Heat Transfer Processes

Abstract

In this chapter a system of uniform temperature in contact with a thermal bath must be heated in a fixed time interval. Several ways of defining the lost work associated with the heating process are presented. The minimum entropy generation is generally not equivalent to the minimum lost available work. Two optimal control problems are defined and solved for various heat transfer mechanisms among which Newtonian heat convection and radiative heat transfer. The optimal paths are obtained for both objective functions (i.e. entropy generation and lost available work).

Appendices

Appendix 8A

A procedure to relate the lost available work with the entropy generation in case of a more complex model has been proposed by Badescu (2004) and is presented here. It follows the main ideas of Hoffmann et al. (1989). The meta-system consists of four systems. They are the heat reservoir 1, the system 2, a work reservoir (denoted by ${\infty }$) and an environment (i.e., a thermodynamic system whose constant intensities, the temperature $T_{0}$ and the pressure $p_{0}$, define the availability scale). Note that the heat reservoir 1 is a particular kind of thermodynamic bath (i.e. a fully controllable environment). Only the case $T_{1} > T_{2} > T_{0}$ is considered here.

In order to evaluate the loss of availability during the heat transfer process one recalls that for closed systems the availability $A$ is defined as:

$$ A \equiv U - T_{0} S + p_{0} V $$

(8.A.1)

Here $U,V$ denote internal energy and volume, respectively. Small changes in internal energy and availability at constant volume are defined as:

$$ \begin{aligned} dU_{i} & = T_{i} dS_{i} - p_{i} dV_{i} = T_{i} dS_{i} , \\ dA_{i} & = \left( {T_{i} - T_{0} } \right)dS_{i} - \left( {p_{i} - p_{0} } \right)dV_{i} = \left( {T_{i} - T_{0} } \right)dS_{i} ,\quad i = 0,1,2,\infty . \\ \end{aligned} $$

(8.A.2a,b)

In deriving Eqs. (8.A.2b), (8.A.1) and (8.A.2a) were used. One uses Eq. (8.A.2b) for a time interval dt. Then, the heat transfer process between systems 1 and 2 determines the availability changes:

$$ \begin{aligned} dA_{1} & = \left( {T_{1} - T_{0} } \right)dS_{1} = \left( {T_{1} - T_{0} } \right)\left( {qdt/T_{1} } \right), \\ dA_{2} & = \left( {T_{2} - T_{0} } \right)dS_{2} = \left( {T_{2} - T_{0} } \right)\left( {qdt/T_{2} } \right), \\ \end{aligned} $$

(8.A.3a,b)

and $dA_{0} = dA_{\infty } = 0$.

The maximum available work $dW_{l,\hbox{max} }$ lost during the time interval $dt$ is defined as the total availability loss associated to the heat transfer process:

$$ dW_{l,\hbox{max} } \equiv \dot{W}_{l,\hbox{max} } dt = \sum\limits_{i} {\left( { - dA_{i} } \right)} = T_{0} \dot{S}_{12} dt $$

(8.A.4)

Here Eqs. (8.A.3a), (8.A.3b) and (8.2.3) were used.

A more involved treatment should take into account that the hypothetical process of generating the work $dW_{l,\hbox{max} }$ is irreversible. Then, (eventually) part of $dW_{l,\hbox{max} }$ goes to the work reservoir and the remaining part is degraded into heat, that is transferred to the other three systems. A vector $\alpha = \left( {\alpha_{0} ,\alpha_{1} ,\alpha_{2} ,\alpha_{\infty } } \right)$, $\sum {\alpha_{i} = 1}$ is used to indicate which fraction of $dW_{l,\hbox{max} }$ is transmitted to each system. The last component of $\alpha$ shows the fraction of $dW_{l,\hbox{max} }$ that goes to the work reservoir. Note that some of the systems where part of $dW_{l,\hbox{max} }$ is transferred as heat appear at temperatures different from the temperature $T_{0}$ of the environment. Therefore, the transferred heat also transfers residual availability (defined as the work produced by an engine while letting that heat move to the environment). From this perspective it is allowed that not all of the availability be lost during the heat transfer process envisaged here. Consequently, different degrees of availability loss are possible. The maximum availability loss is sometime called “work deficiency ”. It is defined as the total loss of availability which would have resulted if all the available work were lost to the environment (Hoffmann et al. 1989).

Some of the work $dW_{l,\hbox{max} }$ is degraded into heat and fractions $\alpha_{i}$ are transferred to each of the four systems. The entropy changes are then:

$$ dS_{i} = \alpha_{i} \frac{{dW_{l,\hbox{max} } }}{{T_{i} }},\quad i = 0,1,2,{\infty }. $$

(8.A.5)

Note that no entropy is transferred to the work reservoir (i.e. $dS_{{\infty }} = 0$) and the associated temperature $T_{{\infty }}$ is taken to be infinite. The corresponding availability changes are obtained using Eqs. (8.A.2b) and (8.A.5):

$$ dA_{i} = \left( {1 - \frac{{T_{0} }}{{T_{i} }}} \right)\alpha_{i} dW_{l,\hbox{max} } ,\quad i = 0,1,2,{\infty }. $$

(8.A.6)

One can easily see that $dA_{0} = 0$. The total available work lost during the time interval $dt$ is given by:

$$ dW_{l} \equiv \sum\limits_{i} {\left( { - dA_{i} } \right) = \left( {\alpha_{0} + \alpha_{1} \frac{{T_{0} }}{{T_{1} }} + \alpha_{2} \frac{{T_{0} }}{{T_{2} }}} \right)dW_{l,\hbox{max} } } $$

(8.A.7)

One divides Eq. (8.A.7) by $dt$ and one finds:

$$ \dot{W}_{l} = \left( {\alpha_{0} + \alpha_{1} \frac{{T_{0} }}{{T_{1} }} + \alpha_{2} \frac{{T_{0} }}{{T_{2} }}} \right)T_{0} \dot{S}_{12} $$

(8.A.8)

Here Eq. (8.A.4) was used. Various particular cases can be obtained from Eq. (8.A.8), depending on the values of the time-dependent coefficients $\alpha_{i} \left( {i = 0,1,2,{\infty }} \right)$. They include the common case $\alpha_{0} = 1$, $a_{1} = a_{2} = a_{\infty } = 0$, when $\dot{W}_{l} = \dot{W}_{l,\hbox{max} } = T_{0} \dot{S}_{12}$.

Appendix 8B

Results presented in Badescu (2004) are summarized here for different heat transfer mechanisms in case of three heating strategies, namely minimum entropy generation, constant heat reservoir temperature and constant heat flux. With two exceptions, these results were also presented in Andresen and Gordon (1992).

For arbitrary n the paths associated to minimum entropy generation are given by Eqs. (7) and (8) of Andresen and Gordon (1992). In the dimensionless form adopted in Sect. 8.2 they are given by:

$$ y^{n} - C_{S} y^{{\frac{n + 1}{2}}} - z^{n} = 0 $$

(8.B.1)

$$ \frac{dy}{d\omega } = nC_{S} A_{n} y^{{\frac{n + 1}{2}}} \frac{{\left( {y^{n} - C_{S} y^{{\frac{n + 1}{2}}} } \right)^{{\frac{n - 1}{n}}} }}{{ny^{n - 1} - \frac{n + 1}{2}C_{S} y^{{\frac{n - 1}{2}}} }} $$

(8.B.2)

where $C_{S}$ is an integration constant.

Equations (8.B.1) and (8.B.2) were solved with values for n, $A_{n}$ and $z_{f}$ as input, taking into account that $z\left( {\omega = 0} \right) = 1$ and $z\left( {\omega = 1} \right) = z_{f}$. First an analytical solution was looked for. When a numerical approach was necessary the following procedure was adopted. A trial value for the integration constant $C_{S}$ was chosen. For that trial value the next steps were performed. First, one assumed $\omega = 0$ (i.e. $z\left( {\omega = 0} \right) = 1$) and Eq. (8.B.1) was solved numerically to find a guess for the value $y\left( {\omega = 0} \right)$. This was subsequently used as an initial value for Eq. (8.B.2). Numerical integration of (8.B.2) allowed to obtaining $y\left( {\omega = 1} \right)$. This last value was replaced in Eq. (8.B.1), which was solved in the unknown $z\left( {\omega = 1} \right)$. Finally, the following quantity was evaluated:

$$ F\left( {C_{S} } \right) \equiv \left[ {z_{f} - z\left( {\omega = 1} \right)} \right]^{2} $$

(8.B.3)

$F\left( {C_{S} } \right)$ vanishes for the right choice of $C_{S}$. In case of a significantly large value of $F\left( {C_{S} } \right)$, another value of $C_{S}$ is chosen and the procedure is repeated. $F\left( {C_{S} } \right)$ was minimized by using the routine FMIN of Kahaner et al. (1989). Once the appropriate value of the integration constant $C_{S}$ was determined, Eqs. (8.B.1) and (8.B.2) are solved for the optimal paths of $z$ and y. The optimal path for u are then obtained as:

$$ u = y/z $$

(8.B.4)

The results are presented in Tables 8.3, 8.4 and 8.5 by using the dimensionless notation of Eqs. (8.2.18), (8.2.19) and (8.2.22).