Rihab Bedoui

† Mai

Modeling Longevity Risk using Consistent Dynamics Affine Mortality Models

Keywords: JEL classification : C02, C14, C65, G13 Longevity Risk, Mortality model,multi-factor,consistent, Affine,Kalman filter, arbitrage-free, USA mortality

Longevity Risk becomes an important challenge in the recent Year because of the decreases in the mortality rates and the rising in the life expectancy through the decades. In this article, we propose a consistent multi-factor dynamics affine mortality model to the longevity risk modeling, we show that this model is an appropriate model to fit the historical mortality rates.To our Knowledge this is the first work that uses a consistent Mortality models to model USA Longevity risk.Indeed the multiple risk factors permitting applications not only to the hedge and price of the longevity risk but also in mortality derivatives and the general problems in the risk management .A state space presentation is used to estimate the model parameters through the kalman filter .To capture the effect of the size of the population sample we include a measurement error variance for each age. We evaluate 2-and 3-factor implementation of the model through the use of the USA mortality data, we employ Bootstrapping method to derive parameter estimated and the Consistent models prove the performance and the stability of the model. We show that the 3 -factor independent model is the best model that can provide a better fit to our survivals curves and especially for the elderly persons.

Introduction

Due to the reduction of the mortality rate around the world amoung the elderly, social security systems and private annuity providers are facing the problem of imbalances between contribution and benefits significantly and partly because of the retirement of baby boomers in the recent years, as result longevity risk become the major problem to many insurance companies and become more important in the presence of interest rate and mortality . The observed constrant improvments in longevity are bringing new issues challenges at various levels : social, political, economics and regulatory ... Many previous studies discuss longevity risk , to start with [START_REF] Blake | Survivor Bonds: Helping to Hedge Mortality Risk[END_REF] who was the first talked about a Mortality-linked securities when they proposed a survivor bond to hedge this risk and evoke the role of the government to issue this kind of bond based on population mortality. [START_REF] Dowd | Survivor Bonds: A Comment on Blake and Burrows[END_REF] suggested others hedging methods like survivor options, swaps and futures ,and in 2006 the longevity bonds and gave some construction which can used it to hedge longevity risk the same thing to [START_REF] Blake | Living with Mortality: Longevity Bonds and Other Mortality-Linked Securities[END_REF].

To manage this risk many previous research find that the most prevalent way is the securitization as result several types of product have been suggested to hedge this risk like longevity bonds survivor swaps and options-type longevity derivatives. Li and COX (2005) suggested a successful securitization of longevity risk through a longevity bond with payments which were based on the loss experience of the insurer to eliminate the others basis risk. [START_REF] Brown | The Political Economy of Goverment Issued Survivor Bonds[END_REF] afforded a discussion of the disadvantages and the advantage of the participation of the government in longevity bonds like the ability to spread the longevity risk across the generations. Longevity risk can be managed through the use of hedging in financial markets or reinsurance blake et al (2006a). D. Blake, A. J. G. Cairns and K. Dowd (2006) discuss some possible ways to manage longevity risk for life companies and pension plans by using Mortality-linked securities.

By time there have been many others innovation to remove the longevity risk to the capital markets containing mortality-linked securities (MLSs), and many others proposal to deal with this risk containing the development of longevity-linked securities and derivatives taking the example of the EINB BNP longevity bond to transfer this risk , without forgetting the first capital market derivative transaction in the world : the q-forward contract between J.P Morgan and the UK company Lucida which is occurred in January 2008, Coughlan et al (2008), and in july 2008 it was the first capital market longevity swap which gave the possibility to canadians to hedge their UK-based annuity policies and last but not least Swiss Re launched the called 'Kortis notes' which is a group eight-year longevity-based insurance-linked security notes , like Denuit et al (2007) when they propose a survivor Bond-design directly issued by insures suffering from longevity risk , which based on public Mortality index using the classical Lee Carter Model to price a risky coupon survivor Bond.

Other studies talked also about longevity bond and analyse the role of longevity risk on the solvency of some portfolio of pension annuities for example Hàri et al (2008) who distinguish between two kind of this risk , a micro longevity risk and a macro longevity risk and precise that longevity bond could be a tool to hedge just the second type of risk ,in other way Hua chen et al (2009) propose a new design of longevity bond through the utilization of a random walk model with drift. Nevertheless [START_REF] Denuit | An index for longevity risk transfer[END_REF] gave a detailed analysis of a longevity index through out the lee carter model to transfer the longevity risk to the market and bring to bear that choosing reinsurance as a cover from longevity risk could be expensive for many life insurance companies.

The increase of life expectancy causes an increase in the duration of pension payments so this can abuse a burden on pension providers , many previous studies discuss this problem and propose an alternative tool to manage the longevity risk in pension annuities by introducing many product design ,Ralph et al (2009) propose in retirement age a defined benefit pension plan which give the choice between two different type of annuities. Horneff et al (2010) propose a deferred annuities as a pure longevity insurance to hedge this risk however Ngai and sherris (2011) examine the capability of static hedging strategies using longevity bonds and many others retail products-derivatives and proved that variable annuities gives a limited protection against longevity risk in comparison with life annuities and indexed annuities. Jo ao F. Cocco and Francisco J. [START_REF] Cocco | Longevity risk, retirement savings, and financial innovation[END_REF] interpret a life cycle model with longevity risk and propose an endogenous savings and retirement decisions as a hedging for this risk also they give an implications for optimal security design.

Pricing was the center of attraction of others studies for example Johny Siu and Hang Li (2010) discuss an alternative pricing model by using the parametric boot strap , however Chen, Zhang and Zhao (2010) use to price a longevity bond three popular method first neutral method sharp ratio rule and the wang transform , Kogue et al (2010) propose a Bayesian approach to pricing longevity risk through out the use of the Lee carter methodology.

Some studies focus on modelling this kind of risk as Loisel and Daniel Serant (2007) when they introduce and discuss a new model so called stochastic logit's Deltas Model which is based on the multivariation studies , also we find Tzuling lin et al (2009) who gave a variables method to modelling longevity risk and [START_REF] Andrew | Modelling and management of longevity risk: Approximations to survivor functions and dynamic hedging[END_REF] who suggested a mortality model with an age shift by using a principal component analysis PCA.

Many others study also discuss a stochastic modeling of mortality to solve the problem of measuring longevity risk , as B. Browne et al (2009) , who suggests a simple model to move from a deterministic Mortality scenario to a stochastic model ,owing to describe the behavior of the longevity enclosing the projected expected values, this model can be used on future mortality rates estimation .Weny Bal et al(2009) also discuss this kind of models by introducing four stochastic mortality models and bring to light the weakness and the strengths of the four models , in addition this research analyze the possibility to manipulate the input data by adjusting the parameters and show that the stochastic modeling is more advantageous than the deterministic one .

Other articles give a comparison discussion between a deterministic shock and a stochastic value at risk in order to propose an analysis of the Solvency II Standard Model Approach to measure longevity Risk Matthias Börger (2010). Sharron S et al (2010) use the Principal Component Approach to modeling longevity risk and compare the PCA model with the existing stochastic Mortality Models , they use this model to examine the ratio of annuity price for both the deferred life annuity and life annuity products and try to forecast the future mortality rates , moreover they prove that this model can resolve the Lee Carter Model's problem by using the PCA approach. More recently, Vesa Rankainen (2012) uses the famous model so called the Lee Carter Model 1992 to develop a stochastic modeling of financing longevity risk in pension insurance by using the Bayesian MCMC methods and prove that the LC model is not completely adaptable with mortality data .

Søren Fiig Jarner and Thomas Møller (2013) Propose currently a simple Partial internal model for longevity risk inside the framework of Solvency II , which have connection with the Danish longevity benchmark mechanisms , this model comprise a component, which is based on the size of a given portfolio , this component can measure for each insurance portfolio the unsystematic longevity risk .

To solve the problem of measuring and modeling the longevity risk many studies use the affine structure with mortality models application like Dahl (2004) who propose for a cohort live an affine mortality structure with the same ages .Shrager(2006) introduces an affine mortality intensity model specially for Thiele and Makeham mortality laws and examines all ages in same time , but They do not take into consideration the property of consistent in their Models.

In this work we propose a consistent multifactor dynamics affine mortality model for modeling the longevity risk which is based on affine term structure model ATSM discussed by Duffi and Khan (1996) on modeling interest rates.To our Knowledge this is the first work that uses a consistent Mortality models to model USA Longevity risk.

We present a similar way to term structure of interest rates to model the mortality survival curve age. The affine framework is advantageous than the others econometrics models because we have the ability to use a different mathematical tools used for risk management and in pricing credit risk .

We introduce in our article a Parametric survival curves that are consistent guaranteeing that the set of the predicted survival curves hold an invariable parametric form as the initial fitted parametric curve . To confirm the model's consistency , we establish different conditions for consistency like it was giving by Bjork and Christensen (1999) also we refer to De Rossi (2004) who gives an application to this conditions in interest rates modeling to derive a 2-factor ATSM which is consistent. To provide a calibration of the Market price of Risk to an admissible an accessible market data , we used an arbitrage-free formulation.Given our Risk neutral process , we use in our work the Gisanov's theorem to move to a real-world measure in order to model the historical mortality rates. As an alternative to presume a constant price of moratlity risk for every factor like it was giving in 'completely affine' model by Duffie and Khan (1996) , we propose in our work an 'Essentially affine ' model as it was giving by Duffie (2002).

To estimate our model , we use a state space formulation which contain the measurement and the transition equations. Unlike the Econometrics models as the famous Lee and Carter model (1992), our transition equations which are the time-series dynamics are presented in the model like a latent stochastic factor , and the measurement equation represents the observed survivor curve ,and it is exponentially affine in the factors .Introducing our ATSM model in a state space formulation in this paper,let us estimate the model parameters using the Kalman filter , [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF] , we include a specific allowance for the age dependent measurement errors , we suppose that the measurement errors are independent between ages to estimates the errors .

We use a USA data to calibrate and evaluate the model fit , we use a state-space re-sampling method to realize a bootstrapped confidence intervals for every paramters estimated.

We present different model assumptions ,these contain 2-and 3-factor independent models.We consider an arbitrage-free type of Nelson Siegel model as presented by [START_REF] Christensen | An arbitrage-free generalized NelsonSiegel term structure model[END_REF],and we compare it with the 3-factor independent model. This article is organized as follow : in Section 2 we outline the general meaning of the longevity risk and its characteristics through the way how to modelling and to transferring the Risk ,in section 3 we present the life insurance with its different products .Section 4 discuss the the ATSM methodology with the derivation of the 2-and 3-factor model, in Section 5 we introduced the empirical study with the data presentation using the mortality survival curve ,the model calibation using the kalman filter and the bootstrap state space model with the differents estimations results , the sample analysis ,the model robustness and finally we find the conclusion.

A presentation of Affine Mortality Model:

In this framework we try to use some techniques taked from the credit Risk securities in Mortality modelling.

To start with the probability space (Ω, F,F,P) where F = G ∨ H represents the information at time t,and we noted that G t which is the sub-filtration includes all the inforamtion (financial and actuarial) other than the current time of death , the other subfiltration H t is the σ-algebra with death information , and the Real world probability space is given by P . let's take N(t) = 1 τ≤t , if we assume that A(t) is an anticipated process of N(t) where: A(t) = t 0 µ(s; x)ds and: dA(t) = µ(t; x)dt so we have: dM(t) = dM(t) -dA(t) M(t) represents a P-martingale . Taking another measure where we considerate dM(t) as a Q-martingale process , under which dA(t) becomes:dA(t) = µ Q (t; x)dt and we can note:µ Q (t; x) = (1 + Φ(t))µ(t; x) where Φ(t) ≥ -1 . We adjust the value of Φ(t) at zero Φ(t) = 0 and we noted that the unsystematic Risk is not priced, this suppose thar we can note µ Q (t; x) = µ(t; x).

Under the abscence of arbitrage assumption,there exists Q which is an equivalent measure where P(t; T; x) is the time t-value of x t (the final payment). x t pretend to be a G adapted process and to the end of time t, x t is conditional on survival, if not the value is zero .So we can note the time t-value after the use of the law of 'iterated expectations'as showing by [START_REF] Bielecki | Credit Risk: Modelling, Valuation and Hedging[END_REF] ,as follow :

P(t; T; x) = E Q e -T t r(s)dsX T 1 τ t |F t = 1 τ t E Q e -T t [r(s)+µ(s;x)]dsX T |g t (1)
by adding the independance assumption between mortality rates µ(x; t) and the interest rates r(s) and suppose that X t = 1 ,we can express the time t-value as :

P(t; T; x) = 1 τ t E Q e -T t r(s)ds|g t E Q e -T t µ(s;x)ds|g t = 1 τ t B(t, T)S(t; T; x) (2)
Where S(t; T; x) is the survival probabilty at time 0 for a person aged X and B(t, T) is the time-t bond price , this description is adaptable to forward rate modelling presentedd in the work of Health-Jarrow -Morton (HJM) Health et al (1992). Because of dealing with the problem of exogeneity of different cohorts and cohorts forecasts, the calibration of affine Models becomes difficult,but Chiarellaand Kwon (2001), prove in their farmework that ATSM is a special case of HJM with showing a determenistic volatility function ,so we apply this characteristic to solve the problem of calibration using our population data. We assume in our framework that the parameters of Risk neutral are not varying so we apply the HJM Model to give a better estimation to the cohorts survival curves. By keeping the filtres space of probability (Ω, F,F,P) we consider µ(t) , the G-adapted process, as an instantaneous measure rate , so under an equivalent martingal measure Q , we can express the survival probability as it was expressed by Black burn and Sherris (2013) by :

S(t, T) = E Q e -T t µ(s)ds|g t (3)
The solution of this equation is given by :

S(t, T) = e -B(t,T) Z t +C(t,T) (4)
Where B(t, T) and C(t, T) are the solution of the ordinary differential equation existing in our given short rate mortality process as it was explained by Duffi and kan (1996).

dB dt = 1 -∆ Q B(t, T), B(0) = 0 (5) dC dt = - 1 2 n j=1 Σ B(t, T)B (t, T)Σ, C(0) = 0 (6)
So µ(t) can be expressed as the sum of latent factors :

µ(t) = 1 Z .
Given our n-factor model when B(T, t) represents the n-factor loadings matrix transpose , C(t, t) a constant and the latent factors Z t .We can show that the curve of the average force of mortality is affine in the factors and it can be presented by :

µ(t, T) = - 1 T -t log[S(t, T)] = B(t, T) Z t -C(t, T) T -t (7)
Under the risk-neutral measure, the n-latent factors can be defined us a gaussian process by :

dZ t = -∆ Q Z t dt + ΣdW Q t Where Σ ∈ R n×n ,∆ Q ∈ R n×n
without a condition about ∆ Q sign, the process of the risk-neutral factors can be: a non -meaning reverting, a random walk or a mean-reverting to zero . Σ, which is the covariance matrix is diagonal and W t are an independant Brownian motions.

Presentation of longevity risk dynamics:

The aim of this paper is to represent a model that is able to fit the survival curve. We have different structures for the dynamics of the latent risk factors which is able to driving our presented survival curve .

We introduce a 2-and 3-factor model dynamics with independence between factors, and the popular 3 factor version of the Nelson Siegel model which is none as a non-consistent interest rate mode to give as a contrast with our consistent model.

2-Factor independent model :

we represent here a model of the Brownian motions with no inter action and no factor dependence , the dynamics of this model is given as bellow :

dZ 1,t dZ 2,t = - δ 1 0 0 δ 2 Z 1,t Z 2,t dt + σ 1 0 0 σ 2       dW Q 1,t dW Q 2,t       (8)
So the instantaneous force of mortality can be written as :

µ x,t = Z 1,t + Z 2,t (9)
The solutions for the factor loadings are given by Dufie and Kan(1996), as we have noted that :

dB(t, T) dt = 1 -∆ Q B(t, T) B(0) = 0
to solve the differential equation for B(t, T) we denote :

d dt [e ∆ Q (T-t) B(t, T)] = e ∆ Q (T-t) dB(t, T) dt -∆ Q e ∆ Q (T-t) B(t, T) = e ∆ Q (T-t) dB(t, T) dt -( dB(t, T) dt -1)e ∆ Q (T-t) = e ∆ Q (T-t)
Weintegratingtheequation f romthebothside :

T t d ds [e ∆ Q (T-s) B(t, T)] = T t e ∆ Q (T-t) ds
by the use of the boundary conditions we find that:

B(t, T) = -e -∆ Q (T-s) T t e ∆ Q (T-s) ds
The matrix exponentials in this equation for our model assumption is given by :

e ∆ Q (T-t) = e δ 11 (T-t) 0 0 e δ 22 (T-t) e -∆ Q (T-t) = e -δ 11 (T-t) 0 0 e -δ 22 (T-t)
After substituing this in the expression of B(t, T) we have :

B(t, T) = - e -δ 11 (T-t) 0 0 e -δ 22 (T-t) × T t e δ 11 (T-s) 0 0 e δ 22 (T-s) ds
After simplifying the integral :

B(t, T) = - e -δ 11 (T-t) 0 0 e -δ 22 (T-t) ×       -1-e δ 11 (T-t) δ 11 -1-e δ 22 (T-t) δ 22       B(t, T) = -       -1-e -δ 11 (T-t) δ 11 -1-e -δ 22 (T-t) δ 22      
So we have finally: [START_REF] Bingzheng Chena | On the robustness of longevity risk pricing[END_REF] and:

B 1 (t, T) = 1 -e -δ 11 (T-t) δ 11 B 2 (t, T) = 1 -e -δ 22 (T-t) δ 22
for the constant term C(t, T) is given by:

C(t, T) = - 1 2 T t n j=1 (Σ B(s, T)B(s, T) Σ) i, j ds = - 1 2 T t (σ 2 11 )B 1 (s, T) 2 + (σ 2 22 )B 2 (s, T) 2 ds = 1 2       σ 2 11 δ 3 11 2e -δ 11 (T-t) - 1 2 e -2δ 11 (T-t) + δ 11 (T -t) - 3 2 σ 2 22 δ 3 22 -2e -δ 22 (T-t) - 1 2 e -2δ 22 (T-t) + δ 22 (T -t) - 3 2
(12)

3-Factor independent model :

We represent here a model when each of the factors are independent , and we have no correlation of the Brownian movements,the dynamics of this model is given by :

        dZ 1,t dZ 2,t dZ 3,t         = -         δ 1 0 0 0 δ 2 0 0 0 δ 3                 Z 1,t Z 2,t Z 3,t         dt +         σ 1 0 0 0 σ 2 0 0 0 σ 3                    dW Q 1,t dW Q 2,t dW Q 3,t            (13)
Where the ordinary differential equations solution is given by :

B i (t, T) = 1 -e -δ i (T-t) δ i ( 14
)
where i = 1, 2, 3 for every factor and we have

C(t, T) = 1 2 T t 3 j=1 (Σ T B(s, T)B(s, T) T Σ) j j ds (15)

The Nelson-Siegel model:

The Nelson siegel model is a model who fits the yield curve at any point in time , (Nelson-Siegel 1987) and in the term sructure modelling this model can ease many empirically observed shapes, and this is the reason of being the most popular one in the financial market. We use the Nelson-Siegel model which is not consistent,to demonstrate the consistency notion in one hand and to give a contrast with our consistent models (2-and 3-factor models) in another hand .

The risk neutral dynamics of the 3-factor Nelason-Siegel model is given as CHRISTENSEN et al (2009) :

        dZ 1,t dZ 2,t dZ 3,t         = -         0 0 0 0 δ -δ 0 0 δ                 Z 1,t Z 2,t Z 3,t         dt +         σ 1 0 0 0 σ 2 0 0 0 σ 3                    dW Q 1,t dW Q 2,t dW Q 3,t            (16)
CHRISTENSEN et al (2009) provide us with the solutions of the factor loadings :

B 1 (t, T) = (T -t) B 2 (t, T) = 1 -e -δ(T-t) δ B 3 (t, T) = 1 -e -δ(T-t) δ -(T -t)e -δ(T-t) (17)
By the use of the previous equations we can now present the expression of the average force of mortality for each models ,that it is estimated in our framework .

Under the risk neutral measure ,we can write the average force of moratlity for the n-factor independant model by combining ( 25),( 29), [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF],and ( 32), [START_REF] Lee | Modeling and forecasting US mortality[END_REF] as:

μ(t, T) = 1 T -t n i=1 [ 1 -e -δ i (T-t) δ i Z i,t - 1 2 σ 2 i δ 3 i × 1 2 1 -e -2δ i (T-t) -2(1 -e -δ i (T-t) ) + δ i (T -t) ] ( 18
)
And the 3-factor Nelson Siegel model by combining [START_REF] Coughlan | Longevity hedging 101: A framework for longevity basis risk analysis and hedge effectiveness[END_REF] and [START_REF] Denuit | An index for longevity risk transfer[END_REF] with C(t,T) given by christensenet al.( 2009)

μ(t, T) = 1 T -t [(T -t)Z 1,t + 1 -e -δ(T-t) δ Z 2,t + 1 -e -δ(T-t) δ -(T -t)e -δ(T-t) Z 3,t -C(t, T)] (19)

Change of measure:

As an alternative of the "'Complete affine" model of duffie and kan (1996)in which they assume that the price of Mortality risk is constant for each factor ,we choose the "essentially affine" model given by Duffie(2002) and we use the Gisanov's theorem to move from the Risk neutral Q-measure dynamics to a real world P measur .Under the P measure , there is no interaction between factors and the moratlity risk price detail of the 'essentially affine' remove the relation ship between real world or historical drift and the factor loadings ,so this let us choose various drift terms for the real word or the risk neutral process. Using Gisanov's theorem the n-factor stochastic process can be written under the real world P-measure as follow :

dZ t = K[Ψ -Z t ]dt + W P t ( 20
)
where

K ∈ R n×n ,Ψ ∈ R n×1 ,Σ ∈ R n×n .
This expression is taking from the 'essentielly affine' model , where the Market price of mortality is given by :

Λ t = λ 0 + λ 1 Z t andΛ t ∈ R n×n ,λ 0 ∈ R n×1 ,λ 1 ∈ R n×n .
The change of measure using Ginasov's theorem from the n-factor real world Brownian movement to the risk-neutral measure is given by : dW Q t = dW P t + Λ t dt Under the P measure , the stochastic process Z t is :

dZ t = [∆Θ -∆Z t ] dt + Σ[λ 0 + λ 1 Z t ]dt + ΣdW P t = [∆Θ + Σλ 0 ] dt -[∆ -Σλ 1 ]Z t dt + ΣdW P t
this can be written the same as [START_REF] Hari | Longevity risk in portfolios of pension annuities[END_REF]. We have to note that we set the Ψ to 0 , which is the long term mean level of the mortality rates , and this can let each factor increase to zero at an axact rate noted K . We can express the real-word dynamics of our given models as bellow : Starting with the 2-factor independent Model :

dZ 1,t dZ 2,t = - k 1 0 0 k 2 Z 1,t Z 2,t dt + σ 1 0 0 σ 2 dW P 1,t dW P 2,t (21)
For the 3-factor independent and the Nelson-Siegel model :

        dZ 1,t dZ 2,t dZ 3,t         = -         k 1 0 0 0 k 2 0 0 0 k 3                 Z 1,t Z 2,t Z 3,t         dt +         σ 1 0 0 0 σ 2 0 0 0 σ 3                  dW P 1,t dW P 2,t dW P 3,t          (22)
Owing to this definition we can estimate K which is the mean reversion real world speed, without affecting the other parameters of the Risk neutral.Duffi (2002) proves that when we add the square root diffusion factors , the detail of risk structure price in the 'essentially affine' model turns to the 'complete affine' model. So we concentrate on the dynamics of the model which is a gaussian dynamics , and we will estimate the real world drift and the Risk-neutral parameter individually using the Kalmna filter.

The Forward mortality rate:

As [START_REF] Chiarella | Classes of interest rate models under the HJM framework[END_REF] show that the ATSM model is a special case of the HJM model ,we use this specification to give a forward mortality rate presentation.

Under the Risk neutral measure ,we can model the forward mortality dynamics as follow:

dµ(t, T) = n i=1       σ 2 1 e -δ i (T-t) δ i (e -δ i (T-t) -1)dt -σ i e -δ i (T-t) dW Q i,t       = υ µ (t, T)dt + σ µ (t, T)dW Q t ( 23
)
Proof :

µ(t, T) = ∂ log[S(t, T)] ∂T = ∂ ∂T [B(t, T) Z t -C(t, T)] = n i=1       e -δ i (T-t) Z i,t - 1 2 σ 2 i δ 3 i (2δ i e -δ i (T-t) -δ i e -2δ i (T-t) + δ i )       (24)
Through the Ito'slemma we can find out the dynamics of µ(t, T) :

dµ(t, T) = ∂µ dt dt + n i=1 ∂µ ∂Z i,t dZ t,t = n i=1             δ i e -δ i (T-t) Z i,t - 1 2 σ 2 i δ 3 i (2δ 2 i e -δ i (T-t) -2δ 2 i e -2δ i (T-t) )       dt + e -δ i (T-t) dZ i,t       (25)
the expression of dZ t of each factors ,is determined as we set in equations.

dZ i,t = -δ i Z i,t dt + σdW Q i,t
After substitution we have:

dµ(t, T) = n i=1       (δ i e -δ i (T-t) Z t - 1 2
σ 2 i δ 3 i (2δ 2 i e -δ i (T-t) -2δ 2 i e -2δ i (T-t) )dt -e -δ i (T-t) δ i Z i,t dt -σ i e -δ i (T-t) dW Q i,t = n i=1       σ 2 i e -δ i (T-t) δ i (e -δ i (T-t) -1)dt -σ i e -δ i (T-t) dW hbbQ i,t       (26)
As we can see, this equation confront the HJM drift given by :

υ µ (t, T) = σ µ (t, T) T t σ µ (t, s)ds = n i=1 σ i e -δ i (T-t) T t σ i e -δ i (s-t) ds = n i=1       σ 2 i e -δ i (T-t) δ i (e -δ i (T-t) -1)       (27)
With σ µ (t, T) represent a volatility functions matrix.

The consistent forward mortality curve :

The main aim of this work is to give a a consistent models for fitting the survival probability curve. These curve have the characteristic of keeping the same form of the survival curves through time. Bjork and Christness (1999) propose a study about the consistent forward curves and we use this study to test if the consistent condition is verified in our 2-and 3-factor independent models and the Nelson siegel model(Christness (2009))

* Consistency Test:

We had shown that the force of mortality is written by :

µ(t, T) = ∂ log[S(

t, T)] ∂T

To determine consistency we can apply the Frechet derivatives to µ(Tt) by using Theorem 4.1 of Christness and [START_REF] Bjork | Interest rate dynamics and consistent forward rate curves[END_REF].The derivatives can be used in the verification of volatility and drift conditions for the dynamics of our risk factor that confirm consistency . The consistent drift condition is given by :

µ x (x, z) + σ(x) x 0 σ(y)dy T ∈ Im µ z (x, z)
Where µ x (x, z) and µ z (x, z) represents respectively the Frechet derivatives with respect to x and z . with x = Tt.

Taking our 2-factor model ,The volatility term in the drift can be given by:

σ(x) x 0 σ(y)dy T = σ 1 e -δ 1 x + σ 2 e -δ 2 x σ 1 ( 1 -e -δ 1 x δ 1 ) + σ 2 ( 1 -e -δ 2 x δ 2 ) T = σ 2 1 e -δ 1 x -e -2δ 1 x δ 1 + σ 2 2 e -δ 2 x -e -2δ 2 x δ 2
We can simplify the time varying parameteres in z and the exponential terms by x so we have :

µ(x, z) = z 1 e -δ 1 x + z 2 e -δ 2 x + z 3 e -2δ 1 x + z 4 e -2δ 2 x
The Frechet derivatives of µ(x, z):

µ z (x, z) = e -δ 1 x e -δ 2 x e -2δ 1 x e -2δ 2 x µ x (x, z) = z 1 e -δ 1 x + z 2 e -δ 2 x -z 3 e -2δ 1 x -z 4 e -2δ 2 x
The consistent drift condition can be given as :

µ x (x, z) + γ 1 e -δ 1 x + γ 2 e -δ 2 x + γ 3 e -2δ 1 x + γ 4 e -2δ 2 x ∈ Im µ z (x, z)
Where γ i are constant. So our 2-factor model is constant since it verify the equation The vectors of volatility for the 2-factor model is given by :

σ 2 f = σ 1 e -δ 1 x σ 2 e -δ 2 x ( 28
)
The same thing for the 3-factor model ,and we can write the consistent drift condition as :

µ x (x, z) + γ 1 e -δ 1 x + γ 2 e -δ 2 x + γ 3 e -2δ x + γ 4 e -2δ 2 x + γ 5 e -δ 3 x + γ 6 e -2δ 3 x ∈ Im µ z (x, z)
The vector of volatilities for the 3-factor model is given by:

σ 3 f (x) = σ 1 e -δ 1 x σ 2 e -δ 2 x σ 3 e -δ 3 x (29)
Taking now the Nelson siegel model , The volatility term in drift can be shown as :

σ(x) x 0 σ(y)dy T = σ 1 σ 2 e -δx + σ 3 e -δx σ 1 x σ 2 (1 -e -δx ) δ + σ 3 (1 -e -δx ) δ -xe -δx T = σ 2 1 x + σ 2 2 (e -δx -e -2δx ) δ + σ 2 3 x (e -δx -e -2δx ) δ -x 2 δe -2δx
So the force of mortality can be written :

µ(x, z) = z 1 + z 2 e -δx + z 3 xe -δx + z 4 e -2δx + z 5 xe -2δx + z 6 x 2 e -2δx + z 7 x 2
Then the frechet derivatives ofµ(x, z) is :

µ z (x, z) = 1 e -δx xe -δx e -2δx xe -2δx x 2 e -2δx x 2 µ x (x, z) = (z 3 -δz 2 )e -δx -δz 3 xe -δx + (z 5 -2δz 4 )e -2δx +(2z 6 -2δz 5 )xe -2δx -2δz 6 x 2 e -2δx + z 7 2x
So we can write the consisteny drift after assembling terms in x,as:

µ x (x, z) + γ 1 e -δx + γ 2 e -δx + γ 3 e -2δ x + γ 4 e -2δx + γ 5 x 2 e -2δx + γ 6 x
Sinc e γ 6 Im µ z (x, z) the Nelson Siegel model is not consistent. The vector of volatilities for the 3-factor Nelson Siegel model is given by :

σ NS (x) = σ 1 e -δx σ 2 e -δx σ 3 e -δx (30)
4 Empirical study:

Data presentation:

To solve the problem of modeling longevity risk, a consistent multi factor dynamics affine mortality model is suggested through the ATSM model.

We prove that ATSMs model are an adaptable model to fit the historical Mortality rates through the use of USA mortality data. Our sample contains the population exposures and the number of deaths for the years 1933-2010 which are taken from the human Mortality data base when the population exposures by age are generally estimated. This mortality data is used to evaluate 2-and 3-factor implementation of the model when the last one is used to give a good fit to the survival curves which is observed especially for the older age.

The United States is considered as the third most populous country in the world since it accounts for about 4.5% of the whole population. Since its 1950 the US population has more than doubled and it was estimated at 308.7 million persons,its just not about size , the US population has become different after 1950 .As it was mentioned by the Population reference Bureau "the US is getting bigger, older and more diverse" .One of most demographic characteristic of the US is the rapid ageing of the population and an important increasing in the persons aged more than 65 which can increase the Longevity risk in this country,and make it one of the most interessent data that we can used for our model fitting . In our work , we refer to the famous Nelson Siegel model especially the way to fit the yield curves in interest rates. So we introduce our model for the survival curves through a functional form which is a sum of exponentials includes a time varying parameters . In this case , the mortality curve of the survival probabilities average at different survival times , is equivalent to the yield curve of the zero coupon bond yield at variance maturities .

Under the risk neutral measure the survival probability at time t for a person aged x to age x + T (it is mean to survive another Tt years ),can be represented through the affine form as :

S(t, T) = E Q t e T-t 0 µ x +u,tdµ |F t = e -B(t,T) Z t +C(t,T) (31)
With µ x+u,t correspond to the force of mortality for a person aged x + u at time t where Z t symbolizes the random factors vectors. We can presented the maturity curve of the survival probabilities S(t, T),when its log is affine in the random factors and able to drive the mortality changing through time,the maturity curve is correspondent, in the interest rates framework ,to the zero coupon yields to maturity and it is shown by:

μ(t, T) = - 1 T -t log[S(t, T)] (32)
μ(t, T) is the average force of mortality for survival Tt years to an individual aged x at year t , we use μ(t, T) to fit our model. We determine the survival probability from our mortality data and it is given by :

S(t, T) = T-t s=0 1 -q(x + s, t) (33)
Where q(x, t) represents the death probability for one year given for an individual aged at year t ,and it is given by: q(x, t) = 1e -m(x,t)

Where m(x, t) corresponds to the central rate of mortality and we can calculate it from our mortality data as follow:

m(x, t) = D(x, t) E(x, t) = #o f deaths aged x in year t Exposure aged x in year t (35)
Figure .1 shows the plot of the average force of mortality given by μ(t, T) for population aged between 50 and 100 for the years 1933-2010. we use this mortality curve data to represent the model fitting.

We can see through this figure the exponential shape of the mortality curve and the improvement in mortality over the periode of the study , this improvements have appeared at various rates for different ages.

Figure .2 shows the mean and the first two principal components of the average of mortality μ(t, T) for USA data set. The mean and first principal component explains 98.58% of the data variation.This proposes that the 2-and 3-factor mortality models should be able to capture parsimoniosly all changes in our mortality survival curves , in the principal component we assume the consistency of the mean and the orthogonality of the principal components. In our work, we suppose a mean equivalent , it represents a dynamic factor with wich we can provide interaction between the model factors under the dynamics of the risk neutral measure or the real world dynamics.

Kalman filter

The kalman Filter is considered as an optimal linear estimator since it minimises the error covariance [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF]).The most financial application used this filter , one of this application is the estimaton of the term structure model as it used by Anderson and Land (1997) and [START_REF] Babbs | Kalman filtering of generalized vasicek term structure models[END_REF] . It bring to us a recursive solution to solve the problem of filtering especially for the linear discreat filtering.

The main advantage of the application of the kalman filter and the state space presentation of the term structure model is that facilitate handling properly the state variable as an unobserved variable by the use the short rate term like a proxy. The kalman flter is presented by a series of state spaace equation with an n-dimensional unobservable factors,Z of the system which is given by a linear stochastic difference equation.

The transition system Z can be written in the discrete form as:

Z t = A × Z T-1 + B + η T-1 (36)
We presume that A ∈ R n×n is a constant matrix and show the relationship between Z t and Z t-1 the actual and the previous state factor respectively ,and B ∈ R n is the controle variable for the factor,η t-1 ∈ R n×n is a random variable presents the transition noise and it is supposed to be independant and normally distributed.

η T-1 ∼ N(0, Q) (37)
where R ∈ R m×m is predicted constant and represents the transition noise covariance of the system .

The measurement equation which is the second equation of the state space illustration , the measurement equation relates the unobserved factors Z t and M t the observed data which has m-dimensions:

M t = H × Z t + C + t (38)
Where H ∈ R m×m is predicted to be a constant matrix and discribes the relationship betwwen Z t the actual factor and M t the measurement of the process with C ∈ R m a constant matrix which gives the external control of the given measurement equation, the random variable t ∈ R m×m is tHe measurement noise with :

t ∼ N(0, R)
Where R ∈ R m×m is predicted to be constant and represents the noise covariance matrix . At every interval time the Z t which is an unobserved factor is estimated with the kalman filter and it updates estimates by using t the feed back from M t the measurement equation .

There is two kind of update equation in kalman filter , the first one is the time update equation and the second is the measurement update equation. The time update equation give a projection to the error covariance and to the actual state to get an estimate for the following term. The measurement update equation integrates an other information to ameliorate the time update equation's estimates. We can express the time update as bellow:

Ẑ- t = A × Ẑt-1 + B × µ t (39)
P- t = A × Pt-1 × A + Q (40)
We initialize the filter by giving the unconditional mean Z 0 and P 0 the variance of the transition system ,with which we gonna begin the recursion. We can calculated its from the historival data as follow :

Ẑ- t = A × Z 0 + B × µ t (41)
P- t = A × P 0 × A + Q (42)
K which is the kalman gain have the ability to minimize the system's covariance error , it represents the difference between M t the actual measurement and the predicted measurement given by H Ẑt + C .Since the value of the measuremnt error covariance is closer to zero the actual value measurement is more weighty . We can express the error term and the residual variance-covariance matrix as :

t = M t -C -H × Ẑ- (43)
S t = H × P- t × H + R (44)
From those two expression we can calculate the standardised residuals as:

e t = t √ S t (45)
So the Kalman gain is gven by:

K t = P- t × H S t (46)
We obtain then the measurement update equation :

Ẑt = Ẑ- t + K t (M t -H × Ẑ- t ) (47) Pt = (I -K t × H) × P- t (48)
Since we are worken with the state space assumptions , the predictions errors are Gaussian , the kalman filter maximzes then a multi variate normal likelihood ,we can compute the log likelihood for each step of recursion as below :

l(θ) = - 1 2 N t=1 ln(2π) + ln |S t | + t S -1 t t (49)
To find the optimal parameter set noted by θ we need a non-linear optimisation which maximise the likelihood function ,it can be calculated from :

θ = argmax θ l(θ)

Estimation method:

As we can see from section 4 our model can be clearly represented as a state space form where the measurement equations for the our independent model and the Nelson Siegel model is given by equations ( 36)and ( 37) respectively.And it is given as follow :

μ = B(t, T) Z t -C(t, T) + Ψ, Ψ → N(0, R) (50)
with R ∈ R m×m is the measurement noise covariance matrix with m represents the number of ages giving in our data set, R is supposed to be time-invariant. We assume that Ψ which is the measurement noise is independent between ages as result R is presented by a diagonal matrix.Instead of estimate every measurement error separately for every age, we use an easy parametric curve to estimate the measurement error thanks to this method the number of the parameters are decreased to three parameters. The diagonal of the covariance matrix can be represented as a parametric form and we can write :

R(t, T) = 1 T -t T-t i=t [r c + r 1 e r 2 ×i ] (51)
And we include the estimation of the different values giving by r c ,r 1 and r 2 with the optimal parameter-set noted by θ. The transition systems of equation are presented in our case with the real world dynamics introduced in (4.5) and we can write :

Z t i = e -K(t i -t i-1 ) Z ti-1 + η t , η t → N(0, Q t i ) (52)
Q t i = E       t i t i-1 e -k(t i -s) ΣdW P s 2       (53)
All the errors are Gaussian since we are under the state-space model assumptions , the kalman filter produces a multivariate normal likelihhod , we can calculate the log-likelihhod as:

l(θ) = - 1 2 N t=1 ln(2π) + ln |S t | + t S -1 t t (54)
t and S t are respectively the error term and the residual variance-covariance matrices.And we can conclude l(θ) the optimal parameter-set that maximizes the likelihood function like: θ = argmax θ l(θ)

Bootstrapping state-space models:

The major advantage of the bootstrap is his simplicity , it is considered as a uncomplicated way to derive estimates of standard errors and confidence intervalles for such a complex paramaters or a complex estimators for any distribution , in addition bootsrap is a suitable way to control and confirm the satbility of the results .Despite the fact that most of the problems we can't know or have the exact confidence interval,Bootstrap is asymptotically more exact than the other standard intervals by the use of a sample variance and the normality assumptions.

In our work we noted that the likelihood is flat around the optimal parameter set for this reason we use a bootstrap method to re-sample the observed data, we use the standardised residuals taking from the kalman filter optimal parameters set, so we can have an asymptotically Gaussian distribution for the parameteres estimate. We use in our framework the re-sample method introduced by [START_REF] Stoffer | Resampling in state space models[END_REF].The process of the state-space bootsrap method is presented by 4 step as below:

1. We define the standardized innovations from the optimal parameter set :

e t = ˆ t Ŝt (55)

Results of the estimation:

In this secion, we represent the results from the three models estimation and evaluation.

Table [START_REF] Andersen | Estimating continuous-time stochastic volatility models of the short-term interest rate[END_REF] shows the maximum log likelihood and RMSE: the Root Mean Squared Error for every model.

We presented here a modified Akaike information Criteria noted by AIC test which is used for state space model as it was given by Cavanaugh and Shumway 1997 . The AICb is noticed by his simplistic form and it would be adapted to calculate as part of the bootstrap-based analysis .

We use the result of the bootstrapped log likelihood in the AICb test to obtain an estimation of the penallty term.We can calculate the AICb test as bellow:

AICb = -2 log L( θ) + 2[ 1 N N i=1 -2 log L( θ * i -(-2 log L( θ))] (59)
Where N presents the number of bootsrap distribution, in our case we choose N = 500 , and L( θ * i ) presents the bootstrapped maximum likelihood for the ith distribution . We note that the 3-factor independent model and the Nelson Siegel model have a comparable loglikelihood and RMSE and both models presented give as a better fit than the two factor model.We noted that the three factor independent model in our framework have 15 parameters and 234 latent factors to be estimated ,(we have 3 factors for everey 78 years of our presented data). We noted that there is no amelioration in the use of the 3-factor independent model in the place of the 3-factor Nelson Siegel model ,when we indicate the rejection of the 2-factor independent model.

Parameter estimation :

Table (2) displays the estimated parameters for the 2-and 3-factor model with 95% confidence intervals from our giving method of state-space bootstrapping introduced in section 5.2.3. As we can see for both model that all the parameter δ are significant.We can note also that the κ parameters for the real-world process have a large confidence intervals , the value of the lower bound which is close to zero dsignates the random walk whereas the wide values show that the factor returns to its long term mean of zero. We can note that the measurement variance parameters r 1 , r 2 and r c ,and the volatilty matrix Σ for both models are significant with a little number of outliers. Even though we have difficults to estimate and therefore to foreccast the real-world drift,we find generally that the risk neutral drift and also the diffusion parameters are stable.

Sample analysis :

The sructural changes in our mortality rates for different models can be appeared through the in-sample analysis. Figure 3 and 4 display the perecentage error of the survival curve already estimated for different models taking the years 1950 and 2000. We note that all the models can fit our data up to the age of 85 , wherever over the age of 85 the percentage error of the 2 factor model continue in increasing in both chosen year and we can conclude that the 2 factor model is unable to fit the data at older ages. Whereas the two other models have a similar percentage error variations. Fig. 5 summarises for the all periods the fit result using the Mean Absolute Relative Error (MARE), we note that the percentage error is law for all models below the age of 80 , whereas after the age of 80 we notice that only the 3-factor independent model is able to capture the whole survival curve variation this indicates that one of the factors is able to fit the population .

Conclusion:

Over the last decades, the humain mortality has globally declined and a rising in the life expectancy has occurred. The mortality improvements presents a big challenge especially for the public retirement systems and also for the life annuities business.When we talk about the long term living benefit , we need an adapted mortality projection to calculate the expected present values using in reserving or in pricing in order to avert the underestimation of the future costs .Therefore the actuary have to employ the life tables or the Project tables containing a future forecasting of the mortality trend .Thus , a new Risk appear , is the Longevity Risk .

In our article , we tried to propose a model who can be used in the longevity risk applications.We prove that the affine term structure model (ATSM) which is used in interest rates applications , is a suitable tool to model the mortality data.

To our Knowledge this is the first work that uses the Consistent dynamics affine mortality models to model USA Longevity risk , indeed we show a closed form representation for the risk adjusted survival curve , we give an estimation for the model parameters according to an essentially affine transformation using the kalman filter and we presented a 2-and 3-factor models in a consistent framework . The aim of this paper is to develop a consistent forecasts that help in pricing and used in risk management applications. We use the USA population between 50 and 100 over 1933 and 2010 to estimate and establish a comparison between models .

Our work shows that the 3-factor independent model is robust during the starting year, and the mean absolute relative error (MARE) for the fitted survival curves are low for all the age range . We prove that the 3-factor independent model is performed better than the other models and we give an appropriate approach which can be used in the longevity risk application through the use of a consistent framework . The ATSM models is able to be used not only for evaluating capital requirements for risk management but also in pricing and hedging longevity risk for pension funds and also for insurers.

Figure 1 : 4 . 1 . 1

 1411 Figure 1: Age-Periode average force of mortality for the USA population aged 50-100 for 1933-2010

Figure 2 :

 2 Figure 2: Principal component analysis

Figure 3 :Figure 4 :

 34 Figure 3: Fit error year 1950

Figure 6

 6 Figure6and 7 display respectively the latent factors and the factor loadings fot the 2-factor independent model .We note that B 1 the first factor loadings have with age an increasing exponential shape and we present through it the general trend of mortality .Z 1 which is the corresponding factor indicates a heavily improving in mortality rates between 1930 and 1950 with a fairly variation between years .After 1950 we see a little change in the factors trend and the mortality rates are increasing but with a lower volatilities between the years. The second latent factors , Z 2 shows a slowly decreasing trend , and after 1950 the volatilty between years became lower , it's corresponding factor loadings B 2 has a declining shape which prove the sensivity to the factor only for the younger ages . Figure8and 9 present respectively the latent factor and the factor loadings for the 3-factor independent model , the first factor loadings have the same shape as the 2-factor independent model the first latent factor shows a little improving in the mortality rate between 1930 and 1950 , since 1950 we note that this trend has changed and the mortality

Figure 8 Figure 6 :Figure 7 :

 867 Figure6and 7 display respectively the latent factors and the factor loadings fot the 2-factor independent model .We note that B 1 the first factor loadings have with age an increasing exponential shape and we present through it the general trend of mortality .Z 1 which is the corresponding factor indicates a heavily improving in mortality rates between 1930 and 1950 with a fairly variation between years .After 1950 we see a little change in the factors trend and the mortality rates are increasing but with a lower volatilities between the years. The second latent factors , Z 2 shows a slowly decreasing trend , and after 1950 the volatilty between years became lower , it's corresponding factor loadings B 2 has a declining shape which prove the sensivity to the factor only for the younger ages . Figure8and 9 present respectively the latent factor and the factor loadings for the 3-factor independent model , the first factor loadings have the same shape as the 2-factor independent model the first latent factor shows a little improving in the mortality rate between 1930 and 1950 , since 1950 we note that this trend has changed and the mortality

Figure 8 :Figure 9 : 6 . 2 . 2

 89622 Figure 8: The Latent Factor of the 3-factor independent model

Figure 10 :

 10 Figure 10: Robustness test:

Table 1 :

 1 Comparison of log likelihood, RMSE and the number of parameters and factors estimated for every model,and an AICb is also presented :

		3-factor	3-factor	2-factor
		independent Nelson Siegel independent
	Log likeihood	24230	24107	20076
	RMSE	0.000251	0.000269	0.000356
	No of factors estimated	234	234	156
	No of model parameters	15	13	11
	∆AIC	-	≈ 0	4404
		20

Table 2 :

 2 the results of 2-and 3-factor model fit with the bootstrap confidence intervals:

	ML	2-factor	95%confidence	3-factor	95%confidence
	estima-
	tion
		independent	Lower	Upper	independent	Lower	Upper
	δ 1	-0.0845	-0.0925	-0.0821	-0.09756	-0.10122	-0.08703
	δ 2	0.0453	0.0320	0.0567	0.00527	0.00457	0.00752
	δ 3	-	-	-	-0.12530	-0.14521	-0.1033
	κ 1	0.0155	0.0025	0.0287	0.00653	0.00320	0.00727
	κ 2	0.0125	0.0108	0.0248	0.00189	0.00127	0.00228
	κ 3	-	-	-	0.00192	0.00103	0.00253
	σ 1 σ 2	4.2012e -05 0.0081	3.705e -05 0.0058	4.835e -05 0.0092	3.857e -04 3.002e -03	3.207e -04 4.007e -04 2.805e -03 3.503e -03
	σ 3	-	-	-	6.428e -04	3.987e -05 2.098e -03
	r 1	6.3169e -12	7.501e -13	8.233e -11	1.657e -11	4.002e -12 2.430e -11
	r 2 r c	0.2589 2.4500e -08	0.1980 1.2801e -07 2.805e -07 03587	0.2908 3.748e -07	0.2365 2.904e -07 4.383e -07 0.3508

Table 3 :

 3 the results of 3-factor Nelson Siegel model fit with the bootstrap confidence

	intervals:
	ML	3-factor	95%confidence
	estimation	Nelson Siegel model	Lower	Upper
	δ	-0.08750	-0.09856	-0.07683
	κ 1	0.10082	0.00586	0.18970
	κ 2	0.00369	0.00287	0.00456
	κ 3	0.00115	0.001085	0.00254
	σ 1 σ 2	4.251e-04 1.328e -04	3.658e -04 5.257e -04 0.982e -04 1.698e -04
	σ 3 r 1	4.921e -04 2.425e -08	3.984e -04 5.327e -04 3.205e -09 2.967e -08
	r 2 r c	0.253 2.845e -06	0.237 1.914e -06 2.968e -06 0.369

Table ( 3

 ( ) displays the estimated parameters of the Nelson Siegel model ,we have similar results of the previous models ,they all have a large confidence intervals and the different parameters of the factor loadings are significant.


Yu Qiu

Ya-Ling He

Wen-Qi Wang

Bao-Cun Du

Kun Wang

Turbulent convective heat transfer of molten salt in the shell side of a prototype rod baffle heat exchanger for concentrating solar power

Keywords: Molten salt, Rod baffle shell-and-tube heat exchanger, Shell-side heat transfer characteristics, Experimental study, Concentrating solar power 1. Introduction

In this paper, a prototype rod baffle shell-and-tube heat exchanger (RB-STHX) was designed and fabricated for concentrating solar power (CSP). In order to investigate the heat transfer characteristics of molten salt in the shell side, the RB-STHX was experimentally studied. A ternary salt Hitec and an oil were adopted as the heat transfer fluids in the shell and tube sides, respectively. Experimental data was firstly obtained with the salt temperature of 212.7~279.6 o C, where the corresponding Prandtl number is in the range of 14.2~23.3. The flow is in the turbulent region with the Reynolds number of 2,697 to 12,517. The average heat transfer rate ranges from 117 kW to 364 kW. Then, the experimental data were compared with two exiting correlations. The results indicate that an average error of -9.1 ~ -12.3% between the Nusselt numbers predicted by the two exiting correlations and the experiment data was found. Finally, for predicting the heat transfer characteristics of the molten salt in the shell side more accurately, two modified heat transfer correlations which show satisfactory agreements with the experimental data were developed. The average deviation between the experiment data and the modified correlations are around ±4.4 ~ ±4.6%. Results from present study will offer the reliable experimental data, useful correlations and helpful guidance for the design and operation of realistic molten salt RB-STHXs in the CSP systems.

The fossil fuel combustion has leaded to serious environmental issues and public health risks for years [START_REF] Spracklen | Global warming: China's contribution to climate change[END_REF][START_REF] Li | A hybrid model for explaining the short-term dynamics of energy efficiency of China's thermal power plants[END_REF][START_REF] Li | Review of methodologies and polices for evaluation of energy efficiency in high energy-consuming industry[END_REF][START_REF] Li | Modeling a hybrid methodology for evaluating and forecasting regional energy efficiency in China[END_REF]. The Concentrating Solar Power (CSP) technology which generates electricity from concentrated solar radiation has been regarded as a candidate for solving these problems [START_REF] He | Non-uniform characteristics of solar flux distribution in the concentrating solar power systems and its corresponding solutions: A review[END_REF][START_REF] Lewis | Research opportunities to advance solar energy utilization[END_REF]. As a result, the CSP has developed quickly in the last few decades. In CSP plants, the thermal energy storage [START_REF] Ma | Investigation on the thermal performance of a high-temperature latent heat storage system[END_REF][START_REF] Wu | Cyclic behaviors of the molten-salt packed-bed thermal storage system filled with cascaded phase change material capsules[END_REF], which usually uses molten salt as energy storage material and allows energy dispatch, is widely adopted. In the thermal storage unit, shell-and-tube heat exchangers (STHXs) are used to exchange thermal energy between this unit and the rest of the plant [START_REF] Kuravi | Thermal energy storage technologies and systems for concentrating solar power plants[END_REF]. For the middle temperature thermal storage in linear Fresnel reflector [START_REF] Qiu | Study on optical and thermal performance of a linear Fresnel solar reflector using molten salt as HTF with MCRT and FVM methods[END_REF][START_REF] Qiu | A comprehensive model for optical and thermal characterization of a linear Fresnel solar reflector with a trapezoidal cavity receiver[END_REF] and parabolic trough collector [START_REF] He | A MCRT and FVM coupled simulation method for energy conversion process in parabolic trough solar collector[END_REF][START_REF] Qiu | Thermal performance analysis of a parabolic trough solar collector using supercritical CO2 as heat transfer fluid under non-uniform solar flux[END_REF], both the Salt-Water/Steam STHXs and the Salt-Oil STHXs are employed. While for high temperature thermal storage in solar power tower [START_REF] Qiu | A comprehensive model for analysis of real-time performance of a solar power tower with a multi-tube cavity receiver[END_REF], only the Salt-Water/Steam STHXs are adopted. In the Salt-Oil STHXs, the salt usually flows in the shell side. In Salt-Water/Steam STHXs, the salt usually flows in the shell sides of salt-water preheater, evaporator and superheater. In a few designs, the salt also flows in the tube side of the salt-water evaporator.

It is well known that the convective heat transfer characteristics (CHTCs) of molten salt in both tube and shell sides of the STHXs directly influence the performance of the thermal storage unit. As a result, they are important for the design and operation of the plants. For these reasons, many experimental studies have focused on investigating the CHTCs of molten salt.

Review of the literature from 1940s to 2017, it is found that most experiments are conducted for studying on the CHTCs in circular tubes. Many of these studies are conducted using different salts under uniform flux. Grele and Gedeon [START_REF] Grele | Forced-convection heat-transfer characteristics of molten Flinak flowing in an inconel X system[END_REF] studied the CHTCs of FLiNaK with Pr=1. 23-4.33 and Re=2000-20,000. Hoffman [START_REF] Hoffman | Fused salt heat transfer. Part II. Forced convection heat transfer in circular tubes containing NaF-KF-LiF eutectic[END_REF] also studied the CHTCs of FLiNaK with Pr=1.6-4 and Re=2428-9,536. He also studied the CHTCs of a ternary nitrate salt Hitec with Pr=3. 3-9.1 and Re=2342-33,493 [START_REF] Hoffman | Fused salt heat transfer: Part III: Forced-convection heat transfer in circular tubes containing the salt mixture NaNO2-NaNO3-KNO3[END_REF]. Cooke and Cox [START_REF] Cooke | Forced-convection heat transfer measurements with a molten fluoride salt mixture flowing in a smooth tube[END_REF] studied the CHTCs of LiF-BeF2-ThF4-UF4 with Pr=4-14 and Re=400~30,600. Silverman et al. [START_REF] Silverman | Heat transfer measurements in a forced convection loop with two molten-fluoride salts LiF-BeF2-ThF2-UF4 and eutectic NaBF4-NaF[END_REF] studied the CHTCs of LiF-BeF2-ThF2-UF4 with Pr=6. 210, and they also studied the CHTCs of NaBF4-NaF with Pr=4.89-5.64 and Re=5,104-44,965. Moreover, some of these studies are conducted using salts in the tube side of a concentric tube heat exchanger (HX). Wu et al. [START_REF] Liu | Turbulent convective heat transfer with molten salt in a circular pipe[END_REF][START_REF] Wu | Convective heat transfer in the laminar-turbulent transition region with molten salt in a circular tube[END_REF] used LiNO 3 with Pr=12.7-14.7, Re=17,000-45,000. They also used Hitec with Pr=8. 861. Chen et al. [START_REF] Chen | Convective heat transfer characteristics in the turbulent region of molten salt in concentric tube[END_REF] used another ternary nitrate salt with Re=10000-50000 and Pr=11-27. In addition, it is also found that several studies have investigated the CHTCs of molten salt in the tube side of STHXs. He et al. [START_REF] He | Experimental investigation on turbulent heat transfer characteristics of molten salt in a shell-and-tube heat exchanger[END_REF] studied the CHTCs of Hitec in the tube-side of the tube bundle in a Salt-Oil STHX with Re=9,000-19,000 and Pr=6.5-10. Jin et al. [START_REF] Qian | Performance of a gas cooled molten salt heat exchanger[END_REF] studied the CHTCs in the tube side of a Salt-Gas HX with Re=4,138 -11,191 and Pr= 9.8-18.9. By summing up the above studies, it is found that most experimental data fall in the deviation of 25% of traditional correlations, such as Gnielinski's correlation and Hansen's correlation. Besides, the CHTCs of Hitec in some noncircular passages, including the enhanced tube, annular passage, etc., have also been studied by Lu et al. [START_REF] Lu | Convective heat transfer in the laminar-turbulent transition region of molten salt in annular passage[END_REF][START_REF] Lu | Transition and turbulent convective heat transfer of molten salt in spirally grooved tube[END_REF][START_REF] Lu | Convective heat transfer of high temperature molten salt in a vertical annular duct with cooled wall[END_REF], and corresponding correlations were obtained.

Recently, two studies also investigated the CHTCs of molten salt in the shell sides of the STHXs. He et al. [START_REF] He | Convective heat transfer of molten salt outside the tube bundle of heat exchanger[END_REF] studied the CHTCs of Hitec outside a tube bundle with seven tubes. It is reported that the Nusselt numbers in the laminar region (Re=400-2300) is 3.5-5.0 times higher than those predicted by the traditional correlation. Du et al. [START_REF] Du | Convective heat transfer of molten salt in the shell-and-tube heat exchanger with segmental baffles[END_REF] studied the CHTCs of Hitec in the shell side of a practical STHX with segmental baffles (STHE-SBs), where the flow is turbulent with Re=6142-9125 and Pr=19. 43-22.33. The testing data was compared with a traditional correlation called Kern's correlation. It is found that this correlation is still appropriate to predict the CHTCs with a maximum deviation of 7.1%.

Based on the above review, on the one hand, it is seen that most studies focus on the CHTCs of molten salt in the tube side, but only one study has focused on the CHTCs in the shell side of a practical STHX. For understanding the heat transfer in the shell sides of the practical STHXs in CSP better, the CHTCs under realistic flow conditions are needed to be further investigated.

In this paper, a prototype rod baffle shell-and-tube heat exchanger (RB-STHX) was designed for CSP. RB-STHX proposed in 1970s [START_REF] Gentry | Rodbaffle heat exchanger technology[END_REF] is a STHX that can eliminate the harmful flow induced vibration in traditional STHE-SBs. This feature can also be helpful for the safe operation in CSP. For testing the CHTCs of molten salt in the shell side of the designed RB-STHX, a large-scale multipurpose molten salt testing facility was built. A ternary nitrate salt Hitec and a synthetic oil were used as the heat transfer fluids in the shell and tube sides, respectively. The experimental data was obtained and compared with two exiting correlations for the ordinary fluids. Based on the comparison, two new correlations were also developed.

RB-STHX and experimental facility 2.1 Prototype rod baffle shell-and-tube heat exchanger

A prototype rod baffle shell-and-tube heat exchanger (RB-STHX) made of 316L stainless steel was designed and constructed. The configuration is presented in Fig. 1, and three photos are shown in Fig. 2. The detailed geometric parameters are given in Table 1. The effective length of the RB-STHX is 2000 mm, and the inner diameter of the shell is 100 mm. A tube bundle with 21 tubes are installed in the shell using a square configuration as shown in Fig. 1 and Fig. 2. Eighteen rod baffles are placed along the flow direction in alternating orientation for disturbing the flow and supporting the tubes. Each baffle is made of a baffle ring welded with several parallel cylindrical rods as shown in Fig. 1 and Fig. 2. A thermal insulation layer of about 150 mm is used to cover the heat exchanger and reduce the heat loss in the experiment.

Large-scale multipurpose molten salt testing facility

The turbulent convective heat transfer performance of the molten salt in the shell side of the RB-STHX is tested in a large-scale multipurpose molten salt testing facility built in Key Laboratory of Thermo-Fluid Science and Engineering of Ministry of Education, China. A photo of the facility is presented in Fig. 3.

Data reduction

Heat transfer rate and energy balance

The heat transfer rates of the shell-side salt (Q s ) and tube-side oil (Q t ) in the RB-STHX are calculated using Eq.( 1) and Eq.( 2), respectively. The average value of Q s and Q t is regarded as the heat transfer rate of the exchanger (Q ave ) in Eq.( 3). The thermal balance deviation (ε) of the exchanger is defined in Eq.( 4).

( )

s ,s s ,s s,in s,out V p Q q c T T ρ = - (1)
( )

t ,t t ,t t,out t,in V p Q q c T T ρ = - (2)
( )

ave s t / 2 Q Q Q = + (3) s t ave -/ 100% Q Q Q e = × (4)
where q V represents the volume flow rate; c p is specific heat; "t","s", "in", and "out" in the subscripts refer to the parameters of the tube side, shell side, inlet, and outlet, respectively.

Overall heat transfer coefficient

The overall heat transfer coefficient (K) between the shell side and tube side in the RB-STHX is defined in Eq.( 5). The logarithmic mean temperature difference of the two sides (△T m ) is calculated by Eq.( 6).

( ) ( ) ave o m ave t o t m / = / K Q A T Q n d L T π = ∆ ⋅ ∆ (5) ( ) ( ) ( ) ( ) s,in t,out s,out t,in m s,in t,out s,out t,in ln T T T T T T T T T - - - ∆ =   - -   (6)
where A o is the heat transfer area based on the outer surfaces of the tubes.

Tube-side heat transfer coefficient

The tube-side convection heat transfer coefficient (h t ) is calculated by Gnielinski's equation in Eq.( 8), where the characteristic dimension is the tube inner diameter. The friction factor (f) is calculated using Filonenko's equation in Eq.( 9) [START_REF] Yang | Heat transfer[END_REF].

  -       + ⋅     + ⋅ ⋅ -        f Re

Shell-side heat transfer coefficient and Nusselt number

The shell-side convection heat transfer coefficient (h s ) is calculated by Eq. [START_REF] Qiu | A comprehensive model for optical and thermal characterization of a linear Fresnel solar reflector with a trapezoidal cavity receiver[END_REF].

s o o o t i w i 1 1 1 ln 2 = - - h d d d K h d d λ (11)
where λ w is the heat conductivity of the tube wall.

Nusselt number of shell side is calculated by Eq. [START_REF] He | A MCRT and FVM coupled simulation method for energy conversion process in parabolic trough solar collector[END_REF]. The Reynolds number of the shell side is expressed in Eq. [START_REF] Qiu | Thermal performance analysis of a parabolic trough solar collector using supercritical CO2 as heat transfer fluid under non-uniform solar flux[END_REF].

s s h s Nu h D λ = ( 12
) s s s h s Re u D ρ m = (13)
( )

2 2 s ,s s t o = 4 V u q D n d π   -   (14)

( ) ( )

2 2 h s t o s t o = D D n d D n d -⋅ + ⋅ (15)
where D h and u s are the hydraulic diameter and velocity of the shell side, respectively.

Uncertainty analysis

The uncertainties of the experimental data were evaluated using the method presented in

Refs. [START_REF] Kline | Describing uncertainties in single-sample experiments[END_REF][START_REF] Moffat | Describing the uncertainties in experimental results[END_REF]. In this method, it assumes that a desired variable (R) can be expressed as a function of n independent measured parameters (x 1 , x 2 , …, x n ). So, the relative uncertainty (δR/R) of R can be calculated by Eq.( 17). Table 2 shows the uncertainties of the measured and calculated parameters. It is seen that the relative uncertainties of Q ave , K, Re s and Nu s are within 8.3%, 8.3%,1.7% and 8.4%, respectively.

( )

1 1 , , , n R f x x x = ⋅⋅⋅ (16) 0.5 2 1 n i i i R f x R x R δ δ =     ∂   =         ∑ ( 17
)
where δx i is the uncertainty of variable x i .

Results and discussion

The turbulent convective heat transfer characteristics of the molten salt in the shell side of the RB-STHX was experimentally studied using the previous described approach. The detailed experimental conditions are given in Table 2. During the experiment, the average temperatures of salt and oil are in the ranges of 212.

Thermal balance check and experimental results

The thermal balance between the shell side and tube side was checked in the experiment.

The check results at different Re are shown in Fig. 6. It can be seen that the thermal balance deviations for the obtained experimental data are within 7%. These results indicate that the present experimental facility and the testing approach are dependable and suitable.

Comparison of experimental data and exiting correlations

For describing the turbulent convective heat transfer of single-phase fluids in a heat exchanger (HX) in a general way, the Nu s is usually expressed as a power function of Re s and Pr s as shown in Eq.( 18), where C 1 is a factor defined by the geometric parameters of the HX.

m and n are constants. m is usually around 0.8 for turbulent flow, and n is usually around 0.4 or 1/3. Moreover, in order to consider the temperature difference between the fluid and the wall, another term (C 2 ) which is usually a function of the viscosity or the Prandtl number is also added.

s 1 s s 2 m n Nu C Re Pr C = ⋅ ⋅ ( 18
)
For the RB-STHX, some correlations which are usually obtained using oil or water as the heat transfer fluid can be found in the literature. The current experimental data are compared with two of the typical existing correlations.

The first one is a correlation provided by the Phillips petroleum company in Ref. [START_REF] Gentry | Rodbaffle heat exchanger technology[END_REF] as shown in Eqs. [START_REF] Silverman | Heat transfer measurements in a forced convection loop with two molten-fluoride salts LiF-BeF2-ThF2-UF4 and eutectic NaBF4-NaF[END_REF] The second correlation is an equation described in Ref. [START_REF] Dong | Numerical and experimental investigation of shellside characteristics for RODbaffle heat exchanger[END_REF] as shown in Eq.(

L L D C C C e D D D D D n d e ξ λ λ λ ξ - - - = ⋅   - + -      - - -    = +  -       +   (20)
(hereafter Correlation B). In this correlation, m of 0.815 and n of 1/3 are adopted, and C 2 is a function of the viscosity. For the tested RB-STHX, C 1 in Eq.( 21) equals to 0.0361. 0.14 0.303 To sum up, comparing with the experimental data, the average underestimates for correlations A and B are 9.1% and 12.3%, respectively. Similar phenomenon has also been reported in the heat transfer of molten salt in the circular tube. For example, it is reported in

0.815 1/3 s b s 1 s s 1 w e = , =0.0589 L Nu C Re Pr C D m m -     ⋅ ⋅         (21)
Ref. [START_REF] Chen | Convective heat transfer characteristics in the turbulent region of molten salt in concentric tube[END_REF] by Chen et al. that the Sieder-Tate correlation and the Hausen correlation underestimate about 8% and 10% compared with the experimental results, respectively.

Therefore, in order to have a better understanding of the heat transfer characteristics in the shell side of RB-STHX, a correlation which can get better predictions of the experimental data would be welcomed by the practitioners in CSP.

Modified heat transfer correlations

In this section, the data of Nusselt number obtained from the experiment for the molten salt in the shell side are fitted into two heat transfer correlations, which are in the forms of Correlation A and Correlation B, respectively.

The first modified correlation fitted in the form of Correlation A is given in Eq. [START_REF] Chen | Convective heat transfer characteristics in the turbulent region of molten salt in concentric tube[END_REF]

m m   ⋅     (23)
where C 1 is the same as that in Eq. [START_REF] Liu | Turbulent convective heat transfer with molten salt in a circular pipe[END_REF]. The examined ranges of key parameters are as follows. Re s =2,697-12,517, Pr s =14.2-23.3, and (μ s /μ w ) 0.14 =0.86-0.93. Fig. 10 presents the fitted correlation along with the experimental data and Correlation A.

It is seen that the current experimental results are in good agreement with the fitted correlation with an average deviation of ±4.6% and a maximum deviation of ±11.5%.

Moreover, an average deviation of -9.3% and a maximum deviation of -11.3% between Correlation A and the modified correlation can also be observed. The second modified correlation fitted in the form of Correlation B is given in Eq. [START_REF] Qian | Performance of a gas cooled molten salt heat exchanger[END_REF].

It has the same n and viscosity term as those in Correlation B. However, a modified m of 0.756 and a revised C 1 is obtained by fitting the data. The goodness of the fit of 0.962 is achieved. For current RB-STHX, the modified correlation can be rewritten as Eq.( 25). It can be observed that the current experimental results agree with the fitted correlation well with an average deviation of ±4.4% and a maximum deviation of ±10.7%. In addition, an average deviation of -12.5% between the modified correlation and Correlation A is observed, and a maximum deviation of -17.5% is also found.

The above results indicate that the two modified correlations can predict the heat transfer of the ternary salt Hitec in the shell side of current RB-STHX more accurately than the two exiting correlations. As a result, we think the modified correlations can be applied in the heat transfer calculation of the RB-STHX in CSP.

Conclusions

In this paper, turbulent convective heat transfer characteristics of a ternary salt Hitec in the shell side of a prototype rod baffle shell-and-tube heat exchanger (RB-STHX) for the CSP was experimentally studied. The conclusions are as follow:

(1) Experimental data was obtained in the salt temperature range of 212. (2) The experimental data was compared with two exiting correlations obtained using ordinary fluids. The comparison indicates that the turbulent convective heat transfer characteristics of Hitec in the shell side of the RB-STHX are close to the exiting correlations.

Nevertheless, the predicted Nusselt numbers of the exiting correlations could not describe the heat transfer characteristics accurately with an average underestimate of 9.1-12.3%.

( The properties of YD-325 synthetic oil are described by equations A(5)-A(8

6 0 K < T T T T k T T - - - - × + ×   = ≤ - + < × ≤  +   A(3)

Fig. 1 .

 1 Fig. 1. Sketch of the rod baffle shell-and-tube HX showing the detailed layout of the baffle.

Fig. 2 .

 2 Fig. 2. Photos of the tube bundle, a rod baffle and an end plate.

Fig. 3 .Fig. 4 .Fig. 1 .Fig. 5 .

 3415 Fig. 3. Photo of the experimental facility.The facility mainly consists of three loops including a salt loop, an oil loop, and a water loop. A detailed schematic of the facility is presented in Fig.4, and a brief introduction of the loops and the experiment procedures are as follows.

  7-279.6 o C and 95.5-154.0 o C, respectively. The volume flow rates of salt and oil are in the ranges of 8.42-24.99 m 3 •h -1 and 19.10-19.51m 3 •h -1 , respectively.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Thermal balance check of the experiment.The testing data were reduced, and the experimental results of the Nusselt number (Nu s ) at different Reynolds numbers (Re s ) are shown in Fig.7. It is seen that Nu s increases with the increase in Re s , which is similar with those of ordinary fluids. This is due to a well-known reason that the larger the Re is, the more intensive the turbulence becomes. In the experiment, the flow is turbulent, and the Re s is in the range of 2,697-12,517. The Nu s ranges between 71 and 190. The Pr s varies from 14.2 to 23.3. The average heat transfer rate ranges from 117 kW to 364 kW. Because the temperature difference between the salt and oil sides is quite large, the average temperature difference of 50.3-105.2 o C between the salt and the tube wall occurs in the test. As a result, the (μ f /μ w ) of 0.35-0.60 can also be found.

Fig. 8 14 ]Fig. 8 .

 8148 Fig. 8 shows the comparison of current experimental data with Correlation A. It can be

Fig. 9

 9 Fig. 9 shows the comparison of current experimental data with Correlation B. It can be observed that the deviations between the experimental results and the predicted values are in the range of +1~+26%. Furthermore, most experimental data are larger than the values predicted by Correlation B. The average deviations of around +14% can be found at the identical Reynolds numbers.

Fig. 9 .

 9 Fig. 9. Comparison of the experimental data with Correlation B.

Fig. 10 .

 10 Fig. 10. Fitting correlation of the experimental data and Correlation A.

Fig. 11 shows

 11 Fig.11shows the second modified correlation, the experimental data, and Correlation B.

Fig. 11 .

 11 Fig. 11. Fitting correlation of the experimental data and Correlation B.

Table 1

 1 Detailed geometric parameters of the RB-STHX.

	Parameters

Table 2

 2 The measured and calculated parameters showing their uncertainties.

				Absolute or relative
	Items	Parameters	Range
				uncertainty
		q V,s / m 3 •h -1	8.42-24.99	±1%
	Measured parameters	T s,in / o C	221.6-290.4	±0.5 o C
	of shell side	T s,out / o C	203.8-268.9	±0.5 o C
		q V,t / m 3 •h -1	19.10-19.51	±1%
	Measured parameters	T t,in / o C	83.4-146.4	±0.5 o C
	of tube side	T t,out / o C	107.5-161.6	±0.5 o C
		Q ave / kW	117-364	8.3%
		K / W•m -2 • o C -1	903-1,559	8.3%
	Desired parameters
		Re s	2,697-12,517	1.7%
		Nu s	71-190	8.4%

  and (20) (hereafter Correlation A). In this equation, m of 0.8 and n of 0.4

			0.8 s Nu C Re Pr s 1 s 0.4 =	  	s w m m	  	0.14	(19)
	1
	( = 0.042 0.0417	) (	0.023 0.0117	)	0.00496	b
	(	2 s	) ( 2 2 o		2 bo 2		2 bi	)	0.1	0.5
			s	t o
	=0.96 0.2437	( 0.01614 / t	bo	) 1	2

are adopted, and C 2 is a function of the viscosity. For current tested RB-STHX, C 1 in Eq.(

19

) equals to 0.0352.

  . It has the same C 1 , n and viscosity term as those in Correlation A. However, a modified m of 0.775 and a new constant term φ of 1.375 are obtained by fitting the data. The goodness of the fit is 0.959. For current RB-STHX, the correlation can be rewritten as Eq.(23).

	s Nu	= ϕ	⋅	0.775 s C Re Pr 1 s 0.4	  	s w m m	  	0.14	, =1.375 ϕ	(22)
										0.14
		s Nu	=0.0484	0.775 s Re Pr s 0.4	s
										w

  7-279.6 o C. The flow is turbulent with the volume flow rate of 8.42-24.99 m 3 •h -1 , and the corresponding Re s ranges from 2,697 to 12,517.

  ) Based on the comparison, two modified heat transfer correlations for Hitec are developed, and they have satisfactory agreements with the experimental data. The first

	correlation is	s Nu	=1.375	( C Re Pr m m 0.775 0.4 1 s s s ⋅ ⋅	w	)	0.14	, where C 1 is a parameter defined in
	Eq.(20). The second correlation is	s Nu	=0.1133	(	b L D e	)	0.303 -	⋅	( Re Pr m m 0.756 1/3 s s s	w	)	0.14	. The
	examined ranges of key parameters are as follows. Re s =2,697-12,517, Pr s =14.2-23.3, and
	(μ s /μ w ) 0.14 =0.86-0.93.
	These results will offer the reliable data and helpful guidance for the design and
	operation of realistic molten salt RB-STHXs in the CSP systems.
			ρ -	+								≤	≤	A(1)
			,s = p c	1560, 420K	T ≤ ≤	800K	A(2)
				1.863 10	8	3	2.551 10	5 2
			s	0.01176			2.2627 ,	420K	536K
				-6.47	10	4		, 0.7663 53 K	8	0

  ). Pr t Prandtl number of shell / tube side Pr w Prandtl number using wall temperature as the qualitative temperature Q s / Q t heat transfer rate of shell / tube side (kW) Q ave heat transfer rate of the heat exchanger (kW) q V,s / q V,t volume flow rate of shell / tube side (m 3 •h -1 ) Re s / Re t Reynolds number of shell / tube side RB-STHX rod baffle shell-and-tube heat exchanger STHX shell-and-tube heat exchanger T temperature (K, o C) u s / u t velocity of the shell / tube side (m•s -1 ) the tube wall (W•m -1 •K -1 ) λ s / λ t heat conductivity of salt / oil (W•m -1 •K -1 ) μ s / μ t dynamic viscosity of salt / oil (Pa•s)

	t -0.6311 1199.13, 30 = + T ,t 3.40 776.0, 300 K 0K = ρ ≤ + p c T 5 t -6.68 0. 10 , 1416 300K< ≤ T ≤ T -= × + T T ≤ 5 K 5 3K 7 73 573K ≤ k 9 3 6 2 3 t 10 3 7 2 4 4.066 10 5.2746 10 2.283 10 0.33065, 323K< 423K = 4.413 10 6.735 10 3.452 10 0.05989, 423K< 523K T T T T T T T T m -------× + ×  -× + ≤   -× + ×   -× + ≤  Nomenclature A o area of the outer surfaces of tubes (m 2 ) c p , s / c p , t specific heat of salt / oil (J•kg -1 •K -1 ) CHTCs convective heat transfer characteristics D h hydraulic diameter of the shell side (m, mm) D s shell inner diameter (m, mm) D o diameter of the circumcircle of tube bundle (m, mm) D bo outer diameter of baffle ring (m, mm) D bi inner diameter of baffle ring (m, mm) d o Tube outer diameter (m, mm) d i Tube inner diameter (m, mm) f friction factor h s / h t heat transfer coefficient of shell / tube side (W•m -2 •K -1 ) K overall heat transfer coefficient (W•m -2 •K -1 ) L t tube effective length (m, mm) L b baffle pitch (m, mm) Nu s / Nu t Nusselt number of shell / tube side n t tube number △T m logarithmic mean temp. difference of two sides (K) ε thermal balance deviation (%) λ w heat conductivity of Subscripts s, t, in, out parameters of tube side / shell side / inlet / outlet Pr s / Greek symbols ave average value	A(5) A(6) A(7) A(8)

Acknowledgements

The study is supported by the funding for Key Project of National Natural Science Foundation of China (No.51436007).

Appendix

The properties of the ternary salt (Hitec) are described by equations A(1)-A(4).