12.5.1 Heat Production

, the degree of reaction. This variable r is equal to the momentary cumulative heat production divided by the total heat production:

r = $\displaystyle {\frac{{ \int_{0}^{t} q_{V.\mathrm{hy}}(r,\tau) \,\mathrm{d}\tau }}{{ \int_{0}^{\infty} q_{V.\mathrm{hy}}(r,\tau) \,\mathrm{d}\tau }}}$

(12.6)

The quantity of the produced heat is a function of temperature history. The momentary heat production rate q can be defined as

q_V.hy(r, T) = $\displaystyle \alpha$ ^.q_r(r)^.q_T(T)

(12.7)

with r the degree of reaction, ( 0 $\leq$ r $\leq$ 1 )T the temperature in °C , $\alpha$ the maximum value of the heat production rate, q_r the degree of reaction dependent heat production (scaled to one), and q_T the temperature dependent heat production. In this context DIANA uses

q_T(T) = e^-

(12.8)

where c_A is the constant of Arrhenius which can be dependent on temperature and/or degree of reaction [§12.5.4].

You may specify the degree of reaction dependent heat production q_r directly in the input file [§12.5.1.2]. However, in most cases it is more convenient to use preprocessing [§12.5.1.1]. In either case you may specify the initial degree of reaction r₀ at initialization of the nonlinear transient analysis [Vol. Analysis Procedures].^12.1

Note that the implementation of (12.8) in DIANA requires that you specify temperatures in degrees Celsius.

As an alternative to direct input or preprocessing you may specify the heat production rate via a user-supplied subroutine [§12.5.1.3].

12.5.1.1 Preprocessing

To determine heat production via preprocessing you must specify the capacitance and a diagram of temperature versus time under adiabatic hydration conditions and let DIANA's Module HEATTR generate q_r from this input.

(syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...t} {]} \\ [.5ex] \>\>$\cdots\;$\>\emph{capacitance} \end{tabbing} \end{figure}$

ADIAB: te1 ...ten are temperature values in °C during adiabatic hydration heat development, corresponding to ages ti1 ...tin.
Arrhenius constant: specification [§12.5.4].
capacitance: specification [§12.5.2].

DIANA derives the heat production q(t) from

q(t) = c $\displaystyle \left(\vphantom{T,r}\right.$ T, r $\displaystyle \left.\vphantom{T,r}\right)$ $\displaystyle {\frac{{\partial T}}{{\partial t}}}$

(12.9)

with c(T, r) the capacitance which can depend on temperature and degree of reaction. DIANA approximates (12.9) and (12.6) numerically at n user-specified time points.

r_m	= $\displaystyle {\frac{{Q_{m}}}{{ Q_{n} }}}$	(12.10)
Q_m	$\displaystyle \approx$ $\displaystyle \sum_{{i=1}}^{{m}}$ c $\displaystyle \left(\vphantom{ T_{i}^{},r_{i}^{} }\right.$ T_i^, r_i^ $\displaystyle \left.\vphantom{ T_{i}^{},r_{i}^{} }\right)$ $\displaystyle \Delta$ T_i, m = 1,..., n	(12.11)

with

$\displaystyle \Delta$ T_i = T_i - T_i-1 ; r_i^* = $\displaystyle {\frac{{ r_{i-1} + r_{i} }}{{ 2 }}}$ ; T_i^* = $\displaystyle {\frac{{ T_{i-1} + T_{i} }}{{ 2 }}}$

(12.12)

If the capacitance depends on degree of reaction, the set of equations is solved iteratively. Finally DIANA approximates $\partial$ T/ $\partial$ t in (12.9) numerically at m = 1,..., n points, and uses (12.7) and (12.8) to find the corresponding q_r.m .

q_m	= c_m $\displaystyle {\frac{{\partial T}}{{\partial t}}}$ $\displaystyle \approx$ c_m $\displaystyle {\frac{{ T_{m+1} - T_{m-1} }}{{ t_{m+1} - t_{m-1} }}}$	(12.13)
$\displaystyle \alpha$ q_r.m	= $\displaystyle {\frac{{q_{m}}}{{ q_{T.m} }}}$	(12.14)

DIANA uses q_r.0 = q_r.n+1 = 0 , assuming q_r.0 is q_r at r = 0 and q_r.n+1 is q_r at r = 1 . This implies that in the actual analysis a small initial degree of reaction will be necessary to start the development of hydration heat.

The following fragment illustrates the input data syntax of adiabatic hydration heat development.

(file.dat)

'MATERI'
   1  ADIAB   0.           25.0
              1.800E+04    2.606E+01
              3.600E+04    3.231E+01
              5.400E+04    3.992E+01
              7.200E+04    4.387E+01
              9.000E+04    4.608E+01
              1.080E+05    4.740E+01
              1.206E+05    4.799E+01
              1.512E+05    4.873E+01
              1.800E+05    4.898E+01

12.5.1.2 Direct Input

For direct input of heat production of the hydration process you must specify the following input data.

(syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ....6mm}{[}\>$\cdots\;$\>\emph{Arrhenius constant} {]} \end{tabbing} \end{figure}$

REACTI: r1 ...rn are the degrees of reaction r_{i=1, n} ( 0 $\leq$ r $\leq$ 1 ) for which the material properties are specified.
PRDKAR: q1 ...qn are values of the normalized heat production q_{r_{i=1, n}} for the corresponding degrees of reaction.
MAXPRD: qtot is the totally produced heat $\int_{{0}}^{{\infty}}$ q_V.r dt per unit volume.
ALPHA: qmax is the maximum value of heat production rate $\alpha$ .
Arrhenius constant: specification [§12.5.4].

(file.dat)

'MATERI'
   1  REACTI      0.0    0.1    1.0
      PRDKAR      0.05   2.0    0.0
      MAXPRD      1.00E3
      ALPHA       0.72E9

In this example input for material 1, the scaled degree of reaction dependent heat production q_r is specified for degrees of reaction 0, 0.1 and 1. The values of q_r for these three degrees of reaction are 0.05, 2 and 0 respectively. The total cumulative heat production is 1000. The multiplication factor for the heat production is 0.72 x 10⁹ .

12.5.1.3 User-supplied Subroutine

As an alternative to the definition of the heat production for potential flow analysis via the degree of reaction (12.7), you may specify the heat production via a user-supplied subroutine. This constitutes a useful mechanism for general specification of heat production, for instance with a mathematical function.

(syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...\>\>\texttt{MAXPRD}\>\texttt{\textit{qtot}}$_{r}\,$ \end{tabbing} \end{figure}$

USRHTP: specifies that the heat generation rate due to hydration is determined via a user-supplied subroutine. DIANA passes the keyword usrkey to the first argument of this subroutine. The heat generation rate can be any function of the temperature and the degree of reaction.
MAXPRD: qtot is the totally produced heat $\int_{{0}}^{{\infty}}$ q_V.hy dt per unit volume.

If USRHTP is specified in table 'MATERI' then subroutine USRHTP must be supplied to set up heat production for potential flow analysis with hydration reaction.

(Fortran)

      SUBROUTINE USRHTP( usrkey, te, re, htp )

$\begin{figure}\centering \begin{tabbing} out~~\=\textsc{dbl}~~\=\texttt{abcdef... ...c{dbl}\>\texttt{htp}\>Heat production rate. \\ [-3ex] \end{tabbing} \end{figure}$

usrkey: is the user-supplied keyword from input table 'MATERI'. This can be used to model various functions for the heat production rate within one subroutine.
te: is the temperature T .
re: is the degree of reaction r .
htp: is the heat production rate generated by hydration.

See also Volume Analysis Procedures for a general description of DIANA's user-supplied subroutine option.

Next: 12.5.2 Conductivity and Capacitance Up: 12.5 Hydration Heat Previous: 12.5 Hydration Heat Contents Index

DIANA-9.3 User's Manual - Material Library
First ed.

Copyright (c) 2008 by TNO DIANA BV.