb. M J _!»-»- «S • • • • • • • • OM BB BBBb. B B B OB Bb. U UB M M MM OB b. U uu

N n e e «4 e MO ee «4 M e «4 MM lOlO •O M

•4 M N M N M MM w n

Mn»inior.«»eMMn MMMMMMMMtOmn-^ mm nminiAininintf) iaMn *tf)<e^»B»eMMn•ln<e^> sae MMn«ir)«^B»eMMm«'in<eNeO'e«<Mn»ir><eNS9 mr)r>nnir>r)fo»»*9'»9'0'»»«irininr>inr>v)ininn«<o<fl<o««««««r-KNKr'KKKNN

lAinininininiAioininMininintf) miAinA tftinminininininininiflinioininintftintftinintnininininininin

Bll
O

X

bi<

bl

O

u

• •

• I •

• I •

• e •

• a • • • •

• a. H •

• t/)UJ •

• • • •

o

K» IliUl bi« IMM

IK3

O b. oco

Bit. bJbi

ZOO O »- Z

MlAbl

tkl ow uuouuuuuu o u

s

e«4^ln•lr>«^'C(^e»«M•n«|n«r>•9>o»«Mn «caicaiC«i(Da)«ao>o>9'a>ao>o>9>(r9eeeeo

lAinminiAiivmtfVinininmin miAiAin ioiAin^««o<oz

kl

w>

B12
B.l CHECK ON COMPUTER PROGRAM EXECUTION

The model presented in this paper has not been validated through field

experimentation. To determine whether the model for predicting the cooling

time of hot bitumen was functioning in a reasonable manner, the computer

program was altered to simulate a condition in which no heat could flow

from the bottom layer (infinitely insulated bottom surface) of the substrate

to the surroundings. The computer program could thus approximate heat

transfer in an infinite flat plate which is initially at a uniform temperature and then subjected to a hot environment such that heat flows from

the surroundings into the flat plate. It is noted that the four-component

model roofing system is equivalent to a flat plate if the physical properties

(i.e., temperature, thermal conductivity, and density) of the four components

are taken to be Identical. Transient heat flow and temperature distribution

in a flat plate have been calculated and the results have been given in

charts. Kreith has presented these charts and examples of their use [Bl].

The treatment here is based on the charts given by Kreith.

The following calculation was performed to compare the output of the computer

program with the charts given by Kreith. A 0.025 ft (7.6 ram) thick plate

containing 100 finite elements and having an initial temperature of 100*F

(38'*C) was subjected to elevated temperatures at the surface. Heat was allowed to flow into the plate for 100 seconds. The temperatures of the

top, twentieth, fourtieth, sixtieth, eightieth, and bottom finite elements

within the plate were calculated from the computer program. The thermal

conductivity and diffusivity of the plate were taken as 0.54 Btu*in/h»ft •**? (0.078 W/m»K) and 0.0187 ft^/h (0.48 mm^/s), respectively. The convection

coefficient was taken as 5.4 Btu/h'ft^'^F (30.6 W/m^*K). Ambient temperatures to which the surface of the plate was subjected were 200, 300, and

400°? (93, 149, and 204°C). The results from the computer program output

and those obtained from the temperature charts are given in table Bl

.

A comparison of the temperature distribution results from the computer

program output and the temperature charts shows differences between 2 and 6

percent, depending upon the temperature to which the plate was subjected and

the position of the finite element within the slab. The percent differences

were considered acceptable, since it was difficult to interpolate the temperature charts given by Kreith because of their small size. Interpolation was considered a major source of error. The results indicated that the computer

program for determining the temperature distribution in the flat plate was functioning as expected. Moreover, since only a minor change was made in

converting the bitumen cooling program into the flat plate program, the

bitumen cooling program was also considered to be functioning in a reasonable

manner

.

B13
B.I.I Reference

Bl. Krelth, Frank, Principles of Heat Transfer, International Textbook

Company, Scranton, Pennsylvania, pp. 148-132, (July 1968).

Table BI. Temperature Distribution in the Finite Elements of the

Flat Plate as Determined from the Computer Program and

the Temperature Charts.

Temperatux;e tjO ,. . Finite Element

Which the Plate

Was Subjected

Finite Element Temperature

Program Output Charts (a)

ZOO'F (93"C) top 114 46 120 49

20th 112 44 118 48

40th 111 44 116 47

60th 110 43 115 46

80th 109 43 115 46

bottom 109 43 114 45

300"*F (149**C) top 132 56 140 60

20th 128 53 136 58

, , 5- ,-^'-MS«- 40 th 125 52 132 56

60th 122 50 IX 54

80th 121 49 129 54

bottom 120 49 128 53

400'F (204*»C) top 157 69 160 71

20th 150 66 154 68

40th 144 62 148 64

60th 139 59 145 63

80th 137 58 144 62

bottom 136 58 142 61

(a) from Kreith [Bl]

B14
APPENDIX C. NOTATION

This appendix lists the notation used in this report in the development of both the nathematical

odel and the coaq>uterized progran for the prediction of the cooling tiiae of bituaen during

the construction of built-up roofing. Table CI includes the algebraic notation, Fortran

code, the description or definition, and the customary an4 S.I. units of measurement for

each term used.

Table CI. Notation and Description of Terms

Algebraic Fortran

Notation Code

Description Units

Customary SI

o **(1)

*(2) DB

* DC

* DF

* DS

C **

'pb

Sc ** Sf **

Ss **

e **

* EMB

* EMS

h ** K

** \

** \ HB

* HBC

* HBR

thermal diffusivity; o - k/C 'p

thermal diffusivity of applied bitumen

thermal diffusivity of bitumen-cement

thermal diffusivity of roofing felt

thermal diffusivity of substrate

specific heat

specific heat of applied bitumen

specific heat of bitumen-cement

specific heat of roofing felt

specific heat of substrate

surface emlttance

surface emlttance of bitumen

surface emlttance of substrate

surface emlttance of heat transmission

surface convection coefficient

surface radiation coefficient

bitumen surface coefficient of heat

transmission

bitumen surface convection coefficient

bitumen surface radiation coefficient

ft^/h m»2/s

ft2/h mm2/s

ft^/h mar/s

ft^/h mm2/8

ft^/h mm^/s

Btu»lbm/*F J/kg»K

Btu»lbm/*F J/kg'K

Btu»lbm/'F J/kg»K

Btu»lbm/«F J/kg»K

Btu»lbm/*F J/kg'K

-(3) -(3)

Btu/h. f t^. 'F W/m^.K

Btu/h.ft2»'F W/m^.K

Btu/h. ft^«'F W/m^.K

Btu/h. ft^.'F W/m^.K

Btu/h. ft^.'F W/m^.K

Btu/h. ft^.'F W/m^.K

(1)

(2)

(3)

The double asterisk indicates that the term was not used in the paper in Fortran code.

The asterisk indicates that the term was not used in algebraic notation.

The dash indicates that the term is dimensionless.

CI
Table CI. (Continued)

Algebraic Fortran

Notation Code

Description Units

Customary S.I.

US

* HSC

* HSR

k **

•^b CB

''c CC

•^f CF

•^8 CS M **

«b AMB

«c

AMC

Mf AMF

«8 AMS

N ** \ ABB

»s ABS

* NB

* NC

* NF

* NS

o **

**

SIGMA

**

TA

substrate surface coefficient of heat

transmission

substrate surface convection coefficient

substrate surface radiation coefficient

thermal conductivity

thermal conductivity of applied bitumen

thermal conductivity of bitumen-cement

thermal conductivity of roofing felt

thermal conductivity of substrate

modulus relating Ay and At

modulus for bitumen

modulus for bitumen-cement

modulus for roofing felt

modulus for substrate

Nusselt or Biot number; N » h'Ay/k

Nusselt or Blot number for bitumen

Nusselt or Biot number for substrate

number of finite-elements in the bitumen

number of finite-elements in the bitumen-cement

number of finite-elements in the felt

number of finite-elements in the substrate

density

density of bitumen

density of bitumen-cement

density of roofing felt

density of substrate

Stefan-Boltzman constant;

temperature

air temperature

Btu/h'ft2*°F W/m^'K

Btu/h»ft^«'F W/m^'K

Btu/h»ft^'°F W/m^«K

Btu^ft/h^ft^^'F W/m»K

Btu^ft/h-ft^'^F W/m»K

Btu«ft/h«ft^»°F W/m«K

Btu'ft/h'ft^-'F W/m«K

Btu^ft/h^ft^'^F W/m»K

Ibm/ft^ kg/m^

Ibm/ft^ kg/m^

Ibm/ft^ kg/m^

Ibm/ft^ kg/m^

Ibm/ft^ kg/m^

.u/h»ft^'° R* W/m2.K^

•f •c

•f •c

C2
Table CI. (Continued)

Algebraic Fortran

Notation Code

Description Units

Customary SI

"F 'c

•f "c

'F »c

»F "c

•f "c

"F •c

°F 'c

°F 'c

"F 'c

"F •c

"F "c

»F

• °c

•f "c

Of "c

'¥ "c

•f •c

"F °c

°F 'c

•f "c

•f 'c

•f "c

"f °c

°F »c

•f •c

°F •c

•f »c

•f 'c

Ts ** t

Tbl TB(1) t

^bl TB(1) f

^bi TB(I) t

^bi TB(I) f

Tbn TB(NB) t

Tin TB(NB) f Td TC(1) t

^cl TC(1) f

Tci TC(I) t

T^l TC(I) f

Ten TC(NC) t

Ten TC(NC) f

Tfl TF(1) t

Tfl TF(1) f

Tfl TF(I) t

Tfi TF(I) f

Tfn TF(NF) t

Tfn TF(NF) f

Tsl TS(1) t

Tsl TS(1) f

Tsl TS(I) t

Ts'i TS(I) f

Tan TS(NS) t

•sn TS(NS) f

* TBO li

* ^CO li

emperature of a surface

emperature of the top bitumen surface element

uture temperature of the top bitumen surface

element

emperature of a bitumen element

uture temperature of a bitumen element

emperature of the bottom bitumen element

uture temperature of the bottom bitumen element

emperature of the top bitumen-cement element

uture temperature of the top bitumen-cement

element

emperature of a bitumen- cement element

uture temperature of a bitumen-cement element

emperature of the bottom bitumen-cement element

uture temperature of the bottom bitumen-cement

element

emperature of the top felt element

uture temperature of the top felt element

emperature of a felt element

uture temperature of a felt element

emperature of the bottom felt element

uture temperature of the bottom felt element

emperature of the top substrate element

uture temperature of the top substrate element

emperature of a substrate element

uture temperature of a substrate element

emperature of the bottom substrate element

uture temperature of the bottom substrate element

nltlal temperature of bitumen

nitlal temperature of bitumen-cement

C3
Table CI . (Continued)

Algebraic Fortran

Notation Code

Description Units

Customary SI

TFO initial temperature of roofing felt •f

* TSO initial temperature of substrate

t ** time

At DELT a small finite time increment or interval

y

** a space coordinate for the heat flow

Ay DELY a small finite distance

"F

h

h

ft

ft

C4
NBS-114A IREV. 2-BC) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS TN 1135

2. Performing Organ. Report No 3. Publication Date

March 1981

4. TITLE AND SUBTITLE

Cooling of Bitiimen During Construction of Built-Up Roofing Systems - A Mathematical Model

5. AUTHOR(S)

Walter J. Rossiter, Jr., Robert G. Mathey, Herbert W. Busching, and William C. Cullen

6. PERFORMING ORGANIZATION (/f joint or other then NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

Department of Civil Engineering

Clemson University

Clemson, S.C. 29613

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City. State. ZIP)

National Bureau of Standards

Department of Commerce

Washington, DC 20234

10. SUPPLEMENTARY NOTES

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

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here) Construction of bituminous built-up roofing systems in the United States generally

involves the application of hot bitumen to the roofing components, including deck,

insulation, and felts_, to adhere them to each other and to form a waterproof membrane.

Adequate adhesion of the bitumen to the roofing component materials may be obtained

only when the hot bitumen is applied at a viscosity sufficient to flow uniformly, to cover the component surfaces or substrate completely, and to provide the proper

thickness. During construction, rapid cooling of hot bitumen increases its viscosity

significantly. If the viscosity becomes too high, poor adhesion between components,

voids within the bitumen, and an excessive and non-uniform thickness of the bitumen

may result.

This report describes a mathematical model based on finite-difference equations for

calculating transient heat flow to estimate the cooling time of hot roofing bitumen.

Estimates of the time required for hot bitumen to cool from its application temperature

to 300°F (149°C) were computed as a function of material and environmental factors

including: quantity of applied bitumen, bitumen application or contact temperature, air

temperature, wind speed, and thermal properties of the bitumen and of the roofing

components. The model was used to predict cooling times expected for hot asphalt

applied to typical substrates with thermal properties representative of those of polyurethane foam and glass fiber insulation boards, insulating concrete, plywood, concrete

and steel decks, and roofing felt on decks or insulations. In addition, the model was

used to predict cooling times for hot coal tar pitch applied to concrete and to felt

adhered to glass fiber insulation. The results of the calculations demonstrate the

widely-varying bitumen cooling times which depend upon the component material to which

the bitumen is applied and the environmental conditions during application. Under

certain environmental conditions, hot bitumen applied to some substrates cools extremal''

rapidly. In these cases,

suffi c ient time for proper applica t ion may no t b a avail abl a

12. KEY WORDS (S/x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Bitumen application temperatures; bituminous roofing; built-up roofing; cooling of

roofing bitumens; mathematical model; roofing.

13. AVAILABILITY

r^ Unlimited

1 I For Official Distribution. Do Not Release to NTIS

r^ Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

I I Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

76

15. Price

irU.S GOVERNMENT PRINTING OFFICE: 1981-340-997:1568 USCOMM-DC 6043-P80



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