Find the Steady State Solution of the Problem

Steady State Problem

The solution of the steady state problem yields the optimum steady-state operating point which usually lies at the intersection of active constraints.

From: Computer Aided Chemical Engineering , 2020

TRANSPORT

Esam M.A. Hussein , in Radiation Mechanics, 2007

4.1.7 Adjoint transport equation

Let us consider a steady-state problem and write the corresponding particle transport equation in the compact form:

(4.31) $T\phi =-Q$

where T is an operator that incorporates all the terms in Eq. (4.5) except for the external source, Q( $\overrightarrow{r}$ , E, $\overrightarrow{\Omega }$ ). Equation (4.31) describes the transport of particles within the domain of the problem. This problem has a natural boundary condition of no incoming particles at the free surfaces defining the external boundaries of the problem's domain. In other words, the domain of the problem is defined by free-surface boundaries where no particles enter into the problem's domain. Let us turn the problem around and rather than considering a source, we will consider a detector with a response function D( $\overrightarrow{r}$ , E, $\overrightarrow{\Omega }$ ). With the detector as a "source", one can write an equation "adjoint" to Eq. (4.31):

(4.32) $T^{+}{\phi }^{+}=-D$

This equation has the boundary condition of no adjoint flux, φ⁺ , at the free surface adjoint to that of Eq. (4.31). Equations (4.31) and (4.32) are related by:

(4.33) $\int Q(\overrightarrow{r},E,\overrightarrow{\Omega }){\phi }^{+}(\overrightarrow{r},E,\overrightarrow{\Omega })\text{d}\overrightarrow{r}\text{d}\overrightarrow{\Omega }\text{d}E=\int D(\overrightarrow{r},E,\overrightarrow{\Omega })\phi (\overrightarrow{r},E,\overrightarrow{\Omega })\text{d}\overrightarrow{r}\text{d}\overrightarrow{\Omega }\text{d}E$

The right-hand side of Eq. (4.33) is obviously the overall response of the detector to the flux induced by the source Q, while φ⁺ in the left-hand side of the equation can be seen as the source component contributing to the detector response.

In the above analysis, so far, no conditions have been imposed on the source Q. Any source value or distribution will result in a detector response that satisfies Eq. (4.33). For a delta-function unit source, δ( ${\overrightarrow{r}}_0$ , ${\overrightarrow{\Omega }}_0$ , E ₀), Eq. (4.33) gives the value of the adjoint flux:

(4.34) ${\phi }^{+}({\overrightarrow{r}}_0,{\overrightarrow{\Omega }}_0,E_0)=\int D(\overrightarrow{r},E,\overrightarrow{\Omega })\phi (\overrightarrow{r},E,\overrightarrow{\Omega })\text{d}\overrightarrow{r}\text{d}\overrightarrow{\Omega }\text{d}E$

If the adjoint flux is zero, then the unit source δ( ${\overrightarrow{r}}_0$ , ${\overrightarrow{\Omega }}_0$ , E ₀) would have a nil contribution to the detector D( $\overrightarrow{r}$ , $\overrightarrow{\Omega }$ , E). The value of the adjoint flux is in turn an indication of the "importance" of a particle in contributing to a detector. A particle at the free surface boundaries of the problem domain have a flux of zero importance, since the outgoing flux at the boundary is naught and cannot in turn contribute to the problem.

We have considered a steady-sate analysis for simplification, but the above analysis can be easily extended to a time-varying problem, leading to the adjoint Boltzmann transport equation:

(4.35) $\begin{array}{c}-\frac{1}{v}\frac{\partial \phi (\overrightarrow{r},E,\overrightarrow{\Omega },t)}{\partial t}=-\overrightarrow{\Omega }ċ\nabla \phi *(\overrightarrow{r},E,\overrightarrow{\Omega },t)-{\Sigma }_{\text{total}}(\overrightarrow{r},E,t)\phi *(\overrightarrow{r},E,\overrightarrow{\Omega },t)\\ +\int {\Sigma }_{\text{scatter}}(\overrightarrow{r},E^{\prime }\to E;{\overrightarrow{\Omega }}^{\prime }\to \overrightarrow{\Omega },t)\phi *(\overrightarrow{r},E^{\prime },{\overrightarrow{\Omega }}^{\prime },t)\text{d}{E}^{\prime }\text{d}{\overrightarrow{\Omega }}^{\prime }\\ +\int v{\Sigma }_{\text{fission}}(\overrightarrow{r},E^{\prime }\to E;{\overrightarrow{\Omega }}^{\prime }\to \overrightarrow{\Omega },t)\phi *(\overrightarrow{r},E^{\prime },{\overrightarrow{\Omega }}^{\prime },t)\text{d}{E}^{\prime }\text{d}{\overrightarrow{\Omega }}^{\prime }\\ +D(\overrightarrow{r},E,\overrightarrow{\Omega },t)\end{array}$

This equation is a "final-value" problem, unlike Eq. (4.5) which is an initial-value problem, in which the flux is determined in subsequent times based on values at a previous time. In Eq. (4.35), the adjoint flux, φ*, is given at a final time, t _f, and the values at earlier times are determined by integrating in reverse time, as evident by the negative sign of the time derivative in Eq. (4.35) compared to that in Eq. (4.5). That is, an adjoint source (detector) affects the importance of particles at earlier times, but has no effect on particles after they are detected. On the other hand, an ordinary source has no effect on the regular flux at earlier times, but changes its value at later times. Similarly, the kinematics of the adjoint transport equation is backward, i.e. instead of a particle losing energy as it moves from a source to detector, it gains energy in adjoint calculations, since they are directed from a detector to a source. This again is physically plausible, since a detector receives particles that have interacted (losing energy in the process), but when acting as "an adjoint source" a detector would "energize" particles to bring them back to the energy of the initial source.

Adjoint calculations are useful, for example, in determining the worth of a control rod in a reactor, where the rod acts as an absorber of radiation, in a manner analogous to that of a detector. However, because of the small volume of a control rod, it is unlikely to receive much radiation, in comparison to the rest of the reactor volume. Another example is in estimating the radiation dose at points around a large distributed source of radiation, such as a cask containing nuclear waste, where adjoint calculations can be used to determine the source regions and energies of most relevance to the magnitude of the recorded dose. In performing Monte Carlo simulations, a rough adjoint calculation can be used to determine the importance-sampling (biasing) parameters that can be used to reduce the variance of the results (see Section 4.4.7).

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Two-Dimensional Problems

Singiresu S. Rao , in The Finite Element Method in Engineering (Sixth Edition), 2018

15.1 Introduction

For a two-dimensional steady-state problem, the governing differential equation is (Fig. 15.1A)

Figure 15.1. Two-dimensional problem.

(15.1) $\frac{\partial }{\partial x}\left(k_x\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left(k_y\frac{\partial T}{\partial y}\right)+\dot{q}=0$

and the boundary conditions are

(15.2) $T=T_o\left(x,y\right)\text{on}{S}_1$

(15.3) $k_x\frac{\partial T}{\partial x}{l}_x+k_y\frac{\partial T}{\partial y}{l}_y+q=0\text{on}{S}_2$

(15.4) $k_x\frac{\partial T}{\partial x}{l}_x+k_y\frac{\partial T}{\partial y}{l}_y+h(T-T_{\infty })=0\text{on}{S}_3$

where k _x and k _y are thermal conductivities in the principal (x and y) directions, $\dot{q}$ is the strength of heat source, q is the magnitude of boundary heat flux, h(T   −   T _∞) is the surface heat flow due to convection, and l _x and l _y are the direction cosines of the outward normal to the surface.

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Erbium-Doped Fiber Amplifiers for Dynamic Optical Networks

Atul Srivastava , Yan Sun , in Guided Wave Optical Components and Devices, 2006

3.1 Gain Dynamics of Single EDFA

After years of extensive study on steady state problems, recently much attention has been focused on dynamics. The dynamics of EDFAs are generally considered to be slow as a result of the long spontaneous lifetime of around 10 ms. For transmission of high-speed data, the gain of EDFAs is undisturbed by signal modulation. Furthermore, EDFAs, in contrast to semiconductor laser amplifiers, do not introduce intersymbol interference in single channel systems or cross-talk among channels in WDM systems. This is one of the chief advantages of EDFAs and the reason why steady-state models have been used to model transmission systems. The results of measurement of cross-talk between two wavelength signals traversing an unsaturated single-stage EDFA is shown in Fig. 12.6. The frequency response of the cross-talk shows a corner frequency to be in the few kHz range with corresponding gain recovery time constant between 110 and 340 μs [18].

FIGURE 12.6. Frequency response of cross-talk between two channels in an EDFA [18].

The speed of gain dynamics in a single EDFA is in general much faster than the spontaneous lifetime because of the gain saturation effect. EDFAs with high output powers for multichannel optical networks are strongly saturated, and the resulting effective time constants are reduced to the order of tens of microseconds. In a recent report, the characteristic transient times were reported to be tens of microseconds in a two-stage EDFA [19]. The measurement of transient behavior of surviving channel power in an EDFA operating under saturation, suitable for an eight-channel multiwavelength network, was presented. In the experiment, the total input power to the EDFA corresponded to 7-dBm input power, as would result from eight WDM channels with an input power of − 2dBm each. The amplifier gain was 9 dB with a corresponding total output power for all channels of 16 dBm. The transient behavior of surviving channels when seven channels are dropped is shown in Fig. 12.7. The upper trace in the figure shows the modulation signal applied to the DFB laser power supply. The transition time for the laser light output to reach 90% of the steady-state value is less than 3 μs. The current supply to the laser diode has a few-microsecond electronic delay in the response of the surviving channels after the channels are dropped. After this short delay, a transient in the power of the surviving channels is observed. During the transient the power rises to 90% of the steady-state value in nearly 80 μs. The transient becomes slower as the number of dropped channels increases. The transient behavior of surviving channel power for the cases of one, four, and seven dropped channels is shown in Fig. 12.8. In the case of seven dropped channels, the transient time constant is nearly 52 μs. As can be seen, the transient becomes faster as the number of dropped channels decreases. The time constant decreases to 29 μs when only one out of eight channels is dropped. The rate equations [20] for the photons and the populations of the upper (⁴I_13/2) and lower (⁴I_15/2) states can be used to derive the following approximate formula for the power transient behavior [21]:

FIGURE 12.7. Oscilloscope trace showing the time dependence of the surviving chanel power (P _SS) along with the dropped channel power (P _SD) [19].

FIGURE 12.8. Measured and calculated surviving power transients for the cases of one-, four-, and seven-channel loss out of eight WDM channels.

(11) $P\left(t\right)=P(\infty ){\left[P\left(0\right)/P(\infty )\right]}^{\left(-t/{\tau }_e\right)}$

where P(0) and P(∞) are the optical powers at time t = 0 and t = ∞, respectively. The characteristic time τ_e is the effective decay time of the upper level averaged over the fiber length. It is used as a fitting parameter to obtain best fit with the experimental data. The experimental data are in good agreement (cf. Fig. 12.8) with the model for the transient response. The model has been used to calculate the fractional power excursions in decibels of the surviving channels for the cases of one, four, and seven dropped channels. The times required to limit the power excursion to 1 dB are 18 and 8μs when four or seven channels are dropped, respectively. The time constant of gain dynamics is a function of the saturation caused by the pump power and the signal power. Present-day WDM systems with 40–100 channels require high-power EDFAs in which the saturation factor becomes higher, leading to a shorter transient time constant. Dynamic gain control of the EDFAs with faster response times is necessary to control the signal power transients.

A model of EDFA dynamics is needed to understand the transient behavior in large system or networks. A simple model has been developed for characterizing the dynamic gain of an EDFA. The time dependent gain is described by a single ordinary differential equation for an EDFA with an arbitrary number of signal channels having arbitrary power levels and propagation directions. Most previous EDFA models are represented by sets of coupled partial differential equations [22, 23], which can be solved only through iterative computationally intensive numerical calculations, especially for multichannel WDM systems with counterpropagating pump or signals. The time-dependent partial differential equations can, however, be dramatically reduced to a single ordinary differential equation. The mathematical details of the model are provided in [24]. Here the simulation results from the model are compared with the measured time-dependent power excursions of surviving channels when one or more input channels to an EDFA are dropped. The structure of the two-stage EDFA used in the experiment [19] and simulation is shown in the inset of Fig. 12.9. The experimentally measured power of the surviving channel when one, four, or seven out of out of eight WDM channels are dropped is plotted in Fig. 12.9. It is seen from the figure that the simulation results agree reasonably well with the experimental data without any fitting parameters. The exception is the 0.9-dB difference at the most for the seven-channel drop case. This discrepancy is believed to arise from pump excited state absorption at high pump intensity. The model can be very useful in the study of power transients in amplified optical networks. A summary of the time constants for both the unsaturated and saturated EDFAs is given in Fig. 12.10. The saturation factor increases with both the signal input power and pump power and gives rise to a shorter gain dynamics time constant as governed by the lifetime of the upper state.

FIGURE 12.9. Comparison between theory and experiment for output power excursions of surviving channels from a two-stage amplifier when one, four, and seven input channels are dropped.

FIGURE 12.10. Lifetimes of saturated and unsaturated EDFAs.

A dynamic behavior study of L-band EDFA has been carried out [25]. In this work, the transient response of the surviving channels in a two-stage L-band EDFA under different channel loading conditions was reported. The observed dynamic behavior of an L-band EDFA shows similar dependence on the output power and the number of dropped channels to that in a C-band EDFA. However, the magnitude of the response time is very different. The response time constants as a function of the number of dropped channels under different saturation conditions is shown in Fig. 12.11. The time constants are about 105 and 260 μs when one and seven channels, respectively, out of eight channels are dropped and the amplifier is well saturated. These values are about four to five times larger than that observed in a C-band EDFA. The difference can be explained by the different intrinsic saturation power in these two bands.

FIGURE 12.11. Response time constant vs. the number of dropped channels under different saturation conditions in a two-stage L-band EDFA [25].

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Compressible High-Speed Gas Flow

O.C. Zienkiewicz , ... P. Nithiarasu , in The Finite Element Method for Fluid Dynamics (Seventh Edition), 2014

7.4 Numerical Approximations and the CBS Algorithm

Various forms of finite element approximation and of solution have been used for compressible flow problems. The first successfully used algorithm here was, as we have already mentioned, the Taylor-Galerkin procedure either in its single-step or two-step form. We have outlined both of these algorithms in Appendix D. However, the most generally applicable and advantageous form is that of the CBS algorithm which we have presented in detail in Chapter 3. In all compressible flows in certain parts of the domain where the velocities are small, the flow is nearly incompressible and without additional damping the direct use of the Taylor-Galerkin method may result in oscillations there. We indeed mentioned an example of such oscillations in Chapter 3 where they are pronounced near the leading edge of an aerofoil even at quite high Mach numbers (Fig. 3.4). With the use of the CBS algorithm such oscillations disappear and the solution is perfectly stable and accurate.

In the same example we also discussed the single-step and two-step forms of the CBS algorithm. We recommend the two-step procedure, which is only slightly more expensive than the single-step version but more stable.

As we have already remarked if the algorithm is used for steady-state problems it is always convenient to use a localized time step rather than proceed with the same time step globally. The full description of the local time step procedure is given in Section 3.4.4 of Chapter 3 and this was invariably used in the examples of this chapter when only the steady state was considered. We now summarize the explicit solution procedure below:

do i = 1,number of time steps

step1: calculation of intermediate momentum

step2: calculation of density

step3: correction of momentum

step4: energy equation

step5: pressure calculation from energy and density

enddo !i

For further details on the algorithm, readers are referred to Chapter 3. If shock capturing or any other form of smoothing is found necessary this is integrated into the first step of the above solution procedure.

One of the additional problems that we need to discuss further for compressible flows is that of the treatment of shocks, which is the subject of the next section.

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Conduction: One-dimensional transient and two-dimensional steady state

C. Balaji , ... Sateesh Gedupudi , in Heat Transfer Engineering, 2021

Abstract

This chapter discusses the analytical approaches to solving one-dimensional transient and two-dimensional steady state problems. We show that the pertinent nondimensional parameters for transient conduction are the Biot number (Bi) and Fourier number (Fo) and discuss their physical meaning. This is followed by a search for simpler solutions to the asymptotic variants of the problem. These are (1) lumped capacitance method and (2) semi-infinite approximation. We discuss the physical and mathematical conditions necessary for these approximations to be valid. The problem in its full strength is then solved using the method of separation of variables, which leads to a series solution. The first term in the series solution is alone enough when the Fourier number is greater than 0.2 (Fo >0.2), and graphical solutions based on this approximation, developed by Heisler and Grober, are presented. This chapter also presents analytical solutions to the two-dimensional steady state conduction problem, using the method of separation of variables.

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Solution of sample problems in classical thermoelasticity

Erasmo Carrera , ... Maria Cinefra , in Thermal Stress Analysis of Beams, Plates and Shells, 2017

Spherical coordinates

The solutions to transient heat conduction problems in spherical coordinates are similarly obtained as those to steady-state problems. In the example that follows a problem of this nature is provided.

Problem

Consider a hot solid sphere of radius b, initially at constant temperature $T_0$ . The surface is exposed to convective cooling as the time increases. It is required to obtain a transient temperature in the sphere.

Solution

Taking $\theta =T(r,t)-T_{\infty }$ , the heat conduction equation is

(2.331) $\frac{\partial \theta }{\partial t}=\kappa (\frac{{\partial }^2\theta }{\partial r^2}+\frac{2}{r}\frac{\partial \theta }{\partial r})$

The boundary and initial conditions are

(2.332) $\begin{array}{cc}\hfill -k\frac{\partial \theta (b,t)}{\partial r}& =h\theta (b,t)\hfill \\ \hfill \frac{\partial \theta (0,t)}{\partial r}& =0\hfill \\ \hfill \theta (r,0)=T_0-T_{\infty }& ={\theta }_0\hfill \end{array}$

A solution to the problem, by interchanging the variable $\theta =\tilde{\theta }/r^{\frac{1}{2}}$ , and employing the method of separation of variables along with the boundary conditions is

(2.333) $\theta =r^{-\frac{1}{2}}\sum _{n=1}\infty A_n{J}_{\frac{1}{2}}\left({\lambda }_{n}r\right)e^{(-\kappa {\lambda }_n^{2}t)}$

which can also be written in the form

(2.334) $\theta =\sum _{n=1}^{\infty }\sqrt{\frac{2}{\pi }}{A}_n\left(\frac{\sin \left({\lambda }_{n}r\right)}{r}\right)e^{(-\kappa {\lambda }_{n}t)}$

where ${\lambda }_n$ is the n-th root of the characteristic equation

(2.335) $\tan \left(b\lambda \right)=-\frac{b\lambda }{\frac{hb}{k}-1}$

Using the initial condition we have

(2.336) ${\theta }_0=\sum _{n=1}^{\infty }\sqrt{\frac{2}{\pi }}{A}_n\left(\frac{\sin \left({\lambda }_{n}r\right)}{r}\right)$

Multiplying Eq. (2.336) by $r\sin \left({\lambda }_{m}r\right)$ and integrating with respect to r from $r=0$ to $r=b$ , when $m=n$ , yields

(2.337) $A_n=\sqrt{\frac{\pi }{2}}\left[\frac{2{\theta }_0(\sin \left(b{\lambda }_n\right)-b{\lambda }_n\cos \left(b{\lambda }_n\right))}{{\lambda }_n(b{\lambda }_n-\sin \left(b{\lambda }_n\right)\cos \left(b{\lambda }_n\right))}\right]$

Substituting for $A_n$ gives

(2.338) ${\theta }_0=2{\theta }_0\sum _{n=1}^{\infty }\left[\frac{2{\theta }_0(\sin \left(b{\lambda }_n\right)-b{\lambda }_n\cos \left(b{\lambda }_n\right))}{{\lambda }_n(b{\lambda }_n-\sin \left(b{\lambda }_n\right)\cos \left(b{\lambda }_n\right))}\right]\left(\frac{\sin \left({\lambda }_{n}r\right)}{r}\right)e^{(-\kappa {\lambda }_{n}t)}$

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Nodal and Mesh Analysis

I.D. Mayergoyz , W. Lawson , in Basic Electric Circuit Theory, 1997

6.5 Summary

The nodal and mesh current techniques allow one to write by inspection matrix equations which can be used to quickly analyze complicated ac steady-state problems. These techniques have been derived from the KCL and KVL equations by expressing them respectively in terms of node potentials and circulating mesh currents. These techniques have been presented for various source configurations, such as current or voltage sources connected in series or parallel with impedances. However, the solution method varies only slightly for each type of source configuration.

In the case of nodal analysis, we have first considered the standard n + 1 node circuit with admittances and current sources. The matrix equation for this circuit is $\tilde{Y}\cdot \overrightarrow{v}={\overrightarrow{I}}_s,$ , where $\tilde{Y}$ is the n × n admittance matrix, $\overrightarrow{v}$ is the column vector of node potentials, and ${\overrightarrow{I}}_s$ , is the column vector of sources. The diagonal elements Y_ii of the admittance matrix are equal to the sum of all the admittances connected to the ith node. The off-diagonal elements Y_ij = Y_ji are the negative sum of all the admittances which are connected to both the ith and jth nodes. The source vector component I _{s (j)} is equal to the algebraic sum of current sources connected to the jth node (the source currents flowing into the node are taken with positive signs).

In the case of mesh current analysis, we have first considered the standard planar circuit with m meshes which has only impedances and voltage sources. The matrix equation for this circuit is $\tilde{Z}\cdot \overrightarrow{ℐ}={\overrightarrow{V}}_s,$ where $\tilde{Z}$ the m × m impedance matrix, $\overrightarrow{ℐ}$ is the column vector of fictitious mesh currents, and ${\overrightarrow{V}}_s$ , is the column vector of sources. The diagonal elements Z _ii of the impedance matrix are equal to the sum of all the impedances in the ith mesh. The off-diagonal elements Z _ij = Z _ji are equal to the negative sum of the impedances which are common to both the ith and jth meshes. The source vector component $V_s^{\left(j\right)}$ is found by taking the algebraic sum of voltage sources in the jth mesh (positive means the mesh current is flowing out of the positive terminal of the source).

The modifications of the described standard forms of nodal and mesh current equations for various source configurations have been discussed in the chapter and illustrated by specific examples.

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31st European Symposium on Computer Aided Process Engineering

Chrysovalantou Ziogou , ... Michael C. Georgiadis , in Computer Aided Chemical Engineering, 2021

2.1.1 Parallel Plate Heat Exchanger Model (PPHE)

Let us assume a parallel plate heat exchanger (PPHE) with mass flow rate of water M _w and mass flow rate of steam M _s . A steady state problem for the countercurrent parallel heat exchanger is considered. The equations can be integrated in closed form and scaled back to the whole heat exchanger to give:

(5) $Q=M_s{C}_{\mathit{ps}}\left(T_{\mathrm{s},\text{in}}-T_{s,\mathit{out}}\right)$

(6) $Q=M_w{C}_{\mathit{pw}}\left(T_{w,\mathit{out}}-T_{w,\mathit{in}}\right)$

(7) $Q=\mathit{UA}\left(T_{w,\mathit{out}}-T_{s,\mathit{in}}-\left(T_{w,\mathit{in}}-T_{s,\mathit{out}}\right)\right)/ln\left(\left(T_{w,\mathit{out}}-T_{s,\mathit{in}}\right)/\left(T_{w,\mathit{in}}-T_{s,\mathit{out}}\right)\right)$

where Q is the rate of heat transferred in the PPHE. The heat transfer coefficient and the (one side) surface area of the plate is U and A respectively. Q can be eliminated between the above equations to leave with a system of two equations with two unknowns (the outlet temperatures). The system of Eq. (4) - (6) is highly non-linear and can be replaced by the linear system:

(8) $T_{w,\mathit{out}}=\frac{T_{w,\mathit{in}}+p2\left(\left(1-p1\right)T_{s,\mathit{in}}-T_{w,\mathit{in}}\right)}{1-p1p2}$

(9) $T_{s,\mathit{out}}=T_{w,\mathit{in}}+p1\left(T_{s,\mathit{in}}-T_{w,\mathit{out}}\right)$

(10) $p1=exp\left(\mathit{UA}\left(\frac{1}{m_w{c}_p}-\frac{1}{M_s{c}_p}\right)\right)$

(11) $p2=\frac{M_s{c}_p}{M_w{c}_p}$

For given inlet temperatures to PHE the outlet temperatures are computed by Eq. (8) and (9) and then Q by Eq. (5).

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Tracer Experiment Design for Metabolic Fluxes Estimation in Steady and Nonsteady State

Andrea Caumo , Claudio Cobelli , in Modelling Methodology for Physiology and Medicine (Second Edition), 2014

10.7 Conclusion

In this chapter, focus has been placed on tracer experiment design strategies needed to quantitate production and utilization fluxes of a substance under both steady- and nonsteady-state conditions. The steady-state problem is easy to tackle and it has been emphasized that, under steady-state conditions, approaches to a rapid attainment of the tracer steady state require attention in selecting the ratio between the priming dose and the constant infusion. The nonsteady-state situation is far more complex and the adoption of "intelligent" infusion strategies becomes a must. In particular, it has been shown that, under nonsteady-state conditions, the tracer infusion should be varied in such a way that the tracer-to-tracee ratio remains as constant as possible to reduce the impact of model error. In this context, the partitioning of the tracee source and concentration into two components, endogenous and exogenous, provides a general framework that helps the investigator to achieve this goal under various experimental conditions.

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Natural Gas Conversion V

L.D. Schmidt , ... C.T. GoralskiJr., in Studies in Surface Science and Catalysis, 1998

2 MODELING PARTIAL OXIDATION REACTORS

Detailed modeling of high temperature oxidation reactors is a difficult problem because the very fast reaction and mass transfer rates generate equations that are stiff both in space and time. The temporal stiffness can be avoided by solving steady state problems, but detailed steady state models often require so much computing time to solve that parameter studies are impractical. We have approached this problem by tackling individual pieces of the problem separately to try to understand the global importance of all parameters in the system. To this end, we have developed two separate computer models of millisecond contact time partial oxidation reactors. The first is a plug flow tubular reactor (PFTR) model to study the interaction of heterogeneous and homogeneous chemistry in the partial oxidation of methane over noble metal coated monoliths. The second is a 2-dimensional model that includes only heterogeneous chemistry to study the importance of heat and mass transfer and radial and axial gradients in high temperature catalytic monolith reactors.

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Find the Steady State Solution of the Problem

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